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In geometric topology, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing (non-trivial) Dehn surgery on the knot is not simply-connected. The conjecture states that all knots, except the unknot, have Property P.
Research on Property P was started by R. H. Bing, who popularized the name and conjecture.
This conjecture can be thought of as a first step to resolving the Poincaré conjecture, since the Lickorish–Wallace theorem says any closed, orientable 3-manifold results from Dehn surgery on a link.
If a knot
K
⊂
S
3
{\displaystyle K\subset \mathbb {S} ^{3}}
has Property P, then one cannot construct a counterexample to the Poincaré conjecture by surgery along
K
{\displaystyle K}
.
A proof was announced in 2004, as the combined result of efforts of mathematicians working in several different fields.
== Algebraic Formulation ==
Let
[
l
]
,
[
m
]
∈
π
1
(
S
3
∖
K
)
{\displaystyle [l],[m]\in \pi _{1}(\mathbb {S} ^{3}\setminus K)}
denote elements corresponding to a preferred longitude and meridian of a tubular neighborhood of
K
{\displaystyle K}
.
K
{\displaystyle K}
has Property P if and only if its Knot group is never trivialised by adjoining a relation of the form
m
=
l
a
{\displaystyle m=l^{a}}
for some
0
≠
a
∈
Z
{\displaystyle 0\neq a\in \mathbb {Z} }
.
== References ==
Eliashberg, Yakov (2004). "A few remarks about symplectic filling". Geometry & Topology. 8: 277–293. arXiv:math.SG/0311459. doi:10.2140/gt.2004.8.277.
Etnyre, John B. (2004). "On symplectic fillings". Algebraic & Geometric Topology. 4: 73–80. arXiv:math.SG/0312091. doi:10.2140/agt.2004.4.73.
Kronheimer, Peter; Mrowka, Tomasz (2004). "Witten's conjecture and Property P". Geometry & Topology. 8: 295–310. arXiv:math.GT/0311489. doi:10.2140/gt.2004.8.295.
Ozsvath, Peter; Szabó, Zoltán (2004). "Holomorphic disks and genus bounds". Geometry & Topology. 8: 311–334. arXiv:math.GT/0311496. doi:10.2140/gt.2004.8.311.
Rolfsen, Dale (1976), "Chapter 9.J", Knots and Links, Mathematics Lecture Series, vol. 7, Berkeley, California: Publish or Perish, pp. 280–283, ISBN 0-914098-16-0, MR 0515288
Adams, Colin (2004). The Knot Book : An elementary introduction to the mathematical theory of knots. American Mathematical Society. p. 262. ISBN 0-8218-3678-1. | Wikipedia/Property_P_conjecture |
In set theory, a tree is a partially ordered set
(
T
,
<
)
{\displaystyle (T,<)}
such that for each
t
∈
T
{\displaystyle t\in T}
, the set
{
s
∈
T
:
s
<
t
}
{\displaystyle \{s\in T:s<t\}}
is well-ordered by the relation
<
{\displaystyle <}
. Frequently trees are assumed to have only one root (i.e. minimal element), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.
== Definition ==
A tree is a partially ordered set (poset)
(
T
,
<
)
{\displaystyle (T,<)}
such that for each
t
∈
T
{\displaystyle t\in T}
, the set
{
s
∈
T
:
s
<
t
}
{\displaystyle \{s\in T:s<t\}}
is well-ordered by the relation
<
{\displaystyle <}
. In particular, each well-ordered set
(
T
,
<
)
{\displaystyle (T,<)}
is a tree. For each
t
∈
T
{\displaystyle t\in T}
, the order type of
{
s
∈
T
:
s
<
t
}
{\displaystyle \{s\in T:s<t\}}
is called the height, rank, or level of
t
{\displaystyle t}
. The height of
T
{\displaystyle T}
itself is the least ordinal greater than the height of each element of
T
{\displaystyle T}
. A root of a tree
T
{\displaystyle T}
is an element of height 0. Frequently trees are assumed to have only one root.
Trees with a single root may be viewed as rooted trees in the sense of graph theory in one of two ways: either as a tree (graph theory) or as a trivially perfect graph. In the first case, the graph is the undirected Hasse diagram of the partially ordered set, and in the second case, the graph is simply the underlying (undirected) graph of the partially ordered set. However, if
T
{\displaystyle T}
is a tree whose height is greater than the smallest infinite ordinal number
ω
{\displaystyle \omega }
, then the Hasse diagram definition does not work. For example, the partially ordered set
ω
+
1
=
{
0
,
1
,
2
,
…
,
ω
}
{\displaystyle \omega +1=\left\{0,1,2,\dots ,\omega \right\}}
does not have a Hasse Diagram, as there is no predecessor to
ω
{\displaystyle \omega }
. Hence a height of at most
ω
{\displaystyle \omega }
is required to define a graph-theoretic tree in this way.
A branch of a tree is a maximal chain in the tree (that is, any two elements of the branch are comparable, and any element of the tree not in the branch is incomparable with at least one element of the branch). The length of a branch is the ordinal that is order isomorphic to the branch. For each ordinal
α
{\displaystyle \alpha }
, the
α
{\displaystyle \alpha }
th level of
T
{\displaystyle T}
is the set of all elements of
T
{\displaystyle T}
of height
α
{\displaystyle \alpha }
. A tree is a
κ
{\displaystyle \kappa }
-tree, for an ordinal number
κ
{\displaystyle \kappa }
, if and only if it has height
κ
{\displaystyle \kappa }
and every level has cardinality less than the cardinality of
κ
{\displaystyle \kappa }
. The width of a tree is the supremum of the cardinalities of its levels.
Any single-rooted tree of height
≤
ω
{\displaystyle \leq \omega }
forms a meet-semilattice, where the meet (common predecessor) is given by the maximal element of the intersection of predecessors; this maximal element exists as the set of predecessors is non-empty and finite. Without a single root, the intersection of predecessors can be empty (two elements need not have common ancestors), for example
{
a
,
b
}
{\displaystyle \left\{a,b\right\}}
where the elements are not comparable; while if there are infinitely many predecessors there need not be a maximal element. An example is the tree
{
0
,
1
,
2
,
…
,
ω
0
,
ω
0
′
}
{\displaystyle \left\{0,1,2,\dots ,\omega _{0},\omega _{0}'\right\}}
where
ω
0
,
ω
0
′
{\displaystyle \omega _{0},\omega _{0}'}
are not comparable.
A subtree of a tree
(
T
,
<
)
{\displaystyle (T,<)}
is a tree
(
T
′
,
<
)
{\displaystyle (T',<)}
where
T
′
⊆
T
{\displaystyle T'\subseteq T}
and
T
′
{\displaystyle T'}
is downward closed under
<
{\displaystyle <}
, i.e., if
s
,
t
∈
T
{\displaystyle s,t\in T}
and
s
<
t
{\displaystyle s<t}
then
t
∈
T
′
⟹
s
∈
T
′
{\displaystyle t\in T'\implies s\in T'}
. The height of each element of a subtree equals its height in the whole tree. This differs from the notion of subtrees in graph theory, which often have different roots than the whole tree.
== Set-theoretic properties ==
There are some fairly simply stated yet hard problems in infinite tree theory. Examples of this are the Kurepa conjecture and the Suslin conjecture. Both of these problems are known to be independent of Zermelo–Fraenkel set theory. By Kőnig's lemma, every
ω
{\displaystyle \omega }
-tree has an infinite branch. On the other hand, it is a theorem of ZFC that there are uncountable trees with no uncountable branches and no uncountable levels; such trees are known as Aronszajn trees. Given a cardinal number
κ
{\displaystyle \kappa }
, a
κ
{\displaystyle \kappa }
-Suslin tree is a tree of height
κ
{\displaystyle \kappa }
which has no chains or antichains of size
κ
{\displaystyle \kappa }
. In particular, if
κ
{\displaystyle \kappa }
is a singular cardinal then there exists a
κ
{\displaystyle \kappa }
-Aronszajn tree and a
κ
{\displaystyle \kappa }
-Suslin tree. In fact, for any infinite cardinal
κ
{\displaystyle \kappa }
, every
κ
{\displaystyle \kappa }
-Suslin tree is a
κ
{\displaystyle \kappa }
-Aronszajn tree (the converse does not hold). One of the equivalent ways to define a weakly compact cardinal is that it is an inaccessible cardinal
κ
{\displaystyle \kappa }
that has the tree property, meaning that there is no
κ
{\displaystyle \kappa }
-Aronszajn tree.
The Suslin conjecture was originally stated as a question about certain total orderings but it is equivalent to the statement: Every tree of whose height is the first uncountable ordinal
ω
1
{\displaystyle \omega _{1}}
has an antichain of cardinality
ω
1
{\displaystyle \omega _{1}}
or a branch of length
ω
1
{\displaystyle \omega _{1}}
.
If
(
T
,
<
)
{\displaystyle (T,<)}
is a tree, then the reflexive closure
≤
{\displaystyle \leq }
of
<
{\displaystyle <}
is a prefix order on
T
{\displaystyle T}
.
The converse does not hold: for example, the usual order
≤
{\displaystyle \leq }
on the set
Z
{\displaystyle \mathbb {Z} }
of integers is a total and hence a prefix order, but
(
Z
,
<
)
{\displaystyle (\mathbb {Z} ,<)}
is not a set-theoretic tree since e.g. the set
{
n
∈
Z
:
n
<
0
}
{\displaystyle \{n\in \mathbb {Z} :n<0\}}
has no least element.
== Examples of infinite trees ==
Let
κ
{\displaystyle \kappa }
be an ordinal number, and let
X
{\displaystyle X}
be a set. Let
T
{\displaystyle T}
be set of all functions
f
:
α
↦
X
{\displaystyle f:\alpha \mapsto X}
where
α
<
κ
{\displaystyle \alpha <\kappa }
. Define
f
<
g
{\displaystyle f<g}
if the domain of
f
{\displaystyle f}
is a proper subset of the domain of
g
{\displaystyle g}
and the two functions agree on the domain of
f
{\displaystyle f}
. Then
(
T
,
<
)
{\displaystyle (T,<)}
is a set-theoretic tree. Its root is the unique function on the empty set, and its height is
κ
{\displaystyle \kappa }
. The union of all functions along a branch yields a function from
κ
{\displaystyle \kappa }
to
X
{\displaystyle X}
, that is, a generalized sequence of members of
X
{\displaystyle X}
. If
κ
{\displaystyle \kappa }
is a limit ordinal, none of the branches has a maximal element ("leaf"). The picture shows an example for
κ
=
ω
⋅
2
{\displaystyle \kappa =\omega \cdot 2}
and
X
=
{
0
,
1
}
{\displaystyle X=\{0,1\}}
.
Each tree data structure in computer science is a set-theoretic tree: for two nodes
m
,
n
{\displaystyle m,n}
, define
m
<
n
{\displaystyle m<n}
if
n
{\displaystyle n}
is a proper descendant of
m
{\displaystyle m}
. The notions of root, node height, and branch length coincide, while the notions of tree height differ by one.
Infinite trees considered in automata theory (see e.g. tree (automata theory)) are also set-theoretic trees, with a tree height of up to
ω
{\displaystyle \omega }
.
A graph-theoretic tree can be turned into a set-theoretic one by choosing a root node
r
{\displaystyle r}
and defining
m
<
n
{\displaystyle m<n}
if
m
≠
n
{\displaystyle m\neq n}
and
m
{\displaystyle m}
lies on the (unique) undirected path from
r
{\displaystyle r}
to
n
{\displaystyle n}
.
Each Cantor tree, each Kurepa tree, and each Laver tree is a set-theoretic tree.
== See also ==
Tree (descriptive set theory)
Continuous graph
== Notes ==
== Further reading ==
== External links ==
Sets, Models and Proofs by Ieke Moerdijk and Jaap van Oosten, see Definition 3.1 and Exercise 56 on pp. 68–69.
tree (set theoretic) by Henry on PlanetMath
branch by Henry on PlanetMath
example of tree (set theoretic) by uzeromay on PlanetMath | Wikipedia/Tree_(set_theory) |
In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers.
== Cardinal functions in set theory ==
The most frequently used cardinal function is the function that assigns to a set A its cardinality, denoted by |A|.
Aleph numbers and beth numbers can both be seen as cardinal functions defined on ordinal numbers.
Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers.
Cardinal characteristics of a (proper) ideal I of subsets of X are:
a
d
d
(
I
)
=
min
{
|
A
|
:
A
⊆
I
∧
⋃
A
∉
I
}
.
{\displaystyle {\rm {add}}(I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}\notin I\}.}
The "additivity" of I is the smallest number of sets from I whose union is not in I any more. As any ideal is closed under finite unions, this number is always at least
ℵ
0
{\displaystyle \aleph _{0}}
; if I is a σ-ideal, then
add
(
I
)
≥
ℵ
1
.
{\displaystyle \operatorname {add} (I)\geq \aleph _{1}.}
cov
(
I
)
=
min
{
|
A
|
:
A
⊆
I
∧
⋃
A
=
X
}
.
{\displaystyle \operatorname {cov} (I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}=X\}.}
The "covering number" of I is the smallest number of sets from I whose union is all of X. As X itself is not in I, we must have add(I ) ≤ cov(I ).
non
(
I
)
=
min
{
|
A
|
:
A
⊆
X
∧
A
∉
I
}
,
{\displaystyle \operatorname {non} (I)=\min\{|A|:A\subseteq X\ \wedge \ A\notin I\},}
The "uniformity number" of I (sometimes also written
u
n
i
f
(
I
)
{\displaystyle {\rm {unif}}(I)}
) is the size of the smallest set not in I. Assuming I contains all singletons, add(I ) ≤ non(I ).
c
o
f
(
I
)
=
min
{
|
B
|
:
B
⊆
I
∧
∀
A
∈
I
(
∃
B
∈
B
)
(
A
⊆
B
)
}
.
{\displaystyle {\rm {cof}}(I)=\min\{|{\mathcal {B}}|:{\mathcal {B}}\subseteq I\wedge \forall A\in I(\exists B\in {\mathcal {B}})(A\subseteq B)\}.}
The "cofinality" of I is the cofinality of the partial order (I, ⊆). It is easy to see that we must have non(I ) ≤ cof(I ) and cov(I ) ≤ cof(I ).
In the case that
I
{\displaystyle I}
is an ideal closely related to the structure of the reals, such as the ideal of Lebesgue null sets or the ideal of meagre sets, these cardinal invariants are referred to as cardinal characteristics of the continuum.
For a preordered set
(
P
,
⊑
)
{\displaystyle (\mathbb {P} ,\sqsubseteq )}
the bounding number
b
(
P
)
{\displaystyle {\mathfrak {b}}(\mathbb {P} )}
and dominating number
d
(
P
)
{\displaystyle {\mathfrak {d}}(\mathbb {P} )}
are defined as
b
(
P
)
=
min
{
|
Y
|
:
Y
⊆
P
∧
(
∀
x
∈
P
)
(
∃
y
∈
Y
)
(
y
⋢
x
)
}
,
{\displaystyle {\mathfrak {b}}(\mathbb {P} )=\min {\big \{}|Y|:Y\subseteq \mathbb {P} \ \wedge \ (\forall x\in \mathbb {P} )(\exists y\in Y)(y\not \sqsubseteq x){\big \}},}
d
(
P
)
=
min
{
|
Y
|
:
Y
⊆
P
∧
(
∀
x
∈
P
)
(
∃
y
∈
Y
)
(
x
⊑
y
)
}
.
{\displaystyle {\mathfrak {d}}(\mathbb {P} )=\min {\big \{}|Y|:Y\subseteq \mathbb {P} \ \wedge \ (\forall x\in \mathbb {P} )(\exists y\in Y)(x\sqsubseteq y){\big \}}.}
In PCF theory the cardinal function
p
p
κ
(
λ
)
{\displaystyle pp_{\kappa }(\lambda )}
is used.
== Cardinal functions in topology ==
Cardinal functions are widely used in topology as a tool for describing various topological properties. Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology", prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the definitions given below, for example by adding "
+
ℵ
0
{\displaystyle \;\;+\;\aleph _{0}}
" to the right-hand side of the definitions, etc.)
Perhaps the simplest cardinal invariants of a topological space
X
{\displaystyle X}
are its cardinality and the cardinality of its topology, denoted respectively by
|
X
|
{\displaystyle |X|}
and
o
(
X
)
.
{\displaystyle o(X).}
The weight
w
(
X
)
{\displaystyle \operatorname {w} (X)}
of a topological space
X
{\displaystyle X}
is the cardinality of the smallest base for
X
.
{\displaystyle X.}
When
w
(
X
)
=
ℵ
0
{\displaystyle \operatorname {w} (X)=\aleph _{0}}
the space
X
{\displaystyle X}
is said to be second countable.
The
π
{\displaystyle \pi }
-weight of a space
X
{\displaystyle X}
is the cardinality of the smallest
π
{\displaystyle \pi }
-base for
X
.
{\displaystyle X.}
(A
π
{\displaystyle \pi }
-base is a set of non-empty open sets whose supersets includes all opens.)
The network weight
nw
(
X
)
{\displaystyle \operatorname {nw} (X)}
of
X
{\displaystyle X}
is the smallest cardinality of a network for
X
.
{\displaystyle X.}
A network is a family
N
{\displaystyle {\mathcal {N}}}
of sets, for which, for all points
x
{\displaystyle x}
and open neighbourhoods
U
{\displaystyle U}
containing
x
,
{\displaystyle x,}
there exists
B
{\displaystyle B}
in
N
{\displaystyle {\mathcal {N}}}
for which
x
∈
B
⊆
U
.
{\displaystyle x\in B\subseteq U.}
The character of a topological space
X
{\displaystyle X}
at a point
x
{\displaystyle x}
is the cardinality of the smallest local base for
x
.
{\displaystyle x.}
The character of space
X
{\displaystyle X}
is
χ
(
X
)
=
sup
{
χ
(
x
,
X
)
:
x
∈
X
}
.
{\displaystyle \chi (X)=\sup \;\{\chi (x,X):x\in X\}.}
When
χ
(
X
)
=
ℵ
0
{\displaystyle \chi (X)=\aleph _{0}}
the space
X
{\displaystyle X}
is said to be first countable.
The density
d
(
X
)
{\displaystyle \operatorname {d} (X)}
of a space
X
{\displaystyle X}
is the cardinality of the smallest dense subset of
X
.
{\displaystyle X.}
When
d
(
X
)
=
ℵ
0
{\displaystyle {\rm {{d}(X)=\aleph _{0}}}}
the space
X
{\displaystyle X}
is said to be separable.
The Lindelöf number
L
(
X
)
{\displaystyle \operatorname {L} (X)}
of a space
X
{\displaystyle X}
is the smallest infinite cardinality such that every open cover has a subcover of cardinality no more than
L
(
X
)
.
{\displaystyle \operatorname {L} (X).}
When
L
(
X
)
=
ℵ
0
{\displaystyle {\rm {{L}(X)=\aleph _{0}}}}
the space
X
{\displaystyle X}
is said to be a Lindelöf space.
The cellularity or Suslin number of a space
X
{\displaystyle X}
is
c
(
X
)
=
sup
{
|
U
|
:
U
is a family of mutually disjoint non-empty open subsets of
X
}
.
{\displaystyle \operatorname {c} (X)=\sup\{|{\mathcal {U}}|:{\mathcal {U}}{\text{ is a family of mutually disjoint non-empty open subsets of }}X\}.}
The hereditary cellularity (sometimes called spread) is the least upper bound of cellularities of its subsets:
s
(
X
)
=
h
c
(
X
)
=
sup
{
c
(
Y
)
:
Y
⊆
X
}
{\displaystyle s(X)={\rm {hc}}(X)=\sup\{{\rm {c}}(Y):Y\subseteq X\}}
or
s
(
X
)
=
sup
{
|
Y
|
:
Y
⊆
X
with the subspace topology is discrete
}
{\displaystyle s(X)=\sup\{|Y|:Y\subseteq X{\text{ with the subspace topology is discrete}}\}}
where "discrete" means that it is a discrete topological space.
The extent of a space
X
{\displaystyle X}
is
e
(
X
)
=
sup
{
|
Y
|
:
Y
⊆
X
is closed and discrete
}
.
{\displaystyle e(X)=\sup\{|Y|:Y\subseteq X{\text{ is closed and discrete}}\}.}
So
X
{\displaystyle X}
has countable extent exactly when it has no uncountable closed discrete subset.
The tightness
t
(
x
,
X
)
{\displaystyle t(x,X)}
of a topological space
X
{\displaystyle X}
at a point
x
∈
X
{\displaystyle x\in X}
is the smallest cardinal number
α
{\displaystyle \alpha }
such that, whenever
x
∈
c
l
X
(
Y
)
{\displaystyle x\in {\rm {cl}}_{X}(Y)}
for some subset
Y
{\displaystyle Y}
of
X
,
{\displaystyle X,}
there exists a subset
Z
{\displaystyle Z}
of
Y
{\displaystyle Y}
with
|
Z
|
≤
α
,
{\displaystyle |Z|\leq \alpha ,}
such that
x
∈
cl
X
(
Z
)
.
{\displaystyle x\in \operatorname {cl} _{X}(Z).}
Symbolically,
t
(
x
,
X
)
=
sup
{
min
{
|
Z
|
:
Z
⊆
Y
∧
x
∈
c
l
X
(
Z
)
}
:
Y
⊆
X
∧
x
∈
c
l
X
(
Y
)
}
.
{\displaystyle t(x,X)=\sup \left\{\min\{|Z|:Z\subseteq Y\ \wedge \ x\in {\rm {cl}}_{X}(Z)\}:Y\subseteq X\ \wedge \ x\in {\rm {cl}}_{X}(Y)\right\}.}
The tightness of a space
X
{\displaystyle X}
is
t
(
X
)
=
sup
{
t
(
x
,
X
)
:
x
∈
X
}
.
{\displaystyle t(X)=\sup\{t(x,X):x\in X\}.}
When
t
(
X
)
=
ℵ
0
{\displaystyle t(X)=\aleph _{0}}
the space
X
{\displaystyle X}
is said to be countably generated or countably tight.
The augmented tightness of a space
X
,
{\displaystyle X,}
t
+
(
X
)
{\displaystyle t^{+}(X)}
is the smallest regular cardinal
α
{\displaystyle \alpha }
such that for any
Y
⊆
X
,
{\displaystyle Y\subseteq X,}
x
∈
c
l
X
(
Y
)
{\displaystyle x\in {\rm {cl}}_{X}(Y)}
there is a subset
Z
{\displaystyle Z}
of
Y
{\displaystyle Y}
with cardinality less than
α
,
{\displaystyle \alpha ,}
such that
x
∈
c
l
X
(
Z
)
.
{\displaystyle x\in {\rm {cl}}_{X}(Z).}
=== Basic inequalities ===
c
(
X
)
≤
d
(
X
)
≤
w
(
X
)
≤
o
(
X
)
≤
2
|
X
|
{\displaystyle c(X)\leq d(X)\leq w(X)\leq o(X)\leq 2^{|X|}}
e
(
X
)
≤
s
(
X
)
{\displaystyle e(X)\leq s(X)}
χ
(
X
)
≤
w
(
X
)
{\displaystyle \chi (X)\leq w(X)}
nw
(
X
)
≤
w
(
X
)
and
o
(
X
)
≤
2
nw
(
X
)
{\displaystyle \operatorname {nw} (X)\leq w(X){\text{ and }}o(X)\leq 2^{\operatorname {nw} (X)}}
== Cardinal functions in Boolean algebras ==
Cardinal functions are often used in the study of Boolean algebras. We can mention, for example, the following functions:
Cellularity
c
(
B
)
{\displaystyle c(\mathbb {B} )}
of a Boolean algebra
B
{\displaystyle \mathbb {B} }
is the supremum of the cardinalities of antichains in
B
{\displaystyle \mathbb {B} }
.
Length
l
e
n
g
t
h
(
B
)
{\displaystyle {\rm {length}}(\mathbb {B} )}
of a Boolean algebra
B
{\displaystyle \mathbb {B} }
is
l
e
n
g
t
h
(
B
)
=
sup
{
|
A
|
:
A
⊆
B
is a chain
}
{\displaystyle {\rm {length}}(\mathbb {B} )=\sup {\big \{}|A|:A\subseteq \mathbb {B} {\text{ is a chain}}{\big \}}}
Depth
d
e
p
t
h
(
B
)
{\displaystyle {\rm {depth}}(\mathbb {B} )}
of a Boolean algebra
B
{\displaystyle \mathbb {B} }
is
d
e
p
t
h
(
B
)
=
sup
{
|
A
|
:
A
⊆
B
is a well-ordered subset
}
{\displaystyle {\rm {depth}}(\mathbb {B} )=\sup {\big \{}|A|:A\subseteq \mathbb {B} {\text{ is a well-ordered subset}}{\big \}}}
.
Incomparability
I
n
c
(
B
)
{\displaystyle {\rm {Inc}}(\mathbb {B} )}
of a Boolean algebra
B
{\displaystyle \mathbb {B} }
is
I
n
c
(
B
)
=
sup
{
|
A
|
:
A
⊆
B
such that
∀
a
,
b
∈
A
(
a
≠
b
⇒
¬
(
a
≤
b
∨
b
≤
a
)
)
}
{\displaystyle {\rm {Inc}}({\mathbb {B} })=\sup {\big \{}|A|:A\subseteq \mathbb {B} {\text{ such that }}\forall a,b\in A{\big (}a\neq b\ \Rightarrow \neg (a\leq b\ \vee \ b\leq a){\big )}{\big \}}}
.
Pseudo-weight
π
(
B
)
{\displaystyle \pi (\mathbb {B} )}
of a Boolean algebra
B
{\displaystyle \mathbb {B} }
is
π
(
B
)
=
min
{
|
A
|
:
A
⊆
B
∖
{
0
}
such that
∀
b
∈
B
∖
{
0
}
(
∃
a
∈
A
)
(
a
≤
b
)
}
.
{\displaystyle \pi (\mathbb {B} )=\min {\big \{}|A|:A\subseteq \mathbb {B} \setminus \{0\}{\text{ such that }}\forall b\in B\setminus \{0\}{\big (}\exists a\in A{\big )}{\big (}a\leq b{\big )}{\big \}}.}
== Cardinal functions in algebra ==
Examples of cardinal functions in algebra are:
Index of a subgroup H of G is the number of cosets.
Dimension of a vector space V over a field K is the cardinality of any Hamel basis of V.
More generally, for a free module M over a ring R we define rank
r
a
n
k
(
M
)
{\displaystyle {\rm {rank}}(M)}
as the cardinality of any basis of this module.
For a linear subspace W of a vector space V we define codimension of W (with respect to V).
For any algebraic structure it is possible to consider the minimal cardinality of generators of the structure.
For algebraic field extensions, algebraic degree and separable degree are often employed (the algebraic degree equals the dimension of the extension as a vector space over the smaller field).
For non-algebraic field extensions, transcendence degree is likewise used.
== External links ==
A Glossary of Definitions from General Topology [1] [2]
== See also ==
Cichoń's diagram
== References ==
Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002. | Wikipedia/Cardinal_function |
In mathematics, a topological space
X
{\displaystyle X}
is called countably generated if the topology of
X
{\displaystyle X}
is determined by the countable sets in a similar way as the topology of a sequential space (or a Fréchet space) is determined by the convergent sequences.
The countably generated spaces are precisely the spaces having countable tightness—therefore the name countably tight is used as well.
== Definition ==
A topological space
X
{\displaystyle X}
is called countably generated if the topology on
X
{\displaystyle X}
is coherent with the family of its countable subspaces.
In other words, any subset
V
⊆
X
{\displaystyle V\subseteq X}
is closed in
X
{\displaystyle X}
whenever for each countable subspace
U
{\displaystyle U}
of
X
{\displaystyle X}
the set
V
∩
U
{\displaystyle V\cap U}
is closed in
U
;
{\displaystyle U;}
or equivalently, any subset
V
⊆
X
{\displaystyle V\subseteq X}
is open in
X
{\displaystyle X}
whenever for each countable subspace
U
{\displaystyle U}
of
X
{\displaystyle X}
the set
V
∩
U
{\displaystyle V\cap U}
is open in
U
.
{\displaystyle U.}
Equivalently,
X
{\displaystyle X}
is countably generated if and only if the closure of any
A
⊆
X
{\displaystyle A\subseteq X}
equals the union of the closures of all countable subsets of
A
.
{\displaystyle A.}
== Countable fan tightness ==
A topological space
X
{\displaystyle X}
has countable fan tightness if for every point
x
∈
X
{\displaystyle x\in X}
and every sequence
A
1
,
A
2
,
…
{\displaystyle A_{1},A_{2},\ldots }
of subsets of the space
X
{\displaystyle X}
such that
x
∈
⋂
n
A
n
¯
=
A
1
¯
∩
A
2
¯
∩
⋯
,
{\displaystyle x\in {\textstyle \bigcap \limits _{n}}\,{\overline {A_{n}}}={\overline {A_{1}}}\cap {\overline {A_{2}}}\cap \cdots ,}
there are finite set
B
1
⊆
A
1
,
B
2
⊆
A
2
,
…
{\displaystyle B_{1}\subseteq A_{1},B_{2}\subseteq A_{2},\ldots }
such that
x
∈
⋃
n
B
n
¯
=
B
1
∪
B
2
∪
⋯
¯
.
{\displaystyle x\in {\overline {{\textstyle \bigcup \limits _{n}}\,B_{n}}}={\overline {B_{1}\cup B_{2}\cup \cdots }}.}
A topological space
X
{\displaystyle X}
has countable strong fan tightness if for every point
x
∈
X
{\displaystyle x\in X}
and every sequence
A
1
,
A
2
,
…
{\displaystyle A_{1},A_{2},\ldots }
of subsets of the space
X
{\displaystyle X}
such that
x
∈
⋂
n
A
n
¯
=
A
1
¯
∩
A
2
¯
∩
⋯
,
{\displaystyle x\in {\textstyle \bigcap \limits _{n}}\,{\overline {A_{n}}}={\overline {A_{1}}}\cap {\overline {A_{2}}}\cap \cdots ,}
there are points
x
1
∈
A
1
,
x
2
∈
A
2
,
…
{\displaystyle x_{1}\in A_{1},x_{2}\in A_{2},\ldots }
such that
x
∈
{
x
1
,
x
2
,
…
}
¯
.
{\displaystyle x\in {\overline {\left\{x_{1},x_{2},\ldots \right\}}}.}
Every strong Fréchet–Urysohn space has strong countable fan tightness.
== Properties ==
A quotient of a countably generated space is again countably generated. Similarly, a topological sum of countably generated spaces is countably generated. Therefore, the countably generated spaces form a coreflective subcategory of the category of topological spaces. They are the coreflective hull of all countable spaces.
Any subspace of a countably generated space is again countably generated.
== Examples ==
Every sequential space (in particular, every metrizable space) is countably generated.
An example of a space which is countably generated but not sequential can be obtained, for instance, as a subspace of Arens–Fort space.
== See also ==
Finitely generated space – topology in which the intersection of any family of open sets is openPages displaying wikidata descriptions as a fallback
Locally closed subset
Tightness (topology) – Function that returns cardinal numbersPages displaying short descriptions of redirect targets − Tightness is a cardinal function related to countably generated spaces and their generalizations.
== References ==
Herrlich, Horst (1968). Topologische Reflexionen und Coreflexionen. Lecture Notes in Math. 78. Berlin: Springer.
== External links ==
A Glossary of Definitions from General Topology [1]
https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf | Wikipedia/Countably_generated_space |
In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers.
== Cardinal functions in set theory ==
The most frequently used cardinal function is the function that assigns to a set A its cardinality, denoted by |A|.
Aleph numbers and beth numbers can both be seen as cardinal functions defined on ordinal numbers.
Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers.
Cardinal characteristics of a (proper) ideal I of subsets of X are:
a
d
d
(
I
)
=
min
{
|
A
|
:
A
⊆
I
∧
⋃
A
∉
I
}
.
{\displaystyle {\rm {add}}(I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}\notin I\}.}
The "additivity" of I is the smallest number of sets from I whose union is not in I any more. As any ideal is closed under finite unions, this number is always at least
ℵ
0
{\displaystyle \aleph _{0}}
; if I is a σ-ideal, then
add
(
I
)
≥
ℵ
1
.
{\displaystyle \operatorname {add} (I)\geq \aleph _{1}.}
cov
(
I
)
=
min
{
|
A
|
:
A
⊆
I
∧
⋃
A
=
X
}
.
{\displaystyle \operatorname {cov} (I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}=X\}.}
The "covering number" of I is the smallest number of sets from I whose union is all of X. As X itself is not in I, we must have add(I ) ≤ cov(I ).
non
(
I
)
=
min
{
|
A
|
:
A
⊆
X
∧
A
∉
I
}
,
{\displaystyle \operatorname {non} (I)=\min\{|A|:A\subseteq X\ \wedge \ A\notin I\},}
The "uniformity number" of I (sometimes also written
u
n
i
f
(
I
)
{\displaystyle {\rm {unif}}(I)}
) is the size of the smallest set not in I. Assuming I contains all singletons, add(I ) ≤ non(I ).
c
o
f
(
I
)
=
min
{
|
B
|
:
B
⊆
I
∧
∀
A
∈
I
(
∃
B
∈
B
)
(
A
⊆
B
)
}
.
{\displaystyle {\rm {cof}}(I)=\min\{|{\mathcal {B}}|:{\mathcal {B}}\subseteq I\wedge \forall A\in I(\exists B\in {\mathcal {B}})(A\subseteq B)\}.}
The "cofinality" of I is the cofinality of the partial order (I, ⊆). It is easy to see that we must have non(I ) ≤ cof(I ) and cov(I ) ≤ cof(I ).
In the case that
I
{\displaystyle I}
is an ideal closely related to the structure of the reals, such as the ideal of Lebesgue null sets or the ideal of meagre sets, these cardinal invariants are referred to as cardinal characteristics of the continuum.
For a preordered set
(
P
,
⊑
)
{\displaystyle (\mathbb {P} ,\sqsubseteq )}
the bounding number
b
(
P
)
{\displaystyle {\mathfrak {b}}(\mathbb {P} )}
and dominating number
d
(
P
)
{\displaystyle {\mathfrak {d}}(\mathbb {P} )}
are defined as
b
(
P
)
=
min
{
|
Y
|
:
Y
⊆
P
∧
(
∀
x
∈
P
)
(
∃
y
∈
Y
)
(
y
⋢
x
)
}
,
{\displaystyle {\mathfrak {b}}(\mathbb {P} )=\min {\big \{}|Y|:Y\subseteq \mathbb {P} \ \wedge \ (\forall x\in \mathbb {P} )(\exists y\in Y)(y\not \sqsubseteq x){\big \}},}
d
(
P
)
=
min
{
|
Y
|
:
Y
⊆
P
∧
(
∀
x
∈
P
)
(
∃
y
∈
Y
)
(
x
⊑
y
)
}
.
{\displaystyle {\mathfrak {d}}(\mathbb {P} )=\min {\big \{}|Y|:Y\subseteq \mathbb {P} \ \wedge \ (\forall x\in \mathbb {P} )(\exists y\in Y)(x\sqsubseteq y){\big \}}.}
In PCF theory the cardinal function
p
p
κ
(
λ
)
{\displaystyle pp_{\kappa }(\lambda )}
is used.
== Cardinal functions in topology ==
Cardinal functions are widely used in topology as a tool for describing various topological properties. Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology", prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the definitions given below, for example by adding "
+
ℵ
0
{\displaystyle \;\;+\;\aleph _{0}}
" to the right-hand side of the definitions, etc.)
Perhaps the simplest cardinal invariants of a topological space
X
{\displaystyle X}
are its cardinality and the cardinality of its topology, denoted respectively by
|
X
|
{\displaystyle |X|}
and
o
(
X
)
.
{\displaystyle o(X).}
The weight
w
(
X
)
{\displaystyle \operatorname {w} (X)}
of a topological space
X
{\displaystyle X}
is the cardinality of the smallest base for
X
.
{\displaystyle X.}
When
w
(
X
)
=
ℵ
0
{\displaystyle \operatorname {w} (X)=\aleph _{0}}
the space
X
{\displaystyle X}
is said to be second countable.
The
π
{\displaystyle \pi }
-weight of a space
X
{\displaystyle X}
is the cardinality of the smallest
π
{\displaystyle \pi }
-base for
X
.
{\displaystyle X.}
(A
π
{\displaystyle \pi }
-base is a set of non-empty open sets whose supersets includes all opens.)
The network weight
nw
(
X
)
{\displaystyle \operatorname {nw} (X)}
of
X
{\displaystyle X}
is the smallest cardinality of a network for
X
.
{\displaystyle X.}
A network is a family
N
{\displaystyle {\mathcal {N}}}
of sets, for which, for all points
x
{\displaystyle x}
and open neighbourhoods
U
{\displaystyle U}
containing
x
,
{\displaystyle x,}
there exists
B
{\displaystyle B}
in
N
{\displaystyle {\mathcal {N}}}
for which
x
∈
B
⊆
U
.
{\displaystyle x\in B\subseteq U.}
The character of a topological space
X
{\displaystyle X}
at a point
x
{\displaystyle x}
is the cardinality of the smallest local base for
x
.
{\displaystyle x.}
The character of space
X
{\displaystyle X}
is
χ
(
X
)
=
sup
{
χ
(
x
,
X
)
:
x
∈
X
}
.
{\displaystyle \chi (X)=\sup \;\{\chi (x,X):x\in X\}.}
When
χ
(
X
)
=
ℵ
0
{\displaystyle \chi (X)=\aleph _{0}}
the space
X
{\displaystyle X}
is said to be first countable.
The density
d
(
X
)
{\displaystyle \operatorname {d} (X)}
of a space
X
{\displaystyle X}
is the cardinality of the smallest dense subset of
X
.
{\displaystyle X.}
When
d
(
X
)
=
ℵ
0
{\displaystyle {\rm {{d}(X)=\aleph _{0}}}}
the space
X
{\displaystyle X}
is said to be separable.
The Lindelöf number
L
(
X
)
{\displaystyle \operatorname {L} (X)}
of a space
X
{\displaystyle X}
is the smallest infinite cardinality such that every open cover has a subcover of cardinality no more than
L
(
X
)
.
{\displaystyle \operatorname {L} (X).}
When
L
(
X
)
=
ℵ
0
{\displaystyle {\rm {{L}(X)=\aleph _{0}}}}
the space
X
{\displaystyle X}
is said to be a Lindelöf space.
The cellularity or Suslin number of a space
X
{\displaystyle X}
is
c
(
X
)
=
sup
{
|
U
|
:
U
is a family of mutually disjoint non-empty open subsets of
X
}
.
{\displaystyle \operatorname {c} (X)=\sup\{|{\mathcal {U}}|:{\mathcal {U}}{\text{ is a family of mutually disjoint non-empty open subsets of }}X\}.}
The hereditary cellularity (sometimes called spread) is the least upper bound of cellularities of its subsets:
s
(
X
)
=
h
c
(
X
)
=
sup
{
c
(
Y
)
:
Y
⊆
X
}
{\displaystyle s(X)={\rm {hc}}(X)=\sup\{{\rm {c}}(Y):Y\subseteq X\}}
or
s
(
X
)
=
sup
{
|
Y
|
:
Y
⊆
X
with the subspace topology is discrete
}
{\displaystyle s(X)=\sup\{|Y|:Y\subseteq X{\text{ with the subspace topology is discrete}}\}}
where "discrete" means that it is a discrete topological space.
The extent of a space
X
{\displaystyle X}
is
e
(
X
)
=
sup
{
|
Y
|
:
Y
⊆
X
is closed and discrete
}
.
{\displaystyle e(X)=\sup\{|Y|:Y\subseteq X{\text{ is closed and discrete}}\}.}
So
X
{\displaystyle X}
has countable extent exactly when it has no uncountable closed discrete subset.
The tightness
t
(
x
,
X
)
{\displaystyle t(x,X)}
of a topological space
X
{\displaystyle X}
at a point
x
∈
X
{\displaystyle x\in X}
is the smallest cardinal number
α
{\displaystyle \alpha }
such that, whenever
x
∈
c
l
X
(
Y
)
{\displaystyle x\in {\rm {cl}}_{X}(Y)}
for some subset
Y
{\displaystyle Y}
of
X
,
{\displaystyle X,}
there exists a subset
Z
{\displaystyle Z}
of
Y
{\displaystyle Y}
with
|
Z
|
≤
α
,
{\displaystyle |Z|\leq \alpha ,}
such that
x
∈
cl
X
(
Z
)
.
{\displaystyle x\in \operatorname {cl} _{X}(Z).}
Symbolically,
t
(
x
,
X
)
=
sup
{
min
{
|
Z
|
:
Z
⊆
Y
∧
x
∈
c
l
X
(
Z
)
}
:
Y
⊆
X
∧
x
∈
c
l
X
(
Y
)
}
.
{\displaystyle t(x,X)=\sup \left\{\min\{|Z|:Z\subseteq Y\ \wedge \ x\in {\rm {cl}}_{X}(Z)\}:Y\subseteq X\ \wedge \ x\in {\rm {cl}}_{X}(Y)\right\}.}
The tightness of a space
X
{\displaystyle X}
is
t
(
X
)
=
sup
{
t
(
x
,
X
)
:
x
∈
X
}
.
{\displaystyle t(X)=\sup\{t(x,X):x\in X\}.}
When
t
(
X
)
=
ℵ
0
{\displaystyle t(X)=\aleph _{0}}
the space
X
{\displaystyle X}
is said to be countably generated or countably tight.
The augmented tightness of a space
X
,
{\displaystyle X,}
t
+
(
X
)
{\displaystyle t^{+}(X)}
is the smallest regular cardinal
α
{\displaystyle \alpha }
such that for any
Y
⊆
X
,
{\displaystyle Y\subseteq X,}
x
∈
c
l
X
(
Y
)
{\displaystyle x\in {\rm {cl}}_{X}(Y)}
there is a subset
Z
{\displaystyle Z}
of
Y
{\displaystyle Y}
with cardinality less than
α
,
{\displaystyle \alpha ,}
such that
x
∈
c
l
X
(
Z
)
.
{\displaystyle x\in {\rm {cl}}_{X}(Z).}
=== Basic inequalities ===
c
(
X
)
≤
d
(
X
)
≤
w
(
X
)
≤
o
(
X
)
≤
2
|
X
|
{\displaystyle c(X)\leq d(X)\leq w(X)\leq o(X)\leq 2^{|X|}}
e
(
X
)
≤
s
(
X
)
{\displaystyle e(X)\leq s(X)}
χ
(
X
)
≤
w
(
X
)
{\displaystyle \chi (X)\leq w(X)}
nw
(
X
)
≤
w
(
X
)
and
o
(
X
)
≤
2
nw
(
X
)
{\displaystyle \operatorname {nw} (X)\leq w(X){\text{ and }}o(X)\leq 2^{\operatorname {nw} (X)}}
== Cardinal functions in Boolean algebras ==
Cardinal functions are often used in the study of Boolean algebras. We can mention, for example, the following functions:
Cellularity
c
(
B
)
{\displaystyle c(\mathbb {B} )}
of a Boolean algebra
B
{\displaystyle \mathbb {B} }
is the supremum of the cardinalities of antichains in
B
{\displaystyle \mathbb {B} }
.
Length
l
e
n
g
t
h
(
B
)
{\displaystyle {\rm {length}}(\mathbb {B} )}
of a Boolean algebra
B
{\displaystyle \mathbb {B} }
is
l
e
n
g
t
h
(
B
)
=
sup
{
|
A
|
:
A
⊆
B
is a chain
}
{\displaystyle {\rm {length}}(\mathbb {B} )=\sup {\big \{}|A|:A\subseteq \mathbb {B} {\text{ is a chain}}{\big \}}}
Depth
d
e
p
t
h
(
B
)
{\displaystyle {\rm {depth}}(\mathbb {B} )}
of a Boolean algebra
B
{\displaystyle \mathbb {B} }
is
d
e
p
t
h
(
B
)
=
sup
{
|
A
|
:
A
⊆
B
is a well-ordered subset
}
{\displaystyle {\rm {depth}}(\mathbb {B} )=\sup {\big \{}|A|:A\subseteq \mathbb {B} {\text{ is a well-ordered subset}}{\big \}}}
.
Incomparability
I
n
c
(
B
)
{\displaystyle {\rm {Inc}}(\mathbb {B} )}
of a Boolean algebra
B
{\displaystyle \mathbb {B} }
is
I
n
c
(
B
)
=
sup
{
|
A
|
:
A
⊆
B
such that
∀
a
,
b
∈
A
(
a
≠
b
⇒
¬
(
a
≤
b
∨
b
≤
a
)
)
}
{\displaystyle {\rm {Inc}}({\mathbb {B} })=\sup {\big \{}|A|:A\subseteq \mathbb {B} {\text{ such that }}\forall a,b\in A{\big (}a\neq b\ \Rightarrow \neg (a\leq b\ \vee \ b\leq a){\big )}{\big \}}}
.
Pseudo-weight
π
(
B
)
{\displaystyle \pi (\mathbb {B} )}
of a Boolean algebra
B
{\displaystyle \mathbb {B} }
is
π
(
B
)
=
min
{
|
A
|
:
A
⊆
B
∖
{
0
}
such that
∀
b
∈
B
∖
{
0
}
(
∃
a
∈
A
)
(
a
≤
b
)
}
.
{\displaystyle \pi (\mathbb {B} )=\min {\big \{}|A|:A\subseteq \mathbb {B} \setminus \{0\}{\text{ such that }}\forall b\in B\setminus \{0\}{\big (}\exists a\in A{\big )}{\big (}a\leq b{\big )}{\big \}}.}
== Cardinal functions in algebra ==
Examples of cardinal functions in algebra are:
Index of a subgroup H of G is the number of cosets.
Dimension of a vector space V over a field K is the cardinality of any Hamel basis of V.
More generally, for a free module M over a ring R we define rank
r
a
n
k
(
M
)
{\displaystyle {\rm {rank}}(M)}
as the cardinality of any basis of this module.
For a linear subspace W of a vector space V we define codimension of W (with respect to V).
For any algebraic structure it is possible to consider the minimal cardinality of generators of the structure.
For algebraic field extensions, algebraic degree and separable degree are often employed (the algebraic degree equals the dimension of the extension as a vector space over the smaller field).
For non-algebraic field extensions, transcendence degree is likewise used.
== External links ==
A Glossary of Definitions from General Topology [1] [2]
== See also ==
Cichoń's diagram
== References ==
Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002. | Wikipedia/Character_(topology) |
PCF theory is the name of a mathematical theory, introduced by Saharon Shelah (1978), that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more applications as well. The abbreviation "PCF" stands for "possible cofinalities".
== Main definitions ==
If A is an infinite set of regular cardinals, D is an ultrafilter on A, then
we let
cf
(
∏
A
/
D
)
{\displaystyle \operatorname {cf} \left(\prod A/D\right)}
denote the cofinality of the ordered set of functions
∏
A
{\displaystyle \prod A}
where the ordering is defined as follows:
f
<
g
{\displaystyle f<g}
if
{
x
∈
A
:
f
(
x
)
<
g
(
x
)
}
∈
D
{\displaystyle \{x\in A:f(x)<g(x)\}\in D}
.
pcf(A) is the set of cofinalities that occur if we consider all ultrafilters on A, that is,
== Main results ==
Obviously, pcf(A) consists of regular cardinals. Considering ultrafilters concentrated on elements of A, we get that
A
⊆
pcf
(
A
)
{\displaystyle A\subseteq \operatorname {pcf} (A)}
. Shelah proved, that if
|
A
|
<
min
(
A
)
{\displaystyle |A|<\min(A)}
, then pcf(A) has a largest element, and there are subsets
{
B
θ
:
θ
∈
pcf
(
A
)
}
{\displaystyle \{B_{\theta }:\theta \in \operatorname {pcf} (A)\}}
of A such that for each ultrafilter D on A,
cf
(
∏
A
/
D
)
{\displaystyle \operatorname {cf} \left(\prod A/D\right)}
is the least element θ of pcf(A) such that
B
θ
∈
D
{\displaystyle B_{\theta }\in D}
. Consequently,
|
pcf
(
A
)
|
≤
2
|
A
|
{\displaystyle \left|\operatorname {pcf} (A)\right|\leq 2^{|A|}}
.
Shelah also proved that if A is an interval of regular cardinals (i.e., A is the set of all regular cardinals between two cardinals), then pcf(A) is also an interval of regular cardinals and |pcf(A)|<|A|+4.
This implies the famous inequality
assuming that ℵω is strong limit.
If λ is an infinite cardinal, then J<λ is the following ideal on A. B∈J<λ if
cf
(
∏
A
/
D
)
<
λ
{\displaystyle \operatorname {cf} \left(\prod A/D\right)<\lambda }
holds for every ultrafilter D with B∈D. Then J<λ is the ideal generated by the sets
{
B
θ
:
θ
∈
pcf
(
A
)
,
θ
<
λ
}
{\displaystyle \{B_{\theta }:\theta \in \operatorname {pcf} (A),\theta <\lambda \}}
. There exist scales, i.e., for every λ∈pcf(A) there is a sequence of length λ of elements of
∏
B
λ
{\displaystyle \prod B_{\lambda }}
which is both increasing and cofinal mod J<λ. This implies that the cofinality of
∏
A
{\displaystyle \prod A}
under pointwise dominance is max(pcf(A)).
Another consequence is that if λ is singular and no regular cardinal less than λ is Jónsson, then also λ+ is not Jónsson. In particular, there is a Jónsson algebra on ℵω+1, which settles an old conjecture.
== Unsolved problems ==
The most notorious conjecture in pcf theory states that |pcf(A)|=|A| holds for every set A of regular cardinals with |A|<min(A). This would imply that if ℵω is strong limit, then the sharp bound
holds. The analogous bound
follows from Chang's conjecture (Magidor) or even from the nonexistence of a Kurepa tree (Shelah).
A weaker, still unsolved conjecture states that if |A|<min(A), then pcf(A) has no inaccessible limit point. This is equivalent to the statement that pcf(pcf(A))=pcf(A).
== Applications ==
The theory has found a great deal of applications, besides cardinal arithmetic.
The original survey by Shelah, Cardinal arithmetic for skeptics, includes the following topics: almost free abelian groups, partition problems, failure of preservation of chain conditions in Boolean algebras under products, existence of Jónsson algebras, existence of entangled linear orders, equivalently narrow Boolean algebras, and the existence of nonisomorphic models equivalent in certain infinitary logics.
In the meantime, many further applications have been found in Set Theory, Model Theory, Algebra and Topology.
== References ==
Saharon Shelah, Cardinal Arithmetic, Oxford Logic Guides, vol. 29. Oxford University Press, 1994.
== External links ==
Menachem Kojman: PCF Theory
Shelah, Saharon (1978), "Jonsson algebras in successor cardinals", Israel Journal of Mathematics, 30 (1): 57–64, doi:10.1007/BF02760829, MR 0505434
Shelah, Saharon (1992), "Cardinal arithmetic for skeptics", Bulletin of the American Mathematical Society, New Series, 26 (2): 197–210, arXiv:math/9201251, doi:10.1090/s0273-0979-1992-00261-6, MR 1112424 | Wikipedia/PCF_theory |
In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space X is a Moore space if the following conditions hold:
Any two distinct points can be separated by neighbourhoods, and any closed set and any point in its complement can be separated by neighbourhoods. (X is a regular Hausdorff space.)
There is a countable collection of open covers of X, such that for any closed set C and any point p in its complement there exists a cover in the collection such that every neighbourhood of p in the cover is disjoint from C. (X is a developable space.)
Moore spaces are generally interesting in mathematics because they may be applied to prove interesting metrization theorems. The concept of a Moore space was formulated by R. L. Moore in the earlier part of the 20th century.
== Examples and properties ==
Every metrizable space, X, is a Moore space. If {A(n)x} is the open cover of X (indexed by x in X) by all balls of radius 1/n, then the collection of all such open covers as n varies over the positive integers is a development of X. Since all metrizable spaces are normal, all metric spaces are Moore spaces.
Moore spaces are a lot like regular spaces and different from normal spaces in the sense that every subspace of a Moore space is also a Moore space.
The image of a Moore space under an injective, continuous open map is always a Moore space. (The image of a regular space under an injective, continuous open map is always regular.)
Both examples 2 and 3 suggest that Moore spaces are similar to regular spaces.
Neither the Sorgenfrey line nor the Sorgenfrey plane are Moore spaces because they are normal and not second countable.
The Moore plane (also known as the Niemytski plane) is an example of a non-metrizable Moore space.
Every metacompact, separable, normal Moore space is metrizable. This theorem is known as Traylor’s theorem.
Every locally compact, locally connected normal Moore space is metrizable. This theorem was proved by Reed and Zenor.
If
2
ℵ
0
<
2
ℵ
1
{\displaystyle 2^{\aleph _{0}}<2^{\aleph _{1}}}
, then every separable normal Moore space is metrizable. This theorem is known as Jones’ theorem.
== Normal Moore space conjecture ==
For a long time, topologists were trying to prove the so-called normal Moore space conjecture: every normal Moore space is metrizable. This was inspired by the fact that all known Moore spaces that were not metrizable were also not normal. This would have been a nice metrization theorem. There were some nice partial results at first; namely properties 7, 8 and 9 as given in the previous section.
With property 9, we see that we can drop metacompactness from Traylor's theorem, but at the cost of a set-theoretic assumption. Another example of this is Fleissner's theorem that the axiom of constructibility implies that locally compact, normal Moore spaces are metrizable.
On the other hand, under the continuum hypothesis (CH) and also under Martin's axiom and not CH, there are several examples of non-metrizable normal Moore spaces. Nyikos proved that, under the so-called PMEA (Product Measure Extension Axiom), which needs a large cardinal, all normal Moore spaces are metrizable. Finally, it was shown later that any model of ZFC in which the conjecture holds, implies the existence of a model with a large cardinal. So large cardinals are needed essentially.
Jones (1937) gave an example of a pseudonormal Moore space that is not metrizable, so the conjecture cannot be strengthened in this way.
Moore himself proved the theorem that a collectionwise normal Moore space is metrizable, so strengthening normality is another way to settle the matter.
== References ==
Lynn Arthur Steen and J. Arthur Seebach, Counterexamples in Topology, Dover Books, 1995. ISBN 0-486-68735-X
Jones, F. B. (1937), "Concerning normal and completely normal spaces" (PDF), Bulletin of the American Mathematical Society, 43 (10): 671–677, doi:10.1090/S0002-9904-1937-06622-5, MR 1563615.
Nyikos, Peter J. (2001), "A history of the normal Moore space problem", Handbook of the History of General Topology, Hist. Topol., vol. 3, Dordrecht: Kluwer Academic Publishers, pp. 1179–1212, ISBN 9780792369707, MR 1900271.
The original definition by R.L. Moore appears here:
MR0150722 (27 #709) Moore, R. L. Foundations of point set theory. Revised edition. American Mathematical Society Colloquium Publications, Vol. XIII American Mathematical Society, Providence, R.I. 1962 xi+419 pp. (Reviewer: F. Burton Jones)
Historical information can be found here:
MR0199840 (33 #7980) Jones, F. Burton "Metrization". American Mathematical Monthly 73 1966 571–576. (Reviewer: R. W. Bagley)
Historical information can be found here:
MR0203661 (34 #3510) Bing, R. H. "Challenging conjectures". American Mathematical Monthly 74 1967 no. 1, part II, 56–64;
Vickery's theorem may be found here:
MR0001909 (1,317f) Vickery, C. W. "Axioms for Moore spaces and metric spaces". Bulletin of the American Mathematical Society 46, (1940). 560–564
This article incorporates material from Moore space on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. | Wikipedia/Normal_Moore_space_conjecture |
The Journal of Mechanics of Materials and Structures is a peer-reviewed scientific journal covering research on the mechanics of materials and deformable structures of all types. It was established by Charles R. Steele, who was also the first editor-in-chief.
== History ==
The journal was established in 2006 after 21 of the 23 members of the editorial board of the International Journal of Solids and Structures resigned in protest of Elsevier's "pressure for increased profits out of the limited institutional resources." In their founding issue, the editors of the new journal indicated several desires for the publication, including, "a low subscription price that will not grow faster than the number of pages and indeed may drop as the subscriber base expands."
== Abstracting and indexing ==
The journal is abstracted and indexed in Current Contents/Engineering, Computing & Technology, Ei Compendex, Science Citation Index Expanded, and Scopus. According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.987.
== References ==
== External links ==
Official website | Wikipedia/Journal_of_Mechanics_of_Materials_and_Structures |
Algebra & Number Theory is a peer-reviewed mathematics journal published by the nonprofit organization Mathematical Sciences Publishers. It was launched on January 17, 2007, with the goal of "providing an alternative to the current range of commercial specialty journals in algebra and number theory, an alternative of higher quality and much lower cost."
The journal publishes original research articles in algebra and number theory, interpreted broadly, including algebraic geometry and arithmetic geometry, for example. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five generalist mathematics journals. Currently, it is regarded as the best journal specializing in number theory.
Issues are published both online and in print.
== Editorial board ==
The Managing Editor is Antoine Chambert-Loir of Paris Cité University, and the Editorial Board Chair is David Eisenbud of U. C. Berkeley.
== See also ==
Jonathan Pila
== References ==
== External links ==
Official website
Mathematical Sciences Publishers | Wikipedia/Algebra_&_Number_Theory |
Mathematics and Mechanics of Complex Systems (MEMOCS) is a quarterly peer-reviewed scientific journal founded by the International Research Center for the Mathematics and Mechanics of Complex Systems (M&MoCS) from Università degli Studi dell'Aquila, in Italy. It is published by Mathematical Sciences Publishers, and first issued in February 2013. The co-chairs of the editorial board are Francesco dell'Isola and Gilles Francfort, and chair managing editor is Martin Ostoja-Starzewski.
MEMOCS is indexed in Scopus, MathSciNet and Zentralblatt MATH.
It is open access, free of author charges (being supported by grants from academic institutions), and available in both printed and electronic forms.
== Contents ==
MEMOCS publishes articles from diverse scientific fields with a specific emphasis on mechanics. Its contents rely on the application or development of rigorous mathematical methods.
The journal also publishes original research in related areas of mathematics of well-established applicability, such as variational methods, numerical methods, and optimization techniques, as well as papers focusing on and clarifying particular aspects of the history of mathematics and science.
Among the contributors are Graeme Milton, Geoffrey Grimmett, David Steigmann, Mario Pulvirenti and Lucio Russo.
== References ==
== External links ==
Official website
Editorial Board
International Research Center on Mathematics and Mechanics of Complex Systems - M&MoCS | Wikipedia/Mathematics_and_Mechanics_of_Complex_Systems |
In molecular physics and chemistry, the van der Waals force (sometimes van der Waals' force) is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and therefore more susceptible to disturbance. The van der Waals force quickly vanishes at longer distances between interacting molecules.
Named after Dutch physicist Johannes Diderik van der Waals, the van der Waals force plays a fundamental role in fields as diverse as supramolecular chemistry, structural biology, polymer science, nanotechnology, surface science, and condensed matter physics. It also underlies many properties of organic compounds and molecular solids, including their solubility in polar and non-polar media.
If no other force is present, the distance between atoms at which the force becomes repulsive rather than attractive as the atoms approach one another is called the van der Waals contact distance; this phenomenon results from the mutual repulsion between the atoms' electron clouds.
The van der Waals forces are usually described as a combination of the London dispersion forces between "instantaneously induced dipoles", Debye forces between permanent dipoles and induced dipoles, and the Keesom force between permanent molecular dipoles whose rotational orientations are dynamically averaged over time.
== Definition ==
Van der Waals forces include attraction and repulsions between atoms, molecules, as well as other intermolecular forces. They differ from covalent and ionic bonding in that they are caused by correlations in the fluctuating polarizations of nearby particles (a consequence of quantum dynamics).
The force results from a transient shift in electron density. Specifically, the electron density may temporarily shift to be greater on one side of the nucleus. This shift generates a transient charge which a nearby atom can be attracted to or repelled by. The force is repulsive at very short distances, reaches zero at an equilibrium distance characteristic for each atom, or molecule, and becomes attractive for distances larger than the equilibrium distance. For individual atoms, the equilibrium distance is between 0.3 nm and 0.5 nm, depending on the atomic-specific diameter. When the interatomic distance is greater than 1.0 nm the force is not strong enough to be easily observed as it decreases as a function of distance r approximately with the 7th power (~r−7).
Van der Waals forces are often among the weakest chemical forces. For example, the pairwise attractive van der Waals interaction energy between H (hydrogen) atoms in different H2 molecules equals 0.06 kJ/mol (0.6 meV) and the pairwise attractive interaction energy between O (oxygen) atoms in different O2 molecules equals 0.44 kJ/mol (4.6 meV). The corresponding vaporization energies of H2 and O2 molecular liquids, which result as a sum of all van der Waals interactions per molecule in the molecular liquids, amount to 0.90 kJ/mol (9.3 meV) and 6.82 kJ/mol (70.7 meV), respectively, and thus approximately 15 times the value of the individual pairwise interatomic interactions (excluding covalent bonds).
The strength of van der Waals bonds increases with higher polarizability of the participating atoms. For example, the pairwise van der Waals interaction energy for more polarizable atoms such as S (sulfur) atoms in H2S and sulfides exceeds 1 kJ/mol (10 meV), and the pairwise interaction energy between even larger, more polarizable Xe (xenon) atoms is 2.35 kJ/mol (24.3 meV). These van der Waals interactions are up to 40 times stronger than in H2, which has only one valence electron, and they are still not strong enough to achieve an aggregate state other than gas for Xe under standard conditions. The interactions between atoms in metals can also be effectively described as van der Waals interactions and account for the observed solid aggregate state with bonding strengths comparable to covalent and ionic interactions. The strength of pairwise van der Waals type interactions is on the order of 12 kJ/mol (120 meV) for low-melting Pb (lead) and on the order of 32 kJ/mol (330 meV) for high-melting Pt (platinum), which is about one order of magnitude stronger than in Xe due to the presence of a highly polarizable free electron gas. Accordingly, van der Waals forces can range from weak to strong interactions, and support integral structural loads when multitudes of such interactions are present.
=== Force contributions ===
More broadly, intermolecular forces have several possible contributions. They are ordered from strongest to weakest:
A repulsive component resulting from the Pauli exclusion principle that prevents close contact of atoms, or the collapse of molecules.
Attractive or repulsive electrostatic interactions between permanent charges (in the case of molecular ions), dipoles (in the case of molecules without inversion centre), quadrupoles (all molecules with symmetry lower than cubic), and in general between permanent multipoles. These interactions also include hydrogen bonds, cation-pi, and pi-stacking interactions. Orientation-averaged contributions from electrostatic interactions are sometimes called the Keesom interaction or Keesom force after Willem Hendrik Keesom.
Induction (also known as polarization), which is the attractive interaction between a permanent multipole on one molecule with an induced multipole on another. This interaction is sometimes called Debye force after Peter J. W. Debye. The interactions (2) and (3) are labelled polar Interactions.
Dispersion (usually named London dispersion interactions after Fritz London), which is the attractive interaction between any pair of molecules, including non-polar atoms, arising from the interactions of instantaneous multipoles.
When to apply the term "van der Waals" force depends on the text. The broadest definitions include all intermolecular forces which are electrostatic in origin, namely (2), (3) and (4). Some authors, whether or not they consider other forces to be of van der Waals type, focus on (3) and (4) as these are the components which act over the longest range.
All intermolecular/van der Waals forces are anisotropic (except those between two noble gas atoms), which means that they depend on the relative orientation of the molecules. The induction and dispersion interactions are always attractive, irrespective of orientation, but the electrostatic interaction changes sign upon rotation of the molecules. That is, the electrostatic force can be attractive or repulsive, depending on the mutual orientation of the molecules. When molecules are in thermal motion, as they are in the gas and liquid phase, the electrostatic force is averaged out to a large extent because the molecules thermally rotate and thus probe both repulsive and attractive parts of the electrostatic force. Random thermal motion can disrupt or overcome the electrostatic component of the van der Waals force but the averaging effect is much less pronounced for the attractive induction and dispersion forces.
The Lennard-Jones potential is often used as an approximate model for the isotropic part of a total (repulsion plus attraction) van der Waals force as a function of distance.
Van der Waals forces are responsible for certain cases of pressure broadening (van der Waals broadening) of spectral lines and the formation of van der Waals molecules. The London–van der Waals forces are related to the Casimir effect for dielectric media, the former being the microscopic description of the latter bulk property. The first detailed calculations of this were done in 1955 by E. M. Lifshitz. A more general theory of van der Waals forces has also been developed.
The main characteristics of van der Waals forces are:
They are weaker than normal covalent and ionic bonds.
The van der Waals forces are additive in nature, consisting of several individual interactions, and cannot be saturated.
They have no directional characteristic.
They are all short-range forces and hence only interactions between the nearest particles need to be considered (instead of all the particles). Van der Waals attraction is greater if the molecules are closer.
Van der Waals forces are independent of temperature except for dipole-dipole interactions.
In low molecular weight alcohols, the hydrogen-bonding properties of their polar hydroxyl group dominate other weaker van der Waals interactions. In higher molecular weight alcohols, the properties of the nonpolar hydrocarbon chain(s) dominate and determine their solubility.
Van der Waals forces are also responsible for the weak hydrogen bond interactions between unpolarized dipoles particularly in acid-base aqueous solution and between biological molecules.
== London dispersion force ==
London dispersion forces, named after the German-American physicist Fritz London, are weak intermolecular forces that arise from the interactive forces between instantaneous multipoles in molecules without permanent multipole moments. In and between organic molecules the multitude of contacts can lead to larger contribution of dispersive attraction, particularly in the presence of heteroatoms. London dispersion forces are also known as 'dispersion forces', 'London forces', or 'instantaneous dipole–induced dipole forces'. The strength of London dispersion forces is proportional to the polarizability of the molecule, which in turn depends on the total number of electrons and the area over which they are spread. Hydrocarbons display small dispersive contributions, the presence of heteroatoms lead to increased LD forces as function of their polarizability, e.g. in the sequence RI>RBr>RCl>RF. In absence of solvents weakly polarizable hydrocarbons form crystals due to dispersive forces; their sublimation heat is a measure of the dispersive interaction.
== Van der Waals forces between macroscopic objects ==
For macroscopic bodies with known volumes and numbers of atoms or molecules per unit volume, the total van der Waals force is often computed based on the "microscopic theory" as the sum over all interacting pairs. It is necessary to integrate over the total volume of the object, which makes the calculation dependent on the objects' shapes. For example, the van der Waals interaction energy between spherical bodies of radii R1 and R2 and with smooth surfaces was approximated in 1937 by Hamaker (using London's famous 1937 equation for the dispersion interaction energy between atoms/molecules as the starting point) by:
where A is the Hamaker coefficient, which is a constant (~10−19 − 10−20 J) that depends on the material properties (it can be positive or negative in sign depending on the intervening medium), and z is the center-to-center distance; i.e., the sum of R1, R2, and r (the distance between the surfaces):
z
=
R
1
+
R
2
+
r
{\displaystyle \ z=R_{1}+R_{2}+r}
.
The van der Waals force between two spheres of constant radii (R1 and R2 are treated as parameters) is then a function of separation since the force on an object is the negative of the derivative of the potential energy function,
F
V
d
W
(
z
)
=
−
d
d
z
U
(
z
)
{\displaystyle \ F_{\rm {VdW}}(z)=-{\frac {d}{dz}}U(z)}
. This yields:
In the limit of close-approach, the spheres are sufficiently large compared to the distance between them; i.e.,
r
≪
R
1
{\displaystyle \ r\ll R_{1}}
or
R
2
{\displaystyle R_{2}}
, so that equation (1) for the potential energy function simplifies to:
with the force:
The van der Waals forces between objects with other geometries using the Hamaker model have been published in the literature.
From the expression above, it is seen that the van der Waals force decreases with decreasing size of bodies (R). Nevertheless, the strength of inertial forces, such as gravity and drag/lift, decrease to a greater extent. Consequently, the van der Waals forces become dominant for collections of very small particles such as very fine-grained dry powders (where there are no capillary forces present) even though the force of attraction is smaller in magnitude than it is for larger particles of the same substance. Such powders are said to be cohesive, meaning they are not as easily fluidized or pneumatically conveyed as their more coarse-grained counterparts. Generally, free-flow occurs with particles greater than about 250 μm.
The van der Waals force of adhesion is also dependent on the surface topography. If there are surface asperities, or protuberances, that result in a greater total area of contact between two particles or between a particle and a wall, this increases the van der Waals force of attraction as well as the tendency for mechanical interlocking.
The microscopic theory assumes pairwise additivity. It neglects many-body interactions and retardation. A more rigorous approach accounting for these effects, called the "macroscopic theory", was developed by Lifshitz in 1956. Langbein derived a much more cumbersome "exact" expression in 1970 for spherical bodies within the framework of the Lifshitz theory while a simpler macroscopic model approximation had been made by Derjaguin as early as 1934. Expressions for the van der Waals forces for many different geometries using the Lifshitz theory have likewise been published.
== Use by geckos and arthropods ==
The ability of geckos – which can hang on a glass surface using only one toe – to climb on sheer surfaces has been for many years mainly attributed to the van der Waals forces between these surfaces and the spatulae, or microscopic projections, which cover the hair-like setae found on their footpads.
There were efforts in 2008 to create a dry glue that exploits the effect, and success was achieved in 2011 to create an adhesive tape on similar grounds (i.e. based on van der Waals forces). In 2011, a paper was published relating the effect to both velcro-like hairs and the presence of lipids in gecko footprints.
A later study suggested that capillary adhesion might play a role, but that hypothesis has been rejected by more recent studies.
A 2014 study has shown that gecko adhesion to smooth Teflon and polydimethylsiloxane surfaces is mainly determined by electrostatic interaction (caused by contact electrification), not van der Waals or capillary forces.
Among the arthropods, some spiders have similar setae on their scopulae or scopula pads, enabling them to climb or hang upside-down from extremely smooth surfaces such as glass or porcelain.
== See also ==
== References ==
== Further reading ==
Brevik, Iver; Marachevsky, V. N.; Milton, Kimball A. (1999). "Identity of the van der Waals Force and the Casimir Effect and the Irrelevance of These Phenomena to Sonoluminescence". Physical Review Letters. 82 (20): 3948–3951. arXiv:hep-th/9810062. Bibcode:1999PhRvL..82.3948B. doi:10.1103/PhysRevLett.82.3948. S2CID 14762105.
Dzyaloshinskii, I. D.; Lifshitz, E. M.; Pitaevskii, Lev P. (1961). "Общая теория ван-дер-ваальсовых сил" [General theory of van der Waals forces] (PDF). Uspekhi Fizicheskikh Nauk (in Russian). 73 (381).
English translation: Dzyaloshinskii, I. D.; Lifshitz, E. M.; Pitaevskii, L. P. (1961). "General theory of van der Waalsforces". Soviet Physics Uspekhi. 4 (2): 153. Bibcode:1961SvPhU...4..153D. doi:10.1070/PU1961v004n02ABEH003330.
Landau, L. D.; Lifshitz, E. M. (1960). Electrodynamics of Continuous Media. Oxford: Pergamon. pp. 368–376.
Langbein, Dieter (1974). Theory of Van der Waals Attraction. Springer Tracts in Modern Physics. Vol. 72. New York, Heidelberg: Springer-Verlag.
Lefers, Mark. "Van der Waals dispersion force". Life Science Glossary. Holmgren Lab. Archived from the original on 24 July 2019. Retrieved 2 October 2017.
Lifshitz, E. M. (1955). "Russian title is missing" [The Theory of Molecular Attractive Forces between Solids]. Zhurnal Éksperimental'noĭ i Teoreticheskoĭ Fiziki (in Russian). 29 (1): 94.
English translation: Lifshitz, E. M. (January 1956). "The Theory of Molecular Attractive Forces between Solids" (PDF). Soviet Physics. 2 (1): 73. Archived from the original (PDF) on 13 July 2019. Retrieved 8 August 2020.
"London force animation". Intermolecular Forces. Western Oregon University.
Lyklema, J. Fundamentals of Interface and Colloid Science. p. 4.43.
Israelachvili, Jacob N. (1992). Intermolecular and Surface Forces. Academic Press. ISBN 9780123751812.
== External links ==
Senese, Fred (1999). "What are van der Waals forces?". Frostburg State University. Retrieved 1 March 2010. An introductory description of the van der Waals force (as a sum of attractive components only)
"Robert Full: Learning from the gecko's tail". TED. 1 February 2009. Retrieved 5 October 2016. TED Talk on biomimicry, including applications of van der Waals force.
Wolff, J. O.; Gorb, S. N. (18 May 2011). "The influence of humidity on the attachment ability of the spider Philodromus dispar (Araneae, Philodromidae)". Proceedings of the Royal Society B: Biological Sciences. 279 (1726): 139–143. doi:10.1098/rspb.2011.0505. PMC 3223641. PMID 21593034. | Wikipedia/Van_der_Waals_force |
In continuum mechanics, the Flamant solution provides expressions for the stresses and displacements in a linear elastic wedge loaded by point forces at its sharp end. This solution was developed by Alfred-Aimé Flamant in 1892 by modifying the three dimensional solutions for linear elasticity of Joseph Valentin Boussinesq.
The stresses predicted by the Flamant solution are (in polar coordinates)
σ
r
r
=
2
C
1
cos
θ
r
+
2
C
3
sin
θ
r
σ
r
θ
=
0
σ
θ
θ
=
0
{\displaystyle {\begin{aligned}\sigma _{rr}&={\frac {2C_{1}\cos \theta }{r}}+{\frac {2C_{3}\sin \theta }{r}}\\\sigma _{r\theta }&=0\\\sigma _{\theta \theta }&=0\end{aligned}}}
where
C
1
,
C
3
{\displaystyle C_{1},C_{3}}
are constants that are determined from the boundary conditions and the geometry of the wedge (i.e., the angles
α
,
β
{\displaystyle \alpha ,\beta }
) and satisfy
F
1
+
2
∫
α
β
(
C
1
cos
θ
+
C
3
sin
θ
)
cos
θ
d
θ
=
0
F
2
+
2
∫
α
β
(
C
1
cos
θ
+
C
3
sin
θ
)
sin
θ
d
θ
=
0
{\displaystyle {\begin{aligned}F_{1}&+2\int _{\alpha }^{\beta }(C_{1}\cos \theta +C_{3}\sin \theta )\,\cos \theta \,d\theta =0\\F_{2}&+2\int _{\alpha }^{\beta }(C_{1}\cos \theta +C_{3}\sin \theta )\,\sin \theta \,d\theta =0\end{aligned}}}
where
F
1
,
F
2
{\displaystyle F_{1},F_{2}}
are the applied forces.
The wedge problem is self-similar and has no inherent length scale. Also, all quantities can be expressed in the separated-variable form
σ
=
f
(
r
)
g
(
θ
)
{\displaystyle \sigma =f(r)g(\theta )}
. The stresses vary as
(
1
/
r
)
{\displaystyle (1/r)}
.
== Forces acting on a half-plane ==
For the special case where
α
=
−
π
{\displaystyle \alpha =-\pi }
,
β
=
0
{\displaystyle \beta =0}
, the wedge is converted into a half-plane with a normal force and a tangential force. In that case
C
1
=
−
F
1
π
,
C
3
=
−
F
2
π
{\displaystyle C_{1}=-{\frac {F_{1}}{\pi }},\quad C_{3}=-{\frac {F_{2}}{\pi }}}
Therefore, the stresses are
σ
r
r
=
−
2
π
r
(
F
1
cos
θ
+
F
2
sin
θ
)
σ
r
θ
=
0
σ
θ
θ
=
0
{\displaystyle {\begin{aligned}\sigma _{rr}&=-{\frac {2}{\pi \,r}}(F_{1}\cos \theta +F_{2}\sin \theta )\\\sigma _{r\theta }&=0\\\sigma _{\theta \theta }&=0\end{aligned}}}
and the displacements are (using Michell's solution)
u
r
=
−
1
4
π
μ
[
F
1
{
(
κ
−
1
)
θ
sin
θ
−
cos
θ
+
(
κ
+
1
)
ln
r
cos
θ
}
+
F
2
{
(
κ
−
1
)
θ
cos
θ
+
sin
θ
−
(
κ
+
1
)
ln
r
sin
θ
}
]
u
θ
=
−
1
4
π
μ
[
F
1
{
(
κ
−
1
)
θ
cos
θ
−
sin
θ
−
(
κ
+
1
)
ln
r
sin
θ
}
−
F
2
{
(
κ
−
1
)
θ
sin
θ
+
cos
θ
+
(
κ
+
1
)
ln
r
cos
θ
}
]
{\displaystyle {\begin{aligned}u_{r}&=-{\cfrac {1}{4\pi \mu }}\left[F_{1}\{(\kappa -1)\theta \sin \theta -\cos \theta +(\kappa +1)\ln r\cos \theta \}+\right.\\&\qquad \qquad \left.F_{2}\{(\kappa -1)\theta \cos \theta +\sin \theta -(\kappa +1)\ln r\sin \theta \}\right]\\u_{\theta }&=-{\cfrac {1}{4\pi \mu }}\left[F_{1}\{(\kappa -1)\theta \cos \theta -\sin \theta -(\kappa +1)\ln r\sin \theta \}-\right.\\&\qquad \qquad \left.F_{2}\{(\kappa -1)\theta \sin \theta +\cos \theta +(\kappa +1)\ln r\cos \theta \}\right]\end{aligned}}}
The
ln
r
{\displaystyle \ln r}
dependence of the displacements implies that the displacement grows the further one moves from the point of application of the force (and is unbounded at infinity). This feature of the Flamant solution is confusing and appears unphysical.
=== Displacements at the surface of the half-plane ===
The displacements in the
x
1
,
x
2
{\displaystyle x_{1},x_{2}}
directions at the surface of the half-plane are given by
u
1
=
F
1
(
κ
+
1
)
ln
|
x
1
|
4
π
μ
+
F
2
(
κ
−
1
)
sign
(
x
1
)
8
μ
u
2
=
F
2
(
κ
+
1
)
ln
|
x
1
|
4
π
μ
+
F
1
(
κ
−
1
)
sign
(
x
1
)
8
μ
{\displaystyle {\begin{aligned}u_{1}&={\frac {F_{1}(\kappa +1)\ln |x_{1}|}{4\pi \mu }}+{\frac {F_{2}(\kappa -1){\text{sign}}(x_{1})}{8\mu }}\\u_{2}&={\frac {F_{2}(\kappa +1)\ln |x_{1}|}{4\pi \mu }}+{\frac {F_{1}(\kappa -1){\text{sign}}(x_{1})}{8\mu }}\end{aligned}}}
where
κ
=
{
3
−
4
ν
plane strain
3
−
ν
1
+
ν
plane stress
{\displaystyle \kappa ={\begin{cases}3-4\nu &\qquad {\text{plane strain}}\\{\cfrac {3-\nu }{1+\nu }}&\qquad {\text{plane stress}}\end{cases}}}
ν
{\displaystyle \nu }
is the Poisson's ratio,
μ
{\displaystyle \mu }
is the shear modulus, and
sign
(
x
)
=
{
+
1
x
>
0
−
1
x
<
0
{\displaystyle {\text{sign}}(x)={\begin{cases}+1&x>0\\-1&x<0\end{cases}}}
== Derivation ==
If we assume the stresses to vary as
(
1
/
r
)
{\displaystyle (1/r)}
, we can pick terms containing
1
/
r
{\displaystyle 1/r}
in the stresses from Michell's solution. Then the Airy stress function can be expressed as
φ
=
C
1
r
θ
sin
θ
+
C
2
r
ln
r
cos
θ
+
C
3
r
θ
cos
θ
+
C
4
r
ln
r
sin
θ
{\displaystyle \varphi =C_{1}r\theta \sin \theta +C_{2}r\ln r\cos \theta +C_{3}r\theta \cos \theta +C_{4}r\ln r\sin \theta }
Therefore, from the tables in Michell's solution, we have
σ
r
r
=
C
1
(
2
cos
θ
r
)
+
C
2
(
cos
θ
r
)
+
C
3
(
2
sin
θ
r
)
+
C
4
(
sin
θ
r
)
σ
r
θ
=
C
2
(
sin
θ
r
)
+
C
4
(
−
cos
θ
r
)
σ
θ
θ
=
C
2
(
cos
θ
r
)
+
C
4
(
sin
θ
r
)
{\displaystyle {\begin{aligned}\sigma _{rr}&=C_{1}\left({\frac {2\cos \theta }{r}}\right)+C_{2}\left({\frac {\cos \theta }{r}}\right)+C_{3}\left({\frac {2\sin \theta }{r}}\right)+C_{4}\left({\frac {\sin \theta }{r}}\right)\\\sigma _{r\theta }&=C_{2}\left({\frac {\sin \theta }{r}}\right)+C_{4}\left({\frac {-\cos \theta }{r}}\right)\\\sigma _{\theta \theta }&=C_{2}\left({\frac {\cos \theta }{r}}\right)+C_{4}\left({\frac {\sin \theta }{r}}\right)\end{aligned}}}
The constants
C
1
,
C
2
,
C
3
,
C
4
{\displaystyle C_{1},C_{2},C_{3},C_{4}}
can then, in principle, be determined from the wedge geometry and the applied boundary conditions.
However, the concentrated loads at the vertex are difficult to express in terms of traction boundary conditions because
the unit outward normal at the vertex is undefined
the forces are applied at a point (which has zero area) and hence the traction at that point is infinite.
To get around this problem, we consider a bounded region of the wedge and consider equilibrium of the bounded wedge. Let the bounded wedge have two traction free surfaces and a third surface in the form of an arc of a circle with radius
a
{\displaystyle a\,}
. Along the arc of the circle, the unit outward normal is
n
=
e
r
{\displaystyle \mathbf {n} =\mathbf {e} _{r}}
where the basis vectors are
(
e
r
,
e
θ
)
{\displaystyle (\mathbf {e} _{r},\mathbf {e} _{\theta })}
. The tractions on the arc are
t
=
σ
⋅
n
⟹
t
r
=
σ
r
r
,
t
θ
=
σ
r
θ
.
{\displaystyle \mathbf {t} ={\boldsymbol {\sigma }}\cdot \mathbf {n} \quad \implies t_{r}=\sigma _{rr},~t_{\theta }=\sigma _{r\theta }~.}
Next, we examine the force and moment equilibrium in the bounded wedge and get
∑
f
1
=
F
1
+
∫
α
β
[
σ
r
r
(
a
,
θ
)
cos
θ
−
σ
r
θ
(
a
,
θ
)
sin
θ
]
a
d
θ
=
0
∑
f
2
=
F
2
+
∫
α
β
[
σ
r
r
(
a
,
θ
)
sin
θ
+
σ
r
θ
(
a
,
θ
)
cos
θ
]
a
d
θ
=
0
∑
m
3
=
∫
α
β
[
a
σ
r
θ
(
a
,
θ
)
]
a
d
θ
=
0
{\displaystyle {\begin{aligned}\sum f_{1}&=F_{1}+\int _{\alpha }^{\beta }\left[\sigma _{rr}(a,\theta )~\cos \theta -\sigma _{r\theta }(a,\theta )~\sin \theta \right]~a~d\theta =0\\\sum f_{2}&=F_{2}+\int _{\alpha }^{\beta }\left[\sigma _{rr}(a,\theta )~\sin \theta +\sigma _{r\theta }(a,\theta )~\cos \theta \right]~a~d\theta =0\\\sum m_{3}&=\int _{\alpha }^{\beta }\left[a~\sigma _{r\theta }(a,\theta )\right]~a~d\theta =0\end{aligned}}}
We require that these equations be satisfied for all values of
a
{\displaystyle a\,}
and thereby satisfy the boundary conditions.
The traction-free boundary conditions on the edges
θ
=
α
{\displaystyle \theta =\alpha }
and
θ
=
β
{\displaystyle \theta =\beta }
also imply that
σ
r
θ
=
σ
θ
θ
=
0
at
θ
=
α
,
θ
=
β
{\displaystyle \sigma _{r\theta }=\sigma _{\theta \theta }=0\qquad {\text{at}}~~\theta =\alpha ,\theta =\beta }
except at the point
r
=
0
{\displaystyle r=0}
.
If we assume that
σ
r
θ
=
0
{\displaystyle \sigma _{r\theta }=0}
everywhere, then the traction-free conditions and the moment equilibrium equation are satisfied and we are left with
F
1
+
∫
α
β
σ
r
r
(
a
,
θ
)
a
cos
θ
d
θ
=
0
F
2
+
∫
α
β
σ
r
r
(
a
,
θ
)
a
sin
θ
d
θ
=
0
{\displaystyle {\begin{aligned}F_{1}&+\int _{\alpha }^{\beta }\sigma _{rr}(a,\theta )~a~\cos \theta ~d\theta =0\\F_{2}&+\int _{\alpha }^{\beta }\sigma _{rr}(a,\theta )~a~\sin \theta ~d\theta =0\end{aligned}}}
and
σ
θ
θ
=
0
{\displaystyle \sigma _{\theta \theta }=0}
along
θ
=
α
,
θ
=
β
{\displaystyle \theta =\alpha ,\theta =\beta }
except at the point
r
=
0
{\displaystyle r=0}
. But the field
σ
θ
θ
=
0
{\displaystyle \sigma _{\theta \theta }=0}
everywhere also satisfies the force equilibrium equations. Hence this must be the solution. Also, the assumption
σ
r
θ
=
0
{\displaystyle \sigma _{r\theta }=0}
implies that
C
2
=
C
4
=
0
{\displaystyle C_{2}=C_{4}=0}
.
Therefore,
σ
r
r
=
2
C
1
cos
θ
r
+
2
C
3
sin
θ
r
;
σ
r
θ
=
0
;
σ
θ
θ
=
0
{\displaystyle \sigma _{rr}={\frac {2C_{1}\cos \theta }{r}}+{\frac {2C_{3}\sin \theta }{r}}~;~~\sigma _{r\theta }=0~;~~\sigma _{\theta \theta }=0}
To find a particular solution for
σ
r
r
{\displaystyle \sigma _{rr}}
we have to plug in the expression for
σ
r
r
{\displaystyle \sigma _{rr}}
into the force equilibrium equations to get a system of two equations which have to be solved for
C
1
,
C
3
{\displaystyle C_{1},C_{3}}
:
F
1
+
2
∫
α
β
(
C
1
cos
θ
+
C
3
sin
θ
)
cos
θ
d
θ
=
0
F
2
+
2
∫
α
β
(
C
1
cos
θ
+
C
3
sin
θ
)
sin
θ
d
θ
=
0
{\displaystyle {\begin{aligned}F_{1}&+2\int _{\alpha }^{\beta }(C_{1}\cos \theta +C_{3}\sin \theta )~\cos \theta ~d\theta =0\\F_{2}&+2\int _{\alpha }^{\beta }(C_{1}\cos \theta +C_{3}\sin \theta )~\sin \theta ~d\theta =0\end{aligned}}}
=== Forces acting on a half-plane ===
If we take
α
=
−
π
{\displaystyle \alpha =-\pi }
and
β
=
0
{\displaystyle \beta =0}
, the problem is converted into one where a normal force
F
2
{\displaystyle F_{2}}
and a tangential force
F
1
{\displaystyle F_{1}}
act on a half-plane. In that case, the force equilibrium equations take the form
F
1
+
2
∫
−
π
0
(
C
1
cos
θ
+
C
3
sin
θ
)
cos
θ
d
θ
=
0
⟹
F
1
+
C
1
π
=
0
F
2
+
2
∫
−
π
0
(
C
1
cos
θ
+
C
3
sin
θ
)
sin
θ
d
θ
=
0
⟹
F
2
+
C
3
π
=
0
{\displaystyle {\begin{aligned}F_{1}&+2\int _{-\pi }^{0}(C_{1}\cos \theta +C_{3}\sin \theta )~\cos \theta ~d\theta =0\qquad \implies F_{1}+C_{1}\pi =0\\F_{2}&+2\int _{-\pi }^{0}(C_{1}\cos \theta +C_{3}\sin \theta )~\sin \theta ~d\theta =0\qquad \implies F_{2}+C_{3}\pi =0\end{aligned}}}
Therefore
C
1
=
−
F
1
π
;
C
3
=
−
F
2
π
.
{\displaystyle C_{1}=-{\cfrac {F_{1}}{\pi }}~;~~C_{3}=-{\cfrac {F_{2}}{\pi }}~.}
The stresses for this situation are
σ
r
r
=
−
2
π
r
(
F
1
cos
θ
+
F
2
sin
θ
)
;
σ
r
θ
=
0
;
σ
θ
θ
=
0
{\displaystyle \sigma _{rr}=-{\frac {2}{\pi r}}(F_{1}\cos \theta +F_{2}\sin \theta )~;~~\sigma _{r\theta }=0~;~~\sigma _{\theta \theta }=0}
Using the displacement tables from the Michell solution, the displacements for this case are given by
u
r
=
−
1
4
π
μ
[
F
1
{
(
κ
−
1
)
θ
sin
θ
−
cos
θ
+
(
κ
+
1
)
ln
r
cos
θ
}
+
F
2
{
(
κ
−
1
)
θ
cos
θ
+
sin
θ
−
(
κ
+
1
)
ln
r
sin
θ
}
]
u
θ
=
−
1
4
π
μ
[
F
1
{
(
κ
−
1
)
θ
cos
θ
−
sin
θ
−
(
κ
+
1
)
ln
r
sin
θ
}
−
F
2
{
(
κ
−
1
)
θ
sin
θ
+
cos
θ
+
(
κ
+
1
)
ln
r
cos
θ
}
]
{\displaystyle {\begin{aligned}u_{r}&=-{\cfrac {1}{4\pi \mu }}\left[F_{1}\{(\kappa -1)\theta \sin \theta -\cos \theta +(\kappa +1)\ln r\cos \theta \}+\right.\\&\qquad \qquad \left.F_{2}\{(\kappa -1)\theta \cos \theta +\sin \theta -(\kappa +1)\ln r\sin \theta \}\right]\\u_{\theta }&=-{\cfrac {1}{4\pi \mu }}\left[F_{1}\{(\kappa -1)\theta \cos \theta -\sin \theta -(\kappa +1)\ln r\sin \theta \}-\right.\\&\qquad \qquad \left.F_{2}\{(\kappa -1)\theta \sin \theta +\cos \theta +(\kappa +1)\ln r\cos \theta \}\right]\end{aligned}}}
==== Displacements at the surface of the half-plane ====
To find expressions for the displacements at the surface of the half plane, we first find the displacements for positive
x
1
{\displaystyle x_{1}}
(
θ
=
0
{\displaystyle \theta =0}
) and negative
x
1
{\displaystyle x_{1}}
(
θ
=
π
{\displaystyle \theta =\pi }
) keeping in mind that
r
=
|
x
1
|
{\displaystyle r=|x_{1}|}
along these locations.
For
θ
=
0
{\displaystyle \theta =0}
we have
u
r
=
u
1
=
F
1
4
π
μ
[
1
−
(
κ
+
1
)
ln
|
x
1
|
]
u
θ
=
u
2
=
F
2
4
π
μ
[
1
+
(
κ
+
1
)
ln
|
x
1
|
]
{\displaystyle {\begin{aligned}u_{r}=u_{1}&={\cfrac {F_{1}}{4\pi \mu }}\left[1-(\kappa +1)\ln |x_{1}|\right]\\u_{\theta }=u_{2}&={\cfrac {F_{2}}{4\pi \mu }}\left[1+(\kappa +1)\ln |x_{1}|\right]\end{aligned}}}
For
θ
=
π
{\displaystyle \theta =\pi }
we have
u
r
=
−
u
1
=
−
F
1
4
π
μ
[
1
−
(
κ
+
1
)
ln
|
x
1
|
]
+
F
2
4
μ
(
κ
−
1
)
u
θ
=
−
u
2
=
F
1
4
μ
(
κ
−
1
)
−
F
2
4
π
μ
[
1
+
(
κ
+
1
)
ln
|
x
1
|
]
{\displaystyle {\begin{aligned}u_{r}=-u_{1}&=-{\cfrac {F_{1}}{4\pi \mu }}\left[1-(\kappa +1)\ln |x_{1}|\right]+{\cfrac {F_{2}}{4\mu }}(\kappa -1)\\u_{\theta }=-u_{2}&={\cfrac {F_{1}}{4\mu }}(\kappa -1)-{\cfrac {F_{2}}{4\pi \mu }}\left[1+(\kappa +1)\ln |x_{1}|\right]\end{aligned}}}
We can make the displacements symmetric around the point of application of the force by adding rigid body displacements (which does not affect the stresses)
u
1
=
F
2
8
μ
(
κ
−
1
)
;
u
2
=
F
1
8
μ
(
κ
−
1
)
{\displaystyle u_{1}={\cfrac {F_{2}}{8\mu }}(\kappa -1)~;~~u_{2}={\cfrac {F_{1}}{8\mu }}(\kappa -1)}
and removing the redundant rigid body displacements
u
1
=
F
1
4
π
μ
;
u
2
=
F
2
4
π
μ
.
{\displaystyle u_{1}={\cfrac {F_{1}}{4\pi \mu }}~;~~u_{2}={\cfrac {F_{2}}{4\pi \mu }}~.}
Then the displacements at the surface can be combined and take the form
u
1
=
F
1
4
π
μ
(
κ
+
1
)
ln
|
x
1
|
+
F
2
8
μ
(
κ
−
1
)
sign
(
x
1
)
u
2
=
F
2
4
π
μ
(
κ
+
1
)
ln
|
x
1
|
+
F
1
8
μ
(
κ
−
1
)
sign
(
x
1
)
{\displaystyle {\begin{aligned}u_{1}&={\cfrac {F_{1}}{4\pi \mu }}(\kappa +1)\ln |x_{1}|+{\cfrac {F_{2}}{8\mu }}(\kappa -1){\text{sign}}(x_{1})\\u_{2}&={\cfrac {F_{2}}{4\pi \mu }}(\kappa +1)\ln |x_{1}|+{\cfrac {F_{1}}{8\mu }}(\kappa -1){\text{sign}}(x_{1})\end{aligned}}}
where
sign
(
x
)
=
{
+
1
x
>
0
−
1
x
<
0
{\displaystyle {\text{sign}}(x)={\begin{cases}+1&x>0\\-1&x<0\end{cases}}}
== References == | Wikipedia/Flamant_solution |
Non-smooth mechanics is a modeling approach in mechanics which does not require the time evolutions of the positions and of the velocities to be smooth functions. Due to possible impacts, the velocities of the mechanical system are allowed to undergo jumps at certain time instants in order to fulfill the kinematical restrictions. Consider for example a rigid model of a ball which falls on the ground. Just before the impact between ball and ground, the ball has non-vanishing pre-impact velocity. At the impact time instant, the velocity must jump to a post-impact velocity which is at least zero, or else penetration would occur. Non-smooth mechanical models are often used in contact dynamics.
== See also ==
Contact dynamics
Unilateral contact
Jean Jacques Moreau
== References ==
Acary V., Brogliato, B. Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. Springer Verlag, LNACM 35, Heidelberg, 2008.
Brogliato B. Nonsmooth Mechanics. Models, Dynamics and Control. Communications and Control Engineering Series, Springer-Verlag, London, 2016 (3rd Ed.)
Demyanov, V.F., Stavroulakis, G.E., Polyakova, L.N., Panagiotopoulos, P.D. "Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics", Springer 1996.
Yang Gao, David, Ogden, Ray W., Stavroulakis, Georgios E. (Eds.) "Nonsmooth/Nonconvex Mechanics Modeling, Analysis and Numerical Methods", Springer 2001
Glocker, Ch. Dynamik von Starrkoerpersystemen mit Reibung und Stoessen, volume 18/182 of VDI Fortschrittsberichte Mechanik/Bruchmechanik. VDI Verlag, Düsseldorf, 1995
Glocker Ch. and Studer C. Formulation and preparation for Numerical Evaluation of Linear Complementarity Systems. Multibody System Dynamics 13(4):447-463, 2005
Jean M. The non-smooth contact dynamics method. Computer Methods in Applied mechanics and Engineering 177(3-4):235-257, 1999
Mistakidis, E.S., Stavroulakis, Georgios E. "Nonconvex Optimization in Mechanics Algorithms, Heuristics and Engineering Applications by the F.E.M.", Springer, 1998
Moreau J.J. Unilateral Contact and Dry Friction in Finite Freedom Dynamics, volume 302 of Non-smooth Mechanics and Applications, CISM Courses and Lectures. Springer, Wien, 1988
Pfeiffer F., Foerg M. and Ulbrich H. Numerical aspects of non-smooth multibody dynamics. Comput. Methods Appl. Mech. Engrg 195(50-51):6891-6908, 2006
Potra F.A., Anitescu M., Gavrea B. and Trinkle J. A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contacts, joints and friction. Int. J. Numer. Meth. Engng 66(7):1079-1124, 2006
Stewart D.E. and Trinkle J.C. An Implicit Time-Stepping Scheme for Rigid Body Dynamics with Inelastic Collisions and Coulomb Friction. Int. J. Numer. Methods Engineering 39(15):2673-2691, 1996 | Wikipedia/Non-smooth_mechanics |
The strength of materials is determined using various methods of calculating the stresses and strains in structural members, such as beams, columns, and shafts. The methods employed to predict the response of a structure under loading and its susceptibility to various failure modes takes into account the properties of the materials such as its yield strength, ultimate strength, Young's modulus, and Poisson's ratio. In addition, the mechanical element's macroscopic properties (geometric properties) such as its length, width, thickness, boundary constraints and abrupt changes in geometry such as holes are considered.
The theory began with the consideration of the behavior of one and two dimensional members of structures, whose states of stress can be approximated as two dimensional, and was then generalized to three dimensions to develop a more complete theory of the elastic and plastic behavior of materials. An important founding pioneer in mechanics of materials was Stephen Timoshenko.
== Definition ==
In the mechanics of materials, the strength of a material is its ability to withstand an applied load without failure or plastic deformation. The field of strength of materials deals with forces and deformations that result from their acting on a material. A load applied to a mechanical member will induce internal forces within the member called stresses when those forces are expressed on a unit basis. The stresses acting on the material cause deformation of the material in various manners including breaking them completely. Deformation of the material is called strain when those deformations too are placed on a unit basis.
The stresses and strains that develop within a mechanical member must be calculated in order to assess the load capacity of that member. This requires a complete description of the geometry of the member, its constraints, the loads applied to the member and the properties of the material of which the member is composed. The applied loads may be axial (tensile or compressive), or rotational (strength shear). With a complete description of the loading and the geometry of the member, the state of stress and state of strain at any point within the member can be calculated. Once the state of stress and strain within the member is known, the strength (load carrying capacity) of that member, its deformations (stiffness qualities), and its stability (ability to maintain its original configuration) can be calculated.
The calculated stresses may then be compared to some measure of the strength of the member such as its material yield or ultimate strength. The calculated deflection of the member may be compared to deflection criteria that are based on the member's use. The calculated buckling load of the member may be compared to the applied load. The calculated stiffness and mass distribution of the member may be used to calculate the member's dynamic response and then compared to the acoustic environment in which it will be used.
Material strength refers to the point on the engineering stress–strain curve (yield stress) beyond which the material experiences deformations that will not be completely reversed upon removal of the loading and as a result, the member will have a permanent deflection. The ultimate strength of the material refers to the maximum value of stress reached. The fracture strength is the stress value at fracture (the last stress value recorded).
=== Types of loadings ===
Transverse loadings – Forces applied perpendicular to the longitudinal axis of a member. Transverse loading causes the member to bend and deflect from its original position, with internal tensile and compressive strains accompanying the change in curvature of the member. Transverse loading also induces shear forces that cause shear deformation of the material and increase the transverse deflection of the member.
Axial loading – The applied forces are collinear with the longitudinal axis of the member. The forces cause the member to either stretch or shorten.
Torsional loading – Twisting action caused by a pair of externally applied equal and oppositely directed force couples acting on parallel planes or by a single external couple applied to a member that has one end fixed against rotation.
=== Stress terms ===
Uniaxial stress is expressed by
σ
=
F
A
,
{\displaystyle \sigma ={\frac {F}{A}},}
where F is the force acting on an area A. The area can be the undeformed area or the deformed area, depending on whether engineering stress or true stress is of interest.
Compressive stress (or compression) is the stress state caused by an applied load that acts to reduce the length of the material (compression member) along the axis of the applied load; it is, in other words, a stress state that causes a squeezing of the material. A simple case of compression is the uniaxial compression induced by the action of opposite, pushing forces. Compressive strength for materials is generally higher than their tensile strength. However, structures loaded in compression are subject to additional failure modes, such as buckling, that are dependent on the member's geometry.
Tensile stress is the stress state caused by an applied load that tends to elongate the material along the axis of the applied load, in other words, the stress caused by pulling the material. The strength of structures of equal cross-sectional area loaded in tension is independent of shape of the cross-section. Materials loaded in tension are susceptible to stress concentrations such as material defects or abrupt changes in geometry. However, materials exhibiting ductile behaviour (many metals for example) can tolerate some defects while brittle materials (such as ceramics and some steels) can fail well below their ultimate material strength.
Shear stress is the stress state caused by the combined energy of a pair of opposing forces acting along parallel lines of action through the material, in other words, the stress caused by faces of the material sliding relative to one another. An example is cutting paper with scissors or stresses due to torsional loading.
=== Stress parameters for resistance ===
Material resistance can be expressed in several mechanical stress parameters. The term material strength is used when referring to mechanical stress parameters. These are physical quantities with dimension homogeneous to pressure and force per unit surface. The traditional measure unit for strength are therefore MPa in the International System of Units, and the psi between the United States customary units. Strength parameters include: yield strength, tensile strength, fatigue strength, crack resistance, and other parameters.
Yield strength is the lowest stress that produces a permanent deformation in a material. In some materials, like aluminium alloys, the point of yielding is difficult to identify, thus it is usually defined as the stress required to cause 0.2% plastic strain. This is called a 0.2% proof stress.
Compressive strength is a limit state of compressive stress that leads to failure in a material in the manner of ductile failure (infinite theoretical yield) or brittle failure (rupture as the result of crack propagation, or sliding along a weak plane – see shear strength).
Tensile strength or ultimate tensile strength is a limit state of tensile stress that leads to tensile failure in the manner of ductile failure (yield as the first stage of that failure, some hardening in the second stage and breakage after a possible "neck" formation) or brittle failure (sudden breaking in two or more pieces at a low-stress state). The tensile strength can be quoted as either true stress or engineering stress, but engineering stress is the most commonly used.
Fatigue strength is a more complex measure of the strength of a material that considers several loading episodes in the service period of an object, and is usually more difficult to assess than the static strength measures. Fatigue strength is quoted here as a simple range (
Δ
σ
=
σ
m
a
x
−
σ
m
i
n
{\displaystyle \Delta \sigma =\sigma _{\mathrm {max} }-\sigma _{\mathrm {min} }}
). In the case of cyclic loading it can be appropriately expressed as an amplitude usually at zero mean stress, along with the number of cycles to failure under that condition of stress.
Impact strength is the capability of the material to withstand a suddenly applied load and is expressed in terms of energy. Often measured with the Izod impact strength test or Charpy impact test, both of which measure the impact energy required to fracture a sample. Volume, modulus of elasticity, distribution of forces, and yield strength affect the impact strength of a material. In order for a material or object to have a high impact strength, the stresses must be distributed evenly throughout the object. It also must have a large volume with a low modulus of elasticity and a high material yield strength.
=== Strain parameters for resistance ===
Deformation of the material is the change in geometry created when stress is applied (as a result of applied forces, gravitational fields, accelerations, thermal expansion, etc.). Deformation is expressed by the displacement field of the material.
Strain, or reduced deformation, is a mathematical term that expresses the trend of the deformation change among the material field. Strain is the deformation per unit length. In the case of uniaxial loading the displacement of a specimen (for example, a bar element) lead to a calculation of strain expressed as the quotient of the displacement and the original length of the specimen. For 3D displacement fields it is expressed as derivatives of displacement functions in terms of a second-order tensor (with 6 independent elements).
Deflection is a term to describe the magnitude to which a structural element is displaced when subject to an applied load.
=== Stress–strain relations ===
Elasticity is the ability of a material to return to its previous shape after stress is released. In many materials, the relation between applied stress is directly proportional to the resulting strain (up to a certain limit), and a graph representing those two quantities is a straight line.
The slope of this line is known as Young's modulus, or the "modulus of elasticity". The modulus of elasticity can be used to determine the stress–strain relationship in the linear-elastic portion of the stress–strain curve. The linear-elastic region is either below the yield point, or if a yield point is not easily identified on the stress–strain plot it is defined to be between 0 and 0.2% strain, and is defined as the region of strain in which no yielding (permanent deformation) occurs.
Plasticity or plastic deformation is the opposite of elastic deformation and is defined as unrecoverable strain. Plastic deformation is retained after the release of the applied stress. Most materials in the linear-elastic category are usually capable of plastic deformation. Brittle materials, like ceramics, do not experience any plastic deformation and will fracture under relatively low strain, while ductile materials such as metallics, lead, or polymers will plastically deform much more before a fracture initiation.
Consider the difference between a carrot and chewed bubble gum. The carrot will stretch very little before breaking. The chewed bubble gum, on the other hand, will plastically deform enormously before finally breaking.
== Design terms ==
Ultimate strength is an attribute related to a material, rather than just a specific specimen made of the material, and as such it is quoted as the force per unit of cross section area (N/m2). The ultimate strength is the maximum stress that a material can withstand before it breaks or weakens. For example, the ultimate tensile strength (UTS) of AISI 1018 Steel is 440 MPa. In Imperial units, the unit of stress is given as lbf/in2 or pounds-force per square inch. This unit is often abbreviated as psi. One thousand psi is abbreviated ksi.
A factor of safety is a design criteria that an engineered component or structure must achieve.
F
S
=
F
/
f
{\displaystyle FS=F/f}
, where FS: the factor of safety, Rf The applied stress, and F: ultimate allowable stress (psi or MPa)
Margin of Safety is the common method for design criteria. It is defined MS = Pu/P − 1.
For example, to achieve a factor of safety of 4, the allowable stress in an AISI 1018 steel component can be calculated to be
F
=
U
T
S
/
F
S
{\displaystyle F=UTS/FS}
= 440/4 = 110 MPa, or
F
{\displaystyle F}
= 110×106 N/m2. Such allowable stresses are also known as "design stresses" or "working stresses".
Design stresses that have been determined from the ultimate or yield point values of the materials give safe and reliable results only for the case of static loading. Many machine parts fail when subjected to a non-steady and continuously varying loads even though the developed stresses are below the yield point. Such failures are called fatigue failure. The failure is by a fracture that appears to be brittle with little or no visible evidence of yielding. However, when the stress is kept below "fatigue stress" or "endurance limit stress", the part will endure indefinitely. A purely reversing or cyclic stress is one that alternates between equal positive and negative peak stresses during each cycle of operation. In a purely cyclic stress, the average stress is zero. When a part is subjected to a cyclic stress, also known as stress range (Sr), it has been observed that the failure of the part occurs after a number of stress reversals (N) even if the magnitude of the stress range is below the material's yield strength. Generally, higher the range stress, the fewer the number of reversals needed for failure.
=== Failure theories ===
There are four failure theories: maximum shear stress theory, maximum normal stress theory, maximum strain energy theory, and maximum distortion energy theory (von Mises criterion of failure). Out of these four theories of failure, the maximum normal stress theory is only applicable for brittle materials, and the remaining three theories are applicable for ductile materials.
Of the latter three, the distortion energy theory provides the most accurate results in a majority of the stress conditions. The strain energy theory needs the value of Poisson's ratio of the part material, which is often not readily available. The maximum shear stress theory is conservative. For simple unidirectional normal stresses all theories are equivalent, which means all theories will give the same result.
Maximum shear stress theory postulates that failure will occur if the magnitude of the maximum shear stress in the part exceeds the shear strength of the material determined from uniaxial testing.
Maximum normal stress theory postulates that failure will occur if the maximum normal stress in the part exceeds the ultimate tensile stress of the material as determined from uniaxial testing. This theory deals with brittle materials only. The maximum tensile stress should be less than or equal to ultimate tensile stress divided by factor of safety. The magnitude of the maximum compressive stress should be less than ultimate compressive stress divided by factor of safety.
Maximum strain energy theory postulates that failure will occur when the strain energy per unit volume due to the applied stresses in a part equals the strain energy per unit volume at the yield point in uniaxial testing.
Maximum distortion energy theory, also known as maximum distortion energy theory of failure or von Mises–Hencky theory. This theory postulates that failure will occur when the distortion energy per unit volume due to the applied stresses in a part equals the distortion energy per unit volume at the yield point in uniaxial testing. The total elastic energy due to strain can be divided into two parts: one part causes change in volume, and the other part causes a change in shape. Distortion energy is the amount of energy that is needed to change the shape.
Fracture mechanics was established by Alan Arnold Griffith and George Rankine Irwin. This important theory is also known as numeric conversion of toughness of material in the case of crack existence.
A material's strength depends on its microstructure. The engineering processes to which a material is subjected can alter its microstructure. Strengthening mechanisms that alter the strength of a material include work hardening, solid solution strengthening, precipitation hardening, and grain boundary strengthening.
Strengthening mechanisms are accompanied by the caveat that some other mechanical properties of the material may degenerate in an attempt to make a material stronger. For example, in grain boundary strengthening, although yield strength is maximized with decreasing grain size, ultimately, very small grain sizes make the material brittle. In general, the yield strength of a material is an adequate indicator of the material's mechanical strength. Considered in tandem with the fact that the yield strength is the parameter that predicts plastic deformation in the material, one can make informed decisions on how to increase the strength of a material depending on its microstructural properties and the desired end effect. Strength is expressed in terms of the limiting values of the compressive stress, tensile stress, and shear stresses that would cause failure. The effects of dynamic loading are probably the most important practical consideration of the theory of elasticity, especially the problem of fatigue. Repeated loading often initiates cracks, which grow until failure occurs at the corresponding residual strength of the structure. Cracks always start at a stress concentrations especially changes in cross-section of the product or defects in manufacturing, near holes and corners at nominal stress levels far lower than those quoted for the strength of the material.
== See also ==
== References ==
== Further reading ==
== External links ==
Failure theories
Case studies in structural failure | Wikipedia/Mechanics_of_materials |
A design is the concept or proposal for an object, process, or system. The word design refers to something that is or has been intentionally created by a thinking agent, and is sometimes used to refer to the inherent nature of something – its design. The verb to design expresses the process of developing a design. In some cases, the direct construction of an object without an explicit prior plan may also be considered to be a design (such as in arts and crafts). A design is expected to have a purpose within a specific context, typically aiming to satisfy certain goals and constraints while taking into account aesthetic, functional and experiential considerations. Traditional examples of designs are architectural and engineering drawings, circuit diagrams, sewing patterns, and less tangible artefacts such as business process models.
== Designing ==
People who produce designs are called designers. The term 'designer' usually refers to someone who works professionally in one of the various design areas. Within the professions, the word 'designer' is generally qualified by the area of practice (for example: a fashion designer, a product designer, a web designer, or an interior designer), but it can also designate other practitioners such as architects and engineers (see below: Types of designing). A designer's sequence of activities to produce a design is called a design process, with some employing designated processes such as design thinking and design methods. The process of creating a design can be brief (a quick sketch) or lengthy and complicated, involving considerable research, negotiation, reflection, modeling, interactive adjustment, and re-design.
Designing is also a widespread activity outside of the professions of those formally recognized as designers. In his influential book The Sciences of the Artificial, the interdisciplinary scientist Herbert A. Simon proposed that, "Everyone designs who devises courses of action aimed at changing existing situations into preferred ones." According to the design researcher Nigel Cross, "Everyone can – and does – design," and "Design ability is something that everyone has, to some extent, because it is embedded in our brains as a natural cognitive function."
== History of design ==
The study of design history is complicated by varying interpretations of what constitutes 'designing'. Many design historians, such as John Heskett, look to the Industrial Revolution and the development of mass production. Others subscribe to conceptions of design that include pre-industrial objects and artefacts, beginning their narratives of design in prehistoric times. Originally situated within art history, the historical development of the discipline of design history coalesced in the 1970s, as interested academics worked to recognize design as a separate and legitimate target for historical research. Early influential design historians include German-British art historian Nikolaus Pevsner and Swiss historian and architecture critic Sigfried Giedion.
== Design education ==
In Western Europe, institutions for design education date back to the nineteenth century. The Norwegian National Academy of Craft and Art Industry was founded in 1818, followed by the United Kingdom's Government School of Design (1837), and Konstfack in Sweden (1844). The Rhode Island School of Design was founded in the United States in 1877. The German art and design school Bauhaus, founded in 1919, greatly influenced modern design education.
Design education covers the teaching of theory, knowledge, and values in the design of products, services, and environments, with a focus on the development of both particular and general skills for designing. Traditionally, its primary orientation has been to prepare students for professional design practice, based on project work and studio, or atelier, teaching methods.
There are also broader forms of higher education in design studies and design thinking. Design is also a part of general education, for example within the curriculum topic, Design and Technology. The development of design in general education in the 1970s created a need to identify fundamental aspects of 'designerly' ways of knowing, thinking, and acting, which resulted in establishing design as a distinct discipline of study.
== Design process ==
Substantial disagreement exists concerning how designers in many fields, whether amateur or professional, alone or in teams, produce designs. Design researchers Dorst and Dijkhuis acknowledged that "there are many ways of describing design processes," and compare and contrast two dominant but different views of the design process: as a rational problem-solving process and as a process of reflection-in-action. They suggested that these two paradigms "represent two fundamentally different ways of looking at the world – positivism and constructionism." The paradigms may reflect differing views of how designing should be done and how it actually is done, and both have a variety of names. The problem-solving view has been called "the rational model," "technical rationality" and "the reason-centric perspective." The alternative view has been called "reflection-in-action," "coevolution" and "the action-centric perspective."
=== Rational model ===
The rational model was independently developed by Herbert A. Simon, an American scientist, and two German engineering design theorists, Gerhard Pahl and Wolfgang Beitz. It posits that:
Designers attempt to optimize a design candidate for known constraints and objectives.
The design process is plan-driven.
The design process is understood in terms of a discrete sequence of stages.
The rational model is based on a rationalist philosophy and underlies the waterfall model, systems development life cycle, and much of the engineering design literature. According to the rationalist philosophy, design is informed by research and knowledge in a predictable and controlled manner.
Typical stages consistent with the rational model include the following:
Pre-production design
Design brief – initial statement of intended outcome.
Analysis – analysis of design goals.
Research – investigating similar designs in the field or related topics.
Specification – specifying requirements of a design for a product (product design specification) or service.
Problem solving – conceptualizing and documenting designs.
Presentation – presenting designs.
Design during production.
Development – continuation and improvement of a design.
Product testing – in situ testing of a design.
Post-production design feedback for future designs.
Implementation – introducing the design into the environment.
Evaluation and conclusion – summary of process and results, including constructive criticism and suggestions for future improvements.
Redesign – any or all stages in the design process repeated (with corrections made) at any time before, during, or after production.
Each stage has many associated best practices.
==== Criticism of the rational model ====
The rational model has been widely criticized on two primary grounds:
Designers do not work this way – extensive empirical evidence has demonstrated that designers do not act as the rational model suggests.
Unrealistic assumptions – goals are often unknown when a design project begins, and the requirements and constraints continue to change.
=== Action-centric model ===
The action-centric perspective is a label given to a collection of interrelated concepts, which are antithetical to the rational model. It posits that:
Designers use creativity and emotion to generate design candidates.
The design process is improvised.
No universal sequence of stages is apparent – analysis, design, and implementation are contemporary and inextricably linked.
The action-centric perspective is based on an empiricist philosophy and broadly consistent with the agile approach and methodical development. Substantial empirical evidence supports the veracity of this perspective in describing the actions of real designers. Like the rational model, the action-centric model sees design as informed by research and knowledge.
At least two views of design activity are consistent with the action-centric perspective. Both involve these three basic activities:
In the reflection-in-action paradigm, designers alternate between "framing", "making moves", and "evaluating moves". "Framing" refers to conceptualizing the problem, i.e., defining goals and objectives. A "move" is a tentative design decision. The evaluation process may lead to further moves in the design.
In the sensemaking–coevolution–implementation framework, designers alternate between its three titular activities. Sensemaking includes both framing and evaluating moves. Implementation is the process of constructing the design object. Coevolution is "the process where the design agent simultaneously refines its mental picture of the design object based on its mental picture of the context, and vice versa".
The concept of the design cycle is understood as a circular time structure, which may start with the thinking of an idea, then expressing it by the use of visual or verbal means of communication (design tools), the sharing and perceiving of the expressed idea, and finally starting a new cycle with the critical rethinking of the perceived idea. Anderson points out that this concept emphasizes the importance of the means of expression, which at the same time are means of perception of any design ideas.
== Philosophies ==
Philosophy of design is the study of definitions, assumptions, foundations, and implications of design. There are also many informal 'philosophies' for guiding design such as personal values or preferred approaches.
=== Approaches to design ===
Some of these values and approaches include:
Critical design uses designed artefacts as an embodied critique or commentary on existing values, morals, and practices in a culture. Critical design can make aspects of the future physically present to provoke a reaction.
Ecological design is a design approach that prioritizes the consideration of the environmental impacts of a product or service, over its whole lifecycle. Ecodesign research focuses primarily on barriers to implementation, ecodesign tools and methods, and the intersection of ecodesign with other research disciplines.
Participatory design (originally co-operative design, now often co-design) is the practice of collective creativity to design, attempting to actively involve all stakeholders (e.g. employees, partners, customers, citizens, end-users) in the design process to help ensure the result meets their needs and is usable. Recent research suggests that designers create more innovative concepts and ideas when working within a co-design environment with others than they do when creating ideas on their own.
Scientific design refers to industrialised design based on scientific knowledge. Science can be used to study the effects and need for a potential or existing product in general and to design products that are based on scientific knowledge. For instance, a scientific design of face masks for COVID-19 mitigation may be based on investigations of filtration performance, mitigation performance, thermal comfort, biodegradability and flow resistance.
Service design is a term that is used for designing or organizing the experience around a product and the service associated with a product's use. The purpose of service design methodologies is to establish the most effective practices for designing services, according to both the needs of users and the competencies and capabilities of service providers.
Sociotechnical system design, a philosophy and tools for participative designing of work arrangements and supporting processes – for organizational purpose, quality, safety, economics, and customer requirements in core work processes, the quality of peoples experience at work, and the needs of society.
Transgenerational design, the practice of making products and environments compatible with those physical and sensory impairments associated with human aging and which limit major activities of daily living.
User-centered design, which focuses on the needs, wants, and limitations of the end-user of the designed artefact. One aspect of user-centered design is ergonomics.
== Relationship with the arts ==
The boundaries between art and design are blurry, largely due to a range of applications both for the term 'art' and the term 'design'. Applied arts can include industrial design, graphic design, fashion design, and the decorative arts which traditionally includes craft objects. In graphic arts (2D image making that ranges from photography to illustration), the distinction is often made between fine art and commercial art, based on the context within which the work is produced and how it is traded.
== Types of designing ==
== See also ==
== References ==
== Further reading ==
Margolin, Victor. World History of Design. New York: Bloomsbury Academic, 2015. (2 vols) ISBN 9781472569288.
Raizman, David Seth (12 November 2003). The History of Modern Design. Pearson. ISBN 978-0131830400. | Wikipedia/Design |
In mechanics and physics, shock is a sudden acceleration caused, for example, by impact, drop, kick, earthquake, or explosion. Shock is a transient physical excitation.
Shock describes matter subject to extreme rates of force with respect to time. Shock is a vector that has units of an acceleration (rate of change of velocity). The unit g (or g) represents multiples of the standard acceleration of gravity and is conventionally used.
A shock pulse can be characterised by its peak acceleration, the duration, and the shape of the shock pulse (half sine, triangular, trapezoidal, etc.). The shock response spectrum is a method for further evaluating a mechanical shock.
== Shock measurement ==
Shock measurement is of interest in several fields such as
Propagation of heel shock through a runner's body
Measure the magnitude of a shock need to cause damage to an item: fragility.
Measure shock attenuation through athletic flooring
Measuring the effectiveness of a shock absorber
Measuring the shock absorbing ability of package cushioning
Measure the ability of an athletic helmet to protect people
Measure the effectiveness of shock mounts
Determining the ability of structures to resist seismic shock: earthquakes, etc.
Determining whether personal protective fabric attenuates or amplifies shocks
Verifying that a Naval ship and its equipment can survive explosive shocks
Shocks are usually measured by accelerometers but other transducers and high speed imaging are also used. A wide variety of laboratory instrumentation is available; stand-alone shock data loggers are also used.
Field shocks are highly variable and often have very uneven shapes. Even laboratory controlled shocks often have uneven shapes and include short duration spikes; Noise can be reduced by appropriate digital or analog filtering.
Governing test methods and specifications provide detail about the conduct of shock tests. Proper placement of measuring instruments is critical. Fragile items and packaged goods respond with variation to uniform laboratory shocks; Replicate testing is often called for. For example, MIL-STD-810G Method 516.6 indicates: at least three times in both directions along each of three orthogonal axes".
== Shock testing ==
Shock testing typically falls into two categories, classical shock testing and pyroshock or ballistic shock testing. Classical shock testing consists of the following shock impulses: half sine, haversine, sawtooth wave, and trapezoid. Pyroshock and ballistic shock tests are specialized and are not considered classical shocks. Classical shocks can be performed on Electro Dynamic (ED) Shakers, Free Fall Drop Tower or Pneumatic Shock Machines. A classical shock impulse is created when the shock machine table changes direction abruptly. This abrupt change in direction causes a rapid velocity change which creates the shock impulse. Testing the effects of shock are sometimes conducted on end-use applications: for example, automobile crash tests.
Use of proper test methods and Verification and validation protocols are important for all phases of testing and evaluation.
== Effects of shock ==
Mechanical shock has the potential for damaging an item (e.g., an entire light bulb) or an element of the item (e.g. a filament in an Incandescent light bulb):
A brittle or fragile item can fracture. For example, two crystal wine glasses may shatter when impacted against each other. A shear pin in an engine is designed to fracture with a specific magnitude of shock. Note that a soft ductile material may sometimes exhibit brittle failure during shock due to time-temperature superposition.
A malleable item can be bent by a shock. For example, a copper pitcher may bend when dropped on the floor.
Some items may appear to be not damaged by a single shock but will experience fatigue failure with numerous repeated low-level shocks.
A shock may result in only minor damage which may not be critical for use. However, cumulative minor damage from several shocks will eventually result in the item being unusable.
A shock may not produce immediate apparent damage but might cause the service life of the product to be shortened: the reliability is reduced.
A shock may cause an item to become out of adjustment. For example, when a precision scientific instrument is subjected to a moderate shock, good metrology practice may be to have it recalibrated before further use.
Some materials such as primary high explosives may detonate with mechanical shock or impact.
When glass bottles of liquid are dropped or subjected to shock, the water hammer effect may cause hydrodynamic glass breakage.
== Considerations ==
When laboratory testing, field experience, or engineering judgement indicates that an item could be damaged by mechanical shock, several courses of action might be considered:
Reduce and control the input shock at the source.
Modify the item to improve its toughness or support it to better handle shocks.
Use shock absorbers, shock mounts, or cushions to control the shock transmitted to the item. Cushioning reduces the peak acceleration by extending the duration of the shock.
Plan for failures: accept certain losses. Have redundant systems available, etc.
== See also ==
== Notes ==
== Further reading ==
DeSilva, C. W., "Vibration and Shock Handbook", CRC, 2005, ISBN 0-8493-1580-8
Harris, C. M., and Peirsol, A. G. "Shock and Vibration Handbook", 2001, McGraw Hill, ISBN 0-07-137081-1
ISO 18431:2007 - Mechanical vibration and shock
ASTM D6537, Standard Practice for Instrumented Package Shock Testing for Determination of Package Performance.
MIL-STD-810G, Environmental Test Methods and Engineering Guidelines, 2000, sect 516.6
Brogliato, B., "Nonsmooth Mechanics. Models, Dynamics and Control", Springer London, 2nd Edition, 1999.
== External links ==
Response to mechanical shock, Department of Energy, [1]
Shock Response Spectrum, a primer, [2]
A Study in the Application of SRS, [3] | Wikipedia/Shock_(mechanics) |
Contact dynamics deals with the motion of multibody systems subjected to unilateral contacts and friction. Such systems are omnipresent in many multibody dynamics applications. Consider for example
Contacts between wheels and ground in vehicle dynamics
Squealing of brakes due to friction induced oscillations
Motion of many particles, spheres which fall in a funnel, mixing processes (granular media)
Clockworks
Walking machines
Arbitrary machines with limit stops, friction.
Anatomic tissues (skin, iris/lens, eyelids/anterior ocular surface, joint cartilages, vascular endothelium/blood cells, muscles/tendons, et cetera)
In the following it is discussed how such mechanical systems with unilateral contacts and friction can be modeled and how the time evolution of such systems can be obtained by numerical integration. In addition, some examples are given.
== Modeling ==
The two main approaches for modeling mechanical systems with unilateral contacts and friction are the regularized and the non-smooth approach. In the following, the two approaches are introduced using a simple example. Consider a block which can slide or stick on a table (see figure 1a). The motion of the block is described by the equation of motion, whereas the friction force is unknown (see figure 1b). In order to obtain the friction force, a separate force law must be specified which links the friction force to the associated velocity of the block.
=== Non-smooth approach ===
A more sophisticated approach is the non-smooth approach, which uses set-valued force laws to model mechanical systems with unilateral contacts and friction. Consider again the block which slides or sticks on the table. The associated set-valued friction law of type Sgn is depicted in figure 3. Regarding the sliding case, the friction force is given. Regarding the sticking case, the friction force is set-valued and determined according to an additional algebraic constraint.
To conclude, the non-smooth approach changes the underlying mathematical structure if required and leads to a proper description of mechanical systems with unilateral contacts and friction. As a consequence of the changing mathematical structure, impacts can occur, and the time evolutions of the positions and the velocities can not be assumed to be smooth anymore. As a consequence, additional impact equations and impact laws have to be defined. In order to handle the changing mathematical structure, the set-valued force laws are commonly written as inequality or inclusion problems. The evaluation of these inequalities/inclusions is commonly done by solving linear (or nonlinear) complementarity problems, by quadratic programming or by transforming the inequality/inclusion problems into projective equations which can be solved iteratively by Jacobi or Gauss–Seidel techniques.
The non-smooth approach provides a new modeling approach for mechanical systems with unilateral contacts and friction, which incorporates also the whole classical mechanics subjected to bilateral constraints. The approach is associated to the classical DAE theory and leads to robust integration schemes.
== Numerical integration ==
The integration of regularized models can be done by standard stiff solvers for ordinary differential equations. However, oscillations induced by the regularization can occur. Considering non-smooth models of mechanical systems with unilateral contacts and friction, two main classes of integrators exist, the event-driven and the so-called time-stepping integrators.
=== Event-driven integrators ===
Event-driven integrators distinguish between smooth parts of the motion in which the underlying structure of the differential equations does not change, and in events or so-called switching points at which this structure changes, i.e. time instants at which a unilateral contact closes or a stick slip transition occurs. At these switching points, the set-valued force (and additional impact) laws are evaluated in order to obtain a new underlying mathematical structure on which the integration can be continued. Event-driven integrators are very accurate but are not suitable for systems with many contacts.
=== Time-stepping integrators ===
Time-stepping integrators are dedicated numerical schemes for mechanical systems with many contacts. The first time-stepping integrator was introduced by J.J. Moreau. The integrators do not aim at resolving switching points and are therefore very robust in application. As the integrators work with the integral of the contact forces and not with the forces itself, the methods can handle both motion and impulsive events like impacts. As a drawback, the accuracy of time-stepping integrators is low. This can be fixed by using a step-size refinement at switching points. Smooth parts of the motion are processed by larger step sizes, and higher order integration methods can be used to increase the integration order.
== Examples ==
This section gives some examples of mechanical systems with unilateral contacts and friction. The results have been obtained by a non-smooth approach using time-stepping integrators.
=== Granular materials ===
Time-stepping methods are especially well suited for the simulation of granular materials. Figure 4 depicts the simulation of mixing 1000 disks.
=== Billiard ===
Consider two colliding spheres in a billiard play. Figure 5a shows some snapshots of two colliding spheres, figure 5b depicts the associated trajectories.
=== Wheely of a motorbike ===
If a motorbike is accelerated too fast, it does a wheelie. Figure 6 shows some snapshots of a simulation.
=== Motion of the woodpecker toy ===
The woodpecker toy is a well known benchmark problem in contact dynamics. The toy consists of a pole, a sleeve with a hole that is slightly larger than the diameter of the pole, a spring and the woodpecker body. In operation, the woodpecker moves down the pole performing some kind of pitching motion, which is controlled by the sleeve. Figure 7 shows some snapshots of a simulation.
A simulation and visualization can be found at https://github.com/gabyx/Woodpecker.
== See also ==
Multibody dynamics
Contact mechanics: Applications with unilateral contacts and friction. Static applications (contact between deformable bodies) and dynamic applications (Contact dynamics).
Lubachevsky-Stillinger algorithm of simulating compression of large assemblies of hard particles
== References ==
== Further reading ==
Acary V. and Brogliato, B. Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. Springer Verlag, LNACM 35, Heidelberg, 2008.
Brogliato B. Nonsmooth Mechanics. Models, Dynamics and Control Communications and Control Engineering Series Springer-Verlag, London, 2016 (third Ed.)
Drumwright, E. and Shell, D. Modeling Contact Friction and Joint Friction in Dynamic Robotic Simulation Using the Principle of Maximum Dissipation. Springer Tracks in Advanced Robotics: Algorithmic Foundations of Robotics IX, 2010
Glocker, Ch. Dynamik von Starrkoerpersystemen mit Reibung und Stoessen, volume 18/182 of VDI Fortschrittsberichte Mechanik/Bruchmechanik. VDI Verlag, Düsseldorf, 1995
Glocker Ch. and Studer C. Formulation and preparation for Numerical Evaluation of Linear Complementarity Systems. Multibody System Dynamics 13(4):447-463, 2005
Jean M. The non-smooth contact dynamics method. Computer Methods in Applied mechanics and Engineering 177(3-4):235-257, 1999
Moreau J.J. Unilateral Contact and Dry Friction in Finite Freedom Dynamics, volume 302 of Non-smooth Mechanics and Applications, CISM Courses and Lectures. Springer, Wien, 1988
Pfeiffer F., Foerg M. and Ulbrich H. Numerical aspects of non-smooth multibody dynamics. Comput. Methods Appl. Mech. Engrg 195(50-51):6891-6908, 2006
Potra F.A., Anitescu M., Gavrea B. and Trinkle J. A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contacts, joints and friction. Int. J. Numer. Meth. Engng 66(7):1079-1124, 2006
Stewart D.E. and Trinkle J.C. An Implicit Time-Stepping Scheme for Rigid Body Dynamics with Inelastic Collisions and Coulomb Friction. Int. J. Numer. Methods Engineering 39(15):2673-2691, 1996
Studer C. Augmented time-stepping integration of non-smooth dynamical systems, PhD Thesis ETH Zurich, ETH E-Collection, to appear 2008
Studer C. Numerics of Unilateral Contacts and Friction—Modeling and Numerical Time Integration in Non-Smooth Dynamics, Lecture Notes in Applied and Computational Mechanics, Volume 47, Springer, Berlin, Heidelberg, 2009
== External links ==
Multibody research group, Center of Mechanics, ETH Zurich.
Lehrstuhl für angewandte Mechanik TU Munich.
BiPoP Team, INRIA Rhone-Alpes, France,
Siconos software. An open-source software dedicated to the modeling and the simulation or nonsmooth dynamical systems, especially mechanical systems with contact and Coulomb's friction
Multibody dynamics, Rensselaer Polytechnic Institute.
dynamY software
LMGC90 software
MigFlow software
Solfec software
GRSFramework Granular Rigid Body Simulation Framework developed at IMES in Ch. Glocker's group (High-Performance Computing with MPI), 2016
Chrono, an open source multi-physics simulation engine, see also project website 2017 | Wikipedia/Contact_dynamics |
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral loads only. By ignoring the effects of shear deformation and rotatory inertia, it is thus a special case of Timoshenko–Ehrenfest beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.
Additional mathematical models have been developed, such as plate theory, but the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering.
== History ==
Prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue that Leonardo da Vinci was the first to make the crucial observations. Da Vinci lacked Hooke's law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made.
The Bernoulli beam is named after Jacob Bernoulli, who made the significant discoveries. Leonhard Euler and Daniel Bernoulli were the first to put together a useful theory circa 1750.
== Static beam equation ==
The Euler–Bernoulli equation describes the relationship between the beam's deflection and the applied load:The curve
w
(
x
)
{\displaystyle w(x)}
describes the deflection of the beam in the
z
{\displaystyle z}
direction at some position
x
{\displaystyle x}
(recall that the beam is modeled as a one-dimensional object).
q
{\displaystyle q}
is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of
x
{\displaystyle x}
,
w
{\displaystyle w}
, or other variables.
E
{\displaystyle E}
is the elastic modulus and
I
{\displaystyle I}
is the second moment of area of the beam's cross section.
I
{\displaystyle I}
must be calculated with respect to the axis which is perpendicular to the applied loading. Explicitly, for a beam whose axis is oriented along
x
{\displaystyle x}
with a loading along
z
{\displaystyle z}
, the beam's cross section is in the
y
z
{\displaystyle yz}
plane, and the relevant second moment of area is
I
=
∬
z
2
d
y
d
z
,
{\displaystyle I=\iint z^{2}\;dy\;dz,}
It can be shown from equilibrium considerations that the centroid of the cross section must be at
y
=
z
=
0
{\displaystyle y=z=0}
.
Often, the product
E
I
{\displaystyle EI}
(known as the flexural rigidity) is a constant, so that
E
I
d
4
w
d
x
4
=
q
(
x
)
.
{\displaystyle EI{\frac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}=q(x).\,}
This equation, describing the deflection of a uniform, static beam, is used widely in engineering practice. Tabulated expressions for the deflection
w
{\displaystyle w}
for common beam configurations can be found in engineering handbooks. For more complicated situations, the deflection can be determined by solving the Euler–Bernoulli equation using techniques such as "direct integration", "Macaulay's method", "moment area method, "conjugate beam method", "the principle of virtual work", "Castigliano's method", "flexibility method", "slope deflection method", "moment distribution method", or "direct stiffness method".
Sign conventions are defined here since different conventions can be found in the literature. In this article, a right-handed coordinate system is used with the
x
{\displaystyle x}
axis to the right, the
z
{\displaystyle z}
axis pointing upwards, and the
y
{\displaystyle y}
axis pointing into the figure. The sign of the bending moment
M
{\displaystyle M}
is taken as positive when the torque vector associated with the bending moment on the right hand side of the section is in the positive
y
{\displaystyle y}
direction, that is, a positive value of
M
{\displaystyle M}
produces compressive stress at the bottom surface. With this choice of bending moment sign convention, in order to have
d
M
=
Q
d
x
{\displaystyle dM=Qdx}
, it is necessary that the shear force
Q
{\displaystyle Q}
acting on the right side of the section be positive in the
z
{\displaystyle z}
direction so as to achieve static equilibrium of moments. If the loading intensity
q
{\displaystyle q}
is taken positive in the positive
z
{\displaystyle z}
direction, then
d
Q
=
−
q
d
x
{\displaystyle dQ=-qdx}
is necessary for force equilibrium.
Successive derivatives of the deflection
w
{\displaystyle w}
have important physical meanings:
d
w
/
d
x
{\displaystyle dw/dx}
is the slope of the beam, which is the anti-clockwise angle of rotation about the
y
{\displaystyle y}
-axis in the limit of small displacements;
M
=
−
E
I
d
2
w
d
x
2
{\displaystyle M=-EI{\frac {d^{2}w}{dx^{2}}}}
is the bending moment in the beam; and
Q
=
−
d
d
x
(
E
I
d
2
w
d
x
2
)
{\displaystyle Q=-{\frac {d}{dx}}\left(EI{\frac {d^{2}w}{dx^{2}}}\right)}
is the shear force in the beam.
The stresses in a beam can be calculated from the above expressions after the deflection due to a given load has been determined.
=== Derivation of the bending equation ===
Because of the fundamental importance of the bending moment equation in engineering, we will provide a short derivation. We change to polar coordinates. The length of the neutral axis in the figure is
ρ
d
θ
.
{\displaystyle \rho d\theta .}
The length of a fiber with a radial distance
z
{\displaystyle z}
below the neutral axis is
(
ρ
+
z
)
d
θ
.
{\displaystyle (\rho +z)d\theta .}
Therefore, the strain of this fiber is
(
ρ
+
z
−
ρ
)
d
θ
ρ
d
θ
=
z
ρ
.
{\displaystyle {\frac {\left(\rho +z-\rho \right)\ d\theta }{\rho \ d\theta }}={\frac {z}{\rho }}.}
The stress of this fiber is
E
z
ρ
{\displaystyle E{\dfrac {z}{\rho }}}
where
E
{\displaystyle E}
is the elastic modulus in accordance with Hooke's law. The differential force vector,
d
F
,
{\displaystyle d\mathbf {F} ,}
resulting from this stress, is given by
d
F
=
E
z
ρ
d
A
e
x
.
{\displaystyle d\mathbf {F} =E{\frac {z}{\rho }}dA\mathbf {e_{x}} .}
This is the differential force vector exerted on the right hand side of the section shown in the figure. We know that it is in the
e
x
{\displaystyle \mathbf {e_{x}} }
direction since the figure clearly shows that the fibers in the lower half are in tension.
d
A
{\displaystyle dA}
is the differential element of area at the location of the fiber. The differential bending moment vector,
d
M
{\displaystyle d\mathbf {M} }
associated with
d
F
{\displaystyle d\mathbf {F} }
is given by
d
M
=
−
z
e
z
×
d
F
=
−
e
y
E
z
2
ρ
d
A
.
{\displaystyle d\mathbf {M} =-z\mathbf {e_{z}} \times d\mathbf {F} =-\mathbf {e_{y}} E{\frac {z^{2}}{\rho }}dA.}
This expression is valid for the fibers in the lower half of the beam. The expression for the fibers in the upper half of the beam will be similar except that the moment arm vector will be in the positive
z
{\displaystyle z}
direction and the force vector will be in the
−
x
{\displaystyle -x}
direction since the upper fibers are in compression. But the resulting bending moment vector will still be in the
−
y
{\displaystyle -y}
direction since
e
z
×
−
e
x
=
−
e
y
.
{\displaystyle \mathbf {e_{z}} \times -\mathbf {e_{x}} =-\mathbf {e_{y}} .}
Therefore, we integrate over the entire cross section of the beam and get for
M
{\displaystyle \mathbf {M} }
the bending moment vector exerted on the right cross section of the beam the expression
M
=
∫
d
M
=
−
e
y
E
ρ
∫
z
2
d
A
=
−
e
y
E
I
ρ
,
{\displaystyle \mathbf {M} =\int d\mathbf {M} =-\mathbf {e_{y}} {\frac {E}{\rho }}\int {z^{2}}\ dA=-\mathbf {e_{y}} {\frac {EI}{\rho }},}
where
I
{\displaystyle I}
is the second moment of area. From calculus, we know that when
d
w
d
x
{\displaystyle {\dfrac {dw}{dx}}}
is small, as it is for an Euler–Bernoulli beam, we can make the approximation
1
ρ
≃
d
2
w
d
x
2
{\displaystyle {\dfrac {1}{\rho }}\simeq {\dfrac {d^{2}w}{dx^{2}}}}
, where
ρ
{\displaystyle \rho }
is the radius of curvature. Therefore,
M
=
−
e
y
E
I
d
2
w
d
x
2
.
{\displaystyle \mathbf {M} =-\mathbf {e_{y}} EI{d^{2}w \over dx^{2}}.}
This vector equation can be separated in the bending unit vector definition (
M
{\displaystyle M}
is oriented as
e
y
{\displaystyle \mathbf {e_{y}} }
), and in the bending equation:
M
=
−
E
I
d
2
w
d
x
2
.
{\displaystyle M=-EI{d^{2}w \over dx^{2}}.}
== Dynamic beam equation ==
The dynamic beam equation is the Euler–Lagrange equation for the following action
S
=
∫
t
1
t
2
∫
0
L
[
1
2
μ
(
∂
w
∂
t
)
2
−
1
2
E
I
(
∂
2
w
∂
x
2
)
2
+
q
(
x
)
w
(
x
,
t
)
]
d
x
d
t
.
{\displaystyle S=\int _{t_{1}}^{t_{2}}\int _{0}^{L}\left[{\frac {1}{2}}\mu \left({\frac {\partial w}{\partial t}}\right)^{2}-{\frac {1}{2}}EI\left({\frac {\partial ^{2}w}{\partial x^{2}}}\right)^{2}+q(x)w(x,t)\right]dxdt.}
The first term represents the kinetic energy where
μ
{\displaystyle \mu }
is the mass per unit length, the second term represents the potential energy due to internal forces (when considered with a negative sign), and the third term represents the potential energy due to the external load
q
(
x
)
{\displaystyle q(x)}
. The Euler–Lagrange equation is used to determine the function that minimizes the functional
S
{\displaystyle S}
. For a dynamic Euler–Bernoulli beam, the Euler–Lagrange equation is
When the beam is homogeneous,
E
{\displaystyle E}
and
I
{\displaystyle I}
are independent of
x
{\displaystyle x}
, and the beam equation is simpler:
E
I
∂
4
w
∂
x
4
=
−
μ
∂
2
w
∂
t
2
+
q
.
{\displaystyle EI{\cfrac {\partial ^{4}w}{\partial x^{4}}}=-\mu {\cfrac {\partial ^{2}w}{\partial t^{2}}}+q\,.}
=== Free vibration ===
In the absence of a transverse load,
q
{\displaystyle q}
, we have the free vibration equation. This equation can be solved using a Fourier decomposition of the displacement into the sum of harmonic vibrations of the form
w
(
x
,
t
)
=
Re
[
w
^
(
x
)
e
−
i
ω
t
]
{\displaystyle w(x,t)={\text{Re}}[{\hat {w}}(x)~e^{-i\omega t}]}
where
ω
{\displaystyle \omega }
is the frequency of vibration. Then, for each value of frequency, we can solve an ordinary differential equation
E
I
d
4
w
^
d
x
4
−
μ
ω
2
w
^
=
0
.
{\displaystyle EI~{\cfrac {\mathrm {d} ^{4}{\hat {w}}}{\mathrm {d} x^{4}}}-\mu \omega ^{2}{\hat {w}}=0\,.}
The general solution of the above equation is
w
^
=
A
1
cosh
(
β
x
)
+
A
2
sinh
(
β
x
)
+
A
3
cos
(
β
x
)
+
A
4
sin
(
β
x
)
with
β
:=
(
μ
ω
2
E
I
)
1
/
4
{\displaystyle {\hat {w}}=A_{1}\cosh(\beta x)+A_{2}\sinh(\beta x)+A_{3}\cos(\beta x)+A_{4}\sin(\beta x)\quad {\text{with}}\quad \beta :=\left({\frac {\mu \omega ^{2}}{EI}}\right)^{1/4}}
where
A
1
,
A
2
,
A
3
,
A
4
{\displaystyle A_{1},A_{2},A_{3},A_{4}}
are constants. These constants are unique for a given set of boundary conditions. However, the solution for the displacement is not unique and depends on the frequency. These solutions are typically written as
w
^
n
=
A
1
cosh
(
β
n
x
)
+
A
2
sinh
(
β
n
x
)
+
A
3
cos
(
β
n
x
)
+
A
4
sin
(
β
n
x
)
with
β
n
:=
(
μ
ω
n
2
E
I
)
1
/
4
.
{\displaystyle {\hat {w}}_{n}=A_{1}\cosh(\beta _{n}x)+A_{2}\sinh(\beta _{n}x)+A_{3}\cos(\beta _{n}x)+A_{4}\sin(\beta _{n}x)\quad {\text{with}}\quad \beta _{n}:=\left({\frac {\mu \omega _{n}^{2}}{EI}}\right)^{1/4}\,.}
The quantities
ω
n
{\displaystyle \omega _{n}}
are called the natural frequencies of the beam. Each of the displacement solutions is called a mode, and the shape of the displacement curve is called a mode shape.
==== Example: Cantilevered beam ====
The boundary conditions for a cantilevered beam of length
L
{\displaystyle L}
(fixed at
x
=
0
{\displaystyle x=0}
) are
w
^
n
=
0
,
d
w
^
n
d
x
=
0
at
x
=
0
d
2
w
^
n
d
x
2
=
0
,
d
3
w
^
n
d
x
3
=
0
at
x
=
L
.
{\displaystyle {\begin{aligned}&{\hat {w}}_{n}=0~,~~{\frac {d{\hat {w}}_{n}}{dx}}=0\quad {\text{at}}~~x=0\\&{\frac {d^{2}{\hat {w}}_{n}}{dx^{2}}}=0~,~~{\frac {d^{3}{\hat {w}}_{n}}{dx^{3}}}=0\quad {\text{at}}~~x=L\,.\end{aligned}}}
If we apply these conditions, non-trivial solutions are found to exist only if
cosh
(
β
n
L
)
cos
(
β
n
L
)
+
1
=
0
.
{\displaystyle \cosh(\beta _{n}L)\,\cos(\beta _{n}L)+1=0\,.}
This nonlinear equation can be solved numerically. The first four roots are
β
1
L
=
0.596864
π
{\displaystyle \beta _{1}L=0.596864\pi }
,
β
2
L
=
1.49418
π
{\displaystyle \beta _{2}L=1.49418\pi }
,
β
3
L
=
2.50025
π
{\displaystyle \beta _{3}L=2.50025\pi }
, and
β
4
L
=
3.49999
π
{\displaystyle \beta _{4}L=3.49999\pi }
.
The corresponding natural frequencies of vibration are
ω
1
=
β
1
2
E
I
μ
=
3.5160
L
2
E
I
μ
,
…
{\displaystyle \omega _{1}=\beta _{1}^{2}{\sqrt {\frac {EI}{\mu }}}={\frac {3.5160}{L^{2}}}{\sqrt {\frac {EI}{\mu }}}~,~~\dots }
The boundary conditions can also be used to determine the mode shapes from the solution for the displacement:
w
^
n
=
A
1
[
(
cosh
β
n
x
−
cos
β
n
x
)
+
cos
β
n
L
+
cosh
β
n
L
sin
β
n
L
+
sinh
β
n
L
(
sin
β
n
x
−
sinh
β
n
x
)
]
{\displaystyle {\hat {w}}_{n}=A_{1}\left[(\cosh \beta _{n}x-\cos \beta _{n}x)+{\frac {\cos \beta _{n}L+\cosh \beta _{n}L}{\sin \beta _{n}L+\sinh \beta _{n}L}}(\sin \beta _{n}x-\sinh \beta _{n}x)\right]}
The unknown constant (actually constants as there is one for each
n
{\displaystyle n}
),
A
1
{\displaystyle A_{1}}
, which in general is complex, is determined by the initial conditions at
t
=
0
{\displaystyle t=0}
on the velocity and displacements of the beam. Typically a value of
A
1
=
1
{\displaystyle A_{1}=1}
is used when plotting mode shapes. Solutions to the undamped forced problem have unbounded displacements when the driving frequency matches a natural frequency
ω
n
{\displaystyle \omega _{n}}
, i.e., the beam can resonate. The natural frequencies of a beam therefore correspond to the frequencies at which resonance can occur.
==== Example: free–free (unsupported) beam ====
A free–free beam is a beam without any supports. The boundary conditions for a free–free beam of length
L
{\displaystyle L}
extending from
x
=
0
{\displaystyle x=0}
to
x
=
L
{\displaystyle x=L}
are given by:
d
2
w
^
n
d
x
2
=
0
,
d
3
w
^
n
d
x
3
=
0
at
x
=
0
and
x
=
L
.
{\displaystyle {\frac {d^{2}{\hat {w}}_{n}}{dx^{2}}}=0~,~~{\frac {d^{3}{\hat {w}}_{n}}{dx^{3}}}=0\quad {\text{at}}~~x=0\,{\text{and}}\,x=L\,.}
If we apply these conditions, non-trivial solutions are found to exist only if
cosh
(
β
n
L
)
cos
(
β
n
L
)
−
1
=
0
.
{\displaystyle \cosh(\beta _{n}L)\,\cos(\beta _{n}L)-1=0\,.}
This nonlinear equation can be solved numerically. The first four roots are
β
1
L
=
1.50562
π
{\displaystyle \beta _{1}L=1.50562\pi }
,
β
2
L
=
2.49975
π
{\displaystyle \beta _{2}L=2.49975\pi }
,
β
3
L
=
3.50001
π
{\displaystyle \beta _{3}L=3.50001\pi }
, and
β
4
L
=
4.50000
π
{\displaystyle \beta _{4}L=4.50000\pi }
.
The corresponding natural frequencies of vibration are:
ω
1
=
β
1
2
E
I
μ
=
22.3733
L
2
E
I
μ
,
…
{\displaystyle \omega _{1}=\beta _{1}^{2}{\sqrt {\frac {EI}{\mu }}}={\frac {22.3733}{L^{2}}}{\sqrt {\frac {EI}{\mu }}}~,~~\dots }
The boundary conditions can also be used to determine the mode shapes from the solution for the displacement:
w
^
n
=
A
1
[
(
cos
β
n
x
+
cosh
β
n
x
)
−
cos
β
n
L
−
cosh
β
n
L
sin
β
n
L
−
sinh
β
n
L
(
sin
β
n
x
+
sinh
β
n
x
)
]
{\displaystyle {\hat {w}}_{n}=A_{1}{\Bigl [}(\cos \beta _{n}x+\cosh \beta _{n}x)-{\frac {\cos \beta _{n}L-\cosh \beta _{n}L}{\sin \beta _{n}L-\sinh \beta _{n}L}}(\sin \beta _{n}x+\sinh \beta _{n}x){\Bigr ]}}
As with the cantilevered beam, the unknown constants are determined by the initial conditions at
t
=
0
{\displaystyle t=0}
on the velocity and displacements of the beam. Also, solutions to the undamped forced problem have unbounded displacements when the driving frequency matches a natural frequency
ω
n
{\displaystyle \omega _{n}}
.
==== Example: hinged-hinged beam ====
The boundary conditions of a hinged-hinged beam of length
L
{\displaystyle L}
(fixed at
x
=
0
{\displaystyle x=0}
and
x
=
L
{\displaystyle x=L}
) are
w
^
n
=
0
,
d
2
w
^
n
d
x
2
=
0
at
x
=
0
and
x
=
L
.
{\displaystyle {\hat {w}}_{n}=0~,~~{\frac {d^{2}{\hat {w}}_{n}}{dx^{2}}}=0\quad {\text{at}}~~x=0\,{\text{and}}\,x=L\,.}
This implies solutions exist for
sin
(
β
n
L
)
sinh
(
β
n
L
)
=
0
.
{\displaystyle \sin(\beta _{n}L)\,\sinh(\beta _{n}L)=0\,.}
Setting
β
n
=
n
π
/
L
{\displaystyle \beta _{n}=n\pi /L}
enforces this condition. Rearranging for natural frequency gives
ω
n
=
n
2
π
2
L
2
E
I
μ
{\displaystyle \omega _{n}={\frac {n^{2}\pi ^{2}}{L^{2}}}{\sqrt {\frac {EI}{\mu }}}}
== Stress ==
Besides deflection, the beam equation describes forces and moments and can thus be used to describe stresses. For this reason, the Euler–Bernoulli beam equation is widely used in engineering, especially civil and mechanical, to determine the strength (as well as deflection) of beams under bending.
Both the bending moment and the shear force cause stresses in the beam. The stress due to shear force is maximum along the neutral axis of the beam (when the width of the beam, t, is constant along the cross section of the beam; otherwise an integral involving the first moment and the beam's width needs to be evaluated for the particular cross section), and the maximum tensile stress is at either the top or bottom surfaces. Thus the maximum principal stress in the beam may be neither at the surface nor at the center but in some general area. However, shear force stresses are negligible in comparison to bending moment stresses in all but the stockiest of beams as well as the fact that stress concentrations commonly occur at surfaces, meaning that the maximum stress in a beam is likely to be at the surface.
=== Simple or symmetrical bending ===
For beam cross-sections that are symmetrical about a plane perpendicular to the neutral plane, it can be shown that the tensile stress experienced by the beam may be expressed as:
σ
=
M
z
I
=
−
z
E
d
2
w
d
x
2
.
{\displaystyle \sigma ={\frac {Mz}{I}}=-zE~{\frac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}.\,}
Here,
z
{\displaystyle z}
is the distance from the neutral axis to a point of interest; and
M
{\displaystyle M}
is the bending moment. Note that this equation implies that pure bending (of positive sign) will cause zero stress at the neutral axis, positive (tensile) stress at the "top" of the beam, and negative (compressive) stress at the bottom of the beam; and also implies that the maximum stress will be at the top surface and the minimum at the bottom. This bending stress may be superimposed with axially applied stresses, which will cause a shift in the neutral (zero stress) axis.
=== Maximum stresses at a cross-section ===
The maximum tensile stress at a cross-section is at the location
z
=
c
1
{\displaystyle z=c_{1}}
and the maximum compressive stress is at the location
z
=
−
c
2
{\displaystyle z=-c_{2}}
where the height of the cross-section is
h
=
c
1
+
c
2
{\displaystyle h=c_{1}+c_{2}}
. These stresses are
σ
1
=
M
c
1
I
=
M
S
1
;
σ
2
=
−
M
c
2
I
=
−
M
S
2
{\displaystyle \sigma _{1}={\cfrac {Mc_{1}}{I}}={\cfrac {M}{S_{1}}}~;~~\sigma _{2}=-{\cfrac {Mc_{2}}{I}}=-{\cfrac {M}{S_{2}}}}
The quantities
S
1
,
S
2
{\displaystyle S_{1},S_{2}}
are the section moduli and are defined as
S
1
=
I
c
1
;
S
2
=
I
c
2
{\displaystyle S_{1}={\cfrac {I}{c_{1}}}~;~~S_{2}={\cfrac {I}{c_{2}}}}
The section modulus combines all the important geometric information about a beam's section into one quantity. For the case where a beam is doubly symmetric,
c
1
=
c
2
{\displaystyle c_{1}=c_{2}}
and we have one section modulus
S
=
I
/
c
{\displaystyle S=I/c}
.
=== Strain in an Euler–Bernoulli beam ===
We need an expression for the strain in terms of the deflection of the neutral surface to relate the stresses in an Euler–Bernoulli beam to the deflection. To obtain that expression we use the assumption that normals to the neutral surface remain normal during the deformation and that deflections are small. These assumptions imply that the beam bends into an arc of a circle of radius
ρ
{\displaystyle \rho }
(see Figure 1) and that the neutral surface does not change in length during the deformation.
Let
d
x
{\displaystyle \mathrm {d} x}
be the length of an element of the neutral surface in the undeformed state. For small deflections, the element does not change its length after bending but deforms into an arc of a circle of radius
ρ
{\displaystyle \rho }
. If
d
θ
{\displaystyle \mathrm {d} \theta }
is the angle subtended by this arc, then
d
x
=
ρ
d
θ
{\displaystyle \mathrm {d} x=\rho ~\mathrm {d} \theta }
.
Let us now consider another segment of the element at a distance
z
{\displaystyle z}
above the neutral surface. The initial length of this element is
d
x
{\displaystyle \mathrm {d} x}
. However, after bending, the length of the element becomes
d
x
′
=
(
ρ
−
z
)
d
θ
=
d
x
−
z
d
θ
{\displaystyle \mathrm {d} x'=(\rho -z)~\mathrm {d} \theta =\mathrm {d} x-z~\mathrm {d} \theta }
. The strain in that segment of the beam is given by
ε
x
=
d
x
′
−
d
x
d
x
=
−
z
ρ
=
−
κ
z
{\displaystyle \varepsilon _{x}={\cfrac {\mathrm {d} x'-\mathrm {d} x}{\mathrm {d} x}}=-{\cfrac {z}{\rho }}=-\kappa ~z}
where
κ
{\displaystyle \kappa }
is the curvature of the beam. This gives us the axial strain in the beam as a function of distance from the neutral surface. However, we still need to find a relation between the radius of curvature and the beam deflection
w
{\displaystyle w}
.
=== Relation between curvature and beam deflection ===
Let P be a point on the neutral surface of the beam at a distance
x
{\displaystyle x}
from the origin of the
(
x
,
z
)
{\displaystyle (x,z)}
coordinate system. The slope of the beam is approximately equal to the angle made by the neutral surface with the
x
{\displaystyle x}
-axis for the small angles encountered in beam theory. Therefore, with this approximation,
θ
(
x
)
=
d
w
d
x
{\displaystyle \theta (x)={\cfrac {\mathrm {d} w}{\mathrm {d} x}}}
Therefore, for an infinitesimal element
d
x
{\displaystyle \mathrm {d} x}
, the relation
d
x
=
ρ
d
θ
{\displaystyle \mathrm {d} x=\rho ~\mathrm {d} \theta }
can be written as
1
ρ
=
d
θ
d
x
=
d
2
w
d
x
2
=
κ
{\displaystyle {\cfrac {1}{\rho }}={\cfrac {\mathrm {d} \theta }{\mathrm {d} x}}={\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}=\kappa }
Hence the strain in the beam may be expressed as
ε
x
=
−
z
κ
{\displaystyle \varepsilon _{x}=-z\kappa }
=== Stress-strain relations ===
For a homogeneous isotropic linear elastic material, the stress is related to the strain by
σ
=
E
ε
{\displaystyle \sigma =E\varepsilon }
, where
E
{\displaystyle E}
is the Young's modulus. Hence the stress in an Euler–Bernoulli beam is given by
σ
x
=
−
z
E
d
2
w
d
x
2
{\displaystyle \sigma _{x}=-zE{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}}
Note that the above relation, when compared with the relation between the axial stress and the bending moment, leads to
M
=
−
E
I
d
2
w
d
x
2
{\displaystyle M=-EI{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}}
Since the shear force is given by
Q
=
d
M
/
d
x
{\displaystyle Q=\mathrm {d} M/\mathrm {d} x}
, we also have
Q
=
−
E
I
d
3
w
d
x
3
{\displaystyle Q=-EI{\cfrac {\mathrm {d} ^{3}w}{\mathrm {d} x^{3}}}}
== Boundary considerations ==
The beam equation contains a fourth-order derivative in
x
{\displaystyle x}
. To find a unique solution
w
(
x
,
t
)
{\displaystyle w(x,t)}
we need four boundary conditions. The boundary conditions usually model supports, but they can also model point loads, distributed loads and moments. The support or displacement boundary conditions are used to fix values of displacement (
w
{\displaystyle w}
) and rotations (
d
w
/
d
x
{\displaystyle \mathrm {d} w/\mathrm {d} x}
) on the boundary. Such boundary conditions are also called Dirichlet boundary conditions. Load and moment boundary conditions involve higher derivatives of
w
{\displaystyle w}
and represent momentum flux. Flux boundary conditions are also called Neumann boundary conditions.
As an example consider a cantilever beam that is built-in at one end and free at the other as shown in the adjacent figure. At the built-in end of the beam there cannot be any displacement or rotation of the beam. This means that at the left end both deflection and slope are zero. Since no external bending moment is applied at the free end of the beam, the bending moment at that location is zero. In addition, if there is no external force applied to the beam, the shear force at the free end is also zero.
Taking the
x
{\displaystyle x}
coordinate of the left end as
0
{\displaystyle 0}
and the right end as
L
{\displaystyle L}
(the length of the beam), these statements translate to the following set of boundary conditions (assume
E
I
{\displaystyle EI}
is a constant):
w
|
x
=
0
=
0
;
∂
w
∂
x
|
x
=
0
=
0
(fixed end)
{\displaystyle w|_{x=0}=0\quad ;\quad {\frac {\partial w}{\partial x}}{\bigg |}_{x=0}=0\qquad {\mbox{(fixed end)}}\,}
∂
2
w
∂
x
2
|
x
=
L
=
0
;
∂
3
w
∂
x
3
|
x
=
L
=
0
(free end)
{\displaystyle {\frac {\partial ^{2}w}{\partial x^{2}}}{\bigg |}_{x=L}=0\quad ;\quad {\frac {\partial ^{3}w}{\partial x^{3}}}{\bigg |}_{x=L}=0\qquad {\mbox{(free end)}}\,}
A simple support (pin or roller) is equivalent to a point force on the beam which is adjusted in such a way as to fix the position of the beam at that point. A fixed support or clamp, is equivalent to the combination of a point force and a point torque which is adjusted in such a way as to fix both the position and slope of the beam at that point. Point forces and torques, whether from supports or directly applied, will divide a beam into a set of segments, between which the beam equation will yield a continuous solution, given four boundary conditions, two at each end of the segment. Assuming that the product EI is a constant, and defining
λ
=
F
/
E
I
{\displaystyle \lambda =F/EI}
where F is the magnitude of a point force, and
τ
=
M
/
E
I
{\displaystyle \tau =M/EI}
where M is the magnitude of a point torque, the boundary conditions appropriate for some common cases is given in the table below. The change in a particular derivative of w across the boundary as x increases is denoted by
Δ
{\displaystyle \Delta }
followed by that derivative. For example,
Δ
w
″
=
w
″
(
x
+
)
−
w
″
(
x
−
)
{\displaystyle \Delta w''=w''(x+)-w''(x-)}
where
w
″
(
x
+
)
{\displaystyle w''(x+)}
is the value of
w
″
{\displaystyle w''}
at the lower boundary of the upper segment, while
w
″
(
x
−
)
{\displaystyle w''(x-)}
is the value of
w
″
{\displaystyle w''}
at the upper boundary of the lower segment. When the values of the particular derivative are not only continuous across the boundary, but fixed as well, the boundary condition is written e.g.,
Δ
w
″
=
0
∗
{\displaystyle \Delta w''=0^{*}}
which actually constitutes two separate equations (e.g.,
w
″
(
x
−
)
=
w
″
(
x
+
)
{\displaystyle w''(x-)=w''(x+)}
= fixed).
Note that in the first cases, in which the point forces and torques are located between two segments, there are four boundary conditions, two for the lower segment, and two for the upper. When forces and torques are applied to one end of the beam, there are two boundary conditions given which apply at that end. The sign of the point forces and torques at an end will be positive for the lower end, negative for the upper end.
== Loading considerations ==
Applied loads may be represented either through boundary conditions or through the function
q
(
x
,
t
)
{\displaystyle q(x,t)}
which represents an external distributed load. Using distributed loading is often favorable for simplicity. Boundary conditions are, however, often used to model loads depending on context; this practice being especially common in vibration analysis.
By nature, the distributed load is very often represented in a piecewise manner, since in practice a load isn't typically a continuous function. Point loads can be modeled with help of the Dirac delta function. For example, consider a static uniform cantilever beam of length
L
{\displaystyle L}
with an upward point load
F
{\displaystyle F}
applied at the free end. Using boundary conditions, this may be modeled in two ways. In the first approach, the applied point load is approximated by a shear force applied at the free end. In that case the governing equation and boundary conditions are:
E
I
d
4
w
d
x
4
=
0
w
|
x
=
0
=
0
;
d
w
d
x
|
x
=
0
=
0
;
d
2
w
d
x
2
|
x
=
L
=
0
;
−
E
I
d
3
w
d
x
3
|
x
=
L
=
F
{\displaystyle {\begin{aligned}&EI{\frac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}=0\\&w|_{x=0}=0\quad ;\quad {\frac {\mathrm {d} w}{\mathrm {d} x}}{\bigg |}_{x=0}=0\quad ;\quad {\frac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}{\bigg |}_{x=L}=0\quad ;\quad -EI{\frac {\mathrm {d} ^{3}w}{\mathrm {d} x^{3}}}{\bigg |}_{x=L}=F\,\end{aligned}}}
Alternatively we can represent the point load as a distribution using the Dirac function. In that case the equation and boundary conditions are
E
I
d
4
w
d
x
4
=
F
δ
(
x
−
L
)
w
|
x
=
0
=
0
;
d
w
d
x
|
x
=
0
=
0
;
d
2
w
d
x
2
|
x
=
L
=
0
{\displaystyle {\begin{aligned}&EI{\frac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}=F\delta (x-L)\\&w|_{x=0}=0\quad ;\quad {\frac {\mathrm {d} w}{\mathrm {d} x}}{\bigg |}_{x=0}=0\quad ;\quad {\frac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}{\bigg |}_{x=L}=0\,\end{aligned}}}
Note that shear force boundary condition (third derivative) is removed, otherwise there would be a contradiction. These are equivalent boundary value problems, and both yield the solution
w
=
F
6
E
I
(
3
L
x
2
−
x
3
)
.
{\displaystyle w={\frac {F}{6EI}}(3Lx^{2}-x^{3})\,~.}
The application of several point loads at different locations will lead to
w
(
x
)
{\displaystyle w(x)}
being a piecewise function. Use of the Dirac function greatly simplifies such situations; otherwise the beam would have to be divided into sections, each with four boundary conditions solved separately. A well organized family of functions called Singularity functions are often used as a shorthand for the Dirac function, its derivative, and its antiderivatives.
Dynamic phenomena can also be modeled using the static beam equation by choosing appropriate forms of the load distribution. As an example, the free vibration of a beam can be accounted for by using the load function:
q
(
x
,
t
)
=
μ
∂
2
w
∂
t
2
{\displaystyle q(x,t)=\mu {\frac {\partial ^{2}w}{\partial t^{2}}}\,}
where
μ
{\displaystyle \mu }
is the linear mass density of the beam, not necessarily a constant. With this time-dependent loading, the beam equation will be a partial differential equation:
∂
2
∂
x
2
(
E
I
∂
2
w
∂
x
2
)
=
−
μ
∂
2
w
∂
t
2
.
{\displaystyle {\frac {\partial ^{2}}{\partial x^{2}}}\left(EI{\frac {\partial ^{2}w}{\partial x^{2}}}\right)=-\mu {\frac {\partial ^{2}w}{\partial t^{2}}}.}
Another interesting example describes the deflection of a beam rotating with a constant angular frequency of
ω
{\displaystyle \omega }
:
q
(
x
)
=
μ
ω
2
w
(
x
)
{\displaystyle q(x)=\mu \omega ^{2}w(x)\,}
This is a centripetal force distribution. Note that in this case,
q
{\displaystyle q}
is a function of the displacement (the dependent variable), and the beam equation will be an autonomous ordinary differential equation.
== Examples ==
=== Three-point bending ===
The three-point bending test is a classical experiment in mechanics. It represents the case of a beam resting on two roller supports and subjected to a concentrated load applied in the middle of the beam. The shear is constant in absolute value: it is half the central load, P / 2. It changes sign in the middle of the beam. The bending moment varies linearly from one end, where it is 0, and the center where its absolute value is PL / 4, is where the risk of rupture is the most important.
The deformation of the beam is described by a polynomial of third degree over a half beam (the other half being symmetrical).
The bending moments (
M
{\displaystyle M}
), shear forces (
Q
{\displaystyle Q}
), and deflections (
w
{\displaystyle w}
) for a beam subjected to a central point load and an asymmetric point load are given in the table below.
=== Cantilever beams ===
Another important class of problems involves cantilever beams. The bending moments (
M
{\displaystyle M}
), shear forces (
Q
{\displaystyle Q}
), and deflections (
w
{\displaystyle w}
) for a cantilever beam subjected to a point load at the free end and a uniformly distributed load are given in the table below.
Solutions for several other commonly encountered configurations are readily available in textbooks on mechanics of materials and engineering handbooks.
=== Statically indeterminate beams ===
The bending moments and shear forces in Euler–Bernoulli beams can often be determined directly using static balance of forces and moments. However, for certain boundary conditions, the number of reactions can exceed the number of independent equilibrium equations. Such beams are called statically indeterminate.
The built-in beams shown in the figure below are statically indeterminate. To determine the stresses and deflections of such beams, the most direct method is to solve the Euler–Bernoulli beam equation with appropriate boundary conditions. But direct analytical solutions of the beam equation are possible only for the simplest cases. Therefore, additional techniques such as linear superposition are often used to solve statically indeterminate beam problems.
The superposition method involves adding the solutions of a number of statically determinate problems which are chosen such that the boundary conditions for the sum of the individual problems add up to those of the original problem.
Another commonly encountered statically indeterminate beam problem is the cantilevered beam with the free end supported on a roller. The bending moments, shear forces, and deflections of such a beam are listed below:
== Extensions ==
The kinematic assumptions upon which the Euler–Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternative constitutive equations can allow for viscoelastic or plastic beam deformation. Euler–Bernoulli beam theory can also be extended to the analysis of curved beams, beam buckling, composite beams, and geometrically nonlinear beam deflection.
Euler–Bernoulli beam theory does not account for the effects of transverse shear strain. As a result, it underpredicts deflections and overpredicts natural frequencies. For thin beams (beam length to thickness ratios of the order 20 or more) these effects are of minor importance. For thick beams, however, these effects can be significant. More advanced beam theories such as the Timoshenko beam theory (developed by the Russian-born scientist Stephen Timoshenko) have been developed to account for these effects.
=== Large deflections ===
The original Euler–Bernoulli theory is valid only for infinitesimal strains and small rotations. The theory can be extended in a straightforward manner to problems involving moderately large rotations provided that the strain remains small by using the von Kármán strains.
The Euler–Bernoulli hypotheses that plane sections remain plane and normal to the axis of the beam lead to displacements of the form
u
1
=
u
0
(
x
)
−
z
d
w
0
d
x
;
u
2
=
0
;
u
3
=
w
0
(
x
)
{\displaystyle u_{1}=u_{0}(x)-z{\cfrac {\mathrm {d} w_{0}}{\mathrm {d} x}}~;~~u_{2}=0~;~~u_{3}=w_{0}(x)}
Using the definition of the Lagrangian Green strain from finite strain theory, we can find the von Kármán strains for the beam that are valid for large rotations but small strains by discarding all the higher-order terms (which contain more than two fields) except
∂
w
∂
x
i
∂
w
∂
x
j
.
{\displaystyle {\frac {\partial {w}}{\partial {x^{i}}}}{\frac {\partial {w}}{\partial {x^{j}}}}.}
The resulting strains take the form:
ε
11
=
d
u
0
d
x
−
z
d
2
w
0
d
x
2
+
1
2
[
(
d
u
0
d
x
−
z
d
2
w
0
d
x
2
)
2
+
(
d
w
0
d
x
)
2
]
≈
d
u
0
d
x
−
z
d
2
w
0
d
x
2
+
1
2
(
d
w
0
d
x
)
2
ε
22
=
0
ε
33
=
1
2
(
d
w
0
d
x
)
2
ε
23
=
0
ε
31
=
−
1
2
[
(
d
u
0
d
x
−
z
d
2
w
0
d
x
2
)
(
d
w
0
d
x
)
]
≈
0
ε
12
=
0.
{\displaystyle {\begin{aligned}\varepsilon _{11}&={\cfrac {\mathrm {d} {u_{0}}}{\mathrm {d} {x}}}-z{\cfrac {\mathrm {d} ^{2}{w_{0}}}{\mathrm {d} {x^{2}}}}+{\frac {1}{2}}\left[\left({\cfrac {\mathrm {d} u_{0}}{\mathrm {d} x}}-z{\cfrac {\mathrm {d} ^{2}w_{0}}{\mathrm {d} x^{2}}}\right)^{2}+\left({\cfrac {\mathrm {d} w_{0}}{\mathrm {d} x}}\right)^{2}\right]\approx {\cfrac {\mathrm {d} {u_{0}}}{\mathrm {d} {x}}}-z{\cfrac {\mathrm {d} ^{2}{w_{0}}}{\mathrm {d} {x^{2}}}}+{\frac {1}{2}}\left({\frac {\mathrm {d} {w_{0}}}{\mathrm {d} {x}}}\right)^{2}\\[0.25em]\varepsilon _{22}&=0\\[0.25em]\varepsilon _{33}&={\frac {1}{2}}\left({\frac {\mathrm {d} {w_{0}}}{\mathrm {d} {x}}}\right)^{2}\\[0.25em]\varepsilon _{23}&=0\\[0.25em]\varepsilon _{31}&=-{\frac {1}{2}}\left[\left({\cfrac {\mathrm {d} u_{0}}{\mathrm {d} x}}-z{\cfrac {\mathrm {d} ^{2}w_{0}}{\mathrm {d} x^{2}}}\right)\left({\cfrac {\mathrm {d} w_{0}}{\mathrm {d} x}}\right)\right]\approx 0\\[0.25em]\varepsilon _{12}&=0.\end{aligned}}}
From the principle of virtual work, the balance of forces and moments in the beams gives us the equilibrium equations
d
N
x
x
d
x
+
f
(
x
)
=
0
d
2
M
x
x
d
x
2
+
q
(
x
)
+
d
d
x
(
N
x
x
d
w
0
d
x
)
=
0
{\displaystyle {\begin{aligned}{\cfrac {\mathrm {d} N_{xx}}{\mathrm {d} x}}+f(x)&=0\\{\cfrac {\mathrm {d} ^{2}M_{xx}}{\mathrm {d} x^{2}}}+q(x)+{\cfrac {\mathrm {d} }{\mathrm {d} x}}\left(N_{xx}{\cfrac {\mathrm {d} w_{0}}{\mathrm {d} x}}\right)&=0\end{aligned}}}
where
f
(
x
)
{\displaystyle f(x)}
is the axial load,
q
(
x
)
{\displaystyle q(x)}
is the transverse load, and
N
x
x
=
∫
A
σ
x
x
d
A
;
M
x
x
=
∫
A
z
σ
x
x
d
A
{\displaystyle N_{xx}=\int _{A}\sigma _{xx}~\mathrm {d} A~;~~M_{xx}=\int _{A}z\sigma _{xx}~\mathrm {d} A}
To close the system of equations we need the constitutive equations that relate stresses to strains (and hence stresses to displacements). For large rotations and small strains these relations are
N
x
x
=
A
x
x
[
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
]
−
B
x
x
d
2
w
0
d
x
2
M
x
x
=
B
x
x
[
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
]
−
D
x
x
d
2
w
0
d
x
2
{\displaystyle {\begin{aligned}N_{xx}&=A_{xx}\left[{\cfrac {\mathrm {d} u_{0}}{\mathrm {d} x}}+{\frac {1}{2}}\left({\cfrac {\mathrm {d} w_{0}}{\mathrm {d} x}}\right)^{2}\right]-B_{xx}{\cfrac {\mathrm {d} ^{2}w_{0}}{\mathrm {d} x^{2}}}\\M_{xx}&=B_{xx}\left[{\cfrac {\mathrm {d} u_{0}}{\mathrm {d} x}}+{\frac {1}{2}}\left({\cfrac {\mathrm {d} w_{0}}{\mathrm {d} x}}\right)^{2}\right]-D_{xx}{\cfrac {\mathrm {d} ^{2}w_{0}}{\mathrm {d} x^{2}}}\end{aligned}}}
where
A
x
x
=
∫
A
E
d
A
;
B
x
x
=
∫
A
z
E
d
A
;
D
x
x
=
∫
A
z
2
E
d
A
.
{\displaystyle A_{xx}=\int _{A}E~\mathrm {d} A~;~~B_{xx}=\int _{A}zE~\mathrm {d} A~;~~D_{xx}=\int _{A}z^{2}E~\mathrm {d} A~.}
The quantity
A
x
x
{\displaystyle A_{xx}}
is the extensional stiffness,
B
x
x
{\displaystyle B_{xx}}
is the coupled extensional-bending stiffness, and
D
x
x
{\displaystyle D_{xx}}
is the bending stiffness.
For the situation where the beam has a uniform cross-section and no axial load, the governing equation for a large-rotation Euler–Bernoulli beam is
E
I
d
4
w
d
x
4
−
3
2
E
A
(
d
w
d
x
)
2
(
d
2
w
d
x
2
)
=
q
(
x
)
{\displaystyle EI~{\cfrac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}-{\frac {3}{2}}~EA~\left({\cfrac {\mathrm {d} w}{\mathrm {d} x}}\right)^{2}\left({\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}\right)=q(x)}
== See also ==
Applied mechanics
Bending
Bending moment
Buckling
Flexural rigidity
Generalised beam theory
Plate theory
Sandwich theory
Shear and moment diagram
Singularity function
Strain (materials science)
Timoshenko beam theory
Theorem of three moments (Clapeyron's theorem)
Three-point flexural test
== References ==
=== Notes ===
=== Citations ===
=== Further reading ===
== External links ==
Beam stress & deflection, beam deflection tables | Wikipedia/Beam_theory |
Geospatial topology is the study and application of qualitative spatial relationships between geographic features, or between representations of such features in geographic information, such as in geographic information systems (GIS). For example, the fact that two regions overlap or that one contains the other are examples of topological relationships. It is thus the application of the mathematics of topology to GIS, and is distinct from, but complementary to the many aspects of geographic information that are based on quantitative spatial measurements through coordinate geometry. Topology appears in many aspects of geographic information science and GIS practice, including the discovery of inherent relationships through spatial query, vector overlay and map algebra; the enforcement of expected relationships as validation rules stored in geospatial data; and the use of stored topological relationships in applications such as network analysis.
Spatial topology is the generalization of geospatial topology for non-geographic domains, e.g., CAD software.
== Topological relationships ==
In keeping with the definition of topology, a topological relationship between two geographic phenomena is any spatial relation that is not sensitive to measurable aspects of space, including transformations of space (e.g. map projection). Thus, it includes most qualitative spatial relations, such as two features being "adjacent," "overlapping," "disjoint," or one being "within" another; conversely, one feature being "5km from" another, or one feature being "due north of" another are metric relations. One of the first developments of Geographic Information Science in the early 1990s was the work of Max Egenhofer, Eliseo Clementini, Peter di Felice, and others to develop a concise theory of such relations commonly called the 9-Intersection Model, which characterizes the range of topological relationships based on the relationships between the interiors, exteriors, and boundaries of features.
These relationships can also be classified semantically:
Inherent relationships are those that are important to the existence or identity of one or both of the related phenomena, such as one expressed in a boundary definition or being a manifestation of a mereological relationship. For example, Nebraska lies within the United States simply because the former was created by the latter as a partition of the territory of the latter. The Missouri River is adjacent to the state of Nebraska because the definition of the boundary of the state says so. These relationships are often stored and enforced in topologically-savvy data.
Coincidental relationships are those that are not crucial to the existence of either, although they can be very important. For example, the fact that the Platte River passes through Nebraska is coincidental because both would still exist unproblematically if the relationship did not exist. These relationships are rarely stored as such, but are usually discovered and documented by spatial analysis methods.
== Topological data structures and validation ==
Topology was a very early concern for GIS. The earliest vector systems, such as the Canadian Geographic Information System, did not manage topological relationships, and problems such as sliver polygons proliferated, especially in operations such as vector overlay. In response, topological vector data models were developed, such as GBF/DIME (U.S. Census Bureau, 1967) and POLYVRT (Harvard University, 1976). The strategy of the topological data model is to store topological relationships (primarily adjacency) between features, and use that information to construct more complex features. Nodes (points) are created where lines intersect and are attributed with a list of the connecting lines. Polygons are constructed from any sequence of lines that forms a closed loop. These structures had three advantages over non-topological vector data (often called "spaghetti data"): First, they were efficient (a crucial factor given the storage and processing capacities of the 1970s), because the shared boundary between two adjacent polygons was only stored once; second, they facilitated the enforcement of data integrity by preventing or highlighting topological errors, such as overlapping polygons, dangling nodes (a line not properly connected to other lines), and sliver polygons (small spurious polygons created where two lines should match but do not); and third, they made the algorithms for operations such as vector overlay simpler. Their primary disadvantage was their complexity, being difficult for many users to understand and requiring extra care during data entry. These became the dominant vector data model of the 1980s.
By the 1990s, the combination of cheaper storage and new users who were not concerned with topology led to a resurgence in spaghetti data structures, such as the shapefile. However, the need for stored topological relationships and integrity enforcement still exists. A common approach in current data is to store such as an extended layer on top of data that is not inherently topological. For example, the Esri geodatabase stores vector data ("feature classes") as spaghetti data, but can build a "network dataset" structure of connections on top of a line feature class. The geodatabase can also store a list of topological rules, constraints on topological relationships within and between layers (e.g., counties cannot have gaps, state boundaries must coincide with county boundaries, counties must collectively cover states) that can be validated and corrected. Other systems, such as PostGIS, take a similar approach. A very different approach is to not store topological information in the data at all, but to construct it dynamically, usually during the editing process, to highlight and correct possible errors; this is a feature of GIS software such as ArcGIS Pro and QGIS.
== Topology in spatial analysis ==
Several spatial analysis tools are ultimately based on the discovery of topological relationships between features:
spatial query, in which one is searching for the features in one dataset based on desired topological relationships to the features of a second dataset. For example, "where are the student locations within the boundaries of School X?"
spatial join, in which the attribute tables of two datasets are combined, with rows being matched based on a desired topological relationship between features in the two datasets, rather than using a stored key as in a normal table join in a relational database. For example, joining the attributes of a schools layer to the table of students based on which school boundary each student resides within.
vector overlay, in which two layers (usually polygons) are merged, with new features being created where features from the two input datasets intersect.
transport network analysis, a large class of tools in which connected lines (e.g., roads, utility infrastructure, streams) are analyzed using the mathematics of graph theory. The most common example is determining the optimal route between two locations through a street network, as implemented in most street web maps.
Oracle and PostGIS provide fundamental topological operators allowing applications to test for "such relationships as contains, inside, covers, covered by, touch, and overlap with boundaries intersecting." Unlike the PostGIS documentation, the Oracle documentation draws a distinction between "topological relationships [which] remain constant when the coordinate space is deformed, such as by twisting or stretching" and "relationships that are not topological [which] include length of, distance between, and area of." These operators are leveraged by applications to ensure that data sets are stored and processed in a topologically correct fashion. However, topological operators are inherently complex and their implementation requires care to be taken with usability and conformance to standards.
== See also ==
Digital topology
DE-9IM (Dimensionally Extended 9-Intersection Model)
== References == | Wikipedia/Geospatial_topology |
Topology of a transmembrane protein refers to locations of N- and C-termini of membrane-spanning polypeptide chain with respect to the inner or outer sides of the biological membrane occupied by the protein.
Several databases provide experimentally determined topologies of membrane proteins. They include Uniprot, TOPDB, OPM, and ExTopoDB. There is also a database of domains located conservatively on a certain side of membranes, TOPDOM.
Several computational methods were developed, with a limited success, for predicting transmembrane alpha-helices and their topology. Pioneer methods utilized the fact that membrane-spanning regions contain more hydrophobic residues than other parts of the protein, however applying different hydrophobic scales altered the prediction results. Later, several statistical methods were developed to improve the topography prediction and a special alignment method was introduced. According to the positive-inside rule, cytosolic loops near the lipid bilayer contain more positively-charged amino acids. Applying this rule resulted in the first topology prediction methods. There is also a negative-outside rule in transmembrane alpha-helices from single-pass proteins, although negatively charged residues are rarer than positively charged residues in transmembrane segments of proteins. As more structures were determined, machine learning algorithms appeared. Supervised learning methods are trained on a set of experimentally determined structures, however, these methods highly depend on the training set. Unsupervised learning methods are based on the principle that topology depends on the maximum divergence of the amino acid distributions in different structural parts. It was also shown that locking a segment location based on prior knowledge about the structure improves the prediction accuracy. This feature has been added to some of the existing prediction methods. The most recent methods use consensus prediction (i.e. they use several algorithms to determine the final topology) and automatically incorporate previously determined experimental informations. HTP database provides a collection of topologies that are computationally predicted for human transmembrane proteins.
Discrimination of signal peptides and transmembrane segments is an additional problem in topology prediction treated with a limited success by different methods. Both signal peptides and transmembrane segments contain hydrophobic regions which form α-helices. This causes the cross-prediction between them, which is a weakness of many transmembrane topology predictors. By predicting signal peptides and transmembrane helices simultaneously (Phobius), the errors caused by cross-prediction are reduced and the performance is substantially increased. Another feature used to increase the accuracy of the prediction is the homology (PolyPhobius).”
It is also possible to predict beta-barrel membrane proteins' topology.
== See also ==
Endomembrane system
Integral membrane protein
Protein topology
Structural biology
Transmembrane domain
== References == | Wikipedia/Membrane_topology |
Network topology is the arrangement of the elements (links, nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and control radio networks, industrial fieldbusses and computer networks.
Network topology is the topological structure of a network and may be depicted physically or logically. It is an application of graph theory wherein communicating devices are modeled as nodes and the connections between the devices are modeled as links or lines between the nodes. Physical topology is the placement of the various components of a network (e.g., device location and cable installation), while logical topology illustrates how data flows within a network. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two different networks, yet their logical topologies may be identical. A network's physical topology is a particular concern of the physical layer of the OSI model.
Examples of network topologies are found in local area networks (LAN), a common computer network installation. Any given node in the LAN has one or more physical links to other devices in the network; graphically mapping these links results in a geometric shape that can be used to describe the physical topology of the network. A wide variety of physical topologies have been used in LANs, including ring, bus, mesh and star. Conversely, mapping the data flow between the components determines the logical topology of the network. In comparison, Controller Area Networks, common in vehicles, are primarily distributed control system networks of one or more controllers interconnected with sensors and actuators over, invariably, a physical bus topology.
== Topologies ==
Two basic categories of network topologies exist, physical topologies and logical topologies.
The transmission medium layout used to link devices is the physical topology of the network. For conductive or fiber optical mediums, this refers to the layout of cabling, the locations of nodes, and the links between the nodes and the cabling. The physical topology of a network is determined by the capabilities of the network access devices and media, the level of control or fault tolerance desired, and the cost associated with cabling or telecommunication circuits.
In contrast, logical topology is the way that the signals act on the network media, or the way that the data passes through the network from one device to the next without regard to the physical interconnection of the devices. A network's logical topology is not necessarily the same as its physical topology. For example, the original twisted pair Ethernet using repeater hubs was a logical bus topology carried on a physical star topology. Token Ring is a logical ring topology, but is wired as a physical star from the media access unit. Physically, Avionics Full-Duplex Switched Ethernet (AFDX) can be a cascaded star topology of multiple dual redundant Ethernet switches; however, the AFDX virtual links are modeled as time-switched single-transmitter bus connections, thus following the safety model of a single-transmitter bus topology previously used in aircraft. Logical topologies are often closely associated with media access control methods and protocols. Some networks are able to dynamically change their logical topology through configuration changes to their routers and switches.
== Links ==
The transmission media (often referred to in the literature as the physical media) used to link devices to form a computer network include electrical cables (Ethernet, HomePNA, power line communication, G.hn), optical fiber (fiber-optic communication), and radio waves (wireless networking). In the OSI model, these are defined at layers 1 and 2 — the physical layer and the data link layer.
A widely adopted family of transmission media used in local area network (LAN) technology is collectively known as Ethernet. The media and protocol standards that enable communication between networked devices over Ethernet are defined by IEEE 802.3. Ethernet transmits data over both copper and fiber cables. Wireless LAN standards (e.g. those defined by IEEE 802.11) use radio waves, or others use infrared signals as a transmission medium. Power line communication uses a building's power cabling to transmit data.
=== Wired technologies ===
The orders of the following wired technologies are, roughly, from slowest to fastest transmission speed.
Coaxial cable is widely used for cable television systems, office buildings, and other work-sites for local area networks. The cables consist of copper or aluminum wire surrounded by an insulating layer (typically a flexible material with a high dielectric constant), which itself is surrounded by a conductive layer. The insulation between the conductors helps maintain the characteristic impedance of the cable which can help improve its performance. Transmission speed ranges from 200 million bits per second to more than 500 million bits per second.
ITU-T G.hn technology uses existing home wiring (coaxial cable, phone lines and power lines) to create a high-speed (up to 1 Gigabit/s) local area network.
Signal traces on printed circuit boards are common for board-level serial communication, particularly between certain types integrated circuits, a common example being SPI.
Ribbon cable (untwisted and possibly unshielded) has been a cost-effective media for serial protocols, especially within metallic enclosures or rolled within copper braid or foil, over short distances, or at lower data rates. Several serial network protocols can be deployed without shielded or twisted pair cabling, that is, with flat or ribbon cable, or a hybrid flat and twisted ribbon cable, should EMC, length, and bandwidth constraints permit: RS-232, RS-422, RS-485, CAN, GPIB, SCSI, etc.
Twisted pair wire is the most widely used medium for all telecommunication. Twisted-pair cabling consist of copper wires that are twisted into pairs. Ordinary telephone wires consist of two insulated copper wires twisted into pairs. Computer network cabling (wired Ethernet as defined by IEEE 802.3) consists of 4 pairs of copper cabling that can be utilized for both voice and data transmission. The use of two wires twisted together helps to reduce crosstalk and electromagnetic induction. The transmission speed ranges from 2 million bits per second to 10 billion bits per second. Twisted pair cabling comes in two forms: unshielded twisted pair (UTP) and shielded twisted pair (STP). Each form comes in several category ratings, designed for use in various scenarios.
An optical fiber is a glass fiber. It carries pulses of light that represent data. Some advantages of optical fibers over metal wires are very low transmission loss and immunity from electrical interference. Optical fibers can simultaneously carry multiple wavelengths of light, which greatly increases the rate that data can be sent, and helps enable data rates of up to trillions of bits per second. Optic fibers can be used for long runs of cable carrying very high data rates, and are used for undersea communications cables to interconnect continents.
Price is a main factor distinguishing wired- and wireless technology options in a business. Wireless options command a price premium that can make purchasing wired computers, printers and other devices a financial benefit. Before making the decision to purchase hard-wired technology products, a review of the restrictions and limitations of the selections is necessary. Business and employee needs may override any cost considerations.
=== Wireless technologies ===
Terrestrial microwave – Terrestrial microwave communication uses Earth-based transmitters and receivers resembling satellite dishes. Terrestrial microwaves are in the low gigahertz range, which limits all communications to line-of-sight. Relay stations are spaced approximately 50 km (30 mi) apart.
Communications satellites – Satellites communicate via microwave radio waves, which are not deflected by the Earth's atmosphere. The satellites are stationed in space, typically in geostationary orbit 35,786 km (22,236 mi) above the equator. These Earth-orbiting systems are capable of receiving and relaying voice, data, and TV signals.
Cellular and PCS systems use several radio communications technologies. The systems divide the region covered into multiple geographic areas. Each area has a low-power transmitter or radio relay antenna device to relay calls from one area to the next area.
Radio and spread spectrum technologies – Wireless local area networks use a high-frequency radio technology similar to digital cellular and a low-frequency radio technology. Wireless LANs use spread spectrum technology to enable communication between multiple devices in a limited area. IEEE 802.11 defines a common flavor of open-standards wireless radio-wave technology known as Wi-Fi.
Free-space optical communication uses visible or invisible light for communications. In most cases, line-of-sight propagation is used, which limits the physical positioning of communicating devices.
=== Exotic technologies ===
There have been various attempts at transporting data over exotic media:
IP over Avian Carriers was a humorous April fool's Request for Comments, issued as RFC 1149. It was implemented in real life in 2001.
Extending the Internet to interplanetary dimensions via radio waves, the Interplanetary Internet.
Both cases have a large round-trip delay time, which gives slow two-way communication, but does not prevent sending large amounts of information.
== Nodes ==
Network nodes are the points of connection of the transmission medium to transmitters and receivers of the electrical, optical, or radio signals carried in the medium. Nodes may be associated with a computer, but certain types may have only a microcontroller at a node or possibly no programmable device at all. In the simplest of serial arrangements, one RS-232 transmitter can be connected by a pair of wires to one receiver, forming two nodes on one link, or a Point-to-Point topology. Some protocols permit a single node to only either transmit or receive (e.g., ARINC 429). Other protocols have nodes that can both transmit and receive into a single channel (e.g., CAN can have many transceivers connected to a single bus). While the conventional system building blocks of a computer network include network interface controllers (NICs), repeaters, hubs, bridges, switches, routers, modems, gateways, and firewalls, most address network concerns beyond the physical network topology and may be represented as single nodes on a particular physical network topology.
=== Network interfaces ===
A network interface controller (NIC) is computer hardware that provides a computer with the ability to access the transmission media, and has the ability to process low-level network information. For example, the NIC may have a connector for accepting a cable, or an aerial for wireless transmission and reception, and the associated circuitry.
The NIC responds to traffic addressed to a network address for either the NIC or the computer as a whole.
In Ethernet networks, each network interface controller has a unique Media Access Control (MAC) address—usually stored in the controller's permanent memory. To avoid address conflicts between network devices, the Institute of Electrical and Electronics Engineers (IEEE) maintains and administers MAC address uniqueness. The size of an Ethernet MAC address is six octets. The three most significant octets are reserved to identify NIC manufacturers. These manufacturers, using only their assigned prefixes, uniquely assign the three least-significant octets of every Ethernet interface they produce.
=== Repeaters and hubs ===
A repeater is an electronic device that receives a network signal, cleans it of unnecessary noise and regenerates it. The signal may be reformed or retransmitted at a higher power level, to the other side of an obstruction possibly using a different transmission medium, so that the signal can cover longer distances without degradation. Commercial repeaters have extended RS-232 segments from 15 meters to over a kilometer. In most twisted pair Ethernet configurations, repeaters are required for cable that runs longer than 100 meters. With fiber optics, repeaters can be tens or even hundreds of kilometers apart.
Repeaters work within the physical layer of the OSI model, that is, there is no end-to-end change in the physical protocol across the repeater, or repeater pair, even if a different physical layer may be used between the ends of the repeater, or repeater pair. Repeaters require a small amount of time to regenerate the signal. This can cause a propagation delay that affects network performance and may affect proper function. As a result, many network architectures limit the number of repeaters that can be used in a row, e.g., the Ethernet 5-4-3 rule.
A repeater with multiple ports is known as hub, an Ethernet hub in Ethernet networks, a USB hub in USB networks.
USB networks use hubs to form tiered-star topologies.
Ethernet hubs and repeaters in LANs have been mostly obsoleted by modern switches.
=== Bridges ===
A network bridge connects and filters traffic between two network segments at the data link layer (layer 2) of the OSI model to form a single network. This breaks the network's collision domain but maintains a unified broadcast domain. Network segmentation breaks down a large, congested network into an aggregation of smaller, more efficient networks.
Bridges come in three basic types:
Local bridges: Directly connect LANs
Remote bridges: Can be used to create a wide area network (WAN) link between LANs. Remote bridges, where the connecting link is slower than the end networks, largely have been replaced with routers.
Wireless bridges: Can be used to join LANs or connect remote devices to LANs.
=== Switches ===
A network switch is a device that forwards and filters OSI layer 2 datagrams (frames) between ports based on the destination MAC address in each frame.
A switch is distinct from a hub in that it only forwards the frames to the physical ports involved in the communication rather than all ports connected. It can be thought of as a multi-port bridge. It learns to associate physical ports to MAC addresses by examining the source addresses of received frames. If an unknown destination is targeted, the switch broadcasts to all ports but the source. Switches normally have numerous ports, facilitating a star topology for devices, and cascading additional switches.
Multi-layer switches are capable of routing based on layer 3 addressing or additional logical levels. The term switch is often used loosely to include devices such as routers and bridges, as well as devices that may distribute traffic based on load or based on application content (e.g., a Web URL identifier).
=== Routers ===
A router is an internetworking device that forwards packets between networks by processing the routing information included in the packet or datagram (Internet protocol information from layer 3). The routing information is often processed in conjunction with the routing table (or forwarding table). A router uses its routing table to determine where to forward packets. A destination in a routing table can include a black hole because data can go into it, however, no further processing is done for said data, i.e. the packets are dropped.
=== Modems ===
Modems (MOdulator-DEModulator) are used to connect network nodes via wire not originally designed for digital network traffic, or for wireless. To do this one or more carrier signals are modulated by the digital signal to produce an analog signal that can be tailored to give the required properties for transmission. Modems are commonly used for telephone lines, using a digital subscriber line technology.
=== Firewalls ===
A firewall is a network device for controlling network security and access rules. Firewalls are typically configured to reject access requests from unrecognized sources while allowing actions from recognized ones. The vital role firewalls play in network security grows in parallel with the constant increase in cyber attacks.
== Classification ==
The study of network topology recognizes eight basic topologies: point-to-point, bus, star, ring or circular, mesh, tree, hybrid, or daisy chain.
=== Point-to-point ===
The simplest topology with a dedicated link between two endpoints. Easiest to understand, of the variations of point-to-point topology, is a point-to-point communication channel that appears, to the user, to be permanently associated with the two endpoints. A child's tin can telephone is one example of a physical dedicated channel.
Using circuit-switching or packet-switching technologies, a point-to-point circuit can be set up dynamically and dropped when no longer needed. Switched point-to-point topologies are the basic model of conventional telephony.
The value of a permanent point-to-point network is unimpeded communications between the two endpoints. The value of an on-demand point-to-point connection is proportional to the number of potential pairs of subscribers and has been expressed as Metcalfe's Law.
=== Daisy chain ===
Daisy chaining is accomplished by connecting each computer in series to the next. If a message is intended for a computer partway down the line, each system bounces it along in sequence until it reaches the destination. A daisy-chained network can take two basic forms: linear and ring.
A linear topology puts a two-way link between one computer and the next. However, this was expensive in the early days of computing, since each computer (except for the ones at each end) required two receivers and two transmitters.
By connecting the computers at each end of the chain, a ring topology can be formed. When a node sends a message, the message is processed by each computer in the ring. An advantage of the ring is that the number of transmitters and receivers can be cut in half. Since a message will eventually loop all of the way around, transmission does not need to go both directions. Alternatively, the ring can be used to improve fault tolerance. If the ring breaks at a particular link then the transmission can be sent via the reverse path thereby ensuring that all nodes are always connected in the case of a single failure.
=== Bus ===
In local area networks using bus topology, each node is connected by interface connectors to a single central cable. This is the 'bus', also referred to as the backbone, or trunk – all data transmission between nodes in the network is transmitted over this common transmission medium and is able to be received by all nodes in the network simultaneously.
A signal containing the address of the intended receiving machine travels from a source machine in both directions to all machines connected to the bus until it finds the intended recipient, which then accepts the data. If the machine address does not match the intended address for the data, the data portion of the signal is ignored. Since the bus topology consists of only one wire it is less expensive to implement than other topologies, but the savings are offset by the higher cost of managing the network. Additionally, since the network is dependent on the single cable, it can be the single point of failure of the network. In this topology data being transferred may be accessed by any node.
==== Linear bus ====
In a linear bus network, all of the nodes of the network are connected to a common transmission medium which has just two endpoints. When the electrical signal reaches the end of the bus, the signal is reflected back down the line, causing unwanted interference. To prevent this, the two endpoints of the bus are normally terminated with a device called a terminator.
==== Distributed bus ====
In a distributed bus network, all of the nodes of the network are connected to a common transmission medium with more than two endpoints, created by adding branches to the main section of the transmission medium – the physical distributed bus topology functions in exactly the same fashion as the physical linear bus topology because all nodes share a common transmission medium.
=== Star ===
In star topology (also called hub-and-spoke), every peripheral node (computer workstation or any other peripheral) is connected to a central node called a hub or switch. The hub is the server and the peripherals are the clients. The network does not necessarily have to resemble a star to be classified as a star network, but all of the peripheral nodes on the network must be connected to one central hub. All traffic that traverses the network passes through the central hub, which acts as a signal repeater.
The star topology is considered the easiest topology to design and implement. One advantage of the star topology is the simplicity of adding additional nodes. The primary disadvantage of the star topology is that the hub represents a single point of failure. Also, since all peripheral communication must flow through the central hub, the aggregate central bandwidth forms a network bottleneck for large clusters.
==== Extended star ====
The extended star network topology extends a physical star topology by one or more repeaters between the central node and the peripheral (or 'spoke') nodes. The repeaters are used to extend the maximum transmission distance of the physical layer, the point-to-point distance between the central node and the peripheral nodes. Repeaters allow greater transmission distance, further than would be possible using just the transmitting power of the central node. The use of repeaters can also overcome limitations from the standard upon which the physical layer is based.
A physical extended star topology in which repeaters are replaced with hubs or switches is a type of hybrid network topology and is referred to as a physical hierarchical star topology, although some texts make no distinction between the two topologies.
A physical hierarchical star topology can also be referred as a tier-star topology. This topology differs from a tree topology in the way star networks are connected together. A tier-star topology uses a central node, while a tree topology uses a central bus and can also be referred as a star-bus network.
==== Distributed star ====
A distributed star is a network topology that is composed of individual networks that are based upon the physical star topology connected in a linear fashion – i.e., 'daisy-chained' – with no central or top level connection point (e.g., two or more 'stacked' hubs, along with their associated star connected nodes or 'spokes').
=== Ring ===
A ring topology is a daisy chain in a closed loop. Data travels around the ring in one direction. When one node sends data to another, the data passes through each intermediate node on the ring until it reaches its destination. The intermediate nodes repeat (retransmit) the data to keep the signal strong. Every node is a peer; there is no hierarchical relationship of clients and servers. If one node is unable to retransmit data, it severs communication between the nodes before and after it in the bus.
Advantages:
When the load on the network increases, its performance is better than bus topology.
There is no need of network server to control the connectivity between workstations.
Disadvantages:
Aggregate network bandwidth is bottlenecked by the weakest link between two nodes.
=== Mesh ===
The value of fully meshed networks is proportional to the exponent of the number of subscribers, assuming that communicating groups of any two endpoints, up to and including all the endpoints, is approximated by Reed's Law.
==== Fully connected network ====
In a fully connected network, all nodes are interconnected. (In graph theory this is called a complete graph.) The simplest fully connected network is a two-node network. A fully connected network doesn't need to use packet switching or broadcasting. However, since the number of connections grows quadratically with the number of nodes:
c
=
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(
n
−
1
)
2
.
{\displaystyle c={\frac {n(n-1)}{2}}.\,}
This makes it impractical for large networks. This kind of topology does not trip and affect other nodes in the network.
==== Partially connected network ====
In a partially connected network, certain nodes are connected to exactly one other node; but some nodes are connected to two or more other nodes with a point-to-point link. This makes it possible to make use of some of the redundancy of mesh topology that is physically fully connected, without the expense and complexity required for a connection between every node in the network.
=== Hybrid ===
Hybrid topology is also known as hybrid network. Hybrid networks combine two or more topologies in such a way that the resulting network does not exhibit one of the standard topologies (e.g., bus, star, ring, etc.). For example, a tree network (or star-bus network) is a hybrid topology in which star networks are interconnected via bus networks. However, a tree network connected to another tree network is still topologically a tree network, not a distinct network type. A hybrid topology is always produced when two different basic network topologies are connected.
A star-ring network consists of two or more ring networks connected using a multistation access unit (MAU) as a centralized hub.
Snowflake topology is meshed at the core, but tree shaped at the edges.
Two other hybrid network types are hybrid mesh and hierarchical star.
== Centralization ==
The star topology reduces the probability of a network failure by connecting all of the peripheral nodes (computers, etc.) to a central node. When the physical star topology is applied to a logical bus network such as Ethernet, this central node (traditionally a hub) rebroadcasts all transmissions received from any peripheral node to all peripheral nodes on the network, sometimes including the originating node. All peripheral nodes may thus communicate with all others by transmitting to, and receiving from, the central node only. The failure of a transmission line linking any peripheral node to the central node will result in the isolation of that peripheral node from all others, but the remaining peripheral nodes will be unaffected. However, the disadvantage is that the failure of the central node will cause the failure of all of the peripheral nodes.
If the central node is passive, the originating node must be able to tolerate the reception of an echo of its own transmission, delayed by the two-way round trip transmission time (i.e. to and from the central node) plus any delay generated in the central node. An active star network has an active central node that usually has the means to prevent echo-related problems.
A tree topology (a.k.a. hierarchical topology) can be viewed as a collection of star networks arranged in a hierarchy. This tree structure has individual peripheral nodes (e.g. leaves) which are required to transmit to and receive from one other node only and are not required to act as repeaters or regenerators. Unlike the star network, the functionality of the central node may be distributed.
As in the conventional star network, individual nodes may thus still be isolated from the network by a single-point failure of a transmission path to the node. If a link connecting a leaf fails, that leaf is isolated; if a connection to a non-leaf node fails, an entire section of the network becomes isolated from the rest.
To alleviate the amount of network traffic that comes from broadcasting all signals to all nodes, more advanced central nodes were developed that are able to keep track of the identities of the nodes that are connected to the network. These network switches will learn the layout of the network by listening on each port during normal data transmission, examining the data packets and recording the address/identifier of each connected node and which port it is connected to in a lookup table held in memory. This lookup table then allows future transmissions to be forwarded to the intended destination only.
Daisy chain topology is a way of connecting network nodes in a linear or ring structure. It is used to transmit messages from one node to the next until they reach the destination node.
A daisy chain network can have two types: linear and ring. A linear daisy chain network is like an electrical series, where the first and last nodes are not connected. A ring daisy chain network is where the first and last nodes are connected, forming a loop.
== Decentralization ==
In a partially connected mesh topology, there are at least two nodes with two or more paths between them to provide redundant paths in case the link providing one of the paths fails. Decentralization is often used to compensate for the single-point-failure disadvantage that is present when using a single device as a central node (e.g., in star and tree networks). A special kind of mesh, limiting the number of hops between two nodes, is a hypercube. The number of arbitrary forks in mesh networks makes them more difficult to design and implement, but their decentralized nature makes them very useful.
This is similar in some ways to a grid network, where a linear or ring topology is used to connect systems in multiple directions. A multidimensional ring has a toroidal topology, for instance.
A fully connected network, complete topology, or full mesh topology is a network topology in which there is a direct link between all pairs of nodes. In a fully connected network with n nodes, there are
n
(
n
−
1
)
2
{\displaystyle {\frac {n(n-1)}{2}}\,}
direct links. Networks designed with this topology are usually very expensive to set up, but provide a high degree of reliability due to the multiple paths for data that are provided by the large number of redundant links between nodes. This topology is mostly seen in military applications.
== See also ==
== References ==
== External links ==
Tetrahedron Core Network: Application of a tetrahedral structure to create a resilient partial-mesh 3-dimensional campus backbone data network | Wikipedia/Logical_topology |
A transmembrane protein is a type of integral membrane protein that spans the entirety of the cell membrane. Many transmembrane proteins function as gateways to permit the transport of specific substances across the membrane. They frequently undergo significant conformational changes to move a substance through the membrane. They are usually highly hydrophobic and aggregate and precipitate in water. They require detergents or nonpolar solvents for extraction, although some of them (beta-barrels) can be also extracted using denaturing agents.
The peptide sequence that spans the membrane, or the transmembrane segment, is largely hydrophobic and can be visualized using the hydropathy plot. Depending on the number of transmembrane segments, transmembrane proteins can be classified as single-pass membrane proteins, or as multipass membrane proteins. Some other integral membrane proteins are called monotopic, meaning that they are also permanently attached to the membrane, but do not pass through it.
== Types ==
=== Classification by structure ===
There are two basic types of transmembrane proteins: alpha-helical and beta barrels. Alpha-helical proteins are present in the inner membranes of bacterial cells or the plasma membrane of eukaryotic cells, and sometimes in the bacterial outer membrane. This is the major category of transmembrane proteins. In humans, 27% of all proteins have been estimated to be alpha-helical membrane proteins.
Beta-barrel proteins are so far found only in outer membranes of gram-negative bacteria, cell walls of gram-positive bacteria, outer membranes of mitochondria and chloroplasts, or can be secreted as pore-forming toxins. All beta-barrel transmembrane proteins have simplest up-and-down topology, which may reflect their common evolutionary origin and similar folding mechanism.
In addition to the protein domains, there are unusual transmembrane elements formed by peptides. A typical example is gramicidin A, a peptide that forms a dimeric transmembrane β-helix. This peptide is secreted by gram-positive bacteria as an antibiotic. A transmembrane polyproline-II helix has not been reported in natural proteins. Nonetheless, this structure was experimentally observed in specifically designed artificial peptides.
=== Classification by topology ===
This classification refers to the position of the protein N- and C-termini on the different sides of the lipid bilayer. Types I, II, III and IV are single-pass molecules. Type I transmembrane proteins are anchored to the lipid membrane with a stop-transfer anchor sequence and have their N-terminal domains targeted to the endoplasmic reticulum (ER) lumen during synthesis (and the extracellular space, if mature forms are located on cell membranes). Type II and III are anchored with a signal-anchor sequence, with type II being targeted to the ER lumen with its C-terminal domain, while type III have their N-terminal domains targeted to the ER lumen. Type IV is subdivided into IV-A, with their N-terminal domains targeted to the cytosol and IV-B, with an N-terminal domain targeted to the lumen. The implications for the division in the four types are especially manifest at the time of translocation and ER-bound translation, when the protein has to be passed through the ER membrane in a direction dependent on the type.
== 3D structure ==
Membrane protein structures can be determined by X-ray crystallography, electron microscopy or NMR spectroscopy. The most common tertiary structures of these proteins are transmembrane helix bundle and beta barrel. The portion of the membrane proteins that are attached to the lipid bilayer (see annular lipid shell) consist mostly of hydrophobic amino acids.
Membrane proteins which have hydrophobic surfaces, are relatively flexible and are expressed at relatively low levels. This creates difficulties in obtaining enough protein and then growing crystals. Hence, despite the significant functional importance of membrane proteins, determining atomic resolution structures for these proteins is more difficult than globular proteins. As of January 2013 less than 0.1% of protein structures determined were membrane proteins despite being 20–30% of the total proteome. Due to this difficulty and the importance of this class of proteins methods of protein structure prediction based on hydropathy plots, the positive inside rule and other methods have been developed.
== Thermodynamic stability and folding ==
=== Stability of alpha-helical transmembrane proteins ===
Transmembrane alpha-helical (α-helical) proteins are unusually stable judging from thermal denaturation studies, because they do not unfold completely within the membranes (the complete unfolding would require breaking down too many α-helical H-bonds in the nonpolar media). On the other hand, these proteins easily misfold, due to non-native aggregation in membranes, transition to the molten globule states, formation of non-native disulfide bonds, or unfolding of peripheral regions and nonregular loops that are locally less stable.
It is also important to properly define the unfolded state. The unfolded state of membrane proteins in detergent micelles is different from that in the thermal denaturation experiments. This state represents a combination of folded hydrophobic α-helices and partially unfolded segments covered by the detergent. For example, the "unfolded" bacteriorhodopsin in SDS micelles has four transmembrane α-helices folded, while the rest of the protein is situated at the micelle-water interface and can adopt different types of non-native amphiphilic structures. Free energy differences between such detergent-denatured and native states are similar to stabilities of water-soluble proteins (< 10 kcal/mol).
=== Folding of α-helical transmembrane proteins ===
Refolding of α-helical transmembrane proteins in vitro is technically difficult. There are relatively few examples of the successful refolding experiments, as for bacteriorhodopsin. In vivo, all such proteins are normally folded co-translationally within the large transmembrane translocon. The translocon channel provides a highly heterogeneous environment for the nascent transmembrane α-helices. A relatively polar amphiphilic α-helix can adopt a transmembrane orientation in the translocon (although it would be at the membrane surface or unfolded in vitro), because its polar residues can face the central water-filled channel of the translocon. Such mechanism is necessary for incorporation of polar α-helices into structures of transmembrane proteins. The amphiphilic helices remain attached to the translocon until the protein is completely synthesized and folded. If the protein remains unfolded and attached to the translocon for too long, it is degraded by specific "quality control" cellular systems.
=== Stability and folding of beta-barrel transmembrane proteins ===
Stability of beta barrel (β-barrel) transmembrane proteins is similar to stability of water-soluble proteins, based on chemical denaturation studies. Some of them are very stable even in chaotropic agents and high temperature. Their folding in vivo is facilitated by water-soluble chaperones, such as protein Skp. It is thought that β-barrel membrane proteins come from one ancestor even having different number of sheets which could be added or doubled during evolution. Some studies show a huge sequence conservation among different organisms and also conserved amino acids which hold the structure and help with folding.
== 3D structures ==
=== Light absorption-driven transporters ===
Bacteriorhodopsin-like proteins including rhodopsin (see also opsin)
Bacterial photosynthetic reaction centres and photosystems I and II
Light-harvesting complexes from bacteria and chloroplasts
=== Oxidoreduction-driven transporters ===
Transmembrane cytochrome b-like proteins: coenzyme Q - cytochrome c reductase (cytochrome bc1 ); cytochrome b6f complex; formate dehydrogenase, respiratory nitrate reductase; succinate - coenzyme Q reductase (fumarate reductase); and succinate dehydrogenase. See electron transport chain.
Cytochrome c oxidases from bacteria and mitochondria
=== Electrochemical potential-driven transporters ===
Proton or sodium translocating F-type and V-type ATPases
=== P-P-bond hydrolysis-driven transporters ===
P-type calcium ATPase (five different conformations)
Calcium ATPase regulators phospholamban and sarcolipin
ABC transporters
General secretory pathway (Sec) translocon (preprotein translocase SecY)
=== Porters (uniporters, symporters, antiporters) ===
Mitochondrial carrier proteins
Major Facilitator Superfamily (Glycerol-3-phosphate transporter, Lactose permease, and Multidrug transporter EmrD)
Resistance-nodulation-cell division (multidrug efflux transporter AcrB, see multidrug resistance)
Dicarboxylate/amino acid:cation symporter (proton glutamate symporter)
Monovalent cation/proton antiporter (Sodium/proton antiporter 1 NhaA)
Neurotransmitter sodium symporter
Ammonia transporters
Drug/Metabolite Transporter (small multidrug resistance transporter EmrE - the structures are retracted as erroneous)
=== Alpha-helical channels including ion channels ===
Voltage-gated ion channel like, including potassium channels KcsA and KvAP, and inward-rectifier potassium ion channel Kirbac
Large-conductance mechanosensitive channel, MscL
Small-conductance mechanosensitive ion channel (MscS)
CorA metal ion transporters
Ligand-gated ion channel of neurotransmitter receptors (acetylcholine receptor)
Aquaporins
Chloride channels
Outer membrane auxiliary proteins (polysaccharide transporter) - α-helical transmembrane proteins from the outer bacterial membrane
=== Enzymes ===
Methane monooxygenase
Rhomboid protease
Disulfide bond formation protein (DsbA-DsbB complex)
=== Proteins with single transmembrane alpha-helices ===
Subunits of T cell receptor complex
Cytochrome c nitrite reductase complex
Glycophorin A dimer
Inovirus (filamentous phage) major coat protein
Pilin
Pulmonary surfactant-associated protein
Monoamine oxidases A and B
Fatty acid amide hydrolase
Cytochrome P450 oxidases
Corticosteroid 11β-dehydrogenases .
Signal Peptide Peptidase
=== Beta-barrels composed of a single polypeptide chain ===
Beta barrels from eight beta-strands and with "shear number" of ten (n=8, S=10). They include:
OmpA-like transmembrane domain (OmpA)
Virulence-related outer membrane protein family (OmpX)
Outer membrane protein W family (OmpW)
Antimicrobial peptide resistance and lipid A acylation protein family (PagP)
Lipid A deacylase PagL
Opacity family porins (NspA)
Autotransporter domain (n=12,S=14)
FadL outer membrane protein transport family, including Fatty acid transporter FadL (n=14,S=14)
General bacterial porin family, known as trimeric porins (n=16,S=20)
Maltoporin, or sugar porins (n=18,S=22)
Nucleoside-specific porin (n=12,S=16)
Outer membrane phospholipase A1(n=12,S=16)
TonB-dependent receptors and their plug domain. They are ligand-gated outer membrane channels (n=22,S=24), including cobalamin transporter BtuB, Fe(III)-pyochelin receptor FptA, receptor FepA, ferric hydroxamate uptake receptor FhuA, transporter FecA, and pyoverdine receptor FpvA
Outer membrane protein OpcA family (n=10,S=12) that includes outer membrane protease OmpT and adhesin/invasin OpcA protein
Outer membrane protein G porin family (n=14,S=16)
Note: n and S are, respectively, the number of beta-strands and the "shear number" of the beta-barrel
=== Beta-barrels composed of several polypeptide chains ===
Trimeric autotransporter (n=12,S=12)
Outer membrane efflux proteins, also known as trimeric outer membrane factors (n=12,S=18) including TolC and multidrug resistance proteins
MspA porin (octamer, n=S=16) and α-hemolysin (heptamer n=S=14) . These proteins are secreted.
== See also ==
Membrane topology
Transmembrane domain
Transmembrane receptors
== References == | Wikipedia/Transmembrane_proteins |
The circuit topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their positions on a circuit diagram; similarly to the mathematical concept of topology, it is only concerned with what connections exist between the components. Numerous physical layouts and circuit diagrams may all amount to the same topology.
Strictly speaking, replacing a component with one of an entirely different type is still the same topology. In some contexts, however, these can loosely be described as different topologies. For instance, interchanging inductors and capacitors in a low-pass filter results in a high-pass filter. These might be described as high-pass and low-pass topologies even though the network topology is identical. A more correct term for these classes of object (that is, a network where the type of component is specified but not the absolute value) is prototype network.
Electronic network topology is related to mathematical topology. In particular, for networks which contain only two-terminal devices, circuit topology can be viewed as an application of graph theory. In a network analysis of such a circuit from a topological point of view, the network nodes are the vertices of graph theory, and the network branches are the edges of graph theory.
Standard graph theory can be extended to deal with active components and multi-terminal devices such as integrated circuits. Graphs can also be used in the analysis of infinite networks.
== Circuit diagrams ==
The circuit diagrams in this article follow the usual conventions in electronics; lines represent conductors, filled small circles represent junctions of conductors, and open small circles represent terminals for connection to the outside world. In most cases, impedances are represented by rectangles. A practical circuit diagram would use the specific symbols for resistors, inductors, capacitors etc., but topology is not concerned with the type of component in the network, so the symbol for a general impedance has been used instead.
The Graph theory section of this article gives an alternative method of representing networks.
== Topology names ==
Many topology names relate to their appearance when drawn diagrammatically. Most circuits can be drawn in a variety of ways and consequently have a variety of names. For instance, the three circuits shown in Figure 1.1 all look different but have identical topologies.
This example also demonstrates a common convention of naming topologies after a letter of the alphabet to which they have a resemblance. Greek alphabet letters can also be used in this way, for example Π (pi) topology and Δ (delta) topology.
== Series and parallel topologies ==
A network with two components or branches has only two possible topologies: series and parallel.
Even for these simplest of topologies, the circuit can be presented in varying ways.
A network with three branches has four possible topologies.
Note that the parallel-series topology is another representation of the Delta topology discussed later.
Series and parallel topologies can continue to be constructed with greater and greater numbers of branches ad infinitum. The number of unique topologies that can be obtained from
n
∈
N
{\displaystyle n\in \mathbb {N} }
series or parallel branches is 1, 2, 4, 10, 24, 66, 180, 522, 1532, 4624,
…
{\displaystyle \dots }
(sequence A000084 in the OEIS).
== Y and Δ topologies ==
Y and Δ are important topologies in linear network analysis due to these being the simplest possible three-terminal networks. A Y-Δ transform is available for linear circuits. This transform is important because some networks cannot be analysed in terms of series and parallel combinations. These networks arise often in 3-phase power circuits as they are the two most common topologies for 3-phase motor or transformer windings. An example of this is the network of figure 1.6, consisting of a Y network connected in parallel with a Δ network. Say it is desired to calculate the impedance between two nodes of the network. In many networks this can be done by successive applications of the rules for combination of series or parallel impedances. This is not, however, possible in this case where the Y-Δ transform is needed in addition to the series and parallel rules.
The Y topology is also called star topology. However, star topology may also refer to the more general case of many branches connected to the same node rather than just three.
== Simple filter topologies ==
The topologies shown in figure 1.7 are commonly used for filter and attenuator designs. The L-section is identical topology to the potential divider topology. The T-section is identical topology to the Y topology. The Π-section is identical topology to the Δ topology.
All these topologies can be viewed as a short section of a ladder topology. Longer sections would normally be described as ladder topology. These kinds of circuits are commonly analysed and characterised in terms of a two-port network.
== Bridge topology ==
Bridge topology is an important topology with many uses in both linear and non-linear applications, including, amongst many others, the bridge rectifier, the Wheatstone bridge and the lattice phase equaliser. Bridge topology is rendered in circuit diagrams in several ways. The first rendering in figure 1.8 is the traditional depiction of a bridge circuit. The second rendering clearly shows the equivalence between the bridge topology and a topology derived by series and parallel combinations. The third rendering is more commonly known as lattice topology. It is not so obvious that this is topologically equivalent. It can be seen that this is indeed so by visualising the top left node moved to the right of the top right node.
It is normal to call a network bridge topology only if it is being used as a two-port network with the input and output ports each consisting of a pair of diagonally opposite nodes. The box topology in figure 1.7 can be seen to be identical to bridge topology but in the case of the filter the input and output ports are each a pair of adjacent nodes. Sometimes the loading (or null indication) component on the output port of the bridge will be included in the bridge topology as shown in figure 1.9.
== Bridged T and twin-T topologies ==
Bridged T topology is derived from bridge topology in a way explained in the Zobel network article. Many derivative topologies are also discussed in the same article.
There is also a twin-T topology, which has practical applications where it is desirable to have the input and output share a common (ground) terminal. This may be, for instance, because the input and output connections are made with co-axial topology. Connecting an input and output terminal is not allowable with normal bridge topology, so Twin-T is used where a bridge would otherwise be used for balance or null measurement applications. The topology is also used in the twin-T oscillator as a sine-wave generator. The lower part of figure 1.11 shows twin-T topology redrawn to emphasise the connection with bridge topology.
== Infinite topologies ==
Ladder topology can be extended without limit and is much used in filter designs. There are many variations on ladder topology, some of which are discussed in the Electronic filter topology and Composite image filter articles.
The balanced form of ladder topology can be viewed as being the graph of the side of a prism of arbitrary order. The side of an antiprism forms a topology which, in this sense, is an anti-ladder. Anti-ladder topology finds an application in voltage multiplier circuits, in particular the Cockcroft-Walton generator. There is also a full-wave version of the Cockcroft-Walton generator which uses a double anti-ladder topology.
Infinite topologies can also be formed by cascading multiple sections of some other simple topology, such as lattice or bridge-T sections. Such infinite chains of lattice sections occur in the theoretical analysis and artificial simulation of transmission lines, but are rarely used as a practical circuit implementation.
== Components with more than two terminals ==
Circuits containing components with three or more terminals greatly increase the number of possible topologies. Conversely, the number of different circuits represented by a topology diminishes and in many cases the circuit is easily recognisable from the topology even when specific components are not identified.
With more complex circuits the description may proceed by specification of a transfer function between the ports of the network rather than the topology of the components.
== Graph theory ==
Graph theory is the branch of mathematics dealing with graphs. In network analysis, graphs are used extensively to represent a network being analysed. The graph of a network captures only certain aspects of a network: those aspects related to its connectivity, or, in other words, its topology. This can be a useful representation and generalisation of a network because many network equations are invariant across networks with the same topology. This includes equations derived from Kirchhoff's laws and Tellegen's theorem.
=== History ===
Graph theory has been used in the network analysis of linear, passive networks almost from the moment that Kirchhoff's laws were formulated. Gustav Kirchhoff himself, in 1847, used graphs as an abstract representation of a network in his loop analysis of resistive circuits. This approach was later generalised to RLC circuits, replacing resistances with impedances. In 1873 James Clerk Maxwell provided the dual of this analysis with node analysis. Maxwell is also responsible for the topological theorem that the determinant of the node-admittance matrix is equal to the sum of all the tree admittance products. In 1900 Henri Poincaré introduced the idea of representing a graph by its incidence matrix, hence founding the field of algebraic topology. In 1916 Oswald Veblen applied the algebraic topology of Poincaré to Kirchhoff's analysis. Veblen is also responsible for the introduction of the spanning tree to aid choosing a compatible set of network variables.
Comprehensive cataloguing of network graphs as they apply to electrical circuits began with Percy MacMahon in 1891 (with an engineer-friendly article in The Electrician in 1892) who limited his survey to series and parallel combinations. MacMahon called these graphs yoke-chains. Ronald M. Foster in 1932 categorised graphs by their nullity or rank and provided charts of all those with a small number of nodes. This work grew out of an earlier survey by Foster while collaborating with George Campbell in 1920 on 4-port telephone repeaters and produced 83,539 distinct graphs.
For a long time topology in electrical circuit theory remained concerned only with linear passive networks. The more recent developments of semiconductor devices and circuits have required new tools in topology to deal with them. Enormous increases in circuit complexity have led to the use of combinatorics in graph theory to improve the efficiency of computer calculation.
=== Graphs and circuit diagrams ===
Networks are commonly classified by the kind of electrical elements making them up. In a circuit diagram these element-kinds are specifically drawn, each with its own unique symbol. Resistive networks are one-element-kind networks, consisting only of R elements. Likewise capacitive or inductive networks are one-element-kind. The RC, RL and LC circuits are simple two-element-kind networks. The RLC circuit is the simplest three-element-kind network. The LC ladder network commonly used for low-pass filters can have many elements but is another example of a two-element-kind network.
Conversely, topology is concerned only with the geometric relationship between the elements of a network, not with the kind of elements themselves. The heart of a topological representation of a network is the graph of the network. Elements are represented as the edges of the graph. An edge is drawn as a line, terminating on dots or small circles from which other edges (elements) may emanate. In circuit analysis, the edges of the graph are called branches. The dots are called the vertices of the graph and represent the nodes of the network. Node and vertex are terms that can be used interchangeably when discussing graphs of networks. Figure 2.2 shows a graph representation of the circuit in figure 2.1.
Graphs used in network analysis are usually, in addition, both directed graphs, to capture the direction of current flow and voltage, and labelled graphs, to capture the uniqueness of the branches and nodes. For instance, a graph consisting of a square of branches would still be the same topological graph if two branches were interchanged unless the branches were uniquely labelled. In directed graphs, the two nodes that a branch connects to are designated the source and target nodes. Typically, these will be indicated by an arrow drawn on the branch.
=== Incidence ===
Incidence is one of the basic properties of a graph. An edge that is connected to a vertex is said to be incident on that vertex. The incidence of a graph can be captured in matrix format with a matrix called an incidence matrix. In fact, the incidence matrix is an alternative mathematical representation of the graph which dispenses with the need for any kind of drawing. Matrix rows correspond to nodes and matrix columns correspond to branches. The elements of the matrix are either zero, for no incidence, or one, for incidence between the node and branch. Direction in directed graphs is indicated by the sign of the element.
=== Equivalence ===
Graphs are equivalent if one can be transformed into the other by deformation. Deformation can include the operations of translation, rotation and reflection; bending and stretching the branches; and crossing or knotting the branches. Two graphs which are equivalent through deformation are said to be congruent.
In the field of electrical networks, two additional transforms are considered to result in equivalent graphs which do not produce congruent graphs. The first of these is the interchange of series-connected branches. This is the dual of interchange of parallel-connected branches which can be achieved by deformation without the need for a special rule. The second is concerned with graphs divided into two or more separate parts, that is, a graph with two sets of nodes which have no branches incident to a node in each set. Two such separate parts are considered an equivalent graph to one where the parts are joined by combining a node from each into a single node. Likewise, a graph that can be split into two separate parts by splitting a node in two is also considered equivalent.
=== Trees and links ===
A tree is a graph in which all the nodes are connected, either directly or indirectly, by branches, but without forming any closed loops. Since there are no closed loops, there are no currents in a tree. In network analysis, we are interested in spanning trees, that is, trees that connect every node in the graph of the network. In this article, spanning tree is meant by an unqualified tree unless otherwise stated. A given network graph can contain a number of different trees. The branches removed from a graph in order to form a tree are called links; the branches remaining in the tree are called twigs. For a graph with n nodes, the number of branches in each tree, t, must be:
t
=
n
−
1
{\displaystyle t=n-1\ }
An important relationship for circuit analysis is:
b
=
ℓ
+
t
{\displaystyle b=\ell +t\ }
where b is the number of branches in the graph and ℓ is the number of links removed to form the tree.
=== Tie sets and cut sets ===
The goal of circuit analysis is to determine all the branch currents and voltages in the network. These network variables are not all independent. The branch voltages are related to the branch currents by the transfer function of the elements of which they are composed. A complete solution of the network can therefore be either in terms of branch currents or branch voltages only. Nor are all the branch currents independent from each other. The minimum number of branch currents required for a complete solution is l. This is a consequence of the fact that a tree has l links removed and there can be no currents in a tree. Since the remaining branches of the tree have zero current they cannot be independent of the link currents. The branch currents chosen as a set of independent variables must be a set associated with the links of a tree: one cannot choose any l branches arbitrarily.
In terms of branch voltages, a complete solution of the network can be obtained with t branch voltages. This is a consequence the fact that short-circuiting all the branches of a tree results in the voltage being zero everywhere. The link voltages cannot, therefore, be independent of the tree branch voltages.
A common analysis approach is to solve for loop currents rather than branch currents. The branch currents are then found in terms of the loop currents. Again, the set of loop currents cannot be chosen arbitrarily. To guarantee a set of independent variables the loop currents must be those associated with a certain set of loops. This set of loops consists of those loops formed by replacing a single link of a given tree of the graph of the circuit to be analysed. Since replacing a single link in a tree forms exactly one unique loop, the number of loop currents so defined is equal to l. The term loop in this context is not the same as the usual meaning of loop in graph theory. The set of branches forming a given loop is called a tie set. The set of network equations are formed by equating the loop currents to the algebraic sum of the tie set branch currents.
It is possible to choose a set of independent loop currents without reference to the trees and tie sets. A sufficient, but not necessary, condition for choosing a set of independent loops is to ensure that each chosen loop includes at least one branch that was not previously included by loops already chosen. A particularly straightforward choice is that used in mesh analysis, in which the loops are all chosen to be meshes. Mesh analysis can only be applied if it is possible to map the graph onto a plane or a sphere without any of the branches crossing over. Such graphs are called planar graphs. Ability to map onto a plane or a sphere are equivalent conditions. Any finite graph mapped onto a plane can be shrunk until it will map onto a small region of a sphere. Conversely, a mesh of any graph mapped onto a sphere can be stretched until the space inside it occupies nearly all of the sphere. The entire graph then occupies only a small region of the sphere. This is the same as the first case, hence the graph will also map onto a plane.
There is an approach to choosing network variables with voltages which is analogous and dual to the loop current method. Here the voltage associated with pairs of nodes are the primary variables and the branch voltages are found in terms of them. In this method also, a particular tree of the graph must be chosen in order to ensure that all the variables are independent. The dual of the tie set is the cut set. A tie set is formed by allowing all but one of the graph links to be open circuit. A cut set is formed by allowing all but one of the tree branches to be short circuit. The cut set consists of the tree branch which was not short-circuited and any of the links which are not short-circuited by the other tree branches. A cut set of a graph produces two disjoint subgraphs, that is, it cuts the graph into two parts, and is the minimum set of branches needed to do so. The set of network equations are formed by equating the node pair voltages to the algebraic sum of the cut set branch voltages. The dual of the special case of mesh analysis is nodal analysis.
=== Nullity and rank ===
The nullity, N, of a graph with s separate parts and b branches is defined by:
N
=
b
−
n
+
s
{\displaystyle N=b-n+s\ }
The nullity of a graph represents the number of degrees of freedom of its set of network equations. For a planar graph, the nullity is equal to the number of meshes in the graph.
The rank, R of a graph is defined by:
R
=
n
−
s
{\displaystyle R=n-s\ }
Rank plays the same role in nodal analysis as nullity plays in mesh analysis. That is, it gives the number of node voltage equations required. Rank and nullity are dual concepts and are related by:
R
+
N
=
b
{\displaystyle R+N=b\ }
=== Solving the network variables ===
Once a set of geometrically independent variables have been chosen the state of the network is expressed in terms of these. The result is a set of independent linear equations which need to be solved simultaneously in order to find the values of the network variables. This set of equations can be expressed in a matrix format which leads to a characteristic parameter matrix for the network. Parameter matrices take the form of an impedance matrix if the equations have been formed on a loop-analysis basis, or as an admittance matrix if the equations have been formed on a node-analysis basis.
These equations can be solved in a number of well-known ways. One method is the systematic elimination of variables. Another method involves the use of determinants. This is known as Cramer's rule and provides a direct expression for the unknown variable in terms of determinants. This is useful in that it provides a compact expression for the solution. However, for anything more than the most trivial networks, a greater calculation effort is required for this method when working manually.
=== Duality ===
Two graphs are dual when the relationship between branches and node pairs in one is the same as the relationship between branches and loops in the other. The dual of a graph can be found entirely by a graphical method.
The dual of a graph is another graph. For a given tree in a graph, the complementary set of branches (i.e., the branches not in the tree) form a tree in the dual graph. The set of current loop equations associated with the tie sets of the original graph and tree is identical to the set of voltage node-pair equations associated with the cut sets of the dual graph.
The following table lists dual concepts in topology related to circuit theory.
The dual of a tree is sometimes called a maze. It consists of spaces connected by links in the same way that the tree consists of nodes connected by tree branches.
Duals cannot be formed for every graph. Duality requires that every tie set has a dual cut set in the dual graph. This condition is met if and only if the graph is mappable on to a sphere with no branches crossing. To see this, note that a tie set is required to "tie off" a graph into two portions and its dual, the cut set, is required to cut a graph into two portions. The graph of a finite network which will not map on to a sphere will require an n-fold torus. A tie set that passes through a hole in a torus will fail to tie the graph into two parts. Consequently, the dual graph will not be cut into two parts and will not contain the required cut set. Consequently, only planar graphs have duals.
Duals also cannot be formed for networks containing mutual inductances since there is no corresponding capacitive element. Equivalent circuits can be developed which do have duals, but the dual cannot be formed of a mutual inductance directly.
=== Node and mesh elimination ===
Operations on a set of network equations have a topological meaning which can aid visualisation of what is happening. Elimination of a node voltage from a set of network equations corresponds topologically to the elimination of that node from the graph. For a node connected to three other nodes, this corresponds to the well known Y-Δ transform. The transform can be extended to greater numbers of connected nodes and is then known as the star-mesh transform.
The inverse of this transform is the Δ-Y transform which analytically corresponds to the elimination of a mesh current and topologically corresponds to the elimination of a mesh. However, elimination of a mesh current whose mesh has branches in common with an arbitrary number of other meshes will not, in general, result in a realisable graph. This is because the graph of the transform of the general star is a graph which will not map on to a sphere (it contains star polygons and hence multiple crossovers). The dual of such a graph cannot exist, but is the graph required to represent a generalised mesh elimination.
=== Mutual coupling ===
In conventional graph representation of circuits, there is no means of explicitly representing mutual inductive couplings, such as occurs in a transformer, and such components may result in a disconnected graph with more than one separate part. For convenience of analysis, a graph with multiple parts can be combined into a single graph by unifying one node in each part into a single node. This makes no difference to the theoretical behaviour of the circuit, so analysis carried out on it is still valid. It would, however, make a practical difference if a circuit were to be implemented this way in that it would destroy the isolation between the parts. An example would be a transformer earthed on both the primary and secondary side. The transformer still functions as a transformer with the same voltage ratio but can now no longer be used as an isolation transformer.
More recent techniques in graph theory are able to deal with active components, which are also problematic in conventional theory. These new techniques are also able to deal with mutual couplings.
=== Active components ===
There are two basic approaches available for dealing with mutual couplings and active components. In the first of these, Samuel Jefferson Mason in 1953 introduced signal-flow graphs. Signal-flow graphs are weighted, directed graphs. He used these to analyse circuits containing mutual couplings and active networks. The weight of a directed edge in these graphs represents a gain, such as possessed by an amplifier. In general, signal-flow graphs, unlike the regular directed graphs described above, do not correspond to the topology of the physical arrangement of components.
The second approach is to extend the classical method so that it includes mutual couplings and active components. Several methods have been proposed for achieving this. In one of these, two graphs are constructed, one representing the currents in the circuit and the other representing the voltages. Passive components will have identical branches in both trees but active components may not. The method relies on identifying spanning trees that are common to both graphs. An alternative method of extending the classical approach which requires only one graph was proposed by Chen in 1965. Chen's method is based on a rooted tree.
==== Hypergraphs ====
Another way of extending classical graph theory for active components is through the use of hypergraphs. Some electronic components are not represented naturally using graphs. The transistor has three connection points, but a normal graph branch may only connect to two nodes. Modern integrated circuits have many more connections than this. This problem can be overcome by using hypergraphs instead of regular graphs.
In a conventional representation components are represented by edges, each of which connects to two nodes. In a hypergraph, components are represented by hyperedges which can connect to an arbitrary number of nodes. Hyperedges have tentacles which connect the hyperedge to the nodes. The graphical representation of a hyperedge may be a box (compared to the edge which is a line) and the representations of its tentacles are lines from the box to the connected nodes. In a directed hypergraph, the tentacles carry labels which are determined by the hyperedge's label. A conventional directed graph can be thought of as a hypergraph with hyperedges each of which has two tentacles. These two tentacles are labelled source and target and usually indicated by an arrow. In a general hypergraph with more tentacles, more complex labelling will be required.
Hypergraphs can be characterised by their incidence matrices. A regular graph containing only two-terminal components will have exactly two non-zero entries in each row. Any incidence matrix with more than two non-zero entries in any row is a representation of a hypergraph. The number of non-zero entries in a row is the rank of the corresponding branch, and the highest branch rank is the rank of the incidence matrix.
=== Non-homogeneous variables ===
Classical network analysis develops a set of network equations whose network variables are homogeneous in either current (loop analysis) or voltage (node analysis). The set of network variables so found is not necessarily the minimum necessary to form a set of independent equations. There may be a difference between the number of variables in a loop analysis to a node analysis. In some cases the minimum number possible may be less than either of these if the requirement for homogeneity is relaxed and a mix of current and voltage variables allowed. A result from Kishi and Katajini in 1967 is that the absolute minimum number of variables required to describe the behaviour of the network is given by the maximum distance between any two spanning forests of the network graph.
=== Network synthesis ===
Graph theory can be applied to network synthesis. Classical network synthesis realises the required network in one of a number of canonical forms. Examples of canonical forms are the realisation of a driving-point impedance by Cauer's canonical ladder network or Foster's canonical form or Brune's realisation of an immittance from his positive-real functions. Topological methods, on the other hand, do not start from a given canonical form. Rather, the form is a result of the mathematical representation. Some canonical forms require mutual inductances for their realisation. A major aim of topological methods of network synthesis has been to eliminate the need for these mutual inductances. One theorem to come out of topology is that a realisation of a driving-point impedance without mutual couplings is minimal if and only if there are no all-inductor or all-capacitor loops.
Graph theory is at its most powerful in network synthesis when the elements of the network can be represented by real numbers (one-element-kind networks such as resistive networks) or binary states (such as switching networks).
=== Infinite networks ===
Perhaps the earliest network with an infinite graph to be studied was the ladder network used to represent transmission lines developed, in its final form, by Oliver Heaviside in 1881. Certainly all early studies of infinite networks were limited to periodic structures such as ladders or grids with the same elements repeated over and over. It was not until the late 20th century that tools for analysing infinite networks with an arbitrary topology became available.
Infinite networks are largely of only theoretical interest and are the plaything of mathematicians. Infinite networks that are not constrained by real-world restrictions can have some very unphysical properties. For instance Kirchhoff's laws can fail in some cases and infinite resistor ladders can be defined which have a driving-point impedance which depends on the termination at infinity. Another unphysical property of theoretical infinite networks is that, in general, they will dissipate infinite power unless constraints are placed on them in addition to the usual network laws such as Ohm's and Kirchhoff's laws. There are, however, some real-world applications. The transmission line example is one of a class of practical problems that can be modelled by infinitesimal elements (the distributed-element model). Other examples are launching waves into a continuous medium, fringing field problems, and measurement of resistance between points of a substrate or down a borehole.
Transfinite networks extend the idea of infinite networks even further. A node at an extremity of an infinite network can have another branch connected to it leading to another network. This new network can itself be infinite. Thus, topologies can be constructed which have pairs of nodes with no finite path between them. Such networks of infinite networks are called transfinite networks.
== Notes ==
== See also ==
Symbolic circuit analysis
Network topology
Topological quantum computer
== References ==
== Bibliography ==
Brittain, James E., The introduction of the loading coil: George A. Campbell and Michael I. Pupin", Technology and Culture, vol. 11, no. 1, pp. 36–57, The Johns Hopkins University Press, January 1970 doi:10.2307/3102809.
Campbell, G. A., "Physical theory of the electric wave-filter", Bell System Technical Journal, November 1922, vol. 1, no. 2, pp. 1–32.
Cederbaum, I., "Some applications of graph theory to network analysis and synthesis", IEEE Transactions on Circuits and Systems, vol.31, iss.1, pp. 64–68, January 1984.
Farago, P. S., An Introduction to Linear Network Analysis, The English Universities Press Ltd, 1961.
Foster, Ronald M., "Geometrical circuits of electrical networks", Transactions of the American Institute of Electrical Engineers, vol.51, iss.2, pp. 309–317, June 1932.
Foster, Ronald M.; Campbell, George A., "Maximum output networks for telephone substation and repeater circuits", Transactions of the American Institute of Electrical Engineers, vol.39, iss.1, pp. 230–290, January 1920.
Guillemin, Ernst A., Introductory Circuit Theory, New York: John Wiley & Sons, 1953 OCLC 535111
Kind, Dieter; Feser, Kurt, High-voltage Test Techniques, translator Y. Narayana Rao, Newnes, 2001 ISBN 0-7506-5183-0.
Kishi, Genya; Kajitani, Yoji, "Maximally distant trees and principal partition of a linear graph", IEEE Transactions on Circuit Theory, vol.16, iss.3, pp. 323–330, August 1969.
MacMahon, Percy A., "Yoke-chains and multipartite compositions in connexion with the analytical forms called “Trees”", Proceedings of the London Mathematical Society, vol.22 (1891), pp.330–346 doi:10.1112/plms/s1-22.1.330.
MacMahon, Percy A., "Combinations of resistances", The Electrician, vol.28, pp. 601–602, 8 April 1892.Reprinted in Discrete Applied Mathematics, vol.54, iss.Iss.2–3, pp. 225–228, 17 October 1994 doi:10.1016/0166-218X(94)90024-8.
Minas, M., "Creating semantic representations of diagrams", Applications of Graph Transformations with Industrial Relevance: international workshop, AGTIVE'99, Kerkrade, The Netherlands, September 1–3, 1999: proceedings, pp. 209–224, Springer, 2000 ISBN 3-540-67658-9.
Redifon Radio Diary, 1970, William Collins Sons & Co, 1969.
Skiena, Steven S., The Algorithm Design Manual, Springer, 2008, ISBN 1-84800-069-3.
Suresh, Kumar K. S., "Introduction to network topology" chapter 11 in Electric Circuits And Networks, Pearson Education India, 2010 ISBN 81-317-5511-8.
Tooley, Mike, BTEC First Engineering: Mandatory and Selected Optional Units for BTEC Firsts in Engineering, Routledge, 2010 ISBN 1-85617-685-1.
Wildes, Karl L.; Lindgren, Nilo A., "Network analysis and synthesis: Ernst A. Guillemin", A Century of Electrical Engineering and Computer Science at MIT, 1882–1982, pp. 154–159, MIT Press, 1985 ISBN 0-262-23119-0.
Zemanian, Armen H., Infinite Electrical Networks, Cambridge University Press, 1991 ISBN 0-521-40153-4. | Wikipedia/Topology_(electronics) |
Topology is an indie classical quintet from Australia, formed in 1997.. They perform throughout Australia and abroad and have to date released 14 albums, including one with rock/electronica band Full Fathom Five and one with contemporary ensemble Loops. They were formerly the resident ensembles at the University of Western Sydney and Brisbane Powerhouse. The group works with composers including Tim Brady, Andrew Poppy, Michael Nyman, Jeremy Peyton Jones, Terry Riley, Steve Reich, Philip Glass, Carl Stone, Pand aul Dresher, as well as with many Australian composers.
In 2009, Topology won the "Outstanding Contribution by an Organization" award at the Australasian Performing Right Association (APRA) Classical Music Awards for their work on the 2008 Brisbane Powerhouse Series.
== Members ==
Bernard Hoey (viola)
Christa Powell (violin)
John Babbage (saxophone)
Kylie Davidson (piano)
Therese Milanovic (piano)
Robert Davidson (bass)
== Discography ==
=== Albums ===
== Awards and nominations ==
=== APRA Awards ===
APRA Awards of 2009: Outstanding Contribution by an Organisation win for the 2008 Brisbane Powerhouse Series by Topology.
=== ARIA Music Awards ===
The ARIA Music Awards are presented annually from 1987 by the Australian Recording Industry Association (ARIA).
== References ==
== External links ==
Official website
Official Facebook Page
Official YouTube Page
Official Twitter Page | Wikipedia/Topology_(musical_ensemble) |
Network topology is the arrangement of the elements (links, nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and control radio networks, industrial fieldbusses and computer networks.
Network topology is the topological structure of a network and may be depicted physically or logically. It is an application of graph theory wherein communicating devices are modeled as nodes and the connections between the devices are modeled as links or lines between the nodes. Physical topology is the placement of the various components of a network (e.g., device location and cable installation), while logical topology illustrates how data flows within a network. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two different networks, yet their logical topologies may be identical. A network's physical topology is a particular concern of the physical layer of the OSI model.
Examples of network topologies are found in local area networks (LAN), a common computer network installation. Any given node in the LAN has one or more physical links to other devices in the network; graphically mapping these links results in a geometric shape that can be used to describe the physical topology of the network. A wide variety of physical topologies have been used in LANs, including ring, bus, mesh and star. Conversely, mapping the data flow between the components determines the logical topology of the network. In comparison, Controller Area Networks, common in vehicles, are primarily distributed control system networks of one or more controllers interconnected with sensors and actuators over, invariably, a physical bus topology.
== Topologies ==
Two basic categories of network topologies exist, physical topologies and logical topologies.
The transmission medium layout used to link devices is the physical topology of the network. For conductive or fiber optical mediums, this refers to the layout of cabling, the locations of nodes, and the links between the nodes and the cabling. The physical topology of a network is determined by the capabilities of the network access devices and media, the level of control or fault tolerance desired, and the cost associated with cabling or telecommunication circuits.
In contrast, logical topology is the way that the signals act on the network media, or the way that the data passes through the network from one device to the next without regard to the physical interconnection of the devices. A network's logical topology is not necessarily the same as its physical topology. For example, the original twisted pair Ethernet using repeater hubs was a logical bus topology carried on a physical star topology. Token Ring is a logical ring topology, but is wired as a physical star from the media access unit. Physically, Avionics Full-Duplex Switched Ethernet (AFDX) can be a cascaded star topology of multiple dual redundant Ethernet switches; however, the AFDX virtual links are modeled as time-switched single-transmitter bus connections, thus following the safety model of a single-transmitter bus topology previously used in aircraft. Logical topologies are often closely associated with media access control methods and protocols. Some networks are able to dynamically change their logical topology through configuration changes to their routers and switches.
== Links ==
The transmission media (often referred to in the literature as the physical media) used to link devices to form a computer network include electrical cables (Ethernet, HomePNA, power line communication, G.hn), optical fiber (fiber-optic communication), and radio waves (wireless networking). In the OSI model, these are defined at layers 1 and 2 — the physical layer and the data link layer.
A widely adopted family of transmission media used in local area network (LAN) technology is collectively known as Ethernet. The media and protocol standards that enable communication between networked devices over Ethernet are defined by IEEE 802.3. Ethernet transmits data over both copper and fiber cables. Wireless LAN standards (e.g. those defined by IEEE 802.11) use radio waves, or others use infrared signals as a transmission medium. Power line communication uses a building's power cabling to transmit data.
=== Wired technologies ===
The orders of the following wired technologies are, roughly, from slowest to fastest transmission speed.
Coaxial cable is widely used for cable television systems, office buildings, and other work-sites for local area networks. The cables consist of copper or aluminum wire surrounded by an insulating layer (typically a flexible material with a high dielectric constant), which itself is surrounded by a conductive layer. The insulation between the conductors helps maintain the characteristic impedance of the cable which can help improve its performance. Transmission speed ranges from 200 million bits per second to more than 500 million bits per second.
ITU-T G.hn technology uses existing home wiring (coaxial cable, phone lines and power lines) to create a high-speed (up to 1 Gigabit/s) local area network.
Signal traces on printed circuit boards are common for board-level serial communication, particularly between certain types integrated circuits, a common example being SPI.
Ribbon cable (untwisted and possibly unshielded) has been a cost-effective media for serial protocols, especially within metallic enclosures or rolled within copper braid or foil, over short distances, or at lower data rates. Several serial network protocols can be deployed without shielded or twisted pair cabling, that is, with flat or ribbon cable, or a hybrid flat and twisted ribbon cable, should EMC, length, and bandwidth constraints permit: RS-232, RS-422, RS-485, CAN, GPIB, SCSI, etc.
Twisted pair wire is the most widely used medium for all telecommunication. Twisted-pair cabling consist of copper wires that are twisted into pairs. Ordinary telephone wires consist of two insulated copper wires twisted into pairs. Computer network cabling (wired Ethernet as defined by IEEE 802.3) consists of 4 pairs of copper cabling that can be utilized for both voice and data transmission. The use of two wires twisted together helps to reduce crosstalk and electromagnetic induction. The transmission speed ranges from 2 million bits per second to 10 billion bits per second. Twisted pair cabling comes in two forms: unshielded twisted pair (UTP) and shielded twisted pair (STP). Each form comes in several category ratings, designed for use in various scenarios.
An optical fiber is a glass fiber. It carries pulses of light that represent data. Some advantages of optical fibers over metal wires are very low transmission loss and immunity from electrical interference. Optical fibers can simultaneously carry multiple wavelengths of light, which greatly increases the rate that data can be sent, and helps enable data rates of up to trillions of bits per second. Optic fibers can be used for long runs of cable carrying very high data rates, and are used for undersea communications cables to interconnect continents.
Price is a main factor distinguishing wired- and wireless technology options in a business. Wireless options command a price premium that can make purchasing wired computers, printers and other devices a financial benefit. Before making the decision to purchase hard-wired technology products, a review of the restrictions and limitations of the selections is necessary. Business and employee needs may override any cost considerations.
=== Wireless technologies ===
Terrestrial microwave – Terrestrial microwave communication uses Earth-based transmitters and receivers resembling satellite dishes. Terrestrial microwaves are in the low gigahertz range, which limits all communications to line-of-sight. Relay stations are spaced approximately 50 km (30 mi) apart.
Communications satellites – Satellites communicate via microwave radio waves, which are not deflected by the Earth's atmosphere. The satellites are stationed in space, typically in geostationary orbit 35,786 km (22,236 mi) above the equator. These Earth-orbiting systems are capable of receiving and relaying voice, data, and TV signals.
Cellular and PCS systems use several radio communications technologies. The systems divide the region covered into multiple geographic areas. Each area has a low-power transmitter or radio relay antenna device to relay calls from one area to the next area.
Radio and spread spectrum technologies – Wireless local area networks use a high-frequency radio technology similar to digital cellular and a low-frequency radio technology. Wireless LANs use spread spectrum technology to enable communication between multiple devices in a limited area. IEEE 802.11 defines a common flavor of open-standards wireless radio-wave technology known as Wi-Fi.
Free-space optical communication uses visible or invisible light for communications. In most cases, line-of-sight propagation is used, which limits the physical positioning of communicating devices.
=== Exotic technologies ===
There have been various attempts at transporting data over exotic media:
IP over Avian Carriers was a humorous April fool's Request for Comments, issued as RFC 1149. It was implemented in real life in 2001.
Extending the Internet to interplanetary dimensions via radio waves, the Interplanetary Internet.
Both cases have a large round-trip delay time, which gives slow two-way communication, but does not prevent sending large amounts of information.
== Nodes ==
Network nodes are the points of connection of the transmission medium to transmitters and receivers of the electrical, optical, or radio signals carried in the medium. Nodes may be associated with a computer, but certain types may have only a microcontroller at a node or possibly no programmable device at all. In the simplest of serial arrangements, one RS-232 transmitter can be connected by a pair of wires to one receiver, forming two nodes on one link, or a Point-to-Point topology. Some protocols permit a single node to only either transmit or receive (e.g., ARINC 429). Other protocols have nodes that can both transmit and receive into a single channel (e.g., CAN can have many transceivers connected to a single bus). While the conventional system building blocks of a computer network include network interface controllers (NICs), repeaters, hubs, bridges, switches, routers, modems, gateways, and firewalls, most address network concerns beyond the physical network topology and may be represented as single nodes on a particular physical network topology.
=== Network interfaces ===
A network interface controller (NIC) is computer hardware that provides a computer with the ability to access the transmission media, and has the ability to process low-level network information. For example, the NIC may have a connector for accepting a cable, or an aerial for wireless transmission and reception, and the associated circuitry.
The NIC responds to traffic addressed to a network address for either the NIC or the computer as a whole.
In Ethernet networks, each network interface controller has a unique Media Access Control (MAC) address—usually stored in the controller's permanent memory. To avoid address conflicts between network devices, the Institute of Electrical and Electronics Engineers (IEEE) maintains and administers MAC address uniqueness. The size of an Ethernet MAC address is six octets. The three most significant octets are reserved to identify NIC manufacturers. These manufacturers, using only their assigned prefixes, uniquely assign the three least-significant octets of every Ethernet interface they produce.
=== Repeaters and hubs ===
A repeater is an electronic device that receives a network signal, cleans it of unnecessary noise and regenerates it. The signal may be reformed or retransmitted at a higher power level, to the other side of an obstruction possibly using a different transmission medium, so that the signal can cover longer distances without degradation. Commercial repeaters have extended RS-232 segments from 15 meters to over a kilometer. In most twisted pair Ethernet configurations, repeaters are required for cable that runs longer than 100 meters. With fiber optics, repeaters can be tens or even hundreds of kilometers apart.
Repeaters work within the physical layer of the OSI model, that is, there is no end-to-end change in the physical protocol across the repeater, or repeater pair, even if a different physical layer may be used between the ends of the repeater, or repeater pair. Repeaters require a small amount of time to regenerate the signal. This can cause a propagation delay that affects network performance and may affect proper function. As a result, many network architectures limit the number of repeaters that can be used in a row, e.g., the Ethernet 5-4-3 rule.
A repeater with multiple ports is known as hub, an Ethernet hub in Ethernet networks, a USB hub in USB networks.
USB networks use hubs to form tiered-star topologies.
Ethernet hubs and repeaters in LANs have been mostly obsoleted by modern switches.
=== Bridges ===
A network bridge connects and filters traffic between two network segments at the data link layer (layer 2) of the OSI model to form a single network. This breaks the network's collision domain but maintains a unified broadcast domain. Network segmentation breaks down a large, congested network into an aggregation of smaller, more efficient networks.
Bridges come in three basic types:
Local bridges: Directly connect LANs
Remote bridges: Can be used to create a wide area network (WAN) link between LANs. Remote bridges, where the connecting link is slower than the end networks, largely have been replaced with routers.
Wireless bridges: Can be used to join LANs or connect remote devices to LANs.
=== Switches ===
A network switch is a device that forwards and filters OSI layer 2 datagrams (frames) between ports based on the destination MAC address in each frame.
A switch is distinct from a hub in that it only forwards the frames to the physical ports involved in the communication rather than all ports connected. It can be thought of as a multi-port bridge. It learns to associate physical ports to MAC addresses by examining the source addresses of received frames. If an unknown destination is targeted, the switch broadcasts to all ports but the source. Switches normally have numerous ports, facilitating a star topology for devices, and cascading additional switches.
Multi-layer switches are capable of routing based on layer 3 addressing or additional logical levels. The term switch is often used loosely to include devices such as routers and bridges, as well as devices that may distribute traffic based on load or based on application content (e.g., a Web URL identifier).
=== Routers ===
A router is an internetworking device that forwards packets between networks by processing the routing information included in the packet or datagram (Internet protocol information from layer 3). The routing information is often processed in conjunction with the routing table (or forwarding table). A router uses its routing table to determine where to forward packets. A destination in a routing table can include a black hole because data can go into it, however, no further processing is done for said data, i.e. the packets are dropped.
=== Modems ===
Modems (MOdulator-DEModulator) are used to connect network nodes via wire not originally designed for digital network traffic, or for wireless. To do this one or more carrier signals are modulated by the digital signal to produce an analog signal that can be tailored to give the required properties for transmission. Modems are commonly used for telephone lines, using a digital subscriber line technology.
=== Firewalls ===
A firewall is a network device for controlling network security and access rules. Firewalls are typically configured to reject access requests from unrecognized sources while allowing actions from recognized ones. The vital role firewalls play in network security grows in parallel with the constant increase in cyber attacks.
== Classification ==
The study of network topology recognizes eight basic topologies: point-to-point, bus, star, ring or circular, mesh, tree, hybrid, or daisy chain.
=== Point-to-point ===
The simplest topology with a dedicated link between two endpoints. Easiest to understand, of the variations of point-to-point topology, is a point-to-point communication channel that appears, to the user, to be permanently associated with the two endpoints. A child's tin can telephone is one example of a physical dedicated channel.
Using circuit-switching or packet-switching technologies, a point-to-point circuit can be set up dynamically and dropped when no longer needed. Switched point-to-point topologies are the basic model of conventional telephony.
The value of a permanent point-to-point network is unimpeded communications between the two endpoints. The value of an on-demand point-to-point connection is proportional to the number of potential pairs of subscribers and has been expressed as Metcalfe's Law.
=== Daisy chain ===
Daisy chaining is accomplished by connecting each computer in series to the next. If a message is intended for a computer partway down the line, each system bounces it along in sequence until it reaches the destination. A daisy-chained network can take two basic forms: linear and ring.
A linear topology puts a two-way link between one computer and the next. However, this was expensive in the early days of computing, since each computer (except for the ones at each end) required two receivers and two transmitters.
By connecting the computers at each end of the chain, a ring topology can be formed. When a node sends a message, the message is processed by each computer in the ring. An advantage of the ring is that the number of transmitters and receivers can be cut in half. Since a message will eventually loop all of the way around, transmission does not need to go both directions. Alternatively, the ring can be used to improve fault tolerance. If the ring breaks at a particular link then the transmission can be sent via the reverse path thereby ensuring that all nodes are always connected in the case of a single failure.
=== Bus ===
In local area networks using bus topology, each node is connected by interface connectors to a single central cable. This is the 'bus', also referred to as the backbone, or trunk – all data transmission between nodes in the network is transmitted over this common transmission medium and is able to be received by all nodes in the network simultaneously.
A signal containing the address of the intended receiving machine travels from a source machine in both directions to all machines connected to the bus until it finds the intended recipient, which then accepts the data. If the machine address does not match the intended address for the data, the data portion of the signal is ignored. Since the bus topology consists of only one wire it is less expensive to implement than other topologies, but the savings are offset by the higher cost of managing the network. Additionally, since the network is dependent on the single cable, it can be the single point of failure of the network. In this topology data being transferred may be accessed by any node.
==== Linear bus ====
In a linear bus network, all of the nodes of the network are connected to a common transmission medium which has just two endpoints. When the electrical signal reaches the end of the bus, the signal is reflected back down the line, causing unwanted interference. To prevent this, the two endpoints of the bus are normally terminated with a device called a terminator.
==== Distributed bus ====
In a distributed bus network, all of the nodes of the network are connected to a common transmission medium with more than two endpoints, created by adding branches to the main section of the transmission medium – the physical distributed bus topology functions in exactly the same fashion as the physical linear bus topology because all nodes share a common transmission medium.
=== Star ===
In star topology (also called hub-and-spoke), every peripheral node (computer workstation or any other peripheral) is connected to a central node called a hub or switch. The hub is the server and the peripherals are the clients. The network does not necessarily have to resemble a star to be classified as a star network, but all of the peripheral nodes on the network must be connected to one central hub. All traffic that traverses the network passes through the central hub, which acts as a signal repeater.
The star topology is considered the easiest topology to design and implement. One advantage of the star topology is the simplicity of adding additional nodes. The primary disadvantage of the star topology is that the hub represents a single point of failure. Also, since all peripheral communication must flow through the central hub, the aggregate central bandwidth forms a network bottleneck for large clusters.
==== Extended star ====
The extended star network topology extends a physical star topology by one or more repeaters between the central node and the peripheral (or 'spoke') nodes. The repeaters are used to extend the maximum transmission distance of the physical layer, the point-to-point distance between the central node and the peripheral nodes. Repeaters allow greater transmission distance, further than would be possible using just the transmitting power of the central node. The use of repeaters can also overcome limitations from the standard upon which the physical layer is based.
A physical extended star topology in which repeaters are replaced with hubs or switches is a type of hybrid network topology and is referred to as a physical hierarchical star topology, although some texts make no distinction between the two topologies.
A physical hierarchical star topology can also be referred as a tier-star topology. This topology differs from a tree topology in the way star networks are connected together. A tier-star topology uses a central node, while a tree topology uses a central bus and can also be referred as a star-bus network.
==== Distributed star ====
A distributed star is a network topology that is composed of individual networks that are based upon the physical star topology connected in a linear fashion – i.e., 'daisy-chained' – with no central or top level connection point (e.g., two or more 'stacked' hubs, along with their associated star connected nodes or 'spokes').
=== Ring ===
A ring topology is a daisy chain in a closed loop. Data travels around the ring in one direction. When one node sends data to another, the data passes through each intermediate node on the ring until it reaches its destination. The intermediate nodes repeat (retransmit) the data to keep the signal strong. Every node is a peer; there is no hierarchical relationship of clients and servers. If one node is unable to retransmit data, it severs communication between the nodes before and after it in the bus.
Advantages:
When the load on the network increases, its performance is better than bus topology.
There is no need of network server to control the connectivity between workstations.
Disadvantages:
Aggregate network bandwidth is bottlenecked by the weakest link between two nodes.
=== Mesh ===
The value of fully meshed networks is proportional to the exponent of the number of subscribers, assuming that communicating groups of any two endpoints, up to and including all the endpoints, is approximated by Reed's Law.
==== Fully connected network ====
In a fully connected network, all nodes are interconnected. (In graph theory this is called a complete graph.) The simplest fully connected network is a two-node network. A fully connected network doesn't need to use packet switching or broadcasting. However, since the number of connections grows quadratically with the number of nodes:
c
=
n
(
n
−
1
)
2
.
{\displaystyle c={\frac {n(n-1)}{2}}.\,}
This makes it impractical for large networks. This kind of topology does not trip and affect other nodes in the network.
==== Partially connected network ====
In a partially connected network, certain nodes are connected to exactly one other node; but some nodes are connected to two or more other nodes with a point-to-point link. This makes it possible to make use of some of the redundancy of mesh topology that is physically fully connected, without the expense and complexity required for a connection between every node in the network.
=== Hybrid ===
Hybrid topology is also known as hybrid network. Hybrid networks combine two or more topologies in such a way that the resulting network does not exhibit one of the standard topologies (e.g., bus, star, ring, etc.). For example, a tree network (or star-bus network) is a hybrid topology in which star networks are interconnected via bus networks. However, a tree network connected to another tree network is still topologically a tree network, not a distinct network type. A hybrid topology is always produced when two different basic network topologies are connected.
A star-ring network consists of two or more ring networks connected using a multistation access unit (MAU) as a centralized hub.
Snowflake topology is meshed at the core, but tree shaped at the edges.
Two other hybrid network types are hybrid mesh and hierarchical star.
== Centralization ==
The star topology reduces the probability of a network failure by connecting all of the peripheral nodes (computers, etc.) to a central node. When the physical star topology is applied to a logical bus network such as Ethernet, this central node (traditionally a hub) rebroadcasts all transmissions received from any peripheral node to all peripheral nodes on the network, sometimes including the originating node. All peripheral nodes may thus communicate with all others by transmitting to, and receiving from, the central node only. The failure of a transmission line linking any peripheral node to the central node will result in the isolation of that peripheral node from all others, but the remaining peripheral nodes will be unaffected. However, the disadvantage is that the failure of the central node will cause the failure of all of the peripheral nodes.
If the central node is passive, the originating node must be able to tolerate the reception of an echo of its own transmission, delayed by the two-way round trip transmission time (i.e. to and from the central node) plus any delay generated in the central node. An active star network has an active central node that usually has the means to prevent echo-related problems.
A tree topology (a.k.a. hierarchical topology) can be viewed as a collection of star networks arranged in a hierarchy. This tree structure has individual peripheral nodes (e.g. leaves) which are required to transmit to and receive from one other node only and are not required to act as repeaters or regenerators. Unlike the star network, the functionality of the central node may be distributed.
As in the conventional star network, individual nodes may thus still be isolated from the network by a single-point failure of a transmission path to the node. If a link connecting a leaf fails, that leaf is isolated; if a connection to a non-leaf node fails, an entire section of the network becomes isolated from the rest.
To alleviate the amount of network traffic that comes from broadcasting all signals to all nodes, more advanced central nodes were developed that are able to keep track of the identities of the nodes that are connected to the network. These network switches will learn the layout of the network by listening on each port during normal data transmission, examining the data packets and recording the address/identifier of each connected node and which port it is connected to in a lookup table held in memory. This lookup table then allows future transmissions to be forwarded to the intended destination only.
Daisy chain topology is a way of connecting network nodes in a linear or ring structure. It is used to transmit messages from one node to the next until they reach the destination node.
A daisy chain network can have two types: linear and ring. A linear daisy chain network is like an electrical series, where the first and last nodes are not connected. A ring daisy chain network is where the first and last nodes are connected, forming a loop.
== Decentralization ==
In a partially connected mesh topology, there are at least two nodes with two or more paths between them to provide redundant paths in case the link providing one of the paths fails. Decentralization is often used to compensate for the single-point-failure disadvantage that is present when using a single device as a central node (e.g., in star and tree networks). A special kind of mesh, limiting the number of hops between two nodes, is a hypercube. The number of arbitrary forks in mesh networks makes them more difficult to design and implement, but their decentralized nature makes them very useful.
This is similar in some ways to a grid network, where a linear or ring topology is used to connect systems in multiple directions. A multidimensional ring has a toroidal topology, for instance.
A fully connected network, complete topology, or full mesh topology is a network topology in which there is a direct link between all pairs of nodes. In a fully connected network with n nodes, there are
n
(
n
−
1
)
2
{\displaystyle {\frac {n(n-1)}{2}}\,}
direct links. Networks designed with this topology are usually very expensive to set up, but provide a high degree of reliability due to the multiple paths for data that are provided by the large number of redundant links between nodes. This topology is mostly seen in military applications.
== See also ==
== References ==
== External links ==
Tetrahedron Core Network: Application of a tetrahedral structure to create a resilient partial-mesh 3-dimensional campus backbone data network | Wikipedia/Network_topology |
Topology is an album by multi-instrumentalist and composer Joe McPhee, recorded in 1981 and first released on the Swiss HatHut label, it was rereleased on CD in 1990.
== Reception ==
AllMusic awarded the album 3 stars.
== Track listing ==
All compositions by Joe McPhee
"Age" – 10:47
"Blues for Chicago" – 5:34
"Pithecanthropus Erectus" (Charles Mingus) – 10:33
"Violets for Pia" – 7:43
"Topology I & II" (André Jaume, Joe McPhee) – 28:40
== Personnel ==
Joe McPhee – tenor saxophone, pocket trumpet
André Jaume – alto saxophone, bass clarinet (tracks 1 & 3–5)
Irène Schweizer – piano
Raymond Boni – guitar (tracks 1–3 & 5)
François Mechali – bass
Radu Malfatti – percussion, trombone, electronics (tracks 1, 3 & 5)
Pierre Favre – percussion (track 5)
Michael Overhage – cello (tracks 1, 2 & 5)
Tamia – vocals (track 5)
== References == | Wikipedia/Topology_(album) |
Topology was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of Topology appeared in 2009.
== Pricing dispute ==
On 10 August 2006, after months of unsuccessful negotiations with Elsevier about the price policy of library subscriptions, the entire editorial board of the journal handed in their resignation, effective 31 December 2006. Subsequently, two more issues appeared in 2007 with papers that had been accepted before the resignation of the editors. In early January the former editors instructed Elsevier to remove their names from the website of the journal, but Elsevier refused to comply, justifying their decision by saying that the editorial board should remain on the journal until all of the papers accepted during its tenure had been published.
In 2007 the former editors of Topology announced the launch of the Journal of Topology, published by Oxford University Press on behalf of the London Mathematical Society at a significantly lower price. Its first issue appeared in January 2008.
== References ==
== External links ==
Official website
Journals declaring independence | Wikipedia/Topology_(journal) |
In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles with non-zero mass and arbitrary spin j, an integer for bosons (j = 1, 2, 3 ...) or half-integer for fermions (j = 1⁄2, 3⁄2, 5⁄2 ...). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields.
They are named after Valentine Bargmann and Eugene Wigner.
== History ==
Paul Dirac first published the Dirac equation in 1928, and later (1936) extended it to particles of any half-integer spin before Fierz and Pauli subsequently found the same equations in 1939, and about a decade before Bargman, and Wigner. Eugene Wigner wrote a paper in 1937 about unitary representations of the inhomogeneous Lorentz group, or the Poincaré group. Wigner notes Ettore Majorana and Dirac used infinitesimal operators applied to functions. Wigner classifies representations as irreducible, factorial, and unitary.
In 1948 Valentine Bargmann and Wigner published the equations now named after them in a paper on a group theoretical discussion of relativistic wave equations.
== Statement of the equations ==
For a free particle of spin j without electric charge, the BW equations are a set of 2j coupled linear partial differential equations, each with a similar mathematical form to the Dirac equation. The full set of equations are:
(
−
γ
μ
P
^
μ
+
m
c
)
α
1
α
1
′
ψ
α
1
′
α
2
α
3
⋯
α
2
j
=
0
(
−
γ
μ
P
^
μ
+
m
c
)
α
2
α
2
′
ψ
α
1
α
2
′
α
3
⋯
α
2
j
=
0
⋮
(
−
γ
μ
P
^
μ
+
m
c
)
α
2
j
α
2
j
′
ψ
α
1
α
2
α
3
⋯
α
2
j
′
=
0
{\displaystyle {\begin{aligned}&\left(-\gamma ^{\mu }{\hat {P}}_{\mu }+mc\right)_{\alpha _{1}\alpha _{1}'}\psi _{\alpha '_{1}\alpha _{2}\alpha _{3}\cdots \alpha _{2j}}=0\\&\left(-\gamma ^{\mu }{\hat {P}}_{\mu }+mc\right)_{\alpha _{2}\alpha _{2}'}\psi _{\alpha _{1}\alpha '_{2}\alpha _{3}\cdots \alpha _{2j}}=0\\&\qquad \vdots \\&\left(-\gamma ^{\mu }{\hat {P}}_{\mu }+mc\right)_{\alpha _{2j}\alpha '_{2j}}\psi _{\alpha _{1}\alpha _{2}\alpha _{3}\cdots \alpha '_{2j}}=0\\\end{aligned}}}
which follow the pattern;
for r = 1, 2, ... 2j. (Some authors e.g. Loide and Saar use n = 2j to remove factors of 2. Also the spin quantum number is usually denoted by s in quantum mechanics, however in this context j is more typical in the literature). The entire wavefunction ψ = ψ(r, t) has components
ψ
α
1
α
2
α
3
⋯
α
2
j
(
r
,
t
)
{\displaystyle \psi _{\alpha _{1}\alpha _{2}\alpha _{3}\cdots \alpha _{2j}}(\mathbf {r} ,t)}
and is a rank-2j 4-component spinor field. Each index takes the values 1, 2, 3, or 4, so there are 42j components of the entire spinor field ψ, although a completely symmetric wavefunction reduces the number of independent components to 2(2j + 1). Further, γμ = (γ0, γ) are the gamma matrices, and
P
^
μ
=
i
ℏ
∂
μ
{\displaystyle {\hat {P}}_{\mu }=i\hbar \partial _{\mu }}
is the 4-momentum operator.
The operator constituting each equation, (−γμPμ + mc) = (−iħγμ∂μ + mc), is a 4 × 4 matrix, because of the γμ matrices, and the mc term scalar-multiplies the 4 × 4 identity matrix (usually not written for simplicity). Explicitly, in the Dirac representation of the gamma matrices:
−
γ
μ
P
^
μ
+
m
c
=
−
γ
0
E
^
c
−
γ
⋅
(
−
p
^
)
+
m
c
=
−
(
I
2
0
0
−
I
2
)
E
^
c
+
(
0
σ
⋅
p
^
−
σ
⋅
p
^
0
)
+
(
I
2
0
0
I
2
)
m
c
=
(
−
E
^
c
+
m
c
0
p
^
z
p
^
x
−
i
p
^
y
0
−
E
^
c
+
m
c
p
^
x
+
i
p
^
y
−
p
^
z
−
p
^
z
−
(
p
^
x
−
i
p
^
y
)
E
^
c
+
m
c
0
−
(
p
^
x
+
i
p
^
y
)
p
^
z
0
E
^
c
+
m
c
)
{\displaystyle {\begin{aligned}-\gamma ^{\mu }{\hat {P}}_{\mu }+mc&=-\gamma ^{0}{\frac {\hat {E}}{c}}-{\boldsymbol {\gamma }}\cdot (-{\hat {\mathbf {p} }})+mc\\[6pt]&=-{\begin{pmatrix}I_{2}&0\\0&-I_{2}\\\end{pmatrix}}{\frac {\hat {E}}{c}}+{\begin{pmatrix}0&{\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }}\\-{\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }}&0\\\end{pmatrix}}+{\begin{pmatrix}I_{2}&0\\0&I_{2}\\\end{pmatrix}}mc\\[8pt]&={\begin{pmatrix}-{\frac {\hat {E}}{c}}+mc&0&{\hat {p}}_{z}&{\hat {p}}_{x}-i{\hat {p}}_{y}\\0&-{\frac {\hat {E}}{c}}+mc&{\hat {p}}_{x}+i{\hat {p}}_{y}&-{\hat {p}}_{z}\\-{\hat {p}}_{z}&-({\hat {p}}_{x}-i{\hat {p}}_{y})&{\frac {\hat {E}}{c}}+mc&0\\-({\hat {p}}_{x}+i{\hat {p}}_{y})&{\hat {p}}_{z}&0&{\frac {\hat {E}}{c}}+mc\\\end{pmatrix}}\\\end{aligned}}}
where σ = (σ1, σ2, σ3) = (σx, σy, σz) is a vector of the Pauli matrices, E is the energy operator, p = (p1, p2, p3) = (px, py, pz) is the 3-momentum operator, I2 denotes the 2 × 2 identity matrix, the zeros (in the second line) are actually 2 × 2 blocks of zero matrices.
The above matrix operator contracts with one bispinor index of ψ at a time (see matrix multiplication), so some properties of the Dirac equation also apply to the BW equations:
the equations are Lorentz covariant,
all components of the solutions ψ also satisfy the Klein–Gordon equation, and hence fulfill the relativistic energy–momentum relation,
E
2
=
(
p
c
)
2
+
(
m
c
2
)
2
{\displaystyle E^{2}=(pc)^{2}+(mc^{2})^{2}}
second quantization is still possible.
Unlike the Dirac equation, which can incorporate the electromagnetic field via minimal coupling, the B–W formalism comprises intrinsic contradictions and difficulties when the electromagnetic field interaction is incorporated. In other words, it is not possible to make the change Pμ → Pμ − eAμ, where e is the electric charge of the particle and Aμ = (A0, A) is the electromagnetic four-potential. An indirect approach to investigate electromagnetic influences of the particle is to derive the electromagnetic four-currents and multipole moments for the particle, rather than include the interactions in the wave equations themselves.
== Lorentz group structure ==
The representation of the Lorentz group for the BW equations is
D
B
W
=
⨂
r
=
1
2
j
[
D
r
(
1
/
2
,
0
)
⊕
D
r
(
0
,
1
/
2
)
]
.
{\displaystyle D^{\mathrm {BW} }=\bigotimes _{r=1}^{2j}\left[D_{r}^{(1/2,0)}\oplus D_{r}^{(0,1/2)}\right]\,.}
where each Dr is an irreducible representation. This representation does not have definite spin unless j equals 1/2 or 0. One may perform a Clebsch–Gordan decomposition to find the irreducible (A, B) terms and hence the spin content. This redundancy necessitates that a particle of definite spin j that transforms under the DBW representation satisfies field equations.
The representations D(j, 0) and D(0, j) can each separately represent particles of spin j. A state or quantum field in such a representation would satisfy no field equation except the Klein–Gordon equation.
== Formulation in curved spacetime ==
Following M. Kenmoku, in local Minkowski space, the gamma matrices satisfy the anticommutation relations:
[
γ
i
,
γ
j
]
+
=
2
η
i
j
I
4
{\displaystyle [\gamma ^{i},\gamma ^{j}]_{+}=2\eta ^{ij}I_{4}}
where ηij = diag(−1, 1, 1, 1) is the Minkowski metric. For the Latin indices here, i, j = 0, 1, 2, 3. In curved spacetime they are similar:
[
γ
μ
,
γ
ν
]
+
=
2
g
μ
ν
{\displaystyle [\gamma ^{\mu },\gamma ^{\nu }]_{+}=2g^{\mu \nu }}
where the spatial gamma matrices are contracted with the vierbein biμ to obtain γμ = biμ γi, and gμν = biμbiν is the metric tensor. For the Greek indices; μ, ν = 0, 1, 2, 3.
A covariant derivative for spinors is given by
D
μ
=
∂
μ
+
Ω
μ
{\displaystyle {\mathcal {D}}_{\mu }=\partial _{\mu }+\Omega _{\mu }}
with the connection Ω given in terms of the spin connection ω by:
Ω
μ
=
1
4
∂
μ
ω
i
j
(
γ
i
γ
j
−
γ
j
γ
i
)
{\displaystyle \Omega _{\mu }={\frac {1}{4}}\partial _{\mu }\omega ^{ij}(\gamma _{i}\gamma _{j}-\gamma _{j}\gamma _{i})}
The covariant derivative transforms like ψ:
D
μ
ψ
→
D
(
Λ
)
D
μ
ψ
{\displaystyle {\mathcal {D}}_{\mu }\psi \rightarrow D(\Lambda ){\mathcal {D}}_{\mu }\psi }
With this setup, equation (1) becomes:
(
−
i
ℏ
γ
μ
D
μ
+
m
c
)
α
1
α
1
′
ψ
α
1
′
α
2
α
3
⋯
α
2
j
=
0
(
−
i
ℏ
γ
μ
D
μ
+
m
c
)
α
2
α
2
′
ψ
α
1
α
2
′
α
3
⋯
α
2
j
=
0
⋮
(
−
i
ℏ
γ
μ
D
μ
+
m
c
)
α
2
j
α
2
j
′
ψ
α
1
α
2
α
3
⋯
α
2
j
′
=
0
.
{\displaystyle {\begin{aligned}&(-i\hbar \gamma ^{\mu }{\mathcal {D}}_{\mu }+mc)_{\alpha _{1}\alpha _{1}'}\psi _{\alpha '_{1}\alpha _{2}\alpha _{3}\cdots \alpha _{2j}}=0\\&(-i\hbar \gamma ^{\mu }{\mathcal {D}}_{\mu }+mc)_{\alpha _{2}\alpha _{2}'}\psi _{\alpha _{1}\alpha '_{2}\alpha _{3}\cdots \alpha _{2j}}=0\\&\qquad \vdots \\&(-i\hbar \gamma ^{\mu }{\mathcal {D}}_{\mu }+mc)_{\alpha _{2j}\alpha '_{2j}}\psi _{\alpha _{1}\alpha _{2}\alpha _{3}\cdots \alpha '_{2j}}=0\,.\\\end{aligned}}}
== See also ==
Two-body Dirac equation
Generalizations of Pauli matrices
Wigner D-matrix
Weyl–Brauer matrices
Higher-dimensional gamma matrices
Joos–Weinberg equation, alternative equations which describe free particles of any spin
Higher-spin theory
== Notes ==
== References ==
== Further reading ==
=== Books ===
=== Selected papers ===
== External links ==
Relativistic wave equations:
Dirac matrices in higher dimensions, Wolfram Demonstrations Project
Learning about spin-1 fields, P. Cahill, K. Cahill, University of New Mexico
Field equations for massless bosons from a Dirac–Weinberg formalism, R.W. Davies, K.T.R. Davies, P. Zory, D.S. Nydick, American Journal of Physics
Quantum field theory I, Martin Mojžiš Archived 2016-03-03 at the Wayback Machine
The Bargmann–Wigner Equation: Field equation for arbitrary spin, FarzadQassemi, IPM School and Workshop on Cosmology, IPM, Tehran, Iran
Lorentz groups in relativistic quantum physics:
Representations of Lorentz Group, indiana.edu
Appendix C: Lorentz group and the Dirac algebra, mcgill.ca
The Lorentz Group, Relativistic Particles, and Quantum Mechanics, D. E. Soper, University of Oregon, 2011
Representations of Lorentz and Poincaré groups, J. Maciejko, Stanford University
Representations of the Symmetry Group of Spacetime, K. Drake, M. Feinberg, D. Guild, E. Turetsky, 2009 | Wikipedia/Bargmann–Wigner_equations |
This article describes the mathematics of the Standard Model of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3) × SU(2) × U(1). The theory is commonly viewed as describing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson.
The Standard Model is renormalizable and mathematically self-consistent; however, despite having huge and continued successes in providing experimental predictions, it does leave some unexplained phenomena. In particular, although the physics of special relativity is incorporated, general relativity is not, and the Standard Model will fail at energies or distances where the graviton is expected to emerge. Therefore, in a modern field theory context, it is seen as an effective field theory.
== Quantum field theory ==
The standard model is a quantum field theory, meaning its fundamental objects are quantum fields, which are defined at all points in spacetime. QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles. These fields are
the fermion fields, ψ, which account for "matter particles";
the electroweak boson fields
W
1
{\displaystyle W_{1}}
,
W
2
{\displaystyle W_{2}}
,
W
3
{\displaystyle W_{3}}
, and B;
the gluon field, Ga; and
the Higgs field, φ.
That these are quantum rather than classical fields has the mathematical consequence that they are operator-valued. In particular, values of the fields generally do not commute. As operators, they act upon a quantum state (ket vector).
== Alternative presentations of the fields ==
As is common in quantum theory, there is more than one way to look at things. At first the basic fields given above may not seem to correspond well with the "fundamental particles" in the chart above, but there are several alternative presentations that, in particular contexts, may be more appropriate than those that are given above.
=== Fermions ===
Rather than having one fermion field ψ, it can be split up into separate components for each type of particle. This mirrors the historical evolution of quantum field theory, since the electron component ψe (describing the electron and its antiparticle the positron) is then the original ψ field of quantum electrodynamics, which was later accompanied by ψμ and ψτ fields for the muon and tauon respectively (and their antiparticles). Electroweak theory added
ψ
ν
e
,
ψ
ν
μ
{\displaystyle \psi _{\nu _{\mathrm {e} }},\psi _{\nu _{\mu }}}
, and
ψ
ν
τ
{\displaystyle \psi _{\nu _{\tau }}}
for the corresponding neutrinos. The quarks add still further components. In order to be four-spinors like the electron and other lepton components, there must be one quark component for every combination of flavor and color, bringing the total to 24 (3 for charged leptons, 3 for neutrinos, and 2·3·3 = 18 for quarks). Each of these is a four component bispinor, for a total of 96 complex-valued components for the fermion field.
An important definition is the barred fermion field
ψ
¯
{\displaystyle {\bar {\psi }}}
, which is defined to be
ψ
†
γ
0
{\displaystyle \psi ^{\dagger }\gamma ^{0}}
, where
†
{\displaystyle \dagger }
denotes the Hermitian adjoint of ψ, and γ0 is the zeroth gamma matrix. If ψ is thought of as an n × 1 matrix then
ψ
¯
{\displaystyle {\bar {\psi }}}
should be thought of as a 1 × n matrix.
==== A chiral theory ====
An independent decomposition of ψ is that into chirality components:
where
γ
5
{\displaystyle \gamma _{5}}
is the fifth gamma matrix. This is very important in the Standard Model because left and right chirality components are treated differently by the gauge interactions.
In particular, under weak isospin SU(2) transformations the left-handed particles are weak-isospin doublets, whereas the right-handed are singlets – i.e. the weak isospin of ψR is zero. Put more simply, the weak interaction could rotate e.g. a left-handed electron into a left-handed neutrino (with emission of a W−), but could not do so with the same right-handed particles. As an aside, the right-handed neutrino originally did not exist in the standard model – but the discovery of neutrino oscillation implies that neutrinos must have mass, and since chirality can change during the propagation of a massive particle, right-handed neutrinos must exist in reality. This does not however change the (experimentally proven) chiral nature of the weak interaction.
Furthermore, U(1) acts differently on
ψ
e
L
{\displaystyle \psi _{\mathrm {e} }^{\rm {L}}}
and
ψ
e
R
{\displaystyle \psi _{\mathrm {e} }^{\rm {R}}}
(because they have different weak hypercharges).
==== Mass and interaction eigenstates ====
A distinction can thus be made between, for example, the mass and interaction eigenstates of the neutrino. The former is the state that propagates in free space, whereas the latter is the different state that participates in interactions. Which is the "fundamental" particle? For the neutrino, it is conventional to define the "flavor" (νe, νμ, or ντ) by the interaction eigenstate, whereas for the quarks we define the flavor (up, down, etc.) by the mass state. We can switch between these states using the CKM matrix for the quarks, or the PMNS matrix for the neutrinos (the charged leptons on the other hand are eigenstates of both mass and flavor).
As an aside, if a complex phase term exists within either of these matrices, it will give rise to direct CP violation, which could explain the dominance of matter over antimatter in our current universe. This has been proven for the CKM matrix, and is expected for the PMNS matrix.
==== Positive and negative energies ====
Finally, the quantum fields are sometimes decomposed into "positive" and "negative" energy parts: ψ = ψ+ + ψ−. This is not so common when a quantum field theory has been set up, but often features prominently in the process of quantizing a field theory.
=== Bosons ===
Due to the Higgs mechanism, the electroweak boson fields
W
1
{\displaystyle W_{1}}
,
W
2
{\displaystyle W_{2}}
,
W
3
{\displaystyle W_{3}}
, and
B
{\displaystyle B}
"mix" to create the states that are physically observable. To retain gauge invariance, the underlying fields must be massless, but the observable states can gain masses in the process. These states are:
The massive neutral (Z) boson:
Z
=
cos
θ
W
W
3
−
sin
θ
W
B
{\displaystyle Z=\cos \theta _{\rm {W}}W_{3}-\sin \theta _{\rm {W}}B}
The massless neutral boson:
A
=
sin
θ
W
W
3
+
cos
θ
W
B
{\displaystyle A=\sin \theta _{\rm {W}}W_{3}+\cos \theta _{\rm {W}}B}
The massive charged W bosons:
W
±
=
1
2
(
W
1
∓
i
W
2
)
{\displaystyle W^{\pm }={\frac {1}{\sqrt {2}}}\left(W_{1}\mp iW_{2}\right)}
where θW is the Weinberg angle.
The A field is the photon, which corresponds classically to the well-known electromagnetic four-potential – i.e. the electric and magnetic fields. The Z field actually contributes in every process the photon does, but due to its large mass, the contribution is usually negligible.
== Perturbative QFT and the interaction picture ==
Much of the qualitative descriptions of the standard model in terms of "particles" and "forces" comes from the perturbative quantum field theory view of the model. In this, the Lagrangian is decomposed as
L
=
L
0
+
L
I
{\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}+{\mathcal {L}}_{\mathrm {I} }}
into separate free field and interaction Lagrangians. The free fields care for particles in isolation, whereas processes involving several particles arise through interactions. The idea is that the state vector should only change when particles interact, meaning a free particle is one whose quantum state is constant. This corresponds to the interaction picture in quantum mechanics.
In the more common Schrödinger picture, even the states of free particles change over time: typically the phase changes at a rate that depends on their energy. In the alternative Heisenberg picture, state vectors are kept constant, at the price of having the operators (in particular the observables) be time-dependent. The interaction picture constitutes an intermediate between the two, where some time dependence is placed in the operators (the quantum fields) and some in the state vector. In QFT, the former is called the free field part of the model, and the latter is called the interaction part. The free field model can be solved exactly, and then the solutions to the full model can be expressed as perturbations of the free field solutions, for example using the Dyson series.
It should be observed that the decomposition into free fields and interactions is in principle arbitrary. For example, renormalization in QED modifies the mass of the free field electron to match that of a physical electron (with an electromagnetic field), and will in doing so add a term to the free field Lagrangian which must be cancelled by a counterterm in the interaction Lagrangian, that then shows up as a two-line vertex in the Feynman diagrams. This is also how the Higgs field is thought to give particles mass: the part of the interaction term that corresponds to the nonzero vacuum expectation value of the Higgs field is moved from the interaction to the free field Lagrangian, where it looks just like a mass term having nothing to do with the Higgs field.
=== Free fields ===
Under the usual free/interaction decomposition, which is suitable for low energies, the free fields obey the following equations:
The fermion field ψ satisfies the Dirac equation;
(
i
ℏ
γ
μ
∂
μ
−
m
f
c
)
ψ
f
=
0
{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-m_{\rm {f}}c)\psi _{\rm {f}}=0}
for each type
f
{\displaystyle f}
of fermion.
The photon field A satisfies the wave equation
∂
μ
∂
μ
A
ν
=
0
{\displaystyle \partial _{\mu }\partial ^{\mu }A^{\nu }=0}
.
The Higgs field φ satisfies the Klein–Gordon equation.
The weak interaction fields Z, W± satisfy the Proca equation.
These equations can be solved exactly. One usually does so by considering first solutions that are periodic with some period L along each spatial axis; later taking the limit: L → ∞ will lift this periodicity restriction.
In the periodic case, the solution for a field F (any of the above) can be expressed as a Fourier series of the form
F
(
x
)
=
β
∑
p
∑
r
E
p
−
1
2
(
a
r
(
p
)
u
r
(
p
)
e
−
i
p
x
ℏ
+
b
r
†
(
p
)
v
r
(
p
)
e
i
p
x
ℏ
)
{\displaystyle F(x)=\beta \sum _{\mathbf {p} }\sum _{r}E_{\mathbf {p} }^{-{\frac {1}{2}}}\left(a_{r}(\mathbf {p} )u_{r}(\mathbf {p} )e^{-{\frac {ipx}{\hbar }}}+b_{r}^{\dagger }(\mathbf {p} )v_{r}(\mathbf {p} )e^{\frac {ipx}{\hbar }}\right)}
where:
β is a normalization factor; for the fermion field
ψ
f
{\displaystyle \psi _{\rm {f}}}
it is
m
f
c
2
/
V
{\textstyle {\sqrt {m_{\rm {f}}c^{2}/V}}}
, where
V
=
L
3
{\displaystyle V=L^{3}}
is the volume of the fundamental cell considered; for the photon field Aμ it is
ℏ
c
/
2
V
{\displaystyle \hbar c/{\sqrt {2V}}}
.
The sum over p is over all momenta consistent with the period L, i.e., over all vectors
2
π
ℏ
L
(
n
1
,
n
2
,
n
3
)
{\displaystyle {\frac {2\pi \hbar }{L}}(n_{1},n_{2},n_{3})}
where
n
1
,
n
2
,
n
3
{\displaystyle n_{1},n_{2},n_{3}}
are integers.
The sum over r covers other degrees of freedom specific for the field, such as polarization or spin; it usually comes out as a sum from 1 to 2 or from 1 to 3.
Ep is the relativistic energy for a momentum p quantum of the field,
=
m
2
c
4
+
c
2
p
2
{\textstyle ={\sqrt {m^{2}c^{4}+c^{2}\mathbf {p} ^{2}}}}
when the rest mass is m.
ar(p) and
b
r
†
(
p
)
{\displaystyle b_{r}^{\dagger }(\mathbf {p} )}
are annihilation and creation operators respectively for "a-particles" and "b-particles" respectively of momentum p; "b-particles" are the antiparticles of "a-particles". Different fields have different "a-" and "b-particles". For some fields, a and b are the same.
ur(p) and vr(p) are non-operators that carry the vector or spinor aspects of the field (where relevant).
p
=
(
E
p
/
c
,
p
)
{\displaystyle p=(E_{\mathbf {p} }/c,\mathbf {p} )}
is the four-momentum for a quantum with momentum p.
p
x
=
p
μ
x
μ
{\displaystyle px=p_{\mu }x^{\mu }}
denotes an inner product of four-vectors.
In the limit L → ∞, the sum would turn into an integral with help from the V hidden inside β. The numeric value of β also depends on the normalization chosen for
u
r
(
p
)
{\displaystyle u_{r}(\mathbf {p} )}
and
v
r
(
p
)
{\displaystyle v_{r}(\mathbf {p} )}
.
Technically,
a
r
†
(
p
)
{\displaystyle a_{r}^{\dagger }(\mathbf {p} )}
is the Hermitian adjoint of the operator ar(p) in the inner product space of ket vectors. The identification of
a
r
†
(
p
)
{\displaystyle a_{r}^{\dagger }(\mathbf {p} )}
and ar(p) as creation and annihilation operators comes from comparing conserved quantities for a state before and after one of these have acted upon it.
a
r
†
(
p
)
{\displaystyle a_{r}^{\dagger }(\mathbf {p} )}
can for example be seen to add one particle, because it will add 1 to the eigenvalue of the a-particle number operator, and the momentum of that particle ought to be p since the eigenvalue of the vector-valued momentum operator increases by that much. For these derivations, one starts out with expressions for the operators in terms of the quantum fields. That the operators with
†
{\displaystyle \dagger }
are creation operators and the one without annihilation operators is a convention, imposed by the sign of the commutation relations postulated for them.
An important step in preparation for calculating in perturbative quantum field theory is to separate the "operator" factors a and b above from their corresponding vector or spinor factors u and v. The vertices of Feynman graphs come from the way that u and v from different factors in the interaction Lagrangian fit together, whereas the edges come from the way that the as and bs must be moved around in order to put terms in the Dyson series on normal form.
=== Interaction terms and the path integral approach ===
The Lagrangian can also be derived without using creation and annihilation operators (the "canonical" formalism) by using a path integral formulation, pioneered by Feynman building on the earlier work of Dirac. Feynman diagrams are pictorial representations of interaction terms. A quick derivation is indeed presented at the article on Feynman diagrams.
== Lagrangian formalism ==
We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model Lagrangian density. Any such term must be both gauge and reference-frame invariant, otherwise the laws of physics would depend on an arbitrary choice or the frame of an observer. Therefore, the global Poincaré symmetry, consisting of translational symmetry, rotational symmetry and the inertial reference frame invariance central to the theory of special relativity must apply. The local SU(3) × SU(2) × U(1) gauge symmetry is the internal symmetry. The three factors of the gauge symmetry together give rise to the three fundamental interactions, after some appropriate relations have been defined, as we shall see.
=== Kinetic terms ===
A free particle can be represented by a mass term, and a kinetic term that relates to the "motion" of the fields.
==== Fermion fields ====
The kinetic term for a Dirac fermion is
i
ψ
¯
γ
μ
∂
μ
ψ
{\displaystyle i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi }
where the notations are carried from earlier in the article. ψ can represent any, or all, Dirac fermions in the standard model. Generally, as below, this term is included within the couplings (creating an overall "dynamical" term).
==== Gauge fields ====
For the spin-1 fields, first define the field strength tensor
F
μ
ν
a
=
∂
μ
A
ν
a
−
∂
ν
A
μ
a
+
g
f
a
b
c
A
μ
b
A
ν
c
{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c}}
for a given gauge field (here we use A), with gauge coupling constant g. The quantity fabc is the structure constant of the particular gauge group, defined by the commutator
[
t
a
,
t
b
]
=
i
f
a
b
c
t
c
,
{\displaystyle [t_{a},t_{b}]=if^{abc}t_{c},}
where ti are the generators of the group. In an abelian (commutative) group (such as the U(1) we use here) the structure constants vanish, since the generators ta all commute with each other. Of course, this is not the case in general – the standard model includes the non-Abelian SU(2) and SU(3) groups (such groups lead to what is called a Yang–Mills gauge theory).
We need to introduce three gauge fields corresponding to each of the subgroups SU(3) × SU(2) × U(1).
The gluon field tensor will be denoted by
G
μ
ν
a
{\displaystyle G_{\mu \nu }^{a}}
, where the index a labels elements of the 8 representation of color SU(3). The strong coupling constant is conventionally labelled gs (or simply g where there is no ambiguity). The observations leading to the discovery of this part of the Standard Model are discussed in the article in quantum chromodynamics.
The notation
W
μ
ν
a
{\displaystyle W_{\mu \nu }^{a}}
will be used for the gauge field tensor of SU(2) where a runs over the 3 generators of this group. The coupling can be denoted gw or again simply g. The gauge field will be denoted by
W
μ
a
{\displaystyle W_{\mu }^{a}}
.
The gauge field tensor for the U(1) of weak hypercharge will be denoted by Bμν, the coupling by g′, and the gauge field by Bμ.
The kinetic term can now be written as
L
k
i
n
=
−
1
4
B
μ
ν
B
μ
ν
−
1
2
t
r
W
μ
ν
W
μ
ν
−
1
2
t
r
G
μ
ν
G
μ
ν
{\displaystyle {\mathcal {L}}_{\rm {kin}}=-{1 \over 4}B_{\mu \nu }B^{\mu \nu }-{1 \over 2}\mathrm {tr} W_{\mu \nu }W^{\mu \nu }-{1 \over 2}\mathrm {tr} G_{\mu \nu }G^{\mu \nu }}
where the traces are over the SU(2) and SU(3) indices hidden in W and G respectively. The two-index objects are the field strengths derived from W and G the vector fields. There are also two extra hidden parameters: the theta angles for SU(2) and SU(3).
=== Coupling terms ===
The next step is to "couple" the gauge fields to the fermions, allowing for interactions.
==== Electroweak sector ====
The electroweak sector interacts with the symmetry group U(1) × SU(2)L, where the subscript L indicates coupling only to left-handed fermions.
L
E
W
=
∑
ψ
ψ
¯
γ
μ
(
i
∂
μ
−
g
′
1
2
Y
W
B
μ
−
g
1
2
τ
W
μ
)
ψ
{\displaystyle {\mathcal {L}}_{\mathrm {EW} }=\sum _{\psi }{\bar {\psi }}\gamma ^{\mu }\left(i\partial _{\mu }-g^{\prime }{1 \over 2}Y_{\mathrm {W} }B_{\mu }-g{1 \over 2}{\boldsymbol {\tau }}\mathbf {W} _{\mu }\right)\psi }
where Bμ is the U(1) gauge field; YW is the weak hypercharge (the generator of the U(1) group); Wμ is the three-component SU(2) gauge field; and the components of τ are the Pauli matrices (infinitesimal generators of the SU(2) group) whose eigenvalues give the weak isospin. Note that we have to redefine a new U(1) symmetry of weak hypercharge, different from QED, in order to achieve the unification with the weak force. The electric charge Q, third component of weak isospin T3 (also called Tz, I3 or Iz) and weak hypercharge YW are related by
Q
=
T
3
+
1
2
Y
W
,
{\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\rm {W}},}
(or by the alternative convention Q = T3 + YW). The first convention, used in this article, is equivalent to the earlier Gell-Mann–Nishijima formula. It makes the hypercharge be twice the average charge of a given isomultiplet.
One may then define the conserved current for weak isospin as
j
μ
=
1
2
ψ
¯
L
γ
μ
τ
ψ
L
{\displaystyle \mathbf {j} _{\mu }={1 \over 2}{\bar {\psi }}_{\rm {L}}\gamma _{\mu }{\boldsymbol {\tau }}\psi _{\rm {L}}}
and for weak hypercharge as
j
μ
Y
=
2
(
j
μ
e
m
−
j
μ
3
)
,
{\displaystyle j_{\mu }^{Y}=2(j_{\mu }^{\rm {em}}-j_{\mu }^{3})~,}
where
j
μ
e
m
{\displaystyle j_{\mu }^{\rm {em}}}
is the electric current and
j
μ
3
{\displaystyle j_{\mu }^{3}}
the third weak isospin current. As explained above, these currents mix to create the physically observed bosons, which also leads to testable relations between the coupling constants.
To explain this in a simpler way, we can see the effect of the electroweak interaction by picking out terms from the Lagrangian. We see that the SU(2) symmetry acts on each (left-handed) fermion doublet contained in ψ, for example
−
g
2
(
ν
¯
e
e
¯
)
τ
+
γ
μ
(
W
+
)
μ
(
ν
e
e
)
=
−
g
2
ν
¯
e
γ
μ
(
W
+
)
μ
e
{\displaystyle -{g \over 2}({\bar {\nu }}_{e}\;{\bar {e}})\tau ^{+}\gamma _{\mu }(W^{+})^{\mu }{\begin{pmatrix}{\nu _{e}}\\e\end{pmatrix}}=-{g \over 2}{\bar {\nu }}_{e}\gamma _{\mu }(W^{+})^{\mu }e}
where the particles are understood to be left-handed, and where
τ
+
≡
1
2
(
τ
1
+
i
τ
2
)
=
(
0
1
0
0
)
{\displaystyle \tau ^{+}\equiv {1 \over 2}(\tau ^{1}{+}i\tau ^{2})={\begin{pmatrix}0&1\\0&0\end{pmatrix}}}
This is an interaction corresponding to a "rotation in weak isospin space" or in other words, a transformation between eL and νeL via emission of a W− boson. The U(1) symmetry, on the other hand, is similar to electromagnetism, but acts on all "weak hypercharged" fermions (both left- and right-handed) via the neutral Z0, as well as the charged fermions via the photon.
==== Quantum chromodynamics sector ====
The quantum chromodynamics (QCD) sector defines the interactions between quarks and gluons, with SU(3) symmetry, generated by Ta. Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by
L
Q
C
D
=
i
U
¯
(
∂
μ
−
i
g
s
G
μ
a
T
a
)
γ
μ
U
+
i
D
¯
(
∂
μ
−
i
g
s
G
μ
a
T
a
)
γ
μ
D
.
{\displaystyle {\mathcal {L}}_{\mathrm {QCD} }=i{\overline {U}}\left(\partial _{\mu }-ig_{s}G_{\mu }^{a}T^{a}\right)\gamma ^{\mu }U+i{\overline {D}}\left(\partial _{\mu }-ig_{s}G_{\mu }^{a}T^{a}\right)\gamma ^{\mu }D.}
where U and D are the Dirac spinors associated with up and down-type quarks, and other notations are continued from the previous section.
=== Mass terms and the Higgs mechanism ===
==== Mass terms ====
The mass term arising from the Dirac Lagrangian (for any fermion ψ) is
−
m
ψ
¯
ψ
{\displaystyle -m{\bar {\psi }}\psi }
, which is not invariant under the electroweak symmetry. This can be seen by writing ψ in terms of left and right-handed components (skipping the actual calculation):
−
m
ψ
¯
ψ
=
−
m
(
ψ
¯
L
ψ
R
+
ψ
¯
R
ψ
L
)
{\displaystyle -m{\bar {\psi }}\psi =-m({\bar {\psi }}_{\rm {L}}\psi _{\rm {R}}+{\bar {\psi }}_{\rm {R}}\psi _{\rm {L}})}
i.e. contribution from
ψ
¯
L
ψ
L
{\displaystyle {\bar {\psi }}_{\rm {L}}\psi _{\rm {L}}}
and
ψ
¯
R
ψ
R
{\displaystyle {\bar {\psi }}_{\rm {R}}\psi _{\rm {R}}}
terms do not appear. We see that the mass-generating interaction is achieved by constant flipping of particle chirality. The spin-half particles have no right/left chirality pair with the same SU(2) representations and equal and opposite weak hypercharges, so assuming these gauge charges are conserved in the vacuum, none of the spin-half particles could ever swap chirality, and must remain massless. Additionally, we know experimentally that the W and Z bosons are massive, but a boson mass term contains the combination e.g. AμAμ, which clearly depends on the choice of gauge. Therefore, none of the standard model fermions or bosons can "begin" with mass, but must acquire it by some other mechanism.
==== Higgs mechanism ====
The solution to both these problems comes from the Higgs mechanism, which involves scalar fields (the number of which depend on the exact form of Higgs mechanism) which (to give the briefest possible description) are "absorbed" by the massive bosons as degrees of freedom, and which couple to the fermions via Yukawa coupling to create what looks like mass terms.
In the Standard Model, the Higgs field is a complex scalar field of the group SU(2)L:
ϕ
=
1
2
(
ϕ
+
ϕ
0
)
,
{\displaystyle \phi ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\phi ^{+}\\\phi ^{0}\end{pmatrix}},}
where the superscripts + and 0 indicate the electric charge (Q) of the components. The weak hypercharge (YW) of both components is 1.
The Higgs part of the Lagrangian is
L
H
=
[
(
∂
μ
−
i
g
W
μ
a
t
a
−
i
g
′
Y
ϕ
B
μ
)
ϕ
]
2
+
μ
2
ϕ
†
ϕ
−
λ
(
ϕ
†
ϕ
)
2
,
{\displaystyle {\mathcal {L}}_{\rm {H}}=\left[\left(\partial _{\mu }-igW_{\mu }^{a}t^{a}-ig'Y_{\phi }B_{\mu }\right)\phi \right]^{2}+\mu ^{2}\phi ^{\dagger }\phi -\lambda (\phi ^{\dagger }\phi )^{2},}
where λ > 0 and μ2 > 0, so that the mechanism of spontaneous symmetry breaking can be used. There is a parameter here, at first hidden within the shape of the potential, that is very important. In a unitarity gauge one can set
ϕ
+
=
0
{\displaystyle \phi ^{+}=0}
and make
ϕ
0
{\displaystyle \phi ^{0}}
real. Then
⟨
ϕ
0
⟩
=
v
{\displaystyle \langle \phi ^{0}\rangle =v}
is the non-vanishing vacuum expectation value of the Higgs field.
v
{\displaystyle v}
has units of mass, and it is the only parameter in the Standard Model that is not dimensionless. It is also much smaller than the Planck scale and about twice the Higgs mass, setting the scale for the mass of all other particles in the Standard Model. This is the only real fine-tuning to a small nonzero value in the Standard Model. Quadratic terms in Wμ and Bμ arise, which give masses to the W and Z bosons:
M
W
=
1
2
v
g
M
Z
=
1
2
v
g
2
+
g
′
2
{\displaystyle {\begin{aligned}M_{\rm {W}}&={\tfrac {1}{2}}vg\\M_{\rm {Z}}&={\tfrac {1}{2}}v{\sqrt {g^{2}+{g'}^{2}}}\end{aligned}}}
The mass of the Higgs boson itself is given by
M
H
=
2
μ
2
≡
2
λ
v
2
.
{\textstyle M_{\rm {H}}={\sqrt {2\mu ^{2}}}\equiv {\sqrt {2\lambda v^{2}}}.}
==== Yukawa interaction ====
The Yukawa interaction terms are
L
Yukawa
=
(
Y
u
)
m
n
(
q
¯
L
)
m
φ
~
(
u
R
)
n
+
(
Y
d
)
m
n
(
q
¯
L
)
m
φ
(
d
R
)
n
+
(
Y
e
)
m
n
(
L
¯
L
)
m
φ
~
(
e
R
)
n
+
h
.
c
.
{\displaystyle {\mathcal {L}}_{\text{Yukawa}}=(Y_{\text{u}})_{mn}({\bar {q}}_{\text{L}})_{m}{\tilde {\varphi }}(u_{\text{R}})_{n}+(Y_{\text{d}})_{mn}({\bar {q}}_{\text{L}})_{m}\varphi (d_{\text{R}})_{n}+(Y_{\text{e}})_{mn}({\bar {L}}_{\text{L}})_{m}{\tilde {\varphi }}(e_{\text{R}})_{n}+\mathrm {h.c.} }
where
Y
u
{\displaystyle Y_{\text{u}}}
,
Y
d
{\displaystyle Y_{\text{d}}}
, and
Y
e
{\displaystyle Y_{\text{e}}}
are 3 × 3 matrices of Yukawa couplings, with the mn term giving the coupling of the generations m and n, and h.c. means Hermitian conjugate of preceding terms. The fields
q
L
{\displaystyle q_{\text{L}}}
and
L
L
{\displaystyle L_{\text{L}}}
are left-handed quark and lepton doublets. Likewise,
u
R
{\displaystyle u_{\text{R}}}
,
d
R
{\displaystyle d_{\text{R}}}
and
e
R
{\displaystyle e_{\text{R}}}
are right-handed up-type quark, down-type quark, and lepton singlets. Finally
φ
{\displaystyle \varphi }
is the Higgs doublet and
φ
~
=
i
τ
2
φ
∗
{\displaystyle {\tilde {\varphi }}=i\tau _{2}\varphi ^{*}}
==== Neutrino masses ====
As previously mentioned, evidence shows neutrinos must have mass. But within the standard model, the right-handed neutrino does not exist, so even with a Yukawa coupling neutrinos remain massless. An obvious solution is to simply add a right-handed neutrino νR, which requires the addition of a new Dirac mass term in the Yukawa sector:
L
ν
Dir
=
(
Y
ν
)
m
n
(
L
¯
L
)
m
φ
(
ν
R
)
n
+
h
.
c
.
{\displaystyle {\mathcal {L}}_{\nu }^{\text{Dir}}=(Y_{\nu })_{mn}({\bar {L}}_{L})_{m}\varphi (\nu _{R})_{n}+\mathrm {h.c.} }
This field however must be a sterile neutrino, since being right-handed it experimentally belongs to an isospin singlet (T3 = 0) and also has charge Q = 0, implying YW = 0 (see above) i.e. it does not even participate in the weak interaction. The experimental evidence for sterile neutrinos is currently inconclusive.
Another possibility to consider is that the neutrino satisfies the Majorana equation, which at first seems possible due to its zero electric charge. In this case a new Majorana mass term is added to the Yukawa sector:
L
ν
Maj
=
−
1
2
m
(
ν
¯
C
ν
+
ν
¯
ν
C
)
{\displaystyle {\mathcal {L}}_{\nu }^{\text{Maj}}=-{\frac {1}{2}}m\left({\overline {\nu }}^{C}\nu +{\overline {\nu }}\nu ^{C}\right)}
where C denotes a charge conjugated (i.e. anti-) particle, and the
ν
{\displaystyle \nu }
terms are consistently all left (or all right) chirality (note that a left-chirality projection of an antiparticle is a right-handed field; care must be taken here due to different notations sometimes used). Here we are essentially flipping between left-handed neutrinos and right-handed anti-neutrinos (it is furthermore possible but not necessary that neutrinos are their own antiparticle, so these particles are the same). However, for left-chirality neutrinos, this term changes weak hypercharge by 2 units – not possible with the standard Higgs interaction, requiring the Higgs field to be extended to include an extra triplet with weak hypercharge = 2 – whereas for right-chirality neutrinos, no Higgs extensions are necessary. For both left and right chirality cases, Majorana terms violate lepton number, but possibly at a level beyond the current sensitivity of experiments to detect such violations.
It is possible to include both Dirac and Majorana mass terms in the same theory, which (in contrast to the Dirac-mass-only approach) can provide a “natural” explanation for the smallness of the observed neutrino masses, by linking the right-handed neutrinos to yet-unknown physics around the GUT scale (see seesaw mechanism).
Since in any case new fields must be postulated to explain the experimental results, neutrinos are an obvious gateway to searching physics beyond the Standard Model.
== Detailed information ==
This section provides more detail on some aspects, and some reference material. Explicit Lagrangian terms are also provided here.
=== Field content in detail ===
The Standard Model has the following fields. These describe one generation of leptons and quarks, and there are three generations, so there are three copies of each fermionic field. By CPT symmetry, there is a set of fermions and antifermions with opposite parity and charges. If a left-handed fermion spans some representation its antiparticle (right-handed antifermion) spans the dual representation (note that
2
¯
=
2
{\displaystyle {\bar {\mathbf {2} }}={\mathbf {2} }}
for SU(2), because it is pseudo-real). The column "representation" indicates under which representations of the gauge groups that each field transforms, in the order (SU(3), SU(2), U(1)) and for the U(1) group, the value of the weak hypercharge is listed. There are twice as many left-handed lepton field components as right-handed lepton field components in each generation, but an equal number of left-handed quark and right-handed quark field components.
=== Fermion content ===
This table is based in part on data gathered by the Particle Data Group.
=== Free parameters ===
Upon writing the most general Lagrangian with massless neutrinos, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment. Straightforward extensions of the Standard Model with massive neutrinos need 7 more parameters (3 masses and 4 PMNS matrix parameters) for a total of 26 parameters. The neutrino parameter values are still uncertain. The 19 certain parameters are summarized here.
The choice of free parameters is somewhat arbitrary. In the table above, gauge couplings are listed as free parameters, therefore with this choice the Weinberg angle is not a free parameter – it is defined as
tan
θ
W
=
g
1
/
g
2
{\displaystyle \tan \theta _{\rm {W}}={g_{1}}/{g_{2}}}
. Likewise, the fine-structure constant of QED is
α
=
1
4
π
(
g
1
g
2
)
2
g
1
2
+
g
2
2
{\displaystyle \alpha ={\frac {1}{4\pi }}{\frac {(g_{1}g_{2})^{2}}{g_{1}^{2}+g_{2}^{2}}}}
. Instead of fermion masses, dimensionless Yukawa couplings can be chosen as free parameters. For example, the electron mass depends on the Yukawa coupling of the electron to the Higgs field, and its value is
m
e
=
y
e
v
/
2
{\displaystyle m_{\rm {e}}=y_{\rm {e}}v/{\sqrt {2}}}
. Instead of the Higgs mass, the Higgs self-coupling strength
λ
=
m
H
2
2
v
2
{\displaystyle \lambda ={\frac {m_{\rm {H}}^{2}}{2v^{2}}}}
, which is approximately 0.129, can be chosen as a free parameter. Instead of the Higgs vacuum expectation value, the
μ
2
{\displaystyle \mu ^{2}}
parameter directly from the Higgs self-interaction term
μ
2
ϕ
†
ϕ
−
λ
(
ϕ
†
ϕ
)
2
{\displaystyle \mu ^{2}\phi ^{\dagger }\phi -\lambda (\phi ^{\dagger }\phi )^{2}}
can be chosen. Its value is
μ
2
=
λ
v
2
=
m
H
2
/
2
{\displaystyle \mu ^{2}=\lambda v^{2}={m_{\rm {H}}^{2}}/2}
, or approximately
μ
{\displaystyle \mu }
= 88.45 GeV.
The value of the vacuum energy (or more precisely, the renormalization scale used to calculate this energy) may also be treated as an additional free parameter. The renormalization scale may be identified with the Planck scale or fine-tuned to match the observed cosmological constant. However, both options are problematic.
=== Additional symmetries of the Standard Model ===
From the theoretical point of view, the Standard Model exhibits four additional global symmetries, not postulated at the outset of its construction, collectively denoted accidental symmetries, which are continuous U(1) global symmetries. The transformations leaving the Lagrangian invariant are:
ψ
q
→
e
i
α
/
3
ψ
q
{\displaystyle \psi _{\text{q}}\to e^{i\alpha /3}\psi _{\text{q}}}
E
L
→
e
i
β
E
L
and
(
e
R
)
c
→
e
i
β
(
e
R
)
c
{\displaystyle E_{\rm {L}}\to e^{i\beta }E_{\rm {L}}{\text{ and }}(e_{\rm {R}})^{\text{c}}\to e^{i\beta }(e_{\rm {R}})^{\text{c}}}
M
L
→
e
i
β
M
L
and
(
μ
R
)
c
→
e
i
β
(
μ
R
)
c
{\displaystyle M_{\rm {L}}\to e^{i\beta }M_{\rm {L}}{\text{ and }}(\mu _{\rm {R}})^{\text{c}}\to e^{i\beta }(\mu _{\rm {R}})^{\text{c}}}
T
L
→
e
i
β
T
L
and
(
τ
R
)
c
→
e
i
β
(
τ
R
)
c
{\displaystyle T_{\rm {L}}\to e^{i\beta }T_{\rm {L}}{\text{ and }}(\tau _{\rm {R}})^{\text{c}}\to e^{i\beta }(\tau _{\rm {R}})^{\text{c}}}
The first transformation rule is shorthand meaning that all quark fields for all generations must be rotated by an identical phase simultaneously. The fields ML, TL and
(
μ
R
)
c
,
(
τ
R
)
c
{\displaystyle (\mu _{\rm {R}})^{\text{c}},(\tau _{\rm {R}})^{\text{c}}}
are the 2nd (muon) and 3rd (tau) generation analogs of EL and
(
e
R
)
c
{\displaystyle (e_{\rm {R}})^{\text{c}}}
fields.
By Noether's theorem, each symmetry above has an associated conservation law: the conservation of baryon number, electron number, muon number, and tau number. Each quark is assigned a baryon number of
1
3
{\textstyle {\frac {1}{3}}}
, while each antiquark is assigned a baryon number of
−
1
3
{\textstyle -{\frac {1}{3}}}
. Conservation of baryon number implies that the number of quarks minus the number of antiquarks is a constant. Within experimental limits, no violation of this conservation law has been found.
Similarly, each electron and its associated neutrino is assigned an electron number of +1, while the anti-electron and the associated anti-neutrino carry a −1 electron number. Similarly, the muons and their neutrinos are assigned a muon number of +1 and the tau leptons are assigned a tau lepton number of +1. The Standard Model predicts that each of these three numbers should be conserved separately in a manner similar to the way baryon number is conserved. These numbers are collectively known as lepton family numbers (LF). (This result depends on the assumption made in Standard Model that neutrinos are massless. Experimentally, neutrino oscillations imply that individual electron, muon and tau numbers are not conserved.)
In addition to the accidental (but exact) symmetries described above, the Standard Model exhibits several approximate symmetries. These are the "SU(2) custodial symmetry" and the "SU(2) or SU(3) quark flavor symmetry".
=== U(1) symmetry ===
For the leptons, the gauge group can be written SU(2)l × U(1)L × U(1)R. The two U(1) factors can be combined into U(1)Y × U(1)l, where l is the lepton number. Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group SU(2)L × U(1)Y. A similar argument in the quark sector also gives the same result for the electroweak theory.
=== Charged and neutral current couplings and Fermi theory ===
The charged currents
j
∓
=
j
1
±
i
j
2
{\displaystyle j^{\mp }=j^{1}\pm ij^{2}}
are
j
μ
−
=
U
¯
i
L
γ
μ
D
i
L
+
ν
¯
i
L
γ
μ
l
i
L
.
{\displaystyle j_{\mu }^{-}={\overline {U}}_{i\mathrm {L} }\gamma _{\mu }D_{i\mathrm {L} }+{\overline {\nu }}_{i\mathrm {L} }\gamma _{\mu }l_{i\mathrm {L} }.}
These charged currents are precisely those that entered the Fermi theory of beta decay. The action contains the charge current piece
L
C
C
=
g
2
(
j
μ
+
W
−
μ
+
j
μ
−
W
+
μ
)
.
{\displaystyle {\mathcal {L}}_{\rm {CC}}={\frac {g}{\sqrt {2}}}(j_{\mu }^{+}W^{-\mu }+j_{\mu }^{-}W^{+\mu }).}
For energy much less than the mass of the W-boson, the effective theory becomes the current–current contact interaction of the Fermi theory,
2
2
G
F
J
μ
+
J
μ
−
{\displaystyle 2{\sqrt {2}}G_{\rm {F}}~~J_{\mu }^{+}J^{\mu ~~-}}
.
However, gauge invariance now requires that the component
W
3
{\displaystyle W^{3}}
of the gauge field also be coupled to a current that lies in the triplet of SU(2). However, this mixes with the U(1), and another current in that sector is needed. These currents must be uncharged in order to conserve charge. So neutral currents are also required,
j
μ
3
=
1
2
(
U
¯
i
L
γ
μ
U
i
L
−
D
¯
i
L
γ
μ
D
i
L
+
ν
¯
i
L
γ
μ
ν
i
L
−
l
¯
i
L
γ
μ
l
i
L
)
{\displaystyle j_{\mu }^{3}={\frac {1}{2}}\left({\overline {U}}_{i\mathrm {L} }\gamma _{\mu }U_{i\mathrm {L} }-{\overline {D}}_{i\mathrm {L} }\gamma _{\mu }D_{i\mathrm {L} }+{\overline {\nu }}_{i\mathrm {L} }\gamma _{\mu }\nu _{i\mathrm {L} }-{\overline {l}}_{i\mathrm {L} }\gamma _{\mu }l_{i\mathrm {L} }\right)}
j
μ
e
m
=
2
3
U
¯
i
γ
μ
U
i
−
1
3
D
¯
i
γ
μ
D
i
−
l
¯
i
γ
μ
l
i
.
{\displaystyle j_{\mu }^{\rm {em}}={\frac {2}{3}}{\overline {U}}_{i}\gamma _{\mu }U_{i}-{\frac {1}{3}}{\overline {D}}_{i}\gamma _{\mu }D_{i}-{\overline {l}}_{i}\gamma _{\mu }l_{i}.}
The neutral current piece in the Lagrangian is then
L
N
C
=
e
j
μ
e
m
A
μ
+
g
cos
θ
W
(
J
μ
3
−
sin
2
θ
W
J
μ
e
m
)
Z
μ
.
{\displaystyle {\mathcal {L}}_{\rm {NC}}=ej_{\mu }^{\rm {em}}A^{\mu }+{\frac {g}{\cos \theta _{\rm {W}}}}(J_{\mu }^{3}-\sin ^{2}\theta _{\rm {W}}J_{\mu }^{\rm {em}})Z^{\mu }.}
== Physics beyond the Standard Model ==
== See also ==
Overview of Standard Model of particle physics
Fundamental interaction
Noncommutative standard model
Open questions: CP violation, Neutrino masses, Quark matter
Physics beyond the Standard Model
Strong interactions
Flavor
Quantum chromodynamics
Quark model
Weak interactions
Electroweak interaction
Fermi's interaction
Weinberg angle
Symmetry in quantum mechanics
Quantum Field Theory in a Nutshell by A. Zee
== References and external links == | Wikipedia/Mathematical_formulation_of_the_Standard_Model |
The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in (1+1) dimensions.
== Definition ==
The Thirring model is given by the Lagrangian density
L
=
ψ
¯
(
i
∂
/
−
m
)
ψ
−
g
2
(
ψ
¯
γ
μ
ψ
)
(
ψ
¯
γ
μ
ψ
)
{\displaystyle {\mathcal {L}}={\overline {\psi }}(i\partial \!\!\!/-m)\psi -{\frac {g}{2}}\left({\overline {\psi }}\gamma ^{\mu }\psi \right)\left({\overline {\psi }}\gamma _{\mu }\psi \right)\ }
where
ψ
=
(
ψ
+
,
ψ
−
)
{\displaystyle \psi =(\psi _{+},\psi _{-})}
is the field, g is the coupling constant, m is the mass, and
γ
μ
{\displaystyle \gamma ^{\mu }}
, for
μ
=
0
,
1
{\displaystyle \mu =0,1}
, are the two-dimensional gamma matrices.
This is the unique model of (1+1)-dimensional, Dirac fermions with a local (self-)interaction. Indeed, since there are only 4 independent fields, because of the Pauli principle, all the quartic, local interactions are equivalent; and all higher power, local interactions vanish. (Interactions containing derivatives, such as
(
ψ
¯
∂
/
ψ
)
2
{\displaystyle ({\bar {\psi }}\partial \!\!\!/\psi )^{2}}
, are not considered because they are non-renormalizable.)
The correlation functions of the Thirring model (massive or massless) verify the Osterwalder–Schrader axioms, and hence the theory makes sense as a quantum field theory.
== Massless case ==
The massless Thirring model is exactly solvable in the sense that a formula for the
n
{\displaystyle n}
-points field correlation is known.
=== Exact solution ===
After it was introduced by Walter Thirring, many authors tried to solve the massless case, with confusing outcomes. The correct formula for the two and four point correlation was finally found by K. Johnson; then C. R. Hagen and B. Klaiber extended the explicit solution to any multipoint correlation function of the fields.
== Massive Thirring model, or MTM ==
The mass spectrum of the model and the scattering matrix was explicitly evaluated by Bethe ansatz. An explicit formula for the correlations is not known. J. I. Cirac, P. Maraner and J. K. Pachos applied the massive Thirring model to the description of optical lattices.
=== Exact solution ===
In one space dimension and one time dimension the model can be solved by the Bethe ansatz. This helps one calculate exactly the mass spectrum
and scattering matrix. Calculation of the scattering matrix reproduces the results published earlier by Alexander Zamolodchikov. The paper with the exact solution of Massive Thirring model by Bethe ansatz was first published in Russian. Ultraviolet renormalization was done in the frame of the Bethe ansatz. The fractional charge appears in the model during renormalization as a repulsion beyond the cutoff.
Multi-particle production cancels on mass shell.
The exact solution shows once again the equivalence of the Thirring model and the quantum sine-Gordon model. The Thirring model is S-dual to the sine-Gordon model. The fundamental fermions of the Thirring model correspond to the solitons of the sine-Gordon model.
== Bosonization ==
S. Coleman discovered an equivalence between the Thirring and the sine-Gordon models. Despite the fact that the latter is a pure boson model, massless Thirring fermions are equivalent to free bosons; besides massive fermions are equivalent to the sine-Gordon bosons. This phenomenon is more general in two dimensions and is called bosonization.
== See also ==
Dirac equation
Gross–Neveu model
Nonlinear Dirac equation
Soler model
== References ==
== External links ==
On the equivalence between sine-Gordon Model and Thirring Model in the chirally broken phase | Wikipedia/Thirring_model |
Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by Rudolf Haag and Daniel Kastler (1964). The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those.
== Haag–Kastler axioms ==
Let
O
{\displaystyle {\mathcal {O}}}
be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a set
{
A
(
O
)
}
O
∈
O
{\displaystyle \{{\mathcal {A}}(O)\}_{O\in {\mathcal {O}}}}
of von Neumann algebras
A
(
O
)
{\displaystyle {\mathcal {A}}(O)}
on a common Hilbert space
H
{\displaystyle {\mathcal {H}}}
satisfying the following axioms:
Isotony:
O
1
⊂
O
2
{\displaystyle O_{1}\subset O_{2}}
implies
A
(
O
1
)
⊂
A
(
O
2
)
{\displaystyle {\mathcal {A}}(O_{1})\subset {\mathcal {A}}(O_{2})}
.
Causality: If
O
1
{\displaystyle O_{1}}
is space-like separated from
O
2
{\displaystyle O_{2}}
, then
[
A
(
O
1
)
,
A
(
O
2
)
]
=
0
{\displaystyle [{\mathcal {A}}(O_{1}),{\mathcal {A}}(O_{2})]=0}
.
Poincaré covariance: A strongly continuous unitary representation
U
(
P
)
{\displaystyle U({\mathcal {P}})}
of the Poincaré group
P
{\displaystyle {\mathcal {P}}}
on
H
{\displaystyle {\mathcal {H}}}
exists such that
A
(
g
O
)
=
U
(
g
)
A
(
O
)
U
(
g
)
∗
,
g
∈
P
.
{\displaystyle {\mathcal {A}}(gO)=U(g){\mathcal {A}}(O)U(g)^{*},\,\,g\in {\mathcal {P}}.}
Spectrum condition: The joint spectrum
S
p
(
P
)
{\displaystyle \mathrm {Sp} (P)}
of the energy-momentum operator
P
{\displaystyle P}
(i.e. the generator of space-time translations) is contained in the closed forward lightcone.
Existence of a vacuum vector: A cyclic and Poincaré-invariant vector
Ω
∈
H
{\displaystyle \Omega \in {\mathcal {H}}}
exists.
The net algebras
A
(
O
)
{\displaystyle {\mathcal {A}}(O)}
are called local algebras and the C* algebra
A
:=
⋃
O
∈
O
A
(
O
)
¯
{\displaystyle {\mathcal {A}}:={\overline {\bigcup _{O\in {\mathcal {O}}}{\mathcal {A}}(O)}}}
is called the quasilocal algebra.
== Category-theoretic formulation ==
Let Mink be the category of open subsets of Minkowski space M with inclusion maps as morphisms. We are given a covariant functor
A
{\displaystyle {\mathcal {A}}}
from Mink to uC*alg, the category of unital C* algebras, such that every morphism in Mink maps to a monomorphism in uC*alg (isotony).
The Poincaré group acts continuously on Mink. There exists a pullback of this action, which is continuous in the norm topology of
A
(
M
)
{\displaystyle {\mathcal {A}}(M)}
(Poincaré covariance).
Minkowski space has a causal structure. If an open set V lies in the causal complement of an open set U, then the image of the maps
A
(
i
U
,
U
∪
V
)
{\displaystyle {\mathcal {A}}(i_{U,U\cup V})}
and
A
(
i
V
,
U
∪
V
)
{\displaystyle {\mathcal {A}}(i_{V,U\cup V})}
commute (spacelike commutativity). If
U
¯
{\displaystyle {\bar {U}}}
is the causal completion of an open set U, then
A
(
i
U
,
U
¯
)
{\displaystyle {\mathcal {A}}(i_{U,{\bar {U}}})}
is an isomorphism (primitive causality).
A state with respect to a C*-algebra is a positive linear functional over it with unit norm. If we have a state over
A
(
M
)
{\displaystyle {\mathcal {A}}(M)}
, we can take the "partial trace" to get states associated with
A
(
U
)
{\displaystyle {\mathcal {A}}(U)}
for each open set via the net monomorphism. The states over the open sets form a presheaf structure.
According to the GNS construction, for each state, we can associate a Hilbert space representation of
A
(
M
)
.
{\displaystyle {\mathcal {A}}(M).}
Pure states correspond to irreducible representations and mixed states correspond to reducible representations. Each irreducible representation (up to equivalence) is called a superselection sector. We assume there is a pure state called the vacuum such that the Hilbert space associated with it is a unitary representation of the Poincaré group compatible with the Poincaré covariance of the net such that if we look at the Poincaré algebra, the spectrum with respect to energy-momentum (corresponding to spacetime translations) lies on and in the positive light cone. This is the vacuum sector.
== QFT in curved spacetime ==
More recently, the approach has been further implemented to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is in particular suitable to generalize the renormalization procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in presence of a black hole have been obtained.
== References ==
== Further reading ==
Haag, Rudolf; Kastler, Daniel (1964), "An Algebraic Approach to Quantum Field Theory", Journal of Mathematical Physics, 5 (7): 848–861, Bibcode:1964JMP.....5..848H, doi:10.1063/1.1704187, ISSN 0022-2488, MR 0165864
Haag, Rudolf (1996) [1992], Local Quantum Physics: Fields, Particles, Algebras, Theoretical and Mathematical Physics (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-61458-3, ISBN 978-3-540-61451-7, MR 1405610
Brunetti, Romeo; Fredenhagen, Klaus; Verch, Rainer (2003). "The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Field Theory". Communications in Mathematical Physics. 237 (1–2): 31–68. arXiv:math-ph/0112041. Bibcode:2003CMaPh.237...31B. doi:10.1007/s00220-003-0815-7. S2CID 13950246.
Brunetti, Romeo; Dütsch, Michael; Fredenhagen, Klaus (2009). "Perturbative Algebraic Quantum Field Theory and the Renormalization Groups". Advances in Theoretical and Mathematical Physics. 13 (5): 1541–1599. arXiv:0901.2038. doi:10.4310/ATMP.2009.v13.n5.a7. S2CID 15493763.
Bär, Christian; Fredenhagen, Klaus, eds. (2009). Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations. Lecture Notes in Physics. Vol. 786. Springer. doi:10.1007/978-3-642-02780-2. ISBN 978-3-642-02780-2.
Brunetti, Romeo; Dappiaggi, Claudio; Fredenhagen, Klaus; Yngvason, Jakob, eds. (2015). Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer. doi:10.1007/978-3-319-21353-8. ISBN 978-3-319-21353-8.
Rejzner, Kasia (2016). Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Mathematical Physics Studies. Springer. arXiv:1208.1428. Bibcode:2016paqf.book.....R. doi:10.1007/978-3-319-25901-7. ISBN 978-3-319-25901-7.
Hack, Thomas-Paul (2016). Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes. SpringerBriefs in Mathematical Physics. Vol. 6. Springer. arXiv:1506.01869. Bibcode:2016caaq.book.....H. doi:10.1007/978-3-319-21894-6. ISBN 978-3-319-21894-6. S2CID 119657309.
Dütsch, Michael (2019). From Classical Field Theory to Perturbative Quantum Field Theory. Progress in Mathematical Physics. Vol. 74. Birkhäuser. doi:10.1007/978-3-030-04738-2. ISBN 978-3-030-04738-2. S2CID 126907045.
Yau, Donald (2019). Homotopical Quantum Field Theory. World Scientific. arXiv:1802.08101. doi:10.1142/11626. ISBN 978-981-121-287-1. S2CID 119168109.
Dedushenko, Mykola (2023). "Snowmass white paper: The quest to define QFT". International Journal of Modern Physics A. 38 (4n05). arXiv:2203.08053. doi:10.1142/S0217751X23300028. S2CID 247450696.
== External links ==
Local Quantum Physics Crossroads 2.0 – A network of scientists working on local quantum physics
Papers – A database of preprints on algebraic QFT
Algebraic Quantum Field Theory – AQFT resources at the University of Hamburg | Wikipedia/Local_quantum_field_theory |
The BF model or BF theory is a topological field, which when quantized, becomes a topological quantum field theory. BF stands for background field B and F, as can be seen below, are also the variables appearing in the Lagrangian of the theory, which is helpful as a mnemonic device.
We have a 4-dimensional differentiable manifold M, a gauge group G, which has as "dynamical" fields a 2-form B taking values in the adjoint representation of G, and a connection form A for G.
The action is given by
S
=
∫
M
K
[
B
∧
F
]
{\displaystyle S=\int _{M}K[\mathbf {B} \wedge \mathbf {F} ]}
where K is an invariant nondegenerate bilinear form over
g
{\displaystyle {\mathfrak {g}}}
(if G is semisimple, the Killing form will do) and F is the curvature form
F
≡
d
A
+
A
∧
A
{\displaystyle \mathbf {F} \equiv d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} }
This action is diffeomorphically invariant and gauge invariant. Its Euler–Lagrange equations are
F
=
0
{\displaystyle \mathbf {F} =0}
(no curvature)
and
d
A
B
=
0
{\displaystyle d_{\mathbf {A} }\mathbf {B} =0}
(the covariant exterior derivative of B is zero).
In fact, it is always possible to gauge away any local degrees of freedom, which is why it is called a topological field theory.
However, if M is topologically nontrivial, A and B can have nontrivial solutions globally.
In fact, BF theory can be used to formulate discrete gauge theory. One can add additional twist terms allowed by group cohomology theory such as Dijkgraaf–Witten topological gauge theory. There are many kinds of modified BF theories as topological field theories, which give rise to link invariants in 3 dimensions, 4 dimensions, and other general dimensions.
== See also ==
Background field method
Barrett–Crane model
Dual graviton
Plebanski action
Spin foam
Tetradic Palatini action
== References ==
== External links ==
http://math.ucr.edu/home/baez/qg-fall2000/qg2.2.html | Wikipedia/BF_model |
In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation.
Liouville theory is defined for all complex values of the central charge
c
{\displaystyle c}
of its Virasoro symmetry algebra, but it is unitary only if
c
∈
(
1
,
+
∞
)
,
{\displaystyle c\in (1,+\infty ),}
and its classical limit is
c
→
+
∞
.
{\displaystyle c\to +\infty .}
Although it is an interacting theory with a continuous spectrum, Liouville theory has been solved. In particular, its three-point function on the sphere has been determined analytically.
== Introduction ==
Liouville theory describes the dynamics of a field
φ
{\displaystyle \varphi }
called the Liouville field, which is defined on a two-dimensional space. This field is not a free field due to the presence of an exponential potential
V
(
φ
)
=
e
2
b
φ
,
{\displaystyle V(\varphi )=e^{2b\varphi }\ ,}
where the parameter
b
{\displaystyle b}
is called the coupling constant. In a free field theory, the energy eigenvectors
e
2
α
φ
{\displaystyle e^{2\alpha \varphi }}
are linearly independent, and the momentum
α
{\displaystyle \alpha }
is conserved in interactions. In Liouville theory, momentum is not conserved.
Moreover, the potential reflects the energy eigenvectors before they reach
φ
=
+
∞
{\displaystyle \varphi =+\infty }
, and two eigenvectors are linearly dependent if their momenta are related by the reflection
α
→
Q
−
α
,
{\displaystyle \alpha \to Q-\alpha \ ,}
where the background charge is
Q
=
b
+
1
b
.
{\displaystyle Q=b+{\frac {1}{b}}\ .}
While the exponential potential breaks momentum conservation, it does not break conformal symmetry, and Liouville theory is a conformal field theory with the central charge
c
=
1
+
6
Q
2
.
{\displaystyle c=1+6Q^{2}\ .}
Under conformal transformations, an energy eigenvector with momentum
α
{\displaystyle \alpha }
transforms as a primary field with the conformal dimension
Δ
{\displaystyle \Delta }
by
Δ
=
α
(
Q
−
α
)
.
{\displaystyle \Delta =\alpha (Q-\alpha )\ .}
The central charge and conformal dimensions are invariant under the duality
b
→
1
b
,
{\displaystyle b\to {\frac {1}{b}}\ ,}
The correlation functions of Liouville theory are covariant under this duality, and under reflections of the momenta. These quantum symmetries of Liouville theory are however not manifest in the Lagrangian formulation, in particular the exponential potential is not invariant under the duality.
== Spectrum and correlation functions ==
=== Spectrum ===
The spectrum
S
{\displaystyle {\mathcal {S}}}
of Liouville theory is a diagonal combination of Verma modules of the Virasoro algebra,
S
=
∫
c
−
1
24
+
R
+
d
Δ
V
Δ
⊗
V
¯
Δ
,
{\displaystyle {\mathcal {S}}=\int _{{\frac {c-1}{24}}+\mathbb {R} _{+}}d\Delta \ {\mathcal {V}}_{\Delta }\otimes {\bar {\mathcal {V}}}_{\Delta }\ ,}
where
V
Δ
{\displaystyle {\mathcal {V}}_{\Delta }}
and
V
¯
Δ
{\displaystyle {\bar {\mathcal {V}}}_{\Delta }}
denote the same Verma module, viewed as a representation of the left- and right-moving Virasoro algebra respectively. In terms of momenta,
Δ
∈
c
−
1
24
+
R
+
{\displaystyle \Delta \in {\frac {c-1}{24}}+\mathbb {R} _{+}}
corresponds to
α
∈
Q
2
+
i
R
+
.
{\displaystyle \alpha \in {\frac {Q}{2}}+i\mathbb {R} _{+}.}
The reflection relation is responsible for the momentum taking values on a half-line, instead of a full line for a free theory.
Liouville theory is unitary if and only if
c
∈
(
1
,
+
∞
)
{\displaystyle c\in (1,+\infty )}
. The spectrum of Liouville theory does not include a vacuum state. A vacuum state can be defined, but it does not contribute to operator product expansions.
=== Fields and reflection relation ===
In Liouville theory, primary fields are usually parametrized by their momentum rather than their conformal dimension, and denoted
V
α
(
z
)
{\displaystyle V_{\alpha }(z)}
.
Both fields
V
α
(
z
)
{\displaystyle V_{\alpha }(z)}
and
V
Q
−
α
(
z
)
{\displaystyle V_{Q-\alpha }(z)}
correspond to the primary state of the representation
V
Δ
⊗
V
¯
Δ
{\displaystyle {\mathcal {V}}_{\Delta }\otimes {\bar {\mathcal {V}}}_{\Delta }}
, and are related by the reflection relation
V
α
(
z
)
=
R
(
α
)
V
Q
−
α
(
z
)
,
{\displaystyle V_{\alpha }(z)=R(\alpha )V_{Q-\alpha }(z)\ ,}
where the reflection coefficient is
R
(
α
)
=
±
λ
Q
−
2
α
Γ
(
b
(
2
α
−
Q
)
)
Γ
(
1
b
(
2
α
−
Q
)
)
Γ
(
b
(
Q
−
2
α
)
)
Γ
(
1
b
(
Q
−
2
α
)
)
.
{\displaystyle R(\alpha )=\pm \lambda ^{Q-2\alpha }{\frac {\Gamma (b(2\alpha -Q))\Gamma ({\frac {1}{b}}(2\alpha -Q))}{\Gamma (b(Q-2\alpha ))\Gamma ({\frac {1}{b}}(Q-2\alpha ))}}\ .}
(The sign is
+
1
{\displaystyle +1}
if
c
∈
(
−
∞
,
1
)
{\displaystyle c\in (-\infty ,1)}
and
−
1
{\displaystyle -1}
otherwise, and the normalization parameter
λ
{\displaystyle \lambda }
is arbitrary.)
=== Correlation functions and DOZZ formula ===
For
c
∉
(
−
∞
,
1
)
{\displaystyle c\notin (-\infty ,1)}
, the three-point structure constant is given by the DOZZ formula (for Dorn–Otto and Zamolodchikov–Zamolodchikov),
C
α
1
,
α
2
,
α
3
=
[
b
2
b
−
2
b
λ
]
Q
−
α
1
−
α
2
−
α
3
Υ
b
′
(
0
)
Υ
b
(
2
α
1
)
Υ
b
(
2
α
2
)
Υ
b
(
2
α
3
)
Υ
b
(
α
1
+
α
2
+
α
3
−
Q
)
Υ
b
(
α
1
+
α
2
−
α
3
)
Υ
b
(
α
2
+
α
3
−
α
1
)
Υ
b
(
α
3
+
α
1
−
α
2
)
,
{\displaystyle C_{\alpha _{1},\alpha _{2},\alpha _{3}}={\frac {\left[b^{{\frac {2}{b}}-2b}\lambda \right]^{Q-\alpha _{1}-\alpha _{2}-\alpha _{3}}\Upsilon _{b}'(0)\Upsilon _{b}(2\alpha _{1})\Upsilon _{b}(2\alpha _{2})\Upsilon _{b}(2\alpha _{3})}{\Upsilon _{b}(\alpha _{1}+\alpha _{2}+\alpha _{3}-Q)\Upsilon _{b}(\alpha _{1}+\alpha _{2}-\alpha _{3})\Upsilon _{b}(\alpha _{2}+\alpha _{3}-\alpha _{1})\Upsilon _{b}(\alpha _{3}+\alpha _{1}-\alpha _{2})}}\ ,}
where the special function
Υ
b
{\displaystyle \Upsilon _{b}}
is a kind of multiple gamma function.
For
c
∈
(
−
∞
,
1
)
{\displaystyle c\in (-\infty ,1)}
, the three-point structure constant is
C
^
α
1
,
α
2
,
α
3
=
[
(
i
b
)
2
b
−
2
b
λ
]
Q
−
α
1
−
α
2
−
α
3
Υ
^
b
(
0
)
Υ
^
b
(
2
α
1
)
Υ
^
b
(
2
α
2
)
Υ
^
b
(
2
α
3
)
Υ
^
b
(
α
1
+
α
2
+
α
3
−
Q
)
Υ
^
b
(
α
1
+
α
2
−
α
3
)
Υ
^
b
(
α
2
+
α
3
−
α
1
)
Υ
^
b
(
α
3
+
α
1
−
α
2
)
,
{\displaystyle {\hat {C}}_{\alpha _{1},\alpha _{2},\alpha _{3}}={\frac {\left[(ib)^{{\frac {2}{b}}-2b}\lambda \right]^{Q-\alpha _{1}-\alpha _{2}-\alpha _{3}}{\hat {\Upsilon }}_{b}(0){\hat {\Upsilon }}_{b}(2\alpha _{1}){\hat {\Upsilon }}_{b}(2\alpha _{2}){\hat {\Upsilon }}_{b}(2\alpha _{3})}{{\hat {\Upsilon }}_{b}(\alpha _{1}+\alpha _{2}+\alpha _{3}-Q){\hat {\Upsilon }}_{b}(\alpha _{1}+\alpha _{2}-\alpha _{3}){\hat {\Upsilon }}_{b}(\alpha _{2}+\alpha _{3}-\alpha _{1}){\hat {\Upsilon }}_{b}(\alpha _{3}+\alpha _{1}-\alpha _{2})}}\ ,}
where
Υ
^
b
(
x
)
=
1
Υ
i
b
(
−
i
x
+
i
b
)
.
{\displaystyle {\hat {\Upsilon }}_{b}(x)={\frac {1}{\Upsilon _{ib}(-ix+ib)}}\ .}
N
{\displaystyle N}
-point functions on the sphere can be expressed in terms of three-point structure constants, and conformal blocks. An
N
{\displaystyle N}
-point function may have several different expressions: that they agree is equivalent to crossing symmetry of the four-point function, which has been checked numerically and proved analytically.
Liouville theory exists not only on the sphere, but also on any Riemann surface of genus
g
≥
1
{\displaystyle g\geq 1}
. Technically, this is equivalent to the modular invariance of the torus one-point function. Due to remarkable identities of conformal blocks and structure constants, this modular invariance property can be deduced from crossing symmetry of the sphere four-point function.
=== Uniqueness of Liouville theory ===
Using the conformal bootstrap approach, Liouville theory can be shown to be the unique conformal field theory such that
the spectrum is a continuum, with no multiplicities higher than one,
the correlation functions depend analytically on
b
{\displaystyle b}
and the momenta,
degenerate fields exist.
== Lagrangian formulation ==
=== Action and equation of motion ===
Liouville theory is defined by the local action
S
[
φ
]
=
1
4
π
∫
d
2
x
g
(
g
μ
ν
∂
μ
φ
∂
ν
φ
+
Q
R
φ
+
λ
′
e
2
b
φ
)
,
{\displaystyle S[\varphi ]={\frac {1}{4\pi }}\int d^{2}x\,{\sqrt {g}}(g^{\mu \nu }\partial _{\mu }\varphi \partial _{\nu }\varphi +QR\varphi +\lambda 'e^{2b\varphi })\ ,}
where
g
μ
ν
{\displaystyle g_{\mu \nu }}
is the metric of the two-dimensional space on which the theory is formulated,
R
{\displaystyle R}
is the Ricci scalar of that space, and
φ
{\displaystyle \varphi }
is the Liouville field. The parameter
λ
′
{\displaystyle \lambda '}
, which is sometimes called the cosmological constant, is related to the parameter
λ
{\displaystyle \lambda }
that appears in correlation functions by
λ
′
=
4
Γ
(
1
−
b
2
)
Γ
(
b
2
)
λ
b
.
{\displaystyle \lambda '=4{\frac {\Gamma (1-b^{2})}{\Gamma (b^{2})}}\lambda ^{b}.}
The equation of motion associated to this action is
Δ
φ
(
x
)
=
1
2
Q
R
(
x
)
+
λ
′
b
e
2
b
φ
(
x
)
,
{\displaystyle \Delta \varphi (x)={\frac {1}{2}}QR(x)+\lambda 'be^{2b\varphi (x)}\ ,}
where
Δ
=
|
g
|
−
1
/
2
∂
μ
(
|
g
|
1
/
2
g
μ
ν
∂
ν
)
{\displaystyle \Delta =|g|^{-1/2}\partial _{\mu }(|g|^{1/2}g^{\mu \nu }\partial _{\nu })}
is the Laplace–Beltrami operator. If
g
μ
ν
{\displaystyle g_{\mu \nu }}
is the Euclidean metric, this equation reduces to
(
∂
2
∂
x
1
2
+
∂
2
∂
x
2
2
)
φ
(
x
1
,
x
2
)
=
λ
′
b
e
2
b
φ
(
x
1
,
x
2
)
,
{\displaystyle \left({\frac {\partial ^{2}}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}}{\partial x_{2}^{2}}}\right)\varphi (x_{1},x_{2})=\lambda 'be^{2b\varphi (x_{1},x_{2})}\ ,}
which is equivalent to Liouville's equation.
Once compactified on a cylinder, Liouville field theory can be equivalently formulated as a worldline theory.
=== Conformal symmetry ===
Using a complex coordinate system
z
{\displaystyle z}
and a Euclidean metric
g
μ
ν
d
x
μ
d
x
ν
=
d
z
d
z
¯
,
{\displaystyle g_{\mu \nu }dx^{\mu }dx^{\nu }=dzd{\bar {z}},}
the energy–momentum tensor's components obey
T
z
z
¯
=
T
z
¯
z
=
0
,
∂
z
¯
T
z
z
=
0
,
∂
z
T
z
¯
z
¯
=
0
.
{\displaystyle T_{z{\bar {z}}}=T_{{\bar {z}}z}=0\;,\quad \partial _{\bar {z}}T_{zz}=0\;,\quad \partial _{z}T_{{\bar {z}}{\bar {z}}}=0\ .}
The non-vanishing components are
T
=
T
z
z
=
(
∂
z
φ
)
2
+
Q
∂
z
2
φ
,
T
¯
=
T
z
¯
z
¯
=
(
∂
z
¯
φ
)
2
+
Q
∂
z
¯
2
φ
.
{\displaystyle T=T_{zz}=(\partial _{z}\varphi )^{2}+Q\partial _{z}^{2}\varphi \;,\quad {\bar {T}}=T_{{\bar {z}}{\bar {z}}}=(\partial _{\bar {z}}\varphi )^{2}+Q\partial _{\bar {z}}^{2}\varphi \ .}
Each one of these two components generates a Virasoro algebra with the central charge
c
=
1
+
6
Q
2
.
{\displaystyle c=1+6Q^{2}.}
For both of these Virasoro algebras, a field
e
2
α
φ
{\displaystyle e^{2\alpha \varphi }}
is a primary field with the conformal dimension
Δ
=
α
(
Q
−
α
)
.
{\displaystyle \Delta =\alpha (Q-\alpha ).}
For the theory to have conformal invariance, the field
e
2
b
φ
{\displaystyle e^{2b\varphi }}
that appears in the action must be marginal, i.e. have the conformal dimension
Δ
(
b
)
=
1.
{\displaystyle \Delta (b)=1.}
This leads to the relation
Q
=
b
+
1
b
{\displaystyle Q=b+{\frac {1}{b}}}
between the background charge and the coupling constant. If this relation is obeyed, then
e
2
b
φ
{\displaystyle e^{2b\varphi }}
is actually exactly marginal, and the theory is conformally invariant.
=== Path integral ===
The path integral representation of an
N
{\displaystyle N}
-point correlation function of primary fields is
⟨
∏
i
=
1
N
V
α
i
(
z
i
)
⟩
=
∫
D
φ
e
−
S
[
φ
]
∏
i
=
1
N
e
2
α
i
φ
(
z
i
)
.
{\displaystyle \left\langle \prod _{i=1}^{N}V_{\alpha _{i}}(z_{i})\right\rangle =\int D\varphi \ e^{-S[\varphi ]}\prod _{i=1}^{N}e^{2\alpha _{i}\varphi (z_{i})}\ .}
It has been difficult to define and to compute this path integral. In the path integral representation, it is not obvious that Liouville theory has exact conformal invariance, and it is not manifest that correlation functions are invariant under
b
→
b
−
1
{\displaystyle b\to b^{-1}}
and obey the reflection relation. Nevertheless, the path integral representation can be used for computing the residues of correlation functions at some of their poles as Dotsenko–Fateev integrals in the Coulomb gas formalism, and this is how the DOZZ formula was first guessed in the 1990s. It is only in the 2010s that a rigorous probabilistic construction of the path integral was found, which led to a proof of the DOZZ formula and the conformal bootstrap.
== Relations with other conformal field theories ==
=== Some limits of Liouville theory ===
When the central charge and conformal dimensions are sent to the relevant discrete values, correlation functions of Liouville theory reduce to correlation functions of diagonal (A-series) Virasoro minimal models.
On the other hand, when the central charge is sent to one while conformal dimensions stay continuous, Liouville theory tends to Runkel–Watts theory, a nontrivial conformal field theory (CFT) with a continuous spectrum whose three-point function is not analytic as a function of the momenta. Generalizations of Runkel-Watts theory are obtained from Liouville theory by taking limits of the type
b
2
∉
R
,
b
2
→
Q
<
0
{\displaystyle b^{2}\notin \mathbb {R} ,b^{2}\to \mathbb {Q} _{<0}}
. So, for
b
2
∈
Q
<
0
{\displaystyle b^{2}\in \mathbb {Q} _{<0}}
, two distinct CFTs with the same spectrum are known: Liouville theory, whose three-point function is analytic, and another CFT with a non-analytic three-point function.
=== WZW models ===
Liouville theory can be obtained from the
S
L
2
(
R
)
{\displaystyle SL_{2}(\mathbb {R} )}
Wess–Zumino–Witten model by a quantum Drinfeld–Sokolov reduction. Moreover, correlation functions of the
H
3
+
{\displaystyle H_{3}^{+}}
model (the Euclidean version of the
S
L
2
(
R
)
{\displaystyle SL_{2}(\mathbb {R} )}
WZW model) can be expressed in terms of correlation functions of Liouville theory. This is also true of correlation functions of the 2d black hole
S
L
2
/
U
1
{\displaystyle SL_{2}/U_{1}}
coset model. Moreover, there exist theories that continuously interpolate between Liouville theory and the
H
3
+
{\displaystyle H_{3}^{+}}
model.
=== Conformal Toda theory ===
Liouville theory is the simplest example of a Toda field theory, associated to the
A
1
{\displaystyle A_{1}}
Cartan matrix. More general conformal Toda theories can be viewed as generalizations of Liouville theory, whose Lagrangians involve several bosons rather than one boson
φ
{\displaystyle \varphi }
, and whose symmetry algebras are W-algebras rather than the Virasoro algebra.
=== Supersymmetric Liouville theory ===
Liouville theory admits two different supersymmetric extensions called
N
=
1
{\displaystyle {\mathcal {N}}=1}
supersymmetric Liouville theory and
N
=
2
{\displaystyle {\mathcal {N}}=2}
supersymmetric Liouville theory.
== Relations with integrable models ==
=== Sinh-Gordon model ===
In flat space, the sinh-Gordon model is defined by the local action:
S
[
φ
]
=
1
4
π
∫
d
2
x
(
∂
μ
φ
∂
μ
φ
+
λ
cosh
(
2
b
φ
)
)
.
{\displaystyle S[\varphi ]={\frac {1}{4\pi }}\int d^{2}x\left(\partial ^{\mu }\varphi \partial _{\mu }\varphi +\lambda \cosh(2b\varphi )\right).}
The corresponding classical equation of motion is the sinh-Gordon equation.
The model can be viewed as a perturbation of Liouville theory. The model's exact S-matrix is known in the weak coupling regime
0
<
b
<
1
{\displaystyle 0<b<1}
, and it is formally invariant under
b
→
b
−
1
{\displaystyle b\to b^{-1}}
. However, it has been argued that the model itself is not invariant.
== Applications ==
=== Liouville gravity ===
In two dimensions, Liouville theory can be used to build a quantum theory of gravity called Liouville gravity. It should not be confused with the CGHS model or Jackiw–Teitelboim gravity.
In two dimensions, the Einstein-Hilbert action is topological, i.e. it is proportional to the Euler characteristic. Nevertheless, after quantization, general relativity is no longer topological, because of the Weyl anomaly: under a rescaling of the metric
g
↦
e
ϕ
g
{\displaystyle g\mapsto e^{\phi }g}
, while the action is invariant, the functional integration measure is not, and gives rise to a term proportional to the Liouville action for
ϕ
{\displaystyle \phi }
. This leads to the construction of Liouville gravity as a product of three CFTs: Liouville theory for the gravitational sector, Faddeev-Popov ghosts for Weyl invariance (viewed as a gauge symmetry), and an arbitrary CFT that describes matter. The central charges of these CFTs must sum to zero in order to cancel the Weyl anomaly, and ensure that the quantum theory is topological.
The observables of Liouville gravity are correlation numbers: correlation functions of the product CFT, integrated over the moduli. Correlation numbers can be computed explicitly in some examples, such as the Virasoro minimal string. Correlation numbers at fixed Euler characteristic are the coefficients of quantum gravity correlators, when expanded in powers of the gravitational constant.
=== String theory ===
Liouville theory appears in the context of string theory when trying to formulate a non-critical version of the theory in the path integral formulation. The theory also appears as the description of bosonic string theory in two spacetime dimensions with a linear dilaton and a tachyon background. The tachyon field equation of motion in the linear dilaton background requires it to take an exponential solution. The Polyakov action in this background is then identical to Liouville field theory, with the linear dilaton being responsible for the background charge term while the tachyon contributing the exponential potential.
=== Random energy models ===
There is an exact mapping between Liouville theory with
c
≥
25
{\displaystyle c\geq 25}
, and certain log-correlated random energy models. These models describe a thermal particle in a random potential that is logarithmically correlated. In two dimensions, such potential coincides with the Gaussian free field. In that case, certain correlation functions between primary fields in the Liouville theory are mapped to correlation functions of the Gibbs measure of the particle. This has applications to extreme value statistics of the two-dimensional Gaussian free field, and allows to predict certain universal properties of the log-correlated random energy models (in two dimensions and beyond).
=== Other applications ===
Liouville theory is related to other subjects in physics and mathematics, such as three-dimensional general relativity in negatively curved spaces, the uniformization problem of Riemann surfaces, and other problems in conformal mapping. It is also related to instanton partition functions in a certain four-dimensional superconformal gauge theories by the AGT correspondence.
== Naming confusion for c ≤ 1 ==
Liouville theory with
c
≤
1
{\displaystyle c\leq 1}
first appeared as a model of time-dependent string theory under the name timelike Liouville theory.
It has also been called a generalized minimal model. It was first called Liouville theory when it was found to actually exist, and to be spacelike rather than timelike. As of 2022, not one of these three names is universally accepted.
== References ==
== External links ==
Mathematicians Prove 2D Version of Quantum Gravity Really Works, Quanta Magazine article by Charlie Wood, June 2021.
An Introduction to Liouville Theory, Talk at Institute for Advanced Study by Antti Kupiainen, May 2018. | Wikipedia/Liouville_field_theory |
Technicolor theories are models of physics beyond the Standard Model that address electroweak gauge symmetry breaking, the mechanism through which W and Z bosons acquire masses. Early technicolor theories were modelled on quantum chromodynamics (QCD), the "color" theory of the strong nuclear force, which inspired their name.
Instead of introducing elementary Higgs bosons to explain observed phenomena, technicolor models were introduced to dynamically generate masses for the W and Z bosons through new gauge interactions. Although asymptotically free at very high energies, these interactions must become strong and confining (and hence unobservable) at lower energies that have been experimentally probed. This dynamical approach is natural and avoids issues of quantum triviality and the hierarchy problem of the Standard Model.
However, since the Higgs boson discovery at the
CERN LHC in 2012, the original models are largely ruled out. Nonetheless, it remains a possibility that the Higgs boson is a composite state.
In order to produce quark and lepton masses, technicolor or composite Higgs models have to be "extended" by additional gauge interactions. Particularly when modelled on QCD, extended technicolor was challenged by experimental constraints on flavor-changing neutral current and precision electroweak measurements. The specific extensions of particle dynamics for technicolor
or composite Higgs bosons are unknown.
Much technicolor research focuses on exploring strongly interacting gauge theories other than QCD, in order to evade some of these challenges. A particularly active framework is "walking" technicolor, which exhibits nearly conformal behavior caused by an infrared fixed point with strength just above that necessary for spontaneous chiral symmetry breaking. Whether walking can occur and lead to agreement with precision electroweak measurements is being studied through non-perturbative lattice simulations.
Experiments at the Large Hadron Collider have discovered the mechanism responsible for electroweak symmetry breaking, i.e., the Higgs boson, with mass approximately 125 GeV/c2; such a particle is not generically predicted by technicolor models. However,
the Higgs boson may be a composite state, e.g., built of top and anti-top quarks
as in the Bardeen–Hill–Lindner theory.
Composite Higgs models are generally solved by the top quark infrared fixed point,
and may require a new dynamics at extremely high energies such as topcolor.
== Introduction ==
The mechanism for the breaking of electroweak gauge symmetry in the Standard Model of elementary particle interactions remains unknown. The breaking must be spontaneous, meaning that the underlying theory manifests the symmetry exactly (the gauge-boson fields are massless in the equations of motion), but the solutions (the ground state and the excited states) do not. In particular, the physical W and Z gauge bosons become massive. This phenomenon, in which the W and Z bosons also acquire an extra polarization state, is called the "Higgs mechanism". Despite the precise agreement of the electroweak theory with experiment at energies accessible so far, the necessary ingredients for the symmetry breaking remain hidden, yet to be revealed at higher energies.
The simplest mechanism of electroweak symmetry breaking introduces a single complex field and predicts the existence of the Higgs boson. Typically, the Higgs boson is "unnatural" in the sense that quantum mechanical fluctuations produce corrections to its mass that lift it to such high values that it cannot play the role for which it was introduced. Unless the Standard Model breaks down at energies less than a few TeV, the Higgs mass can be kept small only by a delicate fine-tuning of parameters.
Technicolor avoids this problem by hypothesizing a new gauge interaction coupled to new massless fermions. This interaction is asymptotically free at very high energies and becomes strong and confining as the energy decreases to the electroweak scale of 246 GeV. These strong forces spontaneously break the massless fermions' chiral symmetries, some of which are weakly gauged as part of the Standard Model. This is the dynamical version of the Higgs mechanism. The electroweak gauge symmetry is thus broken, producing masses for the W and Z bosons.
The new strong interaction leads to a host of new composite, short-lived particles at energies accessible at the Large Hadron Collider (LHC). This framework is natural because there are no elementary Higgs bosons and, hence, no fine-tuning of parameters. Quark and lepton masses also break the electroweak gauge symmetries, so they, too, must arise spontaneously. A mechanism for incorporating this feature is known as extended technicolor. Technicolor and extended technicolor face a number of phenomenological challenges, in particular issues of flavor-changing neutral currents, precision electroweak tests, and the top quark mass. Technicolor models also do not generically predict Higgs-like bosons as light as 125 GeV/c2; such a particle was discovered by experiments at the Large Hadron Collider in 2012. Some of these issues can be addressed with a class of theories known as "walking technicolor".
== Early technicolor ==
Technicolor is the name given to the theory of electroweak symmetry breaking by new strong gauge-interactions whose characteristic energy scale ΛTC is the weak scale itself, ΛTC ≈ FEW ≡ 246 GeV . The guiding principle of technicolor is "naturalness": basic physical phenomena should not require fine-tuning of the parameters in the Lagrangian that describes them. What constitutes fine-tuning is to some extent a subjective matter, but a theory with elementary scalar particles typically is very finely tuned (unless it is supersymmetric). The quadratic divergence in the scalar's mass requires adjustments of a part in
O
(
M
b
a
r
e
2
M
p
h
y
s
i
c
a
l
2
)
{\displaystyle {\mathcal {O}}\left({\frac {M_{\mathrm {bare} }^{2}}{M_{\mathrm {physical} }^{2}}}\right)}
, where Mbare is the cutoff of the theory, the energy scale at which the theory changes in some essential way. In the standard electroweak model with Mbare ~ 1015 GeV (the grand-unification mass scale), and with the Higgs boson mass Mphysical = 100–500 GeV, the mass is tuned to at least a part in 1025.
By contrast, a natural theory of electroweak symmetry breaking is an asymptotically free gauge theory with fermions as the only matter fields. The technicolor gauge group GTC is often assumed to be SU(NTC). Based on analogy with quantum chromodynamics (QCD), it is assumed that there are one or more doublets of massless Dirac "technifermions" transforming vectorially under the same complex representation of GTC,
T
i
L
,
R
=
(
U
i
,
D
i
)
L
,
R
,
for
i
=
1
,
2
,
.
.
.
,
1
2
N
f
{\displaystyle T_{i\,\mathrm {L,R} }=(U_{i},D_{i})_{\mathrm {L,R} }\,,{\text{ for }}i=1,2,...,{\tfrac {1}{2}}N_{\mathrm {f} }}
. Thus, there is a chiral symmetry of these fermions, e.g., SU(Nf)L ⊗ SU(Nf)R, if they all transform according to the same complex representation of GTC. Continuing the analogy with QCD, the running gauge coupling αTC(μ) triggers spontaneous chiral symmetry breaking, the technifermions acquire a dynamical mass, and a number of massless Goldstone bosons result. If the technifermions transform under [SU(2) ⊗ U(1)]EW as left-handed doublets and right-handed singlets, three linear combinations of these Goldstone bosons couple to three of the electroweak gauge currents.
In 1973 Jackiw and Johnson and Cornwall and Norton studied the possibility that a (non-vectorial) gauge interaction of fermions can break itself; i.e., is strong enough to form a Goldstone boson coupled to the gauge current. Using Abelian gauge models, they showed that, if such a Goldstone boson is formed, it is "eaten" by the Higgs mechanism, becoming the longitudinal component of the now massive gauge boson. Technically, the polarization function Π(p2) appearing in the gauge boson propagator,
Δ
μ
ν
=
[
p
μ
p
ν
p
2
−
g
μ
ν
]
p
2
[
1
−
g
2
Π
(
p
2
)
]
{\displaystyle \Delta _{\mu \nu }={\frac {\left[{\frac {p_{\mu }p_{\nu }}{p^{2}}}-g_{\mu \nu }\right]}{~p^{2}\left[1-g^{2}\Pi \left(p^{2}\right)\right]~}}}
develops a pole at p2 = 0 with residue F2, the square of the Goldstone boson's decay constant, and the gauge boson acquires mass M ≈ g F . In 1973, Weinstein showed that composite Goldstone bosons whose constituent fermions transform in the "standard" way under SU(2) ⊗ U(1) generate the weak boson masses
(
1
)
M
W
±
=
1
2
g
F
E
W
and
M
Z
=
1
2
g
2
+
g
′
2
F
E
W
≡
M
W
cos
θ
W
.
{\displaystyle (1)\qquad M_{\mathrm {W^{\pm }} }={\frac {1}{2}}g\,F_{\mathrm {EW} }\quad {\text{ and }}\quad M_{\mathrm {Z} }={\frac {1}{2}}{\sqrt {g^{2}+{g'}^{2}}}F_{\mathrm {EW} }\equiv {\frac {M_{\mathrm {W} }}{\cos \theta _{\mathrm {W} }}}.}
This standard-model relation is achieved with elementary Higgs bosons in electroweak doublets; it is verified experimentally to better than 1%. Here, g and g′ are SU(2) and U(1) gauge couplings and
tan
θ
W
=
g
′
g
{\displaystyle \tan \theta _{\mathrm {W} }={\frac {g'}{g}}}
defines the weak mixing angle.
The important idea of a new strong gauge interaction of massless fermions at the electroweak scale FEW driving the spontaneous breakdown of its global chiral symmetry, of which an SU(2) ⊗ U(1) subgroup is weakly gauged, was first proposed in 1979 by Weinberg. This "technicolor" mechanism is natural in that no fine-tuning of parameters is necessary.
== Extended technicolor ==
Elementary Higgs bosons perform another important task. In the Standard Model, quarks and leptons are necessarily massless because they transform under SU(2) ⊗ U(1) as left-handed doublets and right-handed singlets. The Higgs doublet couples to these fermions. When it develops its vacuum expectation value, it transmits this electroweak breaking to the quarks and leptons, giving them their observed masses. (In general, electroweak-eigenstate fermions are not mass eigenstates, so this process also induces the mixing matrices observed in charged-current weak interactions.)
In technicolor, something else must generate the quark and lepton masses. The only natural possibility, one avoiding the introduction of elementary scalars, is to enlarge GTC to allow technifermions to couple to quarks and leptons. This coupling is induced by gauge bosons of the enlarged group. The picture, then, is that there is a large "extended technicolor" (ETC) gauge group GETC ⊃ GTC in which technifermions, quarks, and leptons live in the same representations. At one or more high scales ΛETC, GETC is broken down to GTC, and quarks and leptons emerge as the TC-singlet fermions. When αTC(μ) becomes strong at scale ΛTC ≈ FEW, the fermionic condensate
⟨
T
¯
T
⟩
TC
≈
4
π
F
EW
3
{\displaystyle \langle {\bar {T}}T\rangle _{\text{TC}}\approx 4\pi F_{\text{EW}}^{3}}
forms. (The condensate is the vacuum expectation value of the technifermion bilinear
T
¯
T
{\displaystyle {\bar {T}}T}
. The estimate here is based on naive dimensional analysis of the quark condensate in QCD, expected to be correct as an order of magnitude.) Then, the transitions
q
L
(
o
r
ℓ
L
)
→
T
L
→
T
R
→
q
R
(
o
r
ℓ
R
)
{\displaystyle q_{\text{L}}(\mathrm {or} \,\,\ell _{\text{L}})\rightarrow T_{\text{L}}\rightarrow T_{\text{R}}\rightarrow q_{\text{R}}\,(\mathrm {or} \,\,\ell _{\text{R}})}
can proceed through the technifermion's dynamical mass by the emission and reabsorption of ETC bosons whose masses METC ≈ gETC ΛETC are much greater than ΛTC. The quarks and leptons develop masses given approximately by
(
2
)
m
q
,
ℓ
(
M
ETC
)
≈
g
ETC
2
⟨
T
¯
T
⟩
ETC
M
ETC
2
≈
4
π
F
EW
3
Λ
ETC
2
.
{\displaystyle (2)\qquad m_{q,\ell }(M_{\text{ETC}})\approx {\frac {g_{\text{ETC}}^{2}\langle {\bar {T}}T\rangle _{\text{ETC}}}{M_{\text{ETC}}^{2}}}\approx {\frac {4\pi F_{\text{EW}}^{3}}{\Lambda _{\text{ETC}}^{2}}}\,.}
Here,
⟨
T
¯
T
⟩
ETC
{\displaystyle \langle {\bar {T}}T\rangle _{\text{ETC}}}
is the technifermion condensate renormalized at the ETC boson mass scale,
(
3
)
⟨
T
¯
T
⟩
ETC
=
exp
(
∫
Λ
TC
M
ETC
d
μ
μ
γ
m
(
μ
)
)
⟨
T
¯
T
⟩
TC
,
{\displaystyle (3)\qquad \langle {\bar {T}}T\rangle _{\text{ETC}}=\exp {\left(\int _{\Lambda _{\text{TC}}}^{M_{\text{ETC}}}{\frac {d\mu }{\mu }}\gamma _{m}(\mu )\right)}\,\langle {\bar {T}}T\rangle _{\text{TC}}\,,}
where γm(μ) is the anomalous dimension of the technifermion bilinear
T
¯
T
{\displaystyle {\bar {T}}T}
at the scale μ. The second estimate in Eq. (2) depends on the assumption that, as happens in QCD, αTC(μ) becomes weak not far above ΛTC, so that the anomalous dimension γm of
T
¯
T
{\displaystyle {\bar {T}}T}
is small there. Extended technicolor was introduced in 1979 by Dimopoulos and Susskind, and by Eichten and Lane. For a quark of mass mq ≈ 1 GeV, and with ΛTC ≈ 246 GeV, one estimates ΛETC ≈ 15 TeV. Therefore, assuming that
g
ETC
2
≳
1
{\displaystyle g_{\text{ETC}}^{2}\gtrsim 1}
, METC will be at least this large.
In addition to the ETC proposal for quark and lepton masses, Eichten and Lane observed that the size of the ETC representations required to generate all quark and lepton masses suggests that there will be more than one electroweak doublet of technifermions. If so, there will be more (spontaneously broken) chiral symmetries and therefore more Goldstone bosons than are eaten by the Higgs mechanism. These must acquire mass by virtue of the fact that the extra chiral symmetries are also explicitly broken, by the standard-model interactions and the ETC interactions. These "pseudo-Goldstone bosons" are called technipions, πT. An application of Dashen's theorem gives for the ETC contribution to their mass
(
4
)
F
EW
2
M
π
T
2
≈
g
ETC
2
⟨
T
¯
T
T
¯
T
⟩
ETC
M
ETC
2
≈
16
π
2
F
E
W
6
Λ
ETC
2
.
{\displaystyle (4)\qquad F_{\text{EW}}^{2}M_{\pi T}^{2}\approx {\frac {g_{\text{ETC}}^{2}\langle {\bar {T}}T{\bar {T}}T\rangle _{\text{ETC}}}{M_{\text{ETC}}^{2}}}\approx {\frac {16\pi ^{2}F_{EW}^{6}}{\Lambda _{\text{ETC}}^{2}}}\,.}
The second approximation in Eq. (4) assumes that
⟨
T
¯
T
T
¯
T
⟩
E
T
C
≈
⟨
T
¯
T
⟩
E
T
C
2
{\displaystyle \langle {\bar {T}}T{\bar {T}}T\rangle _{ETC}\approx \langle {\bar {T}}T\rangle _{ETC}^{2}}
. For FEW ≈ ΛTC ≈ 246 GeV and ΛETC ≈ 15 TeV, this contribution to MπT is about 50 GeV. Since ETC interactions generate
m
q
,
ℓ
{\displaystyle m_{q,\ell }}
and the coupling of technipions to quark and lepton pairs, one expects the couplings to be Higgs-like; i.e., roughly proportional to the masses of the quarks and leptons. This means that technipions are expected to predominately decay to the heaviest possible
q
¯
q
{\displaystyle {\bar {q}}q}
and
ℓ
¯
ℓ
{\displaystyle {\bar {\ell }}\ell }
pairs.
Perhaps the most important restriction on the ETC framework for quark mass generation is that ETC interactions are likely to induce flavor-changing neutral current processes such as μ → e + γ, KL → μ + e, and
|
Δ
S
|
=
2
and
|
Δ
B
′
|
=
2
{\displaystyle \left|\,\operatorname {\Delta } S\,\right|=2{\text{ and }}\left|\,\operatorname {\Delta } B'\,\right|=2}
interactions that induce
K
0
↔
K
¯
0
{\displaystyle {\text{K}}^{0}\leftrightarrow {\bar {\text{K}}}^{0}}
and
B
0
↔
B
¯
0
{\displaystyle {\text{B}}^{0}\leftrightarrow {\bar {\text{B}}}^{0}}
mixing. The reason is that the algebra of the ETC currents involved in
m
q
,
ℓ
{\displaystyle m_{q,\ell }}
generation imply
q
¯
q
′
{\displaystyle {\bar {q}}q^{\prime }}
and
ℓ
¯
ℓ
′
{\displaystyle {\bar {\ell }}\ell ^{\prime }}
ETC currents which, when written in terms of fermion mass eigenstates, have no reason to conserve flavor. The strongest constraint comes from requiring that ETC interactions mediating
K
↔
K
¯
{\displaystyle {\text{K}}\leftrightarrow {\bar {\text{K}}}}
mixing contribute less than the Standard Model. This implies an effective ΛETC greater than 1000 TeV. The actual ΛETC may be reduced somewhat if CKM-like mixing angle factors are present. If these interactions are CP-violating, as they well may be, the constraint from the ε-parameter is that the effective ΛETC > 104 TeV. Such huge ETC mass scales imply tiny quark and lepton masses and ETC contributions to MπT of at most a few GeV, in conflict with LEP searches for πT at the Z0.
Extended technicolor is a very ambitious proposal, requiring that quark and lepton masses and mixing angles arise from experimentally accessible interactions. If there exists a successful model, it would not only predict the masses and mixings of quarks and leptons (and technipions), it would explain why there are three families of each: they are the ones that fit into the ETC representations of q,
ℓ
{\displaystyle \ell }
, and T. It should not be surprising that the construction of a successful model has proven to be very difficult.
== Walking technicolor ==
Since quark and lepton masses are proportional to the bilinear technifermion condensate divided by the ETC mass scale squared, their tiny values can be avoided if the condensate is enhanced above the weak-αTC estimate in Eq. (2),
⟨
T
¯
T
⟩
ETC
≈
⟨
T
¯
T
⟩
TC
≈
4
π
F
EW
3
{\displaystyle \langle {\bar {T}}T\rangle _{\text{ETC}}\approx \langle {\bar {T}}T\rangle _{\text{TC}}\approx 4\pi F_{\text{EW}}^{3}}
.
During the 1980s, several dynamical mechanisms were advanced to do this. In 1981 Holdom suggested that, if the αTC(μ) evolves to a nontrivial fixed point in the ultraviolet, with a large positive anomalous dimension γm for
T
¯
T
{\displaystyle {\bar {T}}T}
, realistic quark and lepton masses could arise with ΛETC large enough to suppress ETC-induced
K
↔
K
¯
{\displaystyle K\leftrightarrow {\bar {K}}}
mixing. However, no example of a nontrivial ultraviolet fixed point in a four-dimensional gauge theory has been constructed. In 1985 Holdom analyzed a technicolor theory in which a "slowly varying" αTC(μ) was envisioned. His focus was to separate the chiral breaking and confinement scales, but he also noted that such a theory could enhance
⟨
T
¯
T
⟩
ETC
{\displaystyle \langle {\bar {T}}T\rangle _{\text{ETC}}}
and thus allow the ETC scale to be raised. In 1986 Akiba and Yanagida also considered enhancing quark and lepton masses, by simply assuming that αTC is constant and strong all the way up to the ETC scale. In the same year Yamawaki, Bando, and Matumoto again imagined an ultraviolet fixed point in a non-asymptotically free theory to enhance the technifermion condensate.
In 1986 Appelquist, Karabali and Wijewardhana discussed the enhancement of fermion masses in an asymptotically free technicolor theory with a slowly running, or "walking", gauge coupling. The slowness arose from the screening effect of a large number of technifermions, with the analysis carried out through two-loop perturbation theory. In 1987 Appelquist and Wijewardhana explored this walking scenario further. They took the analysis to three loops, noted that the walking can lead to a power law enhancement of the technifermion condensate, and estimated the resultant quark, lepton, and technipion masses. The condensate enhancement arises because the associated technifermion mass decreases slowly, roughly linearly, as a function of its renormalization scale. This corresponds to the condensate anomalous dimension γm in Eq. (3) approaching unity (see below).
In the 1990s, the idea emerged more clearly that walking is naturally described by asymptotically free gauge theories dominated in the infrared by an approximate fixed point. Unlike the speculative proposal of ultraviolet fixed points, fixed points in the infrared are known to exist in asymptotically free theories, arising at two loops in the beta function providing that the fermion count Nf is large enough. This has been known since the first two-loop computation in 1974 by Caswell. If Nf is close to the value
N
^
f
{\displaystyle {\hat {N}}_{\text{f}}}
at which asymptotic freedom is lost, the resultant infrared fixed point is weak, of parametric order
N
^
f
−
N
f
{\displaystyle {\hat {N}}_{\text{f}}-N_{\text{f}}}
, and reliably accessible in perturbation theory. This weak-coupling limit was explored by Banks and Zaks in 1982.
The fixed-point coupling αIR becomes stronger as Nf is reduced from
N
^
f
{\displaystyle {\hat {N}}_{\text{f}}}
. Below some critical value Nfc the coupling becomes strong enough (> αχ SB) to break spontaneously the massless technifermions' chiral symmetry. Since the analysis must typically go beyond two-loop perturbation theory, the definition of the running coupling αTC(μ), its fixed point value αIR, and the strength αχ SB necessary for chiral symmetry breaking depend on the particular renormalization scheme adopted. For
0
<
α
IR
−
α
χ
SB
α
IR
≪
1
{\displaystyle 0<{\frac {\alpha _{\text{IR}}-\alpha _{\chi {\text{SB}}}}{\alpha _{\text{IR}}}}\ll 1}
; i.e., for Nf just below Nfc, the evolution of αTC(μ) is governed by the infrared fixed point and it will evolve slowly (walk) for a range of momenta above the breaking scale ΛTC. To overcome the
M
E
T
C
2
{\displaystyle M_{ETC}^{2}}
-suppression of the masses of first and second generation quarks involved in
K
↔
K
¯
{\displaystyle K\leftrightarrow {\bar {K}}}
mixing, this range must extend almost to their ETC scale, of
O
(
10
3
TeV
)
{\displaystyle {\mathcal {O}}(10^{3}{\hbox{ TeV}})}
. Cohen and Georgi argued that γm = 1 is the signal of spontaneous chiral symmetry breaking, i.e., that γm(αχ SB) = 1. Therefore, in the walking-αTC region, γm ≈ 1 and, from Eqs. (2) and (3), the light quark masses are enhanced approximately by
M
ETC
Λ
TC
{\displaystyle {\frac {M_{\text{ETC}}}{\Lambda _{\text{TC}}}}}
.
The idea that αTC(μ) walks for a large range of momenta when αIR lies just above αχ SB was suggested by Lane and Ramana. They made an explicit model, discussed the walking that ensued, and used it in their discussion of walking technicolor phenomenology at hadron colliders. This idea was developed in some detail by Appelquist, Terning, and Wijewardhana. Combining a perturbative computation of the infrared fixed point with an approximation of αχ SB based on the Schwinger–Dyson equation, they estimated the critical value Nfc and explored the resultant electroweak physics. Since the 1990s, most discussions of walking technicolor are in the framework of theories assumed to be dominated in the infrared by an approximate fixed point. Various models have been explored, some with the technifermions in the fundamental representation of the gauge group and some employing higher representations.
The possibility that the technicolor condensate can be enhanced beyond that discussed in the walking literature, has also been considered recently by Luty and Okui under the name "conformal technicolor". They envision an infrared stable fixed point, but with a very large anomalous dimension for the operator
T
¯
T
{\displaystyle {\bar {T}}T}
. It remains to be seen whether this can be realized, for example, in the class of theories currently being examined using lattice techniques.
== Top quark mass ==
The enhancement described above for walking technicolor may not be sufficient to generate the measured top quark mass, even for an ETC scale as low as a few TeV. However, this problem could be addressed if the effective four-technifermion coupling resulting from ETC gauge boson exchange is strong and tuned just above a critical value. The analysis of this strong-ETC possibility is that of a Nambu–Jona–Lasinio model with an additional (technicolor) gauge interaction. The technifermion masses are small compared to the ETC scale (the cutoff on the effective theory), but nearly constant out to this scale, leading to a large top quark mass. No fully realistic ETC theory for all quark masses has yet been developed incorporating these ideas. A related study was carried out by Miransky and Yamawaki. A problem with this approach is that it involves some degree of parameter fine-tuning, in conflict with technicolor's guiding principle of naturalness.
A large body of closely related work in which the Higgs is a composite state, composed of top and anti-top quarks, is the top quark condensate, topcolor and top-color-assisted technicolor models, in which new strong interactions are ascribed to the top quark and other third-generation fermions.
== Technicolor on the lattice ==
Lattice gauge theory is a non-perturbative method applicable to strongly interacting technicolor theories, allowing first-principles exploration of walking and conformal dynamics. In 2007, Catterall and Sannino used lattice gauge theory to study SU(2) gauge theories with two flavors of Dirac fermions in the symmetric representation, finding evidence of conformality that has been confirmed by subsequent studies.
As of 2010, the situation for SU(3) gauge theory with fermions in the fundamental representation is not as clear-cut. In 2007, Appelquist, Fleming, and Neil reported evidence that a non-trivial infrared fixed point develops in such theories when there are twelve flavors, but not when there are eight. While some subsequent studies confirmed these results, others reported different conclusions, depending on the lattice methods used, and there is not yet consensus.
Further lattice studies exploring these issues, as well as considering the consequences of these theories for precision electroweak measurements, are underway by several research groups.
== Technicolor phenomenology ==
Any framework for physics beyond the Standard Model must conform with precision measurements of the electroweak parameters. Its consequences for physics at existing and future high-energy hadron colliders, and for the dark matter of the universe must also be explored.
=== Precision electroweak tests ===
In 1990, the phenomenological parameters S, T, and U were introduced by Peskin and Takeuchi to quantify contributions to electroweak radiative corrections from physics beyond the Standard Model. They have a simple relation to the parameters of the electroweak chiral Lagrangian. The Peskin–Takeuchi analysis was based on the general formalism for weak radiative corrections developed by Kennedy, Lynn, Peskin and Stuart, and alternate formulations also exist.
The S, T, and U-parameters describe corrections to the electroweak gauge boson propagators from physics beyond the Standard Model. They can be written in terms of polarization functions of electroweak currents and their spectral representation as follows:
(
5
)
S
=
16
π
d
d
q
2
[
Π
33
n
e
w
(
q
2
)
−
Π
3
Q
n
e
w
(
q
2
)
]
q
2
=
0
=
4
π
∫
d
m
2
m
4
[
σ
V
3
(
m
2
)
−
σ
A
3
(
m
2
)
]
n
e
w
;
(
6
)
T
=
16
π
M
Z
2
sin
2
2
θ
W
[
Π
11
n
e
w
(
0
)
−
Π
33
n
e
w
(
0
)
]
=
4
π
M
Z
2
sin
2
2
θ
W
∫
0
∞
d
m
2
m
2
[
σ
V
1
(
m
2
)
+
σ
A
1
(
m
2
)
−
σ
V
3
(
m
2
)
−
σ
A
3
(
m
2
)
]
n
e
w
,
{\displaystyle {\begin{aligned}(5)\qquad S&=16\pi {\frac {d}{dq^{2}}}\left[\Pi _{33}^{\mathbf {new} }(q^{2})-\Pi _{3Q}^{\mathbf {new} }(q^{2})\right]_{q^{2}=0}\\&=4\pi \int {\frac {dm^{2}}{m^{4}}}\left[\sigma _{V}^{3}(m^{2})-\sigma _{A}^{3}(m^{2})\right]^{\mathbf {new} };\\\\(6)\qquad T&={\frac {16\pi }{M_{Z}^{2}\sin ^{2}2\theta _{W}}}\;\left[\Pi _{11}^{\mathbf {new} }(0)-\Pi _{33}^{\mathbf {new} }(0)\right]\\&={\frac {4\pi }{M_{Z}^{2}\sin ^{2}2\theta _{W}}}\int _{0}^{\infty }{\frac {dm^{2}}{m^{2}}}\left[\sigma _{V}^{1}(m^{2})+\sigma _{A}^{1}(m^{2})-\sigma _{V}^{3}(m^{2})-\sigma _{A}^{3}(m^{2})\right]^{\mathbf {new} },\end{aligned}}}
where only new, beyond-standard-model physics is included. The quantities are calculated relative to a minimal Standard Model with some chosen reference mass of the Higgs boson, taken to range from the experimental lower bound of 117 GeV to 1000 GeV where its width becomes very large. For these parameters to describe the dominant corrections to the Standard Model, the mass scale of the new physics must be much greater than MW and MZ, and the coupling of quarks and leptons to the new particles must be suppressed relative to their coupling to the gauge bosons. This is the case with technicolor, so long as the lightest technivector mesons, ρT and aT, are heavier than 200–300 GeV. The S-parameter is sensitive to all new physics at the TeV scale, while T is a measure of weak-isospin breaking effects. The U-parameter is generally not useful; most new-physics theories, including technicolor theories, give negligible contributions to it.
The S and T-parameters are determined by global fit to experimental data including Z-pole data from LEP at CERN, top quark and W-mass measurements at Fermilab, and measured levels of atomic parity violation. The resultant bounds on these parameters are given in the Review of Particle Properties. Assuming U = 0, the S and T parameters are small and, in fact, consistent with zero:
(
7
)
S
=
−
0.04
±
0.09
(
−
0.07
)
,
T
=
0.02
±
0.09
(
+
0.09
)
,
{\displaystyle (7)\qquad {\begin{aligned}S&=-0.04\pm 0.09\,(-0.07),\\T&=0.02\pm 0.09\,(+0.09),\end{aligned}}}
where the central value corresponds to a Higgs mass of 117 GeV and the correction to the central value when the Higgs mass is increased to 300 GeV is given in parentheses. These values place tight restrictions on beyond-standard-model theories – when the relevant corrections can be reliably computed.
The S parameter estimated in QCD-like technicolor theories is significantly greater than the experimentally allowed value. The computation was done assuming that the spectral integral for S is dominated by the lightest ρT and aT resonances, or by scaling effective Lagrangian parameters from QCD. In walking technicolor, however, the physics at the TeV scale and beyond must be quite different from that of QCD-like theories. In particular, the vector and axial-vector spectral functions cannot be dominated by just the lowest-lying resonances. It is unknown whether higher energy contributions to
σ
V,A
3
{\displaystyle \sigma _{\text{V,A}}^{3}}
are a tower of identifiable ρT and aT states or a smooth continuum. It has been conjectured that ρT and aT partners could be more nearly degenerate in walking theories (approximate parity doubling), reducing their contribution to S. Lattice calculations are underway or planned to test these ideas and obtain reliable estimates of S in walking theories.
The restriction on the T-parameter poses a problem for the generation of the top-quark mass in the ETC framework. The enhancement from walking can allow the associated ETC scale to be as large as a few TeV, but – since the ETC interactions must be strongly weak-isospin breaking to allow for the large top-bottom mass splitting – the contribution to the T parameter, as well as the rate for the decay
Z
0
→
b
¯
b
{\displaystyle \mathrm {Z^{0}\rightarrow {\bar {b}}b} }
, could be too large.
=== Hadron collider phenomenology ===
Early studies generally assumed the existence of just one electroweak doublet of technifermions, or of one techni-family including one doublet each of color-triplet techniquarks and color-singlet technileptons (four electroweak doublets in total). The number ND of electroweak doublets determines the decay constant F needed to produce the correct electroweak scale, as F = FEW⁄√ND = 246 GeV⁄√ND . In the minimal, one-doublet model, three Goldstone bosons (technipions, πT) have decay constant F = FEW = 246 GeV and are eaten by the electroweak gauge bosons. The most accessible collider signal is the production through
q
¯
q
{\displaystyle {\bar {q}}q}
annihilation in a hadron collider of spin-one
ρ
T
±
,
0
{\displaystyle \mathrm {\rho } _{\text{T}}^{\pm ,0}}
, and their subsequent decay into a pair of longitudinally polarized weak bosons,
W
LP
±
Z
LP
0
{\displaystyle \mathrm {W} _{\text{LP}}^{\pm }\mathrm {Z} _{\text{LP}}^{0}}
and
W
LP
+
W
LP
−
{\displaystyle \mathrm {W} _{\text{LP}}^{+}\mathrm {W} _{\text{LP}}^{-}}
. At an expected mass of 1.5–2.0 TeV and width of 300–400 GeV, such ρT's would be difficult to discover at the LHC. A one-family model has a large number of physical technipions, with F = FEW⁄√4 = 123 GeV. There is a collection of correspondingly lower-mass color-singlet and octet technivectors decaying into technipion pairs. The πT's are expected to decay to the heaviest possible quark and lepton pairs. Despite their lower masses, the ρT's are wider than in the minimal model and the backgrounds to the πT decays are likely to be insurmountable at a hadron collider.
This picture changed with the advent of walking technicolor. A walking gauge coupling occurs if αχ SB lies just below the IR fixed point value αIR, which requires either a large number of electroweak doublets in the fundamental representation of the gauge group, e.g., or a few doublets in higher-dimensional TC representations. In the latter case, the constraints on ETC representations generally imply other technifermions in the fundamental representation as well. In either case, there are technipions πT with decay constant
F
≪
F
E
W
{\displaystyle F\ll F_{EW}}
. This implies
Λ
T
C
≪
F
E
W
{\displaystyle \Lambda _{TC}\ll F_{EW}}
so that the lightest technivectors accessible at the LHC – ρT, ωT, aT (with IG JP C = 1+ 1−−, 0− 1−−, 1− 1++) – have masses well below a TeV. The class of theories with many technifermions and thus
F
≪
F
E
W
{\displaystyle F\ll F_{EW}}
is called low-scale technicolor.
A second consequence of walking technicolor concerns the decays of the spin-one technihadrons. Since technipion masses
M
π
T
2
∝
⟨
T
¯
T
T
¯
T
⟩
M
E
T
C
{\displaystyle M_{\pi _{T}}^{2}\propto \langle {\bar {T}}T{\bar {T}}T\rangle _{M_{ETC}}}
(see Eq. (4)), walking enhances them much more than it does other technihadron masses. Thus, it is very likely that the lightest MρT < 2MπT and that the two and three-πT decay channels of the light technivectors are closed. This further implies that these technivectors are very narrow. Their most probable two-body channels are
W
L
±
,
0
π
T
{\displaystyle \mathrm {W} _{\mathrm {L} }^{\pm ,0}\mathrm {\pi } _{T}}
, WL WL, γ πT and γ WL. The coupling of the lightest technivectors to WL is proportional to F⁄FEW. Thus, all their decay rates are suppressed by powers of
[
F
F
E
W
]
2
≪
1
{\displaystyle \left[{\frac {F}{F_{EW}}}\right]^{2}\ll 1}
or the fine-structure constant, giving total widths of a few GeV (for ρT) to a few tenths of a GeV (for ωT and T).
A more speculative consequence of walking technicolor is motivated by consideration of its contribution to the S-parameter. As noted above, the usual assumptions made to estimate STC are invalid in a walking theory. In particular, the spectral integrals used to evaluate STC cannot be dominated by just the lowest-lying ρT and aT and, if STC is to be small, the masses and weak-current couplings of the ρT and aT could be more nearly equal than they are in QCD.
Low-scale technicolor phenomenology, including the possibility of a more parity-doubled spectrum, has been developed into a set of rules and decay amplitudes. An April 2011 announcement of an excess in jet pairs produced in association with a W boson measured at the Tevatron has been interpreted by Eichten, Lane and Martin as a possible signal of the technipion of low-scale technicolor.
The general scheme of low-scale technicolor makes little sense if the limit on
M
ρ
T
{\displaystyle M_{\rho _{T}}}
is pushed past about 700 GeV. The LHC should be able to discover it or rule it out. Searches there involving decays to technipions and thence to heavy quark jets are hampered by backgrounds from
t
¯
t
{\displaystyle {\bar {t}}t}
production; its rate is 100 times larger than that at the Tevatron. Consequently, the discovery of low-scale technicolor at the LHC relies on all-leptonic final-state channels with favorable signal-to-background ratios:
ρ
T
±
→
W
L
±
Z
L
0
{\displaystyle \rho _{T}^{\pm }\rightarrow W_{L}^{\pm }Z_{L}^{0}}
,
a
T
±
→
γ
W
L
±
{\displaystyle a_{T}^{\pm }\rightarrow \gamma W_{L}^{\pm }}
and
ω
T
→
γ
Z
L
0
{\displaystyle \omega _{T}\rightarrow \gamma Z_{L}^{0}}
.
=== Dark matter ===
Technicolor theories naturally contain dark matter candidates. Almost certainly, models can be built in which the lowest-lying technibaryon, a technicolor-singlet bound state of technifermions, is stable enough to survive the evolution of the universe. If the technicolor theory is low-scale (
F
≪
F
E
W
{\displaystyle F\ll F_{EW}}
), the baryon's mass should be no more than 1–2 TeV. If not, it could be much heavier. The technibaryon must be electrically neutral and satisfy constraints on its abundance. Given the limits on spin-independent dark-matter-nucleon cross sections from dark-matter search experiments (
≲
10
−
42
c
m
2
{\displaystyle \lesssim 10^{-42}\,\mathrm {cm} ^{2}}
for the masses of interest), it may have to be electroweak neutral (weak isospin T3 = 0) as well. These considerations suggest that the "old" technicolor dark matter candidates may be difficult to produce at the LHC.
A different class of technicolor dark matter candidates light enough to be accessible at the LHC was introduced by Francesco Sannino and his collaborators. These states are pseudo Goldstone bosons possessing a global charge that makes them stable against decay.
== See also ==
Higgsless model
Topcolor
Top quark condensate
Infrared fixed point
== References == | Wikipedia/Technicolor_(physics) |
In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.
The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related. (Specifically, only two-point "Green's functions" in the case of a non-interacting system are Green's functions in the mathematical sense; the linear operator that they invert is the Hamiltonian operator, which in the non-interacting case is quadratic in the fields.)
== Spatially uniform case ==
=== Basic definitions ===
We consider a many-body theory with field operator (annihilation operator written in the position basis)
ψ
(
x
)
{\displaystyle \psi (\mathbf {x} )}
.
The Heisenberg operators can be written in terms of Schrödinger operators as
ψ
(
x
,
t
)
=
e
i
K
t
ψ
(
x
)
e
−
i
K
t
,
{\displaystyle \psi (\mathbf {x} ,t)=e^{iKt}\psi (\mathbf {x} )e^{-iKt},}
and the creation operator is
ψ
¯
(
x
,
t
)
=
[
ψ
(
x
,
t
)
]
†
{\displaystyle {\bar {\psi }}(\mathbf {x} ,t)=[\psi (\mathbf {x} ,t)]^{\dagger }}
, where
K
=
H
−
μ
N
{\displaystyle K=H-\mu N}
is the grand-canonical Hamiltonian.
Similarly, for the imaginary-time operators,
ψ
(
x
,
τ
)
=
e
K
τ
ψ
(
x
)
e
−
K
τ
{\displaystyle \psi (\mathbf {x} ,\tau )=e^{K\tau }\psi (\mathbf {x} )e^{-K\tau }}
ψ
¯
(
x
,
τ
)
=
e
K
τ
ψ
†
(
x
)
e
−
K
τ
.
{\displaystyle {\bar {\psi }}(\mathbf {x} ,\tau )=e^{K\tau }\psi ^{\dagger }(\mathbf {x} )e^{-K\tau }.}
[Note that the imaginary-time creation operator
ψ
¯
(
x
,
τ
)
{\displaystyle {\bar {\psi }}(\mathbf {x} ,\tau )}
is not the Hermitian conjugate of the annihilation operator
ψ
(
x
,
τ
)
{\displaystyle \psi (\mathbf {x} ,\tau )}
.]
In real time, the
2
n
{\displaystyle 2n}
-point Green function is defined by
G
(
n
)
(
1
…
n
∣
1
′
…
n
′
)
=
i
n
⟨
T
ψ
(
1
)
…
ψ
(
n
)
ψ
¯
(
n
′
)
…
ψ
¯
(
1
′
)
⟩
,
{\displaystyle G^{(n)}(1\ldots n\mid 1'\ldots n')=i^{n}\langle T\psi (1)\ldots \psi (n){\bar {\psi }}(n')\ldots {\bar {\psi }}(1')\rangle ,}
where we have used a condensed notation in which
j
{\displaystyle j}
signifies
(
x
j
,
t
j
)
{\displaystyle (\mathbf {x} _{j},t_{j})}
and
j
′
{\displaystyle j'}
signifies
(
x
j
′
,
t
j
′
)
{\displaystyle (\mathbf {x} _{j}',t_{j}')}
. The operator
T
{\displaystyle T}
denotes time ordering, and indicates that the field operators that follow it are to be ordered so that their time arguments increase from right to left.
In imaginary time, the corresponding definition is
G
(
n
)
(
1
…
n
∣
1
′
…
n
′
)
=
⟨
T
ψ
(
1
)
…
ψ
(
n
)
ψ
¯
(
n
′
)
…
ψ
¯
(
1
′
)
⟩
,
{\displaystyle {\mathcal {G}}^{(n)}(1\ldots n\mid 1'\ldots n')=\langle T\psi (1)\ldots \psi (n){\bar {\psi }}(n')\ldots {\bar {\psi }}(1')\rangle ,}
where
j
{\displaystyle j}
signifies
x
j
,
τ
j
{\displaystyle \mathbf {x} _{j},\tau _{j}}
. (The imaginary-time variables
τ
j
{\displaystyle \tau _{j}}
are restricted to the range from
0
{\displaystyle 0}
to the inverse temperature
β
=
1
k
B
T
{\textstyle \beta ={\frac {1}{k_{\text{B}}T}}}
.)
Note regarding signs and normalization used in these definitions: The signs of the Green functions have been chosen so that Fourier transform of the two-point (
n
=
1
{\displaystyle n=1}
) thermal Green function for a free particle is
G
(
k
,
ω
n
)
=
1
−
i
ω
n
+
ξ
k
,
{\displaystyle {\mathcal {G}}(\mathbf {k} ,\omega _{n})={\frac {1}{-i\omega _{n}+\xi _{\mathbf {k} }}},}
and the retarded Green function is
G
R
(
k
,
ω
)
=
1
−
(
ω
+
i
η
)
+
ξ
k
,
{\displaystyle G^{\mathrm {R} }(\mathbf {k} ,\omega )={\frac {1}{-(\omega +i\eta )+\xi _{\mathbf {k} }}},}
where
ω
n
=
[
2
n
+
θ
(
−
ζ
)
]
π
β
{\displaystyle \omega _{n}={\frac {[2n+\theta (-\zeta )]\pi }{\beta }}}
is the Matsubara frequency.
Throughout,
ζ
{\displaystyle \zeta }
is
+
1
{\displaystyle +1}
for bosons and
−
1
{\displaystyle -1}
for fermions and
[
…
,
…
]
=
[
…
,
…
]
−
ζ
{\displaystyle [\ldots ,\ldots ]=[\ldots ,\ldots ]_{-\zeta }}
denotes either a commutator or anticommutator as appropriate.
(See below for details.)
=== Two-point functions ===
The Green function with a single pair of arguments (
n
=
1
{\displaystyle n=1}
) is referred to as the two-point function, or propagator. In the presence of both spatial and temporal translational symmetry, it depends only on the difference of its arguments. Taking the Fourier transform with respect to both space and time gives
G
(
x
τ
∣
x
′
τ
′
)
=
∫
k
d
k
1
β
∑
ω
n
G
(
k
,
ω
n
)
e
i
k
⋅
(
x
−
x
′
)
−
i
ω
n
(
τ
−
τ
′
)
,
{\displaystyle {\mathcal {G}}(\mathbf {x} \tau \mid \mathbf {x} '\tau ')=\int _{\mathbf {k} }d\mathbf {k} {\frac {1}{\beta }}\sum _{\omega _{n}}{\mathcal {G}}(\mathbf {k} ,\omega _{n})e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')-i\omega _{n}(\tau -\tau ')},}
where the sum is over the appropriate Matsubara frequencies (and the integral involves an implicit factor of
(
L
/
2
π
)
d
{\displaystyle (L/2\pi )^{d}}
, as usual).
In real time, we will explicitly indicate the time-ordered function with a superscript T:
G
T
(
x
t
∣
x
′
t
′
)
=
∫
k
d
k
∫
d
ω
2
π
G
T
(
k
,
ω
)
e
i
k
⋅
(
x
−
x
′
)
−
i
ω
(
t
−
t
′
)
.
{\displaystyle G^{\mathrm {T} }(\mathbf {x} t\mid \mathbf {x} 't')=\int _{\mathbf {k} }d\mathbf {k} \int {\frac {d\omega }{2\pi }}G^{\mathrm {T} }(\mathbf {k} ,\omega )e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')-i\omega (t-t')}.}
The real-time two-point Green function can be written in terms of 'retarded' and 'advanced' Green functions, which will turn out to have simpler analyticity properties. The retarded and advanced Green functions are defined by
G
R
(
x
t
∣
x
′
t
′
)
=
−
i
⟨
[
ψ
(
x
,
t
)
,
ψ
¯
(
x
′
,
t
′
)
]
ζ
⟩
Θ
(
t
−
t
′
)
{\displaystyle G^{\mathrm {R} }(\mathbf {x} t\mid \mathbf {x} 't')=-i\langle [\psi (\mathbf {x} ,t),{\bar {\psi }}(\mathbf {x} ',t')]_{\zeta }\rangle \Theta (t-t')}
and
G
A
(
x
t
∣
x
′
t
′
)
=
i
⟨
[
ψ
(
x
,
t
)
,
ψ
¯
(
x
′
,
t
′
)
]
ζ
⟩
Θ
(
t
′
−
t
)
,
{\displaystyle G^{\mathrm {A} }(\mathbf {x} t\mid \mathbf {x} 't')=i\langle [\psi (\mathbf {x} ,t),{\bar {\psi }}(\mathbf {x} ',t')]_{\zeta }\rangle \Theta (t'-t),}
respectively.
They are related to the time-ordered Green function by
G
T
(
k
,
ω
)
=
[
1
+
ζ
n
(
ω
)
]
G
R
(
k
,
ω
)
−
ζ
n
(
ω
)
G
A
(
k
,
ω
)
,
{\displaystyle G^{\mathrm {T} }(\mathbf {k} ,\omega )=[1+\zeta n(\omega )]G^{\mathrm {R} }(\mathbf {k} ,\omega )-\zeta n(\omega )G^{\mathrm {A} }(\mathbf {k} ,\omega ),}
where
n
(
ω
)
=
1
e
β
ω
−
ζ
{\displaystyle n(\omega )={\frac {1}{e^{\beta \omega }-\zeta }}}
is the Bose–Einstein or Fermi–Dirac distribution function.
==== Imaginary-time ordering and β-periodicity ====
The thermal Green functions are defined only when both imaginary-time arguments are within the range
0
{\displaystyle 0}
to
β
{\displaystyle \beta }
. The two-point Green function has the following properties. (The position or momentum arguments are suppressed in this section.)
Firstly, it depends only on the difference of the imaginary times:
G
(
τ
,
τ
′
)
=
G
(
τ
−
τ
′
)
.
{\displaystyle {\mathcal {G}}(\tau ,\tau ')={\mathcal {G}}(\tau -\tau ').}
The argument
τ
−
τ
′
{\displaystyle \tau -\tau '}
is allowed to run from
−
β
{\displaystyle -\beta }
to
β
{\displaystyle \beta }
.
Secondly,
G
(
τ
)
{\displaystyle {\mathcal {G}}(\tau )}
is (anti)periodic under shifts of
β
{\displaystyle \beta }
. Because of the small domain within which the function is defined, this means just
G
(
τ
−
β
)
=
ζ
G
(
τ
)
,
{\displaystyle {\mathcal {G}}(\tau -\beta )=\zeta {\mathcal {G}}(\tau ),}
for
0
<
τ
<
β
{\displaystyle 0<\tau <\beta }
. Time ordering is crucial for this property, which can be proved straightforwardly, using the cyclicity of the trace operation.
These two properties allow for the Fourier transform representation and its inverse,
G
(
ω
n
)
=
∫
0
β
d
τ
G
(
τ
)
e
i
ω
n
τ
.
{\displaystyle {\mathcal {G}}(\omega _{n})=\int _{0}^{\beta }d\tau \,{\mathcal {G}}(\tau )\,e^{i\omega _{n}\tau }.}
Finally, note that
G
(
τ
)
{\displaystyle {\mathcal {G}}(\tau )}
has a discontinuity at
τ
=
0
{\displaystyle \tau =0}
; this is consistent with a long-distance behaviour of
G
(
ω
n
)
∼
1
/
|
ω
n
|
{\displaystyle {\mathcal {G}}(\omega _{n})\sim 1/|\omega _{n}|}
.
=== Spectral representation ===
The propagators in real and imaginary time can both be related to the spectral density (or spectral weight), given by
ρ
(
k
,
ω
)
=
1
Z
∑
α
,
α
′
2
π
δ
(
E
α
−
E
α
′
−
ω
)
|
⟨
α
∣
ψ
k
†
∣
α
′
⟩
|
2
(
e
−
β
E
α
′
−
ζ
e
−
β
E
α
)
,
{\displaystyle \rho (\mathbf {k} ,\omega )={\frac {1}{\mathcal {Z}}}\sum _{\alpha ,\alpha '}2\pi \delta (E_{\alpha }-E_{\alpha '}-\omega )|\langle \alpha \mid \psi _{\mathbf {k} }^{\dagger }\mid \alpha '\rangle |^{2}\left(e^{-\beta E_{\alpha '}}-\zeta e^{-\beta E_{\alpha }}\right),}
where |α⟩ refers to a (many-body) eigenstate of the grand-canonical Hamiltonian H − μN, with eigenvalue Eα.
The imaginary-time propagator is then given by
G
(
k
,
ω
n
)
=
∫
−
∞
∞
d
ω
′
2
π
ρ
(
k
,
ω
′
)
−
i
ω
n
+
ω
′
,
{\displaystyle {\mathcal {G}}(\mathbf {k} ,\omega _{n})=\int _{-\infty }^{\infty }{\frac {d\omega '}{2\pi }}{\frac {\rho (\mathbf {k} ,\omega ')}{-i\omega _{n}+\omega '}}~,}
and the retarded propagator by
G
R
(
k
,
ω
)
=
∫
−
∞
∞
d
ω
′
2
π
ρ
(
k
,
ω
′
)
−
(
ω
+
i
η
)
+
ω
′
,
{\displaystyle G^{\mathrm {R} }(\mathbf {k} ,\omega )=\int _{-\infty }^{\infty }{\frac {d\omega '}{2\pi }}{\frac {\rho (\mathbf {k} ,\omega ')}{-(\omega +i\eta )+\omega '}},}
where the limit as
η
→
0
+
{\displaystyle \eta \to 0^{+}}
is implied.
The advanced propagator is given by the same expression, but with
−
i
η
{\displaystyle -i\eta }
in the denominator.
The time-ordered function can be found in terms of
G
R
{\displaystyle G^{\mathrm {R} }}
and
G
A
{\displaystyle G^{\mathrm {A} }}
. As claimed above,
G
R
(
ω
)
{\displaystyle G^{\mathrm {R} }(\omega )}
and
G
A
(
ω
)
{\displaystyle G^{\mathrm {A} }(\omega )}
have simple analyticity properties: the former (latter) has all its poles and discontinuities in the lower (upper) half-plane.
The thermal propagator
G
(
ω
n
)
{\displaystyle {\mathcal {G}}(\omega _{n})}
has all its poles and discontinuities on the imaginary
ω
n
{\displaystyle \omega _{n}}
axis.
The spectral density can be found very straightforwardly from
G
R
{\displaystyle G^{\mathrm {R} }}
, using the Sokhatsky–Weierstrass theorem
lim
η
→
0
+
1
x
±
i
η
=
P
1
x
∓
i
π
δ
(
x
)
,
{\displaystyle \lim _{\eta \to 0^{+}}{\frac {1}{x\pm i\eta }}=P{\frac {1}{x}}\mp i\pi \delta (x),}
where P denotes the Cauchy principal part.
This gives
ρ
(
k
,
ω
)
=
2
Im
G
R
(
k
,
ω
)
.
{\displaystyle \rho (\mathbf {k} ,\omega )=2\operatorname {Im} G^{\mathrm {R} }(\mathbf {k} ,\omega ).}
This furthermore implies that
G
R
(
k
,
ω
)
{\displaystyle G^{\mathrm {R} }(\mathbf {k} ,\omega )}
obeys the following relationship between its real and imaginary parts:
Re
G
R
(
k
,
ω
)
=
−
2
P
∫
−
∞
∞
d
ω
′
2
π
Im
G
R
(
k
,
ω
′
)
ω
−
ω
′
,
{\displaystyle \operatorname {Re} G^{\mathrm {R} }(\mathbf {k} ,\omega )=-2P\int _{-\infty }^{\infty }{\frac {d\omega '}{2\pi }}{\frac {\operatorname {Im} G^{\mathrm {R} }(\mathbf {k} ,\omega ')}{\omega -\omega '}},}
where
P
{\displaystyle P}
denotes the principal value of the integral.
The spectral density obeys a sum rule,
∫
−
∞
∞
d
ω
2
π
ρ
(
k
,
ω
)
=
1
,
{\displaystyle \int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}\rho (\mathbf {k} ,\omega )=1,}
which gives
G
R
(
ω
)
∼
1
|
ω
|
{\displaystyle G^{\mathrm {R} }(\omega )\sim {\frac {1}{|\omega |}}}
as
|
ω
|
→
∞
{\displaystyle |\omega |\to \infty }
.
==== Hilbert transform ====
The similarity of the spectral representations of the imaginary- and real-time Green functions allows us to define the function
G
(
k
,
z
)
=
∫
−
∞
∞
d
x
2
π
ρ
(
k
,
x
)
−
z
+
x
,
{\displaystyle G(\mathbf {k} ,z)=\int _{-\infty }^{\infty }{\frac {dx}{2\pi }}{\frac {\rho (\mathbf {k} ,x)}{-z+x}},}
which is related to
G
{\displaystyle {\mathcal {G}}}
and
G
R
{\displaystyle G^{\mathrm {R} }}
by
G
(
k
,
ω
n
)
=
G
(
k
,
i
ω
n
)
{\displaystyle {\mathcal {G}}(\mathbf {k} ,\omega _{n})=G(\mathbf {k} ,i\omega _{n})}
and
G
R
(
k
,
ω
)
=
G
(
k
,
ω
+
i
η
)
.
{\displaystyle G^{\mathrm {R} }(\mathbf {k} ,\omega )=G(\mathbf {k} ,\omega +i\eta ).}
A similar expression obviously holds for
G
A
{\displaystyle G^{\mathrm {A} }}
.
The relation between
G
(
k
,
z
)
{\displaystyle G(\mathbf {k} ,z)}
and
ρ
(
k
,
x
)
{\displaystyle \rho (\mathbf {k} ,x)}
is referred to as a Hilbert transform.
==== Proof of spectral representation ====
We demonstrate the proof of the spectral representation of the propagator in the case of the thermal Green function, defined as
G
(
x
,
τ
∣
x
′
,
τ
′
)
=
⟨
T
ψ
(
x
,
τ
)
ψ
¯
(
x
′
,
τ
′
)
⟩
.
{\displaystyle {\mathcal {G}}(\mathbf {x} ,\tau \mid \mathbf {x} ',\tau ')=\langle T\psi (\mathbf {x} ,\tau ){\bar {\psi }}(\mathbf {x} ',\tau ')\rangle .}
Due to translational symmetry, it is only necessary to consider
G
(
x
,
τ
∣
0
,
0
)
{\displaystyle {\mathcal {G}}(\mathbf {x} ,\tau \mid \mathbf {0} ,0)}
for
τ
>
0
{\displaystyle \tau >0}
, given by
G
(
x
,
τ
∣
0
,
0
)
=
1
Z
∑
α
′
e
−
β
E
α
′
⟨
α
′
∣
ψ
(
x
,
τ
)
ψ
¯
(
0
,
0
)
∣
α
′
⟩
.
{\displaystyle {\mathcal {G}}(\mathbf {x} ,\tau \mid \mathbf {0} ,0)={\frac {1}{\mathcal {Z}}}\sum _{\alpha '}e^{-\beta E_{\alpha '}}\langle \alpha '\mid \psi (\mathbf {x} ,\tau ){\bar {\psi }}(\mathbf {0} ,0)\mid \alpha '\rangle .}
Inserting a complete set of eigenstates gives
G
(
x
,
τ
∣
0
,
0
)
=
1
Z
∑
α
,
α
′
e
−
β
E
α
′
⟨
α
′
∣
ψ
(
x
,
τ
)
∣
α
⟩
⟨
α
∣
ψ
¯
(
0
,
0
)
∣
α
′
⟩
.
{\displaystyle {\mathcal {G}}(\mathbf {x} ,\tau \mid \mathbf {0} ,0)={\frac {1}{\mathcal {Z}}}\sum _{\alpha ,\alpha '}e^{-\beta E_{\alpha '}}\langle \alpha '\mid \psi (\mathbf {x} ,\tau )\mid \alpha \rangle \langle \alpha \mid {\bar {\psi }}(\mathbf {0} ,0)\mid \alpha '\rangle .}
Since
|
α
⟩
{\displaystyle |\alpha \rangle }
and
|
α
′
⟩
{\displaystyle |\alpha '\rangle }
are eigenstates of
H
−
μ
N
{\displaystyle H-\mu N}
, the Heisenberg operators can be rewritten in terms of Schrödinger operators, giving
G
(
x
,
τ
|
0
,
0
)
=
1
Z
∑
α
,
α
′
e
−
β
E
α
′
e
τ
(
E
α
′
−
E
α
)
⟨
α
′
∣
ψ
(
x
)
∣
α
⟩
⟨
α
∣
ψ
†
(
0
)
∣
α
′
⟩
.
{\displaystyle {\mathcal {G}}(\mathbf {x} ,\tau |\mathbf {0} ,0)={\frac {1}{\mathcal {Z}}}\sum _{\alpha ,\alpha '}e^{-\beta E_{\alpha '}}e^{\tau (E_{\alpha '}-E_{\alpha })}\langle \alpha '\mid \psi (\mathbf {x} )\mid \alpha \rangle \langle \alpha \mid \psi ^{\dagger }(\mathbf {0} )\mid \alpha '\rangle .}
Performing the Fourier transform then gives
G
(
k
,
ω
n
)
=
1
Z
∑
α
,
α
′
e
−
β
E
α
′
1
−
ζ
e
β
(
E
α
′
−
E
α
)
−
i
ω
n
+
E
α
−
E
α
′
∫
k
′
d
k
′
⟨
α
∣
ψ
(
k
)
∣
α
′
⟩
⟨
α
′
∣
ψ
†
(
k
′
)
∣
α
⟩
.
{\displaystyle {\mathcal {G}}(\mathbf {k} ,\omega _{n})={\frac {1}{\mathcal {Z}}}\sum _{\alpha ,\alpha '}e^{-\beta E_{\alpha '}}{\frac {1-\zeta e^{\beta (E_{\alpha '}-E_{\alpha })}}{-i\omega _{n}+E_{\alpha }-E_{\alpha '}}}\int _{\mathbf {k} '}d\mathbf {k} '\langle \alpha \mid \psi (\mathbf {k} )\mid \alpha '\rangle \langle \alpha '\mid \psi ^{\dagger }(\mathbf {k} ')\mid \alpha \rangle .}
Momentum conservation allows the final term to be written as (up to possible factors of the volume)
|
⟨
α
′
∣
ψ
†
(
k
)
∣
α
⟩
|
2
,
{\displaystyle |\langle \alpha '\mid \psi ^{\dagger }(\mathbf {k} )\mid \alpha \rangle |^{2},}
which confirms the expressions for the Green functions in the spectral representation.
The sum rule can be proved by considering the expectation value of the commutator,
1
=
1
Z
∑
α
⟨
α
∣
e
−
β
(
H
−
μ
N
)
[
ψ
k
,
ψ
k
†
]
−
ζ
∣
α
⟩
,
{\displaystyle 1={\frac {1}{\mathcal {Z}}}\sum _{\alpha }\langle \alpha \mid e^{-\beta (H-\mu N)}[\psi _{\mathbf {k} },\psi _{\mathbf {k} }^{\dagger }]_{-\zeta }\mid \alpha \rangle ,}
and then inserting a complete set of eigenstates into both terms of the commutator:
1
=
1
Z
∑
α
,
α
′
e
−
β
E
α
(
⟨
α
∣
ψ
k
∣
α
′
⟩
⟨
α
′
∣
ψ
k
†
∣
α
⟩
−
ζ
⟨
α
∣
ψ
k
†
∣
α
′
⟩
⟨
α
′
∣
ψ
k
∣
α
⟩
)
.
{\displaystyle 1={\frac {1}{\mathcal {Z}}}\sum _{\alpha ,\alpha '}e^{-\beta E_{\alpha }}\left(\langle \alpha \mid \psi _{\mathbf {k} }\mid \alpha '\rangle \langle \alpha '\mid \psi _{\mathbf {k} }^{\dagger }\mid \alpha \rangle -\zeta \langle \alpha \mid \psi _{\mathbf {k} }^{\dagger }\mid \alpha '\rangle \langle \alpha '\mid \psi _{\mathbf {k} }\mid \alpha \rangle \right).}
Swapping the labels in the first term then gives
1
=
1
Z
∑
α
,
α
′
(
e
−
β
E
α
′
−
ζ
e
−
β
E
α
)
|
⟨
α
∣
ψ
k
†
∣
α
′
⟩
|
2
,
{\displaystyle 1={\frac {1}{\mathcal {Z}}}\sum _{\alpha ,\alpha '}\left(e^{-\beta E_{\alpha '}}-\zeta e^{-\beta E_{\alpha }}\right)|\langle \alpha \mid \psi _{\mathbf {k} }^{\dagger }\mid \alpha '\rangle |^{2}~,}
which is exactly the result of the integration of ρ.
==== Non-interacting case ====
In the non-interacting case,
ψ
k
†
∣
α
′
⟩
{\displaystyle \psi _{\mathbf {k} }^{\dagger }\mid \alpha '\rangle }
is an eigenstate with (grand-canonical) energy
E
α
′
+
ξ
k
{\displaystyle E_{\alpha '}+\xi _{\mathbf {k} }}
, where
ξ
k
=
ϵ
k
−
μ
{\displaystyle \xi _{\mathbf {k} }=\epsilon _{\mathbf {k} }-\mu }
is the single-particle dispersion relation measured with respect to the chemical potential. The spectral density therefore becomes
ρ
0
(
k
,
ω
)
=
1
Z
2
π
δ
(
ξ
k
−
ω
)
∑
α
′
⟨
α
′
∣
ψ
k
ψ
k
†
∣
α
′
⟩
(
1
−
ζ
e
−
β
ξ
k
)
e
−
β
E
α
′
.
{\displaystyle \rho _{0}(\mathbf {k} ,\omega )={\frac {1}{\mathcal {Z}}}\,2\pi \delta (\xi _{\mathbf {k} }-\omega )\sum _{\alpha '}\langle \alpha '\mid \psi _{\mathbf {k} }\psi _{\mathbf {k} }^{\dagger }\mid \alpha '\rangle (1-\zeta e^{-\beta \xi _{\mathbf {k} }})e^{-\beta E_{\alpha '}}.}
From the commutation relations,
⟨
α
′
∣
ψ
k
ψ
k
†
∣
α
′
⟩
=
⟨
α
′
∣
(
1
+
ζ
ψ
k
†
ψ
k
)
∣
α
′
⟩
,
{\displaystyle \langle \alpha '\mid \psi _{\mathbf {k} }\psi _{\mathbf {k} }^{\dagger }\mid \alpha '\rangle =\langle \alpha '\mid (1+\zeta \psi _{\mathbf {k} }^{\dagger }\psi _{\mathbf {k} })\mid \alpha '\rangle ,}
with possible factors of the volume again. The sum, which involves the thermal average of the number operator, then gives simply
[
1
+
ζ
n
(
ξ
k
)
]
Z
{\displaystyle [1+\zeta n(\xi _{\mathbf {k} })]{\mathcal {Z}}}
, leaving
ρ
0
(
k
,
ω
)
=
2
π
δ
(
ξ
k
−
ω
)
.
{\displaystyle \rho _{0}(\mathbf {k} ,\omega )=2\pi \delta (\xi _{\mathbf {k} }-\omega ).}
The imaginary-time propagator is thus
G
0
(
k
,
ω
)
=
1
−
i
ω
n
+
ξ
k
{\displaystyle {\mathcal {G}}_{0}(\mathbf {k} ,\omega )={\frac {1}{-i\omega _{n}+\xi _{\mathbf {k} }}}}
and the retarded propagator is
G
0
R
(
k
,
ω
)
=
1
−
(
ω
+
i
η
)
+
ξ
k
.
{\displaystyle G_{0}^{\mathrm {R} }(\mathbf {k} ,\omega )={\frac {1}{-(\omega +i\eta )+\xi _{\mathbf {k} }}}.}
==== Zero-temperature limit ====
As β → ∞, the spectral density becomes
ρ
(
k
,
ω
)
=
2
π
∑
α
[
δ
(
E
α
−
E
0
−
ω
)
|
⟨
α
∣
ψ
k
†
∣
0
⟩
|
2
−
ζ
δ
(
E
0
−
E
α
−
ω
)
|
⟨
0
∣
ψ
k
†
∣
α
⟩
|
2
]
{\displaystyle \rho (\mathbf {k} ,\omega )=2\pi \sum _{\alpha }\left[\delta (E_{\alpha }-E_{0}-\omega )\left|\left\langle \alpha \mid \psi _{\mathbf {k} }^{\dagger }\mid 0\right\rangle \right|^{2}-\zeta \delta (E_{0}-E_{\alpha }-\omega )\left|\left\langle 0\mid \psi _{\mathbf {k} }^{\dagger }\mid \alpha \right\rangle \right|^{2}\right]}
where α = 0 corresponds to the ground state. Note that only the first (second) term contributes when ω is positive (negative).
== General case ==
=== Basic definitions ===
We can use 'field operators' as above, or creation and annihilation operators associated with other single-particle states, perhaps eigenstates of the (noninteracting) kinetic energy. We then use
ψ
(
x
,
τ
)
=
φ
α
(
x
)
ψ
α
(
τ
)
,
{\displaystyle \psi (\mathbf {x} ,\tau )=\varphi _{\alpha }(\mathbf {x} )\psi _{\alpha }(\tau ),}
where
ψ
α
{\displaystyle \psi _{\alpha }}
is the annihilation operator for the single-particle state
α
{\displaystyle \alpha }
and
φ
α
(
x
)
{\displaystyle \varphi _{\alpha }(\mathbf {x} )}
is that state's wavefunction in the position basis. This gives
G
α
1
…
α
n
|
β
1
…
β
n
(
n
)
(
τ
1
…
τ
n
|
τ
1
′
…
τ
n
′
)
=
⟨
T
ψ
α
1
(
τ
1
)
…
ψ
α
n
(
τ
n
)
ψ
¯
β
n
(
τ
n
′
)
…
ψ
¯
β
1
(
τ
1
′
)
⟩
{\displaystyle {\mathcal {G}}_{\alpha _{1}\ldots \alpha _{n}|\beta _{1}\ldots \beta _{n}}^{(n)}(\tau _{1}\ldots \tau _{n}|\tau _{1}'\ldots \tau _{n}')=\langle T\psi _{\alpha _{1}}(\tau _{1})\ldots \psi _{\alpha _{n}}(\tau _{n}){\bar {\psi }}_{\beta _{n}}(\tau _{n}')\ldots {\bar {\psi }}_{\beta _{1}}(\tau _{1}')\rangle }
with a similar expression for
G
(
n
)
{\displaystyle G^{(n)}}
.
=== Two-point functions ===
These depend only on the difference of their time arguments, so that
G
α
β
(
τ
∣
τ
′
)
=
1
β
∑
ω
n
G
α
β
(
ω
n
)
e
−
i
ω
n
(
τ
−
τ
′
)
{\displaystyle {\mathcal {G}}_{\alpha \beta }(\tau \mid \tau ')={\frac {1}{\beta }}\sum _{\omega _{n}}{\mathcal {G}}_{\alpha \beta }(\omega _{n})\,e^{-i\omega _{n}(\tau -\tau ')}}
and
G
α
β
(
t
∣
t
′
)
=
∫
−
∞
∞
d
ω
2
π
G
α
β
(
ω
)
e
−
i
ω
(
t
−
t
′
)
.
{\displaystyle G_{\alpha \beta }(t\mid t')=\int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}\,G_{\alpha \beta }(\omega )\,e^{-i\omega (t-t')}.}
We can again define retarded and advanced functions in the obvious way; these are related to the time-ordered function in the same way as above.
The same periodicity properties as described in above apply to
G
α
β
{\displaystyle {\mathcal {G}}_{\alpha \beta }}
. Specifically,
G
α
β
(
τ
∣
τ
′
)
=
G
α
β
(
τ
−
τ
′
)
{\displaystyle {\mathcal {G}}_{\alpha \beta }(\tau \mid \tau ')={\mathcal {G}}_{\alpha \beta }(\tau -\tau ')}
and
G
α
β
(
τ
)
=
G
α
β
(
τ
+
β
)
,
{\displaystyle {\mathcal {G}}_{\alpha \beta }(\tau )={\mathcal {G}}_{\alpha \beta }(\tau +\beta ),}
for
τ
<
0
{\displaystyle \tau <0}
.
=== Spectral representation ===
In this case,
ρ
α
β
(
ω
)
=
1
Z
∑
m
,
n
2
π
δ
(
E
n
−
E
m
−
ω
)
⟨
m
∣
ψ
α
∣
n
⟩
⟨
n
∣
ψ
β
†
∣
m
⟩
(
e
−
β
E
m
−
ζ
e
−
β
E
n
)
,
{\displaystyle \rho _{\alpha \beta }(\omega )={\frac {1}{\mathcal {Z}}}\sum _{m,n}2\pi \delta (E_{n}-E_{m}-\omega )\;\langle m\mid \psi _{\alpha }\mid n\rangle \langle n\mid \psi _{\beta }^{\dagger }\mid m\rangle \left(e^{-\beta E_{m}}-\zeta e^{-\beta E_{n}}\right),}
where
m
{\displaystyle m}
and
n
{\displaystyle n}
are many-body states.
The expressions for the Green functions are modified in the obvious ways:
G
α
β
(
ω
n
)
=
∫
−
∞
∞
d
ω
′
2
π
ρ
α
β
(
ω
′
)
−
i
ω
n
+
ω
′
{\displaystyle {\mathcal {G}}_{\alpha \beta }(\omega _{n})=\int _{-\infty }^{\infty }{\frac {d\omega '}{2\pi }}{\frac {\rho _{\alpha \beta }(\omega ')}{-i\omega _{n}+\omega '}}}
and
G
α
β
R
(
ω
)
=
∫
−
∞
∞
d
ω
′
2
π
ρ
α
β
(
ω
′
)
−
(
ω
+
i
η
)
+
ω
′
.
{\displaystyle G_{\alpha \beta }^{\mathrm {R} }(\omega )=\int _{-\infty }^{\infty }{\frac {d\omega '}{2\pi }}{\frac {\rho _{\alpha \beta }(\omega ')}{-(\omega +i\eta )+\omega '}}.}
Their analyticity properties are identical to those of
G
(
k
,
ω
n
)
{\displaystyle {\mathcal {G}}(\mathbf {k} ,\omega _{n})}
and
G
R
(
k
,
ω
)
{\displaystyle G^{\mathrm {R} }(\mathbf {k} ,\omega )}
defined in the translationally invariant case. The proof follows exactly the same steps, except that the two matrix elements are no longer complex conjugates.
==== Noninteracting case ====
If the particular single-particle states that are chosen are 'single-particle energy eigenstates', i.e.
[
H
−
μ
N
,
ψ
α
†
]
=
ξ
α
ψ
α
†
,
{\displaystyle [H-\mu N,\psi _{\alpha }^{\dagger }]=\xi _{\alpha }\psi _{\alpha }^{\dagger },}
then for
|
n
⟩
{\displaystyle |n\rangle }
an eigenstate:
(
H
−
μ
N
)
∣
n
⟩
=
E
n
∣
n
⟩
,
{\displaystyle (H-\mu N)\mid n\rangle =E_{n}\mid n\rangle ,}
so is
ψ
α
∣
n
⟩
{\displaystyle \psi _{\alpha }\mid n\rangle }
:
(
H
−
μ
N
)
ψ
α
∣
n
⟩
=
(
E
n
−
ξ
α
)
ψ
α
∣
n
⟩
,
{\displaystyle (H-\mu N)\psi _{\alpha }\mid n\rangle =(E_{n}-\xi _{\alpha })\psi _{\alpha }\mid n\rangle ,}
and so is
ψ
α
†
∣
n
⟩
{\displaystyle \psi _{\alpha }^{\dagger }\mid n\rangle }
:
(
H
−
μ
N
)
ψ
α
†
∣
n
⟩
=
(
E
n
+
ξ
α
)
ψ
α
†
∣
n
⟩
.
{\displaystyle (H-\mu N)\psi _{\alpha }^{\dagger }\mid n\rangle =(E_{n}+\xi _{\alpha })\psi _{\alpha }^{\dagger }\mid n\rangle .}
We therefore have
⟨
m
∣
ψ
α
∣
n
⟩
⟨
n
∣
ψ
β
†
∣
m
⟩
=
δ
ξ
α
,
ξ
β
δ
E
n
,
E
m
+
ξ
α
⟨
m
∣
ψ
α
∣
n
⟩
⟨
n
∣
ψ
β
†
∣
m
⟩
.
{\displaystyle \langle m\mid \psi _{\alpha }\mid n\rangle \langle n\mid \psi _{\beta }^{\dagger }\mid m\rangle =\delta _{\xi _{\alpha },\xi _{\beta }}\delta _{E_{n},E_{m}+\xi _{\alpha }}\langle m\mid \psi _{\alpha }\mid n\rangle \langle n\mid \psi _{\beta }^{\dagger }\mid m\rangle .}
We then rewrite
ρ
α
β
(
ω
)
=
1
Z
∑
m
,
n
2
π
δ
(
ξ
α
−
ω
)
δ
ξ
α
,
ξ
β
⟨
m
∣
ψ
α
∣
n
⟩
⟨
n
∣
ψ
β
†
∣
m
⟩
e
−
β
E
m
(
1
−
ζ
e
−
β
ξ
α
)
,
{\displaystyle \rho _{\alpha \beta }(\omega )={\frac {1}{\mathcal {Z}}}\sum _{m,n}2\pi \delta (\xi _{\alpha }-\omega )\delta _{\xi _{\alpha },\xi _{\beta }}\langle m\mid \psi _{\alpha }\mid n\rangle \langle n\mid \psi _{\beta }^{\dagger }\mid m\rangle e^{-\beta E_{m}}\left(1-\zeta e^{-\beta \xi _{\alpha }}\right),}
therefore
ρ
α
β
(
ω
)
=
1
Z
∑
m
2
π
δ
(
ξ
α
−
ω
)
δ
ξ
α
,
ξ
β
⟨
m
∣
ψ
α
ψ
β
†
e
−
β
(
H
−
μ
N
)
∣
m
⟩
(
1
−
ζ
e
−
β
ξ
α
)
,
{\displaystyle \rho _{\alpha \beta }(\omega )={\frac {1}{\mathcal {Z}}}\sum _{m}2\pi \delta (\xi _{\alpha }-\omega )\delta _{\xi _{\alpha },\xi _{\beta }}\langle m\mid \psi _{\alpha }\psi _{\beta }^{\dagger }e^{-\beta (H-\mu N)}\mid m\rangle \left(1-\zeta e^{-\beta \xi _{\alpha }}\right),}
use
⟨
m
∣
ψ
α
ψ
β
†
∣
m
⟩
=
δ
α
,
β
⟨
m
∣
ζ
ψ
α
†
ψ
α
+
1
∣
m
⟩
{\displaystyle \langle m\mid \psi _{\alpha }\psi _{\beta }^{\dagger }\mid m\rangle =\delta _{\alpha ,\beta }\langle m\mid \zeta \psi _{\alpha }^{\dagger }\psi _{\alpha }+1\mid m\rangle }
and the fact that the thermal average of the number operator gives the Bose–Einstein or Fermi–Dirac distribution function.
Finally, the spectral density simplifies to give
ρ
α
β
=
2
π
δ
(
ξ
α
−
ω
)
δ
α
β
,
{\displaystyle \rho _{\alpha \beta }=2\pi \delta (\xi _{\alpha }-\omega )\delta _{\alpha \beta },}
so that the thermal Green function is
G
α
β
(
ω
n
)
=
δ
α
β
−
i
ω
n
+
ξ
β
{\displaystyle {\mathcal {G}}_{\alpha \beta }(\omega _{n})={\frac {\delta _{\alpha \beta }}{-i\omega _{n}+\xi _{\beta }}}}
and the retarded Green function is
G
α
β
(
ω
)
=
δ
α
β
−
(
ω
+
i
η
)
+
ξ
β
.
{\displaystyle G_{\alpha \beta }(\omega )={\frac {\delta _{\alpha \beta }}{-(\omega +i\eta )+\xi _{\beta }}}.}
Note that the noninteracting Green function is diagonal, but this will not be true in the interacting case.
== See also ==
Fluctuation theorem
Green–Kubo relations
Linear response function
Lindblad equation
Propagator
Correlation function (quantum field theory)
Numerical analytic continuation
== References ==
=== Books ===
Bonch-Bruevich V. L., Tyablikov S. V. (1962): The Green Function Method in Statistical Mechanics. North Holland Publishing Co.
Abrikosov, A. A., Gorkov, L. P. and Dzyaloshinski, I. E. (1963): Methods of Quantum Field Theory in Statistical Physics Englewood Cliffs: Prentice-Hall.
Negele, J. W. and Orland, H. (1988): Quantum Many-Particle Systems AddisonWesley.
Zubarev D. N., Morozov V., Ropke G. (1996): Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory (Vol. 1). John Wiley & Sons. ISBN 3-05-501708-0.
Mattuck Richard D. (1992), A Guide to Feynman Diagrams in the Many-Body Problem, Dover Publications, ISBN 0-486-67047-3.
=== Papers ===
Bogolyubov N. N., Tyablikov S. V. Retarded and advanced Green functions in statistical physics, Soviet Physics Doklady, Vol. 4, p. 589 (1959).
Zubarev D. N., Double-time Green functions in statistical physics, Soviet Physics Uspekhi 3(3), 320–345 (1960).
== External links ==
Linear Response Functions in Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.): DMFT at 25: Infinite Dimensions, Verlag des Forschungszentrum Jülich, 2014 ISBN 978-3-89336-953-9 | Wikipedia/Green's_function_(many-body_theory) |
In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is the mechanism of interaction between subatomic particles that is responsible for the radioactive decay of atoms: The weak interaction participates in nuclear fission and nuclear fusion. The theory describing its behaviour and effects is sometimes called quantum flavordynamics (QFD); however, the term QFD is rarely used, because the weak force is better understood by electroweak theory (EWT).
The effective range of the weak force is limited to subatomic distances and is less than the diameter of a proton.
== Background ==
The Standard Model of particle physics provides a uniform framework for understanding electromagnetic, weak, and strong interactions. An interaction occurs when two particles (typically, but not necessarily, half-integer spin fermions) exchange integer-spin, force-carrying bosons. The fermions involved in such exchanges can be either electric (e.g., electrons or quarks) or composite (e.g. protons or neutrons), although at the deepest levels, all weak interactions ultimately are between elementary particles.
In the weak interaction, fermions can exchange three types of force carriers, namely W+, W−, and Z bosons. The masses of these bosons are far greater than the mass of a proton or neutron, which is consistent with the short range of the weak force. In fact, the force is termed weak because its field strength over any set distance is typically several orders of magnitude less than that of the electromagnetic force, which itself is further orders of magnitude less than the strong nuclear force.
The weak interaction is the only fundamental interaction that breaks parity symmetry, and similarly, but far more rarely, the only interaction to break charge–parity symmetry.
Quarks, which make up composite particles like neutrons and protons, come in six "flavours" – up, down, charm, strange, top and bottom – which give those composite particles their properties. The weak interaction is unique in that it allows quarks to swap their flavour for another. The swapping of those properties is mediated by the force carrier bosons. For example, during beta-minus decay, a down quark within a neutron is changed into an up quark, thus converting the neutron to a proton and resulting in the emission of an electron and an electron antineutrino.
Weak interaction is important in the fusion of hydrogen into helium in a star. This is because it can convert a proton (hydrogen) into a neutron to form deuterium which is important for the continuation of nuclear fusion to form helium. The accumulation of neutrons facilitates the buildup of heavy nuclei in a star.
Most fermions decay by a weak interaction over time. Such decay makes radiocarbon dating possible, as carbon-14 decays through the weak interaction to nitrogen-14. It can also create radioluminescence, commonly used in tritium luminescence, and in the related field of betavoltaics (but not similar to radium luminescence).
The electroweak force is believed to have separated into the electromagnetic and weak forces during the quark epoch of the early universe.
== History ==
In 1933, Enrico Fermi proposed the first theory of the weak interaction, known as Fermi's interaction. He suggested that beta decay could be explained by a four-fermion interaction, involving a contact force with no range.
In the mid-1950s, Chen-Ning Yang and Tsung-Dao Lee first suggested that the handedness of the spins of particles in weak interaction might violate the conservation law or symmetry. In 1957, the Wu experiment, carried by Chien Shiung Wu and collaborators confirmed the symmetry violation.
In the 1960s, Sheldon Glashow, Abdus Salam and Steven Weinberg unified the electromagnetic force and the weak interaction by showing them to be two aspects of a single force, now termed the electroweak force.
The existence of the W and Z bosons was not directly confirmed until 1983.(p8)
== Properties ==
The electrically charged weak interaction is unique in a number of respects:
It is the only interaction that can change the flavour of quarks and leptons (i.e., of changing one type of quark into another).
It is the only interaction that violates P, or parity symmetry. It is also the only one that violates charge–parity (CP) symmetry.
Both the electrically charged and the electrically neutral interactions are mediated (propagated) by force carrier particles that have significant masses, an unusual feature which is explained in the Standard Model by the Higgs mechanism.
Due to their large mass (approximately 90 GeV/c2) these carrier particles, called the W and Z bosons, are short-lived with a lifetime of under 10−24 seconds. The weak interaction has a coupling constant (an indicator of how frequently interactions occur) between 10−7 and 10−6, compared to the electromagnetic coupling constant of about 10−2 and the strong interaction coupling constant of about 1; consequently the weak interaction is "weak" in terms of intensity. The weak interaction has a very short effective range (around 10−17 to 10−16 m (0.01 to 0.1 fm)). At distances around 10−18 meters (0.001 fm), the weak interaction has an intensity of a similar magnitude to the electromagnetic force, but this starts to decrease exponentially with increasing distance. Scaled up by just one and a half orders of magnitude, at distances of around 3×10−17 m, the weak interaction becomes 10,000 times weaker.
The weak interaction affects all the fermions of the Standard Model, as well as the Higgs boson; neutrinos interact only through gravity and the weak interaction. The weak interaction does not produce bound states, nor does it involve binding energy – something that gravity does on an astronomical scale, the electromagnetic force does at the molecular and atomic levels, and the strong nuclear force does only at the subatomic level, inside of nuclei.
Its most noticeable effect is due to its first unique feature: The charged weak interaction causes flavour change. For example, a neutron is heavier than a proton (its partner nucleon) and can decay into a proton by changing the flavour (type) of one of its two down quarks to an up quark. Neither the strong interaction nor electromagnetism permit flavour changing, so this can only proceed by weak decay; without weak decay, quark properties such as strangeness and charm (associated with the strange quark and charm quark, respectively) would also be conserved across all interactions.
All mesons are unstable because of weak decay.(p29)
In the process known as beta decay, a down quark in the neutron can change into an up quark by emitting a virtual W− boson, which then decays into an electron and an electron antineutrino.(p28) Another example is electron capture – a common variant of radioactive decay – wherein a proton and an electron within an atom interact and are changed to a neutron (an up quark is changed to a down quark), and an electron neutrino is emitted.
Due to the large masses of the W bosons, particle transformations or decays (e.g., flavour change) that depend on the weak interaction typically occur much more slowly than transformations or decays that depend only on the strong or electromagnetic forces.
For example, a neutral pion decays electromagnetically, and so has a life of only about 10−16 seconds. In contrast, a charged pion can only decay through the weak interaction, and so lives about 10−8 seconds, or a hundred million times longer than a neutral pion.(p30) A particularly extreme example is the weak-force decay of a free neutron, which takes about 15 minutes.(p28)
=== Weak isospin and weak hypercharge ===
All particles have a property called weak isospin (symbol T3), which serves as an additive quantum number that restricts how the particle can interact with the W± of the weak force. Weak isospin plays the same role in the weak interaction with W± as electric charge does in electromagnetism, and color charge in the strong interaction; a different number with a similar name, weak charge, discussed below, is used for interactions with the Z0. All left-handed fermions have a weak isospin value of either ++1/2 or −+1/2; all right-handed fermions have 0 isospin. For example, the up quark has T3 = ++1/2 and the down quark has T3 = −+1/2. A quark never decays through the weak interaction into a quark of the same T3: Quarks with a T3 of ++1/2 only decay into quarks with a T3 of −+1/2 and conversely.
In any given strong, electromagnetic, or weak interaction, weak isospin is conserved: The sum of the weak isospin numbers of the particles entering the interaction equals the sum of the weak isospin numbers of the particles exiting that interaction. For example, a (left-handed) π+, with a weak isospin of +1 normally decays into a νμ (with T3 = ++1/2) and a μ+ (as a right-handed antiparticle, ++1/2).(p30)
For the development of the electroweak theory, another property, weak hypercharge, was invented, defined as
Y
W
=
2
(
Q
−
T
3
)
,
{\displaystyle Y_{\text{W}}=2\,(Q-T_{3}),}
where YW is the weak hypercharge of a particle with electrical charge Q (in elementary charge units) and weak isospin T3. Weak hypercharge is the generator of the U(1) component of the electroweak gauge group; whereas some particles have a weak isospin of zero, all known spin-1/2 particles have a non-zero weak hypercharge.
== Interaction types ==
There are two types of weak interaction (called vertices). The first type is called the "charged-current interaction" because the weakly interacting fermions form a current with total electric charge that is nonzero. The second type is called the "neutral-current interaction" because the weakly interacting fermions form a current with total electric charge of zero. It is responsible for the (rare) deflection of neutrinos. The two types of interaction follow different selection rules. This naming convention is often misunderstood to label the electric charge of the W and Z bosons, however the naming convention predates the concept of the mediator bosons, and clearly (at least in name) labels the charge of the current (formed from the fermions), not necessarily the bosons.
=== Charged-current interaction ===
In one type of charged current interaction, a charged lepton (such as an electron or a muon, having a charge of −1) can absorb a W+ boson (a particle with a charge of +1) and be thereby converted into a corresponding neutrino (with a charge of 0), where the type ("flavour") of neutrino (electron νe, muon νμ, or tau ντ) is the same as the type of lepton in the interaction, for example:
μ
−
+
W
+
→
ν
μ
{\displaystyle \mu ^{-}+\mathrm {W} ^{+}\to \nu _{\mu }}
Similarly, a down-type quark (d, s, or b, with a charge of −+ 1 /3) can be converted into an up-type quark (u, c, or t, with a charge of ++ 2 /3), by emitting a W− boson or by absorbing a W+ boson. More precisely, the down-type quark becomes a quantum superposition of up-type quarks: that is to say, it has a possibility of becoming any one of the three up-type quarks, with the probabilities given in the CKM matrix tables. Conversely, an up-type quark can emit a W+ boson, or absorb a W− boson, and thereby be converted into a down-type quark, for example:
d
→
u
+
W
−
d
+
W
+
→
u
c
→
s
+
W
+
c
+
W
−
→
s
{\displaystyle {\begin{aligned}\mathrm {d} &\to \mathrm {u} +\mathrm {W} ^{-}\\\mathrm {d} +\mathrm {W} ^{+}&\to \mathrm {u} \\\mathrm {c} &\to \mathrm {s} +\mathrm {W} ^{+}\\\mathrm {c} +\mathrm {W} ^{-}&\to \mathrm {s} \end{aligned}}}
The W boson is unstable so will rapidly decay, with a very short lifetime. For example:
W
−
→
e
−
+
ν
¯
e
W
+
→
e
+
+
ν
e
{\displaystyle {\begin{aligned}\mathrm {W} ^{-}&\to \mathrm {e} ^{-}+{\bar {\nu }}_{\mathrm {e} }~\\\mathrm {W} ^{+}&\to \mathrm {e} ^{+}+\nu _{\mathrm {e} }~\end{aligned}}}
Decay of a W boson to other products can happen, with varying probabilities.
In the so-called beta decay of a neutron (see picture, above), a down quark within the neutron emits a virtual W− boson and is thereby converted into an up quark, converting the neutron into a proton. Because of the limited energy involved in the process (i.e., the mass difference between the down quark and the up quark), the virtual W− boson can only carry sufficient energy to produce an electron and an electron-antineutrino – the two lowest-possible masses among its prospective decay products.
At the quark level, the process can be represented as:
d
→
u
+
e
−
+
ν
¯
e
{\displaystyle \mathrm {d} \to \mathrm {u} +\mathrm {e} ^{-}+{\bar {\nu }}_{\mathrm {e} }~}
=== Neutral-current interaction ===
In neutral current interactions, a quark or a lepton (e.g., an electron or a muon) emits or absorbs a neutral Z boson. For example:
e
−
→
e
−
+
Z
0
{\displaystyle \mathrm {e} ^{-}\to \mathrm {e} ^{-}+\mathrm {Z} ^{0}}
Like the W± bosons, the Z0 boson also decays rapidly, for example:
Z
0
→
b
+
b
¯
{\displaystyle \mathrm {Z} ^{0}\to \mathrm {b} +{\bar {\mathrm {b} }}}
Unlike the charged-current interaction, whose selection rules are strictly limited by chirality, electric charge, and / or weak isospin, the neutral-current Z0 interaction can cause any two fermions in the standard model to deflect: Either particles or anti-particles, with any electric charge, and both left- and right-chirality, although the strength of the interaction differs.
The quantum number weak charge (QW) serves the same role in the neutral current interaction with the Z0 that electric charge (Q, with no subscript) does in the electromagnetic interaction: It quantifies the vector part of the interaction. Its value is given by:
Q
w
=
2
T
3
−
4
Q
sin
2
θ
w
=
2
T
3
−
Q
+
(
1
−
4
sin
2
θ
w
)
Q
.
{\displaystyle Q_{\mathsf {w}}=2\,T_{3}-4\,Q\,\sin ^{2}\theta _{\mathsf {w}}=2\,T_{3}-Q+(1-4\,\sin ^{2}\theta _{\mathsf {w}})\,Q~.}
Since the weak mixing angle
θ
w
≈
29
∘
{\displaystyle \theta _{\mathsf {w}}\approx 29^{\circ }}
, the parenthetic expression
(
1
−
4
sin
2
θ
w
)
≈
0.060
{\displaystyle (1-4\,\sin ^{2}\theta _{\mathsf {w}})\approx 0.060}
, with its value varying slightly with the momentum difference (called "running") between the particles involved. Hence
Q
w
≈
2
T
3
−
Q
=
sgn
(
Q
)
(
1
−
|
Q
|
)
,
{\displaystyle \ Q_{\mathsf {w}}\approx 2\ T_{3}-Q=\operatorname {sgn}(Q)\ {\big (}1-|Q|{\big )}\ ,}
since by convention
sgn
T
3
≡
sgn
Q
{\displaystyle \operatorname {sgn} T_{3}\equiv \operatorname {sgn} Q}
, and for all fermions involved in the weak interaction
T
3
=
±
1
2
{\displaystyle T_{3}=\pm {\tfrac {1}{2}}}
. The weak charge of charged leptons is then close to zero, so these mostly interact with the Z boson through the axial coupling.
== Electroweak theory ==
The Standard Model of particle physics describes the electromagnetic interaction and the weak interaction as two different aspects of a single electroweak interaction. This theory was developed around 1968 by Sheldon Glashow, Abdus Salam, and Steven Weinberg, and they were awarded the 1979 Nobel Prize in Physics for their work. The Higgs mechanism provides an explanation for the presence of three massive gauge bosons (W+, W−, Z0, the three carriers of the weak interaction), and the photon (γ, the massless gauge boson that carries the electromagnetic interaction).
According to the electroweak theory, at very high energies, the universe has four components of the Higgs field whose interactions are carried by four massless scalar bosons forming a complex scalar Higgs field doublet. Likewise, there are four massless electroweak vector bosons, each similar to the photon. However, at low energies, this gauge symmetry is spontaneously broken down to the U(1) symmetry of electromagnetism, since one of the Higgs fields acquires a vacuum expectation value. Naïvely, the symmetry-breaking would be expected to produce three massless bosons, but instead those "extra" three Higgs bosons become incorporated into the three weak bosons, which then acquire mass through the Higgs mechanism. These three composite bosons are the W+, W−, and Z0 bosons actually observed in the weak interaction. The fourth electroweak gauge boson is the photon (γ) of electromagnetism, which does not couple to any of the Higgs fields and so remains massless.
This theory has made a number of predictions, including a prediction of the masses of the Z and W bosons before their discovery and detection in 1983.
On 4 July 2012, the CMS and the ATLAS experimental teams at the Large Hadron Collider independently announced that they had confirmed the formal discovery of a previously unknown boson of mass between 125 and 127 GeV/c2, whose behaviour so far was "consistent with" a Higgs boson, while adding a cautious note that further data and analysis were needed before positively identifying the new boson as being a Higgs boson of some type. By 14 March 2013, a Higgs boson was tentatively confirmed to exist.
In a speculative case where the electroweak symmetry breaking scale were lowered, the unbroken SU(2) interaction would eventually become confining. Alternative models where SU(2) becomes confining above that scale appear quantitatively similar to the Standard Model at lower energies, but dramatically different above symmetry breaking.
== Violation of symmetry ==
The laws of nature were long thought to remain the same under mirror reflection. The results of an experiment viewed via a mirror were expected to be identical to the results of a separately constructed, mirror-reflected copy of the experimental apparatus watched through the mirror. This so-called law of parity conservation was known to be respected by classical gravitation, electromagnetism and the strong interaction; it was assumed to be a universal law. However, in the mid-1950s Chen-Ning Yang and Tsung-Dao Lee suggested that the weak interaction might violate this law. Chien Shiung Wu and collaborators in 1957 discovered that the weak interaction violates parity, earning Yang and Lee the 1957 Nobel Prize in Physics.
Although the weak interaction was once described by Fermi's theory, the discovery of parity violation and renormalization theory suggested that a new approach was needed. In 1957, Robert Marshak and George Sudarshan and, somewhat later, Richard Feynman and Murray Gell-Mann proposed a V − A (vector minus axial vector or left-handed) Lagrangian for weak interactions. In this theory, the weak interaction acts only on left-handed particles (and right-handed antiparticles). Since the mirror reflection of a left-handed particle is right-handed, this explains the maximal violation of parity. The V − A theory was developed before the discovery of the Z boson, so it did not include the right-handed fields that enter in the neutral current interaction.
However, this theory allowed a compound symmetry CP to be conserved. CP combines parity P (switching left to right) with charge conjugation C (switching particles with antiparticles). Physicists were again surprised when in 1964, James Cronin and Val Fitch provided clear evidence in kaon decays that CP symmetry could be broken too, winning them the 1980 Nobel Prize in Physics. In 1973, Makoto Kobayashi and Toshihide Maskawa showed that CP violation in the weak interaction required more than two generations of particles, effectively predicting the existence of a then unknown third generation. This discovery earned them half of the 2008 Nobel Prize in Physics.
Unlike parity violation, CP violation occurs only in rare circumstances. Despite its limited occurrence under present conditions, it is widely believed to be the reason that there is much more matter than antimatter in the universe, and thus forms one of Andrei Sakharov's three conditions for baryogenesis.
== See also ==
Weakless universe – the postulate that weak interactions are not anthropically necessary
Gravity
Strong interaction
Electromagnetism
== Footnotes ==
== References ==
== Sources ==
=== Technical ===
=== For general readers ===
== External links ==
Harry Cheung, The Weak Force @Fermilab
Fundamental Forces @Hyperphysics, Georgia State University.
Brian Koberlein, What is the weak force? | Wikipedia/Quantum_flavordynamics |
Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by Rudolf Haag and Daniel Kastler (1964). The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those.
== Haag–Kastler axioms ==
Let
O
{\displaystyle {\mathcal {O}}}
be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a set
{
A
(
O
)
}
O
∈
O
{\displaystyle \{{\mathcal {A}}(O)\}_{O\in {\mathcal {O}}}}
of von Neumann algebras
A
(
O
)
{\displaystyle {\mathcal {A}}(O)}
on a common Hilbert space
H
{\displaystyle {\mathcal {H}}}
satisfying the following axioms:
Isotony:
O
1
⊂
O
2
{\displaystyle O_{1}\subset O_{2}}
implies
A
(
O
1
)
⊂
A
(
O
2
)
{\displaystyle {\mathcal {A}}(O_{1})\subset {\mathcal {A}}(O_{2})}
.
Causality: If
O
1
{\displaystyle O_{1}}
is space-like separated from
O
2
{\displaystyle O_{2}}
, then
[
A
(
O
1
)
,
A
(
O
2
)
]
=
0
{\displaystyle [{\mathcal {A}}(O_{1}),{\mathcal {A}}(O_{2})]=0}
.
Poincaré covariance: A strongly continuous unitary representation
U
(
P
)
{\displaystyle U({\mathcal {P}})}
of the Poincaré group
P
{\displaystyle {\mathcal {P}}}
on
H
{\displaystyle {\mathcal {H}}}
exists such that
A
(
g
O
)
=
U
(
g
)
A
(
O
)
U
(
g
)
∗
,
g
∈
P
.
{\displaystyle {\mathcal {A}}(gO)=U(g){\mathcal {A}}(O)U(g)^{*},\,\,g\in {\mathcal {P}}.}
Spectrum condition: The joint spectrum
S
p
(
P
)
{\displaystyle \mathrm {Sp} (P)}
of the energy-momentum operator
P
{\displaystyle P}
(i.e. the generator of space-time translations) is contained in the closed forward lightcone.
Existence of a vacuum vector: A cyclic and Poincaré-invariant vector
Ω
∈
H
{\displaystyle \Omega \in {\mathcal {H}}}
exists.
The net algebras
A
(
O
)
{\displaystyle {\mathcal {A}}(O)}
are called local algebras and the C* algebra
A
:=
⋃
O
∈
O
A
(
O
)
¯
{\displaystyle {\mathcal {A}}:={\overline {\bigcup _{O\in {\mathcal {O}}}{\mathcal {A}}(O)}}}
is called the quasilocal algebra.
== Category-theoretic formulation ==
Let Mink be the category of open subsets of Minkowski space M with inclusion maps as morphisms. We are given a covariant functor
A
{\displaystyle {\mathcal {A}}}
from Mink to uC*alg, the category of unital C* algebras, such that every morphism in Mink maps to a monomorphism in uC*alg (isotony).
The Poincaré group acts continuously on Mink. There exists a pullback of this action, which is continuous in the norm topology of
A
(
M
)
{\displaystyle {\mathcal {A}}(M)}
(Poincaré covariance).
Minkowski space has a causal structure. If an open set V lies in the causal complement of an open set U, then the image of the maps
A
(
i
U
,
U
∪
V
)
{\displaystyle {\mathcal {A}}(i_{U,U\cup V})}
and
A
(
i
V
,
U
∪
V
)
{\displaystyle {\mathcal {A}}(i_{V,U\cup V})}
commute (spacelike commutativity). If
U
¯
{\displaystyle {\bar {U}}}
is the causal completion of an open set U, then
A
(
i
U
,
U
¯
)
{\displaystyle {\mathcal {A}}(i_{U,{\bar {U}}})}
is an isomorphism (primitive causality).
A state with respect to a C*-algebra is a positive linear functional over it with unit norm. If we have a state over
A
(
M
)
{\displaystyle {\mathcal {A}}(M)}
, we can take the "partial trace" to get states associated with
A
(
U
)
{\displaystyle {\mathcal {A}}(U)}
for each open set via the net monomorphism. The states over the open sets form a presheaf structure.
According to the GNS construction, for each state, we can associate a Hilbert space representation of
A
(
M
)
.
{\displaystyle {\mathcal {A}}(M).}
Pure states correspond to irreducible representations and mixed states correspond to reducible representations. Each irreducible representation (up to equivalence) is called a superselection sector. We assume there is a pure state called the vacuum such that the Hilbert space associated with it is a unitary representation of the Poincaré group compatible with the Poincaré covariance of the net such that if we look at the Poincaré algebra, the spectrum with respect to energy-momentum (corresponding to spacetime translations) lies on and in the positive light cone. This is the vacuum sector.
== QFT in curved spacetime ==
More recently, the approach has been further implemented to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is in particular suitable to generalize the renormalization procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in presence of a black hole have been obtained.
== References ==
== Further reading ==
Haag, Rudolf; Kastler, Daniel (1964), "An Algebraic Approach to Quantum Field Theory", Journal of Mathematical Physics, 5 (7): 848–861, Bibcode:1964JMP.....5..848H, doi:10.1063/1.1704187, ISSN 0022-2488, MR 0165864
Haag, Rudolf (1996) [1992], Local Quantum Physics: Fields, Particles, Algebras, Theoretical and Mathematical Physics (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-61458-3, ISBN 978-3-540-61451-7, MR 1405610
Brunetti, Romeo; Fredenhagen, Klaus; Verch, Rainer (2003). "The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Field Theory". Communications in Mathematical Physics. 237 (1–2): 31–68. arXiv:math-ph/0112041. Bibcode:2003CMaPh.237...31B. doi:10.1007/s00220-003-0815-7. S2CID 13950246.
Brunetti, Romeo; Dütsch, Michael; Fredenhagen, Klaus (2009). "Perturbative Algebraic Quantum Field Theory and the Renormalization Groups". Advances in Theoretical and Mathematical Physics. 13 (5): 1541–1599. arXiv:0901.2038. doi:10.4310/ATMP.2009.v13.n5.a7. S2CID 15493763.
Bär, Christian; Fredenhagen, Klaus, eds. (2009). Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations. Lecture Notes in Physics. Vol. 786. Springer. doi:10.1007/978-3-642-02780-2. ISBN 978-3-642-02780-2.
Brunetti, Romeo; Dappiaggi, Claudio; Fredenhagen, Klaus; Yngvason, Jakob, eds. (2015). Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer. doi:10.1007/978-3-319-21353-8. ISBN 978-3-319-21353-8.
Rejzner, Kasia (2016). Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Mathematical Physics Studies. Springer. arXiv:1208.1428. Bibcode:2016paqf.book.....R. doi:10.1007/978-3-319-25901-7. ISBN 978-3-319-25901-7.
Hack, Thomas-Paul (2016). Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes. SpringerBriefs in Mathematical Physics. Vol. 6. Springer. arXiv:1506.01869. Bibcode:2016caaq.book.....H. doi:10.1007/978-3-319-21894-6. ISBN 978-3-319-21894-6. S2CID 119657309.
Dütsch, Michael (2019). From Classical Field Theory to Perturbative Quantum Field Theory. Progress in Mathematical Physics. Vol. 74. Birkhäuser. doi:10.1007/978-3-030-04738-2. ISBN 978-3-030-04738-2. S2CID 126907045.
Yau, Donald (2019). Homotopical Quantum Field Theory. World Scientific. arXiv:1802.08101. doi:10.1142/11626. ISBN 978-981-121-287-1. S2CID 119168109.
Dedushenko, Mykola (2023). "Snowmass white paper: The quest to define QFT". International Journal of Modern Physics A. 38 (4n05). arXiv:2203.08053. doi:10.1142/S0217751X23300028. S2CID 247450696.
== External links ==
Local Quantum Physics Crossroads 2.0 – A network of scientists working on local quantum physics
Papers – A database of preprints on algebraic QFT
Algebraic Quantum Field Theory – AQFT resources at the University of Hamburg | Wikipedia/Algebraic_quantum_field_theory |
The Gross–Neveu model (GN) is a quantum field theory model of Dirac fermions interacting via four-fermion interactions in 1 spatial and 1 time dimension. It was introduced in 1974 by David Gross and André Neveu
as a toy model for quantum chromodynamics (QCD), the theory of strong interactions. It shares several features of the QCD: GN theory is asymptotically free thus at strong coupling the strength of the interaction gets weaker and the corresponding
β
{\displaystyle \beta }
function of the interaction coupling is negative, the theory has a dynamical mass generation mechanism with
Z
2
{\displaystyle \mathbb {Z} _{2}}
chiral symmetry breaking, and in the large number of flavor (
N
→
∞
{\displaystyle N\to \infty }
) limit, GN theory behaves as 't Hooft's large
N
c
{\displaystyle N_{c}}
limit in QCD.
It is made using a finite, but possibly large number,
N
,
{\displaystyle \ N\ ,}
of Dirac fermion wave functions
ψ
1
,
ψ
2
,
…
,
ψ
N
{\displaystyle \psi _{1},\psi _{2},\ldots ,\psi _{N}}
, indexed below by Latin letter
a
.
{\displaystyle \ a~.}
The model's Lagrangian density is
L
=
ψ
¯
a
(
i
∂
/
−
m
)
ψ
a
+
g
2
2
N
[
ψ
¯
a
ψ
a
]
2
,
{\displaystyle {\mathcal {L}}={\bar {\psi }}_{a}\left(i\ \partial \!\!\!/\ -\ m\right)\psi ^{a}\ +\ {\frac {\ g^{2}}{\ 2\ N\ }}\left[{\bar {\psi }}_{a}\ \psi ^{a}\right]^{2}\ ,}
where the formula uses Einstein summation notation. Each wave function
ψ
a
{\displaystyle \ \psi ^{a}\ }
is a two component (left / right) spinor and
g
{\displaystyle \ g\ }
is the interaction's coupling constant. If the mass
m
{\displaystyle \ m\ }
is zero, the model is chiral symmetric type, otherwise, for non-zero mass, it is classical mass type.
This model has a U(N) global internal symmetry. If one takes
N
=
1
{\displaystyle \ N=1\ }
(which permits only one quartic interaction) and makes no attempt to analytically continue the dimension, the model reduces to the massive Thirring model (which is completely integrable).
It is a 2 dimensional version of the 4 dimensional Nambu–Jona-Lasinio model (NJL), which was introduced 14 years earlier as a model of dynamical chiral symmetry breaking (but no quark confinement) modeled upon the BCS theory of superconductivity. The 2 dimensional version has the advantage that the 4 fermi interaction is renormalizable, which it is not in any higher number of dimensions.
== Features of the theory ==
Gross and Neveu studied this model in the large
N
{\displaystyle \ N\ }
limit, expanding the relevant parameters in a
1
N
{\displaystyle {\tfrac {1}{\ N\ }}\ }
expansion. After demonstrating that this and related models are asymptotically free, they found that, in the subleading order, for small fermion masses the bifermion condensate
ψ
¯
a
ψ
a
{\displaystyle \ {\overline {\psi }}_{a}\ \psi ^{a}\ }
acquires a vacuum expectation value (VEV) and as a result the fundamental fermions become massive. They find that the mass is not analytic in the coupling constant
g
.
{\displaystyle \ g~.}
The vacuum expectation value spontaneously breaks the chiral symmetry of the theory.
More precisely, expanding about the vacuum with no vacuum expectation value for the bilinear condensate they found a tachyon. To do this they solve the renormalization group equations for the propagator of the bifermion field, using the fact that the only renormalization of the coupling constant comes from the wave function renormalization of the composite field. They then calculated, at leading order in a
1
N
{\displaystyle {\tfrac {1}{\ N\ }}\ }
expansion but to all orders in the coupling constant, the dependence of the potential energy on the condensate using the effective action techniques introduced the previous year by Sidney Coleman at the Erice International Summer School of Physics. They found that this potential is minimized at a nonzero value of the condensate, indicating that this is the true value of the condensate. Expanding the theory about the new vacuum, the tachyon was found to be no longer present and in fact, like the BCS theory of superconductivity, there is a mass gap.
They then made a number of general arguments about dynamical mass generation in quantum field theories. For example, they demonstrated that not all masses may be dynamically generated in theories which are infrared-stable, using this to argue that, at least to leading order in
1
N
{\displaystyle {\tfrac {1}{\ N\ }}\ }
the 4 dimensional
ϕ
4
{\displaystyle \ \phi ^{4}\ }
theory does not exist. They also argued that in asymptotically free theories the dynamically generated masses never depend analytically on the coupling constants.
== Generalizations ==
Gross and Neveu considered several generalizations. First, they considered a Lagrangian with one extra quartic interaction
L
=
ψ
¯
a
(
i
∂
/
−
m
)
ψ
a
+
g
2
2
N
(
[
ψ
¯
a
ψ
a
]
2
−
[
ψ
¯
a
γ
5
ψ
a
]
2
)
{\displaystyle {\mathcal {L}}={\bar {\psi }}_{a}\left(i\ \partial \!\!\!/\ -\ m\right)\psi ^{a}\ +\ {\frac {\ g^{2}}{\ 2\ N\ }}(\left[{\bar {\psi }}_{a}\ \psi ^{a}\right]^{2}\ -\ \left[{\bar {\psi }}_{a}\ \gamma _{5}\ \psi ^{a}\right]^{2})}
chosen so that the discrete chiral symmetry
ψ
↦
γ
5
ψ
{\displaystyle \ \psi \mapsto \gamma _{5}\psi \ }
of the original model is enhanced to a continuous U(1)-valued chiral symmetry
ψ
→
e
i
θ
γ
5
ψ
.
{\displaystyle \psi \rightarrow e^{i\theta \gamma _{5}}\psi ~.}
Chiral symmetry breaking occurs as before, caused by the same VEV. However, as the spontaneously broken symmetry is now continuous, a massless Goldstone boson appears in the spectrum. Although this leads to no problems at the leading order in the
1
N
{\displaystyle {\tfrac {1}{\ N\ }}\ }
expansion, massless particles in 2 dimensional quantum field theories inevitably lead to infrared divergences, and so there appears to be no such theory.
Two further modifications of the modified theory, which remedy this problem, were then considered. In one modification one increases the number of dimensions. As a result, the massless field does not lead to divergences. In the other modification, the chiral symmetry is gauged. Consequently, the Golstone boson is "eaten" by the Higgs mechanism as the photon becomes massive, and so does not lead to any divergences.
== See also ==
Dirac equation
Nonlinear Dirac equation
Thirring model
Nambu–Jona-Lasinio model
== References == | Wikipedia/Gross–Neveu_model |
The Minimal Supersymmetric Standard Model (MSSM) is an extension to the Standard Model that realizes supersymmetry. MSSM is the minimal supersymmetrical model as it considers only "the [minimum] number of new particle states and new interactions consistent with "Reality". Supersymmetry pairs bosons with fermions, so every Standard Model particle has a (yet undiscovered) superpartner. If discovered, such superparticles could be candidates for dark matter, and could provide evidence for grand unification or the viability of string theory. The failure to find evidence for MSSM using the Large Hadron Collider has strengthened an inclination to abandon it.
== Background ==
The MSSM was originally proposed in 1981 to stabilize the weak scale, solving the hierarchy problem. The Higgs boson mass of the Standard Model is unstable to quantum corrections and the theory predicts that weak scale should be much weaker than what is observed to be. In the MSSM, the Higgs boson has a fermionic superpartner, the Higgsino, that has the same mass as it would if supersymmetry were an exact symmetry. Because fermion masses are radiatively stable, the Higgs mass inherits this stability. However, in MSSM there is a need for more than one Higgs field, as described below.
The only unambiguous way to claim discovery of supersymmetry is to produce superparticles in the laboratory. Because superparticles are expected to be 100 to 1000 times heavier than the proton, it requires a huge amount of energy to make these particles that can only be achieved at particle accelerators. The Tevatron was actively looking for evidence of the production of supersymmetric particles before it was shut down on 30 September 2011. Most physicists believe that supersymmetry must be discovered at the LHC if it is responsible for stabilizing the weak scale. There are five classes of particle that superpartners of the Standard Model fall into: squarks, gluinos, charginos, neutralinos, and sleptons. These superparticles have their interactions and subsequent decays described by the MSSM and each has characteristic signatures.
The MSSM imposes R-parity to explain the stability of the proton. It adds supersymmetry breaking by introducing explicit soft supersymmetry breaking operators into the Lagrangian that is communicated to it by some unknown (and unspecified) dynamics. This means that there are 120 new parameters in the MSSM. Most of these parameters lead to unacceptable phenomenology such as large flavor changing neutral currents or large electric dipole moments for the neutron and electron. To avoid these problems, the MSSM takes all of the soft supersymmetry breaking to be diagonal in flavor space and for all of the new CP violating phases to vanish.
== Theoretical motivations ==
There are three principal motivations for the MSSM over other theoretical extensions of the Standard Model, namely:
Naturalness
Gauge coupling unification
Dark Matter
These motivations come out without much effort and they are the primary reasons why the MSSM is the leading candidate for a new theory to be discovered at collider experiments such as the Tevatron or the LHC.
=== Naturalness ===
The original motivation for proposing the MSSM was to stabilize the Higgs mass to radiative corrections that are quadratically divergent in the Standard Model (the hierarchy problem). In supersymmetric models, scalars are related to fermions and have the same mass. Since fermion masses are logarithmically divergent, scalar masses inherit the same radiative stability. The Higgs vacuum expectation value (VEV) is related to the negative scalar mass in the Lagrangian. In order for the radiative corrections to the Higgs mass to not be dramatically larger than the actual value, the mass of the superpartners of the Standard Model should not be significantly heavier than the Higgs VEV – roughly 100 GeV. In 2012, the Higgs particle was discovered at the LHC, and its mass was found to be 125–126 GeV.
=== Gauge-coupling unification ===
If the superpartners of the Standard Model are near the TeV scale, then measured gauge couplings of the three gauge groups unify at high energies. The beta-functions for the MSSM gauge couplings are given by
where
α
1
−
1
{\displaystyle \alpha _{1}^{-1}}
is measured in SU(5) normalization—a factor of 3/5 different
than the Standard Model's normalization and predicted by Georgi–Glashow SU(5) .
The condition for gauge coupling unification at one loop is whether the following expression is satisfied
α
3
−
1
−
α
2
−
1
α
2
−
1
−
α
1
−
1
=
b
0
3
−
b
0
2
b
0
2
−
b
0
1
{\displaystyle {\frac {\alpha _{3}^{-1}-\alpha _{2}^{-1}}{\alpha _{2}^{-1}-\alpha _{1}^{-1}}}={\frac {b_{0\,3}-b_{0\,2}}{b_{0\,2}-b_{0\,1}}}}
.
Remarkably, this is precisely satisfied to experimental errors in the values of
α
−
1
(
M
Z
0
)
{\displaystyle \alpha ^{-1}(M_{Z^{0}})}
. There are two loop corrections and both TeV-scale and GUT-scale threshold corrections that alter this condition on gauge coupling unification, and the results of more extensive calculations reveal that gauge coupling unification occurs to an accuracy of 1%, though this is about 3 standard deviations from the theoretical expectations.
This prediction is generally considered as indirect evidence for both the MSSM and SUSY GUTs. Gauge coupling unification does not necessarily imply grand unification and there exist other mechanisms to reproduce gauge coupling unification. However, if superpartners are found in the near future, the apparent success of gauge coupling unification would suggest that a supersymmetric grand unified theory is a promising candidate for high scale physics.
=== Dark matter ===
If R-parity is preserved, then the lightest superparticle (LSP) of the MSSM is stable and is a Weakly interacting massive particle (WIMP) – i.e. it does not have electromagnetic or strong interactions. This makes the LSP a good dark matter candidate, and falls into the category of cold dark matter (CDM).
== Predictions of the MSSM regarding hadron colliders ==
The Tevatron and LHC have active experimental programs searching for supersymmetric particles. Since both of these machines are hadron colliders – proton antiproton for the Tevatron and proton proton for the LHC – they search best for strongly interacting particles. Therefore, most experimental signature involve production of squarks or gluinos. Since the MSSM has R-parity, the lightest supersymmetric particle is stable and after the squarks and gluinos decay each decay chain will contain one LSP that will leave the detector unseen. This leads to the generic prediction that the MSSM will produce a 'missing energy' signal from these particles leaving the detector.
=== Neutralinos ===
There are four neutralinos that are fermions and are electrically neutral, the lightest of which is typically stable. They are typically labeled N͂01, N͂02, N͂03, N͂04 (although sometimes
χ
~
1
0
,
…
,
χ
~
4
0
{\displaystyle {\tilde {\chi }}_{1}^{0},\ldots ,{\tilde {\chi }}_{4}^{0}}
is used instead). These four states are mixtures of the bino and the neutral wino (which are the neutral electroweak gauginos), and the neutral higgsinos. As the neutralinos are Majorana fermions, each of them is identical with its antiparticle. Because these particles only interact with the weak vector bosons, they are not directly produced at hadron colliders in copious numbers. They primarily appear as particles in cascade decays of heavier particles usually originating from colored supersymmetric particles such as squarks or gluinos.
In R-parity conserving models, the lightest neutralino is stable and all supersymmetric cascade decays end up decaying into this particle which leaves the detector unseen and its existence can only be inferred by looking for unbalanced momentum in a detector.
The heavier neutralinos typically decay through a Z0 to a lighter neutralino or through a W± to chargino. Thus a typical decay is
Note that the “Missing energy” byproduct represents the mass-energy of the neutralino ( N͂01 ) and in the second line, the mass-energy of a neutrino-antineutrino pair ( ν + ν ) produced with the lepton and antilepton in the final decay, all of which are undetectable in individual reactions with current technology.
The mass splittings between the different neutralinos will dictate which patterns of decays are allowed.
=== Charginos ===
There are two charginos that are fermions and are electrically charged. They are typically labeled C͂±1 and C͂±2 (although sometimes
χ
~
1
±
{\displaystyle {\tilde {\chi }}_{1}^{\pm }}
and
χ
~
2
±
{\displaystyle {\tilde {\chi }}_{2}^{\pm }}
is used instead). The heavier chargino can decay through Z0 to the lighter chargino. Both can decay through a W± to neutralino.
=== Squarks ===
The squarks are the scalar superpartners of the quarks and there is one version for each Standard Model quark. Due to phenomenological constraints from flavor changing neutral currents, typically the lighter two generations of squarks have to be nearly the same in mass and therefore are not given distinct names. The superpartners of the top and bottom quark can be split from the lighter squarks and are called stop and sbottom.
In the other direction, there may be a remarkable left-right mixing of the stops
t
~
{\displaystyle {\tilde {t}}}
and of the sbottoms
b
~
{\displaystyle {\tilde {b}}}
because of the high masses of the partner quarks top and bottom:
t
~
1
=
e
+
i
ϕ
cos
(
θ
)
t
L
~
+
sin
(
θ
)
t
R
~
{\displaystyle {\tilde {t}}_{1}=e^{+i\phi }\cos(\theta ){\tilde {t_{L}}}+\sin(\theta ){\tilde {t_{R}}}}
t
~
2
=
e
−
i
ϕ
cos
(
θ
)
t
R
~
−
sin
(
θ
)
t
L
~
{\displaystyle {\tilde {t}}_{2}=e^{-i\phi }\cos(\theta ){\tilde {t_{R}}}-\sin(\theta ){\tilde {t_{L}}}}
A similar story holds for bottom
b
~
{\displaystyle {\tilde {b}}}
with its own parameters
ϕ
{\displaystyle \phi }
and
θ
{\displaystyle \theta }
.
Squarks can be produced through strong interactions and therefore are easily produced at hadron colliders. They decay to quarks and neutralinos or charginos which further decay. In R-parity conserving scenarios, squarks are pair produced and therefore a typical signal is
q
~
q
¯
~
→
q
N
~
1
0
q
¯
N
~
1
0
→
{\displaystyle {\tilde {q}}{\tilde {\bar {q}}}\rightarrow q{\tilde {N}}_{1}^{0}{\bar {q}}{\tilde {N}}_{1}^{0}\rightarrow }
2 jets + missing energy
q
~
q
¯
~
→
q
N
~
2
0
q
¯
N
~
1
0
→
q
N
~
1
0
ℓ
ℓ
¯
q
¯
N
~
1
0
→
{\displaystyle {\tilde {q}}{\tilde {\bar {q}}}\rightarrow q{\tilde {N}}_{2}^{0}{\bar {q}}{\tilde {N}}_{1}^{0}\rightarrow q{\tilde {N}}_{1}^{0}\ell {\bar {\ell }}{\bar {q}}{\tilde {N}}_{1}^{0}\rightarrow }
2 jets + 2 leptons + missing energy
=== Gluinos ===
Gluinos are Majorana fermionic partners of the gluon which means that they are their own antiparticles. They interact strongly and therefore can be produced significantly at the LHC. They can only decay to a quark and a squark and thus a typical gluino signal is
g
~
g
~
→
(
q
q
¯
~
)
(
q
¯
q
~
)
→
(
q
q
¯
N
~
1
0
)
(
q
¯
q
N
~
1
0
)
→
{\displaystyle {\tilde {g}}{\tilde {g}}\rightarrow (q{\tilde {\bar {q}}})({\bar {q}}{\tilde {q}})\rightarrow (q{\bar {q}}{\tilde {N}}_{1}^{0})({\bar {q}}q{\tilde {N}}_{1}^{0})\rightarrow }
4 jets + Missing energy
Because gluinos are Majorana, gluinos can decay to either a quark+anti-squark or an anti-quark+squark with equal probability. Therefore, pairs of gluinos can decay to
g
~
g
~
→
(
q
¯
q
~
)
(
q
¯
q
~
)
→
(
q
q
¯
C
~
1
+
)
(
q
q
¯
C
~
1
+
)
→
(
q
q
¯
W
+
)
(
q
q
¯
W
+
)
→
{\displaystyle {\tilde {g}}{\tilde {g}}\rightarrow ({\bar {q}}{\tilde {q}})({\bar {q}}{\tilde {q}})\rightarrow (q{\bar {q}}{\tilde {C}}_{1}^{+})(q{\bar {q}}{\tilde {C}}_{1}^{+})\rightarrow (q{\bar {q}}W^{+})(q{\bar {q}}W^{+})\rightarrow }
4 jets+
ℓ
+
ℓ
+
{\displaystyle \ell ^{+}\ell ^{+}}
+ Missing energy
This is a distinctive signature because it has same-sign di-leptons and has very little background in the Standard Model.
=== Sleptons ===
Sleptons are the scalar partners of the leptons of the Standard Model. They are not strongly interacting and therefore are not produced very often at hadron colliders unless they are very light.
Because of the high mass of the tau lepton there will be left-right mixing of the stau similar to that of stop and sbottom (see above).
Sleptons will typically be found in decays of a charginos and neutralinos if they are light enough to be a decay product.
C
~
+
→
ℓ
~
+
ν
{\displaystyle {\tilde {C}}^{+}\rightarrow {\tilde {\ell }}^{+}\nu }
N
~
0
→
ℓ
~
+
ℓ
−
{\displaystyle {\tilde {N}}^{0}\rightarrow {\tilde {\ell }}^{+}\ell ^{-}}
== MSSM fields ==
Fermions have bosonic superpartners (called sfermions), and bosons have fermionic superpartners (called bosinos). For most of the Standard Model particles, doubling is very straightforward. However, for the Higgs boson, it is more complicated.
A single Higgsino (the fermionic superpartner of the Higgs boson) would lead to a gauge anomaly and would cause the theory to be inconsistent. However, if two Higgsinos are added, there is no gauge anomaly. The simplest theory is one with two Higgsinos and therefore two scalar Higgs doublets.
Another reason for having two scalar Higgs doublets rather than one is in order to have Yukawa couplings between the Higgs and both down-type quarks and up-type quarks; these are the terms responsible for the quarks' masses. In the Standard Model the down-type quarks couple to the Higgs field (which has Y=−1/2) and the up-type quarks to its complex conjugate (which has Y=+1/2). However, in a supersymmetric theory this is not allowed, so two types of Higgs fields are needed.
=== MSSM superfields ===
In supersymmetric theories, every field and its superpartner can be written together as a superfield. The superfield formulation of supersymmetry is very convenient to write down manifestly supersymmetric theories (i.e. one does not have to tediously check that the theory is supersymmetric term by term in the Lagrangian). The MSSM contains vector superfields associated with the Standard Model gauge groups which contain the vector bosons and associated gauginos. It also contains chiral superfields for the Standard Model fermions and Higgs bosons (and their respective superpartners).
=== MSSM Higgs mass ===
The MSSM Higgs mass is a prediction of the Minimal Supersymmetric Standard Model. The mass of the lightest Higgs boson is set by the Higgs quartic coupling. Quartic couplings are not soft supersymmetry-breaking parameters since they lead to a quadratic divergence of the Higgs mass. Furthermore, there are no supersymmetric parameters to make the Higgs mass a free parameter in the MSSM (though not in non-minimal extensions). This means that Higgs mass is a prediction of the MSSM. The LEP II and the IV experiments placed a lower limit on the Higgs mass of 114.4 GeV. This lower limit is significantly above where the MSSM would typically predict it to be but does not rule out the MSSM; the discovery of the Higgs with a mass of 125 GeV is within the maximal upper bound of approximately 130 GeV that loop corrections within the MSSM would raise the Higgs mass to. Proponents of the MSSM point out that a Higgs mass within the upper bound of the MSSM calculation of the Higgs mass is a successful prediction, albeit pointing to more fine tuning than expected.
==== Formulas ====
The only susy-preserving operator that creates a quartic coupling for the Higgs in the MSSM arise for the D-terms of the SU(2) and U(1) gauge sector and the magnitude of the quartic coupling is set by the size of the gauge couplings.
This leads to the prediction that the Standard Model-like Higgs mass (the scalar that couples approximately to the VEV) is limited to be less than the Z mass:
m
h
0
2
≤
m
Z
0
2
cos
2
2
β
{\displaystyle m_{h^{0}}^{2}\leq m_{Z^{0}}^{2}\cos ^{2}2\beta }
.
Since supersymmetry is broken, there are radiative corrections to the quartic coupling that can increase the Higgs mass. These dominantly arise from the 'top sector':
m
h
0
2
≤
m
Z
0
2
cos
2
2
β
+
3
π
2
m
t
4
sin
4
β
v
2
log
m
t
~
m
t
{\displaystyle m_{h^{0}}^{2}\leq m_{Z^{0}}^{2}\cos ^{2}2\beta +{\frac {3}{\pi ^{2}}}{\frac {m_{t}^{4}\sin ^{4}\beta }{v^{2}}}\log {\frac {m_{\tilde {t}}}{m_{t}}}}
where
m
t
{\displaystyle m_{t}}
is the top mass and
m
t
~
{\displaystyle m_{\tilde {t}}}
is the mass of the top squark. This result can be interpreted as the RG running of the Higgs quartic coupling from the scale of supersymmetry to the top mass—however since the top squark mass should be relatively close to the top mass, this is usually a fairly modest contribution and increases the Higgs mass to roughly the LEP II bound of 114 GeV before the top squark becomes too heavy.
Finally there is a contribution from the top squark A-terms:
L
=
y
t
m
t
~
a
h
u
q
~
3
u
~
3
c
{\displaystyle {\mathcal {L}}=y_{t}\,m_{\tilde {t}}\,a\;h_{u}{\tilde {q}}_{3}{\tilde {u}}_{3}^{c}}
where
a
{\displaystyle a}
is a dimensionless number. This contributes an additional term to the Higgs mass at loop level, but is not logarithmically enhanced
m
h
0
2
≤
m
Z
0
2
cos
2
2
β
+
3
π
2
m
t
4
sin
4
β
v
2
(
log
m
t
~
m
t
+
a
2
(
1
−
a
2
/
12
)
)
{\displaystyle m_{h^{0}}^{2}\leq m_{Z^{0}}^{2}\cos ^{2}2\beta +{\frac {3}{\pi ^{2}}}{\frac {m_{t}^{4}\sin ^{4}\beta }{v^{2}}}\left(\log {\frac {m_{\tilde {t}}}{m_{t}}}+a^{2}(1-a^{2}/12)\right)}
by pushing
a
→
6
{\displaystyle a\rightarrow {\sqrt {6}}}
(known as 'maximal mixing') it is possible to push the Higgs mass to 125 GeV without decoupling the top squark or adding new dynamics to the MSSM.
As the Higgs was found at around 125 GeV (along with no other superparticles) at the LHC, this strongly hints at new dynamics beyond the MSSM, such as the 'Next to Minimal Supersymmetric Standard Model' (NMSSM); and suggests some correlation to the little hierarchy problem.
== MSSM Lagrangian ==
The Lagrangian for the MSSM contains several pieces.
The first is the Kähler potential for the matter and Higgs fields which produces the kinetic terms for the fields.
The second piece is the gauge field superpotential that produces the kinetic terms for the gauge bosons and gauginos.
The next term is the superpotential for the matter and Higgs fields. These produce the Yukawa couplings for the Standard Model fermions and also the mass term for the Higgsinos. After imposing R-parity, the renormalizable, gauge invariant operators in the superpotential are
W
=
μ
H
u
H
d
+
y
u
H
u
Q
U
c
+
y
d
H
d
Q
D
c
+
y
l
H
d
L
E
c
{\displaystyle W_{}^{}=\mu H_{u}H_{d}+y_{u}H_{u}QU^{c}+y_{d}H_{d}QD^{c}+y_{l}H_{d}LE^{c}}
The constant term is unphysical in global supersymmetry (as opposed to supergravity).
=== Soft SUSY breaking ===
The last piece of the MSSM Lagrangian is the soft supersymmetry breaking Lagrangian. The vast majority of the parameters of the MSSM are in the susy breaking Lagrangian. The soft susy breaking are divided into roughly three pieces.
The first are the gaugino masses
L
⊃
m
1
2
λ
~
λ
~
+
h.c.
{\displaystyle {\mathcal {L}}\supset m_{\frac {1}{2}}{\tilde {\lambda }}{\tilde {\lambda }}+{\text{h.c.}}}
where
λ
~
{\displaystyle {\tilde {\lambda }}}
are the gauginos and
m
1
2
{\displaystyle m_{\frac {1}{2}}}
is different for the wino, bino and gluino.
The next are the soft masses for the scalar fields
L
⊃
m
0
2
ϕ
†
ϕ
{\displaystyle {\mathcal {L}}\supset m_{0}^{2}\phi ^{\dagger }\phi }
where
ϕ
{\displaystyle \phi }
are any of the scalars in the MSSM and
m
0
{\displaystyle m_{0}}
are
3
×
3
{\displaystyle 3\times 3}
Hermitian matrices for the squarks and sleptons of a given set of gauge quantum numbers. The eigenvalues of these matrices are actually the masses squared, rather than the masses.
There are the
A
{\displaystyle A}
and
B
{\displaystyle B}
terms which are given by
L
⊃
B
μ
h
u
h
d
+
A
h
u
q
~
u
~
c
+
A
h
d
q
~
d
~
c
+
A
h
d
l
~
e
~
c
+
h.c.
{\displaystyle {\mathcal {L}}\supset B_{\mu }h_{u}h_{d}+Ah_{u}{\tilde {q}}{\tilde {u}}^{c}+Ah_{d}{\tilde {q}}{\tilde {d}}^{c}+Ah_{d}{\tilde {l}}{\tilde {e}}^{c}+{\text{h.c.}}}
The
A
{\displaystyle A}
terms are
3
×
3
{\displaystyle 3\times 3}
complex matrices much as the scalar masses are.
Although not often mentioned with regard to soft terms, to be consistent with observation, one must also include Gravitino and Goldstino soft masses given by
L
⊃
m
3
/
2
Ψ
μ
α
(
σ
μ
ν
)
α
β
Ψ
β
+
m
3
/
2
G
α
G
α
+
h.c.
{\displaystyle {\mathcal {L}}\supset m_{3/2}\Psi _{\mu }^{\alpha }(\sigma ^{\mu \nu })_{\alpha }^{\beta }\Psi _{\beta }+m_{3/2}G^{\alpha }G_{\alpha }+{\text{h.c.}}}
The reason these soft terms are not often mentioned are that they arise through local supersymmetry and not global supersymmetry, although they are required otherwise if the Goldstino were massless it would contradict observation. The Goldstino mode is eaten by the Gravitino to become massive, through a gauge shift, which also absorbs the would-be "mass" term of the Goldstino.
== Problems ==
There are several problems with the MSSM—most of them falling into understanding the parameters.
The mu problem: The Higgsino mass parameter μ appears as the following term in the superpotential: μHuHd. It should have the same order of magnitude as the electroweak scale, many orders of magnitude smaller than that of the Planck scale, which is the natural cutoff scale. The soft supersymmetry breaking terms should also be of the same order of magnitude as the electroweak scale. This brings about a problem of naturalness: why are these scales so much smaller than the cutoff scale yet happen to fall so close to each other?
Flavor universality of soft masses and A-terms: since no flavor mixing additional to that predicted by the standard model has been discovered so far, the coefficients of the additional terms in the MSSM Lagrangian must be, at least approximately, flavor invariant (i.e. the same for all flavors).
Smallness of CP violating phases: since no CP violation additional to that predicted by the standard model has been discovered so far, the additional terms in the MSSM Lagrangian must be, at least approximately, CP invariant, so that their CP violating phases are small.
== Theories of supersymmetry breaking ==
A large amount of theoretical effort has been spent trying to understand the mechanism for soft supersymmetry breaking that produces the desired properties in the superpartner masses and interactions. The three most extensively studied mechanisms are:
=== Gravity-mediated supersymmetry breaking ===
Gravity-mediated supersymmetry breaking is a method of communicating supersymmetry breaking to the supersymmetric Standard Model through gravitational interactions. It was the first method proposed to communicate supersymmetry breaking. In gravity-mediated supersymmetry-breaking models, there is a part of the theory that only interacts with the MSSM through gravitational interaction. This hidden sector of the theory breaks supersymmetry. Through the supersymmetric version of the Higgs mechanism, the gravitino, the supersymmetric version of the graviton, acquires a mass. After the gravitino has a mass, gravitational radiative corrections to soft masses are incompletely cancelled beneath the gravitino's mass.
It is currently believed that it is not generic to have a sector completely decoupled from the MSSM and there should be higher dimension operators that couple different sectors together with the higher dimension operators suppressed by the Planck scale. These operators give as large of a contribution to the soft supersymmetry breaking masses as the gravitational loops; therefore, today people usually consider gravity mediation to be gravitational sized direct interactions between the hidden sector and the MSSM.
mSUGRA stands for minimal supergravity. The construction of a realistic model of interactions within N = 1 supergravity framework where supersymmetry breaking is communicated through the supergravity interactions was carried out by Ali Chamseddine, Richard Arnowitt, and Pran Nath in 1982. mSUGRA is one of the most widely investigated models of particle physics due to its predictive power requiring only 4 input parameters and a sign, to determine the low energy phenomenology from the scale of Grand Unification. The most widely used set of parameters is:
Gravity-Mediated Supersymmetry Breaking was assumed to be flavor universal because of the universality of gravity; however, in 1986 Hall, Kostelecky, and Raby showed that Planck-scale physics that are necessary to generate the Standard-Model Yukawa couplings spoil the universality of the supersymmetry breaking.
=== Gauge-mediated supersymmetry breaking (GMSB) ===
Gauge-mediated supersymmetry breaking is method of communicating supersymmetry breaking to the supersymmetric Standard Model through the Standard Model's gauge interactions. Typically a hidden sector breaks supersymmetry and communicates it to massive messenger fields that are charged under the Standard Model. These messenger fields induce a gaugino mass at one loop and then this is transmitted on to the scalar superpartners at two loops. Requiring stop squarks below 2 TeV, the maximum Higgs boson mass predicted is just 121.5GeV. With the Higgs being discovered at 125GeV - this model requires stops above 2 TeV.
=== Anomaly-mediated supersymmetry breaking (AMSB) ===
Anomaly-mediated supersymmetry breaking is a special type of gravity mediated supersymmetry breaking that results in supersymmetry breaking being communicated to the supersymmetric Standard Model through the conformal anomaly. Requiring stop squarks below 2 TeV, the maximum Higgs boson mass predicted is just 121.0GeV. With the Higgs being discovered at 125GeV - this scenario requires stops heavier than 2 TeV.
== Phenomenological MSSM (pMSSM) ==
The unconstrained MSSM has more than 100 parameters in addition to the Standard Model parameters.
This makes any phenomenological analysis (e.g. finding regions in parameter space consistent
with observed data) impractical. Under the following three assumptions:
no new source of CP-violation
no Flavour Changing Neutral Currents
first and second generation universality
one can reduce the number of additional parameters to the following 19 quantities of the phenomenological MSSM (pMSSM):
The large parameter space of pMSSM makes searches in pMSSM extremely challenging and makes pMSSM difficult to exclude.
== Experimental tests ==
=== Terrestrial detectors ===
XENON1T (a dark matter WIMP detector - being commissioned in 2016) is expected to explore/test supersymmetry candidates such as CMSSM.: Fig 7(a), p15-16
== See also ==
Desert (particle physics)
== References ==
== External links ==
MSSM on arxiv.org
Stephen P. Martin (1997). "A Supersymmetry Primer". Advanced Series on Directions in High Energy Physics. 18: 1–98. arXiv:hep-ph/9709356. doi:10.1142/9789812839657_0001. ISBN 978-981-02-3553-6. S2CID 118973381.
Particle Data Group review of MSSM and search for MSSM predicted particles
Ian J. R. Aitchison (2005). "Supersymmetry and the MSSM: An Elementary Introduction". arXiv:hep-ph/0505105. | Wikipedia/Minimal_Supersymmetric_Standard_Model |
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin modulus, 'a measure'.
Models can be divided into physical models (e.g. a ship model or a fashion model) and abstract models (e.g. a set of mathematical equations describing the workings of the atmosphere for the purpose of weather forecasting). Abstract or conceptual models are central to philosophy of science.
In scholarly research and applied science, a model should not be confused with a theory: while a model seeks only to represent reality with the purpose of better understanding or predicting the world, a theory is more ambitious in that it claims to be an explanation of reality.
== Types of model ==
=== Model in specific contexts ===
As a noun, model has specific meanings in certain fields, derived from its original meaning of "structural design or layout":
Model (art), a person posing for an artist, e.g. a 15th-century criminal representing the biblical Judas in Leonardo da Vinci's painting The Last Supper
Model (person), a person who serves as a template for others to copy, as in a role model, often in the context of advertising commercial products; e.g. the first fashion model, Marie Vernet Worth in 1853, wife of designer Charles Frederick Worth.
Model (product), a particular design of a product as displayed in a catalogue or show room (e.g. Ford Model T, an early car model)
Model (organism) a non-human species that is studied to understand biological phenomena in other organisms, e.g. a guinea pig starved of vitamin C to study scurvy, an experiment that would be immoral to conduct on a person
Model (mimicry), a species that is mimicked by another species
Model (logic), a structure (a set of items, such as natural numbers 1, 2, 3,..., along with mathematical operations such as addition and multiplication, and relations, such as
<
{\displaystyle <}
) that satisfies a given system of axioms (basic truisms), i.e. that satisfies the statements of a given theory
Model (CGI), a mathematical representation of any surface of an object in three dimensions via specialized software
Model (MVC), the information-representing internal component of a software, as distinct from its user interface
=== Physical model ===
A physical model (most commonly referred to simply as a model but in this context distinguished from a conceptual model) is a smaller or larger physical representation of an object, person or system. The object being modelled may be small (e.g., an atom) or large (e.g., the Solar System) or life-size (e.g., a fashion model displaying clothes for similarly-built potential customers).
The geometry of the model and the object it represents are often similar in the sense that one is a rescaling of the other. However, in many cases the similarity is only approximate or even intentionally distorted. Sometimes the distortion is systematic, e.g., a fixed scale horizontally and a larger fixed scale vertically when modelling topography to enhance a region's mountains.
An architectural model permits visualization of internal relationships within the structure or external relationships of the structure to the environment. Another use is as a toy.
Instrumented physical models are an effective way of investigating fluid flows for engineering design. Physical models are often coupled with computational fluid dynamics models to optimize the design of equipment and processes. This includes external flow such as around buildings, vehicles, people, or hydraulic structures. Wind tunnel and water tunnel testing is often used for these design efforts. Instrumented physical models can also examine internal flows, for the design of ductwork systems, pollution control equipment, food processing machines, and mixing vessels. Transparent flow models are used in this case to observe the detailed flow phenomenon. These models are scaled in terms of both geometry and important forces, for example, using Froude number or Reynolds number scaling (see Similitude). In the pre-computer era, the UK economy was modelled with the hydraulic model MONIAC, to predict for example the effect of tax rises on employment.
=== Conceptual model ===
A conceptual model is a theoretical representation of a system, e.g. a set of mathematical equations attempting to describe the workings of the atmosphere for the purpose of weather forecasting. It consists of concepts used to help understand or simulate a subject the model represents.
Abstract or conceptual models are central to philosophy of science, as almost every scientific theory effectively embeds some kind of model of the physical or human sphere. In some sense, a physical model "is always the reification of some conceptual model; the conceptual model is conceived ahead as the blueprint of the physical one", which is then constructed as conceived. Thus, the term refers to models that are formed after a conceptualization or generalization process.
=== Examples ===
Conceptual model (computer science), an agreed representation of entities and their relationships, to assist in developing software
Economic model, a theoretical construct representing economic processes
Language model, a probabilistic model of a natural language, used for speech recognition, language generation, and information retrieval
Large language models are artificial neural networks used for generative artificial intelligence (AI), e.g. ChatGPT
Mathematical model, a description of a system using mathematical concepts and language
Statistical model, a mathematical model that usually specifies the relationship between one or more random variables and other non-random variables
Model (CGI), a mathematical representation of any surface of an object in three dimensions via specialized software
Medical model, a proposed "set of procedures in which all doctors are trained"
Mental model, in psychology, an internal representation of external reality
Model (logic), a set along with a collection of finitary operations, and relations that are defined on it, satisfying a given collection of axioms
Model (MVC), information-representing component of a software, distinct from the user interface (the "view"), both linked by the "controller" component, in the context of the model–view–controller software design
Model act, a law drafted centrally to be disseminated and proposed for enactment in multiple independent legislatures
Standard model (disambiguation)
== Properties of models, according to general model theory ==
According to Herbert Stachowiak, a model is characterized by at least three properties:
1. Mapping
A model always is a model of something—it is an image or representation of some natural or artificial, existing or imagined original, where this original itself could be a model.
2. Reduction
In general, a model will not include all attributes that describe the original but only those that appear relevant to the model's creator or user.
3. Pragmatism
A model does not relate unambiguously to its original. It is intended to work as a replacement for the original
a) for certain subjects (for whom?)
b) within a certain time range (when?)
c) restricted to certain conceptual or physical actions (what for?).
For example, a street map is a model of the actual streets in a city (mapping), showing the course of the streets while leaving out, say, traffic signs and road markings (reduction), made for pedestrians and vehicle drivers for the purpose of finding one's way in the city (pragmatism).
Additional properties have been proposed, like extension and distortion as well as validity. The American philosopher Michael Weisberg differentiates between concrete and mathematical models and proposes computer simulations (computational models) as their own class of models.
== Uses of models ==
According to Bruce Edmonds, there are at least 5 general uses for models:
Prediction: reliably anticipating unknown data, including data within the domain of the training data (interpolation), and outside the domain (extrapolation)
Explanation: establishing plausible chains of causality by proposing mechanisms that can explain patterns seen in data
Theoretical exposition: discovering or proposing new hypotheses, or refuting existing hypotheses about the behaviour of the system being modelled
Description: representing important aspects of the system being modelled
Illustration: communicating an idea or explanation
== See also ==
== References ==
== External links ==
Media related to Physical models at Wikimedia Commons | Wikipedia/Physical_model |
Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. Three of the most common types of decay are alpha, beta, and gamma decay. The weak force is the mechanism that is responsible for beta decay, while the other two are governed by the electromagnetic and nuclear forces.
Radioactive decay is a random process at the level of single atoms. According to quantum theory, it is impossible to predict when a particular atom will decay, regardless of how long the atom has existed. However, for a significant number of identical atoms, the overall decay rate can be expressed as a decay constant or as a half-life. The half-lives of radioactive atoms have a huge range: from nearly instantaneous to far longer than the age of the universe.
The decaying nucleus is called the parent radionuclide (or parent radioisotope), and the process produces at least one daughter nuclide. Except for gamma decay or internal conversion from a nuclear excited state, the decay is a nuclear transmutation resulting in a daughter containing a different number of protons or neutrons (or both). When the number of protons changes, an atom of a different chemical element is created.
There are 28 naturally occurring chemical elements on Earth that are radioactive, consisting of 35 radionuclides (seven elements have two different radionuclides each) that date before the time of formation of the Solar System. These 35 are known as primordial radionuclides. Well-known examples are uranium and thorium, but also included are naturally occurring long-lived radioisotopes, such as potassium-40. Each of the heavy primordial radionuclides participates in one of the four decay chains.
== History of discovery ==
Henri Poincaré laid the seeds for the discovery of radioactivity through his interest in and studies of X-rays, which significantly influenced physicist Henri Becquerel. Radioactivity was discovered in 1896 by Becquerel and independently by Marie Curie, while working with phosphorescent materials. These materials glow in the dark after exposure to light, and Becquerel suspected that the glow produced in cathode-ray tubes by X-rays might be associated with phosphorescence. He wrapped a photographic plate in black paper and placed various phosphorescent salts on it. All results were negative until he used uranium salts. The uranium salts caused a blackening of the plate in spite of the plate being wrapped in black paper. These radiations were given the name "Becquerel Rays".
It soon became clear that the blackening of the plate had nothing to do with phosphorescence, as the blackening was also produced by non-phosphorescent salts of uranium and by metallic uranium. It became clear from these experiments that there was a form of invisible radiation that could pass through paper and was causing the plate to react as if exposed to light.
At first, it seemed as though the new radiation was similar to the then recently discovered X-rays. Further research by Becquerel, Ernest Rutherford, Paul Villard, Pierre Curie, Marie Curie, and others showed that this form of radioactivity was significantly more complicated. Rutherford was the first to realize that all such elements decay in accordance with the same mathematical exponential formula. Rutherford and his student Frederick Soddy were the first to realize that many decay processes resulted in the transmutation of one element to another. Subsequently, the radioactive displacement law of Fajans and Soddy was formulated to describe the products of alpha and beta decay.
The early researchers also discovered that many other chemical elements, besides uranium, have radioactive isotopes. A systematic search for the total radioactivity in uranium ores also guided Pierre and Marie Curie to isolate two new elements: polonium and radium. Except for the radioactivity of radium, the chemical similarity of radium to barium made these two elements difficult to distinguish.
Marie and Pierre Curie's study of radioactivity is an important factor in science and medicine. After their research on Becquerel's rays led them to the discovery of both radium and polonium, they coined the term "radioactivity" to define the emission of ionizing radiation by some heavy elements. (Later the term was generalized to all elements.) Their research on the penetrating rays in uranium and the discovery of radium launched an era of using radium for the treatment of cancer. Their exploration of radium could be seen as the first peaceful use of nuclear energy and the start of modern nuclear medicine.
== Early health dangers ==
The dangers of ionizing radiation due to radioactivity and X-rays were not immediately recognized.
=== X-rays ===
The discovery of X‑rays by Wilhelm Röntgen in 1895 led to widespread experimentation by scientists, physicians, and inventors. Many people began recounting stories of burns, hair loss and worse in technical journals as early as 1896. In February of that year, Professor Daniel and Dr. Dudley of Vanderbilt University performed an experiment involving X-raying Dudley's head that resulted in his hair loss. A report by Dr. H.D. Hawks, of his suffering severe hand and chest burns in an X-ray demonstration, was the first of many other reports in Electrical Review.
Other experimenters, including Elihu Thomson and Nikola Tesla, also reported burns. Thomson deliberately exposed a finger to an X-ray tube over a period of time and suffered pain, swelling, and blistering. Other effects, including ultraviolet rays and ozone, were sometimes blamed for the damage, and many physicians still claimed that there were no effects from X-ray exposure at all.
Despite this, there were some early systematic hazard investigations, and as early as 1902 William Herbert Rollins wrote almost despairingly that his warnings about the dangers involved in the careless use of X-rays were not being heeded, either by industry or by his colleagues. By this time, Rollins had proved that X-rays could kill experimental animals, could cause a pregnant guinea pig to abort, and that they could kill a foetus. He also stressed that "animals vary in susceptibility to the external action of X-light" and warned that these differences be considered when patients were treated by means of X-rays.
=== Radioactive substances ===
However, the biological effects of radiation due to radioactive substances were less easy to gauge. This gave the opportunity for many physicians and corporations to market radioactive substances as patent medicines. Examples were radium enema treatments, and radium-containing waters to be drunk as tonics. Marie Curie protested against this sort of treatment, warning that "radium is dangerous in untrained hands". Curie later died from aplastic anaemia, likely caused by exposure to ionizing radiation. By the 1930s, after a number of cases of bone necrosis and death of radium treatment enthusiasts, radium-containing medicinal products had been largely removed from the market (radioactive quackery).
=== Radiation protection ===
Only a year after Röntgen's discovery of X-rays, the American engineer Wolfram Fuchs (1896) gave what is probably the first protection advice, but it was not until 1925 that the first International Congress of Radiology (ICR) was held and considered establishing international protection standards. The effects of radiation on genes, including the effect of cancer risk, were recognized much later. In 1927, Hermann Joseph Muller published research showing genetic effects and, in 1946, was awarded the Nobel Prize in Physiology or Medicine for his findings.
The second ICR was held in Stockholm in 1928 and proposed the adoption of the röntgen unit, and the International X-ray and Radium Protection Committee (IXRPC) was formed. Rolf Sievert was named chairman, but a driving force was George Kaye of the British National Physical Laboratory. The committee met in 1931, 1934, and 1937.
After World War II, the increased range and quantity of radioactive substances being handled as a result of military and civil nuclear programs led to large groups of occupational workers and the public being potentially exposed to harmful levels of ionising radiation. This was considered at the first post-war ICR convened in London in 1950, when the present International Commission on Radiological Protection (ICRP) was born.
Since then the ICRP has developed the present international system of radiation protection, covering all aspects of radiation hazards.
In 2020, Hauptmann and another 15 international researchers from eight nations (among them: Institutes of Biostatistics, Registry Research, Centers of Cancer Epidemiology, Radiation Epidemiology, and also the U.S. National Cancer Institute (NCI), International Agency for Research on Cancer (IARC) and the Radiation Effects Research Foundation of Hiroshima) studied definitively through meta-analysis the damage resulting from the "low doses" that have afflicted survivors of the atomic bombings of Hiroshima and Nagasaki and also in numerous accidents at nuclear plants that have occurred. These scientists reported, in JNCI Monographs: Epidemiological Studies of Low Dose Ionizing Radiation and Cancer Risk, that the new epidemiological studies directly support excess cancer risks from low-dose ionizing radiation. In 2021, Italian researcher Sebastiano Venturi reported the first correlations between radio-caesium and pancreatic cancer with the role of caesium in biology, in pancreatitis and in diabetes of pancreatic origin.
== Units ==
The International System of Units (SI) unit of radioactive activity is the becquerel (Bq), named in honor of the scientist Henri Becquerel. One Bq is defined as one transformation (or decay or disintegration) per second.
An older unit of radioactivity is the curie, Ci, which was originally defined as "the quantity or mass of radium emanation in equilibrium with one gram of radium (element)". Today, the curie is defined as 3.7×1010 disintegrations per second, so that 1 curie (Ci) = 3.7×1010 Bq.
For radiological protection purposes, although the United States Nuclear Regulatory Commission permits the use of the unit curie alongside SI units, the European Union European units of measurement directives required that its use for "public health ... purposes" be phased out by 31 December 1985.
The effects of ionizing radiation are often measured in units of gray for mechanical or sievert for damage to tissue.
== Types ==
Radioactive decay results in a reduction of summed rest mass, once the released energy (the disintegration energy) has escaped in some way. Although decay energy is sometimes defined as associated with the difference between the mass of the parent nuclide products and the mass of the decay products, this is true only of rest mass measurements, where some energy has been removed from the product system. This is true because the decay energy must always carry mass with it, wherever it appears (see mass in special relativity) according to the formula E = mc2. The decay energy is initially released as the energy of emitted photons plus the kinetic energy of massive emitted particles (that is, particles that have rest mass). If these particles come to thermal equilibrium with their surroundings and photons are absorbed, then the decay energy is transformed to thermal energy, which retains its mass.
Decay energy, therefore, remains associated with a certain measure of the mass of the decay system, called invariant mass, which does not change during the decay, even though the energy of decay is distributed among decay particles. The energy of photons, the kinetic energy of emitted particles, and, later, the thermal energy of the surrounding matter, all contribute to the invariant mass of the system. Thus, while the sum of the rest masses of the particles is not conserved in radioactive decay, the system mass and system invariant mass (and also the system total energy) is conserved throughout any decay process. This is a restatement of the equivalent laws of conservation of energy and conservation of mass.
=== Alpha, beta and gamma decay ===
Early researchers found that an electric or magnetic field could split radioactive emissions into three types of beams. The rays were given the names alpha, beta, and gamma, in increasing order of their ability to penetrate matter. Alpha decay is observed only in heavier elements of atomic number 52 (tellurium) and greater, with the exception of beryllium-8 (which decays to two alpha particles). The other two types of decay are observed in all the elements. Lead, atomic number 82, is the heaviest element to have any isotopes stable (to the limit of measurement) to radioactive decay. Radioactive decay is seen in all isotopes of all elements of atomic number 83 (bismuth) or greater. Bismuth-209, however, is only very slightly radioactive, with a half-life greater than the age of the universe; radioisotopes with extremely long half-lives are considered effectively stable for practical purposes.
In analyzing the nature of the decay products, it was obvious from the direction of the electromagnetic forces applied to the radiations by external magnetic and electric fields that alpha particles carried a positive charge, beta particles carried a negative charge, and gamma rays were neutral. From the magnitude of deflection, it was clear that alpha particles were much more massive than beta particles. Passing alpha particles through a very thin glass window and trapping them in a discharge tube allowed researchers to study the emission spectrum of the captured particles, and ultimately proved that alpha particles are helium nuclei. Other experiments showed beta radiation, resulting from decay and cathode rays, were high-speed electrons. Likewise, gamma radiation and X-rays were found to be high-energy electromagnetic radiation.
The relationship between the types of decays also began to be examined: For example, gamma decay was almost always found to be associated with other types of decay, and occurred at about the same time, or afterwards. Gamma decay as a separate phenomenon, with its own half-life (now termed isomeric transition), was found in natural radioactivity to be a result of the gamma decay of excited metastable nuclear isomers, which were in turn created from other types of decay. Although alpha, beta, and gamma radiations were most commonly found, other types of emission were eventually discovered. Shortly after the discovery of the positron in cosmic ray products, it was realized that the same process that operates in classical beta decay can also produce positrons (positron emission), along with neutrinos (classical beta decay produces antineutrinos).
=== Electron capture ===
In electron capture, some proton-rich nuclides were found to capture their own atomic electrons instead of emitting positrons, and subsequently, these nuclides emit only a neutrino and a gamma ray from the excited nucleus (and often also Auger electrons and characteristic X-rays, as a result of the re-ordering of electrons to fill the place of the missing captured electron). These types of decay involve the nuclear capture of electrons or emission of electrons or positrons, and thus acts to move a nucleus toward the ratio of neutrons to protons that has the least energy for a given total number of nucleons. This consequently produces a more stable (lower energy) nucleus.
A hypothetical process of positron capture, analogous to electron capture, is theoretically possible in antimatter atoms, but has not been observed, as complex antimatter atoms beyond antihelium are not experimentally available. Such a decay would require antimatter atoms at least as complex as beryllium-7, which is the lightest known isotope of normal matter to undergo decay by electron capture.
=== Nucleon emission ===
Shortly after the discovery of the neutron in 1932, Enrico Fermi realized that certain rare beta-decay reactions immediately yield neutrons as an additional decay particle, so called beta-delayed neutron emission. Neutron emission usually happens from nuclei that are in an excited state, such as the excited 17O* produced from the beta decay of 17N. The neutron emission process itself is controlled by the nuclear force and therefore is extremely fast, sometimes referred to as "nearly instantaneous". Isolated proton emission was eventually observed in some elements. It was also found that some heavy elements may undergo spontaneous fission into products that vary in composition. In a phenomenon called cluster decay, specific combinations of neutrons and protons other than alpha particles (helium nuclei) were found to be spontaneously emitted from atoms.
=== More exotic types of decay ===
Other types of radioactive decay were found to emit previously seen particles but via different mechanisms. An example is internal conversion, which results in an initial electron emission, and then often further characteristic X-rays and Auger electrons emissions, although the internal conversion process involves neither beta nor gamma decay. A neutrino is not emitted, and none of the electron(s) and photon(s) emitted originate in the nucleus, even though the energy to emit all of them does originate there. Internal conversion decay, like isomeric transition gamma decay and neutron emission, involves the release of energy by an excited nuclide, without the transmutation of one element into another.
Rare events that involve a combination of two beta-decay-type events happening simultaneously are known (see below). Any decay process that does not violate the conservation of energy or momentum laws (and perhaps other particle conservation laws) is permitted to happen, although not all have been detected. An interesting example discussed in a final section, is bound state beta decay of rhenium-187. In this process, the beta electron-decay of the parent nuclide is not accompanied by beta electron emission, because the beta particle has been captured into the K-shell of the emitting atom. An antineutrino is emitted, as in all negative beta decays.
If energy circumstances are favorable, a given radionuclide may undergo many competing types of decay, with some atoms decaying by one route, and others decaying by another. An example is copper-64, which has 29 protons, and 35 neutrons, which decays with a half-life of 12.7004(13) hours. This isotope has one unpaired proton and one unpaired neutron, so either the proton or the neutron can decay to the other particle, which has opposite isospin. This particular nuclide (though not all nuclides in this situation) is more likely to decay through beta plus decay (61.52(26)%) than through electron capture (38.48(26)%). The excited energy states resulting from these decays which fail to end in a ground energy state, also produce later internal conversion and gamma decay in almost 0.5% of the time.
=== List of decay modes ===
=== Decay chains and multiple modes ===
The daughter nuclide of a decay event may also be unstable (radioactive). In this case, it too will decay, producing radiation. The resulting second daughter nuclide may also be radioactive. This can lead to a sequence of several decay events called a decay chain (see this article for specific details of important natural decay chains). Eventually, a stable nuclide is produced. Any decay daughters that are the result of an alpha decay will also result in helium atoms being created.
Some radionuclides may have several different paths of decay. For example, 35.94(6)% of bismuth-212 decays, through alpha-emission, to thallium-208 while 64.06(6)% of bismuth-212 decays, through beta-emission, to polonium-212. Both thallium-208 and polonium-212 are radioactive daughter products of bismuth-212, and both decay directly to stable lead-208.
== Occurrence and applications ==
According to the Big Bang theory, stable isotopes of the lightest three elements (H, He, and traces of Li) were produced very shortly after the emergence of the universe, in a process called Big Bang nucleosynthesis. These lightest stable nuclides (including deuterium) survive to today, but any radioactive isotopes of the light elements produced in the Big Bang (such as tritium) have long since decayed. Isotopes of elements heavier than boron were not produced at all in the Big Bang, and these first five elements do not have any long-lived radioisotopes. Thus, all radioactive nuclei are, therefore, relatively young with respect to the birth of the universe, having formed later in various other types of nucleosynthesis in stars (in particular, supernovae), and also during ongoing interactions between stable isotopes and energetic particles. For example, carbon-14, a radioactive nuclide with a half-life of only 5700(30) years, is constantly produced in Earth's upper atmosphere due to interactions between cosmic rays and nitrogen.
Nuclides that are produced by radioactive decay are called radiogenic nuclides, whether they themselves are stable or not. There exist stable radiogenic nuclides that were formed from short-lived extinct radionuclides in the early Solar System. The extra presence of these stable radiogenic nuclides (such as xenon-129 from extinct iodine-129) against the background of primordial stable nuclides can be inferred by various means.
Radioactive decay has been put to use in the technique of radioisotopic labeling, which is used to track the passage of a chemical substance through a complex system (such as a living organism). A sample of the substance is synthesized with a high concentration of unstable atoms. The presence of the substance in one or another part of the system is determined by detecting the locations of decay events.
On the premise that radioactive decay is truly random (rather than merely chaotic), it has been used in hardware random-number generators. Because the process is not thought to vary significantly in mechanism over time, it is also a valuable tool in estimating the absolute ages of certain materials. For geological materials, the radioisotopes and some of their decay products become trapped when a rock solidifies, and can then later be used (subject to many well-known qualifications) to estimate the date of the solidification. These include checking the results of several simultaneous processes and their products against each other, within the same sample. In a similar fashion, and also subject to qualification, the rate of formation of carbon-14 in various eras, the date of formation of organic matter within a certain period related to the isotope's half-life may be estimated, because the carbon-14 becomes trapped when the organic matter grows and incorporates the new carbon-14 from the air. Thereafter, the amount of carbon-14 in organic matter decreases according to decay processes that may also be independently cross-checked by other means (such as checking the carbon-14 in individual tree rings, for example).
=== Szilard–Chalmers effect ===
The Szilard–Chalmers effect is the breaking of a chemical bond as a result of a kinetic energy imparted from radioactive decay. It operates by the absorption of neutrons by an atom and subsequent emission of gamma rays, often with significant amounts of kinetic energy. This kinetic energy, by Newton's third law, pushes back on the decaying atom, which causes it to move with enough speed to break a chemical bond. This effect can be used to separate isotopes by chemical means.
The Szilard–Chalmers effect was discovered in 1934 by Leó Szilárd and Thomas A. Chalmers. They observed that after bombardment by neutrons, the breaking of a bond in liquid ethyl iodide allowed radioactive iodine to be removed.
=== Origins of radioactive nuclides ===
Radioactive primordial nuclides found in the Earth are residues from ancient supernova explosions that occurred before the formation of the Solar System. They are the fraction of radionuclides that survived from that time, through the formation of the primordial solar nebula, through planet accretion, and up to the present time. The naturally occurring short-lived radiogenic radionuclides found in today's rocks, are the daughters of those radioactive primordial nuclides. Another minor source of naturally occurring radioactive nuclides are cosmogenic nuclides, that are formed by cosmic ray bombardment of material in the Earth's atmosphere or crust. The decay of the radionuclides in rocks of the Earth's mantle and crust contribute significantly to Earth's internal heat budget.
== Aggregate processes ==
While the underlying process of radioactive decay is subatomic, historically and in most practical cases it is encountered in bulk materials with very large numbers of atoms. This section discusses models that connect events at the atomic level to observations in aggregate.
=== Terminology ===
The decay rate, or activity, of a radioactive substance is characterized by the following time-independent parameters:
The half-life, t1/2, is the time taken for the activity of a given amount of a radioactive substance to decay to half of its initial value.
The decay constant, λ "lambda", the reciprocal of the mean lifetime (in s−1), sometimes referred to as simply decay rate.
The mean lifetime, τ "tau", the average lifetime (1/e life) of a radioactive particle before decay.
Although these are constants, they are associated with the statistical behavior of populations of atoms. In consequence, predictions using these constants are less accurate for minuscule samples of atoms.
In principle a half-life, a third-life, or even a (1/√2)-life, could be used in exactly the same way as half-life; but the mean life and half-life t1/2 have been adopted as standard times associated with exponential decay.
Those parameters can be related to the following time-dependent parameters:
Total activity (or just activity), A, is the number of decays per unit time of a radioactive sample.
Number of particles, N, in the sample.
Specific activity, a, is the number of decays per unit time per amount of substance of the sample at time set to zero (t = 0). "Amount of substance" can be the mass, volume or moles of the initial sample.
These are related as follows:
t
1
/
2
=
ln
(
2
)
λ
=
τ
ln
(
2
)
A
=
−
d
N
d
t
=
λ
N
=
ln
(
2
)
t
1
/
2
N
S
A
a
0
=
−
d
N
d
t
|
t
=
0
=
λ
N
0
{\displaystyle {\begin{aligned}t_{1/2}&={\frac {\ln(2)}{\lambda }}=\tau \ln(2)\\[2pt]A&=-{\frac {\mathrm {d} N}{\mathrm {d} t}}=\lambda N={\frac {\ln(2)}{t_{1/2}}}N\\[2pt]S_{A}a_{0}&=-{\frac {\mathrm {d} N}{\mathrm {d} t}}{\bigg |}_{t=0}=\lambda N_{0}\end{aligned}}}
where N0 is the initial amount of active substance — substance that has the same percentage of unstable particles as when the substance was formed.
=== Assumptions ===
The mathematics of radioactive decay depend on a key assumption that a nucleus of a radionuclide has no "memory" or way of translating its history into its present behavior. A nucleus does not "age" with the passage of time. Thus, the probability of its breaking down does not increase with time but stays constant, no matter how long the nucleus has existed. This constant probability may differ greatly between one type of nucleus and another, leading to the many different observed decay rates. However, whatever the probability is, it does not change over time. This is in marked contrast to complex objects that do show aging, such as automobiles and humans. These aging systems do have a chance of breakdown per unit of time that increases from the moment they begin their existence.
Aggregate processes, like the radioactive decay of a lump of atoms, for which the single-event probability of realization is very small but in which the number of time-slices is so large that there is nevertheless a reasonable rate of events, are modelled by the Poisson distribution, which is discrete. Radioactive decay and nuclear particle reactions are two examples of such aggregate processes. The mathematics of Poisson processes reduce to the law of exponential decay, which describes the statistical behaviour of a large number of nuclei, rather than one individual nucleus. In the following formalism, the number of nuclei or the nuclei population N, is of course a discrete variable (a natural number)—but for any physical sample N is so large that it can be treated as a continuous variable. Differential calculus is used to model the behaviour of nuclear decay.
==== One-decay process ====
Consider the case of a nuclide A that decays into another B by some process A → B (emission of other particles, like electron neutrinos νe and electrons e− as in beta decay, are irrelevant in what follows). The decay of an unstable nucleus is entirely random in time so it is impossible to predict when a particular atom will decay. However, it is equally likely to decay at any instant in time. Therefore, given a sample of a particular radioisotope, the number of decay events −dN expected to occur in a small interval of time dt is proportional to the number of atoms present N, that is
−
d
N
d
t
∝
N
{\displaystyle -{\frac {\mathrm {d} N}{\mathrm {d} t}}\propto N}
Particular radionuclides decay at different rates, so each has its own decay constant λ. The expected decay −dN/N is proportional to an increment of time, dt:
The negative sign indicates that N decreases as time increases, as the decay events follow one after another. The solution to this first-order differential equation is the function:
N
(
t
)
=
N
0
e
−
λ
t
{\displaystyle N(t)=N_{0}\,e^{-{\lambda }t}}
where N0 is the value of N at time t = 0, with the decay constant expressed as λ
We have for all time t:
N
A
+
N
B
=
N
total
=
N
A
0
,
{\displaystyle N_{A}+N_{B}=N_{\text{total}}=N_{A0},}
where Ntotal is the constant number of particles throughout the decay process, which is equal to the initial number of A nuclides since this is the initial substance.
If the number of non-decayed A nuclei is:
N
A
=
N
A
0
e
−
λ
t
{\displaystyle N_{A}=N_{A0}e^{-\lambda t}}
then the number of nuclei of B (i.e. the number of decayed A nuclei) is
N
B
=
N
A
0
−
N
A
=
N
A
0
−
N
A
0
e
−
λ
t
=
N
A
0
(
1
−
e
−
λ
t
)
.
{\displaystyle N_{B}=N_{A0}-N_{A}=N_{A0}-N_{A0}e^{-\lambda t}=N_{A0}\left(1-e^{-\lambda t}\right).}
The number of decays observed over a given interval obeys Poisson statistics. If the average number of decays is ⟨N⟩, the probability of a given number of decays N is
P
(
N
)
=
⟨
N
⟩
N
exp
(
−
⟨
N
⟩
)
N
!
.
{\displaystyle P(N)={\frac {\langle N\rangle ^{N}\exp(-\langle N\rangle )}{N!}}.}
==== Chain-decay processes ====
===== Chain of two decays =====
Now consider the case of a chain of two decays: one nuclide A decaying into another B by one process, then B decaying into another C by a second process, i.e. A → B → C. The previous equation cannot be applied to the decay chain, but can be generalized as follows. Since A decays into B, then B decays into C, the activity of A adds to the total number of B nuclides in the present sample, before those B nuclides decay and reduce the number of nuclides leading to the later sample. In other words, the number of second generation nuclei B increases as a result of the first generation nuclei decay of A, and decreases as a result of its own decay into the third generation nuclei C. The sum of these two terms gives the law for a decay chain for two nuclides:
d
N
B
d
t
=
−
λ
B
N
B
+
λ
A
N
A
.
{\displaystyle {\frac {\mathrm {d} N_{B}}{\mathrm {d} t}}=-\lambda _{B}N_{B}+\lambda _{A}N_{A}.}
The rate of change of NB, that is dNB/dt, is related to the changes in the amounts of A and B, NB can increase as B is produced from A and decrease as B produces C.
Re-writing using the previous results:
The subscripts simply refer to the respective nuclides, i.e. NA is the number of nuclides of type A; NA0 is the initial number of nuclides of type A; λA is the decay constant for A – and similarly for nuclide B. Solving this equation for NB gives:
N
B
=
N
A
0
λ
A
λ
B
−
λ
A
(
e
−
λ
A
t
−
e
−
λ
B
t
)
.
{\displaystyle N_{B}={\frac {N_{A0}\lambda _{A}}{\lambda _{B}-\lambda _{A}}}\left(e^{-\lambda _{A}t}-e^{-\lambda _{B}t}\right).}
In the case where B is a stable nuclide (λB = 0), this equation reduces to the previous solution:
lim
λ
B
→
0
[
N
A
0
λ
A
λ
B
−
λ
A
(
e
−
λ
A
t
−
e
−
λ
B
t
)
]
=
N
A
0
λ
A
0
−
λ
A
(
e
−
λ
A
t
−
1
)
=
N
A
0
(
1
−
e
−
λ
A
t
)
,
{\displaystyle \lim _{\lambda _{B}\rightarrow 0}\left[{\frac {N_{A0}\lambda _{A}}{\lambda _{B}-\lambda _{A}}}\left(e^{-\lambda _{A}t}-e^{-\lambda _{B}t}\right)\right]={\frac {N_{A0}\lambda _{A}}{0-\lambda _{A}}}\left(e^{-\lambda _{A}t}-1\right)=N_{A0}\left(1-e^{-\lambda _{A}t}\right),}
as shown above for one decay. The solution can be found by the integration factor method, where the integrating factor is eλBt. This case is perhaps the most useful since it can derive both the one-decay equation (above) and the equation for multi-decay chains (below) more directly.
===== Chain of any number of decays =====
For the general case of any number of consecutive decays in a decay chain, i.e. A1 → A2 ··· → Ai ··· → AD, where D is the number of decays and i is a dummy index (i = 1, 2, 3, ..., D), each nuclide population can be found in terms of the previous population. In this case N2 = 0, N3 = 0, ..., ND = 0. Using the above result in a recursive form:
d
N
j
d
t
=
−
λ
j
N
j
+
λ
j
−
1
N
(
j
−
1
)
0
e
−
λ
j
−
1
t
.
{\displaystyle {\frac {\mathrm {d} N_{j}}{\mathrm {d} t}}=-\lambda _{j}N_{j}+\lambda _{j-1}N_{(j-1)0}e^{-\lambda _{j-1}t}.}
The general solution to the recursive problem is given by Bateman's equations:
==== Multiple products ====
In all of the above examples, the initial nuclide decays into just one product. Consider the case of one initial nuclide that can decay into either of two products, that is A → B and A → C in parallel. For example, in a sample of potassium-40, 89.3% of the nuclei decay to calcium-40 and 10.7% to argon-40. We have for all time t:
N
=
N
A
+
N
B
+
N
C
{\displaystyle N=N_{A}+N_{B}+N_{C}}
which is constant, since the total number of nuclides remains constant. Differentiating with respect to time:
d
N
A
d
t
=
−
(
d
N
B
d
t
+
d
N
C
d
t
)
−
λ
N
A
=
−
N
A
(
λ
B
+
λ
C
)
{\displaystyle {\begin{aligned}{\frac {\mathrm {d} N_{A}}{\mathrm {d} t}}&=-\left({\frac {\mathrm {d} N_{B}}{\mathrm {d} t}}+{\frac {\mathrm {d} N_{C}}{\mathrm {d} t}}\right)\\-\lambda N_{A}&=-N_{A}\left(\lambda _{B}+\lambda _{C}\right)\\\end{aligned}}}
defining the total decay constant λ in terms of the sum of partial decay constants λB and λC:
λ
=
λ
B
+
λ
C
.
{\displaystyle \lambda =\lambda _{B}+\lambda _{C}.}
Solving this equation for NA:
N
A
=
N
A
0
e
−
λ
t
.
{\displaystyle N_{A}=N_{A0}e^{-\lambda t}.}
where NA0 is the initial number of nuclide A. When measuring the production of one nuclide, one can only observe the total decay constant λ. The decay constants λB and λC determine the probability for the decay to result in products B or C as follows:
N
B
=
λ
B
λ
N
A
0
(
1
−
e
−
λ
t
)
,
{\displaystyle N_{B}={\frac {\lambda _{B}}{\lambda }}N_{A0}\left(1-e^{-\lambda t}\right),}
N
C
=
λ
C
λ
N
A
0
(
1
−
e
−
λ
t
)
.
{\displaystyle N_{C}={\frac {\lambda _{C}}{\lambda }}N_{A0}\left(1-e^{-\lambda t}\right).}
because the fraction λB/λ of nuclei decay into B while the fraction λC/λ of nuclei decay into C.
=== Corollaries of laws ===
The above equations can also be written using quantities related to the number of nuclide particles N in a sample;
The activity: A = λN.
The amount of substance: n = N/NA.
The mass: m = Mn = MN/NA.
where NA = 6.02214076×1023 mol−1 is the Avogadro constant, M is the molar mass of the substance in kg/mol, and the amount of the substance n is in moles.
=== Decay timing: definitions and relations ===
==== Time constant and mean-life ====
For the one-decay solution A → B:
N
=
N
0
e
−
λ
t
=
N
0
e
−
t
/
τ
,
{\displaystyle N=N_{0}\,e^{-{\lambda }t}=N_{0}\,e^{-t/\tau },\,\!}
the equation indicates that the decay constant λ has units of t−1, and can thus also be represented as 1/τ, where τ is a characteristic time of the process called the time constant.
In a radioactive decay process, this time constant is also the mean lifetime for decaying atoms. Each atom "lives" for a finite amount of time before it decays, and it may be shown that this mean lifetime is the arithmetic mean of all the atoms' lifetimes, and that it is τ, which again is related to the decay constant as follows:
τ
=
1
λ
.
{\displaystyle \tau ={\frac {1}{\lambda }}.}
This form is also true for two-decay processes simultaneously A → B + C, inserting the equivalent values of decay constants (as given above)
λ
=
λ
B
+
λ
C
{\displaystyle \lambda =\lambda _{B}+\lambda _{C}\,}
into the decay solution leads to:
1
τ
=
λ
=
λ
B
+
λ
C
=
1
τ
B
+
1
τ
C
{\displaystyle {\frac {1}{\tau }}=\lambda =\lambda _{B}+\lambda _{C}={\frac {1}{\tau _{B}}}+{\frac {1}{\tau _{C}}}\,}
==== Half-life ====
A more commonly used parameter is the half-life T1/2. Given a sample of a particular radionuclide, the half-life is the time taken for half the radionuclide's atoms to decay. For the case of one-decay nuclear reactions:
N
=
N
0
e
−
λ
t
=
N
0
e
−
t
/
τ
,
{\displaystyle N=N_{0}\,e^{-{\lambda }t}=N_{0}\,e^{-t/\tau },\,\!}
the half-life is related to the decay constant as follows: set N = N0/2 and t = T1/2 to obtain
t
1
/
2
=
ln
2
λ
=
τ
ln
2.
{\displaystyle t_{1/2}={\frac {\ln 2}{\lambda }}=\tau \ln 2.}
This relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer. Half-lives of known radionuclides vary by almost 54 orders of magnitude, from more than 2.25(9)×1024 years (6.9×1031 sec) for the very nearly stable nuclide 128Te, to 8.6(6)×10−23 seconds for the highly unstable nuclide 5H.
The factor of ln(2) in the above relations results from the fact that the concept of "half-life" is merely a way of selecting a different base other than the natural base e for the lifetime expression. The time constant τ is the e −1 -life, the time until only 1/e remains, about 36.8%, rather than the 50% in the half-life of a radionuclide. Thus, τ is longer than t1/2. The following equation can be shown to be valid:
N
(
t
)
=
N
0
e
−
t
/
τ
=
N
0
2
−
t
/
t
1
/
2
.
{\displaystyle N(t)=N_{0}\,e^{-t/\tau }=N_{0}\,2^{-t/t_{1/2}}.\,\!}
Since radioactive decay is exponential with a constant probability, each process could as easily be described with a different constant time period that (for example) gave its "(1/3)-life" (how long until only 1/3 is left) or "(1/10)-life" (a time period until only 10% is left), and so on. Thus, the choice of τ and t1/2 for marker-times, are only for convenience, and from convention. They reflect a fundamental principle only in so much as they show that the same proportion of a given radioactive substance will decay, during any time-period that one chooses.
Mathematically, the nth life for the above situation would be found in the same way as above—by setting N = N0/n, t = T1/n and substituting into the decay solution to obtain
t
1
/
n
=
ln
n
λ
=
τ
ln
n
.
{\displaystyle t_{1/n}={\frac {\ln n}{\lambda }}=\tau \ln n.}
=== Example for carbon-14 ===
Carbon-14 has a half-life of 5700(30) years and a decay rate of 14 disintegrations per minute (dpm) per gram of natural carbon.
If an artifact is found to have radioactivity of 4 dpm per gram of its present C, we can find the approximate age of the object using the above equation:
N
=
N
0
e
−
t
/
τ
,
{\displaystyle N=N_{0}\,e^{-t/\tau },}
where:
N
N
0
=
4
/
14
≈
0.286
,
τ
=
T
1
/
2
ln
2
≈
8267
years
,
t
=
−
τ
ln
N
N
0
≈
10356
years
.
{\displaystyle {\begin{aligned}{\frac {N}{N_{0}}}&=4/14\approx 0.286,\\\tau &={\frac {T_{1/2}}{\ln 2}}\approx 8267{\text{ years}},\\t&=-\tau \,\ln {\frac {N}{N_{0}}}\approx 10356{\text{ years}}.\end{aligned}}}
=== Changing rates ===
The radioactive decay modes of electron capture and internal conversion are known to be slightly sensitive to chemical and environmental effects that change the electronic structure of the atom, which in turn affects the presence of 1s and 2s electrons that participate in the decay process. A small number of nuclides are affected. For example, chemical bonds can affect the rate of electron capture to a small degree (in general, less than 1%) depending on the proximity of electrons to the nucleus. In 7Be, a difference of 0.9% has been observed between half-lives in metallic and insulating environments. This relatively large effect is because beryllium is a small atom whose valence electrons are in 2s atomic orbitals, which are subject to electron capture in 7Be because (like all s atomic orbitals in all atoms) they naturally penetrate into the nucleus.
In 1992, Jung et al. of the Darmstadt Heavy-Ion Research group observed an accelerated β− decay of 163Dy66+. Although neutral 163Dy is a stable isotope, the fully ionized 163Dy66+ undergoes β− decay into the K and L shells to 163Ho66+ with a half-life of 47 days.
Rhenium-187 is another spectacular example. 187Re normally undergoes beta decay to 187Os with a half-life of 41.6 × 109 years, but studies using fully ionised 187Re atoms (bare nuclei) have found that this can decrease to only 32.9 years. This is attributed to "bound-state β− decay" of the fully ionised atom – the electron is emitted into the "K-shell" (1s atomic orbital), which cannot occur for neutral atoms in which all low-lying bound states are occupied.
A number of experiments have found that decay rates of other modes of artificial and naturally occurring radioisotopes are, to a high degree of precision, unaffected by external conditions such as temperature, pressure, the chemical environment, and electric, magnetic, or gravitational fields. Comparison of laboratory experiments over the last century, studies of the Oklo natural nuclear reactor (which exemplified the effects of thermal neutrons on nuclear decay), and astrophysical observations of the luminosity decays of distant supernovae (which occurred far away so the light has taken a great deal of time to reach us), for example, strongly indicate that unperturbed decay rates have been constant (at least to within the limitations of small experimental errors) as a function of time as well.
Recent results suggest the possibility that decay rates might have a weak dependence on environmental factors. It has been suggested that measurements of decay rates of silicon-32, manganese-54, and radium-226 exhibit small seasonal variations (of the order of 0.1%). However, such measurements are highly susceptible to systematic errors, and a subsequent paper has found no evidence for such correlations in seven other isotopes (22Na, 44Ti, 108Ag, 121Sn, 133Ba, 241Am, 238Pu), and sets upper limits on the size of any such effects. The decay of radon-222 was once reported to exhibit large 4% peak-to-peak seasonal variations (see plot), which were proposed to be related to either solar flare activity or the distance from the Sun, but detailed analysis of the experiment's design flaws, along with comparisons to other, much more stringent and systematically controlled, experiments refute this claim.
==== GSI anomaly ====
An unexpected series of experimental results for the rate of decay of heavy highly charged radioactive ions circulating in a storage ring has provoked theoretical activity in an effort to find a convincing explanation. The rates of weak decay of two radioactive species with half lives of about 40 s and 200 s are found to have a significant oscillatory modulation, with a period of about 7 s.
The observed phenomenon is known as the GSI anomaly, as the storage ring is a facility at the GSI Helmholtz Centre for Heavy Ion Research in Darmstadt, Germany. As the decay process produces an electron neutrino, some of the proposed explanations for the observed rate oscillation invoke neutrino properties. Initial ideas related to flavour oscillation met with skepticism. A more recent proposal involves mass differences between neutrino mass eigenstates.
== Nuclear processes ==
A nuclide is considered to "exist" if it has a half-life greater than 2x10−14s. This is an arbitrary boundary; shorter half-lives are considered resonances, such as a system undergoing a nuclear reaction. This time scale is characteristic of the strong interaction which creates the nuclear force. Only nuclides are considered to decay and produce radioactivity.: 568
Nuclides can be stable or unstable. Unstable nuclides decay, possibly in several steps, until they become stable. There are 251 known stable nuclides. The number of unstable nuclides discovered has grown, with about 3000 known in 2006.
The most common and consequently historically the most important forms of natural radioactive decay involve the emission of alpha-particles, beta-particles, and gamma rays. Each of these correspond to a fundamental interaction predominantly responsible for the radioactivity:: 142
alpha-decay -> strong interaction,
beta-decay -> weak interaction,
gamma-decay -> electromagnetism.
In alpha decay, a particle containing two protons and two neutrons, equivalent to a He nucleus, breaks out of the parent nucleus. The process represents a competition between the electromagnetic repulsion between the protons in the nucleus and attractive nuclear force, a residual of the strong interaction. The alpha particle is an especially strongly bound nucleus, helping it win the competition more often.: 872 However some nuclei break up or fission into larger particles and artificial nuclei decay with the emission of
single protons, double protons, and other combinations.
Beta decay transforms a neutron into proton or vice versa. When a neutron inside a parent nuclide decays to a proton, an electron, a anti-neutrino, and nuclide with high atomic number results. When a proton in a parent nuclide transforms to a neutron, a positron, a neutrino, and nuclide with a lower atomic number results. These changes are a direct manifestation of the weak interaction.: 874
Gamma decay resembles other kinds of electromagnetic emission: it corresponds to transitions between an excited quantum state and lower energy state. Any of the particle decay mechanisms often leave the daughter in an excited state, which then decays via gamma emission.: 876
Other forms of decay include neutron emission, electron capture, internal conversion, cluster decay.
== Hazard warning signs ==
== See also ==
Nuclear technology portal
Physics portal
== Notes ==
== References ==
== External links ==
The Lund/LBNL Nuclear Data Search – Contains tabulated information on radioactive decay types and energies.
Nomenclature of nuclear chemistry
Specific activity and related topics.
The Live Chart of Nuclides – IAEA
Interactive Chart of Nuclides Archived 10 October 2018 at the Wayback Machine
Health Physics Society Public Education Website
Beach, Chandler B., ed. (1914). "Becquerel Rays" . The New Student's Reference Work . Chicago: F. E. Compton and Co.
Annotated bibliography for radioactivity from the Alsos Digital Library for Nuclear Issues Archived 7 October 2010 at the Wayback Machine
"Henri Becquerel: The Discovery of Radioactivity", Becquerel's 1896 articles online and analyzed on BibNum [click 'à télécharger' for English version].
"Radioactive change", Rutherford & Soddy article (1903), online and analyzed on Bibnum [click 'à télécharger' for English version] | Wikipedia/Decay_rate |
Dirac hole theory is a theory in quantum mechanics, named after English theoretical physicist Paul Dirac, who introduced it in 1929. The theory poses that the continuum of negative energy states, that are solutions to the Dirac equation, are filled with electrons, and the vacancies in this continuum (holes) are manifested as positrons with energy and momentum that are the negative of those of the state. The discovery of the positron in 1929 gave a considerable support to the Dirac hole theory.
While Enrico Fermi, Niels Bohr and Wolfgang Pauli were skeptical about the theory, other physicists, like Guido Beck and Kurt Sitte, made use of Dirac hole theory in alternative theories of beta decay. Gian Wick extended Dirac hole theory to cover neutrinos, introducing the anti-neutrino as a hole in a neutrino Dirac sea.
== Pair production and annihilation ==
Hole theory provides an alternative perspective on the processes of pair production and annihilation – when a photon of sufficient energy is incident upon an occupied state in the negative energy 'sea', it can excite an electron into the positive energy region, creating both an observable electron while creating a vacant state (hole) in the negative energy region – an anti-electron, or more commonly, a positron.
Conversely, due to the principle of least action, the close proximity of an electron and positron presents an opportunity for the electron to de-excite, releasing a photon, reducing the overall energy of the system – this is observationally identical to the process of annihilation.
== References == | Wikipedia/Dirac_hole_theory |
In theoretical physics, more specifically in quantum field theory and supersymmetry, supersymmetric Yang–Mills, also known as super Yang–Mills and abbreviated to SYM, is a supersymmetric generalization of Yang–Mills theory, which is a gauge theory that plays an important part in the mathematical formulation of forces in particle physics. It is a special case of 4D N = 1 global supersymmetry.
Super Yang–Mills was studied by Julius Wess and Bruno Zumino in which they demonstrated the supergauge-invariance of the theory and wrote down its action, alongside the action of the Wess–Zumino model, another early supersymmetric field theory.
The treatment in this article largely follows that of Figueroa-O'Farrill's lectures on supersymmetry and of Tong.
While N = 4 supersymmetric Yang–Mills theory is also a supersymmetric Yang–Mills theory, it has very different properties to
N
=
1
{\displaystyle {\mathcal {N}}=1}
supersymmetric Yang–Mills theory, which is the theory discussed in this article. The
N
=
2
{\displaystyle {\mathcal {N}}=2}
supersymmetric Yang–Mills theory was studied by Seiberg and Witten in Seiberg–Witten theory. All three theories are based in
d
=
4
{\displaystyle d=4}
super Minkowski spaces.
== The supersymmetric Yang–Mills action ==
=== Preliminary treatment ===
A first treatment can be done without defining superspace, instead defining the theory in terms of familiar fields in non-supersymmetric quantum field theory.
==== Spacetime and matter content ====
The base spacetime is flat spacetime (Minkowski space).
SYM is a gauge theory, and there is an associated gauge group
G
{\displaystyle G}
to the theory. The gauge group has associated Lie algebra
g
{\displaystyle {\mathfrak {g}}}
.
The field content then consists of
a
g
{\displaystyle {\mathfrak {g}}}
-valued gauge field
A
μ
{\displaystyle A_{\mu }}
a
g
{\displaystyle {\mathfrak {g}}}
-valued Majorana spinor field
Ψ
{\displaystyle \Psi }
(an adjoint-valued spinor), known as the 'gaugino'
a
g
{\displaystyle {\mathfrak {g}}}
-valued auxiliary scalar field
D
{\displaystyle D}
.
For gauge-invariance, the gauge field
A
μ
{\displaystyle A_{\mu }}
is necessarily massless. This means its superpartner
Ψ
{\displaystyle \Psi }
is also massless if supersymmetry is to hold. Therefore
Ψ
{\displaystyle \Psi }
can be written in terms of two Weyl spinors which are conjugate to one another:
Ψ
=
(
λ
,
λ
¯
)
{\displaystyle \Psi =(\lambda ,{\bar {\lambda }})}
, and the theory can be formulated in terms of the Weyl spinor field
λ
{\displaystyle \lambda }
instead of
Ψ
{\displaystyle \Psi }
.
==== Supersymmetric pure electromagnetic theory ====
When
G
=
U
(
1
)
{\displaystyle G=U(1)}
, the conceptual difficulties simplify somewhat, and this is in some sense the simplest gauge theory. The field content is simply a (co-)vector field
A
μ
{\displaystyle A_{\mu }}
, a Majorana spinor
Ψ
{\displaystyle \Psi }
and a auxiliary real scalar field
D
{\displaystyle D}
.
The field strength tensor is defined as usual as
F
μ
ν
:=
∂
μ
A
ν
−
∂
ν
A
μ
{\displaystyle F_{\mu \nu }:=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}
.
The Lagrangian written down by Wess and Zumino is then
L
=
−
1
4
F
μ
ν
F
μ
ν
−
i
2
Ψ
¯
γ
μ
∂
μ
Ψ
+
1
2
D
2
.
{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {i}{2}}{\bar {\Psi }}\gamma ^{\mu }\partial _{\mu }\Psi +{\frac {1}{2}}D^{2}.}
This can be generalized to include a coupling constant
e
{\displaystyle e}
, and theta term
∝
ϑ
F
μ
ν
∗
F
μ
ν
{\displaystyle \propto \vartheta F_{\mu \nu }*F^{\mu \nu }}
, where
∗
F
μ
ν
{\displaystyle *F^{\mu \nu }}
is the dual field strength tensor
∗
F
μ
ν
=
1
2
ϵ
μ
ν
ρ
σ
F
ρ
σ
.
{\displaystyle *F^{\mu \nu }={\frac {1}{2}}\epsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }.}
and
ϵ
μ
ν
ρ
σ
{\displaystyle \epsilon ^{\mu \nu \rho \sigma }}
is the alternating tensor or totally antisymmetric tensor. If we also replace the field
Ψ
{\displaystyle \Psi }
with the Weyl spinor
λ
{\displaystyle \lambda }
, then a supersymmetric action can be written as
This can be viewed as a supersymmetric generalization of a pure
U
(
1
)
{\displaystyle U(1)}
gauge theory, also known as Maxwell theory or pure electromagnetic theory.
==== Supersymmetric Yang–Mills theory (preliminary treatment) ====
In full generality, we must define the gluon field strength tensor,
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
−
i
[
A
μ
,
A
ν
]
{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }-i[A_{\mu },A_{\nu }]}
and the covariant derivative of the adjoint Weyl spinor,
D
μ
λ
=
∂
μ
λ
−
i
[
A
μ
,
λ
]
.
{\displaystyle D_{\mu }\lambda =\partial _{\mu }\lambda -i[A_{\mu },\lambda ].}
To write down the action, an invariant inner product on
g
{\displaystyle {\mathfrak {g}}}
is needed: the Killing form
B
(
⋅
,
⋅
)
{\displaystyle B(\cdot ,\cdot )}
is such an inner product, and in a typical abuse of notation we write
B
{\displaystyle B}
simply as
Tr
{\displaystyle {\text{Tr}}}
, suggestive of the fact that the invariant inner product arises as the trace in some representation of
g
{\displaystyle {\mathfrak {g}}}
.
Supersymmetric Yang–Mills then readily generalizes from supersymmetric Maxwell theory. A simple version is
S
SYM
=
∫
d
4
x
Tr
[
−
1
4
F
μ
ν
F
μ
ν
−
1
2
Ψ
¯
γ
μ
D
μ
Ψ
]
{\displaystyle S_{\text{SYM}}=\int d^{4}x{\text{Tr}}\left[-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {1}{2}}{\bar {\Psi }}\gamma ^{\mu }D_{\mu }\Psi \right]}
while a more general version is given by
=== Superspace treatment ===
==== Superspace and superfield content ====
The base superspace is
N
=
1
{\displaystyle {\mathcal {N}}=1}
super Minkowski space.
The theory is defined in terms of a single adjoint-valued real superfield
V
{\displaystyle V}
, fixed to be in Wess–Zumino gauge.
==== Supersymmetric Maxwell theory on superspace ====
The theory is defined in terms of a superfield arising from taking covariant derivatives of
V
{\displaystyle V}
:
W
α
=
−
1
4
D
¯
2
D
α
V
{\displaystyle W_{\alpha }=-{\frac {1}{4}}{\mathcal {{\bar {D}}^{2}}}{\mathcal {D}}_{\alpha }V}
.
The supersymmetric action is then written down, with a complex coupling constant
τ
=
ϑ
2
π
+
4
π
i
e
{\displaystyle \tau ={\frac {\vartheta }{2\pi }}+{\frac {4\pi i}{e}}}
, as
where h.c. indicates the Hermitian conjugate of the preceding term.
==== Supersymmetric Yang–Mills on superspace ====
For non-abelian gauge theory, instead define
W
α
=
−
1
8
D
¯
2
(
e
−
2
V
D
α
e
2
V
)
{\displaystyle W_{\alpha }=-{\frac {1}{8}}{\bar {\mathcal {D}}}^{2}(e^{-2V}{\mathcal {D}}_{\alpha }e^{2V})}
and
τ
=
ϑ
2
π
+
4
π
i
g
{\displaystyle \tau ={\frac {\vartheta }{2\pi }}+{\frac {4\pi i}{g}}}
. Then the action is
== Symmetries of the action ==
=== Supersymmetry ===
For the simplified Yang–Mills action on Minkowski space (not on superspace), the supersymmetry transformations are
δ
ϵ
A
μ
=
ϵ
¯
γ
μ
Ψ
{\displaystyle \delta _{\epsilon }A_{\mu }={\bar {\epsilon }}\gamma _{\mu }\Psi }
δ
ϵ
Ψ
=
−
1
2
F
μ
ν
γ
μ
ν
ϵ
{\displaystyle \delta _{\epsilon }\Psi =-{\frac {1}{2}}F_{\mu \nu }\gamma ^{\mu \nu }\epsilon }
where
γ
μ
ν
=
1
2
(
γ
μ
γ
ν
−
γ
ν
γ
μ
)
{\displaystyle \gamma ^{\mu \nu }={\frac {1}{2}}(\gamma ^{\mu }\gamma ^{\nu }-\gamma ^{\nu }\gamma ^{\mu })}
.
For the Yang–Mills action on superspace, since
W
α
{\displaystyle W_{\alpha }}
is chiral, then so are fields built from
W
α
{\displaystyle W_{\alpha }}
. Then integrating over half of superspace,
∫
d
2
θ
{\displaystyle \int d^{2}\theta }
, gives a supersymmetric action.
An important observation is that the Wess–Zumino gauge is not a supersymmetric gauge, that is, it is not preserved by supersymmetry. However, it is possible to do a compensating gauge transformation to return to Wess–Zumino gauge. Then, after a supersymmetry transformation and the compensating gauge transformation, the superfields transform as
δ
A
μ
=
ϵ
σ
μ
λ
¯
+
λ
σ
μ
ϵ
¯
,
{\displaystyle \delta A_{\mu }=\epsilon \sigma _{\mu }{\bar {\lambda }}+\lambda \sigma _{\mu }{\bar {\epsilon }},}
δ
λ
=
ϵ
D
+
(
σ
μ
ν
ϵ
)
F
μ
ν
{\displaystyle \delta \lambda =\epsilon D+(\sigma ^{\mu \nu }\epsilon )F_{\mu \nu }}
δ
D
=
i
ϵ
σ
μ
∂
μ
λ
¯
−
i
∂
μ
λ
σ
¯
μ
ϵ
¯
.
{\displaystyle \delta D=i\epsilon \sigma ^{\mu }\partial _{\mu }{\bar {\lambda }}-i\partial _{\mu }\lambda {\bar {\sigma }}^{\mu }{\bar {\epsilon }}.}
=== Gauge symmetry ===
The preliminary theory defined on spacetime is manifestly gauge invariant as it is built from terms studied in non-supersymmetric gauge theory which are gauge invariant.
The superfield formulation requires a theory of generalized gauge transformations. (Not supergauge transformations, which would be transformations in a theory with local supersymmetry).
==== Generalized abelian gauge transformations ====
Such a transformation is parametrized by a chiral superfield
Ω
{\displaystyle \Omega }
, under which the real superfield transforms as
V
↦
V
+
i
(
Ω
−
Ω
†
)
.
{\displaystyle V\mapsto V+i(\Omega -\Omega ^{\dagger }).}
In particular, upon expanding
V
{\displaystyle V}
and
Ω
{\displaystyle \Omega }
appropriately into constituent superfields, then
V
{\displaystyle V}
contains a vector superfield
A
μ
{\displaystyle A_{\mu }}
while
Ω
{\displaystyle \Omega }
contains a scalar superfield
ω
{\displaystyle \omega }
, such that
A
μ
↦
A
μ
−
2
∂
μ
(
Re
ω
)
=:
A
μ
+
∂
μ
α
.
{\displaystyle A_{\mu }\mapsto A_{\mu }-2\partial _{\mu }({\text{Re}}\,\omega )=:A_{\mu }+\partial _{\mu }\alpha .}
The chiral superfield used to define the action,
W
α
=
−
1
4
D
¯
2
D
α
V
,
{\displaystyle W_{\alpha }=-{\frac {1}{4}}{\bar {\mathcal {D}}}^{2}{\mathcal {D}}_{\alpha }V,}
is gauge invariant.
==== Generalized non-abelian gauge transformations ====
The chiral superfield is adjoint valued. The transformation of
V
{\displaystyle V}
is prescribed by
e
2
V
↦
e
−
2
i
Ω
†
e
2
V
e
2
i
Ω
{\displaystyle e^{2V}\mapsto e^{-2i\Omega ^{\dagger }}e^{2V}e^{2i\Omega }}
,
from which the transformation for
V
{\displaystyle V}
can be derived using the Baker–Campbell–Hausdorff formula.
The chiral superfield
W
α
=
−
1
8
D
¯
2
(
e
−
2
V
D
α
e
2
V
)
{\displaystyle W_{\alpha }=-{\frac {1}{8}}{\bar {\mathcal {D}}}^{2}(e^{-2V}{\mathcal {D}}_{\alpha }e^{2V})}
is not invariant but transforms by conjugation:
W
α
↦
e
2
i
Ω
W
α
e
−
2
i
Ω
{\displaystyle W_{\alpha }\mapsto e^{2i\Omega }W_{\alpha }e^{-2i\Omega }}
,
so that upon tracing in the action, the action is gauge-invariant.
== Extra classical symmetries ==
=== Superconformal symmetry ===
As a classical theory, supersymmetric Yang–Mills theory admits a larger set of symmetries, described at the algebra level by the superconformal algebra. Just as the super Poincaré algebra is a supersymmetric extension of the Poincaré algebra, the superconformal algebra is a supersymmetric extension of the conformal algebra which also contains a spinorial generator of conformal supersymmetry
S
α
{\displaystyle S_{\alpha }}
.
Conformal invariance is broken in the quantum theory by trace and conformal anomalies.
While the quantum
N
=
1
{\displaystyle {\mathcal {N}}=1}
supersymmetric Yang–Mills theory does not have superconformal symmetry, quantum N = 4 supersymmetric Yang–Mills theory does.
=== R-symmetry ===
The
U
(
1
)
{\displaystyle {\text{U}}(1)}
R-symmetry for
N
=
1
{\displaystyle {\mathcal {N}}=1}
supersymmetry is a symmetry of the classical theory, but not of the quantum theory due to an anomaly.
== Adding matter ==
=== Abelian gauge ===
Matter can be added in the form of Wess–Zumino model type superfields
Φ
{\displaystyle \Phi }
. Under a gauge transformation,
Φ
↦
exp
(
−
2
i
q
Ω
)
Φ
{\displaystyle \Phi \mapsto \exp(-2iq\Omega )\Phi }
,
and instead of using just
Φ
†
Φ
{\displaystyle \Phi ^{\dagger }\Phi }
as the Lagrangian as in the Wess–Zumino model, for gauge invariance it must be replaced with
Φ
†
e
2
q
V
Φ
.
{\displaystyle \Phi ^{\dagger }e^{2qV}\Phi .}
This gives a supersymmetric analogue to QED. The action can be written
S
SMaxwell
+
∫
d
4
x
∫
d
4
θ
Φ
†
e
2
q
V
Φ
.
{\displaystyle S_{\text{SMaxwell}}+\int d^{4}x\,\int d^{4}\theta \,\Phi ^{\dagger }e^{2qV}\Phi .}
For
N
f
{\displaystyle N_{f}}
flavours, we instead have
N
f
{\displaystyle N_{f}}
superfields
Φ
i
{\displaystyle \Phi _{i}}
, and the action can be written
S
SMaxwell
+
∫
d
4
x
∫
d
4
θ
Φ
i
†
e
2
q
i
V
Φ
i
.
{\displaystyle S_{\text{SMaxwell}}+\int d^{4}x\,\int d^{4}\theta \,\Phi _{i}^{\dagger }e^{2q_{i}V}\Phi _{i}.}
with implicit summation.
However, for a well-defined quantum theory, a theory such as that defined above suffers a gauge anomaly. We are obliged to add a partner
Φ
~
{\displaystyle {\tilde {\Phi }}}
to each chiral superfield
Φ
{\displaystyle \Phi }
(distinct from the idea of superpartners, and from conjugate superfields), which has opposite charge. This gives the action
S
SQED
=
S
SMaxwell
+
∫
d
4
x
∫
d
4
θ
Φ
i
†
e
2
q
i
V
Φ
i
+
Φ
~
i
†
e
−
2
q
i
V
Φ
~
i
.
{\displaystyle S_{\text{SQED}}=S_{\text{SMaxwell}}+\int d^{4}x\,\int d^{4}\theta \,\Phi _{i}^{\dagger }e^{2q_{i}V}\Phi _{i}+{\tilde {\Phi }}_{i}^{\dagger }e^{-2q_{i}V}{\tilde {\Phi }}_{i}.}
=== Non-Abelian gauge ===
For non-abelian gauge, matter chiral superfields
Φ
{\displaystyle \Phi }
are now valued in a representation
R
{\displaystyle R}
of the gauge group:
Φ
↦
exp
(
−
2
i
Ω
)
Φ
{\displaystyle \Phi \mapsto \exp(-2i\Omega )\Phi }
.
The Wess–Zumino kinetic term must be adjusted to
Φ
†
e
2
V
Φ
{\displaystyle \Phi ^{\dagger }e^{2V}\Phi }
.
Then a simple SQCD action would be to take
R
{\displaystyle R}
to be the fundamental representation, and add the Wess–Zumino term:
S
SYM
+
∫
d
4
x
d
4
θ
Φ
†
e
2
V
Φ
{\displaystyle S_{\text{SYM}}+\int d^{4}x\,d^{4}\theta \,\Phi ^{\dagger }e^{2V}\Phi }
.
More general and detailed forms of the super QCD action are given in that article.
== Fayet–Iliopoulos term ==
When the center of the Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is non-trivial, there is an extra term which can be added to the action known as the Fayet–Iliopoulos term.
== References == | Wikipedia/N_=_1_supersymmetric_Yang–Mills_theory |
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. One GL-type superconductor is the famous YBCO, and generally all cuprates.
Later, a version of Ginzburg–Landau theory was derived from the Bardeen–Cooper–Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters. The theory can also be given a general geometric setting, placing it in the context of Riemannian geometry, where in many cases exact solutions can be given. This general setting then extends to quantum field theory and string theory, again owing to its solvability, and its close relation to other, similar systems.
== Introduction ==
Based on Landau's previously established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy density
f
s
{\displaystyle f_{s}}
of a superconductor near the superconducting transition can be expressed in terms of a complex order parameter field
ψ
(
r
)
=
|
ψ
(
r
)
|
e
i
ϕ
(
r
)
{\displaystyle \psi (r)=|\psi (r)|e^{i\phi (r)}}
, where the quantity
|
ψ
(
r
)
|
2
{\displaystyle |\psi (r)|^{2}}
is a measure of the local density of superconducting electrons
n
s
(
r
)
{\displaystyle n_{s}(r)}
analogous to a quantum mechanical wave function. While
ψ
(
r
)
{\displaystyle \psi (r)}
is nonzero below a phase transition into a superconducting state, no direct interpretation of this parameter was given in the original paper. Assuming smallness of
|
ψ
|
{\displaystyle |\psi |}
and smallness of its gradients, the free energy density has the form of a field theory and exhibits U(1) gauge symmetry:
f
s
=
f
n
+
α
(
T
)
|
ψ
|
2
+
1
2
β
(
T
)
|
ψ
|
4
+
1
2
m
∗
|
(
−
i
ℏ
∇
−
e
∗
c
A
)
ψ
|
2
+
B
2
8
π
,
{\displaystyle f_{s}=f_{n}+\alpha (T)|\psi |^{2}+{\frac {1}{2}}\beta (T)|\psi |^{4}+{\frac {1}{2m^{*}}}\left|\left(-i\hbar \nabla -{\frac {e^{*}}{c}}\mathbf {A} \right)\psi \right|^{2}+{\frac {\mathbf {B} ^{2}}{8\pi }},}
where
f
n
{\displaystyle f_{n}}
is the free energy density of the normal phase,
α
(
T
)
{\displaystyle \alpha (T)}
and
β
(
T
)
{\displaystyle \beta (T)}
are phenomenological parameters that are functions of T (and often written just
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
).
m
∗
{\displaystyle m^{*}}
is an effective mass,
e
∗
{\displaystyle e^{*}}
is an effective charge (usually
2
e
{\displaystyle 2e}
, where e is the charge of an electron),
A
{\displaystyle \mathbf {A} }
is the magnetic vector potential, and
B
=
∇
×
A
{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }
is the magnetic field.
The total free energy is given by
F
=
∫
f
s
d
3
r
{\displaystyle F=\int f_{s}d^{3}r}
. By minimizing
F
{\displaystyle F}
with respect to variations in the order parameter
ψ
{\displaystyle \psi }
and the vector potential
A
{\displaystyle \mathbf {A} }
, one arrives at the Ginzburg–Landau equations
α
ψ
+
β
|
ψ
|
2
ψ
+
1
2
m
∗
(
−
i
ℏ
∇
−
e
∗
c
A
)
2
ψ
=
0
{\displaystyle \alpha \psi +\beta |\psi |^{2}\psi +{\frac {1}{2m^{*}}}\left(-i\hbar \nabla -{\frac {e^{*}}{c}}\mathbf {A} \right)^{2}\psi =0}
∇
×
B
=
4
π
c
J
;
J
=
e
∗
m
∗
Re
{
ψ
∗
(
−
i
ℏ
∇
−
e
∗
c
A
)
ψ
}
,
{\displaystyle \nabla \times \mathbf {B} ={\frac {4\pi }{c}}\mathbf {J} \;\;;\;\;\mathbf {J} ={\frac {e^{*}}{m^{*}}}\operatorname {Re} \left\{\psi ^{*}\left(-i\hbar \nabla -{\frac {e^{*}}{c}}\mathbf {A} \right)\psi \right\},}
where
J
{\displaystyle J}
denotes the dissipation-free electric current density and Re the real part. The first equation — which bears some similarities to the time-independent Schrödinger equation, but is principally different due to a nonlinear term — determines the order parameter,
ψ
{\displaystyle \psi }
. The second equation then provides the superconducting current.
== Simple interpretation ==
Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:
α
ψ
+
β
|
ψ
|
2
ψ
=
0.
{\displaystyle \alpha \psi +\beta |\psi |^{2}\psi =0.}
This equation has a trivial solution: ψ = 0. This corresponds to the normal conducting state, that is for temperatures above the superconducting transition temperature, T > Tc.
Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is
ψ
≠
0
{\displaystyle \psi \neq 0}
). Under this assumption the equation above can be rearranged into:
|
ψ
|
2
=
−
α
β
.
{\displaystyle |\psi |^{2}=-{\frac {\alpha }{\beta }}.}
When the right hand side of this equation is positive, there is a nonzero solution for ψ (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of
α
:
α
(
T
)
=
α
0
(
T
−
T
c
)
{\displaystyle \alpha :\alpha (T)=\alpha _{0}(T-T_{\rm {c}})}
with
α
0
/
β
>
0
{\displaystyle \alpha _{0}/\beta >0}
:
Above the superconducting transition temperature, T > Tc, the expression α(T) / β is positive and the right hand side of the equation above is negative. The magnitude of a complex number must be a non-negative number, so only ψ = 0 solves the Ginzburg–Landau equation.
Below the superconducting transition temperature, T < Tc, the right hand side of the equation above is positive and there is a non-trivial solution for ψ. Furthermore,
|
ψ
|
2
=
−
α
0
(
T
−
T
c
)
β
,
{\displaystyle |\psi |^{2}=-{\frac {\alpha _{0}(T-T_{c})}{\beta }},}
that is ψ approaches zero as T gets closer to Tc from below. Such a behavior is typical for a second order phase transition.
In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to form a superfluid. In this interpretation, |ψ|2 indicates the fraction of electrons that have condensed into a superfluid.
== Coherence length and penetration depth ==
The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor. The first characteristic length was termed coherence length, ξ. For T > Tc (normal phase), it is given by
ξ
=
ℏ
2
2
m
∗
|
α
|
.
{\displaystyle \xi ={\sqrt {\frac {\hbar ^{2}}{2m^{*}|\alpha |}}}.}
while for T < Tc (superconducting phase), where it is more relevant, it is given by
ξ
=
ℏ
2
4
m
∗
|
α
|
.
{\displaystyle \xi ={\sqrt {\frac {\hbar ^{2}}{4m^{*}|\alpha |}}}.}
It sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ψ0. Thus this theory characterized all superconductors by two length scales. The second one is the penetration depth, λ. It was previously introduced by the London brothers in their London theory. Expressed in terms of the parameters of Ginzburg–Landau model it is
λ
=
m
∗
μ
0
e
∗
2
ψ
0
2
=
m
∗
β
μ
0
e
∗
2
|
α
|
,
{\displaystyle \lambda ={\sqrt {\frac {m^{*}}{\mu _{0}e^{*2}\psi _{0}^{2}}}}={\sqrt {\frac {m^{*}\beta }{\mu _{0}e^{*2}|\alpha |}}},}
where ψ0 is the equilibrium value of the order parameter in the absence of an electromagnetic field. The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor.
The original idea on the parameter κ belongs to Landau. The ratio κ = λ/ξ is presently known as the Ginzburg–Landau parameter. It has been proposed by Landau that Type I superconductors are those with 0 < κ < 1/√2, and Type II superconductors those with κ > 1/√2.
== Fluctuations ==
The phase transition from the normal state is of second order for Type II superconductors, taking into account fluctuations, as demonstrated by Dasgupta and Halperin, while for Type I superconductors it is of first order, as demonstrated by Halperin, Lubensky and Ma.
== Classification of superconductors ==
In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending
on the energy of the interface between the normal and superconducting states. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value Hc. Depending on the geometry of the sample, one may obtain an intermediate state consisting of a baroque pattern of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value Hc1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength Hc2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.
The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes of flux vortices.
== Geometric formulation ==
The Ginzburg–Landau functional can be formulated in the general setting of a complex vector bundle over a compact Riemannian manifold. This is the same functional as given above, transposed to the notation commonly used in Riemannian geometry. In multiple interesting cases, it can be shown to exhibit the same phenomena as the above, including Abrikosov vortices (see discussion below).
For a complex vector bundle
E
{\displaystyle E}
over a Riemannian manifold
M
{\displaystyle M}
with fiber
C
n
{\displaystyle \mathbb {C} ^{n}}
, the order parameter
ψ
{\displaystyle \psi }
is understood as a section of the vector bundle
E
{\displaystyle E}
. The Ginzburg–Landau functional is then a Lagrangian for that section:
L
(
ψ
,
A
)
=
∫
M
|
g
|
d
x
1
∧
⋯
∧
d
x
m
[
|
F
|
2
+
|
D
ψ
|
2
+
1
4
(
σ
−
|
ψ
|
2
)
2
]
{\displaystyle {\mathcal {L}}(\psi ,A)=\int _{M}{\sqrt {|g|}}dx^{1}\wedge \dotsm \wedge dx^{m}\left[\vert F\vert ^{2}+\vert D\psi \vert ^{2}+{\frac {1}{4}}\left(\sigma -\vert \psi \vert ^{2}\right)^{2}\right]}
The notation used here is as follows. The fibers
C
n
{\displaystyle \mathbb {C} ^{n}}
are assumed to be equipped with a Hermitian inner product
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
so that the square of the norm is written as
|
ψ
|
2
=
⟨
ψ
,
ψ
⟩
{\displaystyle \vert \psi \vert ^{2}=\langle \psi ,\psi \rangle }
. The phenomenological parameters
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
have been absorbed so that the potential energy term is a quartic mexican hat potential; i.e., exhibiting spontaneous symmetry breaking, with a minimum at some real value
σ
∈
R
{\displaystyle \sigma \in \mathbb {R} }
. The integral is explicitly over the volume form
∗
(
1
)
=
|
g
|
d
x
1
∧
⋯
∧
d
x
m
{\displaystyle *(1)={\sqrt {|g|}}dx^{1}\wedge \dotsm \wedge dx^{m}}
for an
m
{\displaystyle m}
-dimensional manifold
M
{\displaystyle M}
with determinant
|
g
|
{\displaystyle |g|}
of the metric tensor
g
{\displaystyle g}
.
The
D
=
d
+
A
{\displaystyle D=d+A}
is the connection one-form and
F
{\displaystyle F}
is the corresponding curvature 2-form (this is not the same as the free energy
F
{\displaystyle F}
given up top; here,
F
{\displaystyle F}
corresponds to the electromagnetic field strength tensor). The
A
{\displaystyle A}
corresponds to the vector potential, but is in general non-Abelian when
n
>
1
{\displaystyle n>1}
, and is normalized differently. In physics, one conventionally writes the connection as
d
−
i
e
A
{\displaystyle d-ieA}
for the electric charge
e
{\displaystyle e}
and vector potential
A
{\displaystyle A}
; in Riemannian geometry, it is more convenient to drop the
e
{\displaystyle e}
(and all other physical units) and take
A
=
A
μ
d
x
μ
{\displaystyle A=A_{\mu }dx^{\mu }}
to be a one-form taking values in the Lie algebra corresponding to the symmetry group of the fiber. Here, the symmetry group is SU(n), as that leaves the inner product
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
invariant; so here,
A
{\displaystyle A}
is a form taking values in the algebra
s
u
(
n
)
{\displaystyle {\mathfrak {su}}(n)}
.
The curvature
F
{\displaystyle F}
generalizes the electromagnetic field strength to the non-Abelian setting, as the curvature form of an affine connection on a vector bundle . It is conventionally written as
F
=
D
∘
D
=
d
A
+
A
∧
A
=
(
∂
A
ν
∂
x
μ
+
A
μ
A
ν
)
d
x
μ
∧
d
x
ν
=
1
2
(
∂
A
ν
∂
x
μ
−
∂
A
μ
∂
x
ν
+
[
A
μ
,
A
ν
]
)
d
x
μ
∧
d
x
ν
{\displaystyle {\begin{aligned}F=D\circ D=dA+A\wedge A=\left({\frac {\partial A_{\nu }}{\partial x^{\mu }}}+A_{\mu }A_{\nu }\right)dx^{\mu }\wedge dx^{\nu }={\frac {1}{2}}\left({\frac {\partial A_{\nu }}{\partial x^{\mu }}}-{\frac {\partial A_{\mu }}{\partial x^{\nu }}}+[A_{\mu },A_{\nu }]\right)dx^{\mu }\wedge dx^{\nu }\\\end{aligned}}}
That is, each
A
μ
{\displaystyle A_{\mu }}
is an
n
×
n
{\displaystyle n\times n}
skew-symmetric matrix. (See the article on the metric connection for additional articulation of this specific notation.) To emphasize this, note that the first term of the Ginzburg–Landau functional, involving the field-strength only, is
L
(
A
)
=
Y
M
(
A
)
=
∫
M
∗
(
1
)
|
F
|
2
{\displaystyle {\mathcal {L}}(A)=YM(A)=\int _{M}*(1)\vert F\vert ^{2}}
which is just the Yang–Mills action on a compact Riemannian manifold.
The Euler–Lagrange equations for the Ginzburg–Landau functional are the Yang–Mills equations
D
∗
D
ψ
=
1
2
(
σ
−
|
ψ
|
2
)
ψ
{\displaystyle D^{*}D\psi ={\frac {1}{2}}\left(\sigma -\vert \psi \vert ^{2}\right)\psi }
and
D
∗
F
=
−
Re
⟨
D
ψ
,
ψ
⟩
{\displaystyle D^{*}F=-\operatorname {Re} \langle D\psi ,\psi \rangle }
where
D
∗
{\displaystyle D^{*}}
is the adjoint of
D
{\displaystyle D}
, analogous to the codifferential
δ
=
d
∗
{\displaystyle \delta =d^{*}}
. Note that these are closely related to the Yang–Mills–Higgs equations.
=== Specific results ===
In string theory, it is conventional to study the Ginzburg–Landau functional for the manifold
M
{\displaystyle M}
being a Riemann surface, and taking
n
=
1
{\displaystyle n=1}
; i.e., a line bundle. The phenomenon of Abrikosov vortices persists in these general cases, including
M
=
R
2
{\displaystyle M=\mathbb {R} ^{2}}
, where one can specify any finite set of points where
ψ
{\displaystyle \psi }
vanishes, including multiplicity. The proof generalizes to arbitrary Riemann surfaces and to Kähler manifolds. In the limit of weak coupling, it can be shown that
|
ψ
|
{\displaystyle \vert \psi \vert }
converges uniformly to 1, while
D
ψ
{\displaystyle D\psi }
and
d
A
{\displaystyle dA}
converge uniformly to zero, and the curvature becomes a sum over delta-function distributions at the vortices. The sum over vortices, with multiplicity, just equals the degree of the line bundle; as a result, one may write a line bundle on a Riemann surface as a flat bundle, with N singular points and a covariantly constant section.
When the manifold is four-dimensional, possessing a spinc structure, then one may write a very similar functional, the Seiberg–Witten functional, which may be analyzed in a similar fashion, and which possesses many similar properties, including self-duality. When such systems are integrable, they are studied as Hitchin systems.
== Self-duality ==
When the manifold
M
{\displaystyle M}
is a Riemann surface
M
=
Σ
{\displaystyle M=\Sigma }
, the functional can be re-written so as to explicitly show self-duality. One achieves this by writing the exterior derivative as a sum of Dolbeault operators
d
=
∂
+
∂
¯
{\displaystyle d=\partial +{\overline {\partial }}}
. Likewise, the space
Ω
1
{\displaystyle \Omega ^{1}}
of one-forms over a Riemann surface decomposes into a space that is holomorphic, and one that is anti-holomorphic:
Ω
1
=
Ω
1
,
0
⊕
Ω
0
,
1
{\displaystyle \Omega ^{1}=\Omega ^{1,0}\oplus \Omega ^{0,1}}
, so that forms in
Ω
1
,
0
{\displaystyle \Omega ^{1,0}}
are holomorphic in
z
{\displaystyle z}
and have no dependence on
z
¯
{\displaystyle {\overline {z}}}
; and vice-versa for
Ω
0
,
1
{\displaystyle \Omega ^{0,1}}
. This allows the vector potential to be written as
A
=
A
1
,
0
+
A
0
,
1
{\displaystyle A=A^{1,0}+A^{0,1}}
and likewise
D
=
∂
A
+
∂
¯
A
{\displaystyle D=\partial _{A}+{\overline {\partial }}_{A}}
with
∂
A
=
∂
+
A
1
,
0
{\displaystyle \partial _{A}=\partial +A^{1,0}}
and
∂
¯
A
=
∂
¯
+
A
0
,
1
{\displaystyle {\overline {\partial }}_{A}={\overline {\partial }}+A^{0,1}}
.
For the case of
n
=
1
{\displaystyle n=1}
, where the fiber is
C
{\displaystyle \mathbb {C} }
so that the bundle is a line bundle, the field strength can similarly be written as
F
=
−
(
∂
A
∂
¯
A
+
∂
¯
A
∂
A
)
{\displaystyle F=-\left(\partial _{A}{\overline {\partial }}_{A}+{\overline {\partial }}_{A}\partial _{A}\right)}
Note that in the sign-convention being used here, both
A
1
,
0
,
A
0
,
1
{\displaystyle A^{1,0},A^{0,1}}
and
F
{\displaystyle F}
are purely imaginary (viz U(1) is generated by
e
i
θ
{\displaystyle e^{i\theta }}
so derivatives are purely imaginary). The functional then becomes
L
(
ψ
,
A
)
=
2
π
σ
deg
L
+
∫
Σ
i
2
d
z
∧
d
z
¯
[
2
|
∂
¯
A
ψ
|
2
+
(
∗
(
−
i
F
)
−
1
2
(
σ
−
|
ψ
|
2
)
2
]
{\displaystyle {\mathcal {L}}\left(\psi ,A\right)=2\pi \sigma \operatorname {deg} L+\int _{\Sigma }{\frac {i}{2}}dz\wedge d{\overline {z}}\left[2\vert {\overline {\partial }}_{A}\psi \vert ^{2}+\left(*(-iF)-{\frac {1}{2}}(\sigma -\vert \psi \vert ^{2}\right)^{2}\right]}
The integral is understood to be over the volume form
∗
(
1
)
=
i
2
d
z
∧
d
z
¯
{\displaystyle *(1)={\frac {i}{2}}dz\wedge d{\overline {z}}}
,
so that
Area
Σ
=
∫
Σ
∗
(
1
)
{\displaystyle \operatorname {Area} \Sigma =\int _{\Sigma }*(1)}
is the total area of the surface
Σ
{\displaystyle \Sigma }
. The
∗
{\displaystyle *}
is the Hodge star, as before. The degree
deg
L
{\displaystyle \operatorname {deg} L}
of the line bundle
L
{\displaystyle L}
over the surface
Σ
{\displaystyle \Sigma }
is
deg
L
=
c
1
(
L
)
=
1
2
π
∫
Σ
i
F
{\displaystyle \operatorname {deg} L=c_{1}(L)={\frac {1}{2\pi }}\int _{\Sigma }iF}
where
c
1
(
L
)
=
c
1
(
L
)
[
Σ
]
∈
H
2
(
Σ
)
{\displaystyle c_{1}(L)=c_{1}(L)[\Sigma ]\in H^{2}(\Sigma )}
is the first Chern class.
The Lagrangian is minimized (stationary) when
ψ
,
A
{\displaystyle \psi ,A}
solve the Ginzberg–Landau equations
∂
¯
A
ψ
=
0
∗
(
i
F
)
=
1
2
(
σ
−
|
ψ
|
2
)
{\displaystyle {\begin{aligned}{\overline {\partial }}_{A}\psi &=0\\*(iF)&={\frac {1}{2}}\left(\sigma -\vert \psi \vert ^{2}\right)\\\end{aligned}}}
Note that these are both first-order differential equations, manifestly self-dual. Integrating the second of these, one quickly finds that a non-trivial solution must obey
4
π
deg
L
≤
σ
Area
Σ
{\displaystyle 4\pi \operatorname {deg} L\leq \sigma \operatorname {Area} \Sigma }
.
Roughly speaking, this can be interpreted as an upper limit to the density of the Abrikosov vortecies. One can also show that the solutions are bounded; one must have
|
ψ
|
≤
σ
{\displaystyle |\psi |\leq \sigma }
.
== In string theory ==
In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy with a degenerate critical point is called a Landau–Ginzburg theory. The generalization to N = (2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in November 1988; in this generalization one imposes that the superpotential possess a degenerate critical point. The same month, together with Brian Greene they argued that these theories are related by a renormalization group flow to sigma models on Calabi–Yau manifolds. In his 1993 paper "Phases of N = 2 theories in two-dimensions", Edward Witten argued that Landau–Ginzburg theories and sigma models on Calabi–Yau manifolds are different phases of the same theory. A construction of such a duality was given by relating the Gromov–Witten theory of Calabi–Yau orbifolds to FJRW theory an analogous Landau–Ginzburg "FJRW" theory. Witten's sigma models were later used to describe the low energy dynamics of 4-dimensional gauge theories with monopoles as well as brane constructions.
== See also ==
== References ==
=== Papers ===
V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546
A.A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957) (English translation: Sov. Phys. JETP 5 1174 (1957)].) Abrikosov's original paper on vortex structure of Type-II superconductors derived as a solution of G–L equations for κ > 1/√2
L.P. Gor'kov, Sov. Phys. JETP 36, 1364 (1959)
A.A. Abrikosov's 2003 Nobel lecture: pdf file or video
V.L. Ginzburg's 2003 Nobel Lecture: pdf file or video | Wikipedia/Ginzburg–Landau_theory |
In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines its self-energy
Σ
{\displaystyle \Sigma }
. The self-energy represents the contribution to the particle's energy, or effective mass, due to interactions between the particle and its environment. In electrostatics, the energy required to assemble the charge distribution takes the form of self-energy by bringing in the constituent charges from infinity, where the electric force goes to zero. In a condensed matter context, self-energy is used to describe interaction induced renormalization of quasiparticle mass (dispersions) and lifetime. Self-energy is especially used to describe electron-electron interactions in Fermi liquids. Another example of self-energy is found in the context of phonon softening due to electron-phonon coupling.
== Characteristics ==
Mathematically, this energy is equal to the so-called on mass shell value of the proper self-energy operator (or proper mass operator) in the momentum-energy representation (more precisely, to
ℏ
{\displaystyle \hbar }
times this value). In this, or other representations (such as the space-time representation), the self-energy is pictorially (and economically) represented by means of Feynman diagrams, such as the one shown below. In this particular diagram, the three arrowed straight lines represent particles, or particle propagators, and the wavy line a particle-particle interaction; removing (or amputating) the left-most and the right-most straight lines in the diagram shown below (these so-called external lines correspond to prescribed values for, for instance, momentum and energy, or four-momentum), one retains a contribution to the self-energy operator (in, for instance, the momentum-energy representation). Using a small number of simple rules, each Feynman diagram can be readily expressed in its corresponding algebraic form.
In general, the on-the-mass-shell value of the self-energy operator in the momentum-energy representation is complex. In such cases, it is the real part of this self-energy that is identified with the physical self-energy (referred to above as particle's "self-energy"); the inverse of the imaginary part is a measure for the lifetime of the particle under investigation. For clarity, elementary excitations, or dressed particles (see quasi-particle), in interacting systems are distinct from stable particles in vacuum; their state functions consist of complicated superpositions of the eigenstates of the underlying many-particle system, which only momentarily, if at all, behave like those specific to isolated particles; the above-mentioned lifetime is the time over which a dressed particle behaves as if it were a single particle with well-defined momentum and energy.
The self-energy operator (often denoted by
Σ
{\displaystyle \Sigma _{}^{}}
, and less frequently by
M
{\displaystyle M_{}^{}}
) is related to the bare and dressed propagators (often denoted by
G
0
{\displaystyle G_{0}^{}}
and
G
{\displaystyle G_{}^{}}
respectively) via the Dyson equation (named after Freeman Dyson):
G
=
G
0
+
G
0
Σ
G
.
{\displaystyle G=G_{0}^{}+G_{0}\Sigma G.}
Multiplying on the left by the inverse
G
0
−
1
{\displaystyle G_{0}^{-1}}
of the operator
G
0
{\displaystyle G_{0}}
and on the right by
G
−
1
{\displaystyle G^{-1}}
yields
Σ
=
G
0
−
1
−
G
−
1
.
{\displaystyle \Sigma =G_{0}^{-1}-G^{-1}.}
The photon and gluon do not get a mass through renormalization because gauge symmetry protects them from getting a mass. This is a consequence of the Ward identity. The W-boson and the Z-boson get their masses through the Higgs mechanism; they do undergo mass renormalization through the renormalization of the electroweak theory.
Neutral particles with internal quantum numbers can mix with each other through virtual pair production. The primary example of this phenomenon is the mixing of neutral kaons. Under appropriate simplifying assumptions this can be described without quantum field theory.
== Other uses ==
In chemistry, the self-energy or Born energy of an ion is the energy associated with the field of the ion itself.
In solid state and condensed-matter physics self-energies and a myriad of related quasiparticle properties are calculated by Green's function methods and Green's function (many-body theory) of interacting low-energy excitations on the basis of electronic band structure calculations. Self-energies also find extensive application in the calculation of particle transport through open quantum systems and the embedding of sub-regions into larger systems (for example the surface of a semi-infinite crystal).
== See also ==
Quantum field theory
QED vacuum
Renormalization
Self-force
GW approximation
Wheeler–Feynman absorber theory
== References ==
A. L. Fetter, and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971); (Dover, New York, 2003)
J. W. Negele, and H. Orland, Quantum Many-Particle Systems (Westview Press, Boulder, 1998)
A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski (1963): Methods of Quantum Field Theory in Statistical Physics Englewood Cliffs: Prentice-Hall.
Alexei M. Tsvelik (2007). Quantum Field Theory in Condensed Matter Physics (2nd ed.). Cambridge University Press. ISBN 978-0-521-52980-8.
A. N. Vasil'ev The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Routledge Chapman & Hall 2004); ISBN 0-415-31002-4; ISBN 978-0-415-31002-4
John E. Inglesfield (2015). The Embedding Method for Electronic Structure. IOP Publishing. ISBN 978-0-7503-1042-0. | Wikipedia/Self-energy |
In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products of field operators. They are a key object of study in quantum field theory where they can be used to calculate various observables such as S-matrix elements, although they are not themselves observables. This is because they need not be gauge invariant, nor are they unique, with different correlation functions resulting in the same S-matrix and therefore describing the same physics. They are closely related to correlation functions between random variables, although they are nonetheless different objects, being defined in Minkowski spacetime and on quantum operators.
== Definition ==
For a scalar field theory with a single field
ϕ
(
x
)
{\displaystyle \phi (x)}
and a vacuum state
|
Ω
⟩
{\displaystyle |\Omega \rangle }
at every event x in spacetime, the n-point correlation function is the vacuum expectation value of the time-ordered products of n field operators in the Heisenberg picture
G
n
(
x
1
,
…
,
x
n
)
=
⟨
Ω
|
T
{
ϕ
(
x
1
)
…
ϕ
(
x
n
)
}
|
Ω
⟩
.
{\displaystyle G_{n}(x_{1},\dots ,x_{n})=\langle \Omega |T\{{\mathcal {\phi }}(x_{1})\dots {\mathcal {\phi }}(x_{n})\}|\Omega \rangle .}
Here
T
{
⋯
}
{\displaystyle T\{\cdots \}}
is the time-ordering operator for which orders the field operators so that earlier time field operators appear to the right of later time field operators. By transforming the fields and states into the interaction picture, this is rewritten as
G
n
(
x
1
,
…
,
x
n
)
=
⟨
0
|
T
{
ϕ
(
x
1
)
…
ϕ
(
x
n
)
e
i
S
[
ϕ
]
}
|
0
⟩
⟨
0
|
e
i
S
[
ϕ
]
|
0
⟩
,
{\displaystyle G_{n}(x_{1},\dots ,x_{n})={\frac {\langle 0|T\{\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}\}|0\rangle }{\langle 0|e^{iS[\phi ]}|0\rangle }},}
where
|
0
⟩
{\displaystyle |0\rangle }
is the ground state of the free theory and
S
[
ϕ
]
{\displaystyle S[\phi ]}
is the action. Expanding
e
i
S
[
ϕ
]
{\displaystyle e^{iS[\phi ]}}
using its Taylor series, the n-point correlation function becomes a sum of interaction picture correlation functions which can be evaluated using Wick's theorem. A diagrammatic way to represent the resulting sum is via Feynman diagrams, where each term can be evaluated using the position space Feynman rules.
The series of diagrams arising from
⟨
0
|
e
i
S
[
ϕ
]
|
0
⟩
{\displaystyle \langle 0|e^{iS[\phi ]}|0\rangle }
is the set of all vacuum bubble diagrams, which are diagrams with no external legs. Meanwhile,
⟨
0
|
ϕ
(
x
1
)
…
ϕ
(
x
n
)
e
i
S
[
ϕ
]
|
0
⟩
{\displaystyle \langle 0|\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}|0\rangle }
is given by the set of all possible diagrams with exactly n external legs. Since this also includes disconnected diagrams with vacuum bubbles, the sum factorizes into (sum over all bubble diagrams)
×
{\displaystyle \times }
(sum of all diagrams with no bubbles). The first term then cancels with the normalization factor in the denominator meaning that the n-point correlation function is the sum of all Feynman diagrams excluding vacuum bubbles
G
n
(
x
1
,
…
,
x
n
)
=
⟨
0
|
T
{
ϕ
(
x
1
)
…
ϕ
(
x
n
)
e
i
S
[
ϕ
]
}
|
0
⟩
no bubbles
.
{\displaystyle G_{n}(x_{1},\dots ,x_{n})=\langle 0|T\{\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}\}|0\rangle _{\text{no bubbles}}.}
While not including any vacuum bubbles, the sum does include disconnected diagrams, which are diagrams where at least one external leg is not connected to all other external legs through some connected path. Excluding these disconnected diagrams instead defines connected n-point correlation functions
G
n
c
(
x
1
,
…
,
x
n
)
=
⟨
0
|
T
{
ϕ
(
x
1
)
…
ϕ
(
x
n
)
e
i
S
[
ϕ
]
}
|
0
⟩
connected, no bubbles
{\displaystyle G_{n}^{c}(x_{1},\dots ,x_{n})=\langle 0|T\{\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}\}|0\rangle _{\text{connected, no bubbles}}}
It is often preferable to work directly with these as they contain all the information that the full correlation functions contain since any disconnected diagram is merely a product of connected diagrams. By excluding other sets of diagrams one can define other correlation functions such as one-particle irreducible correlation functions.
In the path integral formulation, n-point correlation functions are written as a functional average
G
n
(
x
1
,
…
,
x
n
)
=
∫
D
ϕ
ϕ
(
x
1
)
…
ϕ
(
x
n
)
e
i
S
[
ϕ
]
∫
D
ϕ
e
i
S
[
ϕ
]
.
{\displaystyle G_{n}(x_{1},\dots ,x_{n})={\frac {\int {\mathcal {D}}\phi \ \phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}}{\int {\mathcal {D}}\phi \ e^{iS[\phi ]}}}.}
They can be evaluated using the partition functional
Z
[
J
]
{\displaystyle Z[J]}
which acts as a generating functional, with
J
{\displaystyle J}
being a source-term, for the correlation functions
G
n
(
x
1
,
…
,
x
n
)
=
(
−
i
)
n
1
Z
[
J
]
δ
n
Z
[
J
]
δ
J
(
x
1
)
…
δ
J
(
x
n
)
|
J
=
0
.
{\displaystyle G_{n}(x_{1},\dots ,x_{n})=(-i)^{n}{\frac {1}{Z[J]}}\left.{\frac {\delta ^{n}Z[J]}{\delta J(x_{1})\dots \delta J(x_{n})}}\right|_{J=0}.}
Similarly, connected correlation functions can be generated using
W
[
J
]
=
−
i
ln
Z
[
J
]
{\displaystyle W[J]=-i\ln Z[J]}
as
G
n
c
(
x
1
,
…
,
x
n
)
=
(
−
i
)
n
−
1
δ
n
W
[
J
]
δ
J
(
x
1
)
…
δ
J
(
x
n
)
|
J
=
0
.
{\displaystyle G_{n}^{c}(x_{1},\dots ,x_{n})=(-i)^{n-1}\left.{\frac {\delta ^{n}W[J]}{\delta J(x_{1})\dots \delta J(x_{n})}}\right|_{J=0}.}
== Relation to the S-matrix ==
Scattering amplitudes can be calculated using correlation functions by relating them to the S-matrix through the LSZ reduction formula
⟨
f
|
S
|
i
⟩
=
[
i
∫
d
4
x
1
e
−
i
p
1
x
1
(
∂
x
1
2
+
m
2
)
]
⋯
[
i
∫
d
4
x
n
e
i
p
n
x
n
(
∂
x
n
2
+
m
2
)
]
⟨
Ω
|
T
{
ϕ
(
x
1
)
…
ϕ
(
x
n
)
}
|
Ω
⟩
.
{\displaystyle \langle f|S|i\rangle =\left[i\int d^{4}x_{1}e^{-ip_{1}x_{1}}\left(\partial _{x_{1}}^{2}+m^{2}\right)\right]\cdots \left[i\int d^{4}x_{n}e^{ip_{n}x_{n}}\left(\partial _{x_{n}}^{2}+m^{2}\right)\right]\langle \Omega |T\{\phi (x_{1})\dots \phi (x_{n})\}|\Omega \rangle .}
Here the particles in the initial state
|
i
⟩
{\displaystyle |i\rangle }
have a
−
i
{\displaystyle -i}
sign in the exponential, while the particles in the final state
|
f
⟩
{\displaystyle |f\rangle }
have a
+
i
{\displaystyle +i}
. All terms in the Feynman diagram expansion of the correlation function will have one propagator for each external leg, that is a propagators with one end at
x
i
{\displaystyle x_{i}}
and the other at some internal vertex
x
{\displaystyle x}
. The significance of this formula becomes clear after the application of the Klein–Gordon operators to these external legs using
(
∂
x
i
2
+
m
2
)
Δ
F
(
x
i
,
x
)
=
−
i
δ
4
(
x
i
−
x
)
.
{\displaystyle \left(\partial _{x_{i}}^{2}+m^{2}\right)\Delta _{F}(x_{i},x)=-i\delta ^{4}(x_{i}-x).}
This is said to amputate the diagrams by removing the external leg propagators and putting the external states on-shell. All other off-shell contributions from the correlation function vanish. After integrating the resulting delta functions, what will remain of the LSZ reduction formula is merely a Fourier transformation operation where the integration is over the internal point positions
x
{\displaystyle x}
that the external leg propagators were attached to. In this form the reduction formula shows that the S-matrix is the Fourier transform of the amputated correlation functions with on-shell external states.
It is common to directly deal with the momentum space correlation function
G
~
(
q
1
,
…
,
q
n
)
{\displaystyle {\tilde {G}}(q_{1},\dots ,q_{n})}
, defined through the Fourier transformation of the correlation function
(
2
π
)
4
δ
(
4
)
(
q
1
+
⋯
+
q
n
)
G
~
n
(
q
1
,
…
,
q
n
)
=
∫
d
4
x
1
…
d
4
x
n
(
∏
i
=
1
n
e
−
i
q
i
x
i
)
G
n
(
x
1
,
…
,
x
n
)
,
{\displaystyle (2\pi )^{4}\delta ^{(4)}(q_{1}+\cdots +q_{n}){\tilde {G}}_{n}(q_{1},\dots ,q_{n})=\int d^{4}x_{1}\dots d^{4}x_{n}\left(\prod _{i=1}^{n}e^{-iq_{i}x_{i}}\right)G_{n}(x_{1},\dots ,x_{n}),}
where by convention the momenta are directed inwards into the diagram. A useful quantity to calculate when calculating scattering amplitudes is the matrix element
M
{\displaystyle {\mathcal {M}}}
which is defined from the S-matrix via
⟨
f
|
S
−
1
|
i
⟩
=
i
(
2
π
)
4
δ
4
(
∑
i
p
i
)
M
{\displaystyle \langle f|S-1|i\rangle =i(2\pi )^{4}\delta ^{4}{{\bigg (}\sum _{i}p_{i}{\bigg )}}{\mathcal {M}}}
where
p
i
{\displaystyle p_{i}}
are the external momenta. From the LSZ reduction formula it then follows that the matrix element is equivalent to the amputated connected momentum space correlation function with properly orientated external momenta
i
M
=
G
~
n
c
(
p
1
,
…
,
−
p
n
)
amputated
.
{\displaystyle i{\mathcal {M}}={\tilde {G}}_{n}^{c}(p_{1},\dots ,-p_{n})_{\text{amputated}}.}
For non-scalar theories the reduction formula also introduces external state terms such as polarization vectors for photons or spinor states for fermions. The requirement of using the connected correlation functions arises from the cluster decomposition because scattering processes that occur at large separations do not interfere with each other so can be treated separately.
== See also ==
Effective action
Green's function (many-body theory)
Partition function (mathematics)
Source field
== Notes ==
== References ==
== Further reading ==
Altland, A.; Simons, B. (2006). Condensed Matter Field Theory. Cambridge University Press.
Peskin, M.; Schroeder, D.V. (2018) An Introduction to Quantum Field Theory. Addison-Wesley. | Wikipedia/Correlation_function_(quantum_field_theory) |
Superfluid vacuum theory (SVT), sometimes known as the BEC vacuum theory, is an approach in theoretical physics and quantum mechanics where the fundamental physical vacuum (non-removable background) is considered as a superfluid or as a Bose–Einstein condensate (BEC).
The microscopic structure of this physical vacuum is currently unknown and is a subject of intensive studies in SVT. An ultimate goal of this research is to develop scientific models that unify quantum mechanics (which describes three of the four known fundamental interactions) with gravity, making SVT a derivative of quantum gravity and describes all known interactions in the Universe, at both microscopic and astronomic scales, as different manifestations of the same entity, superfluid vacuum.
== History ==
The concept of a luminiferous aether as a medium sustaining electromagnetic waves was discarded after the advent of the special theory of relativity, as the presence of the concept alongside special relativity results in several contradictions; in particular, aether having a definite velocity at each spacetime point will exhibit a preferred direction. This conflicts with the relativistic requirement that all directions within a light cone are equivalent.
However, as early as in 1951 P.A.M. Dirac published two papers where he pointed out that we should take into account quantum fluctuations in the flow of the aether.
His arguments involve the application of the uncertainty principle to the velocity of aether at any spacetime point, implying that the velocity will not be a well-defined quantity. In fact, it will be distributed over various possible values. At best, one could represent the aether by a wave function representing the perfect vacuum state for which all aether velocities are equally probable.
Inspired by Dirac's ideas, K. P. Sinha, C. Sivaram and E. C. G. Sudarshan published in 1975 a series of papers that suggested a new model for the aether according to which it is a superfluid state of fermion and anti-fermion pairs, describable by a macroscopic wave function.
They noted that particle-like small fluctuations of superfluid background obey the Lorentz symmetry, even if the superfluid itself is non-relativistic.
Nevertheless, they decided to treat the superfluid as the relativistic matter – by putting it into the stress–energy tensor of the Einstein field equations.
This did not allow them to describe the relativistic gravity as a small fluctuation of the superfluid vacuum, as subsequent authors have noted .
Since then, several theories have been proposed within the SVT framework. They differ in how the structure and properties of the background superfluid must look.
In absence of observational data which would rule out some of them, these theories are being pursued independently.
== Relation to other concepts and theories ==
=== Lorentz and Galilean symmetries ===
According to the approach, the background superfluid is assumed to be essentially non-relativistic whereas the Lorentz symmetry is not an exact symmetry of Nature but rather the approximate description valid only for small fluctuations.
An observer who resides inside such vacuum and is capable of creating or measuring the small fluctuations would observe them as relativistic objects – unless their energy and momentum are sufficiently high to make the Lorentz-breaking corrections detectable.
If the energies and momenta are below the excitation threshold then the superfluid background behaves like the ideal fluid, therefore, the Michelson–Morley-type experiments would observe no drag force from such aether.
Further, in the theory of relativity the Galilean symmetry (pertinent to our macroscopic non-relativistic world) arises as the approximate one – when particles' velocities are small compared to speed of light in vacuum.
In SVT one does not need to go through Lorentz symmetry to obtain the Galilean one – the dispersion relations of most non-relativistic superfluids are known to obey the non-relativistic behavior at large momenta.
To summarize, the fluctuations of vacuum superfluid behave like relativistic objects at "small" momenta (a.k.a. the "phononic limit")
E
2
∝
|
p
→
|
2
{\displaystyle E^{2}\propto |{\vec {p}}|^{2}}
and like non-relativistic ones
E
∝
|
p
→
|
2
{\displaystyle E\propto |{\vec {p}}|^{2}}
at large momenta.
The yet unknown nontrivial physics is believed to be located somewhere between these two regimes.
=== Relativistic quantum field theory ===
In the relativistic quantum field theory the physical vacuum is also assumed to be some sort of non-trivial medium to which one can associate certain energy.
This is because the concept of absolutely empty space (or "mathematical vacuum") contradicts the postulates of quantum mechanics.
According to QFT, even in absence of real particles the background is always filled by pairs of creating and annihilating virtual particles.
However, a direct attempt to describe such medium leads to the so-called ultraviolet divergences.
In some QFT models, such as quantum electrodynamics, these problems can be "solved" using the renormalization technique, namely, replacing the diverging physical values by their experimentally measured values.
In other theories, such as the quantum general relativity, this trick does not work, and reliable perturbation theory cannot be constructed.
According to SVT, this is because in the high-energy ("ultraviolet") regime the Lorentz symmetry starts failing so dependent theories cannot be regarded valid for all scales of energies and momenta.
Correspondingly, while the Lorentz-symmetric quantum field models are obviously a good approximation below the vacuum-energy threshold, in its close vicinity the relativistic description becomes more and more "effective" and less and less natural since one will need to adjust the expressions for the covariant field-theoretical actions by hand.
=== Curved spacetime ===
According to general relativity, gravitational interaction is described in terms of spacetime curvature using the mathematical formalism of differential geometry.
This was supported by numerous experiments and observations in the regime of low energies. However, the attempts to quantize general relativity led to various severe problems, therefore, the microscopic structure of gravity is still ill-defined.
There may be a fundamental reason for this—the degrees of freedom of general relativity are based on what may be only approximate and effective. The question of whether general relativity is an effective theory has been raised for a long time.
According to SVT, the curved spacetime arises as the small-amplitude collective excitation mode of the non-relativistic background condensate.
The mathematical description of this is similar to fluid-gravity analogy which is being used also in the analog gravity models.
Thus, relativistic gravity is essentially a long-wavelength theory of the collective modes whose amplitude is small compared to the background one.
Outside this requirement the curved-space description of gravity in terms of the Riemannian geometry becomes incomplete or ill-defined.
=== Cosmological constant ===
The notion of the cosmological constant makes sense in a relativistic theory only, therefore, within the SVT framework this constant can refer at most to the energy of small fluctuations of the vacuum above a background value, but not to the energy of the vacuum itself. Thus, in SVT this constant does not have any fundamental physical meaning, and related problems such as the vacuum catastrophe, simply do not occur in the first place.
=== Gravitational waves and gravitons ===
According to general relativity, the conventional gravitational wave is:
the small fluctuation of curved spacetime which
has been separated from its source and propagates independently.
Superfluid vacuum theory brings into question the possibility that a relativistic object possessing both of these properties exists in nature.
Indeed, according to the approach, the curved spacetime itself is the small collective excitation of the superfluid background, therefore, the property (1) means that the graviton would be in fact the "small fluctuation of the small fluctuation", which does not look like a physically robust concept (as if somebody tried to introduce small fluctuations inside a phonon, for instance).
As a result, it may be not just a coincidence that in general relativity the gravitational field alone has no well-defined stress–energy tensor, only the pseudotensor one.
Therefore, the property (2) cannot be completely justified in a theory with exact Lorentz symmetry which the general relativity is.
Though, SVT does not a priori forbid an existence of the non-localized wave-like excitations of the superfluid background which might be responsible for the astrophysical phenomena which are currently being attributed to gravitational waves, such as the Hulse–Taylor binary. However, such excitations cannot be correctly described within the framework of a fully relativistic theory.
=== Mass generation and Higgs boson ===
The Higgs boson is the spin-0 particle that has been introduced in electroweak theory to give mass to the weak bosons. The origin of mass of the Higgs boson itself is not explained by electroweak theory. Instead, this mass is introduced as a free parameter by means of the Higgs potential, which thus makes it yet another free parameter of the Standard Model. Within the framework of the Standard Model (or its extensions) the theoretical estimates of this parameter's value are possible only indirectly and results differ from each other significantly. Thus, the usage of the Higgs boson (or any other elementary particle with predefined mass) alone is not the most fundamental solution of the mass generation problem but only its reformulation ad infinitum.
Another known issue of the Glashow–Weinberg–Salam model is the wrong sign of mass term in the (unbroken) Higgs sector for
energies above the symmetry-breaking scale.
While SVT does not explicitly forbid the existence of the electroweak Higgs particle, it has its own idea of the fundamental mass generation mechanism – elementary particles acquire mass due to the interaction with the vacuum condensate, similarly to the gap generation mechanism in superconductors or superfluids.
Although this idea is not entirely new, one could recall the relativistic Coleman-Weinberg approach,
SVT gives the meaning to the symmetry-breaking relativistic scalar field as describing small fluctuations of background superfluid which can be interpreted as an elementary particle only under certain conditions. In general, one allows two scenarios to happen:
Higgs boson exists: in this case SVT provides the mass generation mechanism which underlies the electroweak one and explains the origin of mass of the Higgs boson itself;
Higgs boson does not exist: then the weak bosons acquire mass by directly interacting with the vacuum condensate.
Thus, the Higgs boson, even if it exists, would be a by-product of the fundamental mass generation phenomenon rather than its cause.
Also, some versions of SVT favor a wave equation based on the logarithmic potential rather than on the quartic one. The former potential has not only the Mexican-hat shape, necessary for the spontaneous symmetry breaking, but also some other features which make it more suitable for the vacuum's description.
== Logarithmic BEC vacuum theory ==
In this model the physical vacuum is conjectured to be strongly-correlated quantum Bose liquid whose ground-state wavefunction is described by the logarithmic Schrödinger equation. It was shown that the relativistic gravitational interaction arises as the small-amplitude collective excitation mode whereas relativistic elementary particles can be described by the particle-like modes in the limit of low energies and momenta.
The essential difference of this theory from others is that in the logarithmic superfluid the maximal velocity of fluctuations is constant in the leading (classical) order.
This allows to fully recover the relativity postulates in the "phononic" (linearized) limit.
The proposed theory has many observational consequences.
They are based on the fact that at high energies and momenta the behavior of the particle-like modes eventually becomes distinct from the relativistic one – they can reach the speed of light limit at finite energy.
Among other predicted effects is the superluminal propagation and vacuum Cherenkov radiation.
Theory advocates the mass generation mechanism which is supposed to replace or alter the electroweak Higgs one.
It was shown that masses of elementary particles can arise as a result of interaction with the superfluid vacuum, similarly to the gap generation mechanism in superconductors. For instance, the photon propagating in the average interstellar vacuum acquires a tiny mass which is estimated to be about 10−35 electronvolt.
One can also derive an effective potential for the Higgs sector which is different from the one used in the Glashow–Weinberg–Salam model, yet it yields the mass generation and it is free of the imaginary-mass problem appearing in the conventional Higgs potential.
== See also ==
Analog gravity
Acoustic metric
Casimir vacuum
Dilatonic quantum gravity
Hawking radiation
Induced gravity
Logarithmic Schrödinger equation
Hořava–Lifshitz gravity
Sonic black hole
Vacuum energy
Hydrodynamic quantum analogs
Fluid solution
Vacuum solution (general relativity)
== Notes ==
== References == | Wikipedia/Superfluid_vacuum_theory |
N = 4 supersymmetric Yang–Mills (SYM) theory is a relativistic conformally invariant Lagrangian gauge theory describing the interactions of fermions via gauge field exchanges. In D=4 spacetime dimensions, N=4 is the maximal number of supersymmetries or supersymmetry charges.
SYM theory is a toy theory based on Yang–Mills theory; it does not model the real world, but it is useful because it can act as a proving ground for approaches for attacking problems in more complex theories. It describes a universe containing boson fields and fermion fields which are related by four supersymmetries (this means that transforming bosonic and fermionic fields in a certain way leaves the theory invariant). It is one of the simplest (in the sense that it has no free parameters except for the gauge group) and one of the few ultraviolet finite quantum field theories in 4 dimensions. It can be thought of as the most symmetric field theory that does not involve gravity.
Like all supersymmetric field theories, SYM theory may equivalently be formulated as a superfield theory on an extended superspace in which the spacetime variables are augmented by a number of Grassmann variables which, for the case N=4, consist of 4 Dirac spinors, making a total of 16 independent anticommuting generators for the extended ring of superfunctions. The field equations are equivalent to the geometric condition that the supercurvature 2-form vanish identically on all super null lines. This is also known as the super-ambitwistor correspondence.
A similar super-ambitwistor characterization holds for D=10, N=1 dimensional super Yang–Mills theory, and the lower dimensional cases D=6, N=2 and D=4, N=4 may be derived from this via dimensional reduction.
== Meaning of N and numbers of fields ==
In N supersymmetric Yang–Mills theory, N denotes the number of independent supersymmetric operations that transform the spin-1 gauge field into spin-1/2 fermionic fields. In an analogy with symmetries under rotations, N would be the number of independent rotations, N = 1 in a plane, N = 2 in 3D space, etc... That is, in a N = 4 SYM theory, the gauge boson can be "rotated" into N = 4 different supersymmetric fermion partners. In turns, each fermion can be rotated into four different bosons: one corresponds to the rotation back to the spin-1 gauge field, and the three others are spin-0 boson fields. Because in 3D space one may use different rotations to reach a same point (or here the same spin-0 boson), each spin-0 boson is superpartners of two different spin-1/2 fermions, not just one. So in total, one has only 6 spin-0 bosons, not 16.
Therefore, N = 4 SYM has 1 + 4 + 6 = 11 fields, namely: one vector field (the spin-1 gauge boson), four spinor fields (the spin-1/2 fermions) and six scalar fields (the spin-0 bosons). N = 4 is the maximum number of independent supersymmetries: starting from a spin-1 field and using more supersymmetries, e.g., N = 5, only rotates between the 11 fields. To have N > 4 independent supersymmetries, one needs to start from a gauge field of spin higher than 1, e.g., a spin-2 tensor field such as that of the graviton. This is the N = 8 supergravity theory.
== Lagrangian ==
The Lagrangian for the theory is
L
=
tr
{
−
1
2
g
2
F
μ
ν
F
μ
ν
+
θ
I
8
π
2
F
μ
ν
F
¯
μ
ν
−
i
λ
¯
a
σ
¯
μ
D
μ
λ
a
−
D
μ
X
i
D
μ
X
i
+
g
C
i
a
b
λ
a
[
X
i
,
λ
b
]
+
g
C
¯
i
a
b
λ
¯
a
[
X
i
,
λ
¯
b
]
+
g
2
2
[
X
i
,
X
j
]
2
}
,
{\displaystyle L=\operatorname {tr} \left\{-{\frac {1}{2g^{2}}}F_{\mu \nu }F^{\mu \nu }+{\frac {\theta _{I}}{8\pi ^{2}}}F_{\mu \nu }{\bar {F}}^{\mu \nu }-i{\overline {\lambda }}^{a}{\overline {\sigma }}^{\mu }D_{\mu }\lambda _{a}-D_{\mu }X^{i}D^{\mu }X^{i}+gC_{i}^{ab}\lambda _{a}[X^{i},\lambda _{b}]+g{\overline {C}}_{iab}{\overline {\lambda }}^{a}[X^{i},{\overline {\lambda }}^{b}]+{\frac {g^{2}}{2}}[X^{i},X^{j}]^{2}\right\},}
where
g
{\displaystyle g}
and
θ
I
{\displaystyle \theta _{I}}
are coupling constants (specifically
g
{\displaystyle g}
is the gauge coupling and
θ
I
{\displaystyle \theta _{I}}
is the instanton angle),
the field strength is
F
μ
ν
k
=
∂
μ
A
ν
k
−
∂
ν
A
μ
k
+
f
k
l
m
A
μ
l
A
ν
m
{\displaystyle F_{\mu \nu }^{k}=\partial _{\mu }A_{\nu }^{k}-\partial _{\nu }A_{\mu }^{k}+f^{klm}A_{\mu }^{l}A_{\nu }^{m}}
with
A
ν
k
{\displaystyle A_{\nu }^{k}}
the gauge field
and indices i,j = 1, ..., 6 as well as a, b = 1, ..., 4, and
f
{\displaystyle f}
represents the structure constants of the particular gauge group. The
λ
a
{\displaystyle \lambda ^{a}}
are left Weyl fermions,
σ
μ
{\displaystyle \sigma ^{\mu }}
are the Pauli matrices,
D
μ
{\displaystyle D_{\mu }}
is the gauge covariant derivative,
X
i
{\displaystyle X^{i}}
are real scalars, and
C
i
a
b
{\displaystyle C_{i}^{ab}}
represents the structure constants of the R-symmetry group SU(4), which rotates the four supersymmetries. As a consequence of the nonrenormalization theorems, this supersymmetric field theory is in fact a superconformal field theory.
== Ten-dimensional Lagrangian ==
The above Lagrangian can be found by beginning with the simpler ten-dimensional Lagrangian
L
=
tr
{
1
g
2
F
I
J
F
I
J
−
i
λ
¯
Γ
I
D
I
λ
}
,
{\displaystyle L=\operatorname {tr} \left\{{\frac {1}{g^{2}}}F_{IJ}F^{IJ}-i{\bar {\lambda }}\Gamma ^{I}D_{I}\lambda \right\},}
where I and J are now run from 0 through 9 and
Γ
I
{\displaystyle \Gamma ^{I}}
are the 32 by 32 gamma matrices
(
32
=
2
10
/
2
)
{\displaystyle (32=2^{10/2})}
, followed by adding the term with
θ
I
{\displaystyle \theta _{I}}
which is a topological term.
The components
A
i
{\displaystyle A_{i}}
of the gauge field for i = 4 to 9 become scalars upon eliminating the extra dimensions. This also gives an interpretation of the SO(6) R-symmetry as rotations in the extra compact dimensions.
By compactification on a T6, all the supercharges are preserved, giving N = 4 in the 4-dimensional theory.
A Type IIB string theory interpretation of the theory is the worldvolume theory of a stack of D3-branes.
== S-duality ==
The coupling constants
θ
I
{\displaystyle \theta _{I}}
and
g
{\displaystyle g}
naturally pair together into a single coupling constant
τ
:=
θ
I
2
π
+
4
π
i
g
2
.
{\displaystyle \tau :={\frac {\theta _{I}}{2\pi }}+{\frac {4\pi i}{g^{2}}}.}
The theory has symmetries that shift
τ
{\displaystyle \tau }
by integers. The S-duality conjecture says there is also a symmetry which sends
τ
↦
−
1
n
G
τ
{\displaystyle \tau \mapsto {\frac {-1}{n_{G}\tau }}}
as well as switching the group
G
{\displaystyle G}
to its Langlands dual group.
== AdS/CFT correspondence ==
This theory is also important in the context of the holographic principle. There is a duality between Type IIB string theory on AdS5 × S5 space (a product of 5-dimensional AdS space with a 5-dimensional sphere) and N = 4 super Yang–Mills on the 4-dimensional boundary of AdS5. However, this particular realization of the AdS/CFT correspondence is not a realistic model of gravity, since gravity in our universe is 4-dimensional. Despite this, the AdS/CFT correspondence is the most successful realization of the holographic principle, a speculative idea about quantum gravity originally proposed by Gerard 't Hooft, who was expanding on work on black hole thermodynamics, and was improved and promoted in the context of string theory by Leonard Susskind.
== Integrability ==
There is evidence that N = 4 supersymmetric Yang–Mills theory has an integrable structure in the planar large N limit (see below for what "planar" means in the present context). As the number of colors (also denoted N) goes to infinity, the amplitudes scale like
N
2
−
2
g
{\displaystyle N^{2-2g}}
, so that only the genus 0 (planar graph) contribution survives. Planar Feynman diagrams are graphs in which no propagator cross over another one, in contrast to non-planar Feynman graphs where one or more propagator goes over another one. A non-planar graph has a smaller number of possible gauge loops compared to a similar planar graph. Non-planar graphs are thus suppressed by factors
1
/
N
2
−
2
g
{\displaystyle 1/N^{2-2g}}
compared to planar ones which therefore dominate in the large N limit. Consequently, a planar Yang–Mills theory denotes a theory in the large N limit, with N usually the number of colors. Likewise, a planar limit is a limit in which scattering amplitudes are dominated by Feynman diagrams which can be given the structure of planar graphs. In the large N limit, the coupling
g
{\displaystyle g}
vanishes and a perturbative formalism is therefore well-suited for large N calculations. Therefore, planar graphs are associated to the domain where perturbative calculations converge well.
Beisert et al. give a review article demonstrating how in this situation local operators can be expressed via certain states in spin chains (in particular the Heisenberg spin chain), but based on a larger Lie superalgebra rather than
s
u
(
2
)
{\displaystyle {\mathfrak {su}}(2)}
for ordinary spin. These spin chains are integrable in the sense that they can be solved by the Bethe ansatz method. They also construct an action of the associated Yangian on scattering amplitudes.
Nima Arkani-Hamed et al. have also researched this subject. Using twistor theory, they find a description (the amplituhedron formalism) in terms of the positive Grassmannian.
== Relation to 11-dimensional M-theory ==
N = 4 super Yang–Mills can be derived from a simpler 10-dimensional theory, and yet supergravity and M-theory exist in 11 dimensions. The connection is that if the gauge group U(N) of SYM becomes infinite as
N
→
∞
{\displaystyle N\rightarrow \infty }
it becomes equivalent to an 11-dimensional theory known as matrix theory.
== See also ==
4D N = 1 global supersymmetry
6D (2,0) superconformal field theory
Extended supersymmetry
N = 1 supersymmetric Yang–Mills theory
N = 8 supergravity
Seiberg–Witten theory
== References ==
=== Citations ===
=== Sources === | Wikipedia/N_=_4_supersymmetric_Yang–Mills_theory |
In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edward Witten. A WZW model is associated to a Lie group (or supergroup), and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra (or Lie superalgebra). By extension, the name WZW model is sometimes used for any conformal field theory whose symmetry algebra is an affine Lie algebra.
== Action ==
=== Definition ===
For
Σ
{\displaystyle \Sigma }
a Riemann surface,
G
{\displaystyle G}
a Lie group, and
k
{\displaystyle k}
a (generally complex) number, let us define the
G
{\displaystyle G}
-WZW model on
Σ
{\displaystyle \Sigma }
at the level
k
{\displaystyle k}
. The model is a nonlinear sigma model whose action is a functional of a field
γ
:
Σ
→
G
{\displaystyle \gamma :\Sigma \to G}
:
S
k
(
γ
)
=
−
k
8
π
∫
Σ
d
2
x
K
(
γ
−
1
∂
μ
γ
,
γ
−
1
∂
μ
γ
)
+
2
π
k
S
W
Z
(
γ
)
.
{\displaystyle S_{k}(\gamma )=-{\frac {k}{8\pi }}\int _{\Sigma }d^{2}x\,{\mathcal {K}}\left(\gamma ^{-1}\partial ^{\mu }\gamma ,\gamma ^{-1}\partial _{\mu }\gamma \right)+2\pi kS^{\mathrm {W} Z}(\gamma ).}
Here,
Σ
{\displaystyle \Sigma }
is equipped with a flat Euclidean metric,
∂
μ
{\displaystyle \partial _{\mu }}
is the partial derivative, and
K
{\displaystyle {\mathcal {K}}}
is the Killing form on the Lie algebra of
G
{\displaystyle G}
. The Wess–Zumino term of the action is
S
W
Z
(
γ
)
=
−
1
48
π
2
∫
B
3
d
3
y
ϵ
i
j
k
K
(
γ
−
1
∂
i
γ
,
[
γ
−
1
∂
j
γ
,
γ
−
1
∂
k
γ
]
)
.
{\displaystyle S^{\mathrm {W} Z}(\gamma )=-{\frac {1}{48\pi ^{2}}}\int _{\mathbf {B} ^{3}}d^{3}y\,\epsilon ^{ijk}{\mathcal {K}}\left(\gamma ^{-1}\partial _{i}\gamma ,\left[\gamma ^{-1}\partial _{j}\gamma ,\gamma ^{-1}\partial _{k}\gamma \right]\right).}
Here
ϵ
i
j
k
{\displaystyle \epsilon ^{ijk}}
is the completely anti-symmetric tensor, and
[
.
,
.
]
{\displaystyle [.,.]}
is the Lie bracket.
The Wess–Zumino term is an integral over a three-dimensional manifold
B
3
{\displaystyle \mathbf {B} ^{3}}
whose boundary is
∂
B
3
=
Σ
{\displaystyle \partial \mathbf {B} ^{3}=\Sigma }
.
=== Topological properties of the Wess–Zumino term ===
For the Wess–Zumino term to make sense, we need the field
γ
{\displaystyle \gamma }
to have an extension to
B
3
{\displaystyle \mathbf {B} ^{3}}
. This requires the homotopy group
π
2
(
G
)
{\displaystyle \pi _{2}(G)}
to be trivial, which is the case in particular for any compact Lie group
G
{\displaystyle G}
.
The extension of a given
γ
:
Σ
→
G
{\displaystyle \gamma :\Sigma \to G}
to
B
3
{\displaystyle \mathbf {B} ^{3}}
is in general not unique.
For the WZW model to be well-defined,
e
i
S
k
(
γ
)
{\displaystyle e^{iS_{k}(\gamma )}}
should not depend on the choice of the extension.
The Wess–Zumino term is invariant under small deformations of
γ
{\displaystyle \gamma }
, and only depends on its homotopy class.
Possible homotopy classes are controlled by the homotopy group
π
3
(
G
)
{\displaystyle \pi _{3}(G)}
.
For any compact, connected simple Lie group
G
{\displaystyle G}
, we have
π
3
(
G
)
=
Z
{\displaystyle \pi _{3}(G)=\mathbb {Z} }
, and different extensions of
γ
{\displaystyle \gamma }
lead to values of
S
W
Z
(
γ
)
{\displaystyle S^{\mathrm {W} Z}(\gamma )}
that differ by integers. Therefore, they lead to the same value of
e
i
S
k
(
γ
)
{\displaystyle e^{iS_{k}(\gamma )}}
provided the level obeys
k
∈
Z
.
{\displaystyle k\in \mathbb {Z} .}
Integer values of the level also play an important role in the representation theory of the model's symmetry algebra, which is an affine Lie algebra. If the level is a positive integer, the affine Lie algebra has unitary highest weight representations with highest weights that are dominant integral. Such representations decompose into finite-dimensional subrepresentations with respect to the subalgebras spanned by each simple root, the corresponding negative root and their commutator, which is a Cartan generator.
In the case of the noncompact simple Lie group
S
L
(
2
,
R
)
{\displaystyle \mathrm {SL} (2,\mathbb {R} )}
,
the homotopy group
π
3
(
S
L
(
2
,
R
)
)
{\displaystyle \pi _{3}(\mathrm {SL} (2,\mathbb {R} ))}
is trivial, and the level is not constrained to be an integer.
=== Geometrical interpretation of the Wess–Zumino term ===
If ea are the basis vectors for the Lie algebra, then
K
(
e
a
,
[
e
b
,
e
c
]
)
{\displaystyle {\mathcal {K}}(e_{a},[e_{b},e_{c}])}
are the structure constants of the Lie algebra. The structure constants are completely anti-symmetric, and thus they define a 3-form on the group manifold of G. Thus, the integrand above is just the pullback of the harmonic 3-form to the ball
B
3
.
{\displaystyle \mathbf {B} ^{3}.}
Denoting the harmonic 3-form by c and the pullback by
γ
∗
,
{\displaystyle \gamma ^{*},}
one then has
S
W
Z
(
γ
)
=
∫
B
3
γ
∗
c
.
{\displaystyle S^{\mathrm {W} Z}(\gamma )=\int _{\mathbf {B} ^{3}}\gamma ^{*}c.}
This form leads directly to a topological analysis of the WZ term.
Geometrically, this term describes the torsion of the respective manifold. The presence of this torsion compels teleparallelism of the manifold, and thus trivialization of the torsionful curvature tensor; and hence arrest of the renormalization flow, an infrared fixed point of the renormalization group, a phenomenon termed geometrostasis.
== Symmetry algebra ==
=== Generalised group symmetry ===
The Wess–Zumino–Witten model is not only symmetric under global transformations by a group element in
G
{\displaystyle G}
, but also has a much richer symmetry. This symmetry is often called the
G
(
z
)
×
G
(
z
¯
)
{\displaystyle G(z)\times G({\bar {z}})}
symmetry. Namely, given any holomorphic
G
{\displaystyle G}
-valued function
Ω
(
z
)
{\displaystyle \Omega (z)}
, and any other (completely independent of
Ω
(
z
)
{\displaystyle \Omega (z)}
) antiholomorphic
G
{\displaystyle G}
-valued function
Ω
¯
(
z
¯
)
{\displaystyle {\bar {\Omega }}({\bar {z}})}
, where we have identified
z
=
x
+
i
y
{\displaystyle z=x+iy}
and
z
¯
=
x
−
i
y
{\displaystyle {\bar {z}}=x-iy}
in terms of the Euclidean space coordinates
x
,
y
{\displaystyle x,y}
, the following symmetry holds:
S
k
(
γ
)
=
S
k
(
Ω
γ
Ω
¯
−
1
)
{\displaystyle S_{k}(\gamma )=S_{k}(\Omega \gamma {\bar {\Omega }}^{-1})}
One way to prove the existence of this symmetry is through repeated application of the Polyakov–Wiegmann identity regarding products of
G
{\displaystyle G}
-valued fields:
S
k
(
α
β
−
1
)
=
S
k
(
α
)
+
S
k
(
β
−
1
)
+
k
16
π
2
∫
d
2
x
Tr
(
α
−
1
∂
z
¯
α
β
−
1
∂
z
β
)
{\displaystyle S_{k}(\alpha \beta ^{-1})=S_{k}(\alpha )+S_{k}(\beta ^{-1})+{\frac {k}{16\pi ^{2}}}\int d^{2}x{\textrm {Tr}}(\alpha ^{-1}\partial _{\bar {z}}\alpha \beta ^{-1}\partial _{z}\beta )}
The holomorphic and anti-holomorphic currents
J
(
z
)
=
−
1
2
k
(
∂
z
γ
)
γ
−
1
{\displaystyle J(z)=-{\frac {1}{2}}k(\partial _{z}\gamma )\gamma ^{-1}}
and
J
¯
(
z
¯
)
=
−
1
2
k
γ
−
1
∂
z
¯
γ
{\displaystyle {\bar {J}}({\bar {z}})=-{\frac {1}{2}}k\gamma ^{-1}\partial _{\bar {z}}\gamma }
are the conserved currents associated with this symmetry. The singular behaviour of the products of these currents with other quantum fields determine how those fields transform under infinitesimal actions of the
G
(
z
)
×
G
(
z
¯
)
{\displaystyle G(z)\times G({\bar {z}})}
group.
=== Affine Lie algebra ===
Let
z
{\displaystyle z}
be a local complex coordinate on
Σ
{\displaystyle \Sigma }
,
{
t
a
}
{\displaystyle \{t^{a}\}}
an orthonormal basis (with respect to the Killing form) of the Lie algebra of
G
{\displaystyle G}
, and
J
a
(
z
)
{\displaystyle J^{a}(z)}
the quantization of the field
K
(
t
a
,
∂
z
g
g
−
1
)
{\displaystyle {\mathcal {K}}(t^{a},\partial _{z}gg^{-1})}
. We have the following operator product expansion:
J
a
(
z
)
J
b
(
w
)
=
k
δ
a
b
(
z
−
w
)
2
+
i
f
c
a
b
J
c
(
w
)
z
−
w
+
O
(
1
)
,
{\displaystyle J^{a}(z)J^{b}(w)={\frac {k\delta ^{ab}}{(z-w)^{2}}}+{\frac {if_{c}^{ab}J^{c}(w)}{z-w}}+{\mathcal {O}}(1),}
where
f
c
a
b
{\displaystyle f_{c}^{ab}}
are the coefficients such that
[
t
a
,
t
b
]
=
f
c
a
b
t
c
{\displaystyle [t^{a},t^{b}]=f_{c}^{ab}t^{c}}
.
Equivalently, if
J
a
(
z
)
{\displaystyle J^{a}(z)}
is expanded in modes
J
a
(
z
)
=
∑
n
∈
Z
J
n
a
z
−
n
−
1
,
{\displaystyle J^{a}(z)=\sum _{n\in \mathbb {Z} }J_{n}^{a}z^{-n-1},}
then the current algebra generated by
{
J
n
a
}
{\displaystyle \{J_{n}^{a}\}}
is the affine Lie algebra associated to the Lie algebra of
G
{\displaystyle G}
, with a level that coincides with the level
k
{\displaystyle k}
of the WZW model.
If
g
=
L
i
e
(
G
)
{\displaystyle {\mathfrak {g}}=\mathrm {Lie} (G)}
, the notation for the affine Lie algebra is
g
^
k
{\displaystyle {\hat {\mathfrak {g}}}_{k}}
.
The commutation relations of the affine Lie algebra are
[
J
n
a
,
J
m
b
]
=
f
c
a
b
J
m
+
n
c
+
k
n
δ
a
b
δ
n
+
m
,
0
.
{\displaystyle [J_{n}^{a},J_{m}^{b}]=f_{c}^{ab}J_{m+n}^{c}+kn\delta ^{ab}\delta _{n+m,0}.}
This affine Lie algebra is the chiral symmetry algebra associated to the left-moving currents
K
(
t
a
,
∂
z
g
g
−
1
)
{\displaystyle {\mathcal {K}}(t^{a},\partial _{z}gg^{-1})}
. A second copy of the same affine Lie algebra is associated to the right-moving currents
K
(
t
a
,
g
−
1
∂
z
¯
g
)
{\displaystyle {\mathcal {K}}(t^{a},g^{-1}\partial _{\bar {z}}g)}
. The generators
J
¯
a
(
z
)
{\displaystyle {\bar {J}}^{a}(z)}
of that second copy are antiholomorphic. The full symmetry algebra of the WZW model is the product of the two copies of the affine Lie algebra.
=== Sugawara construction ===
The Sugawara construction is an embedding of the Virasoro algebra into the universal enveloping algebra of the affine Lie algebra. The existence of the embedding shows that WZW models are conformal field theories. Moreover, it leads to Knizhnik–Zamolodchikov equations for correlation functions.
The Sugawara construction is most concisely written at the level of the currents:
J
a
(
z
)
{\displaystyle J^{a}(z)}
for the affine Lie algebra, and the energy-momentum tensor
T
(
z
)
{\displaystyle T(z)}
for the Virasoro algebra:
T
(
z
)
=
1
2
(
k
+
h
∨
)
∑
a
:
J
a
J
a
:
(
z
)
,
{\displaystyle T(z)={\frac {1}{2(k+h^{\vee })}}\sum _{a}:J^{a}J^{a}:(z),}
where the
:
{\displaystyle :}
denotes normal ordering, and
h
∨
{\displaystyle h^{\vee }}
is the dual Coxeter number. By using the OPE of the currents and a version of Wick's theorem one may deduce that the OPE of
T
(
z
)
{\displaystyle T(z)}
with itself is given by
T
(
y
)
T
(
z
)
=
c
2
(
y
−
z
)
4
+
2
T
(
z
)
(
y
−
z
)
2
+
∂
T
(
z
)
y
−
z
+
O
(
1
)
,
{\displaystyle T(y)T(z)={\frac {\frac {c}{2}}{(y-z)^{4}}}+{\frac {2T(z)}{(y-z)^{2}}}+{\frac {\partial T(z)}{y-z}}+{\mathcal {O}}(1),}
which is equivalent to the Virasoro algebra's commutation relations. The central charge of the Virasoro algebra is given in terms of the level
k
{\displaystyle k}
of the affine Lie algebra by
c
=
k
d
i
m
(
g
)
k
+
h
∨
.
{\displaystyle c={\frac {k\mathrm {dim} ({\mathfrak {g}})}{k+h^{\vee }}}.}
At the level of the generators of the affine Lie algebra, the Sugawara construction reads
L
n
≠
0
=
1
2
(
k
+
h
∨
)
∑
a
∑
m
∈
Z
J
n
−
m
a
J
m
a
,
{\displaystyle L_{n\neq 0}={\frac {1}{2(k+h^{\vee })}}\sum _{a}\sum _{m\in \mathbb {Z} }J_{n-m}^{a}J_{m}^{a},}
L
0
=
1
2
(
k
+
h
∨
)
(
2
∑
a
∑
m
=
1
∞
J
−
m
a
J
m
a
+
J
a
0
J
a
0
)
.
{\displaystyle L_{0}={\frac {1}{2(k+h^{\vee })}}\left(2\sum _{a}\sum _{m=1}^{\infty }J_{-m}^{a}J_{m}^{a}+J_{a}^{0}J_{a}^{0}\right).}
where the generators
L
n
{\displaystyle L_{n}}
of the Virasoro algebra are the modes of the energy-momentum tensor,
T
(
z
)
=
∑
n
∈
Z
L
n
z
−
n
−
2
{\displaystyle T(z)=\sum _{n\in \mathbb {Z} }L_{n}z^{-n-2}}
.
== Spectrum ==
=== WZW models with compact, simply connected groups ===
If the Lie group
G
{\displaystyle G}
is compact and simply connected, then the WZW model is rational and diagonal: rational because the spectrum is built from a (level-dependent) finite set of irreducible representations of the affine Lie algebra called the integrable highest weight representations, and diagonal because a representation of the left-moving algebra is coupled with the same representation of the right-moving algebra.
For example, the spectrum of the
S
U
(
2
)
{\displaystyle SU(2)}
WZW model at level
k
∈
N
{\displaystyle k\in \mathbb {N} }
is
S
k
=
⨁
j
=
0
,
1
2
,
1
,
…
,
k
2
R
j
⊗
R
¯
j
,
{\displaystyle {\mathcal {S}}_{k}=\bigoplus _{j=0,{\frac {1}{2}},1,\dots ,{\frac {k}{2}}}{\mathcal {R}}_{j}\otimes {\bar {\mathcal {R}}}_{j}\ ,}
where
R
j
{\displaystyle {\mathcal {R}}_{j}}
is the affine highest weight representation of spin
j
{\displaystyle j}
: a representation generated by a state
|
v
⟩
{\displaystyle |v\rangle }
such that
J
n
<
0
a
|
v
⟩
=
J
0
−
|
v
⟩
=
0
,
{\displaystyle J_{n<0}^{a}|v\rangle =J_{0}^{-}|v\rangle =0\ ,}
where
J
−
{\displaystyle J^{-}}
is the current that corresponds to a generator
t
−
{\displaystyle t^{-}}
of the Lie algebra of
S
U
(
2
)
{\displaystyle SU(2)}
.
=== WZW models with other types of groups ===
If the group
G
{\displaystyle G}
is compact but not simply connected, the WZW model is rational but not necessarily diagonal. For example, the
S
O
(
3
)
{\displaystyle SO(3)}
WZW model exists for even integer levels
k
∈
2
N
{\displaystyle k\in 2\mathbb {N} }
, and its spectrum is a non-diagonal combination of finitely many integrable highest weight representations.
If the group
G
{\displaystyle G}
is not compact, the WZW model is non-rational. Moreover, its spectrum may include non highest weight representations. For example, the spectrum of the
S
L
(
2
,
R
)
{\displaystyle SL(2,\mathbb {R} )}
WZW model is built from highest weight representations, plus their images under the spectral flow automorphisms of the affine Lie algebra.
If
G
{\displaystyle G}
is a supergroup, the spectrum may involve representations that do not factorize as tensor products of representations of the left- and right-moving symmetry algebras. This occurs for example in the case
G
=
G
L
(
1
|
1
)
{\displaystyle G=GL(1|1)}
,
and also in more complicated supergroups such as
G
=
P
S
U
(
1
,
1
|
2
)
{\displaystyle G=PSU(1,1|2)}
.
Non-factorizable representations are responsible for the fact that the corresponding WZW models are logarithmic conformal field theories.
=== Other theories based on affine Lie algebras ===
The known conformal field theories based on affine Lie algebras are not limited to WZW models.
For example, in the case of the affine Lie algebra of the
S
U
(
2
)
{\displaystyle SU(2)}
WZW model, modular invariant torus partition functions obey an ADE classification, where the
S
U
(
2
)
{\displaystyle SU(2)}
WZW model accounts for the A series only. The D series corresponds to the
S
O
(
3
)
{\displaystyle SO(3)}
WZW model, and the E series does not correspond to any WZW model.
Another example is the
H
3
+
{\displaystyle H_{3}^{+}}
model. This model is based on the same symmetry algebra as the
S
L
(
2
,
R
)
{\displaystyle SL(2,\mathbb {R} )}
WZW model, to which it is related by Wick rotation. However, the
H
3
+
{\displaystyle H_{3}^{+}}
is not strictly speaking a WZW model, as
H
3
+
=
S
L
(
2
,
C
)
/
S
U
(
2
)
{\displaystyle H_{3}^{+}=SL(2,\mathbb {C} )/SU(2)}
is not a group, but a coset.
== Fields and correlation functions ==
=== Fields ===
Given a simple representation
ρ
{\displaystyle \rho }
of the Lie algebra of
G
{\displaystyle G}
, an affine primary field
Φ
ρ
(
z
)
{\displaystyle \Phi ^{\rho }(z)}
is a field that takes values in the representation space of
ρ
{\displaystyle \rho }
, such that
J
a
(
y
)
Φ
ρ
(
z
)
=
−
ρ
(
t
a
)
Φ
ρ
(
z
)
y
−
z
+
O
(
1
)
.
{\displaystyle J^{a}(y)\Phi ^{\rho }(z)=-{\frac {\rho (t^{a})\Phi ^{\rho }(z)}{y-z}}+O(1)\ .}
An affine primary field is also a primary field for the Virasoro algebra that results from the Sugawara construction. The conformal dimension of the affine primary field is given in terms of the quadratic Casimir
C
2
(
ρ
)
{\displaystyle C_{2}(\rho )}
of the representation
ρ
{\displaystyle \rho }
(i.e. the eigenvalue of the quadratic Casimir element
K
a
b
t
a
t
b
{\displaystyle K_{ab}t^{a}t^{b}}
where
K
a
b
{\displaystyle K_{ab}}
is the inverse of the matrix
K
(
t
a
,
t
b
)
{\displaystyle {\mathcal {K}}(t^{a},t^{b})}
of the Killing form) by
Δ
ρ
=
C
2
(
ρ
)
2
(
k
+
h
∨
)
.
{\displaystyle \Delta _{\rho }={\frac {C_{2}(\rho )}{2(k+h^{\vee })}}\ .}
For example, in the
S
U
(
2
)
{\displaystyle SU(2)}
WZW model, the conformal dimension of a primary field of spin
j
{\displaystyle j}
is
Δ
j
=
j
(
j
+
1
)
k
+
2
.
{\displaystyle \Delta _{j}={\frac {j(j+1)}{k+2}}\ .}
By the state-field correspondence, affine primary fields correspond to affine primary states, which are the highest weight states of highest weight representations of the affine Lie algebra.
=== Correlation functions ===
If the group
G
{\displaystyle G}
is compact, the spectrum of the WZW model is made of highest weight representations, and all correlation functions can be deduced from correlation functions of affine primary fields via Ward identities.
If the Riemann surface
Σ
{\displaystyle \Sigma }
is the Riemann sphere, correlation functions of affine primary fields obey Knizhnik–Zamolodchikov equations. On Riemann surfaces of higher genus, correlation functions obey Knizhnik–Zamolodchikov–Bernard equations, which involve derivatives not only of the fields' positions, but also of the surface's moduli.
== Gauged WZW models ==
Given a Lie subgroup
H
⊂
G
{\displaystyle H\subset G}
, the
G
/
H
{\displaystyle G/H}
gauged WZW model (or coset model) is a nonlinear sigma model whose target space is the quotient
G
/
H
{\displaystyle G/H}
for the adjoint action of
H
{\displaystyle H}
on
G
{\displaystyle G}
. This gauged WZW model is a conformal field theory, whose symmetry algebra is a quotient of the two affine Lie algebras of the
G
{\displaystyle G}
and
H
{\displaystyle H}
WZW models, and whose central charge is the difference of their central charges.
== Applications ==
The WZW model whose Lie group is the universal cover of the group
S
L
(
2
,
R
)
{\displaystyle \mathrm {SL} (2,\mathbb {R} )}
has been used by Juan Maldacena and Hirosi Ooguri to describe bosonic string theory on the three-dimensional anti-de Sitter space
A
d
S
3
{\displaystyle AdS_{3}}
. Superstrings on
A
d
S
3
×
S
3
{\displaystyle AdS_{3}\times S^{3}}
are described by the WZW model on the supergroup
P
S
U
(
1
,
1
|
2
)
{\displaystyle PSU(1,1|2)}
, or a deformation thereof if Ramond-Ramond flux is turned on.
WZW models and their deformations have been proposed for describing
the plateau transition in the integer quantum Hall effect.
The
S
L
(
2
,
R
)
/
U
(
1
)
{\displaystyle SL(2,\mathbb {R} )/U(1)}
gauged WZW model has an interpretation in string theory as Witten's two-dimensional Euclidean black hole.
The same model also describes certain two-dimensional statistical systems at criticality, such as the critical antiferromagnetic Potts model.
== References == | Wikipedia/Wess–Zumino–Witten_model |
In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutative. One commonly studied version of such theories has the "canonical" commutation relation:
[
x
μ
,
x
ν
]
=
i
θ
μ
ν
{\displaystyle [x^{\mu },x^{\nu }]=i\theta ^{\mu \nu }\,\!}
where
x
μ
{\displaystyle x^{\mu }}
and
x
ν
{\displaystyle x^{\nu }}
are the hermitian generators of a noncommutative
C
∗
{\displaystyle C^{*}}
-algebra of "functions on spacetime". That means that (with any given set of axes), it is impossible to accurately measure the position of a particle with respect to more than one axis. In fact, this leads to an uncertainty relation for the coordinates analogous to the Heisenberg uncertainty principle.
Various lower limits have been claimed for the noncommutative scale, (i.e. how accurately positions can be measured) but there is currently no experimental evidence in favour of such a theory or grounds for ruling them out.
One of the novel features of noncommutative field theories is the UV/IR mixing phenomenon in which the physics at high energies affects the physics at low energies which does not occur in quantum field theories in which the coordinates commute.
Other features include violation of Lorentz invariance due to the preferred direction of noncommutativity. Relativistic invariance can however be retained in the sense of twisted Poincaré invariance of the theory. The causality condition is modified from that of the commutative theories.
== History and motivation ==
Heisenberg was the first to suggest extending noncommutativity to the coordinates as a possible way of removing the infinite quantities appearing in field theories before the renormalization procedure was developed and had gained acceptance. The first paper on the subject was published in 1947 by Hartland Snyder. The success of the renormalization method resulted in little attention being paid to the subject for some time. In the 1980s, mathematicians, most notably Alain Connes, developed noncommutative geometry. Among other things, this work generalized the notion of differential structure to a noncommutative setting. This led to an operator algebraic description of noncommutative space-times, with the problem that it classically corresponds to a manifold with positively defined metric tensor, so that there is no description of (noncommutative) causality in this approach. However it also led to the development of a Yang–Mills theory on a noncommutative torus.
The particle physics community became interested in the noncommutative approach because of a paper by Nathan Seiberg and Edward Witten. They argued in the context of string theory that the coordinate functions of the endpoints of open strings constrained to a D-brane in the presence of a constant Neveu–Schwarz B-field—equivalent to a constant magnetic field on the brane—would satisfy the noncommutative algebra set out above. The implication is that a quantum field theory on noncommutative spacetime can be interpreted as a low energy limit of the theory of open strings.
Two papers, one by Sergio Doplicher, Klaus Fredenhagen and John Roberts
and the other by D. V. Ahluwalia,
set out another motivation for the possible noncommutativity of space-time.
The arguments go as follows: According to general relativity, when the energy density grows sufficiently large, a black hole is formed. On the other hand, according to the Heisenberg uncertainty principle, a measurement of a space-time separation causes an uncertainty in momentum inversely proportional to the extent of the separation. Thus energy whose scale corresponds to the uncertainty in momentum is localized in the system within a region corresponding to the uncertainty in position. When the separation is small enough, the Schwarzschild radius of the system is reached and a black hole is formed, which prevents any information from escaping the system. Thus there is a lower bound for the measurement of length. A sufficient condition for preventing gravitational collapse can be expressed as an uncertainty relation for the coordinates. This relation can in turn be derived from a commutation relation for the coordinates.
It is worth stressing that, differently from other approaches, in particular those relying upon Connes' ideas, here the noncommutative spacetime is a proper spacetime, i.e. it extends the idea of a four-dimensional pseudo-Riemannian manifold. On the other hand, differently from Connes' noncommutative geometry, the proposed model turns out to be coordinate-dependent from scratch.
In Doplicher Fredenhagen Roberts' paper noncommutativity of coordinates concerns all four spacetime coordinates and not only spatial ones.
== See also ==
Moyal product
Noncommutative geometry
Noncommutative standard model
Wigner–Weyl transform
== Footnotes ==
== Further reading ==
Grensing, Gerhard (2013). Structural Aspects of Quantum Field Theory and Noncommutative Geometry. World Scientific. doi:10.1142/8771. ISBN 978-981-4472-69-2.
M. R. Douglas and N. A. Nekrasov, (2001). Noncommutative field theory. Rev. Mod. Phys., 73(4), 977.
Richard J. Szabo (2003) "Quantum Field Theory on Noncommutative Spaces," Physics Reports 378: 207-99. An expository article on noncommutative quantum field theories.
Noncommutative quantum field theory, see statistics on arxiv.org
Valter Moretti (2003), "Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes," Rev. Math. Phys. 15: 1171-1218. An expository paper (also) on the difficulties to extend non-commutative geometry to the Lorentzian case describing causality | Wikipedia/Noncommutative_quantum_field_theory |
In theoretical physics, specifically quantum field theory, a beta function, β(g), encodes the dependence of a coupling parameter, g, on the energy scale, μ, of a given physical process described by quantum field theory.
It is defined as
β
(
g
)
=
μ
∂
g
∂
μ
=
∂
g
∂
ln
(
μ
)
,
{\displaystyle \beta (g)=\mu {\frac {\partial g}{\partial \mu }}={\frac {\partial g}{\partial \ln(\mu )}}~,}
and, because of the underlying renormalization group, it has no explicit dependence on μ, so it only depends on μ implicitly through g.
This dependence on the energy scale thus specified is known as the running of the coupling parameter, a fundamental
feature of scale-dependence in quantum field theory, and its explicit computation is achievable through a variety of mathematical techniques. The concept of Beta function was Introduced by Ernst Stueckelberg and André Petermann in 1953.
== Scale invariance ==
If the beta functions of a quantum field theory vanish, usually at particular values of the coupling parameters, then the theory is said to be scale-invariant. Almost all scale-invariant QFTs are also conformally invariant. The study of such theories is conformal field theory.
The coupling parameters of a quantum field theory can run even if the corresponding classical field theory is scale-invariant. In this case, the non-zero beta function tells us that the classical scale invariance is anomalous.
== Examples ==
Beta functions are usually computed in some kind of approximation scheme. An example is perturbation theory, where one assumes that the coupling parameters are small. One can then make an expansion in powers of the coupling parameters and truncate the higher-order terms (also known as higher loop contributions, due to the number of loops in the corresponding Feynman graphs).
Here are some examples of beta functions computed in perturbation theory:
=== Quantum electrodynamics ===
The one-loop beta function in quantum electrodynamics (QED) is
β
(
e
)
=
e
3
12
π
2
,
{\displaystyle \beta (e)={\frac {e^{3}}{12\pi ^{2}}}~,}
or, equivalently,
β
(
α
)
=
2
α
2
3
π
,
{\displaystyle \beta (\alpha )={\frac {2\alpha ^{2}}{3\pi }}~,}
written in terms of the fine structure constant in natural units, α = e2/4π.
This beta function tells us that the coupling increases with increasing energy scale, and QED becomes strongly coupled at high energy. In fact, the coupling apparently becomes infinite at some finite energy, resulting in a Landau pole. However, one cannot expect the perturbative beta function to give accurate results at strong coupling, and so it is likely that the Landau pole is an artifact of applying perturbation theory in a situation where it is no longer valid.
=== Quantum chromodynamics ===
The one-loop beta function in quantum chromodynamics with
n
f
{\displaystyle n_{f}}
flavours and
n
s
{\displaystyle n_{s}}
scalar colored bosons is
β
(
g
)
=
−
(
11
−
n
s
6
−
2
n
f
3
)
g
3
16
π
2
,
{\displaystyle \beta (g)=-\left(11-{\frac {n_{s}}{6}}-{\frac {2n_{f}}{3}}\right){\frac {g^{3}}{16\pi ^{2}}}~,}
or
β
(
α
s
)
=
−
(
11
−
n
s
6
−
2
n
f
3
)
α
s
2
2
π
,
{\displaystyle \beta (\alpha _{s})=-\left(11-{\frac {n_{s}}{6}}-{\frac {2n_{f}}{3}}\right){\frac {\alpha _{s}^{2}}{2\pi }}~,}
written in terms of αs =
g
2
/
4
π
{\displaystyle g^{2}/4\pi }
.
Assuming ns=0, if nf ≤ 16, the ensuing beta function dictates that the coupling decreases with increasing energy scale, a phenomenon known as asymptotic freedom. Conversely, the coupling increases with decreasing energy scale. This means that the coupling becomes large at low energies, and one can no longer rely on perturbation theory.
=== SU(N) Non-Abelian gauge theory ===
While the (Yang–Mills) gauge group of QCD is
S
U
(
3
)
{\displaystyle \mathrm {SU} (3)}
, and determines 3 colors, we can generalize to any number of colors,
N
c
{\displaystyle N_{c}}
, with a gauge group
G
=
S
U
(
N
c
)
{\displaystyle G=\mathrm {SU} (N_{c})}
. Then for this gauge group, with Dirac fermions in a representation
R
f
{\displaystyle R_{f}}
of
G
{\displaystyle G}
and with complex scalars in a representation
R
s
{\displaystyle R_{s}}
, the one-loop beta function is
β
(
g
)
=
−
(
11
3
C
2
(
G
)
−
1
3
n
s
T
(
R
s
)
−
4
3
n
f
T
(
R
f
)
)
g
3
16
π
2
,
{\displaystyle \beta (g)=-\left({\frac {11}{3}}C_{2}(G)-{\frac {1}{3}}n_{s}T(R_{s})-{\frac {4}{3}}n_{f}T(R_{f})\right){\frac {g^{3}}{16\pi ^{2}}}~,}
where
C
2
(
G
)
{\displaystyle C_{2}(G)}
is the quadratic Casimir of
G
{\displaystyle G}
and
T
(
R
)
{\displaystyle T(R)}
is another Casimir invariant defined by
T
r
(
T
R
a
T
R
b
)
=
T
(
R
)
δ
a
b
{\displaystyle Tr(T_{R}^{a}T_{R}^{b})=T(R)\delta ^{ab}}
for generators
T
R
a
,
b
{\displaystyle T_{R}^{a,b}}
of the Lie algebra in the representation R. (For Weyl or Majorana fermions, replace
4
/
3
{\displaystyle 4/3}
by
2
/
3
{\displaystyle 2/3}
, and for real scalars, replace
1
/
3
{\displaystyle 1/3}
by
1
/
6
{\displaystyle 1/6}
.) For gauge fields (i.e. gluons), necessarily in the adjoint of
G
{\displaystyle G}
,
C
2
(
G
)
=
N
c
{\displaystyle C_{2}(G)=N_{c}}
; for fermions in the fundamental (or anti-fundamental) representation of
G
{\displaystyle G}
,
T
(
R
)
=
1
/
2
{\displaystyle T(R)=1/2}
. Then for QCD, with
N
c
=
3
{\displaystyle N_{c}=3}
, the above equation reduces to that listed for the quantum chromodynamics beta function.
This famous result was derived nearly simultaneously in 1973 by Hugh David Politzer, David Gross and Frank Wilczek, for which the three were awarded the Nobel Prize in Physics in 2004.
Unbeknownst to these authors, Gerard 't Hooft had announced the result in a comment following a talk by Kurt Symanzik at a small meeting in Marseille in June 1972, but he never published it.
=== Standard Model Higgs–Yukawa couplings ===
In the Standard Model, quarks and leptons have Yukawa couplings to the Higgs boson. These determine the mass of the particle. Most all of the quarks' and leptons' Yukawa couplings are small compared to the top quark's Yukawa coupling. These Yukawa couplings change their values depending on the energy scale at which they are measured, through running. The dynamics of Yukawa couplings of quarks are determined by the renormalization group equation:
μ
∂
∂
μ
y
≈
y
16
π
2
(
9
2
y
2
−
8
g
3
2
)
{\displaystyle \mu {\frac {\partial }{\partial \mu }}y\approx {\frac {y}{16\pi ^{2}}}\left({\frac {9}{2}}y^{2}-8g_{3}^{2}\right)}
,
where
g
3
{\displaystyle g_{3}}
is the color gauge coupling (which is a function of
μ
{\displaystyle \mu }
and associated with asymptotic freedom) and
y
{\displaystyle y}
is the Yukawa coupling. This equation describes how the Yukawa coupling changes with energy scale
μ
{\displaystyle \mu }
.
The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of grand unification,
μ
≈
10
15
{\displaystyle \mu \approx 10^{15}}
GeV. Therefore, the
y
2
{\displaystyle y^{2}}
term can be neglected in the above equation. Solving, we then find that
y
{\displaystyle y}
is increased slightly at the low energy scales at which the quark masses are generated by the Higgs,
μ
≈
100
{\displaystyle \mu \approx 100}
GeV.
On the other hand, solutions to this equation for large initial values
y
{\displaystyle y}
cause the rhs to quickly approach smaller values as we descend in energy scale. The above equation then locks
y
{\displaystyle y}
to the QCD coupling
g
3
{\displaystyle g_{3}}
. This is known as the (infrared) quasi-fixed point of the renormalization group equation for the Yukawa coupling. No matter what the initial starting value of the coupling is, if it is sufficiently large it will reach this quasi-fixed point value, and the corresponding quark mass is predicted.
=== Minimal supersymmetric Standard Model ===
Renomalization group studies in the minimal supersymmetric Standard Model (MSSM) of grand unification and the Higgs–Yukawa fixed points were very encouraging that the theory was on the right track. So far, however, no evidence of the predicted MSSM particles has emerged in experiment at the Large Hadron Collider.
== See also ==
Banks–Zaks fixed point
Callan–Symanzik equation
Quantum triviality
== References ==
== Further reading ==
Peskin, M and Schroeder, D.; An Introduction to Quantum Field Theory, Westview Press (1995). A standard introductory text, covering many topics in QFT including calculation of beta functions; see especially chapter 16.
Weinberg, Steven; The Quantum Theory of Fields, (3 volumes) Cambridge University Press (1995). A monumental treatise on QFT.
Zinn-Justin, Jean; Quantum Field Theory and Critical Phenomena, Oxford University Press (2002). Emphasis on the renormalization group and related topics. | Wikipedia/Beta_function_(physics) |
In theoretical physics, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory with linearly realised supersymmetry. In 1974, Julius Wess and Bruno Zumino studied, using modern terminology, dynamics of a single chiral superfield (composed of a complex scalar and a spinor fermion) whose cubic superpotential leads to a renormalizable theory. It is a special case of 4D N = 1 global supersymmetry.
The treatment in this article largely follows that of Figueroa-O'Farrill's lectures on supersymmetry, and to some extent of Tong.
The model is an important model in supersymmetric quantum field theory. It is arguably the simplest supersymmetric field theory in four dimensions, and is ungauged.
== The Wess–Zumino action ==
=== Preliminary treatment ===
==== Spacetime and matter content ====
In a preliminary treatment, the theory is defined on flat spacetime (Minkowski space). For this article, the metric has mostly plus signature. The matter content is a real scalar field
S
{\displaystyle S}
, a real pseudoscalar field
P
{\displaystyle P}
, and a real (Majorana) spinor field
ψ
{\displaystyle \psi }
.
This is a preliminary treatment in the sense that the theory is written in terms of familiar scalar and spinor fields which are functions of spacetime, without developing a theory of superspace or superfields, which appear later in the article.
==== Free, massless theory ====
The Lagrangian of the free, massless Wess–Zumino model is
L
kin
=
−
1
2
(
∂
S
)
2
−
1
2
(
∂
P
)
2
−
1
2
ψ
¯
∂
/
ψ
,
{\displaystyle {\mathcal {L}}_{\text{kin}}=-{\frac {1}{2}}(\partial S)^{2}-{\frac {1}{2}}(\partial P)^{2}-{\frac {1}{2}}{\bar {\psi }}\partial \!\!\!/\psi ,}
where
∂
/
=
γ
μ
∂
μ
{\displaystyle \partial \!\!\!/=\gamma ^{\mu }\partial _{\mu }}
ψ
¯
=
ψ
t
C
=
ψ
†
i
γ
0
.
{\displaystyle {\bar {\psi }}=\psi ^{t}C=\psi ^{\dagger }i\gamma ^{0}.}
The corresponding action is
I
kin
=
∫
d
4
x
L
kin
{\displaystyle I_{\text{kin}}=\int d^{4}x{\mathcal {L}}_{\text{kin}}}
.
==== Massive theory ====
Supersymmetry is preserved when adding a mass term of the form
L
m
=
−
1
2
m
2
S
2
−
1
2
m
2
P
2
−
1
2
m
ψ
¯
ψ
{\displaystyle {\mathcal {L}}_{\text{m}}=-{\frac {1}{2}}m^{2}S^{2}-{\frac {1}{2}}m^{2}P^{2}-{\frac {1}{2}}m{\bar {\psi }}\psi }
==== Interacting theory ====
Supersymmetry is preserved when adding an interaction term with coupling constant
λ
{\displaystyle \lambda }
:
L
int
=
−
λ
(
ψ
¯
(
S
−
P
γ
5
)
ψ
+
1
2
λ
(
S
2
+
P
2
)
2
+
m
S
(
S
2
+
P
2
)
)
.
{\displaystyle {\mathcal {L}}_{\text{int}}=-\lambda \left({\bar {\psi }}(S-P\gamma _{5})\psi +{\frac {1}{2}}\lambda (S^{2}+P^{2})^{2}+mS(S^{2}+P^{2})\right).}
The full Wess–Zumino action is then given by putting these Lagrangians together:
==== Alternative expression ====
There is an alternative way of organizing the fields. The real fields
S
{\displaystyle S}
and
P
{\displaystyle P}
are combined into a single complex scalar field
ϕ
:=
1
2
(
S
+
i
P
)
,
{\displaystyle \phi :={\frac {1}{2}}(S+iP),}
while the Majorana spinor is written in terms of two Weyl spinors:
ψ
=
(
χ
α
,
χ
¯
α
˙
)
{\displaystyle \psi =(\chi ^{\alpha },{\bar {\chi }}_{\dot {\alpha }})}
. Defining the superpotential
W
(
ϕ
)
:=
1
2
m
ϕ
2
+
1
3
λ
ϕ
3
,
{\displaystyle W(\phi ):={\frac {1}{2}}m\phi ^{2}+{\frac {1}{3}}\lambda \phi ^{3},}
the Wess–Zumino action can also be written (possibly after relabelling some constant factors)
Upon substituting in
W
(
ϕ
)
{\displaystyle W(\phi )}
, one finds that this is a theory with a massive complex scalar
ϕ
{\displaystyle \phi }
and a massive Majorana spinor
ψ
{\displaystyle \psi }
of the same mass. The interactions are a cubic and quartic
ϕ
{\displaystyle \phi }
interaction, and a Yukawa interaction between
ϕ
{\displaystyle \phi }
and
ψ
{\displaystyle \psi }
, which are all familiar interactions from courses in non-supersymmetric quantum field theory.
=== Using superspace and superfields ===
==== Superspace and superfield content ====
Superspace consists of the direct sum of Minkowski space with 'spin space', a four dimensional space with coordinates
(
θ
α
,
θ
¯
α
˙
)
{\displaystyle (\theta _{\alpha },{\bar {\theta }}^{\dot {\alpha }})}
, where
α
,
α
˙
{\displaystyle \alpha ,{\dot {\alpha }}}
are indices taking values in
1
,
2.
{\displaystyle 1,2.}
More formally, superspace is constructed as the space of right cosets of the Lorentz group in the super-Poincaré group.
The fact there is only 4 'spin coordinates' means that this is a theory with what is known as
N
=
1
{\displaystyle {\mathcal {N}}=1}
supersymmetry, corresponding to an algebra with a single supercharge. The
8
=
4
+
4
{\displaystyle 8=4+4}
dimensional superspace is sometimes written
R
1
,
3
|
4
{\displaystyle \mathbb {R} ^{1,3|4}}
, and called super Minkowski space. The 'spin coordinates' are so called not due to any relation to angular momentum, but because they are treated as anti-commuting numbers, a property typical of spinors in quantum field theory due to the spin statistics theorem.
A superfield
Φ
{\displaystyle \Phi }
is then a function on superspace,
Φ
=
Φ
(
x
,
θ
,
θ
¯
)
{\displaystyle \Phi =\Phi (x,\theta ,{\bar {\theta }})}
.
Defining the supercovariant derivative
D
¯
α
˙
=
∂
¯
α
˙
−
i
(
σ
¯
μ
)
α
˙
β
θ
β
∂
μ
,
{\displaystyle {\bar {D}}_{\dot {\alpha }}={\bar {\partial }}_{\dot {\alpha }}-i({\bar {\sigma }}^{\mu })_{{\dot {\alpha }}\beta }\theta ^{\beta }\partial _{\mu },}
a chiral superfield satisfies
D
¯
α
˙
Φ
=
0.
{\displaystyle {\bar {D}}_{\dot {\alpha }}\Phi =0.}
The field content is then simply a single chiral superfield.
However, the chiral superfield contains fields, in the sense that it admits the expansion
Φ
(
x
,
θ
,
θ
¯
)
=
ϕ
(
y
)
+
θ
χ
(
y
)
+
θ
2
F
(
y
)
{\displaystyle \Phi (x,\theta ,{\bar {\theta }})=\phi (y)+\theta \chi (y)+\theta ^{2}F(y)}
with
y
μ
=
x
μ
−
i
θ
σ
μ
θ
¯
.
{\displaystyle y^{\mu }=x^{\mu }-i\theta \sigma ^{\mu }{\bar {\theta }}.}
Then
ϕ
{\displaystyle \phi }
can be identified as a complex scalar,
χ
{\displaystyle \chi }
is a Weyl spinor and
F
{\displaystyle F}
is an auxiliary complex scalar.
These fields admit a further relabelling, with
ϕ
=
1
2
(
S
+
i
P
)
{\displaystyle \phi ={\frac {1}{2}}(S+iP)}
and
ψ
a
=
(
χ
α
,
χ
¯
α
˙
)
.
{\displaystyle \psi ^{a}=(\chi ^{\alpha },{\bar {\chi }}_{\dot {\alpha }}).}
This allows recovery of the preliminary forms, after eliminating the non-dynamical
F
{\displaystyle F}
using its equation of motion.
==== Free, massless action ====
When written in terms of the chiral superfield
Φ
{\displaystyle \Phi }
, the action (for the free, massless Wess–Zumino model) takes on the simple form
∫
d
4
x
d
2
θ
d
2
θ
¯
2
Φ
¯
Φ
{\displaystyle \int d^{4}xd^{2}\theta d^{2}{\bar {\theta }}\,\,2{\bar {\Phi }}\Phi }
where
∫
d
2
θ
,
∫
d
2
θ
¯
{\displaystyle \int d^{2}\theta ,\int d^{2}{\bar {\theta }}}
are integrals over spinor dimensions of superspace.
==== Superpotential ====
Masses and interactions are added through a superpotential. The Wess–Zumino superpotential is
W
(
Φ
)
=
m
Φ
2
+
4
3
λ
Φ
3
.
{\displaystyle W(\Phi )=m\Phi ^{2}+{\frac {4}{3}}\lambda \Phi ^{3}.}
Since
W
(
Φ
)
{\displaystyle W(\Phi )}
is complex, to ensure the action is real its conjugate must also be added.
The full Wess–Zumino action is written
== Supersymmetry of the action ==
=== Preliminary treatment ===
The action is invariant under the supersymmetry transformations, given in infinitesimal form by
δ
ϵ
S
=
ϵ
¯
ψ
{\displaystyle \delta _{\epsilon }S={\bar {\epsilon }}\psi }
δ
ϵ
P
=
ϵ
¯
γ
5
ψ
{\displaystyle \delta _{\epsilon }P={\bar {\epsilon }}\gamma _{5}\psi }
δ
ϵ
ψ
=
[
∂
/
−
m
−
λ
(
S
+
P
γ
5
)
]
(
S
+
P
γ
5
)
ϵ
{\displaystyle \delta _{\epsilon }\psi =[\partial \!\!\!/-m-\lambda (S+P\gamma _{5})](S+P\gamma _{5})\epsilon }
where
ϵ
{\displaystyle \epsilon }
is a Majorana spinor-valued transformation parameter and
γ
5
{\displaystyle \gamma _{5}}
is the chirality operator.
The alternative form is invariant under the transformation
δ
ϵ
ϕ
=
2
ϵ
χ
{\displaystyle \delta _{\epsilon }\phi ={\sqrt {2}}\epsilon \chi }
δ
ϵ
χ
=
2
i
σ
μ
ϵ
¯
∂
μ
ϕ
−
2
ϵ
∂
W
†
∂
ϕ
†
{\displaystyle \delta _{\epsilon }\chi ={\sqrt {2}}i\sigma ^{\mu }{\bar {\epsilon }}\partial _{\mu }\phi -{\sqrt {2}}\epsilon {\frac {\partial W^{\dagger }}{\partial \phi ^{\dagger }}}}
.
Without developing a theory of superspace transformations, these symmetries appear ad-hoc.
=== Superfield treatment ===
If the action can be written as
S
=
∫
d
4
x
d
4
θ
K
(
x
,
θ
,
θ
¯
)
{\displaystyle S=\int d^{4}xd^{4}\theta K(x,\theta ,{\bar {\theta }})}
where
K
{\displaystyle K}
is a real superfield, that is,
K
†
=
K
{\displaystyle K^{\dagger }=K}
, then the action is invariant under supersymmetry.
Then the reality of
K
=
Φ
¯
Φ
{\displaystyle K={\bar {\Phi }}\Phi }
means it is invariant under supersymmetry.
== Extra classical symmetries ==
=== Superconformal symmetry ===
The massless Wess–Zumino model admits a larger set of symmetries, described at the algebra level by the superconformal algebra. As well as the Poincaré symmetry generators and the supersymmetry translation generators, this contains the conformal algebra as well as a conformal supersymmetry generator
S
α
{\displaystyle S_{\alpha }}
.
The conformal symmetry is broken at the quantum level by trace and conformal anomalies, which break invariance under the conformal generators
D
{\displaystyle D}
for dilatations and
K
μ
{\displaystyle K_{\mu }}
for special conformal transformations respectively.
=== R-symmetry ===
The
U
(
1
)
{\displaystyle {\text{U}}(1)}
R-symmetry of
N
=
1
{\displaystyle {\mathcal {N}}=1}
supersymmetry holds when the superpotential
W
(
Φ
)
{\displaystyle W(\Phi )}
is a monomial. This means either
W
(
ϕ
)
=
1
2
m
ϕ
2
{\displaystyle W(\phi )={\frac {1}{2}}m\phi ^{2}}
, so that the superfield
Φ
{\displaystyle \Phi }
is massive but free (non-interacting), or
W
(
Φ
)
=
1
3
λ
ϕ
3
{\displaystyle W(\Phi )={\frac {1}{3}}\lambda \phi ^{3}}
so the theory is massless but (possibly) interacting.
This is broken at the quantum level by anomalies.
== Action for multiple chiral superfields ==
The action generalizes straightforwardly to multiple chiral superfields
Φ
i
{\displaystyle \Phi ^{i}}
with
i
=
1
,
⋯
,
N
{\displaystyle i=1,\cdots ,N}
. The most general renormalizable theory is
I
=
∫
d
4
x
d
4
θ
K
i
j
¯
Φ
i
Φ
†
j
¯
+
∫
d
4
x
[
∫
d
2
θ
W
(
Φ
)
+
h.c.
]
{\displaystyle I=\int d^{4}x\,d^{4}\theta \,K_{i{\bar {j}}}\Phi ^{i}\Phi ^{\dagger {\bar {j}}}+\int d^{4}x\left[\int d^{2}\theta \,W(\Phi )+{\text{h.c.}}\right]}
where the superpotential is
W
(
Φ
)
=
a
i
Φ
i
+
1
2
m
i
j
Φ
i
Φ
j
+
1
3
λ
i
j
k
Φ
i
Φ
j
Φ
k
{\displaystyle W(\Phi )=a_{i}\Phi ^{i}+{\frac {1}{2}}m_{ij}\Phi ^{i}\Phi ^{j}+{\frac {1}{3}}\lambda _{ijk}\Phi ^{i}\Phi ^{j}\Phi ^{k}}
,
where implicit summation is used.
By a change of coordinates, under which
Φ
i
{\displaystyle \Phi ^{i}}
transforms under
GL
(
N
,
C
)
{\displaystyle {\text{GL}}(N,\mathbb {C} )}
, one can set
K
i
j
¯
=
δ
i
j
¯
{\displaystyle K_{i{\bar {j}}}=\delta _{i{\bar {j}}}}
without loss of generality. With this choice, the expression
K
=
δ
i
j
¯
Φ
i
Φ
†
j
¯
{\displaystyle K=\delta _{i{\bar {j}}}\Phi ^{i}\Phi ^{\dagger {\bar {j}}}}
is known as the canonical Kähler potential. There is residual freedom to make a unitary transformation in order to diagonalise the mass matrix
m
i
j
{\displaystyle m_{ij}}
.
When
N
=
1
{\displaystyle N=1}
, if the multiplet is massive then the Weyl fermion has a Majorana mass. But for
N
=
2
,
{\displaystyle N=2,}
the two Weyl fermions can have a Dirac mass, when the superpotential is taken to be
W
(
Φ
,
Φ
~
)
=
m
Φ
~
Φ
.
{\displaystyle W(\Phi ,{\tilde {\Phi }})=m{\tilde {\Phi }}\Phi .}
This theory has a
U
(
1
)
{\displaystyle {\text{U}}(1)}
symmetry, where
Φ
,
Φ
~
{\displaystyle \Phi ,{\tilde {\Phi }}}
rotate with opposite charges
=== Super QCD ===
For general
N
{\displaystyle N}
, a superpotential of the form
W
(
Φ
a
,
Φ
~
a
)
=
m
Φ
~
a
Φ
a
{\displaystyle W(\Phi _{a},{\tilde {\Phi }}_{a})=m{\tilde {\Phi }}_{a}\Phi _{a}}
has a
SU
(
N
)
{\displaystyle {\text{SU}}(N)}
symmetry when
Φ
a
,
Φ
~
a
{\displaystyle \Phi _{a},{\tilde {\Phi }}_{a}}
rotate with opposite charges, that is under
U
∈
SU
(
N
)
{\displaystyle U\in {\text{SU}}(N)}
Φ
a
↦
U
a
b
Φ
b
{\displaystyle \Phi _{a}\mapsto U_{a}{}^{b}\Phi _{b}}
Φ
~
a
↦
(
U
−
1
)
a
b
Φ
~
b
{\displaystyle {\tilde {\Phi }}_{a}\mapsto (U^{-1})_{a}{}^{b}{\tilde {\Phi }}_{b}}
.
This symmetry can be gauged and coupled to supersymmetric Yang–Mills to form a supersymmetric analogue to quantum chromodynamics, known as super QCD.
=== Supersymmetric sigma models ===
If renormalizability is not insisted upon, then there are two possible generalizations. The first of these is to consider more general superpotentials. The second is to consider
K
{\displaystyle K}
in the kinetic term
S
=
∫
d
4
x
d
2
θ
2
d
2
θ
¯
2
K
(
Φ
,
Φ
¯
)
{\displaystyle S=\int d^{4}x\,d^{2}\theta ^{2}\,d^{2}{\bar {\theta }}^{2}K(\Phi ,{\bar {\Phi }})}
to be a real function
K
=
K
(
Φ
,
Φ
¯
)
{\displaystyle K=K(\Phi ,{\bar {\Phi }})}
of
Φ
i
{\displaystyle \Phi ^{i}}
and
Φ
¯
j
¯
{\displaystyle {\bar {\Phi }}^{\bar {j}}}
.
The action is invariant under transformations
K
(
Φ
,
Φ
†
)
+
Λ
(
Φ
)
+
Λ
¯
(
Φ
¯
)
{\displaystyle K(\Phi ,\Phi ^{\dagger })+\Lambda (\Phi )+{\bar {\Lambda }}({\bar {\Phi }})}
: these are known as Kähler transformations.
Considering this theory gives an intersection of Kähler geometry with supersymmetric field theory.
By expanding the Kähler potential
K
(
Φ
,
Φ
¯
)
{\displaystyle K(\Phi ,{\bar {\Phi }})}
in terms of derivatives of
K
{\displaystyle K}
and the constituent superfields of
Φ
,
Φ
¯
{\displaystyle \Phi ,{\bar {\Phi }}}
, and then eliminating the auxiliary fields
F
,
F
¯
{\displaystyle F,{\bar {F}}}
using the equations of motion, the following expression is obtained:
S
K
=
∫
d
4
x
[
g
i
j
¯
(
∂
μ
ϕ
i
∂
μ
ϕ
¯
j
¯
)
+
g
i
j
¯
i
2
(
∇
μ
ψ
i
σ
μ
ψ
¯
j
¯
−
ψ
i
σ
μ
∇
μ
ψ
¯
j
¯
)
+
1
4
R
i
j
¯
k
l
¯
(
ψ
i
ψ
k
)
(
ψ
¯
j
¯
ψ
¯
l
¯
)
]
{\displaystyle S_{K}=\int d^{4}x\left[g_{i{\bar {j}}}(\partial _{\mu }\phi ^{i}\partial ^{\mu }{\bar {\phi }}^{\bar {j}})+g_{i{\bar {j}}}{\frac {i}{2}}(\nabla _{\mu }\psi ^{i}\sigma ^{\mu }{\bar {\psi }}^{\bar {j}}-\psi ^{i}\sigma ^{\mu }\nabla _{\mu }{\bar {\psi }}^{\bar {j}})+{\frac {1}{4}}R_{i{\bar {j}}k{\bar {l}}}(\psi ^{i}\psi ^{k})({\bar {\psi }}^{\bar {j}}{\bar {\psi }}^{\bar {l}})\right]}
where
g
i
j
¯
{\displaystyle g_{i{\bar {j}}}}
is the Kähler metric. It is invariant under Kähler transformations. If the kinetic term is positive definite, then
g
i
j
¯
{\displaystyle g_{i{\bar {j}}}}
is invertible, allowing the inverse metric
g
i
j
¯
{\displaystyle g^{i{\bar {j}}}}
to be defined.
The Christoffel symbols (adapted for a Kähler metric) are
Γ
i
j
k
=
g
i
l
¯
∂
j
g
k
l
¯
{\displaystyle \Gamma ^{i}{}_{jk}=g^{i{\bar {l}}}\partial _{j}g_{k{\bar {l}}}}
and
Γ
¯
i
¯
j
¯
k
¯
=
g
l
i
¯
∂
j
¯
g
l
k
¯
.
{\displaystyle {\bar {\Gamma }}^{\bar {i}}{}_{{\bar {j}}{\bar {k}}}=g^{l{\bar {i}}}\partial _{\bar {j}}g_{l{\bar {k}}}.}
The covariant derivatives
∇
μ
ψ
i
{\displaystyle \nabla _{\mu }\psi ^{i}}
and
∇
μ
ψ
¯
j
¯
{\displaystyle \nabla _{\mu }{\bar {\psi }}^{\bar {j}}}
are defined
∇
μ
ψ
i
=
∂
μ
ψ
i
+
Γ
i
j
k
ψ
j
∂
μ
ϕ
k
{\displaystyle \nabla _{\mu }\psi ^{i}=\partial _{\mu }\psi ^{i}+\Gamma ^{i}{}_{jk}\psi ^{j}\partial _{\mu }\phi ^{k}}
and
∇
μ
ψ
¯
i
¯
=
∂
μ
ψ
i
¯
+
Γ
¯
i
¯
j
¯
k
¯
ψ
¯
j
¯
∂
μ
ϕ
¯
k
¯
{\displaystyle \nabla _{\mu }{\bar {\psi }}^{\bar {i}}=\partial _{\mu }\psi ^{\bar {i}}+{\bar {\Gamma }}^{\bar {i}}{}_{{\bar {j}}{\bar {k}}}{\bar {\psi }}^{\bar {j}}\partial _{\mu }{\bar {\phi }}^{\bar {k}}}
The Riemann curvature tensor (adapted for a Kähler metric) is defined
R
i
j
¯
k
l
¯
=
g
m
j
¯
∂
l
¯
Γ
m
i
k
=
∂
k
∂
l
¯
g
i
j
¯
−
g
m
n
¯
(
∂
k
g
i
n
¯
)
(
∂
l
¯
g
m
j
¯
)
{\displaystyle R_{i{\bar {j}}k{\bar {l}}}=g_{m{\bar {j}}}\partial _{\bar {l}}\Gamma ^{m}{}_{ik}=\partial _{k}\partial _{\bar {l}}g_{i{\bar {j}}}-g^{m{\bar {n}}}(\partial _{k}g_{i{\bar {n}}})(\partial _{\bar {l}}g_{m{\bar {j}}})}
.
==== Adding a superpotential ====
A superpotential
W
(
Φ
)
{\displaystyle W(\Phi )}
can be added to form the more general action
S
=
S
K
−
∫
d
4
x
[
g
i
j
¯
∂
i
W
∂
j
¯
W
¯
+
1
4
ψ
i
ψ
j
H
i
j
(
W
)
+
1
4
ψ
¯
i
¯
ψ
¯
j
¯
H
i
¯
j
¯
(
W
¯
)
]
{\displaystyle S=S_{K}-\int d^{4}x\left[g^{i{\bar {j}}}\partial _{i}W\partial _{\bar {j}}{\bar {W}}+{\frac {1}{4}}\psi ^{i}\psi ^{j}H_{ij}(W)+{\frac {1}{4}}{\bar {\psi }}^{\bar {i}}{\bar {\psi }}^{\bar {j}}H_{{\bar {i}}{\bar {j}}}({\bar {W}})\right]}
where the Hessians of
W
{\displaystyle W}
are defined
H
i
j
(
W
)
=
∇
i
∂
j
W
=
∂
i
∂
j
W
−
Γ
k
i
j
∂
k
W
{\displaystyle H_{ij}(W)=\nabla _{i}\partial _{j}W=\partial _{i}\partial _{j}W-\Gamma ^{k}{}_{ij}\partial _{k}W}
H
¯
i
¯
j
¯
(
W
¯
)
=
∇
i
¯
∂
j
¯
W
¯
=
∂
i
¯
∂
j
¯
W
¯
−
Γ
k
¯
i
¯
j
¯
∂
k
¯
W
¯
{\displaystyle {\bar {H}}_{{\bar {i}}{\bar {j}}}({\bar {W}})=\nabla _{\bar {i}}\partial _{\bar {j}}{\bar {W}}=\partial _{\bar {i}}\partial _{\bar {j}}{\bar {W}}-\Gamma ^{\bar {k}}{}_{{\bar {i}}{\bar {j}}}\partial _{\bar {k}}{\bar {W}}}
.
== See also ==
N = 4 supersymmetric Yang–Mills theory
Supermultiplet
== References == | Wikipedia/Wess–Zumino_model |
Shinsei Ryu and Tadashi Takayanagi published 2006 a conjecture within holography that posits a quantitative relationship between the entanglement entropy of a conformal field theory and the geometry of an associated anti-de Sitter spacetime. The formula characterizes "holographic screens" in the bulk; that is, it specifies which regions of the bulk geometry are "responsible to particular information in the dual CFT". The authors were awarded the 2015 Breakthrough Prize in Fundamental Physics for "fundamental ideas about entropy in quantum field theory and quantum gravity", and awarded the 2024 Dirac Medal of the ICTP for "their insights on quantum entropy in quantum gravity and quantum field theories". The formula was generalized to a covariant form in 2007.
== Motivation ==
The thermodynamics of black holes suggests certain relationships between the entropy of black holes and their geometry. Specifically, the Bekenstein–Hawking area formula conjectures that the entropy of a black hole is proportional to its surface area:
S
BH
=
k
B
A
4
ℓ
P
2
{\displaystyle S_{\text{BH}}={\frac {k_{\text{B}}A}{4\ell _{\text{P}}^{2}}}}
The Bekenstein–Hawking entropy
S
BH
{\displaystyle S_{\text{BH}}}
is a measure of the information lost to external observers due to the presence of the horizon. The horizon of the black hole acts as a "screen" distinguishing one region of the spacetime (in this case the exterior of the black hole) that is not affected by another region (in this case the interior). The Bekenstein–Hawking area law states that the area of this surface is proportional to the entropy of the information lost behind it.
The Bekenstein–Hawking entropy is a statement about the gravitational entropy of a system; however, there is another type of entropy that is important in quantum information theory, namely the entanglement (or von Neumann) entropy. This form of entropy provides a measure of how far from a pure state a given quantum state is, or, equivalently, how entangled it is. The entanglement entropy is a useful concept in many areas, such as in condensed matter physics and quantum many-body systems. Given its use, and its suggestive similarity to the Bekenstein–Hawking entropy, it is desirable to have a holographic description of entanglement entropy in terms of gravity.
== Holographic preliminaries ==
The holographic principle states that gravitational theories in a given dimension are dual to a gauge theory in one lower dimension. The AdS/CFT correspondence is one example of such duality. Here, the field theory is defined on a fixed background and is equivalent to a quantum gravitational theory whose different states each correspond to a possible spacetime geometry. The conformal field theory is often viewed as living on the boundary of the higher dimensional space whose gravitational theory it defines. The result of such a duality is a dictionary between the two equivalent descriptions. For example, in a CFT defined on
d
{\displaystyle d}
dimensional Minkowski space the vacuum state corresponds to pure AdS space, whereas the thermal state corresponds to a planar black hole. Important for the present discussion is that the thermal state of a CFT defined on the
d
{\displaystyle d}
dimensional sphere corresponds to the
d
+
1
{\displaystyle d+1}
dimensional Schwarzschild black hole in AdS space.
The Bekenstein–Hawking area law, while claiming that the area of the black hole horizon is proportional to the black hole's entropy, fails to provide a sufficient microscopic description of how this entropy arises. The holographic principle provides such a description by relating the black hole system to a quantum system which does admit such a microscopic description. In this case, the CFT has discrete eigenstates and the thermal state is the canonical ensemble of these states. The entropy of this ensemble can be calculated through normal means, and yields the same result as predicted by the area law. This turns out to be a special case of the Ryu–Takayanagi conjecture.
== Conjecture ==
Consider a spatial slice
Σ
{\displaystyle \Sigma }
of an AdS space time on whose boundary we define the dual CFT. The Ryu–Takayanagi formula states:
where
S
A
{\displaystyle S_{A}}
is the entanglement entropy of the CFT in some spatial sub-region
A
⊂
∂
Σ
{\displaystyle A\subset \partial \Sigma }
with its complement
B
{\displaystyle B}
, and
γ
A
{\displaystyle \gamma _{A}}
is the Ryu–Takayanagi surface in the bulk. This surface must satisfy three properties:
γ
A
{\displaystyle \gamma _{A}}
has the same boundary as
A
{\displaystyle A}
.
γ
A
{\displaystyle \gamma _{A}}
is homologous to A.
γ
A
{\displaystyle \gamma _{A}}
extremizes the area. If there are multiple extremal surfaces,
γ
A
{\displaystyle \gamma _{A}}
is the one with the least area.
Because of property (3), this surface is typically called the minimal surface when the context is clear. Furthermore, property (1) ensures that the formula preserves certain features of entanglement entropy, such as
S
A
=
S
B
{\displaystyle S_{A}=S_{B}}
and
S
A
1
+
A
2
≥
S
A
1
∪
A
2
{\displaystyle S_{A_{1}+A_{2}}\geq S_{A_{1}\cup A_{2}}}
. The conjecture provides an explicit geometric interpretation of the entanglement entropy of the boundary CFT, namely as the area of a surface in the bulk.
== Example ==
In their original paper, Ryu and Takayanagi show this result explicitly for an example in
AdS
3
/
CFT
2
{\displaystyle {\text{AdS}}_{3}/{\text{CFT}}_{2}}
where an expression for the entanglement entropy is already known. For an
AdS
3
{\displaystyle {\text{AdS}}_{3}}
space of radius
R
{\displaystyle R}
, the dual CFT has a central charge given by
Furthermore,
AdS
3
{\displaystyle {\text{AdS}}_{3}}
has the metric
d
s
2
=
R
2
(
−
cosh
ρ
2
d
t
2
+
d
ρ
2
+
sinh
ρ
2
d
θ
2
)
{\displaystyle ds^{2}=R^{2}(-\cosh {\rho ^{2}dt^{2}}+d\rho ^{2}+\sinh {\rho ^{2}d\theta ^{2}})}
in
(
t
,
ρ
,
θ
)
{\displaystyle (t,\rho ,\theta )}
(essentially a stack of hyperbolic disks). Since this metric diverges at
ρ
→
∞
{\displaystyle \rho \to \infty }
,
ρ
{\displaystyle \rho }
is restricted to
ρ
≤
ρ
0
{\displaystyle \rho \leq \rho _{0}}
. This act of imposing a maximum
ρ
{\displaystyle \rho }
is analogous to the corresponding CFT having a UV cutoff. If
L
{\displaystyle L}
is the length of the CFT system, in this case the circumference of the cylinder calculated with the appropriate metric, and
a
{\displaystyle a}
is the lattice spacing, we have
e
ρ
0
∼
L
/
a
{\displaystyle e^{\rho _{0}}\sim L/a}
.
In this case, the boundary CFT lives at coordinates
(
t
,
ρ
0
,
θ
)
=
(
t
,
θ
)
{\displaystyle (t,\rho _{0},\theta )=(t,\theta )}
. Consider a fixed
t
{\displaystyle t}
slice and take the subregion A of the boundary to be
θ
∈
[
0
,
2
π
l
/
L
]
{\displaystyle \theta \in [0,2\pi l/L]}
where
l
{\displaystyle l}
is the length of
A
{\displaystyle A}
. The minimal surface is easy to identify in this case, as it is just the geodesic through the bulk that connects
θ
=
0
{\displaystyle \theta =0}
and
θ
=
2
π
l
/
L
{\displaystyle \theta =2\pi l/L}
. Remembering the lattice cutoff, the length of the geodesic can be calculated as
If it is assumed that
e
ρ
0
>>
1
{\displaystyle e^{\rho _{0}}>>1}
, then using the Ryu–Takayanagi formula to compute the entanglement entropy. Plugging in the length of the minimal surface calculated in (3) and recalling the central charge (2), the entanglement entropy is given by
This agrees with the result calculated by usual means.
== References == | Wikipedia/Ryu–Takayanagi_conjecture |
In theoretical physics, ABJM theory is a quantum field theory studied by Ofer Aharony, Oren Bergman, Daniel Jafferis, and Juan Maldacena. It provides a holographic dual to M-theory on
A
d
S
4
×
S
7
{\displaystyle AdS_{4}\times S^{7}}
. The ABJM theory is also closely related to Chern–Simons theory, and it serves as a useful toy model for solving problems that arise in condensed matter physics. It is a theory defined on
d
=
3
,
N
=
6
{\displaystyle d=3,{\mathcal {N}}=6}
superspace.
== See also ==
6D (2,0) superconformal field theory
== Notes ==
== References ==
Aharony, Ofer; Bergman, Oren; Jafferis, Daniel Louis; Maldacena, Juan (2008). "N=6 superconformal Chern–Simons-matter theories, M2-branes and their gravity duals". Journal of High Energy Physics. 2008 (10): 091. arXiv:0806.1218. Bibcode:2008JHEP...10..091A. doi:10.1088/1126-6708/2008/10/091. S2CID 16987793.
Klebanov, Igor (2010). "M2-branes and AdS/CFT". In Peloso, Marco; Vainshtein, Arkady (eds.). Crossing The Boundaries: Gauge Dynamics at Strong Coupling. World Scientific. pp. 158–178. | Wikipedia/ABJM_superconformal_field_theory |
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits.
The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.
Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at a constant speed in vacuum, c (299792458 m/s). Known as electromagnetic radiation, these waves occur at various wavelengths to produce a spectrum of radiation from radio waves to gamma rays.
In partial differential equation form and a coherent system of units, Maxwell's microscopic equations can be written as (top to bottom: Gauss's law, Gauss's law for magnetism, Faraday's law, Ampère-Maxwell law)
∇
⋅
E
=
ρ
ε
0
∇
⋅
B
=
0
∇
×
E
=
−
∂
B
∂
t
∇
×
B
=
μ
0
(
J
+
ε
0
∂
E
∂
t
)
{\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} \,\,\,&={\frac {\rho }{\varepsilon _{0}}}\\\nabla \cdot \mathbf {B} \,\,\,&=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \times \mathbf {B} &=\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\end{aligned}}}
With
E
{\displaystyle \mathbf {E} }
the electric field,
B
{\displaystyle \mathbf {B} }
the magnetic field,
ρ
{\displaystyle \rho }
the electric charge density and
J
{\displaystyle \mathbf {J} }
the current density.
ε
0
{\displaystyle \varepsilon _{0}}
is the vacuum permittivity and
μ
0
{\displaystyle \mu _{0}}
the vacuum permeability.
The equations have two major variants:
The microscopic equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale.
The macroscopic equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic-scale charges and quantum phenomena like spins. However, their use requires experimentally determined parameters for a phenomenological description of the electromagnetic response of materials.
The term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics. The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's equations with special relativity manifest. Maxwell's equations in curved spacetime, commonly used in high-energy and gravitational physics, are compatible with general relativity. In fact, Albert Einstein developed special and general relativity to accommodate the invariant speed of light, a consequence of Maxwell's equations, with the principle that only relative movement has physical consequences.
The publication of the equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation.
Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a classical limit of the more precise theory of quantum electrodynamics.
== History of the equations ==
== Conceptual descriptions ==
=== Gauss's law ===
Gauss's law describes the relationship between an electric field and electric charges: an electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a closed surface is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the permittivity of free space.
=== Gauss's law for magnetism ===
Gauss's law for magnetism states that electric charges have no magnetic analogues, called magnetic monopoles; no north or south magnetic poles exist in isolation. Instead, the magnetic field of a material is attributed to a dipole, and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite "magnetic charges". Precisely, the total magnetic flux through a Gaussian surface is zero, and the magnetic field is a solenoidal vector field.
=== Faraday's law ===
The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds to the negative curl of an electric field. In integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface.
The electromagnetic induction is the operating principle behind many electric generators: for example, a rotating bar magnet creates a changing magnetic field and generates an electric field in a nearby wire.
=== Ampère–Maxwell law ===
The original law of Ampère states that magnetic fields relate to electric current. Maxwell's addition states that magnetic fields also relate to changing electric fields, which Maxwell called displacement current. The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve.
Maxwell's modification of Ampère's circuital law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields. As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field. A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space.
The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, matches the speed of light; indeed, light is one form of electromagnetic radiation (as are X-rays, radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of electromagnetism and optics.
== Formulation in terms of electric and magnetic fields (microscopic or in vacuum version) ==
In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate law of nature, the Lorentz force law, describes how the electric and magnetic fields act on charged particles and currents. By convention, a version of this law in the original equations by Maxwell is no longer included. The vector calculus formalism below, the work of Oliver Heaviside, has become standard. It is rotationally invariant, and therefore mathematically more transparent than Maxwell's original 20 equations in x, y and z components. The relativistic formulations are more symmetric and Lorentz invariant. For the same equations expressed using tensor calculus or differential forms (see § Alternative formulations).
The differential and integral formulations are mathematically equivalent; both are useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential equations are purely local and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using finite element analysis.
=== Key to the notation ===
Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated.
The equations introduce the electric field, E, a vector field, and the magnetic field, B, a pseudovector field, each generally having a time and location dependence.
The sources are
the total electric charge density (total charge per unit volume), ρ, and
the total electric current density (total current per unit area), J.
The universal constants appearing in the equations (the first two ones explicitly only in the SI formulation) are:
the permittivity of free space, ε0, and
the permeability of free space, μ0, and
the speed of light,
c
=
(
ε
0
μ
0
)
−
1
/
2
{\displaystyle c=({\varepsilon _{0}\mu _{0}})^{-1/2}}
==== Differential equations ====
In the differential equations,
the nabla symbol, ∇, denotes the three-dimensional gradient operator, del,
the ∇⋅ symbol (pronounced "del dot") denotes the divergence operator,
the ∇× symbol (pronounced "del cross") denotes the curl operator.
==== Integral equations ====
In the integral equations,
Ω is any volume with closed boundary surface ∂Ω, and
Σ is any surface with closed boundary curve ∂Σ,
The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over a given time interval. For example, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law:
d
d
t
∬
Σ
B
⋅
d
S
=
∬
Σ
∂
B
∂
t
⋅
d
S
,
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =\iint _{\Sigma }{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {S} \,,}
Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using the differential version and using Gauss' and Stokes' theorems as appropriate.
∫
∂
Ω
{\displaystyle {\vphantom {\int }}_{\scriptstyle \partial \Omega }}
is a surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed
∭
Ω
{\displaystyle \iiint _{\Omega }}
is a volume integral over the volume Ω,
∮
∂
Σ
{\displaystyle \oint _{\partial \Sigma }}
is a line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed.
∬
Σ
{\displaystyle \iint _{\Sigma }}
is a surface integral over the surface Σ,
The total electric charge Q enclosed in Ω is the volume integral over Ω of the charge density ρ (see the "macroscopic formulation" section below):
Q
=
∭
Ω
ρ
d
V
,
{\displaystyle Q=\iiint _{\Omega }\rho \ \mathrm {d} V,}
where dV is the volume element.
The net magnetic flux ΦB is the surface integral of the magnetic field B passing through a fixed surface, Σ:
Φ
B
=
∬
Σ
B
⋅
d
S
,
{\displaystyle \Phi _{B}=\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} ,}
The net electric flux ΦE is the surface integral of the electric field E passing through Σ:
Φ
E
=
∬
Σ
E
⋅
d
S
,
{\displaystyle \Phi _{E}=\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} ,}
The net electric current I is the surface integral of the electric current density J passing through Σ:
I
=
∬
Σ
J
⋅
d
S
,
{\displaystyle I=\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} ,}
where dS denotes the differential vector element of surface area S, normal to surface Σ. (Vector area is sometimes denoted by A rather than S, but this conflicts with the notation for magnetic vector potential).
=== Formulation in the SI ===
=== Formulation in the Gaussian system ===
The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing dimensioned factors of ε0 and μ0 into the units (and thus redefining these). With a corresponding change in the values of the quantities for the Lorentz force law this yields the same physics, i.e. trajectories of charged particles, or work done by an electric motor. These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the electromagnetic tensor: the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension.: vii Such modified definitions are conventionally used with the Gaussian (CGS) units. Using these definitions, colloquially "in Gaussian units",
the Maxwell equations become:
The equations simplify slightly when a system of quantities is chosen in the speed of light, c, is used for nondimensionalization, so that, for example, seconds and lightseconds are interchangeable, and c = 1.
Further changes are possible by absorbing factors of 4π. This process, called rationalization, affects whether Coulomb's law or Gauss's law includes such a factor (see Heaviside–Lorentz units, used mainly in particle physics).
== Relationship between differential and integral formulations ==
The equivalence of the differential and integral formulations are a consequence of the Gauss divergence theorem and the Kelvin–Stokes theorem.
=== Flux and divergence ===
According to the (purely mathematical) Gauss divergence theorem, the electric flux through the
boundary surface ∂Ω can be rewritten as
∮
∂
Ω
E
⋅
d
S
=
∭
Ω
∇
⋅
E
d
V
{\displaystyle {\vphantom {\oint }}_{\scriptstyle \partial \Omega }\mathbf {E} \cdot \mathrm {d} \mathbf {S} =\iiint _{\Omega }\nabla \cdot \mathbf {E} \,\mathrm {d} V}
The integral version of Gauss's equation can thus be rewritten as
∭
Ω
(
∇
⋅
E
−
ρ
ε
0
)
d
V
=
0
{\displaystyle \iiint _{\Omega }\left(\nabla \cdot \mathbf {E} -{\frac {\rho }{\varepsilon _{0}}}\right)\,\mathrm {d} V=0}
Since Ω is arbitrary (e.g. an arbitrary small ball with arbitrary center), this is satisfied if and only if the integrand is zero everywhere. This is
the differential equations formulation of Gauss equation up to a trivial rearrangement.
Similarly rewriting the magnetic flux in Gauss's law for magnetism in integral form gives
∮
∂
Ω
B
⋅
d
S
=
∭
Ω
∇
⋅
B
d
V
=
0.
{\displaystyle {\vphantom {\oint }}_{\scriptstyle \partial \Omega }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =\iiint _{\Omega }\nabla \cdot \mathbf {B} \,\mathrm {d} V=0.}
which is satisfied for all Ω if and only if
∇
⋅
B
=
0
{\displaystyle \nabla \cdot \mathbf {B} =0}
everywhere.
=== Circulation and curl ===
By the Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ to an integral of the "circulation of the fields" (i.e. their curls) over a surface it bounds, i.e.
∮
∂
Σ
B
⋅
d
ℓ
=
∬
Σ
(
∇
×
B
)
⋅
d
S
,
{\displaystyle \oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\iint _{\Sigma }(\nabla \times \mathbf {B} )\cdot \mathrm {d} \mathbf {S} ,}
Hence the Ampère–Maxwell law, the modified version of Ampère's circuital law, in integral form can be rewritten as
∬
Σ
(
∇
×
B
−
μ
0
(
J
+
ε
0
∂
E
∂
t
)
)
⋅
d
S
=
0.
{\displaystyle \iint _{\Sigma }\left(\nabla \times \mathbf {B} -\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)\cdot \mathrm {d} \mathbf {S} =0.}
Since Σ can be chosen arbitrarily, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centered disk, we conclude that the integrand is zero if and only if the Ampère–Maxwell law in differential equations form is satisfied.
The equivalence of Faraday's law in differential and integral form follows likewise.
The line integrals and curls are analogous to quantities in classical fluid dynamics: the circulation of a fluid is the line integral of the fluid's flow velocity field around a closed loop, and the vorticity of the fluid is the curl of the velocity field.
== Charge conservation ==
The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the Ampère–Maxwell law has zero divergence by the div–curl identity. Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives:
0
=
∇
⋅
(
∇
×
B
)
=
∇
⋅
(
μ
0
(
J
+
ε
0
∂
E
∂
t
)
)
=
μ
0
(
∇
⋅
J
+
ε
0
∂
∂
t
∇
⋅
E
)
=
μ
0
(
∇
⋅
J
+
∂
ρ
∂
t
)
{\displaystyle 0=\nabla \cdot (\nabla \times \mathbf {B} )=\nabla \cdot \left(\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +\varepsilon _{0}{\frac {\partial }{\partial t}}\nabla \cdot \mathbf {E} \right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +{\frac {\partial \rho }{\partial t}}\right)}
i.e.,
∂
ρ
∂
t
+
∇
⋅
J
=
0.
{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.}
By the Gauss divergence theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary:
d
d
t
Q
Ω
=
d
d
t
∭
Ω
ρ
d
V
=
−
{\displaystyle {\frac {d}{dt}}Q_{\Omega }={\frac {d}{dt}}\iiint _{\Omega }\rho \mathrm {d} V=-}
∮
∂
Ω
J
⋅
d
S
=
−
I
∂
Ω
.
{\displaystyle {\vphantom {\oint }}_{\scriptstyle \partial \Omega }\mathbf {J} \cdot {\rm {d}}\mathbf {S} =-I_{\partial \Omega }.}
In particular, in an isolated system the total charge is conserved.
== Vacuum equations, electromagnetic waves and speed of light ==
In a region with no charges (ρ = 0) and no currents (J = 0), such as in vacuum, Maxwell's equations reduce to:
∇
⋅
E
=
0
,
∇
×
E
+
∂
B
∂
t
=
0
,
∇
⋅
B
=
0
,
∇
×
B
−
μ
0
ε
0
∂
E
∂
t
=
0.
{\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} &=0,&\nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0,\\\nabla \cdot \mathbf {B} &=0,&\nabla \times \mathbf {B} -\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}=0.\end{aligned}}}
Taking the curl (∇×) of the curl equations, and using the curl of the curl identity we obtain
μ
0
ε
0
∂
2
E
∂
t
2
−
∇
2
E
=
0
,
μ
0
ε
0
∂
2
B
∂
t
2
−
∇
2
B
=
0.
{\displaystyle {\begin{aligned}\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} =0,\\\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}-\nabla ^{2}\mathbf {B} =0.\end{aligned}}}
The quantity
μ
0
ε
0
{\displaystyle \mu _{0}\varepsilon _{0}}
has the dimension (T/L)2. Defining
c
=
(
μ
0
ε
0
)
−
1
/
2
{\displaystyle c=(\mu _{0}\varepsilon _{0})^{-1/2}}
, the equations above have the form of the standard wave equations
1
c
2
∂
2
E
∂
t
2
−
∇
2
E
=
0
,
1
c
2
∂
2
B
∂
t
2
−
∇
2
B
=
0.
{\displaystyle {\begin{aligned}{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} =0,\\{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}-\nabla ^{2}\mathbf {B} =0.\end{aligned}}}
Already during Maxwell's lifetime, it was found that the known values for
ε
0
{\displaystyle \varepsilon _{0}}
and
μ
0
{\displaystyle \mu _{0}}
give
c
≈
2.998
×
10
8
m/s
{\displaystyle c\approx 2.998\times 10^{8}~{\text{m/s}}}
, then already known to be the speed of light in free space. This led him to propose that light and radio waves were propagating electromagnetic waves, since amply confirmed. In the old SI system of units, the values of
μ
0
=
4
π
×
10
−
7
{\displaystyle \mu _{0}=4\pi \times 10^{-7}}
and
c
=
299
792
458
m/s
{\displaystyle c=299\,792\,458~{\text{m/s}}}
are defined constants, (which means that by definition
ε
0
=
8.854
187
8...
×
10
−
12
F/m
{\displaystyle \varepsilon _{0}=8.854\,187\,8...\times 10^{-12}~{\text{F/m}}}
) that define the ampere and the metre. In the new SI system, only c keeps its defined value, and the electron charge gets a defined value.
In materials with relative permittivity, εr, and relative permeability, μr, the phase velocity of light becomes
v
p
=
1
μ
0
μ
r
ε
0
ε
r
,
{\displaystyle v_{\text{p}}={\frac {1}{\sqrt {\mu _{0}\mu _{\text{r}}\varepsilon _{0}\varepsilon _{\text{r}}}}},}
which is usually less than c.
In addition, E and B are perpendicular to each other and to the direction of wave propagation, and are in phase with each other. A sinusoidal plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law. In turn, that electric field creates a changing magnetic field through Maxwell's modification of Ampère's circuital law. This perpetual cycle allows these waves, now known as electromagnetic radiation, to move through space at velocity c.
== Macroscopic formulation ==
The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents present. This is sometimes called the "general" form, but the macroscopic version below is equally general, the difference being one of bookkeeping.
The microscopic version is sometimes called "Maxwell's equations in vacuum": this refers to the fact that the material medium is not built into the structure of the equations, but appears only in the charge and current terms. The microscopic version was introduced by Lorentz, who tried to use it to derive the macroscopic properties of bulk matter from its microscopic constituents.: 5
"Maxwell's macroscopic equations", also known as Maxwell's equations in matter, are more similar to those that Maxwell introduced himself.
In the macroscopic equations, the influence of bound charge Qb and bound current Ib is incorporated into the displacement field D and the magnetizing field H, while the equations depend only on the free charges Qf and free currents If. This reflects a splitting of the total electric charge Q and current I (and their densities ρ and J) into free and bound parts:
Q
=
Q
f
+
Q
b
=
∭
Ω
(
ρ
f
+
ρ
b
)
d
V
=
∭
Ω
ρ
d
V
,
I
=
I
f
+
I
b
=
∬
Σ
(
J
f
+
J
b
)
⋅
d
S
=
∬
Σ
J
⋅
d
S
.
{\displaystyle {\begin{aligned}Q&=Q_{\text{f}}+Q_{\text{b}}=\iiint _{\Omega }\left(\rho _{\text{f}}+\rho _{\text{b}}\right)\,\mathrm {d} V=\iiint _{\Omega }\rho \,\mathrm {d} V,\\I&=I_{\text{f}}+I_{\text{b}}=\iint _{\Sigma }\left(\mathbf {J} _{\text{f}}+\mathbf {J} _{\text{b}}\right)\cdot \mathrm {d} \mathbf {S} =\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} .\end{aligned}}}
The cost of this splitting is that the additional fields D and H need to be determined through phenomenological constituent equations relating these fields to the electric field E and the magnetic field B, together with the bound charge and current.
See below for a detailed description of the differences between the microscopic equations, dealing with total charge and current including material contributions, useful in air/vacuum;
and the macroscopic equations, dealing with free charge and current, practical to use within materials.
=== Bound charge and current ===
When an electric field is applied to a dielectric material its molecules respond by forming microscopic electric dipoles – their atomic nuclei move a tiny distance in the direction of the field, while their electrons move a tiny distance in the opposite direction. This produces a macroscopic bound charge in the material even though all of the charges involved are bound to individual molecules. For example, if every molecule responds the same, similar to that shown in the figure, these tiny movements of charge combine to produce a layer of positive bound charge on one side of the material and a layer of negative charge on the other side. The bound charge is most conveniently described in terms of the polarization P of the material, its dipole moment per unit volume. If P is uniform, a macroscopic separation of charge is produced only at the surfaces where P enters and leaves the material. For non-uniform P, a charge is also produced in the bulk.
Somewhat similarly, in all materials the constituent atoms exhibit magnetic moments that are intrinsically linked to the angular momentum of the components of the atoms, most notably their electrons. The connection to angular momentum suggests the picture of an assembly of microscopic current loops. Outside the material, an assembly of such microscopic current loops is not different from a macroscopic current circulating around the material's surface, despite the fact that no individual charge is traveling a large distance. These bound currents can be described using the magnetization M.
The very complicated and granular bound charges and bound currents, therefore, can be represented on the macroscopic scale in terms of P and M, which average these charges and currents on a sufficiently large scale so as not to see the granularity of individual atoms, but also sufficiently small that they vary with location in the material. As such, Maxwell's macroscopic equations ignore many details on a fine scale that can be unimportant to understanding matters on a gross scale by calculating fields that are averaged over some suitable volume.
=== Auxiliary fields, polarization and magnetization ===
The definitions of the auxiliary fields are:
D
(
r
,
t
)
=
ε
0
E
(
r
,
t
)
+
P
(
r
,
t
)
,
H
(
r
,
t
)
=
1
μ
0
B
(
r
,
t
)
−
M
(
r
,
t
)
,
{\displaystyle {\begin{aligned}\mathbf {D} (\mathbf {r} ,t)&=\varepsilon _{0}\mathbf {E} (\mathbf {r} ,t)+\mathbf {P} (\mathbf {r} ,t),\\\mathbf {H} (\mathbf {r} ,t)&={\frac {1}{\mu _{0}}}\mathbf {B} (\mathbf {r} ,t)-\mathbf {M} (\mathbf {r} ,t),\end{aligned}}}
where P is the polarization field and M is the magnetization field, which are defined in terms of microscopic bound charges and bound currents respectively. The macroscopic bound charge density ρb and bound current density Jb in terms of polarization P and magnetization M are then defined as
ρ
b
=
−
∇
⋅
P
,
J
b
=
∇
×
M
+
∂
P
∂
t
.
{\displaystyle {\begin{aligned}\rho _{\text{b}}&=-\nabla \cdot \mathbf {P} ,\\\mathbf {J} _{\text{b}}&=\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}.\end{aligned}}}
If we define the total, bound, and free charge and current density by
ρ
=
ρ
b
+
ρ
f
,
J
=
J
b
+
J
f
,
{\displaystyle {\begin{aligned}\rho &=\rho _{\text{b}}+\rho _{\text{f}},\\\mathbf {J} &=\mathbf {J} _{\text{b}}+\mathbf {J} _{\text{f}},\end{aligned}}}
and use the defining relations above to eliminate D, and H, the "macroscopic" Maxwell's equations reproduce the "microscopic" equations.
=== Constitutive relations ===
In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between displacement field D and the electric field E, as well as the magnetizing field H and the magnetic field B. Equivalently, we have to specify the dependence of the polarization P (hence the bound charge) and the magnetization M (hence the bound current) on the applied electric and magnetic field. The equations specifying this response are called constitutive relations. For real-world materials, the constitutive relations are rarely simple, except approximately, and usually determined by experiment. See the main article on constitutive relations for a fuller description.: 44–45
For materials without polarization and magnetization, the constitutive relations are (by definition): 2
D
=
ε
0
E
,
H
=
1
μ
0
B
,
{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} ,\quad \mathbf {H} ={\frac {1}{\mu _{0}}}\mathbf {B} ,}
where ε0 is the permittivity of free space and μ0 the permeability of free space. Since there is no bound charge, the total and the free charge and current are equal.
An alternative viewpoint on the microscopic equations is that they are the macroscopic equations together with the statement that vacuum behaves like a perfect linear "material" without additional polarization and magnetization.
More generally, for linear materials the constitutive relations are: 44–45
D
=
ε
E
,
H
=
1
μ
B
,
{\displaystyle \mathbf {D} =\varepsilon \mathbf {E} ,\quad \mathbf {H} ={\frac {1}{\mu }}\mathbf {B} ,}
where ε is the permittivity and μ the permeability of the material. For the displacement field D the linear approximation is usually excellent because for all but the most extreme electric fields or temperatures obtainable in the laboratory (high power pulsed lasers) the interatomic electric fields of materials of the order of 1011 V/m are much higher than the external field. For the magnetizing field
H
{\displaystyle \mathbf {H} }
, however, the linear approximation can break down in common materials like iron leading to phenomena like hysteresis. Even the linear case can have various complications, however.
For homogeneous materials, ε and μ are constant throughout the material, while for inhomogeneous materials they depend on location within the material (and perhaps time).: 463
For isotropic materials, ε and μ are scalars, while for anisotropic materials (e.g. due to crystal structure) they are tensors.: 421 : 463
Materials are generally dispersive, so ε and μ depend on the frequency of any incident EM waves.: 625 : 397
Even more generally, in the case of non-linear materials (see for example nonlinear optics), D and P are not necessarily proportional to E, similarly H or M is not necessarily proportional to B. In general D and H depend on both E and B, on location and time, and possibly other physical quantities.
In applications one also has to describe how the free currents and charge density behave in terms of E and B possibly coupled to other physical quantities like pressure, and the mass, number density, and velocity of charge-carrying particles. E.g., the original equations given by Maxwell (see History of Maxwell's equations) included Ohm's law in the form
J
f
=
σ
E
.
{\displaystyle \mathbf {J} _{\text{f}}=\sigma \mathbf {E} .}
== Alternative formulations ==
Following are some of the several other mathematical formalisms of Maxwell's equations, with the columns separating the two homogeneous Maxwell equations from the two inhomogeneous ones. Each formulation has versions directly in terms of the electric and magnetic fields, and indirectly in terms of the electrical potential φ and the vector potential A. Potentials were introduced as a convenient way to solve the homogeneous equations, but it was thought that all observable physics was contained in the electric and magnetic fields (or relativistically, the Faraday tensor). The potentials play a central role in quantum mechanics, however, and act quantum mechanically with observable consequences even when the electric and magnetic fields vanish (Aharonov–Bohm effect).
Each table describes one formalism. See the main article for details of each formulation.
The direct spacetime formulations make manifest that the Maxwell equations are relativistically invariant, where space and time are treated on equal footing. Because of this symmetry, the electric and magnetic fields are treated on equal footing and are recognized as components of the Faraday tensor. This reduces the four Maxwell equations to two, which simplifies the equations, although we can no longer use the familiar vector formulation. Maxwell equations in formulation that do not treat space and time manifestly on the same footing have Lorentz invariance as a hidden symmetry. This was a major source of inspiration for the development of relativity theory. Indeed, even the formulation that treats space and time separately is not a non-relativistic approximation and describes the same physics by simply renaming variables. For this reason the relativistic invariant equations are usually called the Maxwell equations as well.
Each table below describes one formalism.
In the tensor calculus formulation, the electromagnetic tensor Fαβ is an antisymmetric covariant order 2 tensor; the four-potential, Aα, is a covariant vector; the current, Jα, is a vector; the square brackets, [ ], denote antisymmetrization of indices; ∂α is the partial derivative with respect to the coordinate, xα. In Minkowski space coordinates are chosen with respect to an inertial frame; (xα) = (ct, x, y, z), so that the metric tensor used to raise and lower indices is ηαβ = diag(1, −1, −1, −1). The d'Alembert operator on Minkowski space is ◻ = ∂α∂α as in the vector formulation. In general spacetimes, the coordinate system xα is arbitrary, the covariant derivative ∇α, the Ricci tensor, Rαβ and raising and lowering of indices are defined by the Lorentzian metric, gαβ and the d'Alembert operator is defined as ◻ = ∇α∇α. The topological restriction is that the second real cohomology group of the space vanishes (see the differential form formulation for an explanation). This is violated for Minkowski space with a line removed, which can model a (flat) spacetime with a point-like monopole on the complement of the line.
In the differential form formulation on arbitrary space times, F = 1/2Fαβdxα ∧ dxβ is the electromagnetic tensor considered as a 2-form, A = Aαdxα is the potential 1-form,
J
=
−
J
α
⋆
d
x
α
{\displaystyle J=-J_{\alpha }{\star }\mathrm {d} x^{\alpha }}
is the current 3-form, d is the exterior derivative, and
⋆
{\displaystyle {\star }}
is the Hodge star on forms defined (up to its orientation, i.e. its sign) by the Lorentzian metric of spacetime. In the special case of 2-forms such as F, the Hodge star
⋆
{\displaystyle {\star }}
depends on the metric tensor only for its local scale. This means that, as formulated, the differential form field equations are conformally invariant, but the Lorenz gauge condition breaks conformal invariance. The operator
◻
=
(
−
⋆
d
⋆
d
−
d
⋆
d
⋆
)
{\displaystyle \Box =(-{\star }\mathrm {d} {\star }\mathrm {d} -\mathrm {d} {\star }\mathrm {d} {\star })}
is the d'Alembert–Laplace–Beltrami operator on 1-forms on an arbitrary Lorentzian spacetime. The topological condition is again that the second real cohomology group is 'trivial' (meaning that its form follows from a definition). By the isomorphism with the second de Rham cohomology this condition means that every closed 2-form is exact.
Other formalisms include the geometric algebra formulation and a matrix representation of Maxwell's equations. Historically, a quaternionic formulation was used.
== Solutions ==
Maxwell's equations are partial differential equations that relate the electric and magnetic fields to each other and to the electric charges and currents. Often, the charges and currents are themselves dependent on the electric and magnetic fields via the Lorentz force equation and the constitutive relations. These all form a set of coupled partial differential equations which are often very difficult to solve: the solutions encompass all the diverse phenomena of classical electromagnetism. Some general remarks follow.
As for any differential equation, boundary conditions and initial conditions are necessary for a unique solution. For example, even with no charges and no currents anywhere in spacetime, there are the obvious solutions for which E and B are zero or constant, but there are also non-trivial solutions corresponding to electromagnetic waves. In some cases, Maxwell's equations are solved over the whole of space, and boundary conditions are given as asymptotic limits at infinity. In other cases, Maxwell's equations are solved in a finite region of space, with appropriate conditions on the boundary of that region, for example an artificial absorbing boundary representing the rest of the universe, or periodic boundary conditions, or walls that isolate a small region from the outside world (as with a waveguide or cavity resonator).
Jefimenko's equations (or the closely related Liénard–Wiechert potentials) are the explicit solution to Maxwell's equations for the electric and magnetic fields created by any given distribution of charges and currents. It assumes specific initial conditions to obtain the so-called "retarded solution", where the only fields present are the ones created by the charges. However, Jefimenko's equations are unhelpful in situations when the charges and currents are themselves affected by the fields they create.
Numerical methods for differential equations can be used to compute approximate solutions of Maxwell's equations when exact solutions are impossible. These include the finite element method and finite-difference time-domain method. For more details, see Computational electromagnetics.
== Overdetermination of Maxwell's equations ==
Maxwell's equations seem overdetermined, in that they involve six unknowns (the three components of E and B) but eight equations (one for each of the two Gauss's laws, three vector components each for Faraday's and Ampère's circuital laws). (The currents and charges are not unknowns, being freely specifiable subject to charge conservation.) This is related to a certain limited kind of redundancy in Maxwell's equations: It can be proven that any system satisfying Faraday's law and Ampère's circuital law automatically also satisfies the two Gauss's laws, as long as the system's initial condition does, and assuming conservation of charge and the nonexistence of magnetic monopoles.
This explanation was first introduced by Julius Adams Stratton in 1941.
Although it is possible to simply ignore the two Gauss's laws in a numerical algorithm (apart from the initial conditions), the imperfect precision of the calculations can lead to ever-increasing violations of those laws. By introducing dummy variables characterizing these violations, the four equations become not overdetermined after all. The resulting formulation can lead to more accurate algorithms that take all four laws into account.
Both identities
∇
⋅
∇
×
B
≡
0
,
∇
⋅
∇
×
E
≡
0
{\displaystyle \nabla \cdot \nabla \times \mathbf {B} \equiv 0,\nabla \cdot \nabla \times \mathbf {E} \equiv 0}
, which reduce eight equations to six independent ones, are the true reason of overdetermination.
Equivalently, the overdetermination can be viewed as implying conservation of electric and magnetic charge, as they are required in the derivation described above but implied by the two Gauss's laws.
For linear algebraic equations, one can make 'nice' rules to rewrite the equations and unknowns. The equations can be linearly dependent. But in differential equations, and especially partial differential equations (PDEs), one needs appropriate boundary conditions, which depend in not so obvious ways on the equations. Even more, if one rewrites them in terms of vector and scalar potential, then the equations are underdetermined because of gauge fixing.
== Maxwell's equations as the classical limit of QED ==
Maxwell's equations and the Lorentz force law (along with the rest of classical electromagnetism) are extraordinarily successful at explaining and predicting a variety of phenomena. However, they do not account for quantum effects, and so their domain of applicability is limited. Maxwell's equations are thought of as the classical limit of quantum electrodynamics (QED).
Some observed electromagnetic phenomena cannot be explained with Maxwell's equations if the source of the electromagnetic fields are the classical distributions of charge and current. These include photon–photon scattering and many other phenomena related to photons or virtual photons, "nonclassical light" and quantum entanglement of electromagnetic fields (see Quantum optics). E.g. quantum cryptography cannot be described by Maxwell theory, not even approximately. The approximate nature of Maxwell's equations becomes more and more apparent when going into the extremely strong field regime (see Euler–Heisenberg Lagrangian) or to extremely small distances.
Finally, Maxwell's equations cannot explain any phenomenon involving individual photons interacting with quantum matter, such as the photoelectric effect, Planck's law, the Duane–Hunt law, and single-photon light detectors. However, many such phenomena may be explained using a halfway theory of quantum matter coupled to a classical electromagnetic field, either as external field or with the expected value of the charge current and density on the right hand side of Maxwell's equations. This is known as semiclassical theory or self-field QED and was initially discovered by de Broglie and Schrödinger and later fully developed by E.T. Jaynes and A.O. Barut.
== Variations ==
Popular variations on the Maxwell equations as a classical theory of electromagnetic fields are relatively scarce because the standard equations have stood the test of time remarkably well.
=== Magnetic monopoles ===
Maxwell's equations posit that there is electric charge, but no magnetic charge (also called magnetic monopoles), in the universe. Indeed, magnetic charge has never been observed, despite extensive searches, and may not exist. If they did exist, both Gauss's law for magnetism and Faraday's law would need to be modified, and the resulting four equations would be fully symmetric under the interchange of electric and magnetic fields.: 273–275
== See also ==
== Explanatory notes ==
== References ==
== Further reading ==
Imaeda, K. (1995), "Biquaternionic Formulation of Maxwell's Equations and their Solutions", in Ablamowicz, Rafał; Lounesto, Pertti (eds.), Clifford Algebras and Spinor Structures, Springer, pp. 265–280, doi:10.1007/978-94-015-8422-7_16, ISBN 978-90-481-4525-6
=== Historical publications ===
On Faraday's Lines of Force – 1855/56. Maxwell's first paper (Part 1 & 2) – Compiled by Blaze Labs Research (PDF).
On Physical Lines of Force – 1861. Maxwell's 1861 paper describing magnetic lines of force – Predecessor to 1873 Treatise.
James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459–512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
A Dynamical Theory Of The Electromagnetic Field – 1865. Maxwell's 1865 paper describing his 20 equations, link from Google Books.
J. Clerk Maxwell (1873), "A Treatise on Electricity and Magnetism":
Maxwell, J. C., "A Treatise on Electricity And Magnetism" – Volume 1 – 1873 – Posner Memorial Collection – Carnegie Mellon University.
Maxwell, J. C., "A Treatise on Electricity And Magnetism" – Volume 2 – 1873 – Posner Memorial Collection – Carnegie Mellon University.
Developments before the theory of relativity
Larmor Joseph (1897). "On a dynamical theory of the electric and luminiferous medium. Part 3, Relations with material media" . Phil. Trans. R. Soc. 190: 205–300.
Lorentz Hendrik (1899). "Simplified theory of electrical and optical phenomena in moving systems" . Proc. Acad. Science Amsterdam. I: 427–443.
Lorentz Hendrik (1904). "Electromagnetic phenomena in a system moving with any velocity less than that of light" . Proc. Acad. Science Amsterdam. IV: 669–678.
Henri Poincaré (1900) "La théorie de Lorentz et le Principe de Réaction" (in French), Archives Néerlandaises, V, 253–278.
Henri Poincaré (1902) "La Science et l'Hypothèse" (in French).
Henri Poincaré (1905) "Sur la dynamique de l'électron" (in French), Comptes Rendus de l'Académie des Sciences, 140, 1504–1508.
Catt, Walton and Davidson. "The History of Displacement Current" Archived 2008-05-06 at the Wayback Machine. Wireless World, March 1979.
== External links ==
"Maxwell equations", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
maxwells-equations.com — An intuitive tutorial of Maxwell's equations.
The Feynman Lectures on Physics Vol. II Ch. 18: The Maxwell Equations
Wikiversity Page on Maxwell's Equations
=== Modern treatments ===
Electromagnetism (ch. 11), B. Crowell, Fullerton College
Lecture series: Relativity and electromagnetism, R. Fitzpatrick, University of Texas at Austin
Electromagnetic waves from Maxwell's equations on Project PHYSNET.
MIT Video Lecture Series (36 × 50 minute lectures) (in .mp4 format) – Electricity and Magnetism Taught by Professor Walter Lewin.
=== Other ===
Silagadze, Z. K. (2002). "Feynman's derivation of Maxwell equations and extra dimensions". Annales de la Fondation Louis de Broglie. 27: 241–256. arXiv:hep-ph/0106235.
Nature Milestones: Photons – Milestone 2 (1861) Maxwell's equations | Wikipedia/Maxwell's_equation |
In physics, the Callan–Symanzik equation is a differential equation describing the evolution of the n-point correlation functions under variation of the energy scale at which the theory is defined and involves the beta function of the theory and the anomalous dimensions.
The Callan–Symanzik equation was discovered independently by Curtis Callan and Kurt Symanzik in 1970. Later it was used to understand asymptotic freedom.
This equation arises in the framework of renormalization group. It is possible to treat the equation using perturbation theory.
== Example ==
As an example, for a quantum field theory with one massless scalar field and one self-coupling term, denote the bare field strength by
ϕ
0
{\displaystyle \phi _{0}}
and the bare coupling constant by
g
0
{\displaystyle g_{0}}
. In the process of renormalisation, a mass scale M must be chosen. Depending on M, the field strength is rescaled by a constant:
ϕ
=
Z
ϕ
0
{\displaystyle \phi =Z\phi _{0}}
, and as a result the bare coupling constant
g
0
{\displaystyle g_{0}}
is correspondingly shifted to the renormalised coupling constant g.
Of physical importance are the renormalised n-point functions, computed from connected Feynman diagrams, schematically of the form
G
(
n
)
(
x
1
,
x
2
,
…
,
x
n
;
M
,
g
)
=
⟨
ϕ
(
x
1
)
ϕ
(
x
2
)
⋯
ϕ
(
x
n
)
⟩
{\displaystyle G^{(n)}(x_{1},x_{2},\ldots ,x_{n};M,g)=\langle \phi (x_{1})\phi (x_{2})\cdots \phi (x_{n})\rangle }
For a given choice of renormalisation scheme, the computation of this quantity depends on the choice of M, which affects the shift in g and the rescaling of
ϕ
{\displaystyle \phi }
. If the choice of
M
{\displaystyle M}
is slightly altered by
δ
M
{\displaystyle \delta M}
, then the following shifts will occur:
M
→
M
+
δ
M
{\displaystyle M\to M+\delta M}
g
→
g
+
δ
g
{\displaystyle g\to g+\delta g}
ϕ
=
Z
ϕ
0
→
Z
′
ϕ
0
=
(
1
+
δ
η
)
ϕ
{\displaystyle \phi =Z\phi _{0}\to Z'\phi _{0}=(1+\delta \eta )\phi }
G
(
n
)
→
(
1
+
n
δ
η
)
G
(
n
)
{\displaystyle G^{(n)}\to (1+n\,\delta \eta )G^{(n)}}
The Callan–Symanzik equation relates these shifts:
n
δ
η
G
(
n
)
=
∂
G
(
n
)
∂
M
δ
M
+
∂
G
(
n
)
∂
g
δ
g
{\displaystyle n\,\delta \eta G^{(n)}={\frac {\partial G^{(n)}}{\partial M}}\delta M+{\frac {\partial G^{(n)}}{\partial g}}\delta g}
After the following definitions
β
=
M
δ
M
δ
g
{\displaystyle \beta ={\frac {M}{\delta M}}\delta g}
γ
=
−
M
δ
M
δ
η
{\displaystyle \gamma =-{\frac {M}{\delta M}}\delta \eta }
the Callan–Symanzik equation can be put in the conventional form:
[
M
∂
∂
M
+
β
(
g
)
∂
∂
g
+
n
γ
]
G
(
n
)
(
x
1
,
x
2
,
…
,
x
n
;
M
,
g
)
=
0
{\displaystyle \left[M{\frac {\partial }{\partial M}}+\beta (g){\frac {\partial }{\partial g}}+n\gamma \right]G^{(n)}(x_{1},x_{2},\ldots ,x_{n};M,g)=0}
β
(
g
)
{\displaystyle \beta (g)}
being the beta function.
In quantum electrodynamics this equation takes the form
[
M
∂
∂
M
+
β
(
e
)
∂
∂
e
+
n
γ
2
+
m
γ
3
]
G
(
n
,
m
)
(
x
1
,
x
2
,
…
,
x
n
;
y
1
,
y
2
,
…
,
y
m
;
M
,
e
)
=
0
{\displaystyle \left[M{\frac {\partial }{\partial M}}+\beta (e){\frac {\partial }{\partial e}}+n\gamma _{2}+m\gamma _{3}\right]G^{(n,m)}(x_{1},x_{2},\ldots ,x_{n};y_{1},y_{2},\ldots ,y_{m};M,e)=0}
where n and m are the numbers of electron and photon fields, respectively, for which the correlation function
G
(
n
,
m
)
{\displaystyle G^{(n,m)}}
is defined. The renormalised coupling constant is now the renormalised elementary charge e. The electron field and the photon field rescale differently under renormalisation, and thus lead to two separate functions,
γ
2
{\displaystyle \gamma _{2}}
and
γ
3
{\displaystyle \gamma _{3}}
, respectively.
== See also ==
Renormalization group
Beta function
== Notes ==
== References ==
Jean Zinn-Justin, Quantum Field Theory and Critical Phenomena , Oxford University Press, 2003, ISBN 0-19-850923-5
John Clements Collins, Renormalization, Cambridge University Press, 1986, ISBN 0-521-31177-2
Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, 1995. 2nd edition, pbk. Westview Press. 2015. | Wikipedia/Callan–Symanzik_equation |
In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or a symmetric space. The model may or may not be quantized. An example of the non-quantized version is the Skyrme model; it cannot be quantized due to non-linearities of power greater than 4. In general, sigma models admit (classical) topological soliton solutions, for example, the skyrmion for the Skyrme model. When the sigma field is coupled to a gauge field, the resulting model is described by Ginzburg–Landau theory. This article is primarily devoted to the classical field theory of the sigma model; the corresponding quantized theory is presented in the article titled "non-linear sigma model".
== Overview ==
The name has roots in particle physics, where a sigma model describes the interactions of pions. Unfortunately, the "sigma meson" is not described by the sigma-model, but only a component of it.
The sigma model was introduced by Gell-Mann & Lévy (1960, section 5); the name σ-model comes from a field in their model corresponding to a spinless meson called σ, a scalar meson introduced earlier by Julian Schwinger. The model served as the dominant prototype of spontaneous symmetry breaking of O(4) down to O(3): the three axial generators broken are the simplest manifestation of chiral symmetry breaking, the surviving unbroken O(3) representing isospin.
In conventional particle physics settings, the field is generally taken to be SU(N), or the vector subspace of quotient
(
S
U
(
N
)
L
×
S
U
(
N
)
R
)
/
S
U
(
N
)
{\displaystyle (SU(N)_{L}\times SU(N)_{R})/SU(N)}
of the product of left and right chiral fields. In condensed matter theories, the field is taken to be O(N). For the rotation group O(3), the sigma model describes the isotropic ferromagnet; more generally, the O(N) model shows up in the quantum Hall effect, superfluid Helium-3 and spin chains.
In supergravity models, the field is taken to be a symmetric space. Since symmetric spaces are defined in terms of their involution, their tangent space naturally splits into even and odd parity subspaces. This splitting helps propel the dimensional reduction of Kaluza–Klein theories.
In its most basic form, the sigma model can be taken as being purely the kinetic energy of a point particle; as a field, this is just the Dirichlet energy in Euclidean space.
In two spatial dimensions, the O(3) model is completely integrable.
== Definition ==
The Lagrangian density of the sigma model can be written in a variety of different ways, each suitable to a particular type of application. The simplest, most generic definition writes the Lagrangian as the metric trace of the pullback of the metric tensor on a Riemannian manifold. For
ϕ
:
M
→
Φ
{\displaystyle \phi :M\to \Phi }
a field over a spacetime
M
{\displaystyle M}
, this may be written as
L
=
1
2
∑
i
=
1
n
∑
j
=
1
n
g
i
j
(
ϕ
)
∂
μ
ϕ
i
∂
μ
ϕ
j
{\displaystyle {\mathcal {L}}={\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}g_{ij}(\phi )\;\partial ^{\mu }\phi _{i}\partial _{\mu }\phi _{j}}
where the
g
i
j
(
ϕ
)
{\displaystyle g_{ij}(\phi )}
is the metric tensor on the field space
ϕ
∈
Φ
{\displaystyle \phi \in \Phi }
, and the
∂
μ
{\displaystyle \partial _{\mu }}
are the derivatives on the underlying spacetime manifold.
This expression can be unpacked a bit. The field space
Φ
{\displaystyle \Phi }
can be chosen to be any Riemannian manifold. Historically, this is the "sigma" of the sigma model; the historically-appropriate symbol
σ
{\displaystyle \sigma }
is avoided here to prevent clashes with many other common usages of
σ
{\displaystyle \sigma }
in geometry. Riemannian manifolds always come with a metric tensor
g
{\displaystyle g}
. Given an atlas of charts on
Φ
{\displaystyle \Phi }
, the field space can always be locally trivialized, in that given
U
⊂
Φ
{\displaystyle U\subset \Phi }
in the atlas, one may write a map
U
→
R
n
{\displaystyle U\to \mathbb {R} ^{n}}
giving explicit local coordinates
ϕ
=
(
ϕ
1
,
⋯
,
ϕ
n
)
{\displaystyle \phi =(\phi ^{1},\cdots ,\phi ^{n})}
on that patch. The metric tensor on that patch is a matrix having components
g
i
j
(
ϕ
)
.
{\displaystyle g_{ij}(\phi ).}
The base manifold
M
{\displaystyle M}
must be a differentiable manifold; by convention, it is either Minkowski space in particle physics applications, flat two-dimensional Euclidean space for condensed matter applications, or a Riemann surface, the worldsheet in string theory. The
∂
μ
ϕ
=
∂
ϕ
/
∂
x
μ
{\displaystyle \partial _{\mu }\phi =\partial \phi /\partial x^{\mu }}
is just the plain-old covariant derivative on the base spacetime manifold
M
.
{\displaystyle M.}
When
M
{\displaystyle M}
is flat,
∂
μ
ϕ
=
∇
ϕ
{\displaystyle \partial _{\mu }\phi =\nabla \phi }
is just the ordinary gradient of a scalar function (as
ϕ
{\displaystyle \phi }
is a scalar field, from the point of view of
M
{\displaystyle M}
itself.) In more precise language,
∂
μ
ϕ
{\displaystyle \partial _{\mu }\phi }
is a section of the jet bundle of
M
×
Φ
{\displaystyle M\times \Phi }
.
=== Example: O(n) non-linear sigma model ===
Taking
g
i
j
=
δ
i
j
{\displaystyle g_{ij}=\delta _{ij}}
the Kronecker delta, i.e. the scalar dot product in Euclidean space, one gets the
O
(
n
)
{\displaystyle O(n)}
non-linear sigma model. That is, write
ϕ
=
u
^
{\displaystyle \phi ={\hat {u}}}
to be the unit vector in
R
n
{\displaystyle \mathbb {R} ^{n}}
, so that
u
^
⋅
u
^
=
1
{\displaystyle {\hat {u}}\cdot {\hat {u}}=1}
, with
⋅
{\displaystyle \cdot }
the ordinary Euclidean dot product. Then
u
^
∈
S
n
−
1
{\displaystyle {\hat {u}}\in S^{n-1}}
the
(
n
−
1
)
{\displaystyle (n-1)}
-sphere, the isometries of which are the rotation group
O
(
n
)
{\displaystyle O(n)}
. The Lagrangian can then be written as
L
=
1
2
∇
μ
u
^
⋅
∇
μ
u
^
{\displaystyle {\mathcal {L}}={\frac {1}{2}}\nabla _{\mu }{\hat {u}}\cdot \nabla _{\mu }{\hat {u}}}
For
n
=
3
{\displaystyle n=3}
, this is the continuum limit of the isotropic ferromagnet on a lattice, i.e. of the classical Heisenberg model. For
n
=
2
{\displaystyle n=2}
, this is the continuum limit of the classical XY model. See also the n-vector model and the Potts model for reviews of the lattice model equivalents. The continuum limit is taken by writing
δ
h
[
u
^
]
(
i
,
j
)
=
u
^
i
−
u
^
j
h
{\displaystyle \delta _{h}[{\hat {u}}](i,j)={\frac {{\hat {u}}_{i}-{\hat {u}}_{j}}{h}}}
as the finite difference on neighboring lattice locations
i
,
j
.
{\displaystyle i,j.}
Then
δ
h
[
u
^
]
→
∂
μ
u
^
{\displaystyle \delta _{h}[{\hat {u}}]\to \partial _{\mu }{\hat {u}}}
in the limit
h
→
0
{\displaystyle h\to 0}
, and
u
^
i
⋅
u
^
j
→
∂
μ
u
^
⋅
∂
μ
u
^
{\displaystyle {\hat {u}}_{i}\cdot {\hat {u}}_{j}\to \partial _{\mu }{\hat {u}}\cdot \partial _{\mu }{\hat {u}}}
after dropping the constant terms
u
^
i
⋅
u
^
i
=
1
{\displaystyle {\hat {u}}_{i}\cdot {\hat {u}}_{i}=1}
(the "bulk magnetization").
== In geometric notation ==
The sigma model can also be written in a more fully geometric notation, as a fiber bundle with fibers
Φ
{\displaystyle \Phi }
over a differentiable manifold
M
{\displaystyle M}
. Given a section
ϕ
:
M
→
Φ
{\displaystyle \phi :M\to \Phi }
, fix a point
x
∈
M
.
{\displaystyle x\in M.}
The pushforward at
x
{\displaystyle x}
is a map of tangent bundles
d
x
ϕ
:
T
x
M
→
T
ϕ
(
x
)
Φ
{\displaystyle \mathrm {d} _{x}\phi :T_{x}M\to T_{\phi (x)}\Phi \quad }
taking
∂
μ
↦
∂
ϕ
i
∂
x
μ
∂
i
{\displaystyle \quad \partial _{\mu }\mapsto {\frac {\partial \phi ^{i}}{\partial x^{\mu }}}\partial _{i}}
where
∂
μ
=
∂
/
∂
x
μ
{\displaystyle \partial _{\mu }=\partial /\partial x^{\mu }}
is taken to be a local orthonormal vector space basis on
T
M
{\displaystyle TM}
and
∂
i
=
∂
/
∂
q
i
{\displaystyle \partial _{i}=\partial /\partial q^{i}}
the vector space basis on
T
Φ
{\displaystyle T\Phi }
. The
d
ϕ
{\displaystyle \mathrm {d} \phi }
is a differential form. The sigma model action is then just the conventional inner product on vector-valued k-forms
S
=
1
2
∫
M
d
ϕ
∧
⋆
d
ϕ
{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}\mathrm {d} \phi \wedge {\star \mathrm {d} \phi }}
where the
∧
{\displaystyle \wedge }
is the wedge product, and the
⋆
{\displaystyle \star }
is the Hodge star. This is an inner product in two different ways. In the first way, given any two differentiable forms
α
,
β
{\displaystyle \alpha ,\beta }
in
M
{\displaystyle M}
, the Hodge dual defines an invariant inner product on the space of differential forms, commonly written as
⟨
⟨
α
,
β
⟩
⟩
=
∫
M
α
∧
⋆
β
{\displaystyle \langle \!\langle \alpha ,\beta \rangle \!\rangle \ =\ \int _{M}\alpha \wedge \star \beta }
The above is an inner product on the space of square-integrable forms, conventionally taken to be the Sobolev space
L
2
.
{\displaystyle L^{2}.}
In this way, one may write
S
=
1
2
⟨
⟨
d
ϕ
,
d
ϕ
⟩
⟩
{\displaystyle {\mathcal {S}}={\frac {1}{2}}\langle \!\langle \mathrm {d} \phi ,\mathrm {d} \phi \rangle \!\rangle }
.
This makes it explicit and plainly evident that the sigma model is just the kinetic energy of a point particle. From the point of view of the manifold
M
{\displaystyle M}
, the field
ϕ
{\displaystyle \phi }
is a scalar, and so
d
ϕ
{\displaystyle \mathrm {d} \phi }
can be recognized just the ordinary gradient of a scalar function. The Hodge star is merely a fancy device for keeping track of the volume form when integrating on curved spacetime. In the case that
M
{\displaystyle M}
is flat, it can be completely ignored, and so the action is
S
=
1
2
∫
M
‖
∇
ϕ
‖
2
d
m
x
{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}\Vert \nabla \phi \Vert ^{2}d^{m}x}
,
which is the Dirichlet energy of
ϕ
{\displaystyle \phi }
. Classical extrema of the action (the solutions to the Lagrange equations) are then those field configurations that minimize the Dirichlet energy of
ϕ
{\displaystyle \phi }
. Another way to convert this expression into a more easily-recognizable form is to observe that, for a scalar function
f
:
M
→
R
{\displaystyle f:M\to \mathbb {R} }
one has
d
⋆
f
=
0
{\displaystyle \mathrm {d} {\star }f=0}
and so one may also write
S
=
1
2
⟨
⟨
ϕ
,
Δ
ϕ
⟩
⟩
{\displaystyle {\mathcal {S}}={\frac {1}{2}}\langle \!\langle \phi ,\Delta \phi \rangle \!\rangle }
where
Δ
{\displaystyle \Delta }
is the Laplace–Beltrami operator, i.e., the ordinary Laplacian when
M
{\displaystyle M}
is flat.
That there is another, second inner product in play simply requires not forgetting that
d
ϕ
{\displaystyle \mathrm {d} \phi }
is a vector from the point of view of
Φ
{\displaystyle \Phi }
itself. That is, given any two vectors
v
,
w
∈
T
Φ
{\displaystyle v,w\in T\Phi }
, the Riemannian metric
g
i
j
{\displaystyle g_{ij}}
defines an inner product
⟨
v
,
w
⟩
=
g
i
j
v
i
w
j
{\displaystyle \langle v,w\rangle =g_{ij}v^{i}w^{j}}
Since
d
ϕ
{\displaystyle \mathrm {d} \phi }
is vector-valued
d
ϕ
=
(
d
ϕ
1
,
⋯
,
d
ϕ
n
)
{\displaystyle \mathrm {d} \phi =(\mathrm {d} \phi ^{1},\cdots ,\mathrm {d} \phi ^{n})}
on local charts, one also takes the inner product there as well. More verbosely,
S
=
1
2
∫
M
g
i
j
(
ϕ
)
d
ϕ
i
∧
⋆
d
ϕ
j
{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}g_{ij}(\phi )\;\mathrm {d} \phi ^{i}\wedge {\star \mathrm {d} \phi ^{j}}}
.
The tension between these two inner products can be made even more explicit by noting that
B
μ
ν
(
ϕ
)
=
g
i
j
∂
μ
ϕ
i
∂
ν
ϕ
j
{\displaystyle B_{\mu \nu }(\phi )=g_{ij}\partial _{\mu }\phi ^{i}\partial _{\nu }\phi ^{j}}
is a bilinear form; it is a pullback of the Riemann metric
g
i
j
{\displaystyle g_{ij}}
. The individual
∂
μ
ϕ
i
{\displaystyle \partial _{\mu }\phi ^{i}}
can be taken as vielbeins. The Lagrangian density of the sigma model is then
L
=
1
2
g
μ
ν
B
μ
ν
{\displaystyle {\mathcal {L}}={\frac {1}{2}}g^{\mu \nu }B_{\mu \nu }}
for
g
μ
ν
{\displaystyle g_{\mu \nu }}
the metric on
M
.
{\displaystyle M.}
Given this gluing-together, the
d
ϕ
{\displaystyle \mathrm {d} \phi }
can be interpreted as a solder form; this is articulated more fully below.
== Motivations and basic interpretations ==
Several interpretational and foundational remarks can be made about the classical (non-quantized) sigma model. The first of these is that the classical sigma model can be interpreted as a model of non-interacting quantum mechanics. The second concerns the interpretation of energy.
=== Interpretation as quantum mechanics ===
This follows directly from the expression
S
=
1
2
⟨
⟨
ϕ
,
Δ
ϕ
⟩
⟩
{\displaystyle {\mathcal {S}}={\frac {1}{2}}\langle \!\langle \phi ,\Delta \phi \rangle \!\rangle }
given above. Taking
Φ
=
C
{\displaystyle \Phi =\mathbb {C} }
, the function
ϕ
:
M
→
C
{\displaystyle \phi :M\to \mathbb {C} }
can be interpreted as a wave function, and its Laplacian the kinetic energy of that wave function. The
⟨
⟨
⋅
,
⋅
⟩
⟩
{\displaystyle \langle \!\langle \cdot ,\cdot \rangle \!\rangle }
is just some geometric machinery reminding one to integrate over all space. The corresponding quantum mechanical notation is
ϕ
=
|
ψ
⟩
.
{\displaystyle \phi =|\psi \rangle .}
In flat space, the Laplacian is conventionally written as
Δ
=
∇
2
{\displaystyle \Delta =\nabla ^{2}}
. Assembling all these pieces together, the sigma model action is equivalent to
S
=
1
2
∫
M
⟨
ψ
|
∇
2
|
ψ
⟩
d
x
m
=
1
2
∫
M
ψ
†
(
x
)
∇
2
ψ
(
x
)
d
x
m
{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}\langle \psi |\nabla ^{2}|\psi \rangle dx^{m}={\frac {1}{2}}\int _{M}\psi ^{\dagger }(x)\nabla ^{2}\psi (x)dx^{m}}
which is just the grand-total kinetic energy of the wave-function
ψ
(
x
)
{\displaystyle \psi (x)}
, up to a factor of
ℏ
/
m
{\displaystyle \hbar /m}
. To conclude, the classical sigma model on
C
{\displaystyle \mathbb {C} }
can be interpreted as the quantum mechanics of a free, non-interacting quantum particle. Obviously, adding a term of
V
(
ϕ
)
{\displaystyle V(\phi )}
to the Lagrangian results in the quantum mechanics of a wave-function in a potential. Taking
Φ
=
C
n
{\displaystyle \Phi =\mathbb {C} ^{n}}
is not enough to describe the
n
{\displaystyle n}
-particle system, in that
n
{\displaystyle n}
particles require
n
{\displaystyle n}
distinct coordinates, which are not provided by the base manifold. This can be solved by taking
n
{\displaystyle n}
copies of the base manifold.
=== The solder form ===
It is very well-known that the geodesic structure of a Riemannian manifold is described by the Hamilton–Jacobi equations. In thumbnail form, the construction is as follows. Both
M
{\displaystyle M}
and
Φ
{\displaystyle \Phi }
are Riemannian manifolds; the below is written for
Φ
{\displaystyle \Phi }
, the same can be done for
M
{\displaystyle M}
. The cotangent bundle
T
∗
Φ
{\displaystyle T^{*}\Phi }
, supplied with coordinate charts, can always be locally trivialized, i.e.
T
∗
Φ
|
U
≅
U
×
R
n
{\displaystyle \left.T^{*}\Phi \right|_{U}\cong U\times \mathbb {R} ^{n}}
The trivialization supplies canonical coordinates
(
q
1
,
⋯
,
q
n
,
p
1
,
⋯
,
p
n
)
{\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})}
on the cotangent bundle. Given the metric tensor
g
i
j
{\displaystyle g_{ij}}
on
Φ
{\displaystyle \Phi }
, define the Hamiltonian function
H
(
q
,
p
)
=
1
2
g
i
j
(
q
)
p
i
p
j
{\displaystyle H(q,p)={\frac {1}{2}}g^{ij}(q)p_{i}p_{j}}
where, as always, one is careful to note that the inverse of the metric is used in this definition:
g
i
j
g
j
k
=
δ
k
i
.
{\displaystyle g^{ij}g_{jk}=\delta _{k}^{i}.}
Famously, the geodesic flow on
Φ
{\displaystyle \Phi }
is given by the Hamilton–Jacobi equations
q
˙
i
=
∂
H
∂
p
i
{\displaystyle {\dot {q}}^{i}={\frac {\partial H}{\partial p_{i}}}\quad }
and
p
˙
i
=
−
∂
H
∂
q
i
{\displaystyle \quad {\dot {p}}_{i}=-{\frac {\partial H}{\partial q^{i}}}}
The geodesic flow is the Hamiltonian flow; the solutions to the above are the geodesics of the manifold. Note, incidentally, that
d
H
/
d
t
=
0
{\displaystyle dH/dt=0}
along geodesics; the time parameter
t
{\displaystyle t}
is the distance along the geodesic.
The sigma model takes the momenta in the two manifolds
T
∗
Φ
{\displaystyle T^{*}\Phi }
and
T
∗
M
{\displaystyle T^{*}M}
and solders them together, in that
d
ϕ
{\displaystyle \mathrm {d} \phi }
is a solder form. In this sense, the interpretation of the sigma model as an energy functional is not surprising; it is in fact the gluing together of two energy functionals. Caution: the precise definition of a solder form requires it to be an isomorphism; this can only happen if
M
{\displaystyle M}
and
Φ
{\displaystyle \Phi }
have the same real dimension. Furthermore, the conventional definition of a solder form takes
Φ
{\displaystyle \Phi }
to be a Lie group. Both conditions are satisfied in various applications.
== Results on various spaces ==
The space
Φ
{\displaystyle \Phi }
is often taken to be a Lie group, usually SU(N), in the conventional particle physics models, O(N) in condensed matter theories, or as a symmetric space in supergravity models. Since symmetric spaces are defined in terms of their involution, their tangent space (i.e. the place where
d
ϕ
{\displaystyle \mathrm {d} \phi }
lives) naturally splits into even and odd parity subspaces. This splitting helps propel the dimensional reduction of Kaluza–Klein theories.
=== On Lie groups ===
For the special case of
Φ
{\displaystyle \Phi }
being a Lie group, the
g
i
j
{\displaystyle g_{ij}}
is the metric tensor on the Lie group, formally called the Cartan tensor or the Killing form. The Lagrangian can then be written as the pullback of the Killing form. Note that the Killing form can be written as a trace over two matrices from the corresponding Lie algebra; thus, the Lagrangian can also be written in a form involving the trace. With slight re-arrangements, it can also be written as the pullback of the Maurer–Cartan form.
=== On symmetric spaces ===
A common variation of the sigma model is to present it on a symmetric space. The prototypical example is the chiral model, which takes the product
G
=
S
U
(
N
)
×
S
U
(
N
)
{\displaystyle G=SU(N)\times SU(N)}
of the "left" and "right" chiral fields, and then constructs the sigma model on the "diagonal"
Φ
=
S
U
(
N
)
×
S
U
(
N
)
S
U
(
N
)
{\displaystyle \Phi ={\frac {SU(N)\times SU(N)}{SU(N)}}}
Such a quotient space is a symmetric space, and so one can generically take
Φ
=
G
/
H
{\displaystyle \Phi =G/H}
where
H
⊂
G
{\displaystyle H\subset G}
is the maximal subgroup of
G
{\displaystyle G}
that is invariant under the Cartan involution. The Lagrangian is still written exactly as the above, either in terms of the pullback of the metric on
G
{\displaystyle G}
to a metric on
G
/
H
{\displaystyle G/H}
or as a pullback of the Maurer–Cartan form.
=== Trace notation ===
In physics, the most common and conventional statement of the sigma model begins with the definition
L
μ
=
π
m
∘
(
g
−
1
∂
μ
g
)
{\displaystyle L_{\mu }=\pi _{\mathfrak {m}}\circ \left(g^{-1}\partial _{\mu }g\right)}
Here, the
g
−
1
∂
μ
g
{\displaystyle g^{-1}\partial _{\mu }g}
is the pullback of the Maurer–Cartan form, for
g
∈
G
{\displaystyle g\in G}
, onto the spacetime manifold. The
π
m
{\displaystyle \pi _{\mathfrak {m}}}
is a projection onto the odd-parity piece of the Cartan involution. That is, given the Lie algebra
g
{\displaystyle {\mathfrak {g}}}
of
G
{\displaystyle G}
, the involution decomposes the space into odd and even parity components
g
=
m
⊕
h
{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}
corresponding to the two eigenstates of the involution. The sigma model Lagrangian can then be written as
L
=
1
2
t
r
(
L
μ
L
μ
)
{\displaystyle {\mathcal {L}}={\frac {1}{2}}\mathrm {tr} \left(L_{\mu }L^{\mu }\right)}
This is instantly recognizable as the first term of the Skyrme model.
=== Metric form ===
The equivalent metric form of this is to write a group element
g
∈
G
{\displaystyle g\in G}
as the geodesic
g
=
exp
(
θ
i
T
i
)
{\displaystyle g=\exp(\theta ^{i}T_{i})}
of an element
θ
i
T
i
∈
g
{\displaystyle \theta ^{i}T_{i}\in {\mathfrak {g}}}
of the Lie algebra
g
{\displaystyle {\mathfrak {g}}}
. The
[
T
i
,
T
j
]
=
f
i
j
k
T
k
{\displaystyle [T_{i},T_{j}]={f_{ij}}^{k}T_{k}}
are the basis elements for the Lie algebra; the
f
i
j
k
{\displaystyle {f_{ij}}^{k}}
are the structure constants of
g
{\displaystyle {\mathfrak {g}}}
.
Plugging this directly into the above and applying the infinitesimal form of the Baker–Campbell–Hausdorff formula promptly leads to the equivalent expression
L
=
1
2
g
i
j
(
ϕ
)
d
ϕ
i
∧
⋆
d
ϕ
j
=
1
2
W
i
m
W
n
j
d
ϕ
i
∧
⋆
d
ϕ
j
t
r
(
T
m
T
n
)
{\displaystyle {\mathcal {L}}={\frac {1}{2}}g_{ij}(\phi )\;\mathrm {d} \phi _{i}\wedge \star {\mathrm {d} \phi _{j}}={\frac {1}{2}}\;{W_{i}}^{m}{W^{n}}_{j}\;\;\mathrm {d} \phi _{i}\wedge \star {\mathrm {d} \phi _{j}}\;\;\mathrm {tr} (T_{m}T_{n})}
where
t
r
(
T
m
T
n
)
{\displaystyle \mathrm {tr} (T_{m}T_{n})}
is now obviously (proportional to) the Killing form, and the
W
i
m
{\displaystyle {W_{i}}^{m}}
are the vielbeins that express the "curved" metric
g
i
j
{\displaystyle g_{ij}}
in terms of the "flat" metric
t
r
(
T
m
T
n
)
{\displaystyle \mathrm {tr} (T_{m}T_{n})}
. The article on the Baker–Campbell–Hausdorff formula provides an explicit expression for the vielbeins. They can be written as
W
=
∑
n
=
0
∞
(
−
1
)
n
M
n
(
n
+
1
)
!
=
(
I
−
e
−
M
)
M
−
1
{\displaystyle W=\sum _{n=0}^{\infty }{\frac {(-1)^{n}M^{n}}{(n+1)!}}=(I-e^{-M})M^{-1}}
where
M
{\displaystyle M}
is a matrix whose matrix elements are
M
j
k
=
θ
i
f
i
j
k
{\displaystyle {M_{j}}^{k}=\theta ^{i}{f_{ij}}^{k}}
.
For the sigma model on a symmetric space, as opposed to a Lie group, the
T
i
{\displaystyle T_{i}}
are limited to span the subspace
m
{\displaystyle {\mathfrak {m}}}
instead of all of
g
=
m
⊕
h
{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}
. The Lie commutator on
m
{\displaystyle {\mathfrak {m}}}
will not be within
m
{\displaystyle {\mathfrak {m}}}
; indeed, one has
[
m
,
m
]
⊂
h
{\displaystyle [{\mathfrak {m}},{\mathfrak {m}}]\subset {\mathfrak {h}}}
and so a projection is still needed.
== Extensions ==
The model can be extended in a variety of ways. Besides the aforementioned Skyrme model, which introduces quartic terms, the model may be augmented by a torsion term to yield the Wess–Zumino–Witten model.
Another possibility is frequently seen in supergravity models. Here, one notes that the Maurer–Cartan form
g
−
1
d
g
{\displaystyle g^{-1}dg}
looks like "pure gauge". In the construction above for symmetric spaces, one can also consider the other projection
A
μ
=
π
h
∘
(
g
−
1
∂
μ
g
)
{\displaystyle A_{\mu }=\pi _{\mathfrak {h}}\circ \left(g^{-1}\partial _{\mu }g\right)}
where, as before, the symmetric space corresponded to the split
g
=
m
⊕
h
{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}
. This extra term can be interpreted as a connection on the fiber bundle
M
×
H
{\displaystyle M\times H}
(it transforms as a gauge field). It is what is "left over" from the connection on
G
{\displaystyle G}
. It can be endowed with its own dynamics, by writing
L
=
g
i
j
F
i
∧
⋆
F
j
{\displaystyle {\mathcal {L}}=g_{ij}F^{i}\wedge {\star }F^{j}}
with
F
i
=
d
A
i
{\displaystyle F^{i}=dA^{i}}
. Note that the differential here is just "d", and not a covariant derivative; this is not the Yang–Mills stress-energy tensor. This term is not gauge invariant by itself; it must be taken together with the part of the connection that embeds into
L
μ
{\displaystyle L_{\mu }}
, so that taken together, the
L
μ
{\displaystyle L_{\mu }}
, now with the connection as a part of it, together with this term, forms a complete gauge invariant Lagrangian (which does have the Yang–Mills terms in it, when expanded out).
== References ==
Gell-Mann, M.; Lévy, M. (1960), "The axial vector current in beta decay", Il Nuovo Cimento, 16 (4): 705–726, Bibcode:1960NCim...16..705G, doi:10.1007/BF02859738, S2CID 122945049
Ketov, Sergei (2009). "Nonlinear Sigma model". Scholarpedia. 4 (1): 8508. Bibcode:2009SchpJ...4.8508K. doi:10.4249/scholarpedia.8508. | Wikipedia/Sigma_model |
The Wheeler–DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity.
In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called "problem of time". More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergman–Komar "group" (which is the diffeomorphism group on-shell).
== Motivation and background ==
In canonical gravity, spacetime is foliated into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is
γ
i
j
{\displaystyle \gamma _{ij}}
and given by
g
μ
ν
d
x
μ
d
x
ν
=
(
−
N
2
+
β
k
β
k
)
d
t
2
+
2
β
k
d
x
k
d
t
+
γ
i
j
d
x
i
d
x
j
.
{\displaystyle g_{\mu \nu }\,\mathrm {d} x^{\mu }\,\mathrm {d} x^{\nu }=(-N^{2}+\beta _{k}\beta ^{k})\,\mathrm {d} t^{2}+2\beta _{k}\,\mathrm {d} x^{k}\,\mathrm {d} t+\gamma _{ij}\,\mathrm {d} x^{i}\,\mathrm {d} x^{j}.}
In that equation the Latin indices run over the values 1, 2, 3, and the Greek indices run over the values 1, 2, 3, 4. The three-metric
γ
i
j
{\displaystyle \gamma _{ij}}
is the field, and we denote its conjugate momenta as
π
i
j
{\displaystyle \pi ^{ij}}
. The Hamiltonian is a constraint (characteristic of most relativistic systems)
H
=
1
2
γ
G
i
j
k
l
π
i
j
π
k
l
−
γ
(
3
)
R
=
0
,
{\displaystyle {\mathcal {H}}={\frac {1}{2{\sqrt {\gamma }}}}G_{ijkl}\pi ^{ij}\pi ^{kl}-{\sqrt {\gamma }}\,{}^{(3)}\!R=0,}
where
γ
=
det
(
γ
i
j
)
{\displaystyle \gamma =\det(\gamma _{ij})}
, and
G
i
j
k
l
=
(
γ
i
k
γ
j
l
+
γ
i
l
γ
j
k
−
γ
i
j
γ
k
l
)
{\displaystyle G_{ijkl}=(\gamma _{ik}\gamma _{jl}+\gamma _{il}\gamma _{jk}-\gamma _{ij}\gamma _{kl})}
is the Wheeler–DeWitt metric. In index-free notation, the Wheeler–DeWitt metric on the space of positive definite quadratic forms g in three dimensions is
tr
(
(
g
−
1
d
g
)
2
)
−
(
tr
(
g
−
1
d
g
)
)
2
.
{\displaystyle \operatorname {tr} ((g^{-1}dg)^{2})-(\operatorname {tr} (g^{-1}dg))^{2}.}
Quantization "puts hats" on the momenta and field variables; that is, the functions of numbers in the classical case become operators that modify the state function in the quantum case. Thus we obtain the operator
H
^
=
1
2
γ
G
^
i
j
k
l
π
^
i
j
π
^
k
l
−
γ
(
3
)
R
^
.
{\displaystyle {\hat {\mathcal {H}}}={\frac {1}{2{\sqrt {\gamma }}}}{\hat {G}}_{ijkl}{\hat {\pi }}^{ij}{\hat {\pi }}^{kl}-{\sqrt {\gamma }}\,{}^{(3)}\!{\hat {R}}.}
Working in "position space", these operators are
γ
^
i
j
(
t
,
x
k
)
→
γ
i
j
(
t
,
x
k
)
,
π
^
i
j
(
t
,
x
k
)
→
−
i
δ
δ
γ
i
j
(
t
,
x
k
)
.
{\displaystyle {\begin{aligned}{\hat {\gamma }}_{ij}(t,x^{k})&\to \gamma _{ij}(t,x^{k}),\\{\hat {\pi }}^{ij}(t,x^{k})&\to -i{\frac {\delta }{\delta \gamma _{ij}(t,x^{k})}}.\end{aligned}}}
One can apply the operator to a general wave functional of the metric
H
^
Ψ
[
γ
]
=
0
,
{\displaystyle {\hat {\mathcal {H}}}\Psi [\gamma ]=0,}
where
Ψ
[
γ
]
=
a
+
∫
ψ
(
x
)
γ
(
x
)
d
x
3
+
∬
ψ
(
x
,
y
)
γ
(
x
)
γ
(
y
)
d
x
3
d
y
3
+
…
,
{\displaystyle \Psi [\gamma ]=a+\int \psi (x)\gamma (x)\,dx^{3}+\iint \psi (x,y)\gamma (x)\gamma (y)\,dx^{3}\,dy^{3}+\dots ,}
which would give a set of constraints amongst the coefficients
ψ
(
x
,
y
,
…
)
{\displaystyle \psi (x,y,\dots )}
. This means that the amplitudes for
N
{\displaystyle N}
gravitons at certain positions are related to the amplitudes for a different number of gravitons at different positions. Or, one could use the two-field formalism, treating
ω
(
g
)
{\displaystyle \omega (g)}
as an independent field, so that the wave function is
Ψ
[
γ
,
ω
]
{\displaystyle \Psi [\gamma ,\omega ]}
.
== Mathematical formalism ==
The Wheeler–DeWitt equation is a functional differential equation. It is ill-defined in the general case, but very important in theoretical physics, especially in quantum gravity. It is a functional differential equation on the space of three-dimensional spatial metrics. The Wheeler–DeWitt equation has the form of an operator acting on a wave functional; the functional reduces to a function in cosmology. Contrary to the general case, the Wheeler–DeWitt equation is well defined in minisuperspaces like the configuration space of cosmological theories. An example of such a wave function is the Hartle–Hawking state. Bryce DeWitt first published this equation in 1967 under the name "Einstein–Schrödinger equation"; it was later renamed the "Wheeler–DeWitt equation".
=== Hamiltonian constraint ===
Simply speaking, the Wheeler–DeWitt equation says
where
H
^
(
x
)
{\displaystyle {\hat {H}}(x)}
is the Hamiltonian constraint in quantized general relativity, and
|
ψ
⟩
{\displaystyle |\psi \rangle }
stands for the wave function of the universe. Unlike ordinary quantum field theory or quantum mechanics, the Hamiltonian is a first-class constraint on physical states. We also have an independent constraint for each point in space.
Although the symbols
H
^
{\displaystyle {\hat {H}}}
and
|
ψ
⟩
{\displaystyle |\psi \rangle }
may appear familiar, their interpretation in the Wheeler–DeWitt equation is substantially different from non-relativistic quantum mechanics.
|
ψ
⟩
{\displaystyle |\psi \rangle }
is no longer a spatial wave function in the traditional sense of a complex-valued function that is defined on a 3-dimensional space-like surface and normalized to unity. Instead it is a functional of field configurations on all of spacetime. This wave function contains all of the information about the geometry and matter content of the universe.
H
^
{\displaystyle {\hat {H}}}
is still an operator that acts on the Hilbert space of wave functions, but it is not the same Hilbert space as in the nonrelativistic case, and the Hamiltonian no longer determines the evolution of the system, so the Schrödinger equation
H
^
|
ψ
⟩
=
i
ℏ
∂
/
∂
t
|
ψ
⟩
{\displaystyle {\hat {H}}|\psi \rangle =i\hbar \partial /\partial t|\psi \rangle }
no longer applies. This property is known as timelessness. Various attempts to incorporate time in a fully quantum framework have been made, starting with the "Page and Wootters mechanism" and other subsequent proposals. The reemergence of time was also proposed as arising from quantum correlations between an evolving system and a reference quantum clock system, the concept of system-time entanglement is introduced as a quantifier of the actual distinguishable evolution undergone by the system.
=== Momentum constraint ===
We also need to augment the Hamiltonian constraint with momentum constraints
P
→
(
x
)
|
ψ
⟩
=
0
{\displaystyle {\vec {\mathcal {P}}}(x)|\psi \rangle =0}
associated with spatial diffeomorphism invariance.
In minisuperspace approximations, we only have one Hamiltonian constraint (instead of infinitely many of them).
In fact, the principle of general covariance in general relativity implies that global evolution per se does not exist; the time
t
{\displaystyle t}
is just a label we assign to one of the coordinate axes. Thus, what we think about as time evolution of any physical system is just a gauge transformation, similar to that of QED induced by U(1) local gauge transformation
ψ
→
e
i
θ
(
r
→
)
ψ
,
{\displaystyle \psi \to e^{i\theta ({\vec {r}})}\psi ,}
where
θ
(
r
→
)
{\displaystyle \theta ({\vec {r}})}
plays the role of local time. The role of a Hamiltonian is simply to restrict the space of the "kinematic" states of the Universe to that of "physical" states—the ones that follow gauge orbits. For this reason we call it a "Hamiltonian constraint". Upon quantization, physical states become wave functions that lie in the kernel of the Hamiltonian operator.
In general, the Hamiltonian vanishes for a theory with general covariance or time-scaling invariance.
== See also ==
== References ==
Hamber, Herbert W.; Williams, Ruth M. (2011-11-18). "Discrete Wheeler-DeWitt equation". Physical Review D. 84 (10): 104033. arXiv:1109.2530. Bibcode:2011PhRvD..84j4033H. doi:10.1103/PhysRevD.84.104033. ISSN 1550-7998. S2CID 4812404.
Hamber, Herbert W.; Toriumi, Reiko; Williams, Ruth M. (2012-10-02). "Wheeler-DeWitt equation in 2 + 1 dimensions". Physical Review D. 86 (8): 084010. arXiv:1207.3759. Bibcode:2012PhRvD..86h4010H. doi:10.1103/PhysRevD.86.084010. ISSN 1550-7998. S2CID 119229306. | Wikipedia/Wheeler–DeWitt_equation |
A black hole firewall is a hypothetical phenomenon where an observer falling into a black hole encounters high-energy quanta at (or near) the event horizon. The "firewall" phenomenon was proposed in 2012 by physicists Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully as a possible solution to an apparent inconsistency in black hole complementarity. The proposal is sometimes referred to as the AMPS firewall, an acronym for the names of the authors of the 2012 paper. The potential inconsistency pointed out by AMPS had been pointed out earlier by Samir Mathur who used the argument in favour of the fuzzball proposal. The use of a firewall to resolve this inconsistency remains controversial, with physicists divided as to the solution to the paradox.
== The motivating paradox ==
According to quantum field theory in curved spacetime, a single emission of Hawking radiation involves two mutually entangled particles. The outgoing particle escapes and is emitted as a quantum of Hawking radiation; the infalling particle is swallowed by the black hole. Assume that a black hole formed a finite time in the past and will fully evaporate away in some finite time in the future. Then, it will only emit a finite amount of information encoded within its Hawking radiation. For an old black hole that has crossed the half-way point of evaporation, general arguments from quantum-information theory by Page and Lubkin suggest that the new Hawking radiation must be entangled with the old Hawking radiation. However, since the new Hawking radiation must also be entangled with degrees of freedom behind the horizon, this creates a paradox: a principle called "monogamy of entanglement" requires that, like any quantum system, the outgoing particle cannot be fully entangled with two independent systems at the same time; yet here the outgoing particle appears to be entangled with both the infalling particle and, independently, with past Hawking radiation.
AMPS initially argued that to resolve the paradox physicists may eventually be forced to give up one of three time-tested principles: Einstein's equivalence principle, unitarity, or existing quantum field theory. However, it is now accepted that an additional tacit assumption in the monogamy paradox was that of locality. A common view is that theories of quantum gravity do not obey exact locality, which leads to a resolution of the paradox. On the other hand, some physicists argue that such violations of locality cannot resolve the paradox.
=== The "firewall" resolution to the paradox ===
Some scientists suggest that the entanglement must somehow get immediately broken between the infalling particle and the outgoing particle. Breaking this entanglement would release large amounts of energy, thus creating a searing "black hole firewall" at the black hole event horizon. This resolution requires a violation of Einstein's equivalence principle, which states that free-falling is indistinguishable from floating in empty space. This violation has been characterized as "outrageous"; theoretical physicist Raphael Bousso has complained that "a firewall simply can't appear in empty space, any more than a brick wall can suddenly appear in an empty field and smack you in the face."
=== Non-firewall resolutions to the paradox ===
Some scientists suggest that there is in fact no entanglement between the emitted particle and previous Hawking radiation. This resolution would require black hole information loss, a controversial violation of unitarity.
Others, such as Steve Giddings, suggest modifying quantum field theory so that entanglement would be gradually lost as the outgoing and infalling particles separate, resulting in a more gradual release of energy inside the black hole, and consequently no firewall.
The Papadodimas–Raju proposal posited that the interior of the black hole was described by the same degrees of freedom as the Hawking radiation. This resolves the monogamy paradox by identifying the two systems that the late Hawking radiation is entangled with. Since, in this proposal, these systems are the same, there is no contradiction with the monogamy of entanglement. Along similar lines, Juan Maldacena and Leonard Susskind suggested in the ER=EPR proposal that the outgoing and infalling particles are somehow connected by wormholes, and therefore are not independent systems.
The fuzzball picture resolves the dilemma by replacing the 'no-hair' vacuum with a stringy quantum state, thus explicitly coupling any outgoing Hawking radiation with the formation history of the black hole.
Stephen Hawking received widespread mainstream media coverage in January 2014 with an informal proposal to replace the event horizon of a black hole with an "apparent horizon" where infalling matter is suspended and then released; however, some scientists have expressed confusion about what precisely is being proposed and how the proposal would solve the paradox.
== Characteristics and detection ==
The firewall would exist at the black hole's event horizon, and would be invisible to observers outside the event horizon. Matter passing through the event horizon into the black hole would immediately be "burned to a crisp" by an arbitrarily hot "seething maelstrom of particles" at the firewall.
In a merger of two black holes, the characteristics of a firewall (if any) may leave a mark on the outgoing gravitational radiation as "echoes" when waves bounce in the vicinity of the fuzzy event horizon. The expected quantity of such echoes is theoretically unclear, as physicists don't currently have a good physical model of firewalls. In 2016, cosmologist Niayesh Afshordi and others argued there were tentative signs of some such echo in the data from the first black hole merger detected by LIGO; more recent work has argued there is no statistically significant evidence for such echoes in the data.
== See also ==
Black hole information paradox
Black hole thermodynamics
Photon sphere
Gravitational time dilation
Magnetospheric eternally collapsing object
== References == | Wikipedia/Firewall_(physics) |
In theoretical physics, background field method is a useful procedure to calculate the effective action of a quantum field theory by expanding a quantum field around a classical "background" value B:
ϕ
(
x
)
=
B
(
x
)
+
η
(
x
)
{\displaystyle \phi (x)=B(x)+\eta (x)}
.
After this is done, the Green's functions are evaluated as a function of the background. This approach has the advantage that the gauge invariance is manifestly preserved if the approach is applied to gauge theory.
== Method ==
We typically want to calculate expressions like
Z
[
J
]
=
∫
D
ϕ
exp
(
i
∫
d
d
x
(
L
[
ϕ
(
x
)
]
+
J
(
x
)
ϕ
(
x
)
)
)
{\displaystyle Z[J]=\int {\mathcal {D}}\phi \exp \left(\mathrm {i} \int \mathrm {d} ^{d}x({\mathcal {L}}[\phi (x)]+J(x)\phi (x))\right)}
where J(x) is a source,
L
(
x
)
{\displaystyle {\mathcal {L}}(x)}
is the Lagrangian density of the system, d is the number of dimensions and
ϕ
(
x
)
{\displaystyle \phi (x)}
is a field.
In the background field method, one starts by splitting this field into a classical background field B(x) and a field η(x) containing additional quantum fluctuations:
ϕ
(
x
)
=
B
(
x
)
+
η
(
x
)
.
{\displaystyle \phi (x)=B(x)+\eta (x)\,.}
Typically, B(x) will be a solution of the classical equations of motion
δ
S
δ
ϕ
|
ϕ
=
B
=
0
{\displaystyle \left.{\frac {\delta S}{\delta \phi }}\right|_{\phi =B}=0}
where S is the action, i.e. the space integral of the Lagrangian density. Switching on a source J(x) will change the equations into
δ
S
δ
ϕ
|
ϕ
=
B
+
J
=
0
{\displaystyle \left.{\frac {\delta S}{\delta \phi }}\right|_{\phi =B}+J=0}
.
Then the action is expanded around the background B(x):
∫
d
d
x
(
L
[
ϕ
(
x
)
]
+
J
(
x
)
ϕ
(
x
)
)
=
∫
d
d
x
(
L
[
B
(
x
)
]
+
J
(
x
)
B
(
x
)
)
+
∫
d
d
x
(
δ
L
δ
ϕ
(
x
)
[
B
]
+
J
(
x
)
)
η
(
x
)
+
1
2
∫
d
d
x
d
d
y
δ
2
L
δ
ϕ
(
x
)
δ
ϕ
(
y
)
[
B
]
η
(
x
)
η
(
y
)
+
⋯
{\displaystyle {\begin{aligned}\int d^{d}x({\mathcal {L}}[\phi (x)]+J(x)\phi (x))&=\int d^{d}x({\mathcal {L}}[B(x)]+J(x)B(x))\\&+\int d^{d}x\left({\frac {\delta {\mathcal {L}}}{\delta \phi (x)}}[B]+J(x)\right)\eta (x)\\&+{\frac {1}{2}}\int d^{d}xd^{d}y{\frac {\delta ^{2}{\mathcal {L}}}{\delta \phi (x)\delta \phi (y)}}[B]\eta (x)\eta (y)+\cdots \end{aligned}}}
The second term in this expansion is zero by the equations of motion. The first term does not depend on any fluctuating fields, so that it can be brought out of the path integral. The result is
Z
[
J
]
=
e
i
∫
d
d
x
(
L
[
B
(
x
)
]
+
J
(
x
)
B
(
x
)
)
∫
D
η
e
i
2
∫
d
d
x
d
d
y
δ
2
L
δ
ϕ
(
x
)
δ
ϕ
(
y
)
[
B
]
η
(
x
)
η
(
y
)
+
⋯
.
{\displaystyle Z[J]=e^{i\int d^{d}x({\mathcal {L}}[B(x)]+J(x)B(x))}\int {\mathcal {D}}\eta e^{{\frac {i}{2}}\int d^{d}xd^{d}y{\frac {\delta ^{2}{\mathcal {L}}}{\delta \phi (x)\delta \phi (y)}}[B]\eta (x)\eta (y)+\cdots }.}
The path integral which now remains is (neglecting the corrections in the dots) of Gaussian form and can be integrated exactly:
Z
[
J
]
=
C
e
i
∫
d
d
x
(
L
[
B
(
x
)
]
+
J
(
x
)
B
(
x
)
)
(
det
δ
2
L
δ
ϕ
(
x
)
δ
ϕ
(
y
)
[
B
]
)
−
1
/
2
+
⋯
{\displaystyle Z[J]=Ce^{i\int d^{d}x({\mathcal {L}}[B(x)]+J(x)B(x))}\left(\det {\frac {\delta ^{2}{\mathcal {L}}}{\delta \phi (x)\delta \phi (y)}}[B]\right)^{-1/2}+\cdots }
where "det" signifies a functional determinant and C is a constant. The power of minus one half will naturally be plus one for Grassmann fields.
The above derivation gives the Gaussian approximation to the functional integral. Corrections to this can be computed, producing a diagrammatic expansion.
== See also ==
BF theory
Effective action
Source field
== References ==
Peskin, Michael; Schroeder, Daniel (1994). Introduction to Quantum Field Theory. Perseus Publishing. ISBN 0-201-50397-2.
Böhm, Manfred; Denner, Ansgar; Joos, Hans (2001). Gauge Theories of the Strong and Electroweak Interaction (3 ed.). Teubner. ISBN 3-519-23045-3.
Kleinert, Hagen (2009). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (5 ed.). World Scientific.
Abbott, L. F. (1982). "Introduction to the Background Field Method" (PDF). Acta Phys. Pol. B. 13: 33. Archived from the original (PDF) on 2017-05-10. Retrieved 2016-03-10. | Wikipedia/Background_field_method |
The Schwinger–Dyson equations (SDEs) or Dyson–Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between correlation functions in quantum field theories (QFTs). They are also referred to as the Euler–Lagrange equations of quantum field theories, since they are the equations of motion corresponding to the Green's function. They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite tower of SDEs.
In his paper "The S-Matrix in Quantum electrodynamics", Dyson derived relations between different S-matrix elements, or more specific "one-particle Green's functions", in quantum electrodynamics, by summing up infinitely many Feynman diagrams, thus working in a perturbative approach. Starting from his variational principle, Schwinger derived a set of equations for Green's functions non-perturbatively, which generalize Dyson's equations to the Schwinger–Dyson equations for the Green functions of quantum field theories. Today they provide a non-perturbative approach to quantum field theories and applications can be found in many fields of theoretical physics, such as solid-state physics and elementary particle physics.
Schwinger also derived an equation for the two-particle irreducible Green functions, which is nowadays referred to as the inhomogeneous Bethe–Salpeter equation.
== Derivation ==
Given a polynomially bounded functional
F
{\displaystyle F}
over the field configurations, then, for any state vector (which is a solution of the QFT),
|
ψ
⟩
{\displaystyle |\psi \rangle }
, we have
⟨
ψ
|
T
{
δ
δ
φ
F
[
φ
]
}
|
ψ
⟩
=
−
i
⟨
ψ
|
T
{
F
[
φ
]
δ
δ
φ
S
[
φ
]
}
|
ψ
⟩
{\displaystyle \left\langle \psi \left|{\mathcal {T}}\left\{{\frac {\delta }{\delta \varphi }}F[\varphi ]\right\}\right|\psi \right\rangle =-i\left\langle \psi \left|{\mathcal {T}}\left\{F[\varphi ]{\frac {\delta }{\delta \varphi }}S[\varphi ]\right\}\right|\psi \right\rangle }
where
δ
/
δ
φ
{\displaystyle \delta /\delta \varphi }
is the functional derivative with respect to
φ
,
S
{\displaystyle \varphi ,S}
is the action functional and
T
{\displaystyle {\mathcal {T}}}
is the time ordering operation.
Equivalently, in the density state formulation, for any (valid) density state
ρ
{\displaystyle \rho }
, we have
ρ
(
T
{
δ
δ
φ
F
[
φ
]
}
)
=
−
i
ρ
(
T
{
F
[
φ
]
δ
δ
φ
S
[
φ
]
}
)
.
{\displaystyle \rho \left({\mathcal {T}}\left\{{\frac {\delta }{\delta \varphi }}F[\varphi ]\right\}\right)=-i\rho \left({\mathcal {T}}\left\{F[\varphi ]{\frac {\delta }{\delta \varphi }}S[\varphi ]\right\}\right).}
This infinite set of equations can be used to solve for the correlation functions nonperturbatively.
To make the connection to diagrammatic techniques (like Feynman diagrams) clearer, it is often convenient to split the action
S
{\displaystyle S}
as
S
[
φ
]
=
1
2
φ
i
D
i
j
−
1
φ
j
+
S
int
[
φ
]
,
{\displaystyle S[\varphi ]={\frac {1}{2}}\varphi ^{i}D_{ij}^{-1}\varphi ^{j}+S_{\text{int}}[\varphi ],}
where the first term is the quadratic part and
D
−
1
{\displaystyle D^{-1}}
is an invertible symmetric (antisymmetric for fermions) covariant tensor of rank two in the deWitt notation whose inverse,
D
{\displaystyle D}
is called the bare propagator and
S
int
[
φ
]
{\displaystyle S_{\text{int}}[\varphi ]}
is the "interaction action". Then, we can rewrite the SD equations as
⟨
ψ
|
T
{
F
φ
j
}
|
ψ
⟩
=
⟨
ψ
|
T
{
i
F
,
i
D
i
j
−
F
S
int
,
i
D
i
j
}
|
ψ
⟩
.
{\displaystyle \langle \psi |{\mathcal {T}}\{F\varphi ^{j}\}|\psi \rangle =\langle \psi |{\mathcal {T}}\{iF_{,i}D^{ij}-FS_{{\text{int}},i}D^{ij}\}|\psi \rangle .}
If
F
{\displaystyle F}
is a functional of
φ
{\displaystyle \varphi }
, then for an operator
K
{\displaystyle K}
,
F
[
K
]
{\displaystyle F[K]}
is defined to be the operator which substitutes
K
{\displaystyle K}
for
φ
{\displaystyle \varphi }
. For example, if
F
[
φ
]
=
∂
k
1
∂
x
1
k
1
φ
(
x
1
)
⋯
∂
k
n
∂
x
n
k
n
φ
(
x
n
)
{\displaystyle F[\varphi ]={\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}\varphi (x_{1})\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}\varphi (x_{n})}
and
G
{\displaystyle G}
is a functional of
J
{\displaystyle J}
, then
F
[
−
i
δ
δ
J
]
G
[
J
]
=
(
−
i
)
n
∂
k
1
∂
x
1
k
1
δ
δ
J
(
x
1
)
⋯
∂
k
n
∂
x
n
k
n
δ
δ
J
(
x
n
)
G
[
J
]
.
{\displaystyle F\left[-i{\frac {\delta }{\delta J}}\right]G[J]=(-i)^{n}{\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}{\frac {\delta }{\delta J(x_{1})}}\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}{\frac {\delta }{\delta J(x_{n})}}G[J].}
If we have an "analytic" (a function that is locally given by a convergent power series) functional
Z
{\displaystyle Z}
(called the generating functional) of
J
{\displaystyle J}
(called the source field) satisfying
δ
n
Z
δ
J
(
x
1
)
⋯
δ
J
(
x
n
)
[
0
]
=
i
n
Z
[
0
]
⟨
φ
(
x
1
)
⋯
φ
(
x
n
)
⟩
,
{\displaystyle {\frac {\delta ^{n}Z}{\delta J(x_{1})\cdots \delta J(x_{n})}}[0]=i^{n}Z[0]\langle \varphi (x_{1})\cdots \varphi (x_{n})\rangle ,}
then, from the properties of the functional integrals
⟨
δ
S
δ
φ
(
x
)
[
φ
]
+
J
(
x
)
⟩
J
=
0
,
{\displaystyle {\left\langle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}\left[\varphi \right]+J(x)\right\rangle }_{J}=0,}
the Schwinger–Dyson equation for the generating functional is
δ
S
δ
φ
(
x
)
[
−
i
δ
δ
J
]
Z
[
J
]
+
J
(
x
)
Z
[
J
]
=
0.
{\displaystyle {\frac {\delta S}{\delta \varphi (x)}}\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Z[J]=0.}
If we expand this equation as a Taylor series about
J
=
0
{\displaystyle J=0}
, we get the entire set of Schwinger–Dyson equations.
== An example: φ4 ==
To give an example, suppose
S
[
φ
]
=
∫
d
d
x
(
1
2
∂
μ
φ
(
x
)
∂
μ
φ
(
x
)
−
1
2
m
2
φ
(
x
)
2
−
λ
4
!
φ
(
x
)
4
)
{\displaystyle S[\varphi ]=\int d^{d}x\left({\frac {1}{2}}\partial ^{\mu }\varphi (x)\partial _{\mu }\varphi (x)-{\frac {1}{2}}m^{2}\varphi (x)^{2}-{\frac {\lambda }{4!}}\varphi (x)^{4}\right)}
for a real field
φ
{\displaystyle \varphi }
.
Then,
δ
S
δ
φ
(
x
)
=
−
∂
μ
∂
μ
φ
(
x
)
−
m
2
φ
(
x
)
−
λ
3
!
φ
3
(
x
)
.
{\displaystyle {\frac {\delta S}{\delta \varphi (x)}}=-\partial _{\mu }\partial ^{\mu }\varphi (x)-m^{2}\varphi (x)-{\frac {\lambda }{3!}}\varphi ^{3}(x).}
The Schwinger–Dyson equation for this particular example is:
i
∂
μ
∂
μ
δ
δ
J
(
x
)
Z
[
J
]
+
i
m
2
δ
δ
J
(
x
)
Z
[
J
]
−
i
λ
3
!
δ
3
δ
J
(
x
)
3
Z
[
J
]
+
J
(
x
)
Z
[
J
]
=
0
{\displaystyle i\partial _{\mu }\partial ^{\mu }{\frac {\delta }{\delta J(x)}}Z[J]+im^{2}{\frac {\delta }{\delta J(x)}}Z[J]-{\frac {i\lambda }{3!}}{\frac {\delta ^{3}}{\delta J(x)^{3}}}Z[J]+J(x)Z[J]=0}
Note that since
δ
3
δ
J
(
x
)
3
{\displaystyle {\frac {\delta ^{3}}{\delta J(x)^{3}}}}
is not well-defined because
δ
3
δ
J
(
x
1
)
δ
J
(
x
2
)
δ
J
(
x
3
)
Z
[
J
]
{\displaystyle {\frac {\delta ^{3}}{\delta J(x_{1})\delta J(x_{2})\delta J(x_{3})}}Z[J]}
is a distribution in
x
1
{\displaystyle x_{1}}
,
x
2
{\displaystyle x_{2}}
and
x
3
{\displaystyle x_{3}}
,
this equation needs to be regularized.
In this example, the bare propagator D is the Green's function for
−
∂
μ
∂
μ
−
m
2
{\displaystyle -\partial ^{\mu }\partial _{\mu }-m^{2}}
and so, the Schwinger–Dyson set of equations goes as
⟨
ψ
∣
T
{
φ
(
x
0
)
φ
(
x
1
)
}
∣
ψ
⟩
=
i
D
(
x
0
,
x
1
)
+
λ
3
!
∫
d
d
x
2
D
(
x
0
,
x
2
)
⟨
ψ
∣
T
{
φ
(
x
1
)
φ
(
x
2
)
φ
(
x
2
)
φ
(
x
2
)
}
∣
ψ
⟩
{\displaystyle {\begin{aligned}&\langle \psi \mid {\mathcal {T}}\{\varphi (x_{0})\varphi (x_{1})\}\mid \psi \rangle \\[4pt]={}&iD(x_{0},x_{1})+{\frac {\lambda }{3!}}\int d^{d}x_{2}\,D(x_{0},x_{2})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{2})\varphi (x_{2})\varphi (x_{2})\}\mid \psi \rangle \end{aligned}}}
and
⟨
ψ
∣
T
{
φ
(
x
0
)
φ
(
x
1
)
φ
(
x
2
)
φ
(
x
3
)
}
∣
ψ
⟩
=
i
D
(
x
0
,
x
1
)
⟨
ψ
∣
T
{
φ
(
x
2
)
φ
(
x
3
)
}
∣
ψ
⟩
+
i
D
(
x
0
,
x
2
)
⟨
ψ
∣
T
{
φ
(
x
1
)
φ
(
x
3
)
}
∣
ψ
⟩
+
i
D
(
x
0
,
x
3
)
⟨
ψ
∣
T
{
φ
(
x
1
)
φ
(
x
2
)
}
∣
ψ
⟩
+
λ
3
!
∫
d
d
x
4
D
(
x
0
,
x
4
)
⟨
ψ
∣
T
{
φ
(
x
1
)
φ
(
x
2
)
φ
(
x
3
)
φ
(
x
4
)
φ
(
x
4
)
φ
(
x
4
)
}
∣
ψ
⟩
{\displaystyle {\begin{aligned}&\langle \psi \mid {\mathcal {T}}\{\varphi (x_{0})\varphi (x_{1})\varphi (x_{2})\varphi (x_{3})\}\mid \psi \rangle \\[6pt]={}&iD(x_{0},x_{1})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{2})\varphi (x_{3})\}\mid \psi \rangle +iD(x_{0},x_{2})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{3})\}\mid \psi \rangle \\[4pt]&{}+iD(x_{0},x_{3})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{2})\}\mid \psi \rangle \\[4pt]&{}+{\frac {\lambda }{3!}}\int d^{d}x_{4}\,D(x_{0},x_{4})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{2})\varphi (x_{3})\varphi (x_{4})\varphi (x_{4})\varphi (x_{4})\}\mid \psi \rangle \end{aligned}}}
etc.
(Unless there is spontaneous symmetry breaking, the odd correlation functions vanish.)
== See also ==
Functional renormalization group
Dyson equation
Path integral formulation
Source field
== References ==
== Further reading ==
There are not many books that treat the Schwinger–Dyson equations. Here are three standard references:
Claude Itzykson, Jean-Bernard Zuber (1980). Quantum Field Theory. McGraw-Hill. ISBN 9780070320710.
R.J. Rivers (1990). Path Integral Methods in Quantum Field Theories. Cambridge University Press.
V.P. Nair (2005). Quantum Field Theory A Modern Perspective. Springer.
There are some review article about applications of the Schwinger–Dyson equations with applications to special field of physics.
For applications to Quantum Chromodynamics there are
R. Alkofer and L. v.Smekal (2001). "On the infrared behaviour of QCD Green's functions". Phys. Rep. 353 (5–6): 281. arXiv:hep-ph/0007355. Bibcode:2001PhR...353..281A. doi:10.1016/S0370-1573(01)00010-2. S2CID 119411676.
C.D. Roberts and A.G. Williams (1994). "Dyson-Schwinger equations and their applications to hadron physics". Prog. Part. Nucl. Phys. 33: 477–575. arXiv:hep-ph/9403224. Bibcode:1994PrPNP..33..477R. doi:10.1016/0146-6410(94)90049-3. S2CID 119360538. | Wikipedia/Schwinger-Dyson_equation |
The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is a theory of electrodynamics based on a relativistic correct extension of action at a distance electron particles. The theory postulates no independent electromagnetic field. Rather, the whole theory is encapsulated by the Lorentz-invariant action
S
{\displaystyle S}
of particle trajectories
a
μ
(
τ
)
,
b
μ
(
τ
)
,
⋯
{\displaystyle a^{\mu }(\tau ),\,\,b^{\mu }(\tau ),\,\,\cdots }
defined as
S
=
−
∑
a
m
a
c
∫
−
d
a
μ
d
a
μ
+
∑
a
<
b
e
a
e
b
c
∫
∫
δ
(
a
b
μ
a
b
μ
)
d
a
ν
d
b
ν
,
{\displaystyle S=-\sum _{a}m_{a}c\int {\sqrt {-da_{\mu }da^{\mu }}}+\sum _{a<b}{\frac {e_{a}e_{b}}{c}}\int \int \delta (ab_{\mu }ab^{\mu })\,da_{\nu }db^{\nu },}
where
a
b
μ
≡
a
μ
−
b
μ
{\displaystyle ab_{\mu }\equiv a_{\mu }-b_{\mu }}
.
The absorber theory is invariant under time-reversal transformation, consistent with the lack of any physical basis for microscopic time-reversal symmetry breaking. Another key principle resulting from this interpretation, and somewhat reminiscent of Mach's principle and the work of Hugo Tetrode, is that elementary particles are not self-interacting. This immediately removes the problem of electron self-energy giving an infinity in the energy of an electromagnetic field.
== Motivation ==
Wheeler and Feynman begin by observing that classical electromagnetic field theory was designed before the discovery of electrons: charge is a continuous substance in the theory. An electron particle does not naturally fit in to the theory: should a point charge see the effect of its own field? They reconsider the fundamental problem of a collection of point charges, taking up a field-free action at a distance theory developed separately by Karl Schwarzschild, Hugo Tetrode, and Adriaan Fokker. Unlike instantaneous action at a distance theories of the early 1800s these "direct interaction" theories are based on interaction propagation at the speed of light. They differ from the classical field theory in three ways 1) no independent field is postulated; 2) the point charges do not act upon themselves; 3) the equations are time symmetric. Wheeler and Feynman propose to develop these equations into a relativistically correct generalization of electromagnetism based on Newtonian mechanics.
== Problems with previous direct-interaction theories ==
The Tetrode-Fokker work left unsolved two major problems.: 171 First, in a non-instantaneous action at a distance theory, the equal action-reaction of Newton's laws of motion conflicts with causality. If an action propagates forward in time, the reaction would necessarily propagate backwards in time. Second, existing explanations of radiation reaction force or radiation resistance depended upon accelerating electrons interacting with their own field; the direct interaction models explicitly omit self-interaction.
== Absorber and radiation resistance ==
Wheeler and Feynman postulate the "universe" of all other electrons as an absorber of radiation to overcome these issues and extend the direct interaction theories.
Rather than considering an unphysical isolated point charge, they model all charges in the universe with a uniform absorber in a shell around a charge. As the charge moves relative to the absorber, it radiates into the absorber which "pushes back", causing the radiation resistance.
== Key result ==
Feynman and Wheeler obtained their result in a very simple and elegant way. They considered all the charged particles (emitters) present in our universe and assumed all of them to generate time-reversal symmetric waves. The resulting field is
E
tot
(
x
,
t
)
=
∑
n
E
n
ret
(
x
,
t
)
+
E
n
adv
(
x
,
t
)
2
.
{\displaystyle E_{\text{tot}}(\mathbf {x} ,t)=\sum _{n}{\frac {E_{n}^{\text{ret}}(\mathbf {x} ,t)+E_{n}^{\text{adv}}(\mathbf {x} ,t)}{2}}.}
Then they observed that if the relation
E
free
(
x
,
t
)
=
∑
n
E
n
ret
(
x
,
t
)
−
E
n
adv
(
x
,
t
)
2
=
0
{\displaystyle E_{\text{free}}(\mathbf {x} ,t)=\sum _{n}{\frac {E_{n}^{\text{ret}}(\mathbf {x} ,t)-E_{n}^{\text{adv}}(\mathbf {x} ,t)}{2}}=0}
holds, then
E
free
{\displaystyle E_{\text{free}}}
, being a solution of the homogeneous Maxwell equation, can be used to obtain the total field
E
tot
(
x
,
t
)
=
∑
n
E
n
ret
(
x
,
t
)
+
E
n
adv
(
x
,
t
)
2
+
∑
n
E
n
ret
(
x
,
t
)
−
E
n
adv
(
x
,
t
)
2
=
∑
n
E
n
ret
(
x
,
t
)
.
{\displaystyle E_{\text{tot}}(\mathbf {x} ,t)=\sum _{n}{\frac {E_{n}^{\text{ret}}(\mathbf {x} ,t)+E_{n}^{\text{adv}}(\mathbf {x} ,t)}{2}}+\sum _{n}{\frac {E_{n}^{\text{ret}}(\mathbf {x} ,t)-E_{n}^{\text{adv}}(\mathbf {x} ,t)}{2}}=\sum _{n}E_{n}^{\text{ret}}(\mathbf {x} ,t).}
The total field is then the observed pure retarded field.: 173
The assumption that the free field is identically zero is the core of the absorber idea. It means that the radiation emitted by each particle is completely absorbed by all other particles present in the universe. To better understand this point, it may be useful to consider how the absorption mechanism works in common materials. At the microscopic scale, it results from the sum of the incoming electromagnetic wave and the waves generated from the electrons of the material, which react to the external perturbation. If the incoming wave is absorbed, the result is a zero outgoing field. In the absorber theory the same concept is used, however, in presence of both retarded and advanced waves.
== Arrow of time ambiguity ==
The resulting wave appears to have a preferred time direction, because it respects causality. However, this is only an illusion. Indeed, it is always possible to reverse the time direction by simply exchanging the labels emitter and absorber. Thus, the apparently preferred time direction results from the arbitrary labelling.: 52 Wheeler and Feynman claimed that thermodynamics picked the observed direction; cosmological selections have also been proposed.
The requirement of time-reversal symmetry, in general, is difficult to reconcile with the principle of causality. Maxwell's equations and the equations for electromagnetic waves have, in general, two possible solutions: a retarded (delayed) solution and an advanced one. Accordingly, any charged particle generates waves, say at time
t
0
=
0
{\displaystyle t_{0}=0}
and point
x
0
=
0
{\displaystyle x_{0}=0}
, which will arrive at point
x
1
{\displaystyle x_{1}}
at the instant
t
1
=
x
1
/
c
{\displaystyle t_{1}=x_{1}/c}
(here
c
{\displaystyle c}
is the speed of light), after the emission (retarded solution), and other waves, which will arrive at the same place at the instant
t
2
=
−
x
1
/
c
{\displaystyle t_{2}=-x_{1}/c}
, before the emission (advanced solution). The latter, however, violates the causality principle: advanced waves could be detected before their emission. Thus the advanced solutions are usually discarded in the interpretation of electromagnetic waves.
In the absorber theory, instead charged particles are considered as both emitters and absorbers, and the emission process is connected with the absorption process as follows: Both the retarded waves from emitter to absorber and the advanced waves from absorber to emitter are considered. The sum of the two, however, results in causal waves, although the anti-causal (advanced) solutions are not discarded a priori.
Alternatively, the way that Wheeler/Feynman came up with the primary equation is: They assumed that their Lagrangian only interacted when and where the fields for the individual particles were separated by a proper time of zero. So since only massless particles propagate from emission to detection with zero proper time separation, this Lagrangian automatically demands an electromagnetic like interaction.
== New interpretation of radiation damping ==
One of the major results of the absorber theory is the elegant and clear interpretation of the electromagnetic radiation process. A charged particle that experiences acceleration is known to emit electromagnetic waves, i.e., to lose energy. Thus, the Newtonian equation for the particle (
F
=
m
a
{\displaystyle F=ma}
) must contain a dissipative force (damping term), which takes into account this energy loss. In the causal interpretation of electromagnetism, Hendrik Lorentz and Max Abraham proposed that such a force, later called Abraham–Lorentz force, is due to the retarded self-interaction of the particle with its own field. This first interpretation, however, is not completely satisfactory, as it leads to divergences in the theory and needs some assumptions on the structure of charge distribution of the particle. Paul Dirac generalized the formula to make it relativistically invariant. While doing so, he also suggested a different interpretation. He showed that the damping term can be expressed in terms of a free field acting on the particle at its own position:
E
damping
(
x
j
,
t
)
=
E
j
ret
(
x
j
,
t
)
−
E
j
adv
(
x
j
,
t
)
2
.
{\displaystyle E^{\text{damping}}(\mathbf {x} _{j},t)={\frac {E_{j}^{\text{ret}}(\mathbf {x} _{j},t)-E_{j}^{\text{adv}}(\mathbf {x} _{j},t)}{2}}.}
However, Dirac did not propose any physical explanation of this interpretation.
A clear and simple explanation can instead be obtained in the framework of absorber theory, starting from the simple idea that each particle does not interact with itself. This is actually the opposite of the first Abraham–Lorentz proposal. The field acting on the particle
j
{\displaystyle j}
at its own position (the point
x
j
{\displaystyle x_{j}}
) is then
E
tot
(
x
j
,
t
)
=
∑
n
≠
j
E
n
ret
(
x
j
,
t
)
+
E
n
adv
(
x
j
,
t
)
2
.
{\displaystyle E^{\text{tot}}(\mathbf {x} _{j},t)=\sum _{n\neq j}{\frac {E_{n}^{\text{ret}}(\mathbf {x} _{j},t)+E_{n}^{\text{adv}}(\mathbf {x} _{j},t)}{2}}.}
If we sum the free-field term of this expression, we obtain
E
tot
(
x
j
,
t
)
=
∑
n
≠
j
E
n
ret
(
x
j
,
t
)
+
E
n
adv
(
x
j
,
t
)
2
+
∑
n
E
n
ret
(
x
j
,
t
)
−
E
n
adv
(
x
j
,
t
)
2
{\displaystyle E^{\text{tot}}(\mathbf {x} _{j},t)=\sum _{n\neq j}{\frac {E_{n}^{\text{ret}}(\mathbf {x} _{j},t)+E_{n}^{\text{adv}}(\mathbf {x} _{j},t)}{2}}+\sum _{n}{\frac {E_{n}^{\text{ret}}(\mathbf {x} _{j},t)-E_{n}^{\text{adv}}(\mathbf {x} _{j},t)}{2}}}
and, thanks to Dirac's result,
E
tot
(
x
j
,
t
)
=
∑
n
≠
j
E
n
ret
(
x
j
,
t
)
+
E
damping
(
x
j
,
t
)
.
{\displaystyle E^{\text{tot}}(\mathbf {x} _{j},t)=\sum _{n\neq j}E_{n}^{\text{ret}}(\mathbf {x} _{j},t)+E^{\text{damping}}(\mathbf {x} _{j},t).}
Thus, the damping force is obtained without the need for self-interaction, which is known to lead to divergences, and also giving a physical justification to the expression derived by Dirac.
== Developments since original formulation ==
=== Gravity theory ===
Inspired by the Machian nature of the Wheeler–Feynman absorber theory for electrodynamics, Fred Hoyle and Jayant Narlikar proposed their own theory of gravity in the context of general relativity. This model still exists in spite of recent astronomical observations that have challenged the theory. Stephen Hawking had criticized the original Hoyle-Narlikar theory believing that the advanced waves going off to infinity would lead to a divergence, as indeed they would, if the universe were only expanding.
=== Transactional interpretation of quantum mechanics ===
Again inspired by the Wheeler–Feynman absorber theory, the transactional interpretation of quantum mechanics (TIQM) first proposed in 1986 by John G. Cramer, describes quantum interactions in terms of a standing wave formed by retarded (forward-in-time) and advanced (backward-in-time) waves. Cramer claims it avoids the philosophical problems with the Copenhagen interpretation and the role of the observer, and resolves various quantum paradoxes, such as quantum nonlocality, quantum entanglement and retrocausality.
=== Attempted resolution of causality ===
T. C. Scott and R. A. Moore demonstrated that the apparent acausality suggested by the presence of advanced Liénard–Wiechert potentials could be removed by recasting the theory in terms of retarded potentials only, without the complications of the absorber idea.
The Lagrangian describing a particle (
p
1
{\displaystyle p_{1}}
) under the influence of the time-symmetric potential generated by another particle (
p
2
{\displaystyle p_{2}}
) is
L
1
=
T
1
−
1
2
(
(
V
R
)
1
2
+
(
V
A
)
1
2
)
,
{\displaystyle L_{1}=T_{1}-{\frac {1}{2}}\left((V_{R})_{1}^{2}+(V_{A})_{1}^{2}\right),}
where
T
i
{\displaystyle T_{i}}
is the relativistic kinetic energy functional of particle
p
i
{\displaystyle p_{i}}
, and
(
V
R
)
i
j
{\displaystyle (V_{R})_{i}^{j}}
and
(
V
A
)
i
j
{\displaystyle (V_{A})_{i}^{j}}
are respectively the retarded and advanced Liénard–Wiechert potentials acting on particle
p
i
{\displaystyle p_{i}}
and generated by particle
p
j
{\displaystyle p_{j}}
. The corresponding Lagrangian for particle
p
2
{\displaystyle p_{2}}
is
L
2
=
T
2
−
1
2
(
(
V
R
)
2
1
+
(
V
A
)
2
1
)
.
{\displaystyle L_{2}=T_{2}-{\frac {1}{2}}\left((V_{R})_{2}^{1}+(V_{A})_{2}^{1}\right).}
It was originally demonstrated with computer algebra and then proven analytically that
(
V
R
)
j
i
−
(
V
A
)
i
j
{\displaystyle (V_{R})_{j}^{i}-(V_{A})_{i}^{j}}
is a total time derivative, i.e. a divergence in the calculus of variations, and thus it gives no contribution to the Euler–Lagrange equations. Thanks to this result the advanced potentials can be eliminated; here the total derivative plays the same role as the free field. The Lagrangian for the N-body system is therefore
L
=
∑
i
=
1
N
T
i
−
1
2
∑
i
≠
j
N
(
V
R
)
j
i
.
{\displaystyle L=\sum _{i=1}^{N}T_{i}-{\frac {1}{2}}\sum _{i\neq j}^{N}(V_{R})_{j}^{i}.}
The resulting Lagrangian is symmetric under the exchange of
p
i
{\displaystyle p_{i}}
with
p
j
{\displaystyle p_{j}}
. For
N
=
2
{\displaystyle N=2}
this Lagrangian will generate exactly the same equations of motion of
L
1
{\displaystyle L_{1}}
and
L
2
{\displaystyle L_{2}}
. Therefore, from the point of view of an outside observer, everything is causal. This formulation reflects particle-particle symmetry with the variational principle applied to the N-particle system as a whole, and thus Tetrode's Machian principle. Only if we isolate the forces acting on a particular body do the advanced potentials make their appearance. This recasting of the problem comes at a price: the N-body Lagrangian depends on all the time derivatives of the curves traced by all particles, i.e. the Lagrangian is infinite-order. However, much progress was made in examining the unresolved issue of quantizing the theory. Also, this formulation recovers the Darwin Lagrangian, from which the Breit equation was originally derived, but without the dissipative terms. This ensures agreement with theory and experiment, up to but not including the Lamb shift. Numerical solutions for the classical problem were also found. Furthermore, Moore showed that a model by Feynman and Albert Hibbs is amenable to the methods of higher than first-order Lagrangians and revealed chaotic-like solutions. Moore and Scott showed that the radiation reaction can be alternatively derived using the notion that, on average, the net dipole moment is zero for a collection of charged particles, thereby avoiding the complications of the absorber theory.
This apparent acausality may be viewed as merely apparent, and this entire problem goes away. An opposing view was held by Einstein.
=== Alternative Lamb shift calculation ===
As mentioned previously, a serious criticism against the absorber theory is that its Machian assumption that point particles do not act on themselves does not allow (infinite) self-energies and consequently an explanation for the Lamb shift according to quantum electrodynamics (QED). Ed Jaynes proposed an alternate model where the Lamb-like shift is due instead to the interaction with other particles very much along the same notions of the Wheeler–Feynman absorber theory itself. One simple model is to calculate the motion of an oscillator coupled directly with many other oscillators. Jaynes has shown that it is easy to get both spontaneous emission and Lamb shift behavior in classical mechanics. Furthermore, Jaynes' alternative provides a solution to the process of "addition and subtraction of infinities" associated with renormalization.
This model leads to the same type of Bethe logarithm (an essential part of the Lamb shift calculation), vindicating Jaynes' claim that two different physical models can be mathematically isomorphic to each other and therefore yield the same results, a point also apparently made by Scott and Moore on the issue of causality.
== Relationship to quantum field theory ==
This universal absorber theory is mentioned in the chapter titled "Monster Minds" in Feynman's autobiographical work Surely You're Joking, Mr. Feynman! and in Vol. II of the Feynman Lectures on Physics. It led to the formulation of a framework of quantum mechanics using a Lagrangian and action as starting points, rather than a Hamiltonian, namely the formulation using Feynman path integrals, which proved useful in Feynman's earliest calculations in quantum electrodynamics and quantum field theory in general. Both retarded and advanced fields appear respectively as retarded and advanced propagators and also in the Feynman propagator and the Dyson propagator. In hindsight, the relationship between retarded and advanced potentials shown here is not so surprising as, in quantum field theory, the advanced propagator can be obtained from the retarded propagator by exchanging the roles of field source and test particle (usually within the kernel of a Green's function formalism). In quantum field theory, advanced and retarded fields are simply viewed as mathematical solutions of Maxwell's equations whose combinations are decided by the boundary conditions.
== See also ==
Abraham–Lorentz force
Causality
Paradox of radiation of charged particles in a gravitational field
Retrocausality
Symmetry in physics and T-symmetry
Transactional interpretation
Two-state vector formalism
== Notes ==
== Sources ==
Wheeler, J. A.; Feynman, R. P. (April 1945). "Interaction with the Absorber as the Mechanism of Radiation" (PDF). Reviews of Modern Physics. 17 (2–3): 157–181. Bibcode:1945RvMP...17..157W. doi:10.1103/RevModPhys.17.157.
Wheeler, J. A.; Feynman, R. P. (July 1949). "Classical Electrodynamics in Terms of Direct Interparticle Action". Reviews of Modern Physics. 21 (3): 425–433. Bibcode:1949RvMP...21..425W. doi:10.1103/RevModPhys.21.425. | Wikipedia/Wheeler–Feynman_absorber_theory |
In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks that give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Eightfold Way, the successful classification scheme organizing the large number of lighter hadrons that were being discovered starting in the 1950s and continuing through the 1960s. It received experimental verification beginning in the late 1960s and is a valid and effective classification of them to date. The model was independently proposed by physicists Murray Gell-Mann, who dubbed them "quarks" in a concise paper, and George Zweig, who suggested "aces" in a longer manuscript. André Petermann also touched upon the central ideas from 1963 to 1965, without as much quantitative substantiation. Today, the model has essentially been absorbed as a component of the established quantum field theory of strong and electroweak particle interactions, dubbed the Standard Model.
Hadrons are not really "elementary", and can be regarded as bound states of their "valence quarks" and antiquarks, which give rise to the quantum numbers of the hadrons. These quantum numbers are labels identifying the hadrons, and are of two kinds. One set comes from the Poincaré symmetry—JPC, where J, P and C stand for the total angular momentum, P-symmetry, and C-symmetry, respectively.
The other set is the flavor quantum numbers such as the isospin, strangeness, charm, and so on. The strong interactions binding the quarks together are insensitive to these quantum numbers, so variation of them leads to systematic mass and coupling relationships among the hadrons in the same flavor multiplet.
All quarks are assigned a baryon number of 1/3. Up, charm and top quarks have an electric charge of +2/3, while the down, strange, and bottom quarks have an electric charge of −1/3. Antiquarks have the opposite quantum numbers. Quarks are spin-1/2 particles, and thus fermions. Each quark or antiquark obeys the Gell-Mann–Nishijima formula individually, so any additive assembly of them will as well.
Mesons are made of a valence quark–antiquark pair (thus have a baryon number of 0), while baryons are made of three quarks (thus have a baryon number of 1). This article discusses the quark model for the up, down, and strange flavors of quark (which form an approximate flavor SU(3) symmetry). There are generalizations to larger number of flavors.
== History ==
Developing classification schemes for hadrons became a timely question after new experimental techniques uncovered so many of them that it became clear that they could not all be elementary. These discoveries led Wolfgang Pauli to exclaim "Had I foreseen that, I would have gone into botany." and Enrico Fermi to advise his student Leon Lederman: "Young man, if I could remember the names of these particles, I would have been a botanist." These new schemes earned Nobel prizes for experimental particle physicists, including Luis Alvarez, who was at the forefront of many of these developments. Constructing hadrons as bound states of fewer constituents would thus organize the "zoo" at hand. Several early proposals, such as the ones by Enrico Fermi and Chen-Ning Yang (1949), and the Sakata model (1956), ended up satisfactorily covering the mesons, but failed with baryons, and so were unable to explain all the data.
The Gell-Mann–Nishijima formula, developed by Murray Gell-Mann and Kazuhiko Nishijima, led to the Eightfold Way classification, invented by Gell-Mann, with important independent contributions from Yuval Ne'eman, in 1961. The hadrons were organized into SU(3) representation multiplets, octets and decuplets, of roughly the same mass, due to the strong interactions; and smaller mass differences linked to the flavor quantum numbers, invisible to the strong interactions. The Gell-Mann–Okubo mass formula systematized the quantification of these small mass differences among members of a hadronic multiplet, controlled by the explicit symmetry breaking of SU(3).
The spin-3/2 Ω− baryon, a member of the ground-state decuplet, was a crucial prediction of that classification. After it was discovered in an experiment at Brookhaven National Laboratory, Gell-Mann received a Nobel Prize in Physics for his work on the Eightfold Way, in 1969.
Finally, in 1964, Gell-Mann and George Zweig, discerned independently what the Eightfold Way picture encodes: They posited three elementary fermionic constituents—the "up", "down", and "strange" quarks—which are unobserved, and possibly unobservable in a free form. Simple pairwise or triplet combinations of these three constituents and their antiparticles underlie and elegantly encode the Eightfold Way classification, in an economical, tight structure, resulting in further simplicity. Hadronic mass differences were now linked to the different masses of the constituent quarks.
It would take about a decade for the unexpected nature—and physical reality—of these quarks to be appreciated more fully (See Quarks). Counter-intuitively, they cannot ever be observed in isolation (color confinement), but instead always combine with other quarks to form full hadrons, which then furnish ample indirect information on the trapped quarks themselves. Conversely, the quarks serve in the definition of quantum chromodynamics, the fundamental theory fully describing the strong interactions; and the Eightfold Way is now understood to be a consequence of the flavor symmetry structure of the lightest three of them.
== Mesons ==
The Eightfold Way classification is named after the following fact: If we take three flavors of quarks, then the quarks lie in the fundamental representation, 3 (called the triplet) of flavor SU(3). The antiquarks lie in the complex conjugate representation 3. The nine states (nonet) made out of a pair can be decomposed into the trivial representation, 1 (called the singlet), and the adjoint representation, 8 (called the octet). The notation for this decomposition is
3
⊗
3
¯
=
8
⊕
1
.
{\displaystyle \mathbf {3} \otimes \mathbf {\overline {3}} =\mathbf {8} \oplus \mathbf {1} ~.}
Figure 1 shows the application of this decomposition to the mesons. If the flavor symmetry were exact (as in the limit that only the strong interactions operate, but the electroweak interactions are notionally switched off), then all nine mesons would have the same mass. However, the physical content of the full theory includes consideration of the symmetry breaking induced by the quark mass differences, and considerations of mixing between various multiplets (such as the octet and the singlet).
N.B. Nevertheless, the mass splitting between the η and the η′ is larger than the quark model can accommodate, and this "η–η′ puzzle" has its origin in topological peculiarities of the strong interaction vacuum, such as instanton configurations.
Mesons are hadrons with zero baryon number. If the quark–antiquark pair are in an orbital angular momentum L state, and have spin S, then
|L − S| ≤ J ≤ L + S, where S = 0 or 1,
P = (−1)L+1, where the 1 in the exponent arises from the intrinsic parity of the quark–antiquark pair.
C = (−1)L+S for mesons which have no flavor. Flavored mesons have indefinite value of C.
For isospin I = 1 and 0 states, one can define a new multiplicative quantum number called the G-parity such that G = (−1)I+L+S.
If P = (−1)J, then it follows that S = 1, thus PC = 1. States with these quantum numbers are called natural parity states; while all other quantum numbers are thus called exotic (for example, the state JPC = 0−−).
== Baryons ==
Since quarks are fermions, the spin–statistics theorem implies that the wavefunction of a baryon must be antisymmetric under the exchange of any two quarks. This antisymmetric wavefunction is obtained by making it fully antisymmetric in color, discussed below, and symmetric in flavor, spin and space put together. With three flavors, the decomposition in flavor is
3
⊗
3
⊗
3
=
10
S
⊕
8
M
⊕
8
M
⊕
1
A
.
{\displaystyle \mathbf {3} \otimes \mathbf {3} \otimes \mathbf {3} =\mathbf {10} _{S}\oplus \mathbf {8} _{M}\oplus \mathbf {8} _{M}\oplus \mathbf {1} _{A}~.}
The decuplet is symmetric in flavor, the singlet antisymmetric and the two octets have mixed symmetry. The space and spin parts of the states are thereby fixed once the orbital angular momentum is given.
It is sometimes useful to think of the basis states of quarks as the six states of three flavors and two spins per flavor. This approximate symmetry is called spin-flavor SU(6). In terms of this, the decomposition is
6
⊗
6
⊗
6
=
56
S
⊕
70
M
⊕
70
M
⊕
20
A
.
{\displaystyle \mathbf {6} \otimes \mathbf {6} \otimes \mathbf {6} =\mathbf {56} _{S}\oplus \mathbf {70} _{M}\oplus \mathbf {70} _{M}\oplus \mathbf {20} _{A}~.}
The 56 states with symmetric combination of spin and flavour decompose under flavor SU(3) into
56
=
10
3
2
⊕
8
1
2
,
{\displaystyle \mathbf {56} =\mathbf {10} ^{\frac {3}{2}}\oplus \mathbf {8} ^{\frac {1}{2}}~,}
where the superscript denotes the spin, S, of the baryon. Since these states are symmetric in spin and flavor, they should also be symmetric in space—a condition that is easily satisfied by making the orbital angular momentum L = 0. These are the ground-state baryons.
The S = 1/2 octet baryons are the two nucleons (p+, n0), the three Sigmas (Σ+, Σ0, Σ−), the two Xis (Ξ0, Ξ−), and the Lambda (Λ0). The S = 3/2 decuplet baryons are the four Deltas (Δ++, Δ+, Δ0, Δ−), three Sigmas (Σ∗+, Σ∗0, Σ∗−), two Xis (Ξ∗0, Ξ∗−), and the Omega (Ω−).
For example, the constituent quark model wavefunction for the proton is
|
p
↑
⟩
=
1
18
[
2
|
u
↑
d
↓
u
↑
⟩
+
2
|
u
↑
u
↑
d
↓
⟩
+
2
|
d
↓
u
↑
u
↑
⟩
−
|
u
↑
u
↓
d
↑
⟩
−
|
u
↑
d
↑
u
↓
⟩
−
|
u
↓
d
↑
u
↑
⟩
−
|
d
↑
u
↓
u
↑
⟩
−
|
d
↑
u
↑
u
↓
⟩
−
|
u
↓
u
↑
d
↑
⟩
]
.
{\displaystyle |{\text{p}}_{\uparrow }\rangle ={\frac {1}{\sqrt {18}}}[2|{\text{u}}_{\uparrow }{\text{d}}_{\downarrow }{\text{u}}_{\uparrow }\rangle +2|{\text{u}}_{\uparrow }{\text{u}}_{\uparrow }{\text{d}}_{\downarrow }\rangle +2|{\text{d}}_{\downarrow }{\text{u}}_{\uparrow }{\text{u}}_{\uparrow }\rangle -|{\text{u}}_{\uparrow }{\text{u}}_{\downarrow }{\text{d}}_{\uparrow }\rangle -|{\text{u}}_{\uparrow }{\text{d}}_{\uparrow }{\text{u}}_{\downarrow }\rangle -|{\text{u}}_{\downarrow }{\text{d}}_{\uparrow }{\text{u}}_{\uparrow }\rangle -|{\text{d}}_{\uparrow }{\text{u}}_{\downarrow }{\text{u}}_{\uparrow }\rangle -|{\text{d}}_{\uparrow }{\text{u}}_{\uparrow }{\text{u}}_{\downarrow }\rangle -|{\text{u}}_{\downarrow }{\text{u}}_{\uparrow }{\text{d}}_{\uparrow }\rangle ]~.}
Mixing of baryons, mass splittings within and between multiplets, and magnetic moments are some of the other quantities that the model predicts successfully.
The group theory approach described above assumes that the quarks are eight components of a single particle, so the anti-symmetrization applies to all the quarks. A simpler approach is to consider the eight flavored quarks as eight separate, distinguishable, non-identical particles. Then the anti-symmetrization applies only to two identical quarks (like uu, for instance).
Then, the proton wavefunction can be written in a simpler form:
p
(
1
2
,
1
2
)
=
u
u
d
6
[
2
↑↑↓
−
↑↓↑
−
↓↑↑
]
{\displaystyle {\text{p}}\left({\frac {1}{2}},{\frac {1}{2}}\right)={\frac {{\text{u}}{\text{u}}{\text{d}}}{\sqrt {6}}}[2\uparrow \uparrow \downarrow -\uparrow \downarrow \uparrow -\downarrow \uparrow \uparrow ]}
and the
Δ
+
(
3
3
,
3
2
)
=
u
u
d
[
↑↑↑
]
.
{\displaystyle \Delta ^{+}\left({\frac {3}{3}},{\frac {3}{2}}\right)={\text{u}}{\text{u}}{\text{d}}[\uparrow \uparrow \uparrow ]~.}
If quark–quark interactions are limited to two-body interactions, then all the successful quark model predictions, including sum rules for baryon masses and magnetic moments, can be derived.
=== Discovery of color ===
Color quantum numbers are the characteristic charges of the strong force, and are completely uninvolved in electroweak interactions. They were discovered as a consequence of the quark model classification, when it was appreciated that the spin S = 3/2 baryon, the Δ++, required three up quarks with parallel spins and vanishing orbital angular momentum. Therefore, it could not have an antisymmetric wavefunction, (required by the Pauli exclusion principle). Oscar Greenberg noted this problem in 1964, suggesting that quarks should be para-fermions.
Instead, six months later, Moo-Young Han and Yoichiro Nambu suggested the existence of a hidden degree of freedom, they labeled as the group SU(3)' (but later called 'color). This led to three triplets of quarks whose wavefunction was anti-symmetric in the color degree of freedom.
Flavor and color were intertwined in that model: they did not commute.
The modern concept of color completely commuting with all other charges and providing the strong force charge was articulated in 1973, by William Bardeen, Harald Fritzsch, and Murray Gell-Mann.
== States outside the quark model ==
While the quark model is derivable from the theory of quantum chromodynamics, the structure of hadrons is more complicated than this model allows. The full quantum mechanical wavefunction of any hadron must include virtual quark pairs as well as virtual gluons, and allows for a variety of mixings. There may be hadrons which lie outside the quark model. Among these are the glueballs (which contain only valence gluons), hybrids (which contain valence quarks as well as gluons) and exotic hadrons (such as tetraquarks or pentaquarks).
== See also ==
Subatomic particles
Hadrons, baryons, mesons and quarks
Exotic hadrons: exotic mesons and exotic baryons
Quantum chromodynamics, flavor, the QCD vacuum
== Notes ==
== References ==
S. Eidelman et al. Particle Data Group (2004). "Review of Particle Physics" (PDF). Physics Letters B. 592 (1–4): 1. arXiv:astro-ph/0406663. Bibcode:2004PhLB..592....1P. doi:10.1016/j.physletb.2004.06.001. S2CID 118588567.
Lichtenberg, D B (1970). Unitary Symmetry and Elementary Particles. Academic Press. ISBN 978-1483242729.
Thomson, M A (2011), Lecture notes
J.J.J. Kokkedee (1969). The quark model. W. A. Benjamin. ASIN B001RAVDIA. | Wikipedia/Quark_model |
In physics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection):
P
:
(
x
y
z
)
↦
(
−
x
−
y
−
z
)
.
{\displaystyle \mathbf {P} :{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto {\begin{pmatrix}-x\\-y\\-z\end{pmatrix}}.}
It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image.
All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity transformation. As established by the Wu experiment conducted at the US National Bureau of Standards by Chinese-American scientist Chien-Shiung Wu, the weak interaction is chiral and thus provides a means for probing chirality in physics. In her experiment, Wu took advantage of the controlling role of weak interactions in radioactive decay of atomic isotopes to establish the chirality of the weak force.
By contrast, in interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions.
A matrix representation of P (in any number of dimensions) has determinant equal to −1, and hence is distinct from a rotation, which has a determinant equal to 1. In a two-dimensional plane, a simultaneous flip of all coordinates in sign is not a parity transformation; it is the same as a 180° rotation.
In quantum mechanics, wave functions that are unchanged by a parity transformation are described as even functions, while those that change sign under a parity transformation are odd functions.
== Simple symmetry relations ==
Under rotations, classical geometrical objects can be classified into scalars, vectors, and tensors of higher rank. In classical physics, physical configurations need to transform under representations of every symmetry group.
Quantum theory predicts that states in a Hilbert space do not need to transform under representations of the group of rotations, but only under projective representations. The word projective refers to the fact that if one projects out the phase of each state, where we recall that the overall phase of a quantum state is not observable, then a projective representation reduces to an ordinary representation. All representations are also projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states.
The projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. For example, projective representations of the 3-dimensional rotation group, which is the special orthogonal group SO(3), are ordinary representations of the special unitary group SU(2). Projective representations of the rotation group that are not representations are called spinors and so quantum states may transform not only as tensors but also as spinors.
If one adds to this a classification by parity, these can be extended, for example, into notions of
scalars (P = +1) and pseudoscalars (P = −1) which are rotationally invariant.
vectors (P = −1) and axial vectors (also called pseudovectors) (P = +1) which both transform as vectors under rotation.
One can define reflections such as
V
x
:
(
x
y
z
)
↦
(
−
x
y
z
)
,
{\displaystyle V_{x}:{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto {\begin{pmatrix}-x\\y\\z\end{pmatrix}},}
which also have negative determinant and form a valid parity transformation. Then, combining them with rotations (or successively performing x-, y-, and z-reflections) one can recover the particular parity transformation defined earlier. The first parity transformation given does not work in an even number of dimensions, though, because it results in a positive determinant. In even dimensions only the latter example of a parity transformation (or any reflection of an odd number of coordinates) can be used.
Parity forms the abelian group
Z
2
{\displaystyle \mathbb {Z} _{2}}
due to the relation
P
^
2
=
1
^
{\displaystyle {\hat {\mathcal {P}}}^{2}={\hat {1}}}
. All Abelian groups have only one-dimensional irreducible representations. For
Z
2
{\displaystyle \mathbb {Z} _{2}}
, there are two irreducible representations: one is even under parity,
P
^
ϕ
=
+
ϕ
{\displaystyle {\hat {\mathcal {P}}}\phi =+\phi }
, the other is odd,
P
^
ϕ
=
−
ϕ
{\displaystyle {\hat {\mathcal {P}}}\phi =-\phi }
. These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any phase.
=== Representations of O(3) ===
An alternative way to write the above classification of scalars, pseudoscalars, vectors and pseudovectors is in terms of the representation space that each object transforms in. This can be given in terms of the group homomorphism
ρ
{\displaystyle \rho }
which defines the representation. For a matrix
R
∈
O
(
3
)
,
{\displaystyle R\in {\text{O}}(3),}
scalars:
ρ
(
R
)
=
1
{\displaystyle \rho (R)=1}
, the trivial representation
pseudoscalars:
ρ
(
R
)
=
det
(
R
)
{\displaystyle \rho (R)=\det(R)}
vectors:
ρ
(
R
)
=
R
{\displaystyle \rho (R)=R}
, the fundamental representation
pseudovectors:
ρ
(
R
)
=
det
(
R
)
R
.
{\displaystyle \rho (R)=\det(R)R.}
When the representation is restricted to
SO
(
3
)
{\displaystyle {\text{SO}}(3)}
, scalars and pseudoscalars transform identically, as do vectors and pseudovectors.
== Classical mechanics ==
Newton's equation of motion
F
=
m
a
{\displaystyle \mathbf {F} =m\mathbf {a} }
(if the mass is constant) equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity.
However, angular momentum
L
{\displaystyle \mathbf {L} }
is an axial vector,
L
=
r
×
p
P
^
(
L
)
=
(
−
r
)
×
(
−
p
)
=
L
.
{\displaystyle {\begin{aligned}\mathbf {L} &=\mathbf {r} \times \mathbf {p} \\{\hat {P}}\left(\mathbf {L} \right)&=(-\mathbf {r} )\times (-\mathbf {p} )=\mathbf {L} .\end{aligned}}}
In classical electrodynamics, the charge density
ρ
{\displaystyle \rho }
is a scalar, the electric field,
E
{\displaystyle \mathbf {E} }
, and current
j
{\displaystyle \mathbf {j} }
are vectors, but the magnetic field,
B
{\displaystyle \mathbf {B} }
is an axial vector. However, Maxwell's equations are invariant under parity because the curl of an axial vector is a vector.
== Effect of spatial inversion on some variables of classical physics ==
The two major divisions of classical physical variables have either even or odd parity. The way into which particular variables and vectors sort out into either category depends on whether the number of dimensions of space is either an odd or even number. The categories of odd or even given below for the parity transformation is a different, but intimately related issue.
The answers given below are correct for 3 spatial dimensions. In a 2 dimensional space, for example, when constrained to remain on the surface of a planet, some of the variables switch sides.
=== Odd ===
Classical variables whose signs flip under space inversion are predominantly vectors. They include:
=== Even ===
Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include:
== Quantum mechanics ==
=== Possible eigenvalues ===
In quantum mechanics, spacetime transformations act on quantum states. The parity transformation,
P
^
{\displaystyle {\hat {\mathcal {P}}}}
, is a unitary operator, in general acting on a state
ψ
{\displaystyle \psi }
as follows:
P
^
ψ
(
r
)
=
e
i
ϕ
/
2
ψ
(
−
r
)
{\displaystyle {\hat {\mathcal {P}}}\,\psi {\left(r\right)}=e^{{i\phi }/{2}}\psi {\left(-r\right)}}
.
One must then have
P
^
2
ψ
(
r
)
=
e
i
ϕ
ψ
(
r
)
{\displaystyle {\hat {\mathcal {P}}}^{2}\,\psi {\left(r\right)}=e^{i\phi }\psi {\left(r\right)}}
, since an overall phase is unobservable. The operator
P
^
2
{\displaystyle {\hat {\mathcal {P}}}^{2}}
, which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases
e
i
ϕ
{\displaystyle e^{i\phi }}
. If
P
^
2
{\displaystyle {\hat {\mathcal {P}}}^{2}}
is an element
e
i
Q
{\displaystyle e^{iQ}}
of a continuous U(1) symmetry group of phase rotations, then
e
−
i
Q
{\displaystyle e^{-iQ}}
is part of this U(1) and so is also a symmetry. In particular, we can define
P
^
′
≡
P
^
e
−
i
Q
/
2
{\displaystyle {\hat {\mathcal {P}}}'\equiv {\hat {\mathcal {P}}}\,e^{-{iQ}/{2}}}
, which is also a symmetry, and so we can choose to call
P
^
′
{\displaystyle {\hat {\mathcal {P}}}'}
our parity operator, instead of
P
^
{\displaystyle {\hat {\mathcal {P}}}}
. Note that
P
^
′
2
=
1
{\displaystyle {{\hat {\mathcal {P}}}'}^{2}=1}
and so
P
^
′
{\displaystyle {\hat {\mathcal {P}}}'}
has eigenvalues
±
1
{\displaystyle \pm 1}
. Wave functions with eigenvalue
+
1
{\displaystyle +1}
under a parity transformation are even functions, while eigenvalue
−
1
{\displaystyle -1}
corresponds to odd functions. However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than
±
1
{\displaystyle \pm 1}
.
For electronic wavefunctions, even states are usually indicated by a subscript g for gerade (German: even) and odd states by a subscript u for ungerade (German: odd). For example, the lowest energy level of the hydrogen molecule ion (H2+) is labelled
1
σ
g
{\displaystyle 1\sigma _{g}}
and the next-closest (higher) energy level is labelled
1
σ
u
{\displaystyle 1\sigma _{u}}
.
The wave functions of a particle moving into an external potential, which is centrosymmetric (potential energy invariant with respect to a space inversion, symmetric to the origin), either remain invariable or change signs: these two possible states are called the even state or odd state of the wave functions.
The law of conservation of parity of particles states that, if an isolated ensemble of particles has a definite parity, then the parity remains invariable in the process of ensemble evolution. However this is not true for the beta decay of nuclei, because the weak nuclear interaction violates parity.
The parity of the states of a particle moving in a spherically symmetric external field is determined by the angular momentum, and the particle state is defined by three quantum numbers: total energy, angular momentum and the projection of angular momentum.
=== Consequences of parity symmetry ===
When parity generates the Abelian group
Z
2
{\displaystyle \mathbb {Z} _{2}}
, one can always take linear combinations of quantum states such that they are either even or odd under parity (see the figure). Thus the parity of such states is ±1. The parity of a multiparticle state is the product of the parities of each state; in other words parity is a multiplicative quantum number.
In quantum mechanics, Hamiltonians are invariant (symmetric) under a parity transformation if
P
^
{\displaystyle {\hat {\mathcal {P}}}}
commutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any scalar potential, i.e.,
V
=
V
(
r
)
{\displaystyle V=V{\left(r\right)}}
, hence the potential is spherically symmetric. The following facts can be easily proven:
If
|
φ
⟩
{\displaystyle |\varphi \rangle }
and
|
ψ
⟩
{\displaystyle |\psi \rangle }
have the same parity, then
⟨
φ
|
X
^
|
ψ
⟩
=
0
{\displaystyle \langle \varphi |{\hat {X}}|\psi \rangle =0}
where
X
^
{\displaystyle {\hat {X}}}
is the position operator.
For a state
|
L
→
,
L
z
⟩
{\displaystyle {\bigl |}{\vec {L}},L_{z}{\bigr \rangle }}
of orbital angular momentum
L
→
{\displaystyle {\vec {L}}}
with z-axis projection
L
z
{\displaystyle L_{z}}
, then
P
^
|
L
→
,
L
z
⟩
=
(
−
1
)
L
|
L
→
,
L
z
⟩
{\displaystyle {\hat {\mathcal {P}}}{\bigl |}{\vec {L}},L_{z}{\bigr \rangle }=\left(-1\right)^{L}{\bigl |}{\vec {L}},L_{z}{\bigr \rangle }}
.
If
[
H
^
,
P
^
]
=
0
{\displaystyle {\bigl [}{\hat {H}},{\hat {\mathcal {P}}}{\bigr ]}=0}
, then atomic dipole transitions only occur between states of opposite parity.
If
[
H
^
,
P
^
]
=
0
{\displaystyle {\bigl [}{\hat {H}},{\hat {\mathcal {P}}}{\bigr ]}=0}
, then a non-degenerate eigenstate of
H
^
{\displaystyle {\hat {H}}}
is also an eigenstate of the parity operator; i.e., a non-degenerate eigenfunction of
H
^
{\displaystyle {\hat {H}}}
is either invariant to
P
^
{\displaystyle {\hat {\mathcal {P}}}}
or is changed in sign by
P
^
{\displaystyle {\hat {\mathcal {P}}}}
.
Some of the non-degenerate eigenfunctions of
H
^
{\displaystyle {\hat {H}}}
are unaffected (invariant) by parity
P
^
{\displaystyle {\hat {\mathcal {P}}}}
and the others are merely reversed in sign when the Hamiltonian operator and the parity operator commute:
P
^
|
ψ
⟩
=
c
|
ψ
⟩
,
{\displaystyle {\hat {\mathcal {P}}}|\psi \rangle =c\left|\psi \right\rangle ,}
where
c
{\displaystyle c}
is a constant, the eigenvalue of
P
^
{\displaystyle {\hat {\mathcal {P}}}}
,
P
^
2
|
ψ
⟩
=
c
P
^
|
ψ
⟩
.
{\displaystyle {\hat {\mathcal {P}}}^{2}\left|\psi \right\rangle =c\,{\hat {\mathcal {P}}}\left|\psi \right\rangle .}
== Many-particle systems: atoms, molecules, nuclei ==
The overall parity of a many-particle system is the product of the parities of the one-particle states. It is −1 if an odd number of particles are in odd-parity states, and +1 otherwise. Different notations are in use to denote the parity of nuclei, atoms, and molecules.
=== Atoms ===
Atomic orbitals have parity (−1)ℓ, where the exponent ℓ is the azimuthal quantum number. The parity is odd for orbitals p, f, ... with ℓ = 1, 3, ..., and an atomic state has odd parity if an odd number of electrons occupy these orbitals. For example, the ground state of the nitrogen atom has the electron configuration 1s22s22p3, and is identified by the term symbol 4So, where the superscript o denotes odd parity. However the third excited term at about 83,300 cm−1 above the ground state has electron configuration 1s22s22p23s has even parity since there are only two 2p electrons, and its term symbol is 4P (without an o superscript).
=== Molecules ===
The complete (rotational-vibrational-electronic-nuclear spin) electromagnetic Hamiltonian of any molecule commutes with (or is invariant to) the parity operation P (or E*, in the notation introduced by Longuet-Higgins) and its eigenvalues can be given the parity symmetry label + or − as they are even or odd, respectively. The parity operation involves the inversion of electronic and nuclear spatial coordinates at the molecular center of mass.
Centrosymmetric molecules at equilibrium have a centre of symmetry at their midpoint (the nuclear center of mass). This includes all homonuclear diatomic molecules as well as certain symmetric molecules such as ethylene, benzene, xenon tetrafluoride and sulphur hexafluoride. For centrosymmetric molecules, the point group contains the operation i which is not to be confused with the parity operation. The operation i involves the inversion of the electronic and vibrational displacement coordinates at the nuclear centre of mass. For centrosymmetric molecules the operation i commutes with the rovibronic (rotation-vibration-electronic) Hamiltonian and can be used to label such states. Electronic and vibrational states of centrosymmetric molecules are either unchanged by the operation i, or they are changed in sign by i. The former are denoted by the subscript g and are called gerade, while the latter are denoted by the subscript u and are called ungerade. The complete electromagnetic Hamiltonian of a centrosymmetric molecule
does not commute with the point group inversion operation i because of the effect of the nuclear hyperfine Hamiltonian. The nuclear hyperfine Hamiltonian can mix the rotational levels of g and u vibronic states (called ortho-para mixing) and give rise to ortho-para transitions
=== Nuclei ===
In atomic nuclei, the state of each nucleon (proton or neutron) has even or odd parity, and nucleon configurations can be predicted using the nuclear shell model. As for electrons in atoms, the nucleon state has odd overall parity if and only if the number of nucleons in odd-parity states is odd. The parity is usually written as a + (even) or − (odd) following the nuclear spin value. For example, the isotopes of oxygen include 17O(5/2+), meaning that the spin is 5/2 and the parity is even. The shell model explains this because the first 16 nucleons are paired so that each pair has spin zero and even parity, and the last nucleon is in the 1d5/2 shell, which has even parity since ℓ = 2 for a d orbital.
== Quantum field theory ==
If one can show that the vacuum state is invariant under parity,
P
^
|
0
⟩
=
|
0
⟩
{\displaystyle {\hat {\mathcal {P}}}\left|0\right\rangle =\left|0\right\rangle }
, the Hamiltonian is parity invariant
[
H
^
,
P
^
]
{\displaystyle \left[{\hat {H}},{\hat {\mathcal {P}}}\right]}
and the quantization conditions remain unchanged under parity, then it follows that every state has good parity, and this parity is conserved in any reaction.
To show that quantum electrodynamics is invariant under parity, we have to prove that the action is invariant and the quantization is also invariant. For simplicity we will assume that canonical quantization is used; the vacuum state is then invariant under parity by construction. The invariance of the action follows from the classical invariance of Maxwell's equations. The invariance of the canonical quantization procedure can be worked out, and turns out to depend on the transformation of the annihilation operator:
P
a
(
p
,
±
)
P
+
=
a
(
−
p
,
±
)
{\displaystyle \mathbf {Pa} (\mathbf {p} ,\pm )\mathbf {P} ^{+}=\mathbf {a} (-\mathbf {p} ,\pm )}
where
p
{\displaystyle \mathbf {p} }
denotes the momentum of a photon and
±
{\displaystyle \pm }
refers to its polarization state. This is equivalent to the statement that the photon has odd intrinsic parity. Similarly all vector bosons can be shown to have odd intrinsic parity, and all axial-vectors to have even intrinsic parity.
A straightforward extension of these arguments to scalar field theories shows that scalars have even parity. That is,
P
ϕ
(
−
x
,
t
)
P
−
1
=
ϕ
(
x
,
t
)
{\displaystyle {\mathsf {P}}\phi (-\mathbf {x} ,t){\mathsf {P}}^{-1}=\phi (\mathbf {x} ,t)}
, since
P
a
(
p
)
P
+
=
a
(
−
p
)
{\displaystyle \mathbf {Pa} (\mathbf {p} )\mathbf {P} ^{+}=\mathbf {a} (-\mathbf {p} )}
This is true even for a complex scalar field. (Details of spinors are dealt with in the article on the Dirac equation, where it is shown that fermions and antifermions have opposite intrinsic parity.)
With fermions, there is a slight complication because there is more than one spin group.
== Parity in the Standard Model ==
=== Fixing the global symmetries ===
Applying the parity operator twice leaves the coordinates unchanged, meaning that P2 must act as one of the internal symmetries of the theory, at most changing the phase of a state. For example, the Standard Model has three global U(1) symmetries with charges equal to the baryon number B, the lepton number L, and the electric charge Q. Therefore, the parity operator satisfies P2 = eiαB+iβL+iγQ for some choice of α, β, and γ. This operator is also not unique in that a new parity operator P' can always be constructed by multiplying it by an internal symmetry such as P' = P eiαB for some α.
To see if the parity operator can always be defined to satisfy P2 = 1, consider the general case when P2 = Q for some internal symmetry Q present in the theory. The desired parity operator would be P' = PQ−1/2. If Q is part of a continuous symmetry group then Q−1/2 exists, but if it is part of a discrete symmetry then this element need not exist and such a redefinition may not be possible.
The Standard Model exhibits a (−1)F symmetry, where F is the fermion number operator counting how many fermions are in a state. Since all particles in the Standard Model satisfy F = B + L, the discrete symmetry is also part of the eiα(B + L) continuous symmetry group. If the parity operator satisfied P2 = (−1)F, then it can be redefined to give a new parity operator satisfying P2 = 1. But if the Standard Model is extended by incorporating Majorana neutrinos, which have F = 1 and B + L = 0, then the discrete symmetry (−1)F is no longer part of the continuous symmetry group and the desired redefinition of the parity operator cannot be performed. Instead it satisfies P4 = 1 so the Majorana neutrinos would have intrinsic parities of ±i.
=== Parity of the pion ===
In 1954, a paper by William Chinowsky and Jack Steinberger demonstrated that the pion has negative parity.
They studied the decay of an "atom" made from a deuteron (21H+) and a negatively charged pion (π− ) in a state with zero orbital angular momentum
L
=
0
{\displaystyle ~\mathbf {L} ={\boldsymbol {0}}~}
into two neutrons (
n
{\displaystyle n}
).
Neutrons are fermions and so obey Fermi–Dirac statistics, which implies that the final state is antisymmetric. Using the fact that the deuteron has spin one and the pion spin zero together with the antisymmetry of the final state they concluded that the two neutrons must have orbital angular momentum
L
=
1
.
{\displaystyle ~L=1~.}
The total parity is the product of the intrinsic parities of the particles and the extrinsic parity of the spherical harmonic function
(
−
1
)
L
.
{\displaystyle ~\left(-1\right)^{L}~.}
Since the orbital momentum changes from zero to one in this process, if the process is to conserve the total parity then the products of the intrinsic parities of the initial and final particles must have opposite sign. A deuteron nucleus is made from a proton and a neutron, and so using the aforementioned convention that protons and neutrons have intrinsic parities equal to
+
1
{\displaystyle ~+1~}
they argued that the parity of the pion is equal to minus the product of the parities of the two neutrons divided by that of the proton and neutron in the deuteron, explicitly
(
−
1
)
(
1
)
2
(
1
)
2
=
−
1
,
{\textstyle {\frac {(-1)(1)^{2}}{(1)^{2}}}=-1~,}
from which they concluded that the pion is a pseudoscalar particle.
=== Parity violation ===
Although parity is conserved in electromagnetism and gravity, it is violated in weak interactions, and perhaps, to some degree, in strong interactions. The Standard Model incorporates parity violation by expressing the weak interaction as a chiral gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in charged weak interactions in the Standard Model. This implies that parity is not a symmetry of our universe, unless a hidden mirror sector exists in which parity is violated in the opposite way.
An obscure 1928 experiment, undertaken by R. T. Cox, G. C. McIlwraith, and B. Kurrelmeyer, had in effect reported parity violation in weak decays, but, since the appropriate concepts had not yet been developed, those results had no impact. In 1929, Hermann Weyl explored, without any evidence, the existence of a two-component massless particle of spin one-half. This idea was rejected by Pauli, because it implied parity violation.
By the mid-20th century, it had been suggested by several scientists that parity might not be conserved (in different contexts), but without solid evidence these suggestions were not considered important. Then, in 1956, a careful review and analysis by theoretical physicists Tsung-Dao Lee and Chen-Ning Yang went further, showing that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction. They proposed several possible direct experimental tests. They were mostly ignored, but Lee was able to convince his Columbia colleague Chien-Shiung Wu to try it. She needed special cryogenic facilities and expertise, so the experiment was done at the National Bureau of Standards.
Wu, Ambler, Hayward, Hoppes, and Hudson (1957) found a clear violation of parity conservation in the beta decay of cobalt-60. As the experiment was winding down, with double-checking in progress, Wu informed Lee and Yang of their positive results, and saying the results need further examination, she asked them not to publicize the results first. However, Lee revealed the results to his Columbia colleagues on 4 January 1957 at a "Friday lunch" gathering of the Physics Department of Columbia. Three of them, R. L. Garwin, L. M. Lederman, and R. M. Weinrich, modified an existing cyclotron experiment, and immediately verified the parity violation. They delayed publication of their results until after Wu's group was ready, and the two papers appeared back-to-back in the same physics journal.
The discovery of parity violation explained the outstanding τ–θ puzzle in the physics of kaons.
In 2010, it was reported that physicists working with the Relativistic Heavy Ion Collider had created a short-lived parity symmetry-breaking bubble in quark–gluon plasmas. An experiment conducted by several physicists in the STAR collaboration, suggested that parity may also be violated in the strong interaction. It is predicted that this local parity violation manifests itself by chiral magnetic effect.
=== Intrinsic parity of hadrons ===
To every particle one can assign an intrinsic parity as long as nature preserves parity. Although weak interactions do not, one can still assign a parity to any hadron by examining the strong interaction reaction that produces it, or through decays not involving the weak interaction, such as rho meson decay to pions.
== See also ==
C-symmetry
CP violation
Electroweak theory
Mirror matter
Molecular symmetry
T-symmetry
== References ==
Footnotes
Citations
=== Sources ===
Perkins, Donald H. (2000). Introduction to High Energy Physics. Cambridge University Press. ISBN 9780521621960.
Sozzi, M. S. (2008). Discrete symmetries and CP violation. Oxford University Press. ISBN 978-0-19-929666-8.
Bigi, I. I.; Sanda, A. I. (2000). CP Violation. Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Cambridge University Press. ISBN 0-521-44349-0.
Weinberg, S. (1995). The Quantum Theory of Fields. Cambridge University Press. ISBN 0-521-67053-5. | Wikipedia/Parity_(physics) |
In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory.
In classical physics, a classical anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking parameter goes to zero. Perhaps the first known anomaly was the dissipative anomaly in turbulence: time-reversibility remains broken (and energy dissipation rate finite) at the limit of vanishing viscosity.
In quantum theory, the first anomaly discovered was the Adler–Bell–Jackiw anomaly, wherein the axial vector current is conserved as a classical symmetry of electrodynamics, but is broken by the quantized theory. The relationship of this anomaly to the Atiyah–Singer index theorem was one of the celebrated achievements of the theory. Technically, an anomalous symmetry in a quantum theory is a symmetry of the action, but not of the measure, and so not of the partition function as a whole.
== Global anomalies ==
A global anomaly is the quantum violation of a global symmetry current conservation.
A global anomaly can also mean that a non-perturbative global anomaly cannot be captured by one loop or any loop perturbative Feynman diagram calculations—examples include the Witten anomaly and Wang–Wen–Witten anomaly.
=== Scaling and renormalization ===
The most prevalent global anomaly in physics is associated with the violation of scale invariance by quantum corrections, quantified in renormalization.
Since regulators generally introduce a distance scale, the classically scale-invariant theories are subject to renormalization group flow, i.e., changing behavior with energy scale. For example, the large strength of the strong nuclear force results from a theory that is weakly coupled at short distances flowing to a strongly coupled theory at long distances, due to this scale anomaly.
=== Rigid symmetries ===
Anomalies in abelian global symmetries pose no problems in a quantum field theory, and are often encountered (see the example of the chiral anomaly). In particular the corresponding anomalous symmetries can be fixed by fixing the boundary conditions of the path integral.
=== Large gauge transformations ===
Global anomalies in symmetries that approach the identity sufficiently quickly at infinity do, however, pose problems. In known examples such symmetries correspond to disconnected components of gauge symmetries. Such symmetries and possible anomalies occur, for example, in theories with chiral fermions or self-dual differential forms coupled to gravity in 4k + 2 dimensions, and also in the Witten anomaly in an ordinary 4-dimensional SU(2) gauge theory.
As these symmetries vanish at infinity, they cannot be constrained by boundary conditions and so must be summed over in the path integral. The sum of the gauge orbit of a state is a sum of phases which form a subgroup of U(1). As there is an anomaly, not all of these phases are the same, therefore it is not the identity subgroup. The sum of the phases in every other subgroup of U(1) is equal to zero, and so all path integrals are equal to zero when there is such an anomaly and a theory does not exist.
An exception may occur when the space of configurations is itself disconnected, in which case one may have the freedom to choose to integrate over any
subset of the components. If the disconnected gauge symmetries map the system between disconnected configurations, then there is in general a consistent truncation of a theory in which one integrates only over those connected components that are not related by large gauge transformations. In this case the large gauge transformations do not act on the system and do not cause the path integral to vanish.
==== Witten anomaly and Wang–Wen–Witten anomaly ====
In SU(2) gauge theory in 4 dimensional Minkowski space, a gauge transformation corresponds to a choice of an element of the special unitary group SU(2) at each point in spacetime. The group of such gauge transformations is connected.
However, if we are only interested in the subgroup of gauge transformations that vanish at infinity, we may consider the 3-sphere at infinity to be a single point, as the gauge transformations vanish there anyway. If the 3-sphere at infinity is identified with a point, our Minkowski space is identified with the 4-sphere. Thus we see that the group of gauge transformations vanishing at infinity in Minkowski 4-space is isomorphic to the group of all gauge transformations on the 4-sphere.
This is the group which consists of a continuous choice of a gauge transformation in SU(2) for each point on the 4-sphere. In other words, the gauge symmetries are in one-to-one correspondence with maps from the 4-sphere to the 3-sphere, which is the group manifold of SU(2). The space of such maps is not connected, instead the connected components are classified by the fourth homotopy group of the 3-sphere which is the cyclic group of order two. In particular, there are two connected components. One contains the identity and is called the identity component, the other is called the disconnected component.
When a theory contains an odd number of flavors of chiral fermions, the actions of gauge symmetries in the identity component and the disconnected component of the gauge group on a physical state differ by a sign. Thus when one sums over all physical configurations in the path integral, one finds that contributions come in pairs with opposite signs. As a result, all path integrals vanish and a theory does not exist.
The above description of a global anomaly is for the SU(2) gauge theory coupled to an odd number of (iso-)spin-1/2 Weyl fermion in 4 spacetime dimensions. This is known as the Witten SU(2) anomaly. In 2018, it is found by Wang, Wen and Witten that the SU(2) gauge theory coupled to an odd number of (iso-)spin-3/2 Weyl fermion in 4 spacetime dimensions has a further subtler non-perturbative global anomaly detectable on certain non-spin manifolds without spin structure. This new anomaly is called the new SU(2) anomaly. Both types of anomalies have analogs of (1) dynamical gauge anomalies for dynamical gauge theories and (2) the 't Hooft anomalies of global symmetries. In addition, both types of anomalies are mod 2 classes (in terms of classification, they are both finite groups Z2 of order 2 classes), and have analogs in 4 and 5 spacetime dimensions. More generally, for any natural integer N, it can be shown that an odd number of fermion multiplets in representations of (iso)-spin 2N+1/2 can have the SU(2) anomaly; an odd number of fermion multiplets in representations of (iso)-spin 4N+3/2 can have the new SU(2) anomaly. For fermions in the half-integer spin representation, it is shown that there are only these two types of SU(2) anomalies and the linear combinations of these two anomalies; these classify all global SU(2) anomalies. This new SU(2) anomaly also plays an important rule for confirming the consistency of SO(10) grand unified theory, with a Spin(10) gauge group and chiral fermions in the 16-dimensional spinor representations, defined on non-spin manifolds.
=== Higher anomalies involving higher global symmetries: Pure Yang–Mills gauge theory as an example ===
The concept of global symmetries can be generalized to higher global symmetries, such that the charged object for the ordinary 0-form symmetry is a particle, while the charged object for the n-form symmetry is an n-dimensional extended operator. It is found that the 4 dimensional pure Yang–Mills theory with only SU(2) gauge fields with a topological theta term
θ
=
π
,
{\displaystyle \theta =\pi ,}
can have a mixed higher 't Hooft anomaly between the 0-form time-reversal symmetry and 1-form Z2 center symmetry. The 't Hooft anomaly of 4 dimensional pure Yang–Mills theory can be precisely written as a 5 dimensional invertible topological field theory or mathematically a 5 dimensional bordism invariant, generalizing the anomaly inflow picture to this Z2 class of global anomaly involving higher symmetries. In other words, we can regard the 4 dimensional pure Yang–Mills theory with a topological theta term
θ
=
π
{\displaystyle \theta =\pi }
live as a boundary condition of a certain Z2 class invertible topological field theory, in order to match their higher anomalies on the 4 dimensional boundary.
== Gauge anomalies ==
Anomalies in gauge symmetries lead to an inconsistency, since a gauge symmetry is required in order to cancel unphysical degrees of freedom with a negative norm (such as a photon polarized in the time direction). An attempt to cancel them—i.e., to build theories consistent with the gauge symmetries—often leads to extra constraints on the theories (such is the case of the gauge anomaly in the Standard Model of particle physics). Anomalies in gauge theories have important connections to the topology and geometry of the gauge group.
Anomalies in gauge symmetries can be calculated exactly at the one-loop level. At tree level (zero loops), one reproduces the classical theory. Feynman diagrams with more than one loop always contain internal boson propagators. As bosons may always be given a mass without breaking gauge invariance, a Pauli–Villars regularization of such diagrams is possible while preserving the symmetry. Whenever the regularization of a diagram is consistent with a given symmetry, that diagram does not generate an anomaly with respect to the symmetry.
Vector gauge anomalies are always chiral anomalies. Another type of gauge anomaly is the gravitational anomaly.
== At different energy scales ==
Quantum anomalies were discovered via the process of renormalization, when some divergent integrals cannot be regularized in such a way that all the symmetries are preserved simultaneously. This is related to the high energy physics. However, due to Gerard 't Hooft's anomaly matching condition, any chiral anomaly can be described either by the UV degrees of freedom (those relevant at high energies) or by the IR degrees of freedom (those relevant at low energies). Thus one cannot cancel an anomaly by a UV completion of a theory—an anomalous symmetry is simply not a symmetry of a theory, even though classically it appears to be.
== Anomaly cancellation ==
Since cancelling anomalies is necessary for the consistency of gauge theories, such cancellations are of central importance in constraining the fermion content of the standard model, which is a chiral gauge theory.
For example, the vanishing of the mixed anomaly involving two SU(2) generators and one U(1) hypercharge constrains all charges in a fermion generation to add up to zero, and thereby dictates that the sum of the proton plus the sum of the electron vanish: the charges of quarks and leptons must be commensurate.
Specifically, for two external gauge fields Wa, Wb and one hypercharge B at the vertices of the triangle diagram, cancellation of the triangle requires
∑
a
l
l
d
o
u
b
l
e
t
s
T
r
T
a
T
b
Y
∝
δ
a
b
∑
a
l
l
d
o
u
b
l
e
t
s
Y
=
∑
a
l
l
d
o
u
b
l
e
t
s
Q
=
0
,
{\displaystyle \sum _{all~doublets}\!\!\!\!\mathrm {Tr} ~T^{a}T^{b}Y\propto \delta ^{ab}\sum _{all~doublets}Y=\sum _{all~doublets}Q=0~,}
so, for each generation, the charges of the leptons and quarks are balanced,
−
1
+
3
×
2
−
1
3
=
0
{\displaystyle -1+3\times {\frac {2-1}{3}}=0}
, whence Qp + Qe = 0.
The anomaly cancelation in SM was also used to predict a quark from 3rd generation, the top quark.
Further such mechanisms include:
Axion
Chern–Simons
Green–Schwarz mechanism
Liouville action
== Anomalies and cobordism ==
In the modern description of anomalies classified by cobordism theory, the Feynman-Dyson graphs only captures the perturbative local anomalies classified by integer Z classes also known as the free part. There exists nonperturbative global anomalies classified by cyclic groups Z/nZ classes also known as the torsion part.
It is widely known and checked in the late 20th century that the standard model and chiral gauge theories are free from perturbative local anomalies (captured by Feynman diagrams). However, it is not entirely clear whether there are any nonperturbative global anomalies for the standard model and chiral gauge theories.
Recent developments
based on the cobordism theory examine this problem, and several additional nontrivial global anomalies found can further constrain these gauge theories. There is also a formulation of both perturbative local and nonperturbative global description of anomaly inflow in terms of Atiyah, Patodi, and Singer
eta invariant in one higher dimension. This eta invariant is a cobordism invariant whenever the perturbative local anomalies vanish.
== Examples ==
Chiral anomaly
Conformal anomaly (anomaly of scale invariance)
Gauge anomaly
Global anomaly
Gravitational anomaly (also known as diffeomorphism anomaly)
Konishi anomaly
Mixed anomaly
Parity anomaly
't Hooft anomaly
== See also ==
Anomalons, a topic of some debate in the 1980s, anomalons were found in the results of some high-energy physics experiments that seemed to point to the existence of anomalously highly interactive states of matter. The topic was controversial throughout its history.
== References ==
Citations
General
Gravitational Anomalies by Luis Alvarez-Gaumé: This classic paper, which introduces pure gravitational anomalies, contains a good general introduction to anomalies and their relation to regularization and to conserved currents. All occurrences of the number 388 should be read "384". Originally at: ccdb4fs.kek.jp/cgi-bin/img_index?8402145. Springer https://link.springer.com/chapter/10.1007%2F978-1-4757-0280-4_1 | Wikipedia/Anomaly_(physics) |
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3-form.
In condensed-matter physics, Chern–Simons theory describes composite fermions and the topological order in fractional quantum Hall effect states. In mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the Jones polynomial.
Particularly, Chern–Simons theory is specified by a choice of simple Lie group G known as the gauge group of the theory and also a number referred to as the level of the theory, which is a constant that multiplies the action. The action is gauge dependent, however the partition function of the quantum theory is well-defined when the level is an integer and the gauge field strength vanishes on all boundaries of the 3-dimensional spacetime.
It is also the central mathematical object in theoretical models for topological quantum computers (TQC). Specifically, an SU(2) Chern–Simons theory describes the simplest non-abelian anyonic model of a TQC, the Yang–Lee–Fibonacci model.
The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to fusion rules and conformal blocks in conformal field theory, and in particular WZW theory.
== The classical theory ==
=== Mathematical origin ===
In the 1940s S. S. Chern and A. Weil studied the global curvature properties of smooth manifolds M as de Rham cohomology (Chern–Weil theory), which is an important step in the theory of characteristic classes in differential geometry. Given a flat G-principal bundle P on M there exists a unique homomorphism, called the Chern–Weil homomorphism, from the algebra of G-adjoint invariant polynomials on g (Lie algebra of G) to the cohomology
H
∗
(
M
,
R
)
{\displaystyle H^{*}(M,\mathbb {R} )}
. If the invariant polynomial is homogeneous one can write down concretely any k-form of the closed connection ω as some 2k-form of the associated curvature form Ω of ω.
In 1974 S. S. Chern and J. H. Simons had concretely constructed a (2k − 1)-form df(ω) such that
d
T
f
(
ω
)
=
f
(
Ω
k
)
,
{\displaystyle dTf(\omega )=f(\Omega ^{k}),}
where T is the Chern–Weil homomorphism. This form is called Chern–Simons form. If df(ω) is closed one can integrate the above formula
T
f
(
ω
)
=
∫
C
f
(
Ω
k
)
,
{\displaystyle Tf(\omega )=\int _{C}f(\Omega ^{k}),}
where C is a (2k − 1)-dimensional cycle on M. This invariant is called Chern–Simons invariant. As pointed out in the introduction of the Chern–Simons paper, the Chern–Simons invariant CS(M) is the boundary term that cannot be determined by any pure combinatorial formulation. It also can be defined as
CS
(
M
)
=
∫
s
(
M
)
1
2
T
p
1
∈
R
/
Z
,
{\displaystyle \operatorname {CS} (M)=\int _{s(M)}{\tfrac {1}{2}}Tp_{1}\in \mathbb {R} /\mathbb {Z} ,}
where
p
1
{\displaystyle p_{1}}
is the first Pontryagin number and s(M) is the section of the normal orthogonal bundle P. Moreover, the Chern–Simons term is described as the eta invariant defined by Atiyah, Patodi and Singer.
The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory. The action integral (path integral) of the field theory in physics is viewed as the Lagrangian integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle on M. These explain why the Chern–Simons theory is closely related to topological field theory.
=== Configurations ===
Chern–Simons theories can be defined on any topological 3-manifold M, with or without boundary. As these theories are Schwarz-type topological theories, no metric needs to be introduced on M.
Chern–Simons theory is a gauge theory, which means that a classical configuration in the Chern–Simons theory on M with gauge group G is described by a principal G-bundle on M. The connection of this bundle is characterized by a connection one-form A which is valued in the Lie algebra g of the Lie group G. In general the connection A is only defined on individual coordinate patches, and the values of A on different patches are related by maps known as gauge transformations. These are characterized by the assertion that the covariant derivative, which is the sum of the exterior derivative operator d and the connection A, transforms in the adjoint representation of the gauge group G. The square of the covariant derivative with itself can be interpreted as a g-valued 2-form F called the curvature form or field strength. It also transforms in the adjoint representation.
=== Dynamics ===
The action S of Chern–Simons theory is proportional to the integral of the Chern–Simons 3-form
S
=
k
4
π
∫
M
tr
(
A
∧
d
A
+
2
3
A
∧
A
∧
A
)
.
{\displaystyle S={\frac {k}{4\pi }}\int _{M}{\text{tr}}\,(A\wedge dA+{\tfrac {2}{3}}A\wedge A\wedge A).}
The constant k is called the level of the theory. The classical physics of Chern–Simons theory is independent of the choice of level k.
Classically the system is characterized by its equations of motion which are the extrema of the action with respect to variations of the field A. In terms of the field curvature
F
=
d
A
+
A
∧
A
{\displaystyle F=dA+A\wedge A\,}
the field equation is explicitly
0
=
δ
S
δ
A
=
k
2
π
F
.
{\displaystyle 0={\frac {\delta S}{\delta A}}={\frac {k}{2\pi }}F.}
The classical equations of motion are therefore satisfied if and only if the curvature vanishes everywhere, in which case the connection is said to be flat. Thus the classical solutions to G Chern–Simons theory are the flat connections of principal G-bundles on M. Flat connections are determined entirely by holonomies around noncontractible cycles on the base M. More precisely, they are in one-to-one correspondence with equivalence classes of homomorphisms from the fundamental group of M to the gauge group G up to conjugation.
If M has a boundary N then there is additional data which describes a choice of trivialization of the principal G-bundle on N. Such a choice characterizes a map from N to G. The dynamics of this map is described by the Wess–Zumino–Witten (WZW) model on N at level k.
== Quantization ==
To canonically quantize Chern–Simons theory one defines a state on each 2-dimensional surface Σ in M. As in any quantum field theory, the states correspond to rays in a Hilbert space. There is no preferred notion of time in a Schwarz-type topological field theory and so one can require that Σ be a Cauchy surface, in fact, a state can be defined on any surface.
Σ is of codimension one, and so one may cut M along Σ. After such a cutting M will be a manifold with boundary and in particular classically the dynamics of Σ will be described by a WZW model. Witten has shown that this correspondence holds even quantum mechanically. More precisely, he demonstrated that the Hilbert space of states is always finite-dimensional and can be canonically identified with the space of conformal blocks of the G WZW model at level k.
For example, when Σ is a 2-sphere, this Hilbert space is one-dimensional and so there is only one state. When Σ is a 2-torus the states correspond to the integrable representations of the affine Lie algebra corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern–Simons theory.
== Observables ==
=== Wilson loops ===
The observables of Chern–Simons theory are the n-point correlation functions of gauge-invariant operators. The most often studied class of gauge invariant operators are Wilson loops. A Wilson loop is the holonomy around a loop in M, traced in a given representation R of G. As we will be interested in products of Wilson loops, without loss of generality we may restrict our attention to irreducible representations R.
More concretely, given an irreducible representation R and a loop K in M, one may define the Wilson loop
W
R
(
K
)
{\displaystyle W_{R}(K)}
by
W
R
(
K
)
=
Tr
R
P
exp
(
i
∮
K
A
)
{\displaystyle W_{R}(K)=\operatorname {Tr} _{R}\,{\mathcal {P}}\exp \left(i\oint _{K}A\right)}
where A is the connection 1-form and we take the Cauchy principal value of the contour integral and
P
exp
{\displaystyle {\mathcal {P}}\exp }
is the path-ordered exponential.
=== HOMFLY and Jones polynomials ===
Consider a link L in M, which is a collection of ℓ disjoint loops. A particularly interesting observable is the ℓ-point correlation function formed from the product of the Wilson loops around each disjoint loop, each traced in the fundamental representation of G. One may form a normalized correlation function by dividing this observable by the partition function Z(M), which is just the 0-point correlation function.
In the special case in which M is the 3-sphere, Witten has shown that these normalized correlation functions are proportional to known knot polynomials. For example, in G = U(N) Chern–Simons theory at level k the normalized correlation function is, up to a phase, equal to
sin
(
π
/
(
k
+
N
)
)
sin
(
π
N
/
(
k
+
N
)
)
{\displaystyle {\frac {\sin(\pi /(k+N))}{\sin(\pi N/(k+N))}}}
times the HOMFLY polynomial. In particular when N = 2 the HOMFLY polynomial reduces to the Jones polynomial. In the SO(N) case, one finds a similar expression with the Kauffman polynomial.
The phase ambiguity reflects the fact that, as Witten has shown, the quantum correlation functions are not fully defined by the classical data. The linking number of a loop with itself enters into the calculation of the partition function, but this number is not invariant under small deformations and in particular, is not a topological invariant. This number can be rendered well defined if one chooses a framing for each loop, which is a choice of preferred nonzero normal vector at each point along which one deforms the loop to calculate its self-linking number. This procedure is an example of the point-splitting regularization procedure introduced by Paul Dirac and Rudolf Peierls to define apparently divergent quantities in quantum field theory in 1934.
Sir Michael Atiyah has shown that there exists a canonical choice of 2-framing, which is generally used in the literature today and leads to a well-defined linking number. With the canonical framing the above phase is the exponential of 2πi/(k + N) times the linking number of L with itself.
Problem (Extension of Jones polynomial to general 3-manifolds)
"The original Jones polynomial was defined for 1-links in the 3-sphere (the 3-ball, the 3-space R3). Can you define the Jones polynomial for 1-links in any 3-manifold?"
See section 1.1 of this paper for the background and the history of this problem. Kauffman submitted a solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots. It is open in the other cases. Witten's path integral for Jones polynomial is written for links in any compact 3-manifold formally, but the calculus is not done even in physics level in any case other than the 3-sphere (the 3-ball, the 3-space R3). This problem is also open in physics level. In the case of Alexander polynomial, this problem is solved.
== Relationships with other theories ==
=== Topological string theories ===
In the context of string theory, a U(N) Chern–Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifold X arises as the string field theory of open strings ending on a D-brane wrapping X in the A-model topological string theory on X. The B-model topological open string field theory on the spacefilling worldvolume of a stack of D5-branes is a 6-dimensional variant of Chern–Simons theory known as holomorphic Chern–Simons theory.
=== WZW and matrix models ===
Chern–Simons theories are related to many other field theories. For example, if one considers a Chern–Simons theory with gauge group G on a manifold with boundary then all of the 3-dimensional propagating degrees of freedom may be gauged away, leaving a two-dimensional conformal field theory known as a G Wess–Zumino–Witten model on the boundary. In addition the U(N) and SO(N) Chern–Simons theories at large N are well approximated by matrix models.
=== Chern–Simons gravity theory ===
In 1982, S. Deser, R. Jackiw and S. Templeton proposed the Chern–Simons gravity theory in three dimensions, in which the Einstein–Hilbert action in gravity theory is modified by adding the Chern–Simons term. (Deser, Jackiw & Templeton (1982))
In 2003, R. Jackiw and S. Y. Pi extended this theory to four dimensions (Jackiw & Pi (2003)) and Chern–Simons gravity theory has some considerable effects not only to fundamental physics but also condensed matter theory and astronomy.
The four-dimensional case is very analogous to the three-dimensional case. In three dimensions, the gravitational Chern–Simons term is
CS
(
Γ
)
=
1
2
π
2
∫
d
3
x
ε
i
j
k
(
Γ
i
q
p
∂
j
Γ
k
p
q
+
2
3
Γ
i
q
p
Γ
j
r
q
Γ
k
p
r
)
.
{\displaystyle \operatorname {CS} (\Gamma )={\frac {1}{2\pi ^{2}}}\int d^{3}x\varepsilon ^{ijk}{\biggl (}\Gamma _{iq}^{p}\partial _{j}\Gamma _{kp}^{q}+{\frac {2}{3}}\Gamma _{iq}^{p}\Gamma _{jr}^{q}\Gamma _{kp}^{r}{\biggr )}.}
This variation gives the Cotton tensor
=
−
1
2
g
(
ε
m
i
j
D
i
R
j
n
+
ε
n
i
j
D
i
R
j
m
)
.
{\displaystyle =-{\frac {1}{2{\sqrt {g}}}}{\bigl (}\varepsilon ^{mij}D_{i}R_{j}^{n}+\varepsilon ^{nij}D_{i}R_{j}^{m}).}
Then, Chern–Simons modification of three-dimensional gravity is made by adding the above Cotton tensor to the field equation, which can be obtained as the vacuum solution by varying the Einstein–Hilbert action.
=== Chern–Simons matter theories ===
In 2013 Kenneth A. Intriligator and Nathan Seiberg solved these 3d Chern–Simons gauge theories and their phases using monopoles carrying extra degrees of freedom. The Witten index of the many vacua discovered was computed by compactifying the space by turning on mass parameters and then computing the index. In some vacua, supersymmetry was computed to be broken. These monopoles were related to condensed matter vortices. (Intriligator & Seiberg (2013))
The N = 6 Chern–Simons matter theory is the holographic dual of M-theory on
A
d
S
4
×
S
7
{\displaystyle AdS_{4}\times S_{7}}
.
=== Four-dimensional Chern–Simons theory ===
In 2013 Kevin Costello defined a closely related theory defined on a four-dimensional manifold consisting of the product of a two-dimensional 'topological plane' and a two-dimensional (or one complex dimensional) complex curve. He later studied the theory in more detail together with Witten and Masahito Yamazaki, demonstrating how the gauge theory could be related to many notions in integrable systems theory, including exactly solvable lattice models (like the six-vertex model or the XXZ spin chain), integrable quantum field theories (such as the Gross–Neveu model, principal chiral model and symmetric space coset sigma models), the Yang–Baxter equation and quantum groups such as the Yangian which describe symmetries underpinning the integrability of the aforementioned systems.
The action on the 4-manifold
M
=
Σ
×
C
{\displaystyle M=\Sigma \times C}
where
Σ
{\displaystyle \Sigma }
is a two-dimensional manifold and
C
{\displaystyle C}
is a complex curve is
S
=
∫
M
ω
∧
C
S
(
A
)
{\displaystyle S=\int _{M}\omega \wedge CS(A)}
where
ω
{\displaystyle \omega }
is a meromorphic one-form on
C
{\displaystyle C}
.
== Chern–Simons terms in other theories ==
The Chern–Simons term can also be added to models which aren't topological quantum field theories. In 3D, this gives rise to a massive photon if this term is added to the action of Maxwell's theory of electrodynamics. This term can be induced by integrating over a massive charged Dirac field. It also appears for example in the quantum Hall effect. The addition of the Chern–Simons term to various theories gives rise to vortex- or soliton-type solutions Ten- and eleven-dimensional generalizations of Chern–Simons terms appear in the actions of all ten- and eleven-dimensional supergravity theories.
=== One-loop renormalization of the level ===
If one adds matter to a Chern–Simons gauge theory then, in general it is no longer topological. However, if one adds n Majorana fermions then, due to the parity anomaly, when integrated out they lead to a pure Chern–Simons theory with a one-loop renormalization of the Chern–Simons level by −n/2, in other words the level k theory with n fermions is equivalent to the level k − n/2 theory without fermions.
== See also ==
Gauge theory (mathematics)
Chern–Simons form
Topological quantum field theory
Alexander polynomial
Jones polynomial
2+1D topological gravity
Skyrmion
∞-Chern–Simons theory
== References ==
Arthur, K.; Tchrakian, D.H.; Y.-S., Yang (1996). "Topological and nontopological selfdual Chern-Simons solitons in a gauged O(3) sigma model". Physical Review D. 54 (8): 5245–5258. Bibcode:1996PhRvD..54.5245A. doi:10.1103/PhysRevD.54.5245. PMID 10021215.
Chern, S.-S. & Simons, J. (1974). "Characteristic forms and geometric invariants". Annals of Mathematics. 99 (1): 48–69. doi:10.2307/1971013. JSTOR 1971013.
Deser, Stanley; Jackiw, Roman; Templeton, S. (1982). "Three-Dimensional Massive Gauge Theories" (PDF). Physical Review Letters. 48 (15): 975–978. Bibcode:1982PhRvL..48..975D. doi:10.1103/PhysRevLett.48.975. S2CID 122537043.
Intriligator, Kenneth; Seiberg, Nathan (2013). "Aspects of 3d N = 2 Chern–Simons Matter Theories". Journal of High Energy Physics. 2013: 79. arXiv:1305.1633. Bibcode:2013JHEP...07..079I. doi:10.1007/JHEP07(2013)079. S2CID 119106931.
Jackiw, Roman; Pi, S.-Y (2003). "Chern–Simons modification of general relativity". Physical Review D. 68 (10): 104012. arXiv:gr-qc/0308071. Bibcode:2003PhRvD..68j4012J. doi:10.1103/PhysRevD.68.104012. S2CID 2243511.
Kulshreshtha, Usha; Kulshreshtha, D.S.; Mueller-Kirsten, H. J. W.; Vary, J. P. (2009). "Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under appropriate gauge fixing". Physica Scripta . 79 (4): 045001. Bibcode:2009PhyS...79d5001K. doi:10.1088/0031-8949/79/04/045001. S2CID 120594654.
Kulshreshtha, Usha; Kulshreshtha, D.S.; Vary, J. P. (2010). "Light-front Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under appropriate gauge fixing". Physica Scripta. 82 (5): 055101. Bibcode:2010PhyS...82e5101K. doi:10.1088/0031-8949/82/05/055101. S2CID 54602971.
Lopez, Ana; Fradkin, Eduardo (1991). "Fractional quantum Hall effect and Chern-Simons gauge theories". Physical Review B. 44 (10): 5246–5262. Bibcode:1991PhRvB..44.5246L. doi:10.1103/PhysRevB.44.5246. PMID 9998334.
Marino, Marcos (2005). "Chern–Simons Theory and Topological Strings". Reviews of Modern Physics. 77 (2): 675–720. arXiv:hep-th/0406005. Bibcode:2005RvMP...77..675M. doi:10.1103/RevModPhys.77.675. S2CID 6207500.
Marino, Marcos (2005). Chern–Simons Theory, Matrix Models, And Topological Strings. International Series of Monographs on Physics. Oxford University Press.
Witten, Edward (1988). "Topological Quantum Field Theory". Communications in Mathematical Physics. 117 (3): 353–386. Bibcode:1988CMaPh.117..353W. doi:10.1007/BF01223371. S2CID 43230714.
Witten, Edward (1995). "Chern–Simons Theory as a String Theory". Progress in Mathematics. 133: 637–678. arXiv:hep-th/9207094. Bibcode:1992hep.th....7094W.
Specific
== External links ==
"Chern-Simons functional". Encyclopedia of Mathematics. EMS Press. 2001 [1994]. | Wikipedia/Chern–Simons_theory |
Quantum hadrodynamics (QHD) is an effective field theory pertaining to interactions between hadrons, that is, hadron-hadron interactions or the inter-hadron force. It is "a framework for describing the nuclear many-body problem as a relativistic system of baryons and mesons". Quantum hadrodynamics is closely related and partly derived from quantum chromodynamics, which is the theory of interactions between quarks and gluons that bind them together to form hadrons, via the strong force.
An important phenomenon in quantum hadrodynamics is the nuclear force, or residual strong force. It is the force operating between those hadrons which are nucleons – protons and neutrons – as it binds them together to form the atomic nucleus. The bosons which mediate the nuclear force are three types of mesons: pions, rho mesons and omega mesons. Since mesons are themselves hadrons, quantum hadrodynamics also deals with the interaction between the carriers of the nuclear force itself, alongside the nucleons bound by it. The hadrodynamic force keeps nuclei bound, against the electrodynamic force which operates to break them apart (due to the mutual repulsion between protons in the nucleus).
Quantum hadrodynamics, dealing with the nuclear force and its mediating mesons, can be compared to other quantum field theories which describe fundamental forces and their associated bosons: quantum chromodynamics, dealing with the strong interaction and gluons; quantum electrodynamics, dealing with electromagnetism and photons; quantum flavordynamics, dealing with the weak interaction and W and Z bosons.
== See also ==
Atomic nucleus
Hadron
Nuclear force
Quantum chromodynamics and strong interaction
Quantum electrodynamics and electromagnetism
Quantum flavordynamics and weak interaction
== References == | Wikipedia/Quantum_hadrodynamics |
In mathematics and physics, specifically the study of field theory and partial differential equations, a Toda field theory, named after Morikazu Toda, is specified by a choice of Lie algebra and a specific Lagrangian.
== Formulation ==
Fixing the Lie algebra to have rank
r
{\displaystyle r}
, that is, the Cartan subalgebra of the algebra has dimension
r
{\displaystyle r}
, the Lagrangian can be written
L
=
1
2
⟨
∂
μ
ϕ
,
∂
μ
ϕ
⟩
−
m
2
β
2
∑
i
=
1
r
n
i
exp
(
β
⟨
α
i
,
ϕ
⟩
)
.
{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left\langle \partial _{\mu }\phi ,\partial ^{\mu }\phi \right\rangle -{\frac {m^{2}}{\beta ^{2}}}\sum _{i=1}^{r}n_{i}\exp(\beta \langle \alpha _{i},\phi \rangle ).}
The background spacetime is 2-dimensional Minkowski space, with space-like coordinate
x
{\displaystyle x}
and timelike coordinate
t
{\displaystyle t}
. Greek indices indicate spacetime coordinates.
For some choice of root basis,
α
i
{\displaystyle \alpha _{i}}
is the
i
{\displaystyle i}
th simple root. This provides a basis for the Cartan subalgebra, allowing it to be identified with
R
r
{\displaystyle \mathbb {R} ^{r}}
.
Then the field content is a collection of
r
{\displaystyle r}
scalar fields
ϕ
i
{\displaystyle \phi _{i}}
, which are scalar in the sense that they transform trivially under Lorentz transformations of the underlying spacetime.
The inner product
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
is the restriction of the Killing form to the Cartan subalgebra.
The
n
i
{\displaystyle n_{i}}
are integer constants, known as Kac labels or Dynkin labels.
The physical constants are the mass
m
{\displaystyle m}
and the coupling constant
β
{\displaystyle \beta }
.
== Classification of Toda field theories ==
Toda field theories are classified according to their associated Lie algebra.
Toda field theories usually refer to theories with a finite Lie algebra. If the Lie algebra is an affine Lie algebra, it is called an affine Toda field theory (after the component of φ which decouples is removed). If it is hyperbolic, it is called a hyperbolic Toda field theory.
Toda field theories are integrable models and their solutions describe solitons.
== Examples ==
Liouville field theory is associated to the A1 Cartan matrix, which corresponds to the Lie algebra
s
u
(
2
)
{\displaystyle {\mathfrak {su}}(2)}
in the classification of Lie algebras by Cartan matrices. The algebra
s
u
(
2
)
{\displaystyle {\mathfrak {su}}(2)}
has only a single simple root.
The sinh-Gordon model is the affine Toda field theory with the generalized Cartan matrix
(
2
−
2
−
2
2
)
{\displaystyle {\begin{pmatrix}2&-2\\-2&2\end{pmatrix}}}
and a positive value for β after we project out a component of φ which decouples.
The sine-Gordon model is the model with the same Cartan matrix but an imaginary β. This Cartan matrix corresponds to the Lie algebra
s
u
(
2
)
{\displaystyle {\mathfrak {su}}(2)}
. This has a single simple root,
α
1
=
1
{\displaystyle \alpha _{1}=1}
and Coxeter label
n
1
=
1
{\displaystyle n_{1}=1}
, but the Lagrangian is modified for the affine theory: there is also an affine root
α
0
=
−
1
{\displaystyle \alpha _{0}=-1}
and Coxeter label
n
0
=
1
{\displaystyle n_{0}=1}
. One can expand
ϕ
{\displaystyle \phi }
as
ϕ
0
α
0
+
ϕ
1
α
1
{\displaystyle \phi _{0}\alpha _{0}+\phi _{1}\alpha _{1}}
, but for the affine root
⟨
α
0
,
α
0
⟩
=
0
{\displaystyle \langle \alpha _{0},\alpha _{0}\rangle =0}
, so the
ϕ
0
{\displaystyle \phi _{0}}
component decouples.
The sum is
∑
i
=
0
1
n
i
exp
(
β
α
i
ϕ
)
=
exp
(
β
ϕ
)
+
exp
(
−
β
ϕ
)
.
{\displaystyle \sum _{i=0}^{1}n_{i}\exp(\beta \alpha _{i}\phi )=\exp(\beta \phi )+\exp(-\beta \phi ).}
Then if
β
{\displaystyle \beta }
is purely imaginary,
β
=
i
b
{\displaystyle \beta =ib}
with
b
{\displaystyle b}
real and, without loss of generality, positive, then this is
2
cos
(
b
ϕ
)
{\displaystyle 2\cos(b\phi )}
. The Lagrangian is then
L
=
1
2
∂
μ
ϕ
∂
μ
ϕ
+
2
m
2
b
2
cos
(
b
ϕ
)
,
{\displaystyle {\mathcal {L}}={\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi +{\frac {2m^{2}}{b^{2}}}\cos(b\phi ),}
which is the sine-Gordon Lagrangian.
== References ==
Mussardo, Giuseppe (2009), Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics, Oxford University Press, ISBN 978-0-199-54758-6 | Wikipedia/Toda_field_theory |
In quantum field theory, a nonlinear σ model describes a field Σ that takes on values in a nonlinear manifold called the target manifold T. The non-linear σ-model was introduced by Gell-Mann & Lévy (1960, §6), who named it after a field corresponding to a spinless meson called σ in their model. This article deals primarily with the quantization of the non-linear sigma model; please refer to the base article on the sigma model for general definitions and classical (non-quantum) formulations and results.
== Description ==
The target manifold T is equipped with a Riemannian metric g. Σ is a differentiable map from Minkowski space M (or some other space) to T.
The Lagrangian density in contemporary chiral form is given by
L
=
1
2
g
(
∂
μ
Σ
,
∂
μ
Σ
)
−
V
(
Σ
)
{\displaystyle {\mathcal {L}}={1 \over 2}g(\partial ^{\mu }\Sigma ,\partial _{\mu }\Sigma )-V(\Sigma )}
where we have used a + − − − metric signature and the partial derivative ∂Σ is given by a section of the jet bundle of T×M and V is the potential.
In the coordinate notation, with the coordinates Σa, a = 1, ..., n where n is the dimension of T,
L
=
1
2
g
a
b
(
Σ
)
(
∂
μ
Σ
a
)
(
∂
μ
Σ
b
)
−
V
(
Σ
)
.
{\displaystyle {\mathcal {L}}={1 \over 2}g_{ab}(\Sigma )(\partial ^{\mu }\Sigma ^{a})(\partial _{\mu }\Sigma ^{b})-V(\Sigma ).}
In more than two dimensions, nonlinear σ models contain a dimensionful coupling constant and are thus not perturbatively renormalizable.
Nevertheless, they exhibit a non-trivial ultraviolet fixed point of the renormalization group both in the lattice formulation and in the double expansion originally proposed by Kenneth G. Wilson.
In both approaches, the non-trivial renormalization-group fixed point found for the O(n)-symmetric model is seen to simply describe, in dimensions greater than two, the critical point separating the ordered from the disordered phase. In addition, the improved lattice or quantum field theory predictions can then be compared to laboratory experiments on critical phenomena, since the O(n) model describes physical Heisenberg ferromagnets and related systems. The above results point therefore to a failure of naive perturbation theory in describing correctly the physical behavior of the O(n)-symmetric model above two dimensions, and to the need for more sophisticated non-perturbative methods such as the lattice formulation.
This means they can only arise as effective field theories. New physics is needed at around the distance scale where the two point connected correlation function is of the same order as the curvature of the target manifold. This is called the UV completion of the theory. There is a special class of nonlinear σ models with the internal symmetry group G *. If G is a Lie group and H is a Lie subgroup, then the quotient space G/H is a manifold (subject to certain technical restrictions like H being a closed subset) and is also a homogeneous space of G or in other words, a nonlinear realization of G. In many cases, G/H can be equipped with a Riemannian metric which is G-invariant. This is always the case, for example, if G is compact. A nonlinear σ model with G/H as the target manifold with a G-invariant Riemannian metric and a zero potential is called a quotient space (or coset space) nonlinear σ model.
When computing path integrals, the functional measure needs to be "weighted" by the square root of the determinant of g,
det
g
D
Σ
.
{\displaystyle {\sqrt {\det g}}{\mathcal {D}}\Sigma .}
== Renormalization ==
This model proved to be relevant in string theory where the two-dimensional manifold is named worldsheet. Appreciation of its generalized renormalizability was provided by Daniel Friedan. He showed that the theory admits a renormalization group equation, at the leading order of perturbation theory, in the form
λ
∂
g
a
b
∂
λ
=
β
a
b
(
T
−
1
g
)
=
R
a
b
+
O
(
T
2
)
,
{\displaystyle \lambda {\frac {\partial g_{ab}}{\partial \lambda }}=\beta _{ab}(T^{-1}g)=R_{ab}+O(T^{2})~,}
Rab being the Ricci tensor of the target manifold.
This represents a Ricci flow, obeying Einstein field equations for the target manifold as a fixed point. The existence of such a fixed point is relevant, as it grants, at this order of perturbation theory, that conformal invariance is not lost due to quantum corrections, so that the quantum field theory of this model is sensible (renormalizable).
Further adding nonlinear interactions representing flavor-chiral anomalies results in the Wess–Zumino–Witten model, which
augments the geometry of the flow to include torsion, preserving renormalizability and leading to an infrared fixed point as well, on account of teleparallelism ("geometrostasis").
== O(3) non-linear sigma model ==
A celebrated example, of particular interest due to its topological properties, is the O(3) nonlinear σ-model in 1 + 1 dimensions, with the Lagrangian density
L
=
1
2
∂
μ
n
^
⋅
∂
μ
n
^
{\displaystyle {\mathcal {L}}={\tfrac {1}{2}}\ \partial ^{\mu }{\hat {n}}\cdot \partial _{\mu }{\hat {n}}}
where n̂=(n1, n2, n3) with the constraint n̂⋅n̂=1 and μ=1,2.
This model allows for topological finite action solutions, as at infinite space-time the Lagrangian density must vanish, meaning n̂ = constant at infinity. Therefore, in the class of finite-action solutions, one may identify the points at infinity as a single point, i.e. that space-time can be identified with a Riemann sphere.
Since the n̂-field lives on a sphere as well, the mapping S2→ S2 is in evidence, the solutions of which are classified by the second homotopy group of a 2-sphere: These solutions are called the O(3) Instantons.
This model can also be considered in 1+2 dimensions, where the topology now comes only from the spatial slices. These are modelled as R^2 with a point at infinity, and hence have the same topology as the O(3) instantons in 1+1 dimensions. They are called sigma model lumps.
== See also ==
Sigma model
Chiral model
Little Higgs
Skyrmion, a soliton in non-linear sigma models
Polyakov action
WZW model
Fubini–Study metric, a metric often used with non-linear sigma models
Ricci flow
Scale invariance
== References ==
== External links ==
Ketov, Sergei (2009). "Nonlinear Sigma model". Scholarpedia. 4 (1): 8508. Bibcode:2009SchpJ...4.8508K. doi:10.4249/scholarpedia.8508.
Kulshreshtha, U.; Kulshreshtha, D. S. (2002). "Front-Form Hamiltonian, Path Integral, and BRST Formulations of the Nonlinear Sigma Model". International Journal of Theoretical Physics. 41 (10): 1941–1956. doi:10.1023/A:1021009008129. S2CID 115710780. | Wikipedia/Non-linear_sigma_model |
The Russo–Susskind–Thorlacius model or RST model in short is a modification of the CGHS model to take care of conformal anomalies and render it analytically soluble. In the CGHS model, if we include Faddeev–Popov ghosts to gauge-fix diffeomorphisms in the conformal gauge, they contribute an anomaly of -24. Each matter field contributes an anomaly of 1. So, unless N=24, we will have gravitational anomalies.
To the CGHS action
S
CGHS
=
1
2
π
∫
d
2
x
−
g
{
e
−
2
ϕ
[
R
+
4
(
∇
ϕ
)
2
+
4
λ
2
]
−
∑
i
=
1
N
1
2
(
∇
f
i
)
2
}
{\displaystyle S_{\text{CGHS}}={\frac {1}{2\pi }}\int d^{2}x\,{\sqrt {-g}}\left\{e^{-2\phi }\left[R+4\left(\nabla \phi \right)^{2}+4\lambda ^{2}\right]-\sum _{i=1}^{N}{\frac {1}{2}}\left(\nabla f_{i}\right)^{2}\right\}}
, the following term
S
RST
=
−
κ
8
π
∫
d
2
x
−
g
[
R
1
∇
2
R
−
2
ϕ
R
]
{\displaystyle S_{\text{RST}}=-{\frac {\kappa }{8\pi }}\int d^{2}x\,{\sqrt {-g}}\left[R{\frac {1}{\nabla ^{2}}}R-2\phi R\right]}
is added, where κ is either
(
N
−
24
)
/
12
{\displaystyle (N-24)/12}
or
N
/
12
{\displaystyle N/12}
depending upon whether ghosts are considered. The nonlocal term leads to nonlocality.
In the conformal gauge,
S
RST
=
−
κ
π
∫
d
x
+
d
x
−
[
∂
+
ρ
∂
−
ρ
+
ϕ
∂
+
∂
−
ρ
]
{\displaystyle S_{\text{RST}}=-{\frac {\kappa }{\pi }}\int dx^{+}\,dx^{-}\left[\partial _{+}\rho \partial _{-}\rho +\phi \partial _{+}\partial _{-}\rho \right]}
.
It might appear as if the theory is local in the conformal gauge, but this overlooks the fact that the Raychaudhuri equations are still nonlocal.
== References == | Wikipedia/RST_model |
The soler model is a quantum field theory model of Dirac fermions interacting via four fermion interactions in 3 spatial and 1 time dimension. It was introduced in 1938 by Dmitri Ivanenko
and re-introduced and investigated in 1970 by Mario Soler as a toy model of self-interacting electron.
This model is described by the Lagrangian density
L
=
ψ
¯
(
i
∂
/
−
m
)
ψ
+
g
2
(
ψ
¯
ψ
)
2
{\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +{\frac {g}{2}}\left({\overline {\psi }}\psi \right)^{2}}
where
g
{\displaystyle g}
is the coupling constant,
∂
/
=
∑
μ
=
0
3
γ
μ
∂
∂
x
μ
{\displaystyle \partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}}
in the Feynman slash notations,
ψ
¯
=
ψ
∗
γ
0
{\displaystyle {\overline {\psi }}=\psi ^{*}\gamma ^{0}}
.
Here
γ
μ
{\displaystyle \gamma ^{\mu }}
,
0
≤
μ
≤
3
{\displaystyle 0\leq \mu \leq 3}
, are Dirac gamma matrices.
The corresponding equation can be written as
i
∂
∂
t
ψ
=
−
i
∑
j
=
1
3
α
j
∂
∂
x
j
ψ
+
m
β
ψ
−
g
(
ψ
¯
ψ
)
β
ψ
{\displaystyle i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi }
,
where
α
j
{\displaystyle \alpha ^{j}}
,
1
≤
j
≤
3
{\displaystyle 1\leq j\leq 3}
,
and
β
{\displaystyle \beta }
are the Dirac matrices.
In one dimension,
this model is known as the massive Gross–Neveu model.
== Generalizations ==
A commonly considered generalization is
L
=
ψ
¯
(
i
∂
/
−
m
)
ψ
+
g
(
ψ
¯
ψ
)
k
+
1
k
+
1
{\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +g{\frac {\left({\overline {\psi }}\psi \right)^{k+1}}{k+1}}}
with
k
>
0
{\displaystyle k>0}
, or even
L
=
ψ
¯
(
i
∂
/
−
m
)
ψ
+
F
(
ψ
¯
ψ
)
{\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +F\left({\overline {\psi }}\psi \right)}
,
where
F
{\displaystyle F}
is a smooth function.
== Features ==
=== Internal symmetry ===
Besides the unitary symmetry U(1),
in dimensions 1, 2, and 3
the equation has SU(1,1) global internal symmetry.
=== Renormalizability ===
The Soler model is renormalizable by the power counting for
k
=
1
{\displaystyle k=1}
and in one dimension only,
and non-renormalizable for higher values of
k
{\displaystyle k}
and in higher dimensions.
=== Solitary wave solutions ===
The Soler model admits solitary wave solutions
of the form
ϕ
(
x
)
e
−
i
ω
t
,
{\displaystyle \phi (x)e^{-i\omega t},}
where
ϕ
{\displaystyle \phi }
is localized (becomes small when
x
{\displaystyle x}
is large)
and
ω
{\displaystyle \omega }
is a real number.
=== Reduction to the massive Thirring model ===
In spatial dimension 2, the Soler model coincides with the massive Thirring model,
due to the relation
(
ψ
¯
ψ
)
2
=
J
μ
J
μ
{\displaystyle ({\bar {\psi }}\psi )^{2}=J_{\mu }J^{\mu }}
,
with
ψ
¯
ψ
=
ψ
∗
σ
3
ψ
{\displaystyle {\bar {\psi }}\psi =\psi ^{*}\sigma _{3}\psi }
the relativistic scalar
and
J
μ
=
(
ψ
∗
ψ
,
ψ
∗
σ
1
ψ
,
ψ
∗
σ
2
ψ
)
{\displaystyle J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )}
the charge-current density.
The relation follows from the identity
(
ψ
∗
σ
1
ψ
)
2
+
(
ψ
∗
σ
2
ψ
)
2
+
(
ψ
∗
σ
3
ψ
)
2
=
(
ψ
∗
ψ
)
2
{\displaystyle (\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}}
,
for any
ψ
∈
C
2
{\displaystyle \psi \in \mathbb {C} ^{2}}
.
== See also ==
Dirac equation
Gross–Neveu model
Nonlinear Dirac equation
Thirring model
== References == | Wikipedia/Soler_model |
Group field theory (GFT) is a quantum field theory in which the base manifold is taken to be a Lie group. It is closely related to background independent quantum gravity approaches such as loop quantum gravity, the spin foam formalism and causal dynamical triangulation. Its perturbative expansion can be interpreted as spin foams and simplicial pseudo-manifolds (depending on the representation of the fields). Thus, its partition function defines a non-perturbative sum over all simplicial topologies and geometries, giving a path integral formulation of quantum spacetime.
== See also ==
Shape dynamics
Causal Sets
Fractal cosmology
Loop quantum gravity
Planck scale
Quantum gravity
Regge calculus
Simplex
Simplicial manifold
Spin foam
== References ==
Wayback Machine see Sec 6.8 Dynamics: III. Group field theory
Freidel, L. (2005). "Group Field Theory: An Overview". International Journal of Theoretical Physics. 44 (10): 1769–1783. arXiv:hep-th/0505016. Bibcode:2005IJTP...44.1769F. doi:10.1007/s10773-005-8894-1. S2CID 119099369.
Oriti, Daniele (2006). "The group field theory approach to quantum gravity". arXiv:gr-qc/0607032. Bibcode:2006gr.qc.....7032O. {{cite journal}}: Cite journal requires |journal= (help)
Oriti, Daniele (2009). "The Group Field Theory Approach to Quantum Gravity: A QFT for the Microstructure of Spacetime" (PDF). arXiv:0912.2441. {{cite journal}}: Cite journal requires |journal= (help)
Geloun, Joseph Ben; Krajewski, Thomas; Magnen, Jacques; Rivasseau, Vincent (2010). "Linearized group field theory and power-counting theorems". Classical and Quantum Gravity. 27 (15): 155012. arXiv:1002.3592. Bibcode:2010CQGra..27o5012B. doi:10.1088/0264-9381/27/15/155012. S2CID 29020457.
Ben Geloun, J.; Gurau, R.; Rivasseau, V. (2010). "EPRL/FK group field theory". Europhysics Letters. 92 (6): 60008. arXiv:1008.0354. Bibcode:2010EL.....9260008B. doi:10.1209/0295-5075/92/60008. S2CID 119247896.
Ashtekar, Abhay; Campiglia, Miguel; Henderson, Adam (2009). "Loop quantum cosmology and spin foams". Physics Letters B. 681 (4): 347–352. arXiv:0909.4221. Bibcode:2009PhLB..681..347A. doi:10.1016/j.physletb.2009.10.042. S2CID 56281948.
Fairbairn, Winston J.; Livine, Etera R. (2007). "3D spinfoam quantum gravity: Matter as a phase of the group field theory". Classical and Quantum Gravity. 24 (20): 5277–5297. arXiv:gr-qc/0702125. Bibcode:2007CQGra..24.5277F. doi:10.1088/0264-9381/24/20/021. S2CID 119369221.
Alexandrov, Sergei; Roche, Philippe (2011). "Critical overview of loops and foams". Physics Reports. 506 (3–4): 41–86. arXiv:1009.4475. Bibcode:2011PhR...506...41A. doi:10.1016/j.physrep.2011.05.002. S2CID 118543391.
Gielen, Steffen; Oriti, Daniele; Sindoni, Lorenzo (2013). "Cosmology from Group Field Theory Formalism for Quantum Gravity". Physical Review Letters. 111 (3): 031301. arXiv:1303.3576. Bibcode:2013PhRvL.111c1301G. doi:10.1103/PhysRevLett.111.031301. PMID 23909305. S2CID 14203682. | Wikipedia/Group_field_theory |
In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle (specifically, the adjoint bundle). These equations are
D
A
∗
F
A
+
∗
[
Φ
,
D
A
Φ
]
=
0
,
D
A
∗
D
A
Φ
=
0
{\displaystyle {\begin{aligned}D_{A}*F_{A}+*[\Phi ,D_{A}\Phi ]&=0,\\D_{A}*D_{A}\Phi &=0\end{aligned}}}
with a boundary condition
lim
|
x
|
→
∞
|
Φ
|
(
x
)
=
1
{\displaystyle \lim _{|x|\rightarrow \infty }|\Phi |(x)=1}
where
A is a connection on a vector bundle,
DA is the exterior covariant derivative,
FA is the curvature of that connection,
Φ is a section of that vector bundle,
∗ is the Hodge star, and
[·,·] is the natural, graded bracket.
These equations are named after Chen Ning Yang, Robert Mills, and Peter Higgs. They are very closely related to the Ginzburg–Landau equations, when these are expressed in a general geometric setting.
M.V. Goganov and L.V. Kapitanskii have shown that the Cauchy problem for hyperbolic Yang–Mills–Higgs equations in Hamiltonian gauge on 4-dimensional Minkowski space have a unique global solution with no restrictions at the spatial infinity. Furthermore, the solution has the finite propagation speed property.
== Lagrangian ==
The equations arise as the equations of motion of the Lagrangian density
where
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
is an invariant symmetric bilinear form on the adjoint bundle. This is sometimes written as
tr
{\displaystyle {\text{tr}}}
due to the fact that such a form can arise from the trace on
g
{\displaystyle {\mathfrak {g}}}
under some representation; in particular here we are concerned with the adjoint representation, and the trace on this representation is the Killing form.
For the particular form of the Yang–Mills–Higgs equations given above, the potential
V
(
ϕ
)
{\displaystyle V(\phi )}
is vanishing. Another common choice is
V
(
ϕ
)
=
1
2
m
2
⟨
ϕ
,
ϕ
⟩
{\displaystyle V(\phi )={\frac {1}{2}}m^{2}\langle \phi ,\phi \rangle }
, corresponding to a massive Higgs field.
This theory is a particular case of scalar chromodynamics where the Higgs field
ϕ
{\displaystyle \phi }
is valued in the adjoint representation as opposed to a general representation.
== See also ==
Yang–Mills equations
Stable Yang–Mills–Higgs pair
Scalar chromodynamics
== References ==
M.V. Goganov and L.V. Kapitansii, "Global solvability of the initial problem for Yang-Mills-Higgs equations", Zapiski LOMI 147,18–48, (1985); J. Sov. Math, 37, 802–822 (1987). | Wikipedia/Yang–Mills–Higgs_equations |
The sine-Gordon equation is a second-order nonlinear partial differential equation for a function
φ
{\displaystyle \varphi }
dependent on two variables typically denoted
x
{\displaystyle x}
and
t
{\displaystyle t}
, involving the wave operator and the sine of
φ
{\displaystyle \varphi }
.
It was originally introduced by Edmond Bour (1862) in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of constant Gaussian curvature −1 in 3-dimensional space. The equation was rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova model.
This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions, and is an example of an integrable PDE. Among well-known integrable PDEs, the sine-Gordon equation is the only relativistic system due to its Lorentz invariance.
== Realizations of the sine-Gordon equation ==
=== Differential geometry ===
This is the first derivation of the equation, by Bour (1862).
There are two equivalent forms of the sine-Gordon equation. In the (real) space-time coordinates, denoted
(
x
,
t
)
{\displaystyle (x,t)}
, the equation reads:
φ
t
t
−
φ
x
x
+
sin
φ
=
0
,
{\displaystyle \varphi _{tt}-\varphi _{xx}+\sin \varphi =0,}
where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (u, v), akin to asymptotic coordinates where
u
=
x
+
t
2
,
v
=
x
−
t
2
,
{\displaystyle u={\frac {x+t}{2}},\quad v={\frac {x-t}{2}},}
the equation takes the form
φ
u
v
=
sin
φ
.
{\displaystyle \varphi _{uv}=\sin \varphi .}
This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of surfaces of constant Gaussian curvature K = −1, also called pseudospherical surfaces.
Consider an arbitrary pseudospherical surface. Across every point on the surface there are two asymptotic curves. This allows us to construct a distinguished coordinate system for such a surface, in which u = constant, v = constant are the asymptotic lines, and the coordinates are incremented by the arc length on the surface. At every point on the surface, let
φ
{\displaystyle \varphi }
be the angle between the asymptotic lines.
The first fundamental form of the surface is
d
s
2
=
d
u
2
+
2
cos
φ
d
u
d
v
+
d
v
2
,
{\displaystyle ds^{2}=du^{2}+2\cos \varphi \,du\,dv+dv^{2},}
and the second fundamental form is
L
=
N
=
0
,
M
=
sin
φ
{\displaystyle L=N=0,M=\sin \varphi }
and the Gauss–Codazzi equation is
φ
u
v
=
sin
φ
.
{\displaystyle \varphi _{uv}=\sin \varphi .}
Thus, any pseudospherical surface gives rise to a solution of the sine-Gordon equation, although with some caveats: if the surface is complete, it is necessarily singular due to the Hilbert embedding theorem. In the simplest case, the pseudosphere, also known as the tractroid, corresponds to a static one-soliton, but the tractroid has a singular cusp at its equator.
Conversely, one can start with a solution to the sine-Gordon equation to obtain a pseudosphere uniquely up to rigid transformations. There is a theorem, sometimes called the fundamental theorem of surfaces, that if a pair of matrix-valued bilinear forms satisfy the Gauss–Codazzi equations, then they are the first and second fundamental forms of an embedded surface in 3-dimensional space. Solutions to the sine-Gordon equation can be used to construct such matrices by using the forms obtained above.
=== New solutions from old ===
The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by Bianchi and Bäcklund led to the discovery of Bäcklund transformations. Another transformation of pseudospherical surfaces is the Lie transform introduced by Sophus Lie in 1879, which corresponds to Lorentz boosts for solutions of the sine-Gordon equation.
There are also some more straightforward ways to construct new solutions but which do not give new surfaces. Since the sine-Gordon equation is odd, the negative of any solution is another solution. However this does not give a new surface, as the sign-change comes down to a choice of direction for the normal to the surface. New solutions can be found by translating the solution: if
φ
{\displaystyle \varphi }
is a solution, then so is
φ
+
2
n
π
{\displaystyle \varphi +2n\pi }
for
n
{\displaystyle n}
an integer.
=== Frenkel–Kontorova model ===
=== A mechanical model ===
Consider a line of pendula, hanging on a straight line, in constant gravity. Connect the bobs of the pendula together by a string in constant tension. Let the angle of the pendulum at location
x
{\displaystyle x}
be
φ
{\displaystyle \varphi }
, then schematically, the dynamics of the line of pendulum follows Newton's second law:
m
φ
t
t
⏟
mass times acceleration
=
T
φ
x
x
⏟
tension
−
m
g
sin
φ
⏟
gravity
{\displaystyle \underbrace {m\varphi _{tt}} _{\text{mass times acceleration}}=\underbrace {T\varphi _{xx}} _{\text{tension}}-\underbrace {mg\sin \varphi } _{\text{gravity}}}
and this is the sine-Gordon equation, after scaling time and distance appropriately.
Note that this is not exactly correct, since the net force on a pendulum due to the tension is not precisely
T
φ
x
x
{\displaystyle T\varphi _{xx}}
, but more accurately
T
φ
x
x
(
1
+
φ
x
2
)
−
3
/
2
{\displaystyle T\varphi _{xx}(1+\varphi _{x}^{2})^{-3/2}}
. However this does give an intuitive picture for the sine-gordon equation. One can produce exact mechanical realizations of the sine-gordon equation by more complex methods.
== Naming ==
The name "sine-Gordon equation" is a pun on the well-known Klein–Gordon equation in physics:
φ
t
t
−
φ
x
x
+
φ
=
0.
{\displaystyle \varphi _{tt}-\varphi _{xx}+\varphi =0.}
The sine-Gordon equation is the Euler–Lagrange equation of the field whose Lagrangian density is given by
L
SG
(
φ
)
=
1
2
(
φ
t
2
−
φ
x
2
)
−
1
+
cos
φ
.
{\displaystyle {\mathcal {L}}_{\text{SG}}(\varphi )={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-1+\cos \varphi .}
Using the Taylor series expansion of the cosine in the Lagrangian,
cos
(
φ
)
=
∑
n
=
0
∞
(
−
φ
2
)
n
(
2
n
)
!
,
{\displaystyle \cos(\varphi )=\sum _{n=0}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}},}
it can be rewritten as the Klein–Gordon Lagrangian plus higher-order terms:
L
SG
(
φ
)
=
1
2
(
φ
t
2
−
φ
x
2
)
−
φ
2
2
+
∑
n
=
2
∞
(
−
φ
2
)
n
(
2
n
)
!
=
L
KG
(
φ
)
+
∑
n
=
2
∞
(
−
φ
2
)
n
(
2
n
)
!
.
{\displaystyle {\begin{aligned}{\mathcal {L}}_{\text{SG}}(\varphi )&={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-{\frac {\varphi ^{2}}{2}}+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}\\&={\mathcal {L}}_{\text{KG}}(\varphi )+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}.\end{aligned}}}
== Soliton solutions ==
An interesting feature of the sine-Gordon equation is the existence of soliton and multisoliton solutions.
=== 1-soliton solutions ===
The sine-Gordon equation has the following 1-soliton solutions:
φ
soliton
(
x
,
t
)
:=
4
arctan
(
e
m
γ
(
x
−
v
t
)
+
δ
)
,
{\displaystyle \varphi _{\text{soliton}}(x,t):=4\arctan \left(e^{m\gamma (x-vt)+\delta }\right),}
where
γ
2
=
1
1
−
v
2
,
{\displaystyle \gamma ^{2}={\frac {1}{1-v^{2}}},}
and the slightly more general form of the equation is assumed:
φ
t
t
−
φ
x
x
+
m
2
sin
φ
=
0.
{\displaystyle \varphi _{tt}-\varphi _{xx}+m^{2}\sin \varphi =0.}
The 1-soliton solution for which we have chosen the positive root for
γ
{\displaystyle \gamma }
is called a kink and represents a twist in the variable
φ
{\displaystyle \varphi }
which takes the system from one constant solution
φ
=
0
{\displaystyle \varphi =0}
to an adjacent constant solution
φ
=
2
π
{\displaystyle \varphi =2\pi }
. The states
φ
≅
2
π
n
{\displaystyle \varphi \cong 2\pi n}
are known as vacuum states, as they are constant solutions of zero energy. The 1-soliton solution in which we take the negative root for
γ
{\displaystyle \gamma }
is called an antikink. The form of the 1-soliton solutions can be obtained through application of a Bäcklund transform to the trivial (vacuum) solution and the integration of the resulting first-order differentials:
φ
u
′
=
φ
u
+
2
β
sin
φ
′
+
φ
2
,
{\displaystyle \varphi '_{u}=\varphi _{u}+2\beta \sin {\frac {\varphi '+\varphi }{2}},}
φ
v
′
=
−
φ
v
+
2
β
sin
φ
′
−
φ
2
with
φ
=
φ
0
=
0
{\displaystyle \varphi '_{v}=-\varphi _{v}+{\frac {2}{\beta }}\sin {\frac {\varphi '-\varphi }{2}}{\text{ with }}\varphi =\varphi _{0}=0}
for all time.
The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model introduced by Julio Rubinstein in 1970. Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge
θ
K
=
−
1
{\displaystyle \theta _{\text{K}}=-1}
. The alternative counterclockwise (right-handed) twist with topological charge
θ
AK
=
+
1
{\displaystyle \theta _{\text{AK}}=+1}
will be an antikink.
=== 2-soliton solutions ===
Multi-soliton solutions can be obtained through continued application of the Bäcklund transform to the 1-soliton solution, as prescribed by a Bianchi lattice relating the transformed results. The 2-soliton solutions of the sine-Gordon equation show some of the characteristic features of the solitons. The traveling sine-Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a phase shift. Since the colliding solitons recover their velocity and shape, such an interaction is called an elastic collision.
The kink-kink solution is given by
φ
K
/
K
(
x
,
t
)
=
4
arctan
(
v
sinh
x
1
−
v
2
cosh
v
t
1
−
v
2
)
{\displaystyle \varphi _{K/K}(x,t)=4\arctan \left({\frac {v\sinh {\frac {x}{\sqrt {1-v^{2}}}}}{\cosh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}
while the kink-antikink solution is given by
φ
K
/
A
K
(
x
,
t
)
=
4
arctan
(
v
cosh
x
1
−
v
2
sinh
v
t
1
−
v
2
)
{\displaystyle \varphi _{K/AK}(x,t)=4\arctan \left({\frac {v\cosh {\frac {x}{\sqrt {1-v^{2}}}}}{\sinh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}
Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as a breather. There are known three types of breathers: standing breather, traveling large-amplitude breather, and traveling small-amplitude breather.
The standing breather solution is given by
φ
(
x
,
t
)
=
4
arctan
(
1
−
ω
2
cos
(
ω
t
)
ω
cosh
(
1
−
ω
2
x
)
)
.
{\displaystyle \varphi (x,t)=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right).}
=== 3-soliton solutions ===
3-soliton collisions between a traveling kink and a standing breather or a traveling antikink and a standing breather results in a phase shift of the standing breather. In the process of collision between a moving kink and a standing breather,
the shift of the breather
Δ
B
{\displaystyle \Delta _{\text{B}}}
is given by
Δ
B
=
2
artanh
(
1
−
ω
2
)
(
1
−
v
K
2
)
1
−
ω
2
,
{\displaystyle \Delta _{\text{B}}={\frac {2\operatorname {artanh} {\sqrt {(1-\omega ^{2})(1-v_{\text{K}}^{2})}}}{\sqrt {1-\omega ^{2}}}},}
where
v
K
{\displaystyle v_{\text{K}}}
is the velocity of the kink, and
ω
{\displaystyle \omega }
is the breather's frequency. If the old position of the standing breather is
x
0
{\displaystyle x_{0}}
, after the collision the new position will be
x
0
+
Δ
B
{\displaystyle x_{0}+\Delta _{\text{B}}}
.
== Bäcklund transformation ==
Suppose that
φ
{\displaystyle \varphi }
is a solution of the sine-Gordon equation
φ
u
v
=
sin
φ
.
{\displaystyle \varphi _{uv}=\sin \varphi .\,}
Then the system
ψ
u
=
φ
u
+
2
a
sin
(
ψ
+
φ
2
)
ψ
v
=
−
φ
v
+
2
a
sin
(
ψ
−
φ
2
)
{\displaystyle {\begin{aligned}\psi _{u}&=\varphi _{u}+2a\sin {\Bigl (}{\frac {\psi +\varphi }{2}}{\Bigr )}\\\psi _{v}&=-\varphi _{v}+{\frac {2}{a}}\sin {\Bigl (}{\frac {\psi -\varphi }{2}}{\Bigr )}\end{aligned}}\,\!}
where a is an arbitrary parameter, is solvable for a function
ψ
{\displaystyle \psi }
which will also satisfy the sine-Gordon equation. This is an example of an auto-Bäcklund transform, as both
φ
{\displaystyle \varphi }
and
ψ
{\displaystyle \psi }
are solutions to the same equation, that is, the sine-Gordon equation.
By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation.
For example, if
φ
{\displaystyle \varphi }
is the trivial solution
φ
≡
0
{\displaystyle \varphi \equiv 0}
, then
ψ
{\displaystyle \psi }
is the one-soliton solution with
a
{\displaystyle a}
related to the boost applied to the soliton.
== Topological charge and energy ==
The topological charge or winding number of a solution
φ
{\displaystyle \varphi }
is
N
=
1
2
π
∫
R
d
φ
=
1
2
π
[
φ
(
x
=
∞
,
t
)
−
φ
(
x
=
−
∞
,
t
)
]
.
{\displaystyle N={\frac {1}{2\pi }}\int _{\mathbb {R} }d\varphi ={\frac {1}{2\pi }}\left[\varphi (x=\infty ,t)-\varphi (x=-\infty ,t)\right].}
The energy of a solution
φ
{\displaystyle \varphi }
is
E
=
∫
R
d
x
(
1
2
(
φ
t
2
+
φ
x
2
)
+
m
2
(
1
−
cos
φ
)
)
{\displaystyle E=\int _{\mathbb {R} }dx\left({\frac {1}{2}}(\varphi _{t}^{2}+\varphi _{x}^{2})+m^{2}(1-\cos \varphi )\right)}
where a constant energy density has been added so that the potential is non-negative. With it the first two terms in the Taylor expansion of the potential coincide with the potential of a massive scalar field, as mentioned in the naming section; the higher order terms can be thought of as interactions.
The topological charge is conserved if the energy is finite. The topological charge does not determine the solution, even up to Lorentz boosts. Both the trivial solution and the soliton-antisoliton pair solution have
N
=
0
{\displaystyle N=0}
.
== Zero-curvature formulation ==
The sine-Gordon equation is equivalent to the curvature of a particular
s
u
(
2
)
{\displaystyle {\mathfrak {su}}(2)}
-connection on
R
2
{\displaystyle \mathbb {R} ^{2}}
being equal to zero.
Explicitly, with coordinates
(
u
,
v
)
{\displaystyle (u,v)}
on
R
2
{\displaystyle \mathbb {R} ^{2}}
, the connection components
A
μ
{\displaystyle A_{\mu }}
are given by
A
u
=
(
i
λ
i
2
φ
u
i
2
φ
u
−
i
λ
)
=
1
2
φ
u
i
σ
1
+
λ
i
σ
3
,
{\displaystyle A_{u}={\begin{pmatrix}i\lambda &{\frac {i}{2}}\varphi _{u}\\{\frac {i}{2}}\varphi _{u}&-i\lambda \end{pmatrix}}={\frac {1}{2}}\varphi _{u}i\sigma _{1}+\lambda i\sigma _{3},}
A
v
=
(
−
i
4
λ
cos
φ
−
1
4
λ
sin
φ
1
4
λ
sin
φ
i
4
λ
cos
φ
)
=
−
1
4
λ
i
sin
φ
σ
2
−
1
4
λ
i
cos
φ
σ
3
,
{\displaystyle A_{v}={\begin{pmatrix}-{\frac {i}{4\lambda }}\cos \varphi &-{\frac {1}{4\lambda }}\sin \varphi \\{\frac {1}{4\lambda }}\sin \varphi &{\frac {i}{4\lambda }}\cos \varphi \end{pmatrix}}=-{\frac {1}{4\lambda }}i\sin \varphi \sigma _{2}-{\frac {1}{4\lambda }}i\cos \varphi \sigma _{3},}
where the
σ
i
{\displaystyle \sigma _{i}}
are the Pauli matrices.
Then the zero-curvature equation
∂
v
A
u
−
∂
u
A
v
+
[
A
u
,
A
v
]
=
0
{\displaystyle \partial _{v}A_{u}-\partial _{u}A_{v}+[A_{u},A_{v}]=0}
is equivalent to the sine-Gordon equation
φ
u
v
=
sin
φ
{\displaystyle \varphi _{uv}=\sin \varphi }
. The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined
F
μ
ν
=
[
∂
μ
−
A
μ
,
∂
ν
−
A
ν
]
{\displaystyle F_{\mu \nu }=[\partial _{\mu }-A_{\mu },\partial _{\nu }-A_{\nu }]}
.
The pair of matrices
A
u
{\displaystyle A_{u}}
and
A
v
{\displaystyle A_{v}}
are also known as a Lax pair for the sine-Gordon equation, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation.
== Related equations ==
The sinh-Gordon equation is given by
φ
x
x
−
φ
t
t
=
sinh
φ
.
{\displaystyle \varphi _{xx}-\varphi _{tt}=\sinh \varphi .}
This is the Euler–Lagrange equation of the Lagrangian
L
=
1
2
(
φ
t
2
−
φ
x
2
)
−
cosh
φ
.
{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-\cosh \varphi .}
Another closely related equation is the elliptic sine-Gordon equation or Euclidean sine-Gordon equation, given by
φ
x
x
+
φ
y
y
=
sin
φ
,
{\displaystyle \varphi _{xx}+\varphi _{yy}=\sin \varphi ,}
where
φ
{\displaystyle \varphi }
is now a function of the variables x and y. This is no longer a soliton equation, but it has many similar properties, as it is related to the sine-Gordon equation by the analytic continuation (or Wick rotation) y = it.
The elliptic sinh-Gordon equation may be defined in a similar way.
Another similar equation comes from the Euler–Lagrange equation for Liouville field theory
φ
x
x
−
φ
t
t
=
2
e
2
φ
.
{\displaystyle \varphi _{xx}-\varphi _{tt}=2e^{2\varphi }.}
A generalization is given by Toda field theory. More precisely, Liouville field theory is the Toda field theory for the finite Kac–Moody algebra
s
l
2
{\displaystyle {\mathfrak {sl}}_{2}}
, while sin(h)-Gordon is the Toda field theory for the affine Kac–Moody algebra
s
l
^
2
{\displaystyle {\hat {\mathfrak {sl}}}_{2}}
.
== Infinite volume and on a half line ==
One can also consider the sine-Gordon model on a circle, on a line segment, or on a half line. It is possible to find boundary conditions which preserve the integrability of the model. On a half line the spectrum contains boundary bound states in addition to the solitons and breathers.
== Quantum sine-Gordon model ==
In quantum field theory the sine-Gordon model contains a parameter that can be identified with the Planck constant. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of breathers. The number of breathers depends on the value of the parameter. Multiparticle production cancels on mass shell.
Semi-classical quantization of the sine-Gordon model was done by Ludwig Faddeev and Vladimir Korepin. The exact quantum scattering matrix was discovered by Alexander Zamolodchikov.
This model is S-dual to the Thirring model, as discovered by Coleman. This is sometimes known as the Coleman correspondence and serves as an example of boson-fermion correspondence in the interacting case. This article also showed that the constants appearing in the model behave nicely under renormalization: there are three parameters
α
0
,
β
{\displaystyle \alpha _{0},\beta }
and
γ
0
{\displaystyle \gamma _{0}}
. Coleman showed
α
0
{\displaystyle \alpha _{0}}
receives only a multiplicative correction,
γ
0
{\displaystyle \gamma _{0}}
receives only an additive correction, and
β
{\displaystyle \beta }
is not renormalized. Further, for a critical, non-zero value
β
=
4
π
{\displaystyle \beta ={\sqrt {4\pi }}}
, the theory is in fact dual to a free massive Dirac field theory.
The quantum sine-Gordon equation should be modified so the exponentials become vertex operators
L
Q
s
G
=
1
2
∂
μ
φ
∂
μ
φ
+
1
2
m
0
2
φ
2
−
α
(
V
β
+
V
−
β
)
{\displaystyle {\mathcal {L}}_{QsG}={\frac {1}{2}}\partial _{\mu }\varphi \partial ^{\mu }\varphi +{\frac {1}{2}}m_{0}^{2}\varphi ^{2}-\alpha (V_{\beta }+V_{-\beta })}
with
V
β
=:
e
i
β
φ
:
{\displaystyle V_{\beta }=:e^{i\beta \varphi }:}
, where the semi-colons denote normal ordering. A possible mass term is included.
=== Regimes of renormalizability ===
For different values of the parameter
β
2
{\displaystyle \beta ^{2}}
, the renormalizability properties of the sine-Gordon theory change. The identification of these regimes is attributed to Jürg Fröhlich.
The finite regime is
β
2
<
4
π
{\displaystyle \beta ^{2}<4\pi }
, where no counterterms are needed to render the theory well-posed. The super-renormalizable regime is
4
π
<
β
2
<
8
π
{\displaystyle 4\pi <\beta ^{2}<8\pi }
, where a finite number of counterterms are needed to render the theory well-posed. More counterterms are needed for each threshold
n
n
+
1
8
π
{\displaystyle {\frac {n}{n+1}}8\pi }
passed. For
β
2
>
8
π
{\displaystyle \beta ^{2}>8\pi }
, the theory becomes ill-defined (Coleman 1975). The boundary values are
β
2
=
4
π
{\displaystyle \beta ^{2}=4\pi }
and
β
2
=
8
π
{\displaystyle \beta ^{2}=8\pi }
, which are respectively the free fermion point, as the theory is dual to a free fermion via the Coleman correspondence, and the self-dual point, where the vertex operators form an affine sl2 subalgebra, and the theory becomes strictly renormalizable (renormalizable, but not super-renormalizable).
== Stochastic sine-Gordon model ==
The stochastic or dynamical sine-Gordon model has been studied by Martin Hairer and Hao Shen
allowing heuristic results from the quantum sine-Gordon theory to be proven in a statistical setting.
The equation is
∂
t
u
=
1
2
Δ
u
+
c
sin
(
β
u
+
θ
)
+
ξ
,
{\displaystyle \partial _{t}u={\frac {1}{2}}\Delta u+c\sin(\beta u+\theta )+\xi ,}
where
c
,
β
,
θ
{\displaystyle c,\beta ,\theta }
are real-valued constants, and
ξ
{\displaystyle \xi }
is space-time white noise. The space dimension is fixed to 2. In the proof of existence of solutions, the thresholds
β
2
=
n
n
+
1
8
π
{\displaystyle \beta ^{2}={\frac {n}{n+1}}8\pi }
again play a role in determining convergence of certain terms.
== Supersymmetric sine-Gordon model ==
A supersymmetric extension of the sine-Gordon model also exists. Integrability preserving boundary conditions for this extension can be found as well.
== Physical applications ==
The sine-Gordon model arises as the continuum limit of the Frenkel–Kontorova model which models crystal dislocations.
Dynamics in long Josephson junctions are well-described by the sine-Gordon equations, and conversely provide a useful experimental system for studying the sine-Gordon model.
The sine-Gordon model is in the same universality class as the effective action for a Coulomb gas of vortices and anti-vortices in the continuous classical XY model, which is a model of magnetism. The Kosterlitz–Thouless transition for vortices can therefore be derived from a renormalization group analysis of the sine-Gordon field theory.
The sine-Gordon equation also arises as the formal continuum limit of a different model of magnetism, the quantum Heisenberg model, in particular the XXZ model.
== See also ==
Josephson effect
Fluxon
Shape waves
== References ==
== External links ==
sine-Gordon equation at EqWorld: The World of Mathematical Equations.
Sinh-Gordon Equation at EqWorld: The World of Mathematical Equations.
sine-Gordon equation Archived 2012-03-16 at the Wayback Machine at NEQwiki, the nonlinear equations encyclopedia. | Wikipedia/Sine-Gordon_equation |
In theoretical physics, a minimal model or Virasoro minimal model is a two-dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro algebra.
Minimal models have been classified, giving rise to an ADE classification. Most minimal models have been solved, i.e. their 3-point structure constants have been computed analytically.
The term minimal model can also refer to a rational CFT based on an algebra that is larger than the Virasoro algebra, such as a W-algebra.
== Relevant representations of the Virasoro algebra ==
=== Representations ===
In minimal models, the central charge of the Virasoro algebra takes values of the type
c
p
,
q
=
1
−
6
(
p
−
q
)
2
p
q
.
{\displaystyle c_{p,q}=1-6{(p-q)^{2} \over pq}\ .}
where
p
,
q
{\displaystyle p,q}
are coprime integers such that
p
,
q
≥
2
{\displaystyle p,q\geq 2}
.
Then the conformal dimensions of degenerate representations are
h
r
,
s
=
(
p
r
−
q
s
)
2
−
(
p
−
q
)
2
4
p
q
,
with
r
,
s
∈
N
∗
,
{\displaystyle h_{r,s}={\frac {(pr-qs)^{2}-(p-q)^{2}}{4pq}}\ ,\quad {\text{with}}\ r,s\in \mathbb {N} ^{*}\ ,}
and they obey the identities
h
r
,
s
=
h
q
−
r
,
p
−
s
=
h
r
+
q
,
s
+
p
.
{\displaystyle h_{r,s}=h_{q-r,p-s}=h_{r+q,s+p}\ .}
The spectrums of minimal models are made of irreducible, degenerate lowest-weight representations of the Virasoro algebra, whose conformal dimensions are of the type
h
r
,
s
{\displaystyle h_{r,s}}
with
1
≤
r
≤
q
−
1
,
1
≤
s
≤
p
−
1
.
{\displaystyle 1\leq r\leq q-1\quad ,\quad 1\leq s\leq p-1\ .}
Such a representation
R
r
,
s
{\displaystyle {\mathcal {R}}_{r,s}}
is a coset of a Verma module by its infinitely many nontrivial submodules. It is unitary if and only if
|
p
−
q
|
=
1
{\displaystyle |p-q|=1}
. At a given central charge, there are
1
2
(
p
−
1
)
(
q
−
1
)
{\displaystyle {\frac {1}{2}}(p-1)(q-1)}
distinct representations of this type. The set of these representations, or of their conformal dimensions, is called the Kac table with parameters
(
p
,
q
)
{\displaystyle (p,q)}
. The Kac table is usually drawn as a rectangle of size
(
q
−
1
)
×
(
p
−
1
)
{\displaystyle (q-1)\times (p-1)}
, where each representation appears twice
due to the relation
R
r
,
s
=
R
q
−
r
,
p
−
s
.
{\displaystyle {\mathcal {R}}_{r,s}={\mathcal {R}}_{q-r,p-s}\ .}
=== Fusion rules ===
The fusion rules of the multiply degenerate representations
R
r
,
s
{\displaystyle {\mathcal {R}}_{r,s}}
encode constraints from all their null vectors. They can therefore be deduced from the fusion rules of simply degenerate representations, which encode constraints from individual null vectors. Explicitly, the fusion rules are
R
r
1
,
s
1
×
R
r
2
,
s
2
=
∑
r
3
=
2
|
r
1
−
r
2
|
+
1
min
(
r
1
+
r
2
,
2
q
−
r
1
−
r
2
)
−
1
∑
s
3
=
2
|
s
1
−
s
2
|
+
1
min
(
s
1
+
s
2
,
2
p
−
s
1
−
s
2
)
−
1
R
r
3
,
s
3
,
{\displaystyle {\mathcal {R}}_{r_{1},s_{1}}\times {\mathcal {R}}_{r_{2},s_{2}}=\sum _{r_{3}{\overset {2}{=}}|r_{1}-r_{2}|+1}^{\min(r_{1}+r_{2},2q-r_{1}-r_{2})-1}\ \sum _{s_{3}{\overset {2}{=}}|s_{1}-s_{2}|+1}^{\min(s_{1}+s_{2},2p-s_{1}-s_{2})-1}{\mathcal {R}}_{r_{3},s_{3}}\ ,}
where the sums run by increments of two.
== Classification and spectrums ==
Minimal models are the only 2d CFTs that are consistent on any Riemann surface, and are built from finitely many representations of the Virasoro algebra. There are many more rational CFTs that are consistent on the sphere only: these CFTs are submodels of minimal models, built from subsets of the Kac table that are closed under fusion. Such submodels can also be classified.
=== A-series minimal models: the diagonal case ===
For any coprime integers
p
,
q
{\displaystyle p,q}
such that
p
,
q
≥
2
{\displaystyle p,q\geq 2}
, there exists a diagonal minimal model whose spectrum contains one copy of each distinct representation in the Kac table:
S
p
,
q
A-series
=
1
2
⨁
r
=
1
q
−
1
⨁
s
=
1
p
−
1
R
r
,
s
⊗
R
¯
r
,
s
.
{\displaystyle {\mathcal {S}}_{p,q}^{\text{A-series}}={\frac {1}{2}}\bigoplus _{r=1}^{q-1}\bigoplus _{s=1}^{p-1}{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{r,s}\ .}
The
(
p
,
q
)
{\displaystyle (p,q)}
and
(
q
,
p
)
{\displaystyle (q,p)}
models are the same.
The OPE of two fields involves all the fields that are allowed by the fusion rules of the corresponding representations.
=== D-series minimal models ===
A D-series minimal model with the central charge
c
p
,
q
{\displaystyle c_{p,q}}
exists if
p
{\displaystyle p}
or
q
{\displaystyle q}
is even and at least
6
{\displaystyle 6}
. Using the symmetry
p
↔
q
{\displaystyle p\leftrightarrow q}
we assume that
q
{\displaystyle q}
is even, then
p
{\displaystyle p}
is odd. The spectrum is
S
p
,
q
D-series
=
q
≡
0
mod
4
,
q
≥
8
1
2
⨁
r
=
2
1
q
−
1
⨁
s
=
1
p
−
1
R
r
,
s
⊗
R
¯
r
,
s
⊕
1
2
⨁
r
=
2
2
q
−
2
⨁
s
=
1
p
−
1
R
r
,
s
⊗
R
¯
q
−
r
,
s
,
{\displaystyle {\mathcal {S}}_{p,q}^{\text{D-series}}\ \ {\underset {q\equiv 0\operatorname {mod} 4,\ q\geq 8}{=}}\ \ {\frac {1}{2}}\bigoplus _{r{\overset {2}{=}}1}^{q-1}\bigoplus _{s=1}^{p-1}{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{r,s}\oplus {\frac {1}{2}}\bigoplus _{r{\overset {2}{=}}2}^{q-2}\bigoplus _{s=1}^{p-1}{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{q-r,s}\ ,}
S
p
,
q
D-series
=
q
≡
2
mod
4
,
q
≥
6
1
2
⨁
r
=
2
1
q
−
1
⨁
s
=
1
p
−
1
R
r
,
s
⊗
R
¯
r
,
s
⊕
1
2
⨁
r
=
2
1
q
−
1
⨁
s
=
1
p
−
1
R
r
,
s
⊗
R
¯
q
−
r
,
s
,
{\displaystyle {\mathcal {S}}_{p,q}^{\text{D-series}}\ \ {\underset {q\equiv 2\operatorname {mod} 4,\ q\geq 6}{=}}\ \ {\frac {1}{2}}\bigoplus _{r{\overset {2}{=}}1}^{q-1}\bigoplus _{s=1}^{p-1}{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{r,s}\oplus {\frac {1}{2}}\bigoplus _{r{\overset {2}{=}}1}^{q-1}\bigoplus _{s=1}^{p-1}{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{q-r,s}\ ,}
where the sums over
r
{\displaystyle r}
run by increments of two.
In any given spectrum, each representation has multiplicity one, except the representations of the type
R
q
2
,
s
⊗
R
¯
q
2
,
s
{\displaystyle {\mathcal {R}}_{{\frac {q}{2}},s}\otimes {\bar {\mathcal {R}}}_{{\frac {q}{2}},s}}
if
q
≡
2
m
o
d
4
{\displaystyle q\equiv 2\ \mathrm {mod} \ 4}
, which have multiplicity two. These representations indeed appear in both terms in our formula for the spectrum.
The OPE of two fields involves all the fields that are allowed by the fusion rules of the corresponding representations, and that respect the conservation of diagonality: the OPE of one diagonal and one non-diagonal field yields only non-diagonal fields, and the OPE of two fields of the same type yields only diagonal fields.
For this rule, one copy of the representation
R
q
2
,
s
⊗
R
¯
q
2
,
s
{\displaystyle {\mathcal {R}}_{{\frac {q}{2}},s}\otimes {\bar {\mathcal {R}}}_{{\frac {q}{2}},s}}
counts as diagonal, and the other copy as non-diagonal.
=== E-series minimal models ===
There are three series of E-series minimal models. Each series exists for a given value of
q
∈
{
12
,
18
,
30
}
,
{\displaystyle q\in \{12,18,30\},}
for any
p
≥
2
{\displaystyle p\geq 2}
that is coprime with
q
{\displaystyle q}
. (This actually implies
p
≥
5
{\displaystyle p\geq 5}
.) Using the notation
|
R
|
2
=
R
⊗
R
¯
{\displaystyle |{\mathcal {R}}|^{2}={\mathcal {R}}\otimes {\bar {\mathcal {R}}}}
, the spectrums read:
S
p
,
12
E-series
=
1
2
⨁
s
=
1
p
−
1
{
|
R
1
,
s
⊕
R
7
,
s
|
2
⊕
|
R
4
,
s
⊕
R
8
,
s
|
2
⊕
|
R
5
,
s
⊕
R
11
,
s
|
2
}
,
{\displaystyle {\mathcal {S}}_{p,12}^{\text{E-series}}={\frac {1}{2}}\bigoplus _{s=1}^{p-1}\left\{\left|{\mathcal {R}}_{1,s}\oplus {\mathcal {R}}_{7,s}\right|^{2}\oplus \left|{\mathcal {R}}_{4,s}\oplus {\mathcal {R}}_{8,s}\right|^{2}\oplus \left|{\mathcal {R}}_{5,s}\oplus {\mathcal {R}}_{11,s}\right|^{2}\right\}\ ,}
S
p
,
18
E-series
=
1
2
⨁
s
=
1
p
−
1
{
|
R
9
,
s
⊕
2
R
3
,
s
|
2
⊖
4
|
R
3
,
s
|
2
⊕
⨁
r
∈
{
1
,
5
,
7
}
|
R
r
,
s
⊕
R
18
−
r
,
s
|
2
}
,
{\displaystyle {\mathcal {S}}_{p,18}^{\text{E-series}}={\frac {1}{2}}\bigoplus _{s=1}^{p-1}\left\{\left|{\mathcal {R}}_{9,s}\oplus 2{\mathcal {R}}_{3,s}\right|^{2}\ominus 4\left|{\mathcal {R}}_{3,s}\right|^{2}\oplus \bigoplus _{r\in \{1,5,7\}}\left|{\mathcal {R}}_{r,s}\oplus {\mathcal {R}}_{18-r,s}\right|^{2}\right\}\ ,}
S
p
,
30
E-series
=
1
2
⨁
s
=
1
p
−
1
{
|
⨁
r
∈
{
1
,
11
,
19
,
29
}
R
r
,
s
|
2
⊕
|
⨁
r
∈
{
7
,
13
,
17
,
23
}
R
r
,
s
|
2
}
.
{\displaystyle {\mathcal {S}}_{p,30}^{\text{E-series}}={\frac {1}{2}}\bigoplus _{s=1}^{p-1}\left\{\left|\bigoplus _{r\in \{1,11,19,29\}}{\mathcal {R}}_{r,s}\right|^{2}\oplus \left|\bigoplus _{r\in \{7,13,17,23\}}{\mathcal {R}}_{r,s}\right|^{2}\right\}\ .}
== Examples ==
The following A-series minimal models are related to well-known physical systems:
(
p
,
q
)
=
(
3
,
2
)
{\displaystyle (p,q)=(3,2)}
: trivial CFT,
(
p
,
q
)
=
(
5
,
2
)
{\displaystyle (p,q)=(5,2)}
: Yang-Lee edge singularity,
(
p
,
q
)
=
(
4
,
3
)
{\displaystyle (p,q)=(4,3)}
: critical Ising model,
(
p
,
q
)
=
(
5
,
4
)
{\displaystyle (p,q)=(5,4)}
: tricritical Ising model,
(
p
,
q
)
=
(
6
,
5
)
{\displaystyle (p,q)=(6,5)}
: tetracritical Ising model.
The following D-series minimal models are related to well-known physical systems:
(
p
,
q
)
=
(
6
,
5
)
{\displaystyle (p,q)=(6,5)}
: 3-state Potts model at criticality,
(
p
,
q
)
=
(
7
,
6
)
{\displaystyle (p,q)=(7,6)}
: tricritical 3-state Potts model.
The Kac tables of these models, together with a few other Kac tables with
2
≤
q
≤
6
{\displaystyle 2\leq q\leq 6}
, are:
1
0
0
1
2
c
3
,
2
=
0
1
0
−
1
5
−
1
5
0
1
2
3
4
c
5
,
2
=
−
22
5
{\displaystyle {\begin{array}{c}{\begin{array}{c|cc}1&0&0\\\hline &1&2\end{array}}\\c_{3,2}=0\end{array}}\qquad {\begin{array}{c}{\begin{array}{c|cccc}1&0&-{\frac {1}{5}}&-{\frac {1}{5}}&0\\\hline &1&2&3&4\end{array}}\\c_{5,2}=-{\frac {22}{5}}\end{array}}}
2
1
2
1
16
0
1
0
1
16
1
2
1
2
3
c
4
,
3
=
1
2
2
3
4
1
5
−
1
20
0
1
0
−
1
20
1
5
3
4
1
2
3
4
c
5
,
3
=
−
3
5
{\displaystyle {\begin{array}{c}{\begin{array}{c|ccc}2&{\frac {1}{2}}&{\frac {1}{16}}&0\\1&0&{\frac {1}{16}}&{\frac {1}{2}}\\\hline &1&2&3\end{array}}\\c_{4,3}={\frac {1}{2}}\end{array}}\qquad {\begin{array}{c}{\begin{array}{c|cccc}2&{\frac {3}{4}}&{\frac {1}{5}}&-{\frac {1}{20}}&0\\1&0&-{\frac {1}{20}}&{\frac {1}{5}}&{\frac {3}{4}}\\\hline &1&2&3&4\end{array}}\\c_{5,3}=-{\frac {3}{5}}\end{array}}}
3
3
2
3
5
1
10
0
2
7
16
3
80
3
80
7
16
1
0
1
10
3
5
3
2
1
2
3
4
c
5
,
4
=
7
10
3
5
2
10
7
9
14
1
7
−
1
14
0
2
13
16
27
112
−
5
112
−
5
112
27
112
13
16
1
0
−
1
14
1
7
9
14
10
7
5
2
1
2
3
4
5
6
c
7
,
4
=
−
13
14
{\displaystyle {\begin{array}{c}{\begin{array}{c|cccc}3&{\frac {3}{2}}&{\frac {3}{5}}&{\frac {1}{10}}&0\\2&{\frac {7}{16}}&{\frac {3}{80}}&{\frac {3}{80}}&{\frac {7}{16}}\\1&0&{\frac {1}{10}}&{\frac {3}{5}}&{\frac {3}{2}}\\\hline &1&2&3&4\end{array}}\\c_{5,4}={\frac {7}{10}}\end{array}}\qquad {\begin{array}{c}{\begin{array}{c|cccccc}3&{\frac {5}{2}}&{\frac {10}{7}}&{\frac {9}{14}}&{\frac {1}{7}}&-{\frac {1}{14}}&0\\2&{\frac {13}{16}}&{\frac {27}{112}}&-{\frac {5}{112}}&-{\frac {5}{112}}&{\frac {27}{112}}&{\frac {13}{16}}\\1&0&-{\frac {1}{14}}&{\frac {1}{7}}&{\frac {9}{14}}&{\frac {10}{7}}&{\frac {5}{2}}\\\hline &1&2&3&4&5&6\end{array}}\\c_{7,4}=-{\frac {13}{14}}\end{array}}}
4
3
13
8
2
3
1
8
0
3
7
5
21
40
1
15
1
40
2
5
2
2
5
1
40
1
15
21
40
7
5
1
0
1
8
2
3
13
8
3
1
2
3
4
5
c
6
,
5
=
4
5
4
15
4
16
7
33
28
3
7
1
28
0
3
9
5
117
140
8
35
−
3
140
3
35
11
20
2
11
20
3
35
−
3
140
8
35
117
140
9
5
1
0
1
28
3
7
33
28
16
7
15
4
1
2
3
4
5
6
c
7
,
5
=
11
35
{\displaystyle {\begin{array}{c}{\begin{array}{c|ccccc}4&3&{\frac {13}{8}}&{\frac {2}{3}}&{\frac {1}{8}}&0\\3&{\frac {7}{5}}&{\frac {21}{40}}&{\frac {1}{15}}&{\frac {1}{40}}&{\frac {2}{5}}\\2&{\frac {2}{5}}&{\frac {1}{40}}&{\frac {1}{15}}&{\frac {21}{40}}&{\frac {7}{5}}\\1&0&{\frac {1}{8}}&{\frac {2}{3}}&{\frac {13}{8}}&3\\\hline &1&2&3&4&5\end{array}}\\c_{6,5}={\frac {4}{5}}\end{array}}\qquad {\begin{array}{c}{\begin{array}{c|cccccc}4&{\frac {15}{4}}&{\frac {16}{7}}&{\frac {33}{28}}&{\frac {3}{7}}&{\frac {1}{28}}&0\\3&{\frac {9}{5}}&{\frac {117}{140}}&{\frac {8}{35}}&-{\frac {3}{140}}&{\frac {3}{35}}&{\frac {11}{20}}\\2&{\frac {11}{20}}&{\frac {3}{35}}&-{\frac {3}{140}}&{\frac {8}{35}}&{\frac {117}{140}}&{\frac {9}{5}}\\1&0&{\frac {1}{28}}&{\frac {3}{7}}&{\frac {33}{28}}&{\frac {16}{7}}&{\frac {15}{4}}\\\hline &1&2&3&4&5&6\end{array}}\\c_{7,5}={\frac {11}{35}}\end{array}}}
5
5
22
7
12
7
5
7
1
7
0
4
23
8
85
56
33
56
5
56
1
56
3
8
3
4
3
10
21
1
21
1
21
10
21
4
3
2
3
8
1
56
5
56
33
56
85
56
23
8
1
0
1
7
5
7
12
7
22
7
5
1
2
3
4
5
6
c
7
,
6
=
6
7
{\displaystyle {\begin{array}{c}{\begin{array}{c|cccccc}5&5&{\frac {22}{7}}&{\frac {12}{7}}&{\frac {5}{7}}&{\frac {1}{7}}&0\\4&{\frac {23}{8}}&{\frac {85}{56}}&{\frac {33}{56}}&{\frac {5}{56}}&{\frac {1}{56}}&{\frac {3}{8}}\\3&{\frac {4}{3}}&{\frac {10}{21}}&{\frac {1}{21}}&{\frac {1}{21}}&{\frac {10}{21}}&{\frac {4}{3}}\\2&{\frac {3}{8}}&{\frac {1}{56}}&{\frac {5}{56}}&{\frac {33}{56}}&{\frac {85}{56}}&{\frac {23}{8}}\\1&0&{\frac {1}{7}}&{\frac {5}{7}}&{\frac {12}{7}}&{\frac {22}{7}}&5\\\hline &1&2&3&4&5&6\end{array}}\\c_{7,6}={\frac {6}{7}}\end{array}}}
== Solution of minimal models ==
The 3-point structure constants of minimal models take different forms depending on the series:
For A-series minimal models, an expression in terms of the Gamma function was obtained using Coulomb gas techniques in the 1980s.
For D-series minimal models, an expression in terms of the fusing matrix is known.
For E-series minimal models with
q
=
12
{\displaystyle q=12}
, an expression in terms of the double Gamma function is known. The A-series and D-series structure constants can also be rewritten in terms of the same special function.
== Related conformal field theories ==
=== Coset realizations ===
The A-series minimal model with indices
(
p
,
q
)
{\displaystyle (p,q)}
coincides with the following coset of WZW models:
S
U
(
2
)
k
×
S
U
(
2
)
1
S
U
(
2
)
k
+
1
,
where
k
=
q
p
−
q
−
2
.
{\displaystyle {\frac {SU(2)_{k}\times SU(2)_{1}}{SU(2)_{k+1}}}\ ,\quad {\text{where}}\quad k={\frac {q}{p-q}}-2\ .}
Assuming
p
>
q
{\displaystyle p>q}
, the level
k
{\displaystyle k}
is integer if and only if
p
=
q
+
1
{\displaystyle p=q+1}
i.e. if and only if the minimal model is unitary.
There exist other realizations of certain minimal models, diagonal or not, as cosets of WZW models, not necessarily based on the group
S
U
(
2
)
{\displaystyle SU(2)}
.
=== Generalized minimal models ===
For any central charge
c
∈
C
{\displaystyle c\in \mathbb {C} }
, there is a diagonal CFT whose spectrum is made of all degenerate representations,
S
=
⨁
r
,
s
=
1
∞
R
r
,
s
⊗
R
¯
r
,
s
.
{\displaystyle {\mathcal {S}}=\bigoplus _{r,s=1}^{\infty }{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{r,s}\ .}
When the central charge tends to
c
p
,
q
{\displaystyle c_{p,q}}
, the generalized minimal models tend to the corresponding A-series minimal model. This means in particular that the degenerate representations that are not in the Kac table decouple.
=== Liouville theory ===
Since Liouville theory reduces to a generalized minimal model when the fields are taken to be degenerate, it further reduces to an A-series minimal model when the central charge is then sent to
c
p
,
q
{\displaystyle c_{p,q}}
.
Moreover, A-series minimal models have a well-defined limit as
c
→
1
{\displaystyle c\to 1}
: a diagonal CFT with a continuous spectrum called Runkel–Watts theory, which coincides with the limit of Liouville theory when
c
→
1
+
{\displaystyle c\to 1^{+}}
.
=== Products of minimal models ===
There are three cases of minimal models that are products of two minimal models.
At the level of their spectrums, the relations are:
S
2
,
5
A-series
⊗
S
2
,
5
A-series
=
S
3
,
10
D-series
,
{\displaystyle {\mathcal {S}}_{2,5}^{\text{A-series}}\otimes {\mathcal {S}}_{2,5}^{\text{A-series}}={\mathcal {S}}_{3,10}^{\text{D-series}}\ ,}
S
2
,
5
A-series
⊗
S
3
,
4
A-series
=
S
5
,
12
E-series
,
{\displaystyle {\mathcal {S}}_{2,5}^{\text{A-series}}\otimes {\mathcal {S}}_{3,4}^{\text{A-series}}={\mathcal {S}}_{5,12}^{\text{E-series}}\ ,}
S
2
,
5
A-series
⊗
S
2
,
7
A-series
=
S
7
,
30
E-series
.
{\displaystyle {\mathcal {S}}_{2,5}^{\text{A-series}}\otimes {\mathcal {S}}_{2,7}^{\text{A-series}}={\mathcal {S}}_{7,30}^{\text{E-series}}\ .}
=== Fermionic extensions of minimal models ===
If
q
≡
0
mod
4
{\displaystyle q\equiv 0{\bmod {4}}}
, the A-series and the D-series
(
p
,
q
)
{\displaystyle (p,q)}
minimal models each have a fermionic extension. These two fermionic extensions involve fields with half-integer spins, and they are related to one another by a parity-shift operation.
== References == | Wikipedia/Minimal_model_(physics) |
In the physics of electromagnetism, the Abraham–Lorentz force (also known as the Lorentz–Abraham force) is the reaction force on an accelerating charged particle caused by the particle emitting electromagnetic radiation by self-interaction. It is also called the radiation reaction force, the radiation damping force, or the self-force. It is named after the physicists Max Abraham and Hendrik Lorentz.
The formula, although predating the theory of special relativity, was initially calculated for non-relativistic velocity approximations. It was extended to arbitrary velocities by Max Abraham and was shown to be physically consistent by George Adolphus Schott. The non-relativistic form is called Lorentz self-force while the relativistic version is called the Lorentz–Dirac force or collectively known as Abraham–Lorentz–Dirac force. The equations are in the domain of classical physics, not quantum physics, and therefore may not be valid at distances of roughly the Compton wavelength or below. There are, however, two analogs of the formula that are both fully quantum and relativistic: one is called the "Abraham–Lorentz–Dirac–Langevin equation", the other is the self-force on a moving mirror.
The force is proportional to the square of the object's charge, multiplied by the jerk that it is experiencing. (Jerk is the rate of change of acceleration.) The force points in the direction of the jerk. For example, in a cyclotron, where the jerk points opposite to the velocity, the radiation reaction is directed opposite to the velocity of the particle, providing a braking action. The Abraham–Lorentz force is the source of the radiation resistance of a radio antenna radiating radio waves.
There are pathological solutions of the Abraham–Lorentz–Dirac equation in which a particle accelerates in advance of the application of a force, so-called pre-acceleration solutions. Since this would represent an effect occurring before its cause (retrocausality), some theories have speculated that the equation allows signals to travel backward in time, thus challenging the physical principle of causality. One resolution of this problem was discussed by Arthur D. Yaghjian and was further discussed by Fritz Rohrlich and Rodrigo Medina. Furthermore, some authors argue that a radiation reaction force is unnecessary, introducing a corresponding stress-energy tensor that naturally conserves energy and momentum in Minkowski space and other suitable spacetimes.
== Definition and description ==
The Lorentz self-force derived for non-relativistic velocity approximation
v
≪
c
{\displaystyle v\ll c}
, is given in SI units by:
F
r
a
d
=
μ
0
q
2
6
π
c
a
˙
=
q
2
6
π
ε
0
c
3
a
˙
=
2
3
q
2
4
π
ε
0
c
3
a
˙
{\displaystyle \mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} ={\frac {q^{2}}{6\pi \varepsilon _{0}c^{3}}}\mathbf {\dot {a}} ={\frac {2}{3}}{\frac {q^{2}}{4\pi \varepsilon _{0}c^{3}}}\mathbf {\dot {a}} }
or in Gaussian units by
F
r
a
d
=
2
3
q
2
c
3
a
˙
.
{\displaystyle \mathbf {F} _{\mathrm {rad} }={2 \over 3}{\frac {q^{2}}{c^{3}}}\mathbf {\dot {a}} .}
where
F
r
a
d
{\displaystyle \mathbf {F} _{\mathrm {rad} }}
is the force,
a
˙
{\displaystyle \mathbf {\dot {a}} }
is the derivative of acceleration, or the third derivative of displacement, also called jerk, μ0 is the magnetic constant, ε0 is the electric constant, c is the speed of light in free space, and q is the electric charge of the particle.
Physically, an accelerating charge emits radiation (according to the Larmor formula), which carries momentum away from the charge. Since momentum is conserved, the charge is pushed in the direction opposite the direction of the emitted radiation. In fact the formula above for radiation force can be derived from the Larmor formula, as shown below.
The Abraham–Lorentz force, a generalization of Lorentz self-force for arbitrary velocities is given by:
F
r
a
d
=
μ
0
q
2
6
π
c
(
γ
2
a
˙
+
γ
4
v
(
v
⋅
a
˙
)
c
2
+
3
γ
4
a
(
v
⋅
a
)
c
2
+
3
γ
6
v
(
v
⋅
a
)
2
c
4
)
{\displaystyle \mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\left(\gamma ^{2}{\dot {a}}+{\frac {\gamma ^{4}v(v\cdot {\dot {a}})}{c^{2}}}+{\frac {3\gamma ^{4}a(v\cdot a)}{c^{2}}}+{\frac {3\gamma ^{6}v(v\cdot a)^{2}}{c^{4}}}\right)}
Where
γ
{\displaystyle \gamma }
is the Lorentz factor associated with
v
{\displaystyle v}
, the velocity of particle. The formula is consistent with special relativity and reduces to Lorentz's self-force expression for low velocity limit.
The covariant form of radiation reaction deduced by Dirac for arbitrary shape of elementary charges is found to be:
F
μ
r
a
d
=
μ
0
q
2
6
π
m
c
[
d
2
p
μ
d
τ
2
−
p
μ
m
2
c
2
(
d
p
ν
d
τ
d
p
ν
d
τ
)
]
{\displaystyle F_{\mu }^{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi mc}}\left[{\frac {d^{2}p_{\mu }}{d\tau ^{2}}}-{\frac {p_{\mu }}{m^{2}c^{2}}}\left({\frac {dp_{\nu }}{d\tau }}{\frac {dp^{\nu }}{d\tau }}\right)\right]}
== History ==
The first calculation of electromagnetic radiation energy due to current was given by George Francis FitzGerald in 1883, in which radiation resistance appears. However, dipole antenna experiments by Heinrich Hertz made a bigger impact and gathered commentary by Poincaré on the amortissement or damping of the oscillator due to the emission of radiation. Qualitative discussions surrounding damping effects of radiation emitted by accelerating charges was sparked by Henry Poincaré in 1891. In 1892, Hendrik Lorentz derived the self-interaction force of charges for low velocities but did not relate it to radiation losses. Suggestion of a relationship between radiation energy loss and self-force was first made by Max Planck. Planck's concept of the damping force, which did not assume any particular shape for elementary charged particles, was applied by Max Abraham to find the radiation resistance of an antenna in 1898, which remains the most practical application of the phenomenon.
In the early 1900s, Abraham formulated a generalization of the Lorentz self-force to arbitrary velocities, the physical consistency of which was later shown by George Adolphus Schott. Schott was able to derive the Abraham equation and attributed "acceleration energy" to be the source of energy of the electromagnetic radiation. Originally submitted as an essay for the 1908 Adams Prize, he won the competition and had the essay published as a book in 1912. The relationship between self-force and radiation reaction became well-established at this point. Wolfgang Pauli first obtained the covariant form of the radiation reaction and in 1938, Paul Dirac found that the equation of motion of charged particles, without assuming the shape of the particle, contained Abraham's formula within reasonable approximations. The equations derived by Dirac are considered exact within the limits of classical theory.
== Background ==
In classical electrodynamics, problems are typically divided into two classes:
Problems in which the charge and current sources of fields are specified and the fields are calculated, and
The reverse situation, problems in which the fields are specified and the motion of particles are calculated.
In some fields of physics, such as plasma physics and the calculation of transport coefficients (conductivity, diffusivity, etc.), the fields generated by the sources and the motion of the sources are solved self-consistently. In such cases, however, the motion of a selected source is calculated in response to fields generated by all other sources. Rarely is the motion of a particle (source) due to the fields generated by that same particle calculated. The reason for this is twofold:
Neglect of the "self-fields" usually leads to answers that are accurate enough for many applications, and
Inclusion of self-fields leads to problems in physics such as renormalization, some of which are still unsolved, that relate to the very nature of matter and energy.
These conceptual problems created by self-fields are highlighted in a standard graduate text. [Jackson]
The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of the elementary particle. Although partial solutions, workable within limited areas, can be given, the basic problem remains unsolved. One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties. While there is still hope that this may eventually occur, the present quantum-mechanical discussions are beset with even more elaborate troubles than the classical ones. It is one of the triumphs of comparatively recent years (~ 1948–1950) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment. From a fundamental point of view, however, the difficulties remain.
The Abraham–Lorentz force is the result of the most fundamental calculation of the effect of self-generated fields. It arises from the observation that accelerating charges emit radiation. The Abraham–Lorentz force is the average force that an accelerating charged particle feels in the recoil from the emission of radiation. The introduction of quantum effects leads one to quantum electrodynamics. The self-fields in quantum electrodynamics generate a finite number of infinities in the calculations that can be removed by the process of renormalization. This has led to a theory that is able to make the most accurate predictions that humans have made to date. (See precision tests of QED.) The renormalization process fails, however, when applied to the gravitational force. The infinities in that case are infinite in number, which causes the failure of renormalization. Therefore, general relativity has an unsolved self-field problem. String theory and loop quantum gravity are current attempts to resolve this problem, formally called the problem of radiation reaction or the problem of self-force.
== Derivation ==
The simplest derivation for the self-force is found for periodic motion from the Larmor formula for the power radiated from a point charge that moves with velocity much lower than that of speed of light:
P
=
μ
0
q
2
6
π
c
a
2
.
{\displaystyle P={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {a} ^{2}.}
If we assume the motion of a charged particle is periodic, then the average work done on the particle by the Abraham–Lorentz force is the negative of the Larmor power integrated over one period from
τ
1
{\displaystyle \tau _{1}}
to
τ
2
{\displaystyle \tau _{2}}
:
∫
τ
1
τ
2
F
r
a
d
⋅
v
d
t
=
∫
τ
1
τ
2
−
P
d
t
=
−
∫
τ
1
τ
2
μ
0
q
2
6
π
c
a
2
d
t
=
−
∫
τ
1
τ
2
μ
0
q
2
6
π
c
d
v
d
t
⋅
d
v
d
t
d
t
.
{\displaystyle \int _{\tau _{1}}^{\tau _{2}}\mathbf {F} _{\mathrm {rad} }\cdot \mathbf {v} dt=\int _{\tau _{1}}^{\tau _{2}}-Pdt=-\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {a} ^{2}dt=-\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d\mathbf {v} }{dt}}\cdot {\frac {d\mathbf {v} }{dt}}dt.}
The above expression can be integrated by parts. If we assume that there is periodic motion, the boundary term in the integral by parts disappears:
∫
τ
1
τ
2
F
r
a
d
⋅
v
d
t
=
−
μ
0
q
2
6
π
c
d
v
d
t
⋅
v
|
τ
1
τ
2
+
∫
τ
1
τ
2
μ
0
q
2
6
π
c
d
2
v
d
t
2
⋅
v
d
t
=
−
0
+
∫
τ
1
τ
2
μ
0
q
2
6
π
c
a
˙
⋅
v
d
t
.
{\displaystyle \int _{\tau _{1}}^{\tau _{2}}\mathbf {F} _{\mathrm {rad} }\cdot \mathbf {v} dt=-{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d\mathbf {v} }{dt}}\cdot \mathbf {v} {\bigg |}_{\tau _{1}}^{\tau _{2}}+\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d^{2}\mathbf {v} }{dt^{2}}}\cdot \mathbf {v} dt=-0+\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} \cdot \mathbf {v} dt.}
Clearly, we can identify the Lorentz self-force equation which is applicable to slow moving particles as:
F
r
a
d
=
μ
0
q
2
6
π
c
a
˙
.
{\displaystyle \mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {{\dot {a}}.} }
A more rigorous derivation, which does not require periodic motion, was found using an effective field theory formulation.
A generalized equation for arbitrary velocities was formulated by Max Abraham, which is found to be consistent with special relativity. An alternative derivation, making use of theory of relativity which was well established at that time, was found by Dirac without any assumption of the shape of the charged particle.
== Signals from the future ==
Below is an illustration of how a classical analysis can lead to surprising results. The classical theory can be seen to challenge standard pictures of causality, thus signaling either a breakdown or a need for extension of the theory. In this case the extension is to quantum mechanics and its relativistic counterpart quantum field theory. See the quote from Rohrlich in the introduction concerning "the importance of obeying the validity limits of a physical theory".
For a particle in an external force
F
e
x
t
{\displaystyle \mathbf {F} _{\mathrm {ext} }}
, we have
m
v
˙
=
F
r
a
d
+
F
e
x
t
=
m
t
0
v
¨
+
F
e
x
t
.
{\displaystyle m{\dot {\mathbf {v} }}=\mathbf {F} _{\mathrm {rad} }+\mathbf {F} _{\mathrm {ext} }=mt_{0}{\ddot {\mathbf {v} }}+\mathbf {F} _{\mathrm {ext} }.}
where
t
0
=
μ
0
q
2
6
π
m
c
.
{\displaystyle t_{0}={\frac {\mu _{0}q^{2}}{6\pi mc}}.}
This equation can be integrated once to obtain
m
v
˙
=
1
t
0
∫
t
∞
exp
(
−
t
′
−
t
t
0
)
F
e
x
t
(
t
′
)
d
t
′
.
{\displaystyle m{\dot {\mathbf {v} }}={1 \over t_{0}}\int _{t}^{\infty }\exp \left(-{t'-t \over t_{0}}\right)\,\mathbf {F} _{\mathrm {ext} }(t')\,dt'.}
The integral extends from the present to infinitely far in the future. Thus future values of the force affect the acceleration of the particle in the present. The future values are weighted by the factor
exp
(
−
t
′
−
t
t
0
)
{\displaystyle \exp \left(-{t'-t \over t_{0}}\right)}
which falls off rapidly for times greater than
t
0
{\displaystyle t_{0}}
in the future. Therefore, signals from an interval approximately
t
0
{\displaystyle t_{0}}
into the future affect the acceleration in the present. For an electron, this time is approximately
10
−
24
{\displaystyle 10^{-24}}
sec, which is the time it takes for a light wave to travel across the "size" of an electron, the classical electron radius. One way to define this "size" is as follows: it is (up to some constant factor) the distance
r
{\displaystyle r}
such that two electrons placed at rest at a distance
r
{\displaystyle r}
apart and allowed to fly apart, would have sufficient energy to reach half the speed of light. In other words, it forms the length (or time, or energy) scale where something as light as an electron would be fully relativistic. It is worth noting that this expression does not involve the Planck constant at all, so although it indicates something is wrong at this length scale, it does not directly relate to quantum uncertainty, or to the frequency–energy relation of a photon. Although it is common in quantum mechanics to treat
ℏ
→
0
{\displaystyle \hbar \to 0}
as a "classical limit", some speculate that even the classical theory needs renormalization, no matter how the Planck constant would be fixed.
== Abraham–Lorentz–Dirac force ==
To find the relativistic generalization, Dirac renormalized the mass in the equation of motion with the Abraham–Lorentz force in 1938. This renormalized equation of motion is called the Abraham–Lorentz–Dirac equation of motion.
=== Definition ===
The expression derived by Dirac is given in signature (− + + +) by
F
μ
r
a
d
=
μ
0
q
2
6
π
m
c
[
d
2
p
μ
d
τ
2
−
p
μ
m
2
c
2
(
d
p
ν
d
τ
d
p
ν
d
τ
)
]
.
{\displaystyle F_{\mu }^{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi mc}}\left[{\frac {d^{2}p_{\mu }}{d\tau ^{2}}}-{\frac {p_{\mu }}{m^{2}c^{2}}}\left({\frac {dp_{\nu }}{d\tau }}{\frac {dp^{\nu }}{d\tau }}\right)\right].}
With Liénard's relativistic generalization of Larmor's formula in the co-moving frame,
P
=
μ
0
q
2
a
2
γ
6
6
π
c
,
{\displaystyle P={\frac {\mu _{0}q^{2}a^{2}\gamma ^{6}}{6\pi c}},}
one can show this to be a valid force by manipulating the time average equation for power:
1
Δ
t
∫
0
t
P
d
t
=
1
Δ
t
∫
0
t
F
⋅
v
d
t
.
{\displaystyle {\frac {1}{\Delta t}}\int _{0}^{t}Pdt={\frac {1}{\Delta t}}\int _{0}^{t}{\textbf {F}}\cdot {\textbf {v}}\,dt.}
== Paradoxes ==
=== Pre-acceleration ===
Similar to the non-relativistic case, there are pathological solutions using the Abraham–Lorentz–Dirac equation that anticipate a change in the external force and according to which the particle accelerates in advance of the application of a force, so-called preacceleration solutions. One resolution of this problem was discussed by Yaghjian, and is further discussed by Rohrlich and Medina.
=== Runaway solutions ===
Runaway solutions are solutions to ALD equations that suggest the force on objects will increase exponential over time. It is considered as an unphysical solution.
=== Hyperbolic motion ===
The ALD equations are known to be zero for constant acceleration or hyperbolic motion in Minkowski spacetime diagram. The subject of whether in such condition electromagnetic radiation exists was matter of debate until Fritz Rohrlich resolved the problem by showing that hyperbolically moving charges do emit radiation. Subsequently, the issue is discussed in context of energy conservation and equivalence principle which is classically resolved by considering "acceleration energy" or Schott energy.
== Self-interactions ==
However the antidamping mechanism resulting from the Abraham–Lorentz force can be compensated by other nonlinear terms, which are frequently disregarded in the expansions of the retarded Liénard–Wiechert potential.
== Landau–Lifshitz radiation damping force ==
The Abraham–Lorentz–Dirac force leads to some pathological solutions. In order to avoid this, Lev Landau and Evgeny Lifshitz came with the following formula for radiation damping force, which is valid when the radiation damping force is small compared with the Lorentz force in some frame of reference (assuming it exists),
g
i
=
2
e
3
3
m
c
3
{
∂
F
i
k
∂
x
l
u
k
u
l
−
e
m
c
2
[
F
i
l
F
k
l
u
k
−
(
F
k
l
u
l
)
(
F
k
m
u
m
)
u
i
]
}
{\displaystyle g^{i}={\frac {2e^{3}}{3mc^{3}}}\left\{{\frac {\partial F^{ik}}{\partial x^{l}}}u_{k}u^{l}-{\frac {e}{mc^{2}}}\left[F^{il}F_{kl}u^{k}-(F_{kl}u^{l})(F^{km}u_{m})u^{i}\right]\right\}}
so that the equation of motion of the charge
e
{\displaystyle e}
in an external field
F
i
k
{\displaystyle F^{ik}}
can be written as
m
c
d
u
i
d
s
=
e
c
F
i
k
u
k
+
g
i
.
{\displaystyle mc{\frac {du^{i}}{ds}}={\frac {e}{c}}F^{ik}u_{k}+g^{i}.}
Here
u
i
=
(
γ
,
γ
v
/
c
)
{\displaystyle u^{i}=(\gamma ,\gamma \mathbf {v} /c)}
is the four-velocity of the particle,
γ
=
1
/
1
−
v
2
/
c
2
{\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}
is the Lorentz factor and
v
{\displaystyle \mathbf {v} }
is the three-dimensional velocity vector. The three-dimensional Landau–Lifshitz radiation damping force can be written as
F
r
a
d
=
2
e
3
γ
3
m
c
3
{
D
E
D
t
+
1
c
v
×
D
H
D
t
}
+
2
e
4
3
m
2
c
4
[
E
×
H
+
1
c
H
×
(
H
×
v
)
+
1
c
E
(
v
⋅
E
)
]
−
2
e
4
γ
2
v
3
m
2
c
5
[
(
E
+
1
c
v
×
H
)
2
−
1
c
2
(
E
⋅
v
)
2
]
{\displaystyle \mathbf {F} _{\mathrm {rad} }={\frac {2e^{3}\gamma }{3mc^{3}}}\left\{{\frac {D\mathbf {E} }{Dt}}+{\frac {1}{c}}\mathbf {v} \times {\frac {D\mathbf {H} }{Dt}}\right\}+{\frac {2e^{4}}{3m^{2}c^{4}}}\left[\mathbf {E} \times \mathbf {H} +{\frac {1}{c}}\mathbf {H} \times (\mathbf {H} \times \mathbf {v} )+{\frac {1}{c}}\mathbf {E} (\mathbf {v} \cdot \mathbf {E} )\right]-{\frac {2e^{4}\gamma ^{2}\mathbf {v} }{3m^{2}c^{5}}}\left[\left(\mathbf {E} +{\frac {1}{c}}\mathbf {v} \times \mathbf {H} \right)^{2}-{\frac {1}{c^{2}}}(\mathbf {E} \cdot \mathbf {v} )^{2}\right]}
where
D
/
D
t
=
∂
/
∂
t
+
v
⋅
∇
{\displaystyle D/Dt=\partial /\partial t+\mathbf {v} \cdot \nabla }
is the total derivative.
== Experimental observations ==
While the Abraham–Lorentz force is largely neglected for many experimental considerations, it gains importance for plasmonic excitations in larger nanoparticles due to large local field enhancements. Radiation damping acts as a limiting factor for the plasmonic excitations in surface-enhanced Raman scattering. The damping force was shown to broaden surface plasmon resonances in gold nanoparticles, nanorods and clusters.
The effects of radiation damping on nuclear magnetic resonance were also observed by Nicolaas Bloembergen and Robert Pound, who reported its dominance over spin–spin and spin–lattice relaxation mechanisms for certain cases.
The Abraham–Lorentz force has been observed in the semiclassical regime in experiments which involve the scattering of a relativistic beam of electrons with a high intensity laser. In the experiments, a supersonic jet of helium gas is intercepted by a high-intensity (1018–1020 W/cm2) laser. The laser ionizes the helium gas and accelerates the electrons via what is known as the “laser-wakefield” effect. A second high-intensity laser beam is then propagated counter to this accelerated electron beam. In a small number of cases, inverse-Compton scattering occurs between the photons and the electron beam, and the spectra of the scattered electrons and photons are measured. The photon spectra are then compared with spectra calculated from Monte Carlo simulations that use either the QED or classical LL equations of motion.
== Collective effects ==
The effects of radiation reaction are often considered within the framework of single-particle dynamics. However, interesting phenomena arise when a collection of charged particles is subjected to strong electromagnetic fields, such as in a plasma. In such scenarios, the collective behavior of the plasma can significantly modify its properties due to radiation reaction effects.
Theoretical studies have shown that in environments with strong magnetic fields, like those found around pulsars and magnetars, radiation reaction cooling can alter the collective dynamics of the plasma. This modification can lead to instabilities within the plasma. Specifically, in the high magnetic fields typical of these astrophysical objects, the momentum distribution of particles is bunched and becomes anisotropic due to radiation reaction forces, potentially driving plasma instabilities and affecting overall plasma behavior. Among these instabilities, the firehose instability can arise due to the anisotropic pressure, and electron cyclotron maser due to population inversion in the rings.
== See also ==
Lorentz force
Cyclotron radiation
Synchrotron radiation
Electromagnetic mass
Radiation resistance
Radiation damping
Wheeler–Feynman absorber theory
Magnetic radiation reaction force
== References ==
== Further reading ==
Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 978-0-13-805326-0. See sections 11.2.2 and 11.2.3
Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 978-0-471-30932-1.
Donald H. Menzel (1960) Fundamental Formulas of Physics, Dover Publications Inc., ISBN 0-486-60595-7, vol. 1, p. 345.
Stephen Parrott (1987) Relativistic Electrodynamics and Differential Geometry, § 4.3 Radiation reaction and the Lorentz–Dirac equation, pages 136–45, and § 5.5 Peculiar solutions of the Lorentz–Dirac equation, pp. 195–204, Springer-Verlag ISBN 0-387-96435-5 .
== External links ==
MathPages – Does A Uniformly Accelerating Charge Radiate?
Feynman: The Development of the Space-Time View of Quantum Electrodynamics
EC. del Río: Radiation of an accelerated charge | Wikipedia/Abraham–Lorentz_force |
In theoretical physics, the Born–Infeld model or the Dirac–Born–Infeld action is a particular example of what is usually known as a nonlinear electrodynamics. It was historically introduced in the 1930s to remove the divergence of the electron's self-energy in classical electrodynamics by introducing an upper bound of the electric field at the origin. It was introduced by Max Born and Leopold Infeld in 1934, with further work by Paul Dirac in 1962.
== Overview ==
Born–Infeld electrodynamics is named after physicists Max Born and Leopold Infeld, who first proposed it. The model possesses a whole series of physically interesting properties.
In analogy to a relativistic limit on velocity, Born–Infeld theory proposes a limiting force via limited electric field strength. A maximum electric field strength produces a finite electric field self-energy, which when attributed entirely to electron mass-produces maximum field.
E
B
I
=
1.187
×
10
20
V
/
m
.
{\displaystyle E_{\rm {BI}}=1.187\times 10^{20}\,\mathrm {V} /\mathrm {m} .}
Born–Infeld electrodynamics displays good physical properties concerning wave propagation, such as the absence of shock waves and birefringence. A field theory showing this property is usually called completely exceptional, and Born–Infeld theory is the only completely exceptional regular nonlinear electrodynamics.
This theory can be seen as a covariant generalization of Mie's theory and very close to Albert Einstein's idea of introducing a nonsymmetric metric tensor with the symmetric part corresponding to the usual metric tensor and the antisymmetric to the electromagnetic field tensor.
The compatibility of Born–Infeld theory with high-precision atomic experimental data requires a value of a limiting field some 200 times higher than that introduced in the original formulation of the theory.
Since 1985 there was a revival of interest on Born–Infeld theory and its nonabelian extensions, as they were found in some limits of string theory. It was discovered by E.S. Fradkin and A.A. Tseytlin that the Born–Infeld action is the leading term in the low-energy effective action of the open string theory expanded in powers of derivatives of gauge field strength.
== Equations ==
We will use the relativistic notation here, as this theory is fully relativistic.
The Lagrangian density is
L
=
−
b
2
−
det
(
η
+
F
b
)
+
b
2
,
{\displaystyle {\mathcal {L}}=-b^{2}{\sqrt {-\det \left(\eta +{\frac {F}{b}}\right)}}+b^{2},}
where η is the Minkowski metric, F is the Faraday tensor (both are treated as square matrices, so that we can take the determinant of their sum), and b is a scale parameter. The maximal possible value of the electric field in this theory is b, and the self-energy of point charges is finite. For electric and magnetic fields much smaller than b, the theory reduces to Maxwell electrodynamics.
In 4-dimensional spacetime the Lagrangian can be written as
L
=
−
b
2
1
−
E
2
−
B
2
b
2
−
(
E
⋅
B
)
2
b
4
+
b
2
,
{\displaystyle {\mathcal {L}}=-b^{2}{\sqrt {1-{\frac {E^{2}-B^{2}}{b^{2}}}-{\frac {(\mathbf {E} \cdot \mathbf {B} )^{2}}{b^{4}}}}}+b^{2},}
where E is the electric field, and B is the magnetic field.
In string theory, gauge fields on a D-brane (that arise from attached open strings) are described by the same type of Lagrangian:
L
=
−
T
−
det
(
η
+
2
π
α
′
F
)
,
{\displaystyle {\mathcal {L}}=-T{\sqrt {-\det(\eta +2\pi \alpha 'F)}},}
where T is the tension of the D-brane and
2
π
α
′
{\displaystyle 2\pi \alpha '}
is the invert of the string tension.
== References == | Wikipedia/Born–Infeld_model |
In particle physics, flavour or flavor refers to the species of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with flavour quantum numbers that are assigned to all subatomic particles. They can also be described by some of the family symmetries proposed for the quark-lepton generations.
== Quantum numbers ==
In classical mechanics, a force acting on a point-like particle can only alter the particle's dynamical state, i.e., its momentum, angular momentum, etc. Quantum field theory, however, allows interactions that can alter other facets of a particle's nature described by non-dynamical, discrete quantum numbers. In particular, the action of the weak force is such that it allows the conversion of quantum numbers describing mass and electric charge of both quarks and leptons from one discrete type to another. This is known as a flavour change, or flavour transmutation. Due to their quantum description, flavour states may also undergo quantum superposition.
In atomic physics the principal quantum number of an electron specifies the electron shell in which it resides, which determines the energy level of the whole atom. Analogously, the five flavour quantum numbers (isospin, strangeness, charm, bottomness or topness) can characterize the quantum state of quarks, by the degree to which it exhibits six distinct flavours (u, d, c, s, t, b).
Composite particles can be created from multiple quarks, forming hadrons, such as mesons and baryons, each possessing unique aggregate characteristics, such as different masses, electric charges, and decay modes. A hadron's overall flavour quantum numbers depend on the numbers of constituent quarks of each particular flavour.
=== Conservation laws ===
All of the various charges discussed above are conserved by the fact that the corresponding charge operators can be understood as generators of symmetries that commute with the Hamiltonian. Thus, the eigenvalues of the various charge operators are conserved.
Absolutely conserved quantum numbers in the Standard Model are:
electric charge (Q)
weak isospin (T3)
baryon number (B)
lepton number (L)
In some theories, such as the grand unified theory, the individual baryon and lepton number conservation can be violated, if the difference between them (B − L) is conserved (see Chiral anomaly).
Strong interactions conserve all flavours, but all flavour quantum numbers are violated (changed, non-conserved) by electroweak interactions.
== Flavour symmetry ==
If there are two or more particles which have identical interactions, then they may be interchanged without affecting the physics. All (complex) linear combinations of these two particles give the same physics, as long as the combinations are orthogonal, or perpendicular, to each other.
In other words, the theory possesses symmetry transformations such as
M
(
u
d
)
{\displaystyle M\left({u \atop d}\right)}
, where u and d are the two fields (representing the various generations of leptons and quarks, see below), and M is any 2×2 unitary matrix with a unit determinant. Such matrices form a Lie group called SU(2) (see special unitary group). This is an example of flavour symmetry.
In quantum chromodynamics, flavour is a conserved global symmetry. In the electroweak theory, on the other hand, this symmetry is broken, and flavour changing processes exist, such as quark decay or neutrino oscillations.
== Flavour quantum numbers ==
=== Leptons ===
All leptons carry a lepton number L = 1. In addition, leptons carry weak isospin, T3, which is −1/2 for the three charged leptons (i.e. electron, muon and tau) and +1/2 for the three associated neutrinos. Each doublet of a charged lepton and a neutrino consisting of opposite T3 are said to constitute one generation of leptons. In addition, one defines a quantum number called weak hypercharge, YW, which is −1 for all left-handed leptons. Weak isospin and weak hypercharge are gauged in the Standard Model.
Leptons may be assigned the six flavour quantum numbers: electron number, muon number, tau number, and corresponding numbers for the neutrinos (electron neutrino, muon neutrino and tau neutrino). These are conserved in strong and electromagnetic interactions, but violated by weak interactions. Therefore, such flavour quantum numbers are not of great use. A separate quantum number for each generation is more useful: electronic lepton number (+1 for electrons and electron neutrinos), muonic lepton number (+1 for muons and muon neutrinos), and tauonic lepton number (+1 for tau leptons and tau neutrinos). However, even these numbers are not absolutely conserved, as neutrinos of different generations can mix; that is, a neutrino of one flavour can transform into another flavour. The strength of such mixings is specified by a matrix called the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix).
=== Quarks ===
All quarks carry a baryon number B = ++1/3 , and all anti-quarks have B = −+1/3 . They also all carry weak isospin, T3 = ±+1/2 . The positively charged quarks (up, charm, and top quarks) are called up-type quarks and have T3 = ++1/2 ; the negatively charged quarks (down, strange, and bottom quarks) are called down-type quarks and have T3 = −+1/2 . Each doublet of up and down type quarks constitutes one generation of quarks.
For all the quark flavour quantum numbers listed below, the convention is that the flavour charge and the electric charge of a quark have the same sign. Thus any flavour carried by a charged meson has the same sign as its charge. Quarks have the following flavour quantum numbers:
The third component of isospin (usually just "isospin") (I3), which has value I3 = 1/2 for the up quark and I3 = −1/2 for the down quark.
Strangeness (S): Defined as S = −n s + n s̅ , where ns represents the number of strange quarks (s) and ns̅ represents the number of strange antiquarks (s). This quantum number was introduced by Murray Gell-Mann. This definition gives the strange quark a strangeness of −1 for the above-mentioned reason.
Charm (C): Defined as C = n c − n c̅ , where nc represents the number of charm quarks (c) and nc̅ represents the number of charm antiquarks. The charm quark's value is +1.
Bottomness (or beauty) (B′): Defined as B′ = −n b + n b̅ , where nb represents the number of bottom quarks (b) and nb̅ represents the number of bottom antiquarks.
Topness (or truth) (T): Defined as T = n t − n t̅ , where nt represents the number of top quarks (t) and nt̅ represents the number of top antiquarks. However, because of the extremely short half-life of the top quark (predicted lifetime of only 5×10−25 s), by the time it can interact strongly it has already decayed to another flavour of quark (usually to a bottom quark). For that reason the top quark doesn't hadronize, that is it never forms any meson or baryon.
These five quantum numbers, together with baryon number (which is not a flavour quantum number), completely specify numbers of all 6 quark flavours separately (as n q − n q̅ , i.e. an antiquark is counted with the minus sign). They are conserved by both the electromagnetic and strong interactions (but not the weak interaction). From them can be built the derived quantum numbers:
Hypercharge (Y): Y = B + S + C + B′ + T
Electric charge (Q): Q = I3 + 1/2Y (see Gell-Mann–Nishijima formula)
The terms "strange" and "strangeness" predate the discovery of the quark, but continued to be used after its discovery for the sake of continuity (i.e. the strangeness of each type of hadron remained the same); strangeness of anti-particles being referred to as +1, and particles as −1 as per the original definition. Strangeness was introduced to explain the rate of decay of newly discovered particles, such as the kaon, and was used in the Eightfold Way classification of hadrons and in subsequent quark models. These quantum numbers are preserved under strong and electromagnetic interactions, but not under weak interactions.
For first-order weak decays, that is processes involving only one quark decay, these quantum numbers (e.g. charm) can only vary by 1, that is, for a decay involving a charmed quark or antiquark either as the incident particle or as a decay byproduct, ΔC = ±1 ; likewise, for a decay involving a bottom quark or antiquark ΔB′ = ±1 . Since first-order processes are more common than second-order processes (involving two quark decays), this can be used as an approximate "selection rule" for weak decays.
A special mixture of quark flavours is an eigenstate of the weak interaction part of the Hamiltonian, so will interact in a particularly simple way with the W bosons (charged weak interactions violate flavour). On the other hand, a fermion of a fixed mass (an eigenstate of the kinetic and strong interaction parts of the Hamiltonian) is an eigenstate of flavour. The transformation from the former basis to the flavour-eigenstate/mass-eigenstate basis for quarks underlies the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix). This matrix is analogous to the PMNS matrix for neutrinos, and quantifies flavour changes under charged weak interactions of quarks.
The CKM matrix allows for CP violation if there are at least three generations.
=== Antiparticles and hadrons ===
Flavour quantum numbers are additive. Hence antiparticles have flavour equal in magnitude to the particle but opposite in sign. Hadrons inherit their flavour quantum number from their valence quarks: this is the basis of the classification in the quark model. The relations between the hypercharge, electric charge and other flavour quantum numbers hold for hadrons as well as quarks.
== Flavour problem ==
The flavour problem (also known as the flavour puzzle) is the inability of current Standard Model flavour physics to explain why the free parameters of particles in the Standard Model have the values they have, and why there are specified values for mixing angles in the PMNS and CKM matrices. These free parameters - the fermion masses and their mixing angles - appear to be specifically tuned. Understanding the reason for such tuning would be the solution to the flavor puzzle. There are very fundamental questions involved in this puzzle such as why there are three generations of quarks (up-down, charm-strange, and top-bottom quarks) and leptons (electron, muon and tau neutrino), as well as how and why the mass and mixing hierarchy arises among different flavours of these fermions.
== Quantum chromodynamics ==
Quantum chromodynamics (QCD) contains six flavours of quarks. However, their masses differ and as a result they are not strictly interchangeable with each other. The up and down flavours are close to having equal masses, and the theory of these two quarks possesses an approximate SU(2) symmetry (isospin symmetry).
=== Chiral symmetry description ===
Under some circumstances (for instance when the quark masses are much smaller than the chiral symmetry breaking scale of 250 MeV), the masses of quarks do not substantially contribute to the system's behavior, and to zeroth approximation the masses of the lightest quarks can be ignored for most purposes, as if they had zero mass. The simplified behavior of flavour transformations can then be successfully modeled as acting independently on the left- and right-handed parts of each quark field. This approximate description of the flavour symmetry is described by a chiral group SUL(Nf) × SUR(Nf).
=== Vector symmetry description ===
If all quarks had non-zero but equal masses, then this chiral symmetry is broken to the vector symmetry of the "diagonal flavour group" SU(Nf), which applies the same transformation to both helicities of the quarks. This reduction of symmetry is a form of explicit symmetry breaking. The strength of explicit symmetry breaking is controlled by the current quark masses in QCD.
Even if quarks are massless, chiral flavour symmetry can be spontaneously broken if the vacuum of the theory contains a chiral condensate (as it does in low-energy QCD). This gives rise to an effective mass for the quarks, often identified with the valence quark mass in QCD.
=== Symmetries of QCD ===
Analysis of experiments indicate that the current quark masses of the lighter flavours of quarks are much smaller than the QCD scale, ΛQCD, hence chiral flavour symmetry is a good approximation to QCD for the up, down and strange quarks. The success of chiral perturbation theory and the even more naive chiral models spring from this fact. The valence quark masses extracted from the quark model are much larger than the current quark mass. This indicates that QCD has spontaneous chiral symmetry breaking with the formation of a chiral condensate. Other phases of QCD may break the chiral flavour symmetries in other ways.
== History ==
=== Isospin ===
Isospin, strangeness and hypercharge predate the quark model. The first of those quantum numbers, Isospin, was introduced as a concept in 1932 by Werner Heisenberg, to explain symmetries of the then newly discovered neutron (symbol n):
The mass of the neutron and the proton (symbol p) are almost identical: They are nearly degenerate, and both are thus often referred to as “nucleons”, a term that ignores their intrinsic differences. Although the proton has a positive electric charge, and the neutron is neutral, they are almost identical in all other aspects, and their nuclear binding-force interactions (old name for the residual color force) are so strong compared to the electrical force between some, that there is very little point in paying much attention to their differences.
The strength of the strong interaction between any pair of nucleons is the same, independent of whether they are interacting as protons or as neutrons.
Protons and neutrons were grouped together as nucleons and treated as different states of the same particle, because they both have nearly the same mass and interact in nearly the same way, if the (much weaker) electromagnetic interaction is neglected.
Heisenberg noted that the mathematical formulation of this symmetry was in certain respects similar to the mathematical formulation of non-relativistic spin, whence the name "isospin" derives. The neutron and the proton are assigned to the doublet (the spin-1⁄2, 2, or fundamental representation) of SU(2), with the proton and neutron being then associated with different isospin projections I3 = ++1⁄2 and −+1⁄2 respectively. The pions are assigned to the triplet (the spin-1, 3, or adjoint representation) of SU(2). Though there is a difference from the theory of spin: The group action does not preserve flavor (in fact, the group action is specifically an exchange of flavour).
When constructing a physical theory of nuclear forces, one could simply assume that it does not depend on isospin, although the total isospin should be conserved. The concept of isospin proved useful in classifying hadrons discovered in the 1950s and 1960s (see particle zoo), where particles with similar mass are assigned an SU(2) isospin multiplet.
=== Strangeness and hypercharge ===
The discovery of strange particles like the kaon led to a new quantum number that was conserved by the strong interaction: strangeness (or equivalently hypercharge). The Gell-Mann–Nishijima formula was identified in 1953, which relates strangeness and hypercharge with isospin and electric charge.
=== The eightfold way and quark model ===
Once the kaons and their property of strangeness became better understood, it started to become clear that these, too, seemed to be a part of an enlarged symmetry that contained isospin as a subgroup. The larger symmetry was named the Eightfold Way by Murray Gell-Mann, and was promptly recognized to correspond to the adjoint representation of SU(3). To better understand the origin of this symmetry, Gell-Mann proposed the existence of up, down and strange quarks which would belong to the fundamental representation of the SU(3) flavor symmetry.
=== GIM-Mechanism and charm ===
To explain the observed absence of flavor-changing neutral currents, the GIM mechanism was proposed in 1970, which introduced the charm quark and predicted the J/psi meson. The J/psi meson was indeed found in 1974, which confirmed the existence of charm quarks. This discovery is known as the November Revolution. The flavor quantum number associated with the charm quark became known as charm.
=== Bottomness and topness ===
The bottom and top quarks were predicted in 1973 in order to explain CP violation, which also implied two new flavor quantum numbers: bottomness and topness.
== See also ==
Standard Model (mathematical formulation)
Cabibbo–Kobayashi–Maskawa matrix
Strong CP problem and chirality (physics)
Chiral symmetry breaking and quark matter
Quark flavour tagging, such as B-tagging, is an example of particle identification in experimental particle physics.
== References ==
== Further reading ==
Lessons in Particle Physics Luis Anchordoqui and Francis Halzen, University of Wisconsin, 18th Dec. 2009
== External links ==
The particle data group. | Wikipedia/Flavour_(particle_physics) |
The Thirring–Wess model or Vector Meson model
is an exactly solvable quantum field theory, describing the interaction of a Dirac field with a vector field in dimension two.
== Definition ==
The Lagrangian density is made of three terms:
the free vector field
A
μ
{\displaystyle A^{\mu }}
is described by
(
F
μ
ν
)
2
4
+
μ
2
2
(
A
μ
)
2
{\displaystyle {(F^{\mu \nu })^{2} \over 4}+{\mu ^{2} \over 2}(A^{\mu })^{2}}
for
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
{\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }}
and the boson mass
μ
{\displaystyle \mu }
must be
strictly positive;
the free fermion field
ψ
{\displaystyle \psi }
is described by
ψ
¯
(
i
∂
/
−
m
)
ψ
{\displaystyle {\overline {\psi }}(i\partial \!\!\!/-m)\psi }
where the fermion mass
m
{\displaystyle m}
can be positive or zero.
And the interaction term is
q
A
μ
(
ψ
¯
γ
μ
ψ
)
{\displaystyle qA^{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )}
Although not required to define the massive vector field, there can be also a gauge-fixing term
α
2
(
∂
μ
A
μ
)
2
{\displaystyle {\alpha \over 2}(\partial ^{\mu }A^{\mu })^{2}}
for
α
≥
0
{\displaystyle \alpha \geq 0}
There is a remarkable difference between the case
α
>
0
{\displaystyle \alpha >0}
and the case
α
=
0
{\displaystyle \alpha =0}
: the latter requires a field renormalization to absorb divergences of the two point correlation.
== History ==
This model was introduced by Thirring and Wess as a version of the Schwinger model with a vector mass term in the Lagrangian .
When the fermion is massless (
m
=
0
{\displaystyle m=0}
), the model is exactly solvable. One solution was found, for
α
=
1
{\displaystyle \alpha =1}
, by Thirring and Wess
using a method introduced by Johnson for the Thirring model; and, for
α
=
0
{\displaystyle \alpha =0}
, two different solutions were given by Brown and Sommerfield. Subsequently Hagen showed (for
α
=
0
{\displaystyle \alpha =0}
, but it turns out to be true for
α
≥
0
{\displaystyle \alpha \geq 0}
) that there is a one parameter family of solutions.
== References ==
== External links == | Wikipedia/Thirring–Wess_model |
Many first principles in quantum field theory are explained, or get further insight, in string theory.
== From quantum field theory to string theory ==
Emission and absorption: one of the most basic building blocks of quantum field theory, is the notion that particles (such as electrons) can emit and absorb other particles (such as photons). Thus, an electron may just "split" into an electron plus a photon, with a certain probability (which is roughly the coupling constant). This is described in string theory as one string splitting into two. This process is an integral part of the theory. The mode on the original string also "splits" between its two parts, resulting in two strings which possibly have different modes, representing two different particles.
Coupling constant: in quantum field theory this is, roughly, the probability for one particle to emit or absorb another particle, the latter typically being a gauge boson (a particle carrying a force). In string theory, the coupling constant is no longer a constant, but is rather determined by the abundance of strings in a particular mode, the dilaton. Strings in this mode couple to the worldsheet curvature of other strings, so their abundance through space-time determines the measure by which an average string worldsheet will be curved. This determines its probability to split or connect to other strings: the more a worldsheet is curved, the higher a chance of splitting and reconnecting it has.
Spin: each particle in quantum field theory has a particular spin s, which is an internal angular momentum. Classically, the particle rotates in a fixed frequency, but this cannot be understood if particles are point-like. In string theory, spin is understood by the rotation of the string; For example, a photon with well-defined spin components (i.e. in circular polarization) looks like a tiny straight line revolving around its center.
Gauge symmetry: in quantum field theory, the mathematical description of physical fields include non-physical states. In order to omit these states from the description of every physical process, a mechanism called gauge symmetry is used. This is true for string theory as well, but in string theory it is often more intuitive to understand why the non-physical states should be disposed of. The simplest example is the photon: a photon is a vector particle (it has an inner "arrow" which points to some direction, its polarization). Mathematically, it can point towards any direction in space-time. Suppose the photon is moving in the z direction; then it may either point towards the x, y or z spatial directions, or towards the t (time) direction (or any diagonal direction). Physically, however, the photon may not point towards the z or t directions (longitudinal polarization), but only in the x-y plane (transverse polarization). A gauge symmetry is used to dispose of the non-physical states. In string theory, a photon is described by a tiny oscillating line, with the axis of the line being the direction of the polarization (i.e. the inner direction of the photon is the axis of the string which the photon is made of). If we look at the worldsheet, the photon will look like a long strip which stretches along the time direction with an angle towards the z-direction (because it is moving along the z-direction as time goes by); its short dimension is therefore in the x-y plane. The short dimension of this strip is precisely the direction of the photon (its polarization) in a certain moment in time. Thus the photon cannot point towards the z or t directions, and its polarization must be transverse.
Note: formally, gauge symmetries in string theory are (at least in most cases) a result of the existence of a global symmetry together with the profound gauge symmetry of string theory, which is the symmetry of the worldsheet under a local change of coordinates and scales.
Renormalization: in particle physics the behaviour of particles in the smallest scales is largely unknown. In order to avoid this difficulty, the particles are treated as fields behaving according to an "effective field theory" at low energy scales, and a mathematical tool known as renormalization is used to describe the unknown aspects of this effective theory using only a few parameters. These parameters can be adjusted so that calculations give adequate results. In string theory, this is unnecessary since the behaviour of the strings is presumed to be known to every scale.
Fermions: in the bosonic string, a string can be described as an elastic one-dimensional object (i.e. a line) "living" in spacetime. In superstring theory, every point of the string is not only located at some point in spacetime, but it may also have a small arrow "drawn" on it, pointing at some direction in spacetime. These arrows are described by a field "living" on the string. This is a fermionic field, because at each point of the string there is only one arrow; thus one cannot bring two arrows to the same point. This fermionic field (which is a field on the worldsheet) is ultimately responsible for the appearance of fermions in spacetime: roughly, two strings with arrows drawn on them cannot coexist at the same point in spacetime, because then one would effectively have one string with two sets of arrows at the same point, which is not allowed, as explained above. Therefore two such strings are fermions in spacetime.
== Notes == | Wikipedia/Relationship_between_string_theory_and_quantum_field_theory |
In particle physics, NMSSM is an acronym for Next-to-Minimal Supersymmetric Standard Model.
It is a supersymmetric extension to the Standard Model that adds an additional singlet chiral superfield to the MSSM and can be used to dynamically generate the
μ
{\displaystyle \mu }
term, solving the
μ
{\displaystyle \mu }
-problem. Articles about the NMSSM are available for review.
The Minimal Supersymmetric Standard Model does not explain why the
μ
{\displaystyle \mu }
parameter in the superpotential term
μ
H
u
H
d
{\displaystyle \mu H_{u}H_{d}}
is at the electroweak scale. The idea behind the Next-to-Minimal Supersymmetric Standard Model is to promote the
μ
{\displaystyle \mu }
term to a gauge singlet, chiral superfield
S
{\displaystyle S}
. Note that the scalar superpartner of the singlino
S
{\displaystyle S}
is denoted by
S
^
{\displaystyle {\hat {S}}}
and the spin-1/2 singlino superpartner by
S
~
{\displaystyle {\tilde {S}}}
in the following. The superpotential for the NMSSM is given by
W
NMSSM
=
W
Yuk
+
λ
S
H
u
H
d
+
κ
3
S
3
{\displaystyle W_{\text{NMSSM}}=W_{\text{Yuk}}+\lambda SH_{u}H_{d}+{\frac {\kappa }{3}}S^{3}}
where
W
Yuk
{\displaystyle W_{\text{Yuk}}}
gives the Yukawa couplings for the Standard Model fermions. Since the superpotential has a mass dimension of 3, the couplings
λ
{\displaystyle \lambda }
and
κ
{\displaystyle \kappa }
are dimensionless; hence the
μ
{\displaystyle \mu }
-problem of the MSSM is solved in the NMSSM, the superpotential of the NMSSM being scale-invariant. The role of the
λ
{\displaystyle \lambda }
term is to generate an effective
μ
{\displaystyle \mu }
term. This is done with the scalar component of the singlet
S
^
{\displaystyle {\hat {S}}}
getting a vacuum-expectation value of
⟨
S
^
⟩
{\displaystyle \langle {\hat {S}}\rangle }
; that is, we have
μ
eff
=
λ
⟨
S
^
⟩
{\displaystyle \mu _{\text{eff}}=\lambda \langle {\hat {S}}\rangle }
Without the
κ
{\displaystyle \kappa }
term the superpotential would have a U(1)' symmetry, so-called Peccei–Quinn symmetry; see Peccei–Quinn theory. This additional symmetry would alter the phenomenology completely. The role of the
κ
{\displaystyle \kappa }
term is to break this U(1)' symmetry. The
κ
{\displaystyle \kappa }
term is introduced trilinearly such that
κ
{\displaystyle \kappa }
is dimensionless. However, there remains a discrete
Z
3
{\displaystyle \mathbb {Z} _{3}}
symmetry, which is moreover broken spontaneously. In principle this leads to the domain wall problem. Introducing additional but suppressed terms, the
Z
3
{\displaystyle \mathbb {Z} _{3}}
symmetry can be broken without changing phenomenology at the electroweak scale.
It is assumed that the domain wall problem is circumvented in this way without any modifications except far beyond the electroweak scale.
Other models have been proposed which solve the
μ
{\displaystyle \mu }
-problem of the MSSM. One idea is to keep the
κ
{\displaystyle \kappa }
term in the superpotential and take the U(1)' symmetry into account. Assuming this symmetry to be local, an additional,
Z
′
{\displaystyle Z'}
gauge boson is predicted in this model, called the UMSSM.
== Phenomenology ==
Due to the additional singlet
S
{\displaystyle S}
, the NMSSM alters in general the phenomenology of both the Higgs sector and the neutralino sector compared with the MSSM.
=== Higgs phenomenology ===
In the Standard Model we have one physical Higgs boson. In the MSSM we encounter five physical Higgs bosons. Due to the additional singlet
S
^
{\displaystyle {\hat {S}}}
in the NMSSM we have two more Higgs bosons; that is, in total seven physical Higgs bosons. Its Higgs sector is therefore much richer than that of the MSSM. In particular, the Higgs potential is in general no longer invariant under CP transformations; see CP violation. Typically, the Higgs bosons in the NMSSM are denoted in an order with increasing masses; that is, by
H
1
,
H
2
,
.
.
.
,
H
7
{\displaystyle H_{1},H_{2},...,H_{7}}
, with
H
1
{\displaystyle H_{1}}
the lightest Higgs boson. In the special case of a CP-conserving Higgs potential we have three CP even Higgs bosons,
H
1
,
H
2
,
H
3
{\displaystyle H_{1},H_{2},H_{3}}
, two CP odd ones,
A
1
,
A
2
{\displaystyle A_{1},A_{2}}
, and a pair of charged Higgs bosons,
H
+
,
H
−
{\displaystyle H^{+},H^{-}}
. In the MSSM, the lightest Higgs boson is always Standard Model-like, and therefore its production and decays are roughly known. In the NMSSM, the lightest Higgs can be very light (even of the order of 1 GeV), and thus may have escaped detection so far. In addition, in the CP-conserving case, the lightest CP even Higgs boson turns out to have an enhanced lower bound compared with the MSSM. This is one of the reasons why the NMSSM has been the focus of much attention in recent years.
=== Neutralino phenomenology ===
The spin-1/2 singlino
S
~
{\displaystyle {\tilde {S}}}
gives a fifth neutralino, compared with the four neutralinos of the MSSM. The singlino does not couple with any gauge bosons, gauginos (the superpartners of the gauge bosons), leptons, sleptons (the superpartners of the leptons), quarks or squarks (the superpartners of the quarks). Suppose that a supersymmetric partner particle is produced at a collider, for instance at the LHC, the singlino is omitted in cascade decays and therefore escapes detection. However, if the singlino is the lightest supersymmetric particle (LSP), all supersymmetric partner particles eventually decay into the singlino. Due to R parity conservation this LSP is stable. In this way the singlino could be detected via missing transverse energy in a detector.
== References == | Wikipedia/Next-to-Minimal_Supersymmetric_Standard_Model |
In theoretical physics, Seiberg–Witten theory is an
N
=
2
{\displaystyle {\mathcal {N}}=2}
supersymmetric gauge theory with an exact low-energy effective action (for massless degrees of freedom), of which the kinetic part coincides with the Kähler potential of the moduli space of vacua. Before taking the low-energy effective action, the theory is known as
N
=
2
{\displaystyle {\mathcal {N}}=2}
supersymmetric Yang–Mills theory, as the field content is a single
N
=
2
{\displaystyle {\mathcal {N}}=2}
vector supermultiplet, analogous to the field content of Yang–Mills theory being a single vector gauge field (in particle theory language) or connection (in geometric language).
The theory was studied in detail by Nathan Seiberg and Edward Witten (Seiberg & Witten 1994).
== Seiberg–Witten curves ==
In general, effective Lagrangians of supersymmetric gauge theories are largely determined by their holomorphic (really, meromorphic) properties and their behavior near the singularities. In gauge theory with
N
=
2
{\displaystyle {\mathcal {N}}=2}
extended supersymmetry, the moduli space of vacua is a special Kähler manifold and its Kähler potential is constrained by above conditions.
In the original approach, by Seiberg and Witten, holomorphy and electric-magnetic duality constraints are strong enough to almost uniquely
constrain the prepotential
F
{\displaystyle {\mathcal {F}}}
(a holomorphic function which defines the theory), and therefore the metric of the moduli space of vacua, for theories with SU(2) gauge group.
More generally, consider the example with gauge group SU(n). The classical potential is
where
ϕ
{\displaystyle \phi }
is a scalar field appearing in an expansion of superfields in the theory. The potential must vanish on the moduli space of vacua by definition, but the
ϕ
{\displaystyle \phi }
need not. The vacuum expectation value of
ϕ
{\displaystyle \phi }
can be gauge rotated into the Cartan subalgebra, making it a traceless diagonal complex matrix
a
{\displaystyle a}
.
Because the fields
ϕ
{\displaystyle \phi }
no longer have vanishing vacuum expectation value, other fields become massive due to the Higgs mechanism (spontaneous symmetry breaking). They are integrated out in order to find the effective
N
=
2
{\displaystyle {\mathcal {N}}=2}
U(1) gauge theory. Its two-derivative, four-fermions low-energy action is given by a Lagrangian which can be expressed in terms of a single holomorphic function
F
{\displaystyle {\mathcal {F}}}
on
N
=
1
{\displaystyle {\mathcal {N}}=1}
superspace as follows:
where
and
A
{\displaystyle A}
is a chiral superfield on
N
=
1
{\displaystyle {\mathcal {N}}=1}
superspace which fits inside the
N
=
2
{\displaystyle {\mathcal {N}}=2}
chiral multiplet
A
{\displaystyle {\mathcal {A}}}
.
The first term is a perturbative loop calculation and the second is the instanton part where
k
{\displaystyle k}
labels fixed instanton numbers. In theories whose gauge groups are products of unitary groups,
F
{\displaystyle {\mathcal {F}}}
can be computed exactly using localization and the limit shape techniques.
The Kähler potential is the kinetic part of the low energy action, and explicitly is written in terms of
F
{\displaystyle {\mathcal {F}}}
as
From
F
{\displaystyle {\mathcal {F}}}
we can get the mass of the BPS particles.
One way to interpret this is that these variables
a
{\displaystyle a}
and its dual can be expressed as periods of a meromorphic differential on a Riemann surface called the Seiberg–Witten curve.
== N = 2 supersymmetric Yang–Mills theory ==
Before the low energy, or infrared, limit is taken, the action can be given in terms of a Lagrangian over
N
=
2
{\displaystyle {\mathcal {N}}=2}
superspace with field content
Ψ
{\displaystyle \Psi }
, which is a single
N
=
2
{\displaystyle {\mathcal {N}}=2}
vector/chiral superfield in the adjoint representation of the gauge group, and a holomorphic function
F
{\displaystyle {\mathcal {F}}}
of
Ψ
{\displaystyle \Psi }
called the prepotential. Then the Lagrangian is given by
L
S
Y
M
2
=
I
m
T
r
(
1
4
π
∫
d
2
θ
d
2
ϑ
F
(
Ψ
)
)
{\displaystyle {\mathcal {L}}_{SYM2}=\mathrm {Im} \mathrm {Tr} \left({\frac {1}{4\pi }}\int d^{2}\theta d^{2}\vartheta {\mathcal {F}}(\Psi )\right)}
where
θ
,
ϑ
{\displaystyle \theta ,\vartheta }
are coordinates for the spinor directions of superspace. Once the low energy limit is taken, the
N
=
2
{\displaystyle {\mathcal {N}}=2}
superfield
Ψ
{\displaystyle \Psi }
is typically labelled by
A
{\displaystyle {\mathcal {A}}}
instead.
The so called minimal theory is given by a specific choice of
F
{\displaystyle {\mathcal {F}}}
,
F
(
ψ
)
=
1
2
τ
Ψ
2
,
{\displaystyle {\mathcal {F}}(\psi )={\frac {1}{2}}\tau \Psi ^{2},}
where
τ
{\displaystyle \tau }
is the complex coupling constant.
The minimal theory can be written on Minkowski spacetime as
L
=
1
g
2
T
r
(
−
1
4
F
μ
ν
F
μ
ν
+
g
2
θ
32
π
2
F
μ
ν
∗
F
μ
ν
+
(
D
μ
ϕ
)
†
(
D
μ
ϕ
)
−
1
2
[
ϕ
,
ϕ
†
]
2
−
i
λ
σ
μ
D
μ
λ
¯
−
i
ψ
¯
σ
¯
μ
D
μ
ψ
−
i
2
[
λ
,
ψ
]
ϕ
†
−
i
2
[
λ
¯
,
ψ
¯
]
ϕ
)
{\displaystyle {\mathcal {L}}={\frac {1}{g^{2}}}\mathrm {Tr} \left(-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+g^{2}{\frac {\theta }{32\pi ^{2}}}F_{\mu \nu }*F^{\mu \nu }+(D_{\mu }\phi )^{\dagger }(D^{\mu }\phi )-{\frac {1}{2}}[\phi ,\phi ^{\dagger }]^{2}-i\lambda \sigma ^{\mu }D_{\mu }{\bar {\lambda }}-i{\bar {\psi }}{\bar {\sigma }}^{\mu }D_{\mu }\psi -i{\sqrt {2}}[\lambda ,\psi ]\phi ^{\dagger }-i{\sqrt {2}}[{\bar {\lambda }},{\bar {\psi }}]\phi \right)}
with
A
μ
,
λ
,
ψ
,
ϕ
{\displaystyle A_{\mu },\lambda ,\psi ,\phi }
making up the
N
=
2
{\displaystyle {\mathcal {N}}=2}
chiral multiplet.
== Geometry of the moduli space ==
For this section fix the gauge group as
S
U
(
2
)
{\displaystyle \mathrm {SU(2)} }
. A low-energy vacuum solution is an
N
=
2
{\displaystyle {\mathcal {N}}=2}
vector superfield
A
{\displaystyle {\mathcal {A}}}
solving the equations of motion of the low-energy Lagrangian, for which the scalar part
ϕ
{\displaystyle \phi }
has vanishing potential, which as mentioned earlier holds if
[
ϕ
,
ϕ
†
]
=
0
{\displaystyle [\phi ,\phi ^{\dagger }]=0}
(which exactly means
ϕ
{\displaystyle \phi }
is a normal operator, and therefore diagonalizable). The scalar
ϕ
{\displaystyle \phi }
transforms in the adjoint, that is, it can be identified as an element of
s
u
(
2
)
C
≅
s
l
(
2
,
C
)
{\displaystyle {\mathfrak {su}}(2)_{\mathbb {C} }\cong {\mathfrak {sl}}(2,\mathbb {C} )}
, the complexification of
s
u
(
2
)
{\displaystyle {\mathfrak {su}}(2)}
. Thus
ϕ
{\displaystyle \phi }
is traceless and diagonalizable so can be gauge rotated to (is in the conjugacy class of) a matrix of the form
1
2
a
σ
3
{\displaystyle {\frac {1}{2}}a\sigma _{3}}
(where
σ
3
{\displaystyle \sigma _{3}}
is the third Pauli matrix) for
a
∈
C
{\displaystyle a\in \mathbb {C} }
. However,
a
{\displaystyle a}
and
−
a
{\displaystyle -a}
give conjugate matrices (corresponding to the fact the Weyl group of
S
U
(
2
)
{\displaystyle \mathrm {SU} (2)}
is
Z
2
{\displaystyle \mathbb {Z} _{2}}
) so both label the same vacuum. Thus the gauge invariant quantity labelling inequivalent vacua is
u
=
a
2
/
2
=
T
r
ϕ
2
{\displaystyle u=a^{2}/2=\mathrm {Tr} \phi ^{2}}
. The (classical) moduli space of vacua is a one-dimensional complex manifold (Riemann surface) parametrized by
u
{\displaystyle u}
, although the Kähler metric is given in terms of
a
{\displaystyle a}
as
d
s
2
=
I
m
∂
2
F
∂
a
2
d
a
d
a
¯
=
I
m
d
a
D
d
a
¯
=
−
i
2
(
d
a
D
d
a
¯
−
d
a
d
a
¯
D
)
=:
I
m
τ
(
a
)
d
a
d
a
¯
,
{\displaystyle ds^{2}=\mathrm {Im} {\frac {\partial ^{2}{\mathcal {F}}}{\partial a^{2}}}dad{\bar {a}}=\mathrm {Im} da_{D}d{\bar {a}}=-{\frac {i}{2}}(da_{D}d{\bar {a}}-dad{\bar {a}}_{D})=:\mathrm {Im} \tau (a)dad{\bar {a}},}
where
a
D
=
∂
F
∂
a
{\displaystyle a_{D}={\frac {\partial {\mathcal {F}}}{\partial a}}}
. This is not invariant under an arbitrary change of coordinates, but due to symmetry in
a
{\displaystyle a}
and
a
D
{\displaystyle a_{D}}
, switching to local coordinate
a
D
{\displaystyle a_{D}}
gives a metric similar to the final form but with a different harmonic function replacing
I
m
τ
(
a
)
{\displaystyle \mathrm {Im} \tau (a)}
. The switching of the two coordinates can be interpreted as an instance of electric-magnetic duality (Seiberg & Witten 1994).
Under a minimal assumption of assuming there are only three singularities in the moduli space at
u
=
−
1
,
+
1
{\displaystyle u=-1,+1}
and
∞
{\displaystyle \infty }
, with prescribed monodromy data at each point derived from quantum field theoretic arguments, the moduli space
M
{\displaystyle {\mathcal {M}}}
was found to be
H
/
Γ
(
2
)
{\displaystyle H/\Gamma (2)}
, where
H
{\displaystyle H}
is the hyperbolic half-plane and
Γ
(
2
)
<
S
L
(
2
,
Z
)
{\displaystyle \Gamma (2)<\mathrm {SL} (2,\mathbb {Z} )}
is the second principal congruence subgroup, the subgroup of matrices congruent to 1 mod 2, generated by
M
∞
=
(
−
1
2
0
−
1
)
,
M
1
=
(
1
0
−
2
1
)
,
M
−
1
=
(
−
1
2
−
2
3
)
.
{\displaystyle M_{\infty }={\begin{pmatrix}-1&2\\0&-1\end{pmatrix}},M_{1}={\begin{pmatrix}1&0\\-2&1\end{pmatrix}},M_{-1}={\begin{pmatrix}-1&2\\-2&3\end{pmatrix}}.}
This space is a six-fold cover of the fundamental domain of the modular group and admits an explicit description as parametrizing a space of elliptic curves
E
u
{\displaystyle E_{u}}
given by the vanishing of
y
2
=
(
x
−
1
)
(
x
+
1
)
(
x
−
u
)
,
{\displaystyle y^{2}=(x-1)(x+1)(x-u),}
which are the Seiberg–Witten curves. The curve becomes singular precisely when
u
=
−
1
,
+
1
{\displaystyle u=-1,+1}
or
∞
{\displaystyle \infty }
.
== Monopole condensation and confinement ==
The theory exhibits physical phenomena involving and linking magnetic monopoles, confinement, an attained mass gap and strong-weak duality, described in section 5.6 of Seiberg and Witten (1994). The study of these physical phenomena also motivated the theory of Seiberg–Witten invariants.
The low-energy action is described by the
N
=
2
{\displaystyle {\mathcal {N}}=2}
chiral multiplet
A
{\displaystyle {\mathcal {A}}}
with gauge group
U
(
1
)
{\displaystyle \mathrm {U} (1)}
, the residual unbroken gauge from the original
S
U
(
2
)
{\displaystyle \mathrm {SU} (2)}
symmetry. This description is weakly coupled for large
u
{\displaystyle u}
, but strongly coupled for small
u
{\displaystyle u}
. However, at the strongly coupled point the theory admits a dual description which is weakly coupled. The dual theory has different field content, with two
N
=
1
{\displaystyle {\mathcal {N}}=1}
chiral superfields
M
,
M
~
{\displaystyle M,{\tilde {M}}}
, and gauge field the dual photon
A
D
{\displaystyle {\mathcal {A}}_{D}}
, with a potential that gives equations of motion which are Witten's monopole equations, also known as the Seiberg–Witten equations at the critical points
u
=
±
u
0
{\displaystyle u=\pm u_{0}}
where the monopoles become massless.
In the context of Seiberg–Witten invariants, one can view Donaldson invariants as coming from a twist of the original theory at
u
=
∞
{\displaystyle u=\infty }
giving a topological field theory. On the other hand, Seiberg–Witten invariants come from twisting the dual theory at
u
=
±
u
0
{\displaystyle u=\pm u_{0}}
. In theory, such invariants should receive contributions from all finite
u
{\displaystyle u}
but in fact can be localized to the two critical points, and topological invariants can be read off from solution spaces to the monopole equations.
== Relation to integrable systems ==
The special Kähler geometry on the moduli space of vacua in Seiberg–Witten theory can be identified with the geometry of the base of complex completely integrable system. The total phase of this complex completely integrable system can be identified with the moduli space of vacua of the 4d theory compactified on a circle. The relation between Seiberg–Witten theory and integrable systems has been reviewed by Eric D'Hoker and D. H. Phong. See Hitchin system.
== Seiberg–Witten prepotential via instanton counting ==
Using supersymmetric localisation techniques, one can explicitly determine the instanton partition function of
N
=
2
{\displaystyle {\mathcal {N}}=2}
super Yang–Mills theory. The Seiberg–Witten prepotential can then be extracted using the localization approach of Nikita Nekrasov. It arises in the flat space limit
ε
1
{\displaystyle \varepsilon _{1}}
,
ε
2
→
0
{\displaystyle \varepsilon _{2}\to 0}
, of the partition function of the theory subject to the so-called
Ω
{\displaystyle \Omega }
-background.
The latter is a specific background of four dimensional
N
=
2
{\displaystyle {\mathcal {N}}=2}
supergravity. It can be engineered, formally by lifting the super Yang–Mills theory to six dimensions, then compactifying on 2-torus, while twisting the four dimensional spacetime around the two non-contractible cycles. In addition, one twists fermions so as to produce covariantly constant spinors generating unbroken supersymmetries. The two parameters
ε
1
{\displaystyle \varepsilon _{1}}
,
ε
2
{\displaystyle \varepsilon _{2}}
of the
Ω
{\displaystyle \Omega }
-background correspond to the angles of the spacetime rotation.
In Ω-background, all the non-zero modes can be integrated out, so the path integral with the boundary condition
ϕ
→
a
{\displaystyle \phi \to a}
at
x
→
∞
{\displaystyle x\to \infty }
can be expressed as a sum over instanton number of the products and ratios of fermionic and bosonic determinants, producing the so-called Nekrasov partition function.
In the limit where
ε
1
{\displaystyle \varepsilon _{1}}
,
ε
2
{\displaystyle \varepsilon _{2}}
approach 0, this sum is dominated by a unique saddle point.
On the other hand, when
ε
1
{\displaystyle \varepsilon _{1}}
,
ε
2
{\displaystyle \varepsilon _{2}}
approach 0,
holds.
== See also ==
Ginzburg–Landau theory
Donaldson theory
== References ==
Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Springer-Verlag. ISBN 3-540-42627-2. (See Section 7.2)
Hunter-Jones, Nicholas R. (September 2012). Seiberg–Witten Theory and Duality in N = 2 Supersymmetric Gauge Theories (PDF) (Masters). Imperial College London. | Wikipedia/Seiberg–Witten_theory |
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In contrast to other types of conformal field theories, two-dimensional conformal field theories have infinite-dimensional symmetry algebras. In some cases, this allows them to be solved exactly, using the conformal bootstrap method.
Notable two-dimensional conformal field theories include minimal models, Liouville theory, massless free bosonic theories, Wess–Zumino–Witten models, and certain sigma models.
== Basic structures ==
=== Geometry ===
Two-dimensional conformal field theories (CFTs) are defined on Riemann surfaces, where local conformal maps are holomorphic functions.
While a CFT might conceivably exist only on a given Riemann surface, its existence on any surface other than the sphere implies its existence on all surfaces.
Given a CFT, it is indeed possible to glue two Riemann surfaces where it exists, and obtain the CFT on the glued surface.
On the other hand, some CFTs exist only on the sphere.
Unless stated otherwise, we consider CFT on the sphere in this article.
=== Symmetries and integrability ===
Given a local complex coordinate
z
{\displaystyle z}
, the real vector space of infinitesimal conformal maps
has the basis
(
ℓ
n
+
ℓ
¯
n
)
n
∈
Z
∪
(
i
(
ℓ
n
−
ℓ
¯
n
)
)
n
∈
Z
{\displaystyle (\ell _{n}+{\bar {\ell }}_{n})_{n\in \mathbb {Z} }\cup (i(\ell _{n}-{\bar {\ell }}_{n}))_{n\in \mathbb {Z} }}
, with
ℓ
n
=
−
z
n
+
1
∂
∂
z
{\displaystyle \ell _{n}=-z^{n+1}{\frac {\partial }{\partial z}}}
. (For example,
ℓ
−
1
+
ℓ
¯
−
1
{\displaystyle \ell _{-1}+{\bar {\ell }}_{-1}}
and
i
(
ℓ
−
1
−
ℓ
¯
−
1
)
{\displaystyle i(\ell _{-1}-{\bar {\ell }}_{-1})}
generate translations.) Relaxing the assumption that
z
¯
{\displaystyle {\bar {z}}}
is the complex conjugate of
z
{\displaystyle z}
, i.e. complexifying the space of infinitesimal conformal maps, one obtains a complex vector space with the basis
(
ℓ
n
)
n
∈
Z
∪
(
ℓ
¯
n
)
n
∈
Z
{\displaystyle (\ell _{n})_{n\in \mathbb {Z} }\cup ({\bar {\ell }}_{n})_{n\in \mathbb {Z} }}
.
With their natural commutators,
the differential operators
ℓ
n
{\displaystyle \ell _{n}}
generate a Witt algebra.
Unfortunately the Witt algebra on its own always generates a space of particle states which has infinitely many negative energy states with the energy of each state getting progressively lower. For a physical theory to make sensible predictions in the sense of having a stationary phase approximation of the action to expand about, there must be a lowest energy state called the vacuum. The energy of the vacuum is completely arbitrary since a central scalar constant may be added to the Hamiltonian to globally shift the phase without changing the observable dynamics, and so the vacuum energy may take negative values so long as it is bounded below. This requirement is for instance what prompted the Dirac sea interpretation to address the Dirac equation's prediction of negative energy solutions, precisely because they generate an algebra of creation operators that can lower the energy ad infinitum.
To rectify this situation, the Witt algebra is centrally extended to provide a richer variety of Hilbert space modules to choose from, including the so-called positive energy representations, while leaving intact almost all of the Lie bracket relations between operators. This new algebra is called the Virasoro algebra, whose generators are
(
L
n
)
n
∈
Z
{\displaystyle (L_{n})_{n\in \mathbb {Z} }}
, plus a central generator. The central generator takes a constant value
c
{\displaystyle c}
, called the central charge, and the values of
c
{\displaystyle c}
for which there is a positive energy representation is known (either
c
≥
1
{\displaystyle c\geq 1}
or
c
=
1
−
6
(
m
+
2
)
(
m
+
3
)
{\displaystyle c=1-{\frac {6}{\left(m+2\right)\left(m+3\right)}}}
for
m
∈
N
{\displaystyle m\in \mathbb {N} }
).
The symmetry algebra of a CFT is the product of two commuting copies of the Virasoro algebra: the left-moving or holomorphic algebra, with generators
L
n
{\displaystyle L_{n}}
, and the right-moving or antiholomorphic algebra, with generators
L
¯
n
{\displaystyle {\bar {L}}_{n}}
. These two copies are also known as the chiral algebras.
In the universal enveloping algebra of the Virasoro algebra, it is possible to construct an infinite set of mutually commuting charges. The first charge is
L
0
−
c
24
{\displaystyle L_{0}-{\frac {c}{24}}}
, the second charge is quadratic in the Virasoro generators, the third charge is cubic, and so on. This shows that any two-dimensional conformal field theory is also a quantum integrable system.
=== Space of states ===
The space of states, also called the spectrum, of a CFT, is a representation of the product of the two Virasoro algebras.
For a state that is an eigenvector of
L
0
{\displaystyle L_{0}}
and
L
¯
0
{\displaystyle {\bar {L}}_{0}}
with the eigenvalues
Δ
{\displaystyle \Delta }
and
Δ
¯
{\displaystyle {\bar {\Delta }}}
,
Δ
{\displaystyle \Delta }
is the left conformal dimension,
Δ
¯
{\displaystyle {\bar {\Delta }}}
is the right conformal dimension,
Δ
+
Δ
¯
{\displaystyle \Delta +{\bar {\Delta }}}
is the total conformal dimension or the energy,
Δ
−
Δ
¯
{\displaystyle \Delta -{\bar {\Delta }}}
is the conformal spin.
A CFT is called rational if its space of states decomposes into finitely many irreducible representations of the product of the two Virasoro algebras. In a rational CFT that is defined on all Riemann surfaces, the central charge and conformal dimensions are rational numbers.
A CFT is called diagonal if its space of states is a direct sum of representations of the type
R
⊗
R
¯
{\displaystyle R\otimes {\bar {R}}}
, where
R
{\displaystyle R}
is an indecomposable representation of the left Virasoro algebra, and
R
¯
{\displaystyle {\bar {R}}}
is the same representation of the right Virasoro algebra.
The CFT is called unitary if the space of states has a positive definite Hermitian form such that
L
0
{\displaystyle L_{0}}
and
L
¯
0
{\displaystyle {\bar {L}}_{0}}
are self-adjoint,
L
0
†
=
L
0
{\displaystyle L_{0}^{\dagger }=L_{0}}
and
L
¯
0
†
=
L
¯
0
{\displaystyle {\bar {L}}_{0}^{\dagger }={\bar {L}}_{0}}
. This implies in particular that
L
n
†
=
L
−
n
{\displaystyle L_{n}^{\dagger }=L_{-n}}
, and that the central charge is real. The space of states is then a Hilbert space. While unitarity is necessary for a CFT to be a proper quantum system with a probabilistic interpretation, many interesting CFTs are nevertheless non-unitary, including minimal models and Liouville theory for most allowed values of the central charge.
=== Fields and correlation functions ===
The state-field correspondence is a linear map
v
↦
V
v
(
z
)
{\displaystyle v\mapsto V_{v}(z)}
from the space of states to the space of fields, which commutes with the action of the symmetry algebra.
In particular, the image of a primary state of a lowest weight representation of the Virasoro algebra is a primary field
V
Δ
(
z
)
{\displaystyle V_{\Delta }(z)}
, such that
L
n
>
0
V
Δ
(
z
)
=
0
,
L
0
V
Δ
(
z
)
=
Δ
V
Δ
(
z
)
.
{\displaystyle L_{n>0}V_{\Delta }(z)=0\quad ,\quad L_{0}V_{\Delta }(z)=\Delta V_{\Delta }(z)\ .}
Descendant fields are obtained from primary fields by acting with creation modes
L
n
<
0
{\displaystyle L_{n<0}}
. Degenerate fields correspond to primary states of degenerate representations. For example, the degenerate field
V
1
,
1
(
z
)
{\displaystyle V_{1,1}(z)}
obeys
L
−
1
V
1
,
1
(
z
)
=
0
{\displaystyle L_{-1}V_{1,1}(z)=0}
, due to the presence of a null vector in the corresponding degenerate representation.
An
N
{\displaystyle N}
-point correlation function is a number that depends linearly on
N
{\displaystyle N}
fields, denoted as
⟨
V
1
(
z
1
)
⋯
V
N
(
z
N
)
⟩
{\displaystyle \left\langle V_{1}(z_{1})\cdots V_{N}(z_{N})\right\rangle }
with
i
≠
j
⇒
z
i
≠
z
j
{\displaystyle i\neq j\Rightarrow z_{i}\neq z_{j}}
.
In the path integral formulation of conformal field theory, correlation functions are defined as functional integrals. In the conformal bootstrap approach, correlation functions are defined by axioms. In particular, it is assumed that there exists an operator product expansion (OPE),
V
1
(
z
1
)
V
2
(
z
2
)
=
∑
i
C
12
v
i
(
z
1
,
z
2
)
V
v
i
(
z
2
)
,
{\displaystyle V_{1}(z_{1})V_{2}(z_{2})=\sum _{i}C_{12}^{v_{i}}(z_{1},z_{2})V_{v_{i}}(z_{2})\ ,}
where
{
v
i
}
{\displaystyle \{v_{i}\}}
is a basis of the space of states, and the numbers
C
12
v
i
(
z
1
,
z
2
)
{\displaystyle C_{12}^{v_{i}}(z_{1},z_{2})}
are called OPE coefficients. Moreover, correlation functions are assumed to be invariant under permutations on the fields, in other words the OPE is assumed to be associative and commutative. (OPE commutativity
V
1
(
z
1
)
V
2
(
z
2
)
=
V
2
(
z
2
)
V
1
(
z
1
)
{\displaystyle V_{1}(z_{1})V_{2}(z_{2})=V_{2}(z_{2})V_{1}(z_{1})}
does not imply that OPE coefficients are invariant under
1
↔
2
{\displaystyle 1\leftrightarrow 2}
, because expanding on fields
V
v
i
(
z
2
)
{\displaystyle V_{v_{i}}(z_{2})}
breaks that symmetry.)
OPE commutativity implies that primary fields have integer conformal spins
S
∈
Z
{\displaystyle S\in \mathbb {Z} }
. A primary field with zero conformal spin is called a diagonal field.
There also exist fermionic CFTs that include fermionic fields with half-integer conformal spins
S
∈
1
2
+
Z
{\displaystyle S\in {\tfrac {1}{2}}+\mathbb {Z} }
, which anticommute.
There also exist parafermionic CFTs that include fields with more general rational spins
S
∈
Q
{\displaystyle S\in \mathbb {Q} }
. Not only parafermions do not commute, but also their correlation functions are multivalued.
The torus partition function is a particular correlation function that depends solely on the spectrum
S
{\displaystyle {\mathcal {S}}}
, and not on the OPE coefficients. For a complex torus
C
Z
+
τ
Z
{\displaystyle {\frac {\mathbb {C} }{\mathbb {Z} +\tau \mathbb {Z} }}}
with modulus
τ
{\displaystyle \tau }
, the partition function is
Z
(
τ
)
=
Tr
S
q
L
0
−
c
24
q
¯
L
¯
0
−
c
24
{\displaystyle Z(\tau )=\operatorname {Tr} _{\mathcal {S}}q^{L_{0}-{\frac {c}{24}}}{\bar {q}}^{{\bar {L}}_{0}-{\frac {c}{24}}}}
where
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
. The torus partition function coincides with the character of the spectrum, considered as a representation of the symmetry algebra.
== Chiral conformal field theory ==
In a two-dimensional conformal field theory, properties are called chiral if they follow from the action of one of the two Virasoro algebras. If the space of states can be decomposed into factorized representations of the product of the two Virasoro algebras, then all consequences of conformal symmetry are chiral. In other words, the actions of the two Virasoro algebras can be studied separately.
=== Energy–momentum tensor ===
The dependence of a field
V
(
z
)
{\displaystyle V(z)}
on its position is assumed to be determined by
∂
∂
z
V
(
z
)
=
L
−
1
V
(
z
)
.
{\displaystyle {\frac {\partial }{\partial z}}V(z)=L_{-1}V(z).}
It follows that the OPE
T
(
y
)
V
(
z
)
=
∑
n
∈
Z
L
n
V
(
z
)
(
y
−
z
)
n
+
2
,
{\displaystyle T(y)V(z)=\sum _{n\in \mathbb {Z} }{\frac {L_{n}V(z)}{(y-z)^{n+2}}},}
defines a locally holomorphic field
T
(
y
)
{\displaystyle T(y)}
that does not depend on
z
.
{\displaystyle z.}
This field is identified with (a component of) the energy–momentum tensor. In particular, the OPE of the energy–momentum tensor with a primary field is
T
(
y
)
V
Δ
(
z
)
=
Δ
(
y
−
z
)
2
V
Δ
(
z
)
+
1
y
−
z
∂
∂
z
V
Δ
(
z
)
+
O
(
1
)
.
{\displaystyle T(y)V_{\Delta }(z)={\frac {\Delta }{(y-z)^{2}}}V_{\Delta }(z)+{\frac {1}{y-z}}{\frac {\partial }{\partial z}}V_{\Delta }(z)+O(1).}
The OPE of the energy–momentum tensor with itself is
T
(
y
)
T
(
z
)
=
c
2
(
y
−
z
)
4
+
2
T
(
z
)
(
y
−
z
)
2
+
∂
T
(
z
)
y
−
z
+
O
(
1
)
,
{\displaystyle T(y)T(z)={\frac {\frac {c}{2}}{(y-z)^{4}}}+{\frac {2T(z)}{(y-z)^{2}}}+{\frac {\partial T(z)}{y-z}}+O(1),}
where
c
{\displaystyle c}
is the central charge. (This OPE is equivalent to the commutation relations of the Virasoro algebra.)
=== Conformal Ward identities ===
Conformal Ward identities are linear equations that correlation functions obey as a consequence of conformal symmetry. They can be derived by studying correlation functions that involve insertions of the energy–momentum tensor. Their solutions are conformal blocks.
For example, consider conformal Ward identities on the sphere. Let
z
{\displaystyle z}
be a global complex coordinate on the sphere, viewed as
C
∪
{
∞
}
.
{\displaystyle \mathbb {C} \cup \{\infty \}.}
Holomorphy of the energy–momentum tensor at
z
=
∞
{\displaystyle z=\infty }
is equivalent to
T
(
z
)
=
z
→
∞
O
(
1
z
4
)
.
{\displaystyle T(z){\underset {z\to \infty }{=}}O\left({\frac {1}{z^{4}}}\right).}
Moreover, inserting
T
(
z
)
{\displaystyle T(z)}
in an
N
{\displaystyle N}
-point function of primary fields yields
⟨
T
(
z
)
∏
i
=
1
N
V
Δ
i
(
z
i
)
⟩
=
∑
i
=
1
N
(
Δ
i
(
z
−
z
i
)
2
+
1
z
−
z
i
∂
∂
z
i
)
⟨
∏
i
=
1
N
V
Δ
i
(
z
i
)
⟩
.
{\displaystyle \left\langle T(z)\prod _{i=1}^{N}V_{\Delta _{i}}(z_{i})\right\rangle =\sum _{i=1}^{N}\left({\frac {\Delta _{i}}{(z-z_{i})^{2}}}+{\frac {1}{z-z_{i}}}{\frac {\partial }{\partial z_{i}}}\right)\left\langle \prod _{i=1}^{N}V_{\Delta _{i}}(z_{i})\right\rangle .}
From the last two equations, it is possible to deduce local Ward identities that express
N
{\displaystyle N}
-point functions of descendant fields in terms of
N
{\displaystyle N}
-point functions of primary fields. Moreover, it is possible to deduce three differential equations for any
N
{\displaystyle N}
-point function of primary fields, called global conformal Ward identities:
∑
i
=
1
N
(
z
i
k
∂
∂
z
i
+
Δ
i
k
z
i
k
−
1
)
⟨
∏
i
=
1
N
V
Δ
i
(
z
i
)
⟩
=
0
,
(
k
∈
{
0
,
1
,
2
}
)
.
{\displaystyle \sum _{i=1}^{N}\left(z_{i}^{k}{\frac {\partial }{\partial z_{i}}}+\Delta _{i}kz_{i}^{k-1}\right)\left\langle \prod _{i=1}^{N}V_{\Delta _{i}}(z_{i})\right\rangle =0,\qquad (k\in \{0,1,2\}).}
These identities determine how two- and three-point functions depend on
z
,
{\displaystyle z,}
⟨
V
Δ
1
(
z
1
)
V
Δ
2
(
z
2
)
⟩
{
=
0
(
Δ
1
≠
Δ
2
)
∝
(
z
1
−
z
2
)
−
2
Δ
1
(
Δ
1
=
Δ
2
)
{\displaystyle \left\langle V_{\Delta _{1}}(z_{1})V_{\Delta _{2}}(z_{2})\right\rangle {\begin{cases}=0&\ \ (\Delta _{1}\neq \Delta _{2})\\\propto (z_{1}-z_{2})^{-2\Delta _{1}}&\ \ (\Delta _{1}=\Delta _{2})\end{cases}}}
⟨
V
Δ
1
(
z
1
)
V
Δ
2
(
z
2
)
V
Δ
3
(
z
3
)
⟩
∝
(
z
1
−
z
2
)
Δ
3
−
Δ
1
−
Δ
2
(
z
2
−
z
3
)
Δ
1
−
Δ
2
−
Δ
3
(
z
1
−
z
3
)
Δ
2
−
Δ
1
−
Δ
3
,
{\displaystyle \left\langle V_{\Delta _{1}}(z_{1})V_{\Delta _{2}}(z_{2})V_{\Delta _{3}}(z_{3})\right\rangle \propto (z_{1}-z_{2})^{\Delta _{3}-\Delta _{1}-\Delta _{2}}(z_{2}-z_{3})^{\Delta _{1}-\Delta _{2}-\Delta _{3}}(z_{1}-z_{3})^{\Delta _{2}-\Delta _{1}-\Delta _{3}},}
where the undetermined proportionality coefficients are functions of
z
¯
.
{\displaystyle {\bar {z}}.}
=== BPZ equations ===
A correlation function that involves a degenerate field satisfies a linear partial differential equation called a Belavin–Polyakov–Zamolodchikov equation after Alexander Belavin, Alexander Polyakov and Alexander Zamolodchikov. The order of this equation is the level of the null vector in the corresponding degenerate representation.
A trivial example is the order one BPZ equation
∂
∂
z
1
⟨
V
1
,
1
(
z
1
)
V
2
(
z
2
)
⋯
V
N
(
z
N
)
⟩
=
0.
{\displaystyle {\frac {\partial }{\partial z_{1}}}\left\langle V_{1,1}(z_{1})V_{2}(z_{2})\cdots V_{N}(z_{N})\right\rangle =0.}
which follows from
∂
∂
z
1
V
1
,
1
(
z
1
)
=
L
−
1
V
1
,
1
(
z
1
)
=
0.
{\displaystyle {\frac {\partial }{\partial z_{1}}}V_{1,1}(z_{1})=L_{-1}V_{1,1}(z_{1})=0.}
The first nontrivial example involves a degenerate field
V
2
,
1
{\displaystyle V_{2,1}}
with a vanishing null vector at the level two,
(
L
−
1
2
+
b
2
L
−
2
)
V
2
,
1
=
0
,
{\displaystyle \left(L_{-1}^{2}+b^{2}L_{-2}\right)V_{2,1}=0,}
where
b
{\displaystyle b}
is related to the central charge by
c
=
1
+
6
(
b
+
b
−
1
)
2
.
{\displaystyle c=1+6\left(b+b^{-1}\right)^{2}.}
Then an
N
{\displaystyle N}
-point function of
V
2
,
1
{\displaystyle V_{2,1}}
and
N
−
1
{\displaystyle N-1}
other primary fields obeys:
(
1
b
2
∂
2
∂
z
1
2
+
∑
i
=
2
N
(
1
z
1
−
z
i
∂
∂
z
i
+
Δ
i
(
z
1
−
z
i
)
2
)
)
⟨
V
2
,
1
(
z
1
)
∏
i
=
2
N
V
Δ
i
(
z
i
)
⟩
=
0.
{\displaystyle \left({\frac {1}{b^{2}}}{\frac {\partial ^{2}}{\partial z_{1}^{2}}}+\sum _{i=2}^{N}\left({\frac {1}{z_{1}-z_{i}}}{\frac {\partial }{\partial z_{i}}}+{\frac {\Delta _{i}}{(z_{1}-z_{i})^{2}}}\right)\right)\left\langle V_{2,1}(z_{1})\prod _{i=2}^{N}V_{\Delta _{i}}(z_{i})\right\rangle =0.}
A BPZ equation of order
r
s
{\displaystyle rs}
for a correlation function that involve the degenerate field
V
r
,
s
{\displaystyle V_{r,s}}
can be deduced from the vanishing of the null vector, and the local Ward identities. Thanks to global Ward identities, four-point functions can be written in terms of one variable instead of four, and BPZ equations for four-point functions can be reduced to ordinary differential equations.
=== Fusion rules ===
In an OPE that involves a degenerate field, the vanishing of the null vector (plus conformal symmetry) constrains which primary fields can appear. The resulting constraints are called fusion rules. Using the momentum
α
{\displaystyle \alpha }
such that
Δ
=
α
(
b
+
b
−
1
−
α
)
{\displaystyle \Delta =\alpha \left(b+b^{-1}-\alpha \right)}
instead of the conformal dimension
Δ
{\displaystyle \Delta }
for parametrizing primary fields, the fusion rules are
V
r
,
s
×
V
α
=
∑
i
=
0
r
−
1
∑
j
=
0
s
−
1
V
α
+
(
i
−
r
−
1
2
)
b
+
(
j
−
s
−
1
2
)
b
−
1
{\displaystyle V_{r,s}\times V_{\alpha }=\sum _{i=0}^{r-1}\sum _{j=0}^{s-1}V_{\alpha +\left(i-{\frac {r-1}{2}}\right)b+\left(j-{\frac {s-1}{2}}\right)b^{-1}}}
in particular
V
1
,
1
×
V
α
=
V
α
V
2
,
1
×
V
α
=
V
α
−
b
2
+
V
α
+
b
2
V
1
,
2
×
V
α
=
V
α
−
1
2
b
+
V
α
+
1
2
b
{\displaystyle {\begin{aligned}V_{1,1}\times V_{\alpha }&=V_{\alpha }\\[6pt]V_{2,1}\times V_{\alpha }&=V_{\alpha -{\frac {b}{2}}}+V_{\alpha +{\frac {b}{2}}}\\[6pt]V_{1,2}\times V_{\alpha }&=V_{\alpha -{\frac {1}{2b}}}+V_{\alpha +{\frac {1}{2b}}}\end{aligned}}}
Alternatively, fusion rules have an algebraic definition in terms of an associative fusion product of representations of the Virasoro algebra at a given central charge. The fusion product differs from the tensor product of representations. (In a tensor product, the central charges add.) In certain finite cases, this leads to the structure of a fusion category.
A conformal field theory is quasi-rational if the fusion product of two indecomposable representations is a sum of finitely many indecomposable representations. For example, generalized minimal models are quasi-rational without being rational.
== Conformal bootstrap ==
The conformal bootstrap method consists in defining and solving CFTs using only symmetry and consistency assumptions, by reducing all correlation functions to combinations of structure constants and conformal blocks.
In two dimensions, this method leads to exact solutions of certain CFTs, and to classifications of rational theories.
=== Structure constants ===
Let
V
i
{\displaystyle V_{i}}
be a left- and right-primary field with left- and right-conformal dimensions
Δ
i
{\displaystyle \Delta _{i}}
and
Δ
¯
i
{\displaystyle {\bar {\Delta }}_{i}}
. According to the left and right global Ward identities, three-point functions of such fields are of the type
⟨
V
1
(
z
1
)
V
2
(
z
2
)
V
3
(
z
3
)
⟩
=
C
123
×
(
z
1
−
z
2
)
Δ
3
−
Δ
1
−
Δ
2
(
z
2
−
z
3
)
Δ
1
−
Δ
2
−
Δ
3
(
z
1
−
z
3
)
Δ
2
−
Δ
1
−
Δ
3
×
(
z
¯
1
−
z
¯
2
)
Δ
¯
3
−
Δ
¯
1
−
Δ
¯
2
(
z
¯
2
−
z
¯
3
)
Δ
¯
1
−
Δ
¯
2
−
Δ
¯
3
(
z
¯
1
−
z
¯
3
)
Δ
¯
2
−
Δ
¯
1
−
Δ
¯
3
,
{\displaystyle {\begin{aligned}&\left\langle V_{1}(z_{1})V_{2}(z_{2})V_{3}(z_{3})\right\rangle =C_{123}\\&\qquad \times (z_{1}-z_{2})^{\Delta _{3}-\Delta _{1}-\Delta _{2}}(z_{2}-z_{3})^{\Delta _{1}-\Delta _{2}-\Delta _{3}}(z_{1}-z_{3})^{\Delta _{2}-\Delta _{1}-\Delta _{3}}\\&\qquad \times ({\bar {z}}_{1}-{\bar {z}}_{2})^{{\bar {\Delta }}_{3}-{\bar {\Delta }}_{1}-{\bar {\Delta }}_{2}}({\bar {z}}_{2}-{\bar {z}}_{3})^{{\bar {\Delta }}_{1}-{\bar {\Delta }}_{2}-{\bar {\Delta }}_{3}}({\bar {z}}_{1}-{\bar {z}}_{3})^{{\bar {\Delta }}_{2}-{\bar {\Delta }}_{1}-{\bar {\Delta }}_{3}}\ ,\end{aligned}}}
where the
z
i
{\displaystyle z_{i}}
-independent number
C
123
{\displaystyle C_{123}}
is called a three-point structure constant. For the three-point function to be single-valued, the left- and right-conformal dimensions of primary fields must obey
Δ
i
−
Δ
¯
i
∈
1
2
Z
.
{\displaystyle \Delta _{i}-{\bar {\Delta }}_{i}\in {\frac {1}{2}}\mathbb {Z} \ .}
This condition is satisfied by bosonic (
Δ
i
−
Δ
¯
i
∈
Z
{\displaystyle \Delta _{i}-{\bar {\Delta }}_{i}\in \mathbb {Z} }
) and fermionic (
Δ
i
−
Δ
¯
i
∈
Z
+
1
2
{\displaystyle \Delta _{i}-{\bar {\Delta }}_{i}\in \mathbb {Z} +{\frac {1}{2}}}
) fields. It is however violated by parafermionic fields (
Δ
i
−
Δ
¯
i
∈
Q
{\displaystyle \Delta _{i}-{\bar {\Delta }}_{i}\in \mathbb {Q} }
), whose correlation functions are therefore not single-valued on the Riemann sphere.
Three-point structure constants also appear in OPEs,
V
1
(
z
1
)
V
2
(
z
2
)
=
∑
i
C
12
i
(
z
1
−
z
2
)
Δ
i
−
Δ
1
−
Δ
2
(
z
¯
1
−
z
¯
2
)
Δ
¯
i
−
Δ
¯
1
−
Δ
¯
2
(
V
i
(
z
2
)
+
⋯
)
.
{\displaystyle V_{1}(z_{1})V_{2}(z_{2})=\sum _{i}C_{12i}(z_{1}-z_{2})^{\Delta _{i}-\Delta _{1}-\Delta _{2}}({\bar {z}}_{1}-{\bar {z}}_{2})^{{\bar {\Delta }}_{i}-{\bar {\Delta }}_{1}-{\bar {\Delta }}_{2}}{\Big (}V_{i}(z_{2})+\cdots {\Big )}\ .}
The contributions of descendant fields, denoted by the dots, are completely determined by conformal symmetry.
=== Conformal blocks ===
Any correlation function can be written as a linear combination of conformal blocks: functions that are determined by conformal symmetry, and labelled by representations of the symmetry algebra. The coefficients of the linear combination are products of structure constants.
In two-dimensional CFT, the symmetry algebra is factorized into two copies of the Virasoro algebra, and a conformal block that involves primary fields has a holomorphic factorization: it is a product of a locally holomorphic factor that is determined by the left-moving Virasoro algebra, and a locally antiholomorphic factor that is determined by the right-moving Virasoro algebra. These factors are themselves called conformal blocks.
For example, using the OPE of the first two fields in a four-point function of primary fields yields
⟨
∏
i
=
1
4
V
i
(
z
i
)
⟩
=
∑
s
C
12
s
C
s
34
F
Δ
s
(
s
)
(
{
Δ
i
}
,
{
z
i
}
)
F
Δ
¯
s
(
s
)
(
{
Δ
¯
i
}
,
{
z
¯
i
}
)
,
{\displaystyle \left\langle \prod _{i=1}^{4}V_{i}(z_{i})\right\rangle =\sum _{s}C_{12s}C_{s34}{\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\},\{z_{i}\}){\mathcal {F}}_{{\bar {\Delta }}_{s}}^{(s)}(\{{\bar {\Delta }}_{i}\},\{{\bar {z}}_{i}\})\ ,}
where
F
Δ
s
(
s
)
(
{
Δ
i
}
,
{
z
i
}
)
{\displaystyle {\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\},\{z_{i}\})}
is an s-channel four-point conformal block. Four-point conformal blocks are complicated functions that can be efficiently computed using Alexei Zamolodchikov's recursion relations. If one of the four fields is degenerate, then the corresponding conformal blocks obey BPZ equations. If in particular one of the four fields is
V
2
,
1
{\displaystyle V_{2,1}}
, then the corresponding conformal blocks can be written in terms of the hypergeometric function.
As first explained by Witten, the space of conformal blocks of a two-dimensional CFT can be identified with the quantum Hilbert space of a 2+1 dimensional Chern-Simons theory, which is an example of a topological field theory. This connection has been very fruitful in the theory of the fractional quantum Hall effect.
=== Conformal bootstrap equations ===
When a correlation function can be written in terms of conformal blocks in several different ways, the equality of the resulting expressions provides constraints on the space of states and on three-point structure constants. These constraints are called the conformal bootstrap equations. While the Ward identities are linear equations for correlation functions, the conformal bootstrap equations depend non-linearly on the three-point structure constants.
For example, a four-point function
⟨
V
1
V
2
V
3
V
4
⟩
{\displaystyle \left\langle V_{1}V_{2}V_{3}V_{4}\right\rangle }
can be written in terms of conformal blocks in three inequivalent ways, corresponding to using the OPEs
V
1
V
2
{\displaystyle V_{1}V_{2}}
(s-channel),
V
1
V
4
{\displaystyle V_{1}V_{4}}
(t-channel) or
V
1
V
3
{\displaystyle V_{1}V_{3}}
(u-channel). The equality of the three resulting expressions is called crossing symmetry of the four-point function, and is equivalent to the associativity of the OPE.
For example, the torus partition function is invariant under the action of the modular group on the modulus of the torus, equivalently
Z
(
τ
)
=
Z
(
τ
+
1
)
=
Z
(
−
1
τ
)
{\displaystyle Z(\tau )=Z(\tau +1)=Z(-{\frac {1}{\tau }})}
. This invariance is a constraint on the space of states. The study of modular invariant torus partition functions is sometimes called the modular bootstrap.
The consistency of a CFT on the sphere is equivalent to crossing symmetry of the four-point function. The consistency of a CFT on all Riemann surfaces also requires modular invariance of the torus one-point function. Modular invariance of the torus partition function is therefore neither necessary, nor sufficient, for a CFT to exist. It has however been widely studied in rational CFTs, because characters of representations are simpler than other kinds of conformal blocks, such as sphere four-point conformal blocks.
== Examples ==
=== Minimal models ===
A minimal model is a CFT whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models only exist for particular values of the central charge,
c
p
,
q
=
1
−
6
(
p
−
q
)
2
p
q
,
p
>
q
∈
{
2
,
3
,
…
}
.
{\displaystyle c_{p,q}=1-6{\frac {(p-q)^{2}}{pq}},\qquad p>q\in \{2,3,\ldots \}.}
There is an ADE classification of minimal models. In particular, the A-series minimal model with the central charge
c
=
c
p
,
q
{\displaystyle c=c_{p,q}}
is a diagonal CFT whose spectrum is built from
1
2
(
p
−
1
)
(
q
−
1
)
{\displaystyle {\tfrac {1}{2}}(p-1)(q-1)}
degenerate lowest weight representations of the Virasoro algebra. These degenerate representations are labelled by pairs of integers that form the Kac table,
(
r
,
s
)
∈
{
1
,
…
,
p
−
1
}
×
{
1
,
…
,
q
−
1
}
with
(
r
,
s
)
≃
(
p
−
r
,
q
−
s
)
.
{\displaystyle (r,s)\in \{1,\ldots ,p-1\}\times \{1,\ldots ,q-1\}\qquad {\text{with}}\qquad (r,s)\simeq (p-r,q-s).}
For example, the A-series minimal model with
c
=
c
4
,
3
=
1
2
{\displaystyle c=c_{4,3}={\tfrac {1}{2}}}
describes spin and energy correlators of the two-dimensional critical Ising model.
=== Liouville theory ===
For any
c
∈
C
,
{\displaystyle c\in \mathbb {C} ,}
Liouville theory is a diagonal CFT whose spectrum is built from Verma modules with conformal dimensions
Δ
∈
c
−
1
24
+
R
+
{\displaystyle \Delta \in {\frac {c-1}{24}}+\mathbb {R} _{+}}
Liouville theory has been solved, in the sense that its three-point structure constants are explicitly known. Liouville theory has applications to string theory, and to two-dimensional quantum gravity.
=== Extended symmetry algebras ===
In some CFTs, the symmetry algebra is not just the Virasoro algebra, but an associative algebra (i.e. not necessarily a Lie algebra) that contains the Virasoro algebra. The spectrum is then decomposed into representations of that algebra, and the notions of diagonal and rational CFTs are defined with respect to that algebra.
==== Massless free bosonic theories ====
In two dimensions, massless free bosonic theories are conformally invariant. Their symmetry algebra is the affine Lie algebra
u
^
1
{\displaystyle {\hat {\mathfrak {u}}}_{1}}
built from the abelian, rank one Lie algebra. The fusion product of any two representations of this symmetry algebra yields only one representation, and this makes correlation functions very simple.
Viewing minimal models and Liouville theory as perturbed free bosonic theories leads to the Coulomb gas method for computing their correlation functions. Moreover, for
c
=
1
,
{\displaystyle c=1,}
there is a one-parameter family of free bosonic theories with infinite discrete spectrums, which describe compactified free bosons, with the parameter being the compactification radius.
==== Wess–Zumino–Witten models ====
Given a Lie group
G
,
{\displaystyle G,}
the corresponding Wess–Zumino–Witten model is a CFT whose symmetry algebra is the affine Lie algebra built from the Lie algebra of
G
.
{\displaystyle G.}
If
G
{\displaystyle G}
is compact, then this CFT is rational, its central charge takes discrete values, and its spectrum is known.
==== Superconformal field theories ====
The symmetry algebra of a supersymmetric CFT is a super Virasoro algebra, or a larger algebra. Supersymmetric CFTs are in particular relevant to superstring theory.
==== Theories based on W-algebras ====
W-algebras are natural extensions of the Virasoro algebra. CFTs based on W-algebras include generalizations of minimal models and Liouville theory, respectively called W-minimal models and conformal Toda theories. Conformal Toda theories are more complicated than Liouville theory, and less well understood.
=== Sigma models ===
In two dimensions, classical sigma models are conformally invariant, but only some target manifolds lead to quantum sigma models that are conformally invariant. Examples of such target manifolds include toruses, and Calabi–Yau manifolds.
=== Logarithmic conformal field theories ===
Logarithmic conformal field theories are two-dimensional CFTs such that the action of the Virasoro algebra generator
L
0
{\displaystyle L_{0}}
on the spectrum is not diagonalizable. In particular, the spectrum cannot be built solely from lowest weight representations. As a consequence, the dependence of correlation functions on the positions of the fields can be logarithmic. This contrasts with the power-like dependence of the two- and three-point functions that are associated to lowest weight representations.
=== Critical Q-state Potts model ===
The critical
Q
{\displaystyle Q}
-state Potts model or critical random cluster model is a conformal field theory that generalizes and unifies the critical Ising model, Potts model, and percolation. The model has a parameter
Q
{\displaystyle Q}
, which must be integer in the Potts model, but which can take any complex value in the random cluster model. This parameter is related to the central charge by
Q
=
4
cos
2
(
π
β
2
)
with
c
=
13
−
6
β
2
−
6
β
−
2
.
{\displaystyle Q=4\cos ^{2}(\pi \beta ^{2})\qquad {\text{with}}\qquad c=13-6\beta ^{2}-6\beta ^{-2}\ .}
Special values of
Q
{\displaystyle Q}
include:
The known torus partition function suggests that the model is non-rational with a discrete spectrum.
== References ==
== Further reading ==
P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer-Verlag, New York, 1997. ISBN 0-387-94785-X.
Conformal Field Theory page in String Theory Wiki lists books and reviews.
Ribault, Sylvain (2014). "Conformal field theory on the plane". arXiv:1406.4290 [hep-th]. | Wikipedia/Two-dimensional_conformal_field_theory |
In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory uses a semi-classical approach; it treats spacetime as a fixed, classical background, while giving a quantum-mechanical description of the matter and energy propagating through that spacetime. A general prediction of this theory is that particles can be created by time-dependent gravitational fields (multigraviton pair production), or by time-independent gravitational fields that contain horizons. The most famous example of the latter is the phenomenon of Hawking radiation emitted by black holes.
== Overview ==
Ordinary quantum field theories, which form the basis of standard model, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth. In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.
For non-zero cosmological constants, on curved spacetimes quantum fields lose their interpretation as asymptotic particles. Only in certain situations, such as in asymptotically flat spacetimes (zero cosmological curvature), can the notion of incoming and outgoing particle be recovered, thus enabling one to define an S-matrix. Even then, as in flat spacetime, the asymptotic particle interpretation depends on the observer (i.e., different observers may measure different numbers of asymptotic particles on a given spacetime).
Another observation is that unless the background metric tensor has a global timelike Killing vector, there is no way to define a vacuum or ground state canonically. The concept of a vacuum is not invariant under diffeomorphisms. This is because a mode decomposition of a field into positive and negative frequency modes is not invariant under diffeomorphisms. If t′(t) is a diffeomorphism, in general, the Fourier transform of exp[ikt′(t)] will contain negative frequencies even if k > 0. Creation operators correspond to positive frequencies, while annihilation operators correspond to negative frequencies. This is why a state which looks like a vacuum to one observer cannot look like a vacuum state to another observer; it could even appear as a heat bath under suitable hypotheses.
Since the end of the 1980s, the local quantum field theory approach due to Rudolf Haag and Daniel Kastler has been implemented in order to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is suitable to generalize the renormalization procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in the presence of a black hole have been obtained. In particular the algebraic approach allows one to deal with the problems mentioned above arising from the absence of a preferred reference vacuum state, the absence of a natural notion of particle and the appearance of unitarily inequivalent representations of the algebra of observables.
== Applications ==
Using perturbation theory in quantum field theory in curved spacetime geometry is known as the semiclassical approach to quantum gravity. This approach studies the interaction of quantum fields in a fixed classical spacetime and among other thing predicts the creation of particles by time-varying spacetimes and Hawking radiation. The latter can be understood as a manifestation of the Unruh effect where an accelerating observer observes black body radiation. Other prediction of quantum fields in curved spaces include, for example, the radiation emitted by a particle moving along a geodesic and the interaction of Hawking radiation with particles outside black holes.
This formalism is also used to predict the primordial density perturbation spectrum arising in different models of cosmic inflation. These predictions are calculated using the Bunch–Davies vacuum or modifications thereto.
== Approximation to quantum gravity ==
The theory of quantum field theory in curved spacetime may be considered as an intermediate step towards quantum gravity. QFT in curved spacetime is expected to be a viable approximation to the theory of quantum gravity when spacetime curvature is not significant on the Planck scale. However, the fact that the true theory of quantum gravity remains unknown means that the precise criteria for when QFT on curved spacetime is a good approximation are also unknown.: 1
Gravity is not renormalizable in QFT, so merely formulating QFT in curved spacetime is not a true theory of quantum gravity.
== See also ==
== References ==
== Further reading ==
Birrell, N. D.; Davies, P. C. W. (1982). Quantum fields in curved space. CUP. ISBN 0-521-23385-2.
Fulling, S. A. (1989). Aspects of quantum field theory in curved space-time. CUP. ISBN 0-521-34400-X.
Mukhanov, V.; Winitzki, S. (2007). Introduction to Quantum Effects in Gravity. CUP. ISBN 978-0-521-86834-1.
Parker, L.; Toms, D. (2009). Quantum Field Theory in Curved Spacetime. Cambridge University Press. ISBN 978-0-521-87787-9.
== External links ==
Summary Chart of Intro Steps to Quantum Fields in Curved Spacetime A two-page chart outline of the basic principles governing the behavior of quantum fields in general relativity. | Wikipedia/Quantum_field_theory_in_curved_spacetime |
In elementary particle physics and mathematical physics, in particular in effective field theory, a form factor is a function that encapsulates the properties of a certain particle interaction without including all of the underlying physics, but instead, providing the momentum dependence of suitable matrix elements. It is further measured experimentally in confirmation or specification of a theory—see experimental particle physics.
== Photon–nucleon example ==
For example, at low energies the interaction of a photon with a nucleon is a very complicated calculation involving interactions between the photon and a sea of quarks and gluons, and often the calculation cannot be fully performed from first principles. Often in this context, form factors are also called "structure functions", since they can be used to describe the structure of the nucleon.
However, the generic Lorentz-invariant form of the matrix element for the electromagnetic current interaction is known,
ε
μ
N
¯
(
α
(
q
2
)
γ
μ
+
β
(
q
2
)
q
μ
+
κ
(
q
2
)
σ
μ
ν
q
ν
)
N
{\displaystyle \varepsilon _{\mu }{\bar {N}}\left(\alpha (q^{2})\gamma ^{\mu }+\beta (q^{2})q^{\mu }+\kappa (q^{2})\sigma ^{\mu \nu }q_{\nu }\right)N\,}
where
q
μ
{\displaystyle q^{\mu }}
represents the photon momentum (equal in magnitude to E/c, where E is the energy of the photon). The three functions:
α
,
β
,
κ
{\displaystyle \alpha ,\beta ,\kappa }
are associated to the electric and magnetic form factors for this interaction, and are routinely measured experimentally; these three effective vertices can then be used to check, or perform calculations that would otherwise be too difficult to perform from first principles. This matrix element then serves to determine the transition amplitude involved in the scattering interaction or the respective particle decay—cf. Fermi's golden rule.
In general, the Fourier transforms of form factor components correspond to electric charge or magnetic profile space distributions (such as the charge radius) of the hadron involved. The analogous QCD structure functions are a probe of the quark and gluon distributions of nucleons.
== See also ==
Structure function
Atomic form factor
Electric form factor
Magnetic form factor
Photon structure function
Quantum field theory
Standard model
Quantum mechanics
Special relativity
Charge radius
== References ==
Brown, Lowell S. (1994). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3. p 400
Gasiorowicz, Stephen (1966), Elementary Particle Physics, John Wiley & Sons, ISBN 978-0471292876
Wilson, R. (1969). "Form factors of elementary particles", Physics today 22 p 47, doi:10.1063/1.3035356
Charles Perdrisat and Vina Punjabi (2010). "Nucleon Form factors", Scholarpedia 5(8): 10204. online article | Wikipedia/Form_factor_(quantum_field_theory) |
In condensed matter physics, quantum hydrodynamics (QHD) is most generally the study of hydrodynamic-like systems which demonstrate quantum mechanical behavior. They arise in semiclassical mechanics in the study of metal and semiconductor devices, in which case being derived from the Boltzmann transport equation combined with Wigner quasiprobability distribution. In quantum chemistry they arise as solutions to chemical kinetic systems, in which case they are derived from the Schrödinger equation by way of Madelung equations.
An important system of study in quantum hydrodynamics is that of superfluidity. Some other topics of interest in quantum hydrodynamics are quantum turbulence, quantized vortices, second and third sound, and quantum solvents. The quantum hydrodynamic equation is an equation in Bohmian mechanics, which, it turns out, has a mathematical relationship to classical fluid dynamics (see Madelung equations).
Some common experimental applications of these studies are in liquid helium (3He and 4He), and of the interior of neutron stars and the quark–gluon plasma. Many famous scientists have worked in quantum hydrodynamics, including Richard Feynman, Lev Landau, and Pyotr Kapitsa.
== See also ==
Quantum turbulence
Hydrodynamic quantum analogs
== References ==
Robert E. Wyatt (2005). Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics. Springer. ISBN 978-0-387-22964-5. | Wikipedia/Quantum_hydrodynamics |
Communications in Mathematical Physics is a peer-reviewed academic journal published by Springer. The journal publishes papers in all fields of mathematical physics, but focuses particularly in analysis related to condensed matter physics, statistical mechanics and quantum field theory, and in operator algebras, quantum information and relativity.
== History ==
Rudolf Haag conceived this journal with Res Jost, and Haag became the Founding Chief Editor. The first issue of Communications in Mathematical Physics appeared in 1965. Haag guided the journal for the next eight years. Then Klaus Hepp succeeded him for three years, followed by James Glimm, for another three years. Arthur Jaffe began as chief editor in 1979 and served for 21 years. Michael Aizenman became the fifth chief editor in the year 2000 and served in this role until 2012. The current editor-in-chief is Horng-Tzer Yau.
== Archives ==
Articles from 1965 to 1997 are available in electronic form free of charge, via Project Euclid, a non-profit organization initiated by Cornell University Library. This portion of the journal is provided through the Electronic Mathematical Archiving Network Initiative (EMANI) to support the long-term electronic preservation of mathematical publications.
== See also ==
List of mathematics journals
List of physics journals
Res Jost
Rudolf Haag
== References ==
== Further reading ==
Jaffe, Arthur. "Haag's visit in honor of 40 years of Communications in Mathematical Physics". arthurjaffe.com. Retrieved February 9, 2021.
Jaffe, Arthur (2015). "50 Years of Communications in Mathematical Physics" (PDF). News Bulletin, International Association of Mathematical Physics: 15–26.
== External links ==
"Journal website". Springer. Retrieved March 28, 2021.
"Communications in Mathematical Physics in Project Euclid". Project Euclid. Retrieved March 28, 2021. | Wikipedia/Communications_in_Mathematical_Physics |
The Schwinger–Dyson equations (SDEs) or Dyson–Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between correlation functions in quantum field theories (QFTs). They are also referred to as the Euler–Lagrange equations of quantum field theories, since they are the equations of motion corresponding to the Green's function. They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite tower of SDEs.
In his paper "The S-Matrix in Quantum electrodynamics", Dyson derived relations between different S-matrix elements, or more specific "one-particle Green's functions", in quantum electrodynamics, by summing up infinitely many Feynman diagrams, thus working in a perturbative approach. Starting from his variational principle, Schwinger derived a set of equations for Green's functions non-perturbatively, which generalize Dyson's equations to the Schwinger–Dyson equations for the Green functions of quantum field theories. Today they provide a non-perturbative approach to quantum field theories and applications can be found in many fields of theoretical physics, such as solid-state physics and elementary particle physics.
Schwinger also derived an equation for the two-particle irreducible Green functions, which is nowadays referred to as the inhomogeneous Bethe–Salpeter equation.
== Derivation ==
Given a polynomially bounded functional
F
{\displaystyle F}
over the field configurations, then, for any state vector (which is a solution of the QFT),
|
ψ
⟩
{\displaystyle |\psi \rangle }
, we have
⟨
ψ
|
T
{
δ
δ
φ
F
[
φ
]
}
|
ψ
⟩
=
−
i
⟨
ψ
|
T
{
F
[
φ
]
δ
δ
φ
S
[
φ
]
}
|
ψ
⟩
{\displaystyle \left\langle \psi \left|{\mathcal {T}}\left\{{\frac {\delta }{\delta \varphi }}F[\varphi ]\right\}\right|\psi \right\rangle =-i\left\langle \psi \left|{\mathcal {T}}\left\{F[\varphi ]{\frac {\delta }{\delta \varphi }}S[\varphi ]\right\}\right|\psi \right\rangle }
where
δ
/
δ
φ
{\displaystyle \delta /\delta \varphi }
is the functional derivative with respect to
φ
,
S
{\displaystyle \varphi ,S}
is the action functional and
T
{\displaystyle {\mathcal {T}}}
is the time ordering operation.
Equivalently, in the density state formulation, for any (valid) density state
ρ
{\displaystyle \rho }
, we have
ρ
(
T
{
δ
δ
φ
F
[
φ
]
}
)
=
−
i
ρ
(
T
{
F
[
φ
]
δ
δ
φ
S
[
φ
]
}
)
.
{\displaystyle \rho \left({\mathcal {T}}\left\{{\frac {\delta }{\delta \varphi }}F[\varphi ]\right\}\right)=-i\rho \left({\mathcal {T}}\left\{F[\varphi ]{\frac {\delta }{\delta \varphi }}S[\varphi ]\right\}\right).}
This infinite set of equations can be used to solve for the correlation functions nonperturbatively.
To make the connection to diagrammatic techniques (like Feynman diagrams) clearer, it is often convenient to split the action
S
{\displaystyle S}
as
S
[
φ
]
=
1
2
φ
i
D
i
j
−
1
φ
j
+
S
int
[
φ
]
,
{\displaystyle S[\varphi ]={\frac {1}{2}}\varphi ^{i}D_{ij}^{-1}\varphi ^{j}+S_{\text{int}}[\varphi ],}
where the first term is the quadratic part and
D
−
1
{\displaystyle D^{-1}}
is an invertible symmetric (antisymmetric for fermions) covariant tensor of rank two in the deWitt notation whose inverse,
D
{\displaystyle D}
is called the bare propagator and
S
int
[
φ
]
{\displaystyle S_{\text{int}}[\varphi ]}
is the "interaction action". Then, we can rewrite the SD equations as
⟨
ψ
|
T
{
F
φ
j
}
|
ψ
⟩
=
⟨
ψ
|
T
{
i
F
,
i
D
i
j
−
F
S
int
,
i
D
i
j
}
|
ψ
⟩
.
{\displaystyle \langle \psi |{\mathcal {T}}\{F\varphi ^{j}\}|\psi \rangle =\langle \psi |{\mathcal {T}}\{iF_{,i}D^{ij}-FS_{{\text{int}},i}D^{ij}\}|\psi \rangle .}
If
F
{\displaystyle F}
is a functional of
φ
{\displaystyle \varphi }
, then for an operator
K
{\displaystyle K}
,
F
[
K
]
{\displaystyle F[K]}
is defined to be the operator which substitutes
K
{\displaystyle K}
for
φ
{\displaystyle \varphi }
. For example, if
F
[
φ
]
=
∂
k
1
∂
x
1
k
1
φ
(
x
1
)
⋯
∂
k
n
∂
x
n
k
n
φ
(
x
n
)
{\displaystyle F[\varphi ]={\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}\varphi (x_{1})\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}\varphi (x_{n})}
and
G
{\displaystyle G}
is a functional of
J
{\displaystyle J}
, then
F
[
−
i
δ
δ
J
]
G
[
J
]
=
(
−
i
)
n
∂
k
1
∂
x
1
k
1
δ
δ
J
(
x
1
)
⋯
∂
k
n
∂
x
n
k
n
δ
δ
J
(
x
n
)
G
[
J
]
.
{\displaystyle F\left[-i{\frac {\delta }{\delta J}}\right]G[J]=(-i)^{n}{\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}{\frac {\delta }{\delta J(x_{1})}}\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}{\frac {\delta }{\delta J(x_{n})}}G[J].}
If we have an "analytic" (a function that is locally given by a convergent power series) functional
Z
{\displaystyle Z}
(called the generating functional) of
J
{\displaystyle J}
(called the source field) satisfying
δ
n
Z
δ
J
(
x
1
)
⋯
δ
J
(
x
n
)
[
0
]
=
i
n
Z
[
0
]
⟨
φ
(
x
1
)
⋯
φ
(
x
n
)
⟩
,
{\displaystyle {\frac {\delta ^{n}Z}{\delta J(x_{1})\cdots \delta J(x_{n})}}[0]=i^{n}Z[0]\langle \varphi (x_{1})\cdots \varphi (x_{n})\rangle ,}
then, from the properties of the functional integrals
⟨
δ
S
δ
φ
(
x
)
[
φ
]
+
J
(
x
)
⟩
J
=
0
,
{\displaystyle {\left\langle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}\left[\varphi \right]+J(x)\right\rangle }_{J}=0,}
the Schwinger–Dyson equation for the generating functional is
δ
S
δ
φ
(
x
)
[
−
i
δ
δ
J
]
Z
[
J
]
+
J
(
x
)
Z
[
J
]
=
0.
{\displaystyle {\frac {\delta S}{\delta \varphi (x)}}\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Z[J]=0.}
If we expand this equation as a Taylor series about
J
=
0
{\displaystyle J=0}
, we get the entire set of Schwinger–Dyson equations.
== An example: φ4 ==
To give an example, suppose
S
[
φ
]
=
∫
d
d
x
(
1
2
∂
μ
φ
(
x
)
∂
μ
φ
(
x
)
−
1
2
m
2
φ
(
x
)
2
−
λ
4
!
φ
(
x
)
4
)
{\displaystyle S[\varphi ]=\int d^{d}x\left({\frac {1}{2}}\partial ^{\mu }\varphi (x)\partial _{\mu }\varphi (x)-{\frac {1}{2}}m^{2}\varphi (x)^{2}-{\frac {\lambda }{4!}}\varphi (x)^{4}\right)}
for a real field
φ
{\displaystyle \varphi }
.
Then,
δ
S
δ
φ
(
x
)
=
−
∂
μ
∂
μ
φ
(
x
)
−
m
2
φ
(
x
)
−
λ
3
!
φ
3
(
x
)
.
{\displaystyle {\frac {\delta S}{\delta \varphi (x)}}=-\partial _{\mu }\partial ^{\mu }\varphi (x)-m^{2}\varphi (x)-{\frac {\lambda }{3!}}\varphi ^{3}(x).}
The Schwinger–Dyson equation for this particular example is:
i
∂
μ
∂
μ
δ
δ
J
(
x
)
Z
[
J
]
+
i
m
2
δ
δ
J
(
x
)
Z
[
J
]
−
i
λ
3
!
δ
3
δ
J
(
x
)
3
Z
[
J
]
+
J
(
x
)
Z
[
J
]
=
0
{\displaystyle i\partial _{\mu }\partial ^{\mu }{\frac {\delta }{\delta J(x)}}Z[J]+im^{2}{\frac {\delta }{\delta J(x)}}Z[J]-{\frac {i\lambda }{3!}}{\frac {\delta ^{3}}{\delta J(x)^{3}}}Z[J]+J(x)Z[J]=0}
Note that since
δ
3
δ
J
(
x
)
3
{\displaystyle {\frac {\delta ^{3}}{\delta J(x)^{3}}}}
is not well-defined because
δ
3
δ
J
(
x
1
)
δ
J
(
x
2
)
δ
J
(
x
3
)
Z
[
J
]
{\displaystyle {\frac {\delta ^{3}}{\delta J(x_{1})\delta J(x_{2})\delta J(x_{3})}}Z[J]}
is a distribution in
x
1
{\displaystyle x_{1}}
,
x
2
{\displaystyle x_{2}}
and
x
3
{\displaystyle x_{3}}
,
this equation needs to be regularized.
In this example, the bare propagator D is the Green's function for
−
∂
μ
∂
μ
−
m
2
{\displaystyle -\partial ^{\mu }\partial _{\mu }-m^{2}}
and so, the Schwinger–Dyson set of equations goes as
⟨
ψ
∣
T
{
φ
(
x
0
)
φ
(
x
1
)
}
∣
ψ
⟩
=
i
D
(
x
0
,
x
1
)
+
λ
3
!
∫
d
d
x
2
D
(
x
0
,
x
2
)
⟨
ψ
∣
T
{
φ
(
x
1
)
φ
(
x
2
)
φ
(
x
2
)
φ
(
x
2
)
}
∣
ψ
⟩
{\displaystyle {\begin{aligned}&\langle \psi \mid {\mathcal {T}}\{\varphi (x_{0})\varphi (x_{1})\}\mid \psi \rangle \\[4pt]={}&iD(x_{0},x_{1})+{\frac {\lambda }{3!}}\int d^{d}x_{2}\,D(x_{0},x_{2})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{2})\varphi (x_{2})\varphi (x_{2})\}\mid \psi \rangle \end{aligned}}}
and
⟨
ψ
∣
T
{
φ
(
x
0
)
φ
(
x
1
)
φ
(
x
2
)
φ
(
x
3
)
}
∣
ψ
⟩
=
i
D
(
x
0
,
x
1
)
⟨
ψ
∣
T
{
φ
(
x
2
)
φ
(
x
3
)
}
∣
ψ
⟩
+
i
D
(
x
0
,
x
2
)
⟨
ψ
∣
T
{
φ
(
x
1
)
φ
(
x
3
)
}
∣
ψ
⟩
+
i
D
(
x
0
,
x
3
)
⟨
ψ
∣
T
{
φ
(
x
1
)
φ
(
x
2
)
}
∣
ψ
⟩
+
λ
3
!
∫
d
d
x
4
D
(
x
0
,
x
4
)
⟨
ψ
∣
T
{
φ
(
x
1
)
φ
(
x
2
)
φ
(
x
3
)
φ
(
x
4
)
φ
(
x
4
)
φ
(
x
4
)
}
∣
ψ
⟩
{\displaystyle {\begin{aligned}&\langle \psi \mid {\mathcal {T}}\{\varphi (x_{0})\varphi (x_{1})\varphi (x_{2})\varphi (x_{3})\}\mid \psi \rangle \\[6pt]={}&iD(x_{0},x_{1})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{2})\varphi (x_{3})\}\mid \psi \rangle +iD(x_{0},x_{2})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{3})\}\mid \psi \rangle \\[4pt]&{}+iD(x_{0},x_{3})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{2})\}\mid \psi \rangle \\[4pt]&{}+{\frac {\lambda }{3!}}\int d^{d}x_{4}\,D(x_{0},x_{4})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{2})\varphi (x_{3})\varphi (x_{4})\varphi (x_{4})\varphi (x_{4})\}\mid \psi \rangle \end{aligned}}}
etc.
(Unless there is spontaneous symmetry breaking, the odd correlation functions vanish.)
== See also ==
Functional renormalization group
Dyson equation
Path integral formulation
Source field
== References ==
== Further reading ==
There are not many books that treat the Schwinger–Dyson equations. Here are three standard references:
Claude Itzykson, Jean-Bernard Zuber (1980). Quantum Field Theory. McGraw-Hill. ISBN 9780070320710.
R.J. Rivers (1990). Path Integral Methods in Quantum Field Theories. Cambridge University Press.
V.P. Nair (2005). Quantum Field Theory A Modern Perspective. Springer.
There are some review article about applications of the Schwinger–Dyson equations with applications to special field of physics.
For applications to Quantum Chromodynamics there are
R. Alkofer and L. v.Smekal (2001). "On the infrared behaviour of QCD Green's functions". Phys. Rep. 353 (5–6): 281. arXiv:hep-ph/0007355. Bibcode:2001PhR...353..281A. doi:10.1016/S0370-1573(01)00010-2. S2CID 119411676.
C.D. Roberts and A.G. Williams (1994). "Dyson-Schwinger equations and their applications to hadron physics". Prog. Part. Nucl. Phys. 33: 477–575. arXiv:hep-ph/9403224. Bibcode:1994PrPNP..33..477R. doi:10.1016/0146-6410(94)90049-3. S2CID 119360538. | Wikipedia/Schwinger–Dyson_equation |
Particle physics or high-energy physics is the study of fundamental particles and forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the scale of protons and neutrons, while the study of combinations of protons and neutrons is called nuclear physics.
The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and bosons (force-carrying particles). There are three generations of fermions, although ordinary matter is made only from the first fermion generation. The first generation consists of up and down quarks which form protons and neutrons, and electrons and electron neutrinos. The three fundamental interactions known to be mediated by bosons are electromagnetism, the weak interaction, and the strong interaction.
Quarks cannot exist on their own but form hadrons. Hadrons that contain an odd number of quarks are called baryons and those that contain an even number are called mesons. Two baryons, the proton and the neutron, make up most of the mass of ordinary matter. Mesons are unstable and the longest-lived last for only a few hundredths of a microsecond. They occur after collisions between particles made of quarks, such as fast-moving protons and neutrons in cosmic rays. Mesons are also produced in cyclotrons or other particle accelerators.
Particles have corresponding antiparticles with the same mass but with opposite electric charges. For example, the antiparticle of the electron is the positron. The electron has a negative electric charge, the positron has a positive charge. These antiparticles can theoretically form a corresponding form of matter called antimatter. Some particles, such as the photon, are their own antiparticle.
These elementary particles are excitations of the quantum fields that also govern their interactions. The dominant theory explaining these fundamental particles and fields, along with their dynamics, is called the Standard Model. The reconciliation of gravity to the current particle physics theory is not solved; many theories have addressed this problem, such as loop quantum gravity, string theory and supersymmetry theory.
Experimental particle physics is the study of these particles in radioactive processes and in particle accelerators such as the Large Hadron Collider. Theoretical particle physics is the study of these particles in the context of cosmology and quantum theory. The two are closely interrelated: the Higgs boson was postulated theoretically before being confirmed by experiments.
== History ==
The idea that all matter is fundamentally composed of elementary particles dates from at least the 6th century BC. In the 19th century, John Dalton, through his work on stoichiometry, concluded that each element of nature was composed of a single, unique type of particle. The word atom, after the Greek word atomos meaning "indivisible", has since then denoted the smallest particle of a chemical element, but physicists later discovered that atoms are not, in fact, the fundamental particles of nature, but are conglomerates of even smaller particles, such as the electron. The early 20th century explorations of nuclear physics and quantum physics led to proofs of nuclear fission in 1939 by Lise Meitner (based on experiments by Otto Hahn), and nuclear fusion by Hans Bethe in that same year; both discoveries also led to the development of nuclear weapons. Bethe's 1947 calculation of the Lamb shift is credited with having "opened the way to the modern era of particle physics".
Throughout the 1950s and 1960s, a bewildering variety of particles was found in collisions of particles from beams of increasingly high energy. It was referred to informally as the "particle zoo". Important discoveries such as the CP violation by James Cronin and Val Fitch brought new questions to matter-antimatter imbalance. After the formulation of the Standard Model during the 1970s, physicists clarified the origin of the particle zoo. The large number of particles was explained as combinations of a (relatively) small number of more fundamental particles and framed in the context of quantum field theories. This reclassification marked the beginning of modern particle physics.
== Standard Model ==
The current state of the classification of all elementary particles is explained by the Standard Model, which gained widespread acceptance in the mid-1970s after experimental confirmation of the existence of quarks. It describes the strong, weak, and electromagnetic fundamental interactions, using mediating gauge bosons. The species of gauge bosons are eight gluons, W−, W+ and Z bosons, and the photon. The Standard Model also contains 24 fundamental fermions (12 particles and their associated anti-particles), which are the constituents of all matter. Finally, the Standard Model also predicted the existence of a type of boson known as the Higgs boson. On 4 July 2012, physicists with the Large Hadron Collider at CERN announced they had found a new particle that behaves similarly to what is expected from the Higgs boson.
The Standard Model, as currently formulated, has 61 elementary particles. Those elementary particles can combine to form composite particles, accounting for the hundreds of other species of particles that have been discovered since the 1960s. The Standard Model has been found to agree with almost all the experimental tests conducted to date. However, most particle physicists believe that it is an incomplete description of nature and that a more fundamental theory awaits discovery (See Theory of Everything). In recent years, measurements of neutrino mass have provided the first experimental deviations from the Standard Model, since neutrinos do not have mass in the Standard Model.
== Subatomic particles ==
Modern particle physics research is focused on subatomic particles, including atomic constituents, such as electrons, protons, and neutrons (protons and neutrons are composite particles called baryons, made of quarks), that are produced by radioactive and scattering processes; such particles are photons, neutrinos, and muons, as well as a wide range of exotic particles. All particles and their interactions observed to date can be described almost entirely by the Standard Model.
Dynamics of particles are also governed by quantum mechanics; they exhibit wave–particle duality, displaying particle-like behaviour under certain experimental conditions and wave-like behaviour in others. In more technical terms, they are described by quantum state vectors in a Hilbert space, which is also treated in quantum field theory. Following the convention of particle physicists, the term elementary particles is applied to those particles that are, according to current understanding, presumed to be indivisible and not composed of other particles.
=== Quarks and leptons ===
Ordinary matter is made from first-generation quarks (up, down) and leptons (electron, electron neutrino). Collectively, quarks and leptons are called fermions, because they have a quantum spin of half-integers (−1/2, 1/2, 3/2, etc.). This causes the fermions to obey the Pauli exclusion principle, where no two particles may occupy the same quantum state. Quarks have fractional elementary electric charge (−1/3 or 2/3) and leptons have whole-numbered electric charge (0 or -1). Quarks also have color charge, which is labeled arbitrarily with no correlation to actual light color as red, green and blue. Because the interactions between the quarks store energy which can convert to other particles when the quarks are far apart enough, quarks cannot be observed independently. This is called color confinement.
There are three known generations of quarks (up and down, strange and charm, top and bottom) and leptons (electron and its neutrino, muon and its neutrino, tau and its neutrino), with strong indirect evidence that a fourth generation of fermions does not exist.
=== Bosons ===
Bosons are the mediators or carriers of fundamental interactions, such as electromagnetism, the weak interaction, and the strong interaction. Electromagnetism is mediated by the photon, the quanta of light.: 29–30 The weak interaction is mediated by the W and Z bosons. The strong interaction is mediated by the gluon, which can link quarks together to form composite particles. Due to the aforementioned color confinement, gluons are never observed independently. The Higgs boson gives mass to the W and Z bosons via the Higgs mechanism – the gluon and photon are expected to be massless. All bosons have an integer quantum spin (0 and 1) and can have the same quantum state.
=== Antiparticles and color charge ===
Most aforementioned particles have corresponding antiparticles, which compose antimatter. Normal particles have positive lepton or baryon number, and antiparticles have these numbers negative. Most properties of corresponding antiparticles and particles are the same, with a few gets reversed; the electron's antiparticle, positron, has an opposite charge. To differentiate between antiparticles and particles, a plus or negative sign is added in superscript. For example, the electron and the positron are denoted e− and e+. However, in the case that the particle has a charge of 0 (equal to that of the antiparticle), the antiparticle is denoted with a line above the symbol. As such, an electron neutrino is νe, whereas its antineutrino is νe. When a particle and an antiparticle interact with each other, they are annihilated and convert to other particles. Some particles, such as the photon or gluon, have no antiparticles.
Quarks and gluons additionally have color charges, which influences the strong interaction. Quark's color charges are called red, green and blue (though the particle itself have no physical color), and in antiquarks are called antired, antigreen and antiblue. The gluon can have eight color charges, which are the result of quarks' interactions to form composite particles (gauge symmetry SU(3)).
=== Composite ===
The neutrons and protons in the atomic nuclei are baryons – the neutron is composed of two down quarks and one up quark, and the proton is composed of two up quarks and one down quark. A baryon is composed of three quarks, and a meson is composed of two quarks (one normal, one anti). Baryons and mesons are collectively called hadrons. Quarks inside hadrons are governed by the strong interaction, thus are subjected to quantum chromodynamics (color charges). The bounded quarks must have their color charge to be neutral, or "white" for analogy with mixing the primary colors. More exotic hadrons can have other types, arrangement or number of quarks (tetraquark, pentaquark).
An atom is made from protons, neutrons and electrons. By modifying the particles inside a normal atom, exotic atoms can be formed. A simple example would be the hydrogen-4.1, which has one of its electrons replaced with a muon.
=== Hypothetical ===
The graviton is a hypothetical particle that can mediate the gravitational interaction, but it has not been detected or completely reconciled with current theories. Many other hypothetical particles have been proposed to address the limitations of the Standard Model. Notably, supersymmetric particles aim to solve the hierarchy problem, axions address the strong CP problem, and various other particles are proposed to explain the origins of dark matter and dark energy.
== Experimental laboratories ==
The world's major particle physics laboratories are:
Brookhaven National Laboratory (Long Island, New York, United States). Its main facility is the Relativistic Heavy Ion Collider (RHIC), which collides heavy ions such as gold ions and polarized protons. It is the world's first heavy ion collider, and the world's only polarized proton collider.
Budker Institute of Nuclear Physics (Novosibirsk, Russia). Its main projects are now the electron-positron colliders VEPP-2000, operated since 2006, and VEPP-4, started experiments in 1994. Earlier facilities include the first electron–electron beam–beam collider VEP-1, which conducted experiments from 1964 to 1968; the electron-positron colliders VEPP-2, operated from 1965 to 1974; and, its successor VEPP-2M, performed experiments from 1974 to 2000.
CERN (European Organization for Nuclear Research) (Franco-Swiss border, near Geneva, Switzerland). Its main project is now the Large Hadron Collider (LHC), which had its first beam circulation on 10 September 2008, and is now the world's most energetic collider of protons. It also became the most energetic collider of heavy ions after it began colliding lead ions. Earlier facilities include the Large Electron–Positron Collider (LEP), which was stopped on 2 November 2000 and then dismantled to give way for LHC; and the Super Proton Synchrotron, which is being reused as a pre-accelerator for the LHC and for fixed-target experiments.
DESY (Deutsches Elektronen-Synchrotron) (Hamburg, Germany). Its main facility was the Hadron Elektron Ring Anlage (HERA), which collided electrons and positrons with protons. The accelerator complex is now focused on the production of synchrotron radiation with PETRA III, FLASH and the European XFEL.
Fermi National Accelerator Laboratory (Fermilab) (Batavia, Illinois, United States). Its main facility until 2011 was the Tevatron, which collided protons and antiprotons and was the highest-energy particle collider on earth until the Large Hadron Collider surpassed it on 29 November 2009.
Institute of High Energy Physics (IHEP) (Beijing, China). IHEP manages a number of China's major particle physics facilities, including the Beijing Electron–Positron Collider II(BEPC II), the Beijing Spectrometer (BES), the Beijing Synchrotron Radiation Facility (BSRF), the International Cosmic-Ray Observatory at Yangbajing in Tibet, the Daya Bay Reactor Neutrino Experiment, the China Spallation Neutron Source, the Hard X-ray Modulation Telescope (HXMT), and the Accelerator-driven Sub-critical System (ADS) as well as the Jiangmen Underground Neutrino Observatory (JUNO).
KEK (Tsukuba, Japan). It is the home of a number of experiments such as the K2K experiment and its successor T2K experiment, a neutrino oscillation experiment and Belle II, an experiment measuring the CP violation of B mesons.
SLAC National Accelerator Laboratory (Menlo Park, California, United States). Its 2-mile-long linear particle accelerator began operating in 1962 and was the basis for numerous electron and positron collision experiments until 2008. Since then the linear accelerator is being used for the Linac Coherent Light Source X-ray laser as well as advanced accelerator design research. SLAC staff continue to participate in developing and building many particle detectors around the world.
== Theory ==
Theoretical particle physics attempts to develop the models, theoretical framework, and mathematical tools to understand current experiments and make predictions for future experiments (see also theoretical physics). There are several major interrelated efforts being made in theoretical particle physics today.
One important branch attempts to better understand the Standard Model and its tests. Theorists make quantitative predictions of observables at collider and astronomical experiments, which along with experimental measurements is used to extract the parameters of the Standard Model with less uncertainty. This work probes the limits of the Standard Model and therefore expands scientific understanding of nature's building blocks. Those efforts are made challenging by the difficulty of calculating high precision quantities in quantum chromodynamics. Some theorists working in this area use the tools of perturbative quantum field theory and effective field theory, referring to themselves as phenomenologists. Others make use of lattice field theory and call themselves lattice theorists.
Another major effort is in model building where model builders develop ideas for what physics may lie beyond the Standard Model (at higher energies or smaller distances). This work is often motivated by the hierarchy problem and is constrained by existing experimental data. It may involve work on supersymmetry, alternatives to the Higgs mechanism, extra spatial dimensions (such as the Randall–Sundrum models), Preon theory, combinations of these, or other ideas. Vanishing-dimensions theory is a particle physics theory suggesting that systems with higher energy have a smaller number of dimensions.
A third major effort in theoretical particle physics is string theory. String theorists attempt to construct a unified description of quantum mechanics and general relativity by building a theory based on small strings, and branes rather than particles. If the theory is successful, it may be considered a "Theory of Everything", or "TOE".
There are also other areas of work in theoretical particle physics ranging from particle cosmology to loop quantum gravity.
== Practical applications ==
In principle, all physics (and practical applications developed therefrom) can be derived from the study of fundamental particles. In practice, even if "particle physics" is taken to mean only "high-energy atom smashers", many technologies have been developed during these pioneering investigations that later find wide uses in society. Particle accelerators are used to produce medical isotopes for research and treatment (for example, isotopes used in PET imaging), or used directly in external beam radiotherapy. The development of superconductors has been pushed forward by their use in particle physics. The World Wide Web and touchscreen technology were initially developed at CERN. Additional applications are found in medicine, national security, industry, computing, science, and workforce development, illustrating a long and growing list of beneficial practical applications with contributions from particle physics.
== Future ==
Major efforts to look for physics beyond the Standard Model include the Future Circular Collider proposed for CERN and the Particle Physics Project Prioritization Panel (P5) in the US that will update the 2014 P5 study that recommended the Deep Underground Neutrino Experiment, among other experiments.
== See also ==
== References ==
== External links == | Wikipedia/High-energy_physics |
In nuclear physics, the chiral model, introduced by Feza Gürsey in 1960, is a phenomenological model describing effective interactions of mesons in the chiral limit (where the masses of the quarks go to zero), but without necessarily mentioning quarks at all. It is a nonlinear sigma model with the principal homogeneous space of a Lie group
G
{\displaystyle G}
as its target manifold. When the model was originally introduced, this Lie group was the SU(N), where N is the number of quark flavors. The Riemannian metric of the target manifold is given by a positive constant multiplied by the Killing form acting upon the Maurer–Cartan form of SU(N).
The internal global symmetry of this model is
G
L
×
G
R
{\displaystyle G_{L}\times G_{R}}
, the left and right copies, respectively; where the left copy acts as the left action upon the target space, and the right copy acts as the right action. Phenomenologically, the left copy represents flavor rotations among the left-handed quarks, while the right copy describes rotations among the right-handed quarks, while these, L and R, are completely independent of each other. The axial pieces of these symmetries are spontaneously broken so that the corresponding scalar fields are the requisite Nambu−Goldstone bosons.
The model was later studied in the two-dimensional case as an integrable system, in particular an integrable field theory. Its integrability was shown by Faddeev and Reshetikhin in 1982 through the quantum inverse scattering method. The two-dimensional principal chiral model exhibits signatures of integrability such as a Lax pair/zero-curvature formulation, an infinite number of symmetries, and an underlying quantum group symmetry (in this case, Yangian symmetry).
This model admits topological solitons called skyrmions.
Departures from exact chiral symmetry are dealt with in chiral perturbation theory.
== Mathematical formulation ==
On a manifold (considered as the spacetime) M and a choice of compact Lie group G, the field content is a function
U
:
M
→
G
{\displaystyle U:M\rightarrow G}
. This defines a related field
j
μ
=
U
−
1
∂
μ
U
{\displaystyle j_{\mu }=U^{-1}\partial _{\mu }U}
, a
g
{\displaystyle {\mathfrak {g}}}
-valued vector field (really, covector field) which is the Maurer–Cartan form. The principal chiral model is defined by the Lagrangian density
L
=
κ
2
t
r
(
∂
μ
U
−
1
∂
μ
U
)
=
−
κ
2
t
r
(
j
μ
j
μ
)
,
{\displaystyle {\mathcal {L}}={\frac {\kappa }{2}}\mathrm {tr} (\partial _{\mu }U^{-1}\partial ^{\mu }U)=-{\frac {\kappa }{2}}\mathrm {tr} (j_{\mu }j^{\mu }),}
where
κ
{\displaystyle \kappa }
is a dimensionless coupling. In differential-geometric language, the field
U
{\displaystyle U}
is a section of a principal bundle
π
:
P
→
M
{\displaystyle \pi :P\rightarrow M}
with fibres isomorphic to the principal homogeneous space for M (hence why this defines the principal chiral model).
== Phenomenology ==
=== An outline of the original, 2-flavor model ===
The chiral model of Gürsey (1960; also see Gell-Mann and Lévy) is now appreciated to be an effective theory of QCD with two light quarks, u, and d. The QCD Lagrangian is approximately invariant under independent global flavor rotations of the left- and right-handed quark fields,
{
q
L
↦
q
L
′
=
L
q
L
=
exp
(
−
i
θ
L
⋅
1
2
τ
)
q
L
q
R
↦
q
R
′
=
R
q
R
=
exp
(
−
i
θ
R
⋅
1
2
τ
)
q
R
{\displaystyle {\begin{cases}q_{\mathsf {L}}\mapsto q_{\mathsf {L}}'=L\ q_{\mathsf {L}}=\exp {\left(-i{\boldsymbol {\theta }}_{\mathsf {L}}\cdot {\tfrac {1}{2}}{\boldsymbol {\tau }}\right)}q_{\mathsf {L}}\\q_{\mathsf {R}}\mapsto q_{\mathsf {R}}'=R\ q_{\mathsf {R}}=\exp {\left(-i{\boldsymbol {\theta }}_{\mathsf {R}}\cdot {\tfrac {1}{2}}{\boldsymbol {\tau }}\right)}q_{\mathsf {R}}\end{cases}}}
where τ denote the Pauli matrices in the flavor space and θL , θR are the corresponding rotation angles.
The corresponding symmetry group
SU
(
2
)
L
×
SU
(
2
)
R
{\displaystyle \ {\text{SU}}(2)_{\mathsf {L}}\times {\text{SU}}(2)_{\mathsf {R}}\ }
is the chiral group, controlled by the six conserved currents
L
μ
i
=
q
¯
L
γ
μ
τ
i
2
q
L
,
R
μ
i
=
q
¯
R
γ
μ
τ
i
2
q
R
,
{\displaystyle L_{\mu }^{i}={\bar {q}}_{\mathsf {L}}\gamma _{\mu }{\tfrac {\tau ^{i}}{2}}q_{\mathsf {L}},\qquad R_{\mu }^{i}={\bar {q}}_{\mathsf {R}}\gamma _{\mu }{\tfrac {\tau ^{i}}{2}}q_{\mathsf {R}}\ ,}
which can equally well be expressed in terms of the vector and axial-vector currents
V
μ
i
=
L
μ
i
+
R
μ
i
,
A
μ
i
=
R
μ
i
−
L
μ
i
.
{\displaystyle V_{\mu }^{i}=L_{\mu }^{i}+R_{\mu }^{i},\qquad A_{\mu }^{i}=R_{\mu }^{i}-L_{\mu }^{i}~.}
The corresponding conserved charges generate the algebra of the chiral group,
[
Q
I
i
,
Q
I
j
]
=
i
ϵ
i
j
k
Q
I
k
[
Q
L
i
,
Q
R
j
]
=
0
,
{\displaystyle \left[Q_{I}^{i},Q_{I}^{j}\right]=i\epsilon ^{ijk}Q_{I}^{k}\qquad \qquad \left[Q_{\mathsf {L}}^{i},Q_{\mathsf {R}}^{j}\right]=0,}
with I = L, R , or, equivalently,
[
Q
V
i
,
Q
V
j
]
=
i
ϵ
i
j
k
Q
V
k
,
[
Q
A
i
,
Q
A
j
]
=
i
ϵ
i
j
k
Q
V
k
,
[
Q
V
i
,
Q
A
j
]
=
i
ϵ
i
j
k
Q
A
k
.
{\displaystyle \left[Q_{V}^{i},Q_{V}^{j}\right]=i\epsilon ^{ijk}Q_{V}^{k},\qquad \left[Q_{A}^{i},Q_{A}^{j}\right]=i\epsilon ^{ijk}Q_{V}^{k},\qquad \left[Q_{V}^{i},Q_{A}^{j}\right]=i\epsilon ^{ijk}Q_{A}^{k}.}
Application of these commutation relations to hadronic reactions dominated current algebra calculations in the early 1970s.
At the level of hadrons, pseudoscalar mesons, the ambit of the chiral model, the chiral
SU
(
2
)
L
×
SU
(
2
)
R
{\displaystyle \ {\text{SU}}(2)_{\mathsf {L}}\times {\text{SU}}(2)_{\mathsf {R}}\ }
group is spontaneously broken down to
SU
(
2
)
V
,
{\displaystyle {\text{SU}}(2)_{V}\ ,}
by the QCD vacuum. That is, it is realized nonlinearly, in the Nambu–Goldstone mode: The QV annihilate the vacuum, but the QA do not! This is visualized nicely through a geometrical argument based on the fact that the Lie algebra of
SU
(
2
)
L
×
SU
(
2
)
R
{\displaystyle {\text{SU}}(2)_{\mathsf {L}}\times {\text{SU}}(2)_{\mathsf {R}}\ }
is isomorphic to that of SO(4). The unbroken subgroup, realized in the linear Wigner–Weyl mode, is
SO
(
3
)
⊂
SO
(
4
)
{\displaystyle \ {\text{SO}}(3)\subset {\text{SO}}(4)\ }
which is locally isomorphic to SU(2) (V: isospin).
To construct a non-linear realization of SO(4), the representation describing four-dimensional rotations of a vector
(
π
σ
)
≡
(
π
1
π
2
π
3
σ
)
,
{\displaystyle {\begin{pmatrix}{\boldsymbol {\pi }}\\\sigma \end{pmatrix}}\equiv {\begin{pmatrix}\pi _{1}\\\pi _{2}\\\pi _{3}\\\sigma \end{pmatrix}},}
for an infinitesimal rotation parametrized by six angles
{
θ
i
V
,
A
}
,
i
=
1
,
2
,
3
,
{\displaystyle \left\{\theta _{i}^{V,A}\right\},\qquad i=1,2,3,}
is given by
(
π
σ
)
⟶
S
O
(
4
)
(
π
′
σ
′
)
=
[
1
4
+
∑
i
=
1
3
θ
i
V
V
i
+
∑
i
=
1
3
θ
i
A
A
i
]
(
π
σ
)
{\displaystyle {\begin{pmatrix}{\boldsymbol {\pi }}\\\sigma \end{pmatrix}}{\stackrel {SO(4)}{\longrightarrow }}{\begin{pmatrix}{{\boldsymbol {\pi }}'}\\\sigma '\end{pmatrix}}=\left[\mathbf {1} _{4}+\sum _{i=1}^{3}\theta _{i}^{V}\ V_{i}+\sum _{i=1}^{3}\theta _{i}^{A}\ A_{i}\right]{\begin{pmatrix}{\boldsymbol {\pi }}\\\sigma \end{pmatrix}}}
where
∑
i
=
1
3
θ
i
V
V
i
=
(
0
−
θ
3
V
θ
2
V
0
θ
3
V
0
−
θ
1
V
0
−
θ
2
V
θ
1
V
0
0
0
0
0
0
)
∑
i
=
1
3
θ
i
A
A
i
=
(
0
0
0
θ
1
A
0
0
0
θ
2
A
0
0
0
θ
3
A
−
θ
1
A
−
θ
2
A
−
θ
3
A
0
)
.
{\displaystyle \sum _{i=1}^{3}\theta _{i}^{V}\ V_{i}={\begin{pmatrix}0&-\theta _{3}^{V}&\theta _{2}^{V}&0\\\theta _{3}^{V}&0&-\theta _{1}^{V}&0\\-\theta _{2}^{V}&\theta _{1}^{V}&0&0\\0&0&0&0\end{pmatrix}}\qquad \qquad \sum _{i=1}^{3}\theta _{i}^{A}\ A_{i}={\begin{pmatrix}0&0&0&\theta _{1}^{A}\\0&0&0&\theta _{2}^{A}\\0&0&0&\theta _{3}^{A}\\-\theta _{1}^{A}&-\theta _{2}^{A}&-\theta _{3}^{A}&0\end{pmatrix}}.}
The four real quantities (π, σ) define the smallest nontrivial chiral multiplet and represent the field content of the linear sigma model.
To switch from the above linear realization of SO(4) to the nonlinear one, we observe that, in fact, only three of the four components of (π, σ) are independent with respect to four-dimensional rotations. These three independent components
correspond to coordinates on a hypersphere S3, where π and σ are subjected to the constraint
π
2
+
σ
2
=
F
2
,
{\displaystyle {\boldsymbol {\pi }}^{2}+\sigma ^{2}=F^{2}\ ,}
with F a pion decay constant with dimension = mass.
Utilizing this to eliminate σ yields the following transformation properties of π under SO(4),
{
θ
V
:
π
↦
π
′
=
π
+
θ
V
×
π
θ
A
:
π
↦
π
′
=
π
+
θ
A
F
2
−
π
2
θ
V
,
A
≡
{
θ
i
V
,
A
}
,
i
=
1
,
2
,
3.
{\displaystyle {\begin{cases}\theta ^{V}:{\boldsymbol {\pi }}\mapsto {\boldsymbol {\pi }}'={\boldsymbol {\pi }}+{\boldsymbol {\theta }}^{V}\times {\boldsymbol {\pi }}\\\theta ^{A}:{\boldsymbol {\pi }}\mapsto {\boldsymbol {\pi }}'={\boldsymbol {\pi }}+{\boldsymbol {\theta }}^{A}{\sqrt {F^{2}-{\boldsymbol {\pi }}^{2}}}\end{cases}}\qquad {\boldsymbol {\theta }}^{V,A}\equiv \left\{\theta _{i}^{V,A}\right\},\qquad i=1,2,3.}
The nonlinear terms (shifting π) on the right-hand side of the second equation underlie the nonlinear realization of SO(4). The chiral group
SU
(
2
)
L
×
SU
(
2
)
R
≃
SO
(
4
)
{\displaystyle \ {\text{SU}}(2)_{\mathsf {L}}\times {\text{SU}}(2)_{\mathsf {R}}\simeq {\text{SO}}(4)\ }
is realized nonlinearly on the triplet of pions – which, however, still transform linearly under isospin
SU
(
2
)
V
≃
SO
(
3
)
{\displaystyle \ {\text{SU}}(2)_{V}\simeq {\text{SO}}(3)\ }
rotations parametrized through the angles
{
θ
V
}
.
{\displaystyle \ \left\{{\boldsymbol {\theta }}_{V}\right\}~.}
By contrast, the
{
θ
A
}
{\displaystyle \ \left\{{\boldsymbol {\theta }}_{A}\right\}\ }
represent the nonlinear "shifts" (spontaneous breaking).
Through the spinor map, these four-dimensional rotations of (π, σ) can also be conveniently written using 2×2 matrix notation by introducing the unitary matrix
U
=
1
F
(
σ
1
2
+
i
π
⋅
τ
)
,
{\displaystyle U={\frac {1}{F}}\left(\sigma \mathbf {1} _{2}+i{\boldsymbol {\pi }}\cdot {\boldsymbol {\tau }}\right)\ ,}
and requiring the transformation properties of U under chiral rotations to be
U
⟶
U
′
=
L
U
R
†
,
{\displaystyle U\longrightarrow U'=LUR^{\dagger }\ ,}
where
θ
L
=
θ
V
−
θ
A
,
θ
R
=
θ
V
+
θ
A
.
{\displaystyle ~\theta _{\mathsf {L}}=\theta _{V}-\theta _{A}\ ,\quad \theta _{\mathsf {R}}=\theta _{V}+\theta _{A}~.}
The transition to the nonlinear realization follows,
U
=
1
F
(
F
2
−
π
2
1
2
+
i
π
⋅
τ
)
,
L
π
(
2
)
=
1
4
F
2
⟨
∂
μ
U
∂
μ
U
†
⟩
t
r
,
{\displaystyle U={\frac {1}{F}}\left({\sqrt {F^{2}-{\boldsymbol {\pi }}^{2}\ }}\ \mathbf {1} _{2}+i{\boldsymbol {\pi }}\cdot {\boldsymbol {\tau }}\right)\ ,\qquad {\mathcal {L}}_{\pi }^{(2)}={\tfrac {1}{4}}F^{2}\ \langle \ \partial _{\mu }U\ \partial ^{\mu }U^{\dagger }\ \rangle _{\mathsf {tr}}\ ,}
where
⟨
…
⟩
t
r
{\displaystyle \ \langle \ldots \rangle _{\mathsf {tr}}\ }
denotes the trace in the flavor space. This is a non-linear sigma model.
Terms involving
∂
μ
∂
μ
U
{\displaystyle \textstyle \ \partial _{\mu }\partial ^{\mu }\ U\ }
or
∂
μ
∂
μ
U
†
{\displaystyle \textstyle \ \partial _{\mu }\partial ^{\mu }\ U^{\dagger }\ }
are not independent and can be brought to this form through partial integration.
The constant 1/4F2 is chosen in such a way that the Lagrangian matches the usual free term for massless scalar fields when written in terms of the pions,
L
π
(
2
)
=
1
2
∂
μ
π
⋅
∂
μ
π
+
1
2
(
∂
μ
π
⋅
π
F
)
2
+
O
(
π
6
)
.
{\displaystyle \ {\mathcal {L}}_{\pi }^{(2)}={\frac {1}{2}}\partial _{\mu }{\boldsymbol {\pi }}\cdot \partial ^{\mu }{\boldsymbol {\pi }}+{\frac {1}{2}}\left({\frac {\partial _{\mu }{\boldsymbol {\pi }}\cdot {\boldsymbol {\pi }}}{F}}\right)^{2}+{\mathcal {O}}(\pi ^{6})~.}
=== Alternate Parametrization ===
An alternative, equivalent (Gürsey, 1960), parameterization
π
↦
π
sin
(
|
π
/
F
|
)
|
π
/
F
|
,
{\displaystyle {\boldsymbol {\pi }}\mapsto {\boldsymbol {\pi }}~{\frac {\sin(|\pi /F|)}{|\pi /F|}},}
yields a simpler expression for U,
U
=
1
cos
|
π
/
F
|
+
i
π
^
⋅
τ
sin
|
π
/
F
|
=
e
i
τ
⋅
π
/
F
.
{\displaystyle U=\mathbf {1} \cos |\pi /F|+i{\widehat {\pi }}\cdot {\boldsymbol {\tau }}\sin |\pi /F|=e^{i~{\boldsymbol {\tau }}\cdot {\boldsymbol {\pi }}/F}.}
Note the reparameterized π transform under
L
U
R
†
=
exp
(
i
θ
A
⋅
τ
/
2
−
i
θ
V
⋅
τ
/
2
)
exp
(
i
π
⋅
τ
/
F
)
exp
(
i
θ
A
⋅
τ
/
2
+
i
θ
V
⋅
τ
/
2
)
{\displaystyle LUR^{\dagger }=\exp(i{\boldsymbol {\theta }}_{A}\cdot {\boldsymbol {\tau }}/2-i{\boldsymbol {\theta }}_{V}\cdot {\boldsymbol {\tau }}/2)\exp(i{\boldsymbol {\pi }}\cdot {\boldsymbol {\tau }}/F)\exp(i{\boldsymbol {\theta }}_{A}\cdot {\boldsymbol {\tau }}/2+i{\boldsymbol {\theta }}_{V}\cdot {\boldsymbol {\tau }}/2)}
so, then, manifestly identically to the above under isorotations, V; and similarly to the above, as
π
⟶
π
+
θ
A
F
+
⋯
=
π
+
θ
A
F
(
|
π
/
F
|
cot
|
π
/
F
|
)
{\displaystyle {\boldsymbol {\pi }}\longrightarrow {\boldsymbol {\pi }}+{\boldsymbol {\theta }}_{A}F+\cdots ={\boldsymbol {\pi }}+{\boldsymbol {\theta }}_{A}F(|\pi /F|\cot |\pi /F|)}
under the broken symmetries, A, the shifts. This simpler expression generalizes readily (Cronin, 1967) to N light quarks, so
SU
(
N
)
L
×
SU
(
N
)
R
/
SU
(
N
)
V
.
{\displaystyle \textstyle {\text{SU}}(N)_{L}\times {\text{SU}}(N)_{R}/{\text{SU}}(N)_{V}.}
== Integrability ==
=== Integrable chiral model ===
Introduced by Richard S. Ward, the integrable chiral model or Ward model is described in terms of a matrix-valued field
J
:
R
3
→
U
(
n
)
{\displaystyle J:\mathbb {R} ^{3}\rightarrow U(n)}
and is given by the partial differential equation
∂
t
(
J
−
1
J
t
)
−
∂
x
(
J
−
1
J
x
)
−
∂
y
(
J
−
1
J
y
)
−
[
J
−
1
J
t
,
J
−
1
J
y
]
=
0.
{\displaystyle \partial _{t}(J^{-1}J_{t})-\partial _{x}(J^{-1}J_{x})-\partial _{y}(J^{-1}J_{y})-[J^{-1}J_{t},J^{-1}J_{y}]=0.}
It has a Lagrangian formulation with the expected kinetic term together with a term which resembles a Wess–Zumino–Witten term. It also has a formulation which is formally identical to the Bogomolny equations but with Lorentz signature. The relation between these formulations can be found in Dunajski (2010).
Many exact solutions are known.
=== Two-dimensional principal chiral model ===
Here the underlying manifold
M
{\displaystyle M}
is taken to be a Riemann surface, in particular the cylinder
C
∗
{\displaystyle \mathbb {C} ^{*}}
or plane
C
{\displaystyle \mathbb {C} }
, conventionally given real coordinates
τ
,
σ
{\displaystyle \tau ,\sigma }
, where on the cylinder
σ
∼
σ
+
2
π
{\displaystyle \sigma \sim \sigma +2\pi }
is a periodic coordinate. For application to string theory, this cylinder is the world sheet swept out by the closed string.
==== Global symmetries ====
The global symmetries act as internal symmetries on the group-valued field
g
(
x
)
{\displaystyle g(x)}
as
ρ
L
(
g
′
)
g
(
x
)
=
g
′
g
(
x
)
{\displaystyle \rho _{L}(g')g(x)=g'g(x)}
and
ρ
R
(
g
)
g
(
x
)
=
g
(
x
)
g
′
{\displaystyle \rho _{R}(g)g(x)=g(x)g'}
. The corresponding conserved currents from Noether's theorem are
L
α
=
g
−
1
∂
α
g
,
R
α
=
∂
α
g
g
−
1
.
{\displaystyle L_{\alpha }=g^{-1}\partial _{\alpha }g,\qquad R_{\alpha }=\partial _{\alpha }gg^{-1}.}
The equations of motion turn out to be equivalent to conservation of these currents,
∂
α
L
α
=
∂
α
R
α
=
0
,
or, in coordinate-free form,
d
∗
L
=
d
∗
R
=
0.
{\displaystyle \partial _{\alpha }L^{\alpha }=\partial _{\alpha }R^{\alpha }=0,~{\text{ or, in coordinate-free form, }}~d*L=d*R=0.}
The currents additionally satisfy the flatness condition,
d
L
+
1
2
[
L
,
L
]
=
0
or, in coordinates,
∂
α
L
β
−
∂
β
L
α
+
[
L
α
,
L
β
]
=
0
,
{\displaystyle dL+{\frac {1}{2}}[L,L]=0~~~{\text{ or, in coordinates, }}~~~\partial _{\alpha }L_{\beta }-\partial _{\beta }L_{\alpha }+[L_{\alpha },L_{\beta }]=0,}
and therefore the equations of motion can be formulated entirely in terms of the currents.
Upon quantization, the axial combination of these currents develop chiral anomalies, summarized in the above-mentioned topological WZWN term.
==== Lax formulation ====
Consider the worldsheet in light-cone coordinates
x
±
=
t
±
x
{\displaystyle x^{\pm }=t\pm x}
. The components of the appropriate Lax matrix are
L
±
(
x
+
,
x
−
;
λ
)
=
j
±
1
∓
λ
.
{\displaystyle L_{\pm }(x^{+},x^{-};\lambda )={\frac {j_{\pm }}{1\mp \lambda }}.}
The requirement that the zero-curvature condition on
L
±
{\displaystyle L_{\pm }}
for all
λ
{\displaystyle \lambda }
is equivalent to the conservation of current and flatness of the current
j
=
(
j
+
,
j
−
)
{\displaystyle j=(j_{+},j_{-})}
, that is, the equations of motion from the principal chiral model (PCM).
== See also ==
Sigma model
Chirality (physics)
== References ==
Gürsey, F. (1960). "On the symmetries of strong and weak interactions". Il Nuovo Cimento. 16 (2): 230–240. Bibcode:1960NCim...16..230G. doi:10.1007/BF02860276. S2CID 122270607.
Gürsey, Feza (1961). "On the structure and parity of weak interaction currents". Annals of Physics. 12 (1). Elsevier BV: 91–117. Bibcode:1961AnPhy..12...91G. doi:10.1016/0003-4916(61)90147-6. ISSN 0003-4916.
Coleman, S.; Wess, J.; Zumino, B. (1969). "Structure of Phenomenological Lagrangians. I". Physical Review. 177 (5): 2239. Bibcode:1969PhRv..177.2239C. doi:10.1103/PhysRev.177.2239.; Callan, C.; Coleman, S.; Wess, J.; Zumino, B. (1969). "Structure of Phenomenological Lagrangians. II". Physical Review. 177 (5): 2247. Bibcode:1969PhRv..177.2247C. doi:10.1103/PhysRev.177.2247.
Georgi, H. (1984, 2009). Weak Interactions and Modern Particle Theory (Dover Books on Physics) ISBN 0486469042 online .
Fry, M. P. (2000). "Chiral limit of the two-dimensional fermionic determinant in a general magnetic field". Journal of Mathematical Physics. 41 (4): 1691–1710. arXiv:hep-th/9911131. Bibcode:2000JMP....41.1691F. doi:10.1063/1.533204. S2CID 14302881.
Gell-Mann, M.; Lévy, M. (1960), "The axial vector current in beta decay", Il Nuovo Cimento, 16 (4), Italian Physical Society: 705–726, Bibcode:1960NCim...16..705G, doi:10.1007/BF02859738, ISSN 1827-6121, S2CID 122945049
Cronin, Jeremiah A. (1967-09-25). "Phenomenological Model of Strong and Weak Interactions in ChiralU(3)⊗U(3)". Physical Review. 161 (5). American Physical Society (APS): 1483–1494. Bibcode:1967PhRv..161.1483C. doi:10.1103/physrev.161.1483. ISSN 0031-899X. | Wikipedia/Chiral_model |
The Bousso bound captures a fundamental relation between quantum information and the geometry of space and time. It appears to be an imprint of a unified theory that combines quantum mechanics with Einstein's general relativity. The study of black hole thermodynamics and the information paradox led to the idea of the holographic principle: the entropy of matter and radiation in a spatial region cannot exceed the Bekenstein–Hawking entropy of the boundary of the region, which is proportional to the boundary area. However, this "spacelike" entropy bound fails in cosmology; for example, it does not hold true in our universe.
Raphael Bousso showed that the spacelike entropy bound is violated more broadly in many dynamical settings. For example, the entropy of a collapsing star, once inside a black hole, will eventually exceed its surface area. Due to relativistic length contraction, even ordinary thermodynamic systems can be enclosed in an arbitrarily small area.
To preserve the holographic principle, Bousso proposed a different law, which does not follow from black hole physics: the covariant entropy bound or Bousso bound. Its central geometric object is a lightsheet, defined as a region traced out by non-expanding light-rays emitted orthogonally from an arbitrary surface B. For example, if B is a sphere at a moment of time in Minkowski space, then there are two lightsheets, generated by the past or future directed light-rays emitted towards the interior of the sphere at that time. If B is a sphere surrounding a large region in an expanding universe (an anti-trapped sphere), then there are again two light-sheets that can be considered. Both are directed towards the past, to the interior or the exterior. If B is a trapped surface, such as the surface of a star in its final stages of gravitational collapse, then the lightsheets are directed to the future.
The Bousso bound evades all known counterexamples to the spacelike bound. It was proven to hold when the entropy is approximately a local current, under weak assumptions. In weakly gravitating settings, the Bousso bound implies the Bekenstein bound and admits a formulation that can be proven to hold in any relativistic quantum field theory. The lightsheet construction can be inverted to construct holographic screens for arbitrary spacetimes.
A more recent proposal, the quantum focusing conjecture, implies the original Bousso bound and so can be viewed as a stronger version of it. In the limit where gravity is negligible, the quantum focusing conjecture predicts the quantum null energy condition, which relates the local energy density to a derivative of the entropy. This relation was later proven to hold in any relativistic quantum field theory, such as the Standard Model.
== References == | Wikipedia/Bousso's_holographic_bound |
S-matrix theory was a proposal for replacing local quantum field theory as the basic principle of elementary particle physics.
It avoided the notion of space and time by replacing it with abstract mathematical properties of the S-matrix. In S-matrix theory, the S-matrix relates the infinite past to the infinite future in one step, without being decomposable into intermediate steps corresponding to time-slices.
This program was very influential in the 1960s, because it was a plausible substitute for quantum field theory, which was plagued with the zero interaction phenomenon at strong coupling. Applied to the strong interaction, it led to the development of string theory.
S-matrix theory was largely abandoned by physicists in the 1970s, as quantum chromodynamics was recognized to solve the problems of strong interactions within the framework of field theory. But in the guise of string theory, S-matrix theory is still a popular approach to the problem of quantum gravity.
The S-matrix theory is related to the holographic principle and the AdS/CFT correspondence by a flat space limit. The analog of the S-matrix relations in AdS space is the boundary conformal theory.
The most lasting legacy of the theory is string theory. Other notable achievements are the Froissart bound, and the prediction of the pomeron.
== History ==
S-matrix theory was proposed as a principle of particle interactions by Werner Heisenberg in 1943, following John Archibald Wheeler's 1937 introduction of the S-matrix.
It was developed heavily by Geoffrey Chew, Steven Frautschi, Stanley Mandelstam, Vladimir Gribov, and Tullio Regge. Some aspects of the theory were promoted by Lev Landau in the Soviet Union, and by Murray Gell-Mann in the United States.
== Basic principles ==
The basic principles are:
Relativity: The S-matrix is a representation of the Poincaré group;
Unitarity:
S
S
†
=
1
{\displaystyle SS^{\dagger }=1}
;
Analyticity: integral relations and singularity conditions.
The basic analyticity principles were also called analyticity of the first kind, and they were never fully enumerated, but they include
Crossing: The amplitudes for antiparticle scattering are the analytic continuation of particle scattering amplitudes.
Dispersion relations: the values of the S-matrix can be calculated by integrals over internal energy variables of the imaginary part of the same values.
Causality conditions: the singularities of the S-matrix can only occur in ways that don't allow the future to influence the past (motivated by Kramers–Kronig relations)
Landau principle: Any singularity of the S-matrix corresponds to production thresholds of physical particles.
These principles were to replace the notion of microscopic causality in field theory, the idea that field operators exist at each spacetime point, and that spacelike separated operators commute with one another.
== Bootstrap models ==
The basic principles were too general to apply directly, because they are satisfied automatically by any field theory. So to apply to the real world, additional principles were added.
The phenomenological way in which this was done was by taking experimental data and using the dispersion relations to compute new limits. This led to the discovery of some particles, and to successful parameterizations of the interactions of pions and nucleons.
This path was mostly abandoned because the resulting equations, devoid of any space-time interpretation, were very difficult to understand and solve.
== Regge theory ==
The principle behind the Regge theory hypothesis (also called analyticity of the second kind or the bootstrap principle) is that all strongly interacting particles lie on Regge trajectories. This was considered the definitive sign that all the hadrons are composite particles, but within S-matrix theory, they are not thought of as being made up of elementary constituents.
The Regge theory hypothesis allowed for the construction of string theories, based on bootstrap principles. The additional assumption was the narrow resonance approximation, which started with stable particles on Regge trajectories, and added interaction loop by loop in a perturbation series.
String theory was given a Feynman path-integral interpretation a little while later. The path integral in this case is the analog of a sum over particle paths, not of a sum over field configurations. Feynman's original path integral formulation of field theory also had little need for local fields, since Feynman derived the propagators and interaction rules largely using Lorentz invariance and unitarity.
== See also ==
Landau pole
Regge trajectory
Bootstrap model
Pomeron
Dual resonance model
History of string theory
== Notes ==
== References ==
Steven Frautschi, Regge Poles and S-matrix Theory, New York: W. A. Benjamin, Inc., 1963. | Wikipedia/S-matrix_theory |
The Callan–Giddings–Harvey–Strominger (CGHS) model is a toy model of general relativity in 1 spatial and 1 time dimension. It is named after Curtis Callan, Steven Giddings, Jeffrey A. Harvey and Andrew Strominger, who published on it in 1992.
== Overview ==
General relativity is a highly nonlinear model, and as such, its 3+1D version is usually too complicated to analyze in detail. In 3+1D and higher, propagating gravitational waves exist, but not in 2+1D or 1+1D. In 2+1D, general relativity becomes a topological field theory with no local degrees of freedom, and all 1+1D models are locally flat. However, a slightly more complicated generalization of general relativity which includes dilatons will turn the 2+1D model into one admitting mixed propagating dilaton-gravity waves, as well as making the 1+1D model geometrically nontrivial locally. The 1+1D model still does not admit any propagating gravitational (or dilaton) degrees of freedom, but with the addition of matter fields, it becomes a simplified, but still nontrivial model. With other numbers of dimensions, a dilaton-gravity coupling can always be rescaled away by a conformal rescaling of the metric, converting the Jordan frame to the Einstein frame. But not in two dimensions, because the conformal weight of the dilaton is now 0. The metric in this case is more amenable to analytical solutions than the general 3+1D case. And of course, 0+1D models cannot capture any nontrivial aspect of relativity because there is no space at all.
This class of models retains just enough complexity to include among its solutions black holes, their formation, FRW cosmological models, gravitational singularities, etc. In the quantized version of such models with matter fields, Hawking radiation also shows up, just as in higher-dimensional models.
== Action ==
A very specific choice of couplings and interactions leads to the CGHS model.
S
=
1
2
π
∫
d
2
x
−
g
{
e
−
2
ϕ
[
R
+
4
(
∇
ϕ
)
2
+
4
λ
2
]
−
∑
i
=
1
N
1
2
(
∇
f
i
)
2
}
{\displaystyle S={\frac {1}{2\pi }}\int d^{2}x\,{\sqrt {-g}}\left\{e^{-2\phi }\left[R+4\left(\nabla \phi \right)^{2}+4\lambda ^{2}\right]-\sum _{i=1}^{N}{\frac {1}{2}}\left(\nabla f_{i}\right)^{2}\right\}}
where g is the metric tensor,
ϕ
{\displaystyle \phi }
is the dilaton field, fi are the matter fields, and λ2 is the cosmological constant. In particular, the cosmological constant is nonzero, and the matter fields are massless real scalars.
This specific choice is classically integrable, but still not amenable to an exact quantum solution. It is also the action for Non-critical string theory and dimensional reduction of higher-dimensional model. It also distinguishes it from Jackiw–Teitelboim gravity and Liouville gravity, which are entirely different models.
The matter field only couples to the causal structure, and in the light-cone gauge ds2 = − e2ρ du,dv, has the simple generic form
f
i
(
u
,
v
)
=
A
i
(
u
)
+
B
i
(
v
)
{\displaystyle f_{i}\left(u,v\right)=A_{i}\left(u\right)+B_{i}\left(v\right)}
,
with a factorization between left- and right-movers.
The Raychaudhuri equations are
e
−
2
ϕ
(
−
2
ϕ
,
v
v
+
4
ρ
,
v
ϕ
,
v
)
+
f
i
,
v
f
i
,
v
/
2
=
0
{\displaystyle e^{-2\phi }\left(-2\phi _{,vv}+4\rho _{,v}\phi _{,v}\right)+f_{i,v}f_{i,v}/2=0}
and
e
−
2
ϕ
(
−
2
ϕ
,
u
u
+
4
ρ
,
u
ϕ
,
u
)
+
f
i
,
u
f
i
,
u
/
2
=
0
{\displaystyle e^{-2\phi }\left(-2\phi _{,uu}+4\rho _{,u}\phi _{,u}\right)+f_{i,u}f_{i,u}/2=0}
.
The dilaton evolves according to
(
e
−
2
ϕ
)
,
u
v
=
−
λ
2
e
−
2
ϕ
e
2
ρ
{\displaystyle \left(e^{-2\phi }\right)_{,uv}=-\lambda ^{2}e^{-2\phi }e^{2\rho }}
,
while the metric evolves according to
2
ρ
,
u
v
−
4
ϕ
,
u
v
+
4
ϕ
,
u
ϕ
,
v
+
λ
2
e
2
ρ
=
0
{\displaystyle 2\rho _{,uv}-4\phi _{,uv}+4\phi _{,u}\phi _{,v}+\lambda ^{2}e^{2\rho }=0}
.
The conformal anomaly due to matter induces a Liouville term in the effective action.
=== Black hole ===
A vacuum black hole solution is given by
d
s
2
=
−
(
M
λ
−
λ
2
u
v
)
−
1
d
u
d
v
{\displaystyle ds^{2}=-\left({\frac {M}{\lambda }}-\lambda ^{2}uv\right)^{-1}du\,dv}
e
−
2
ϕ
=
M
λ
−
λ
2
u
v
{\displaystyle e^{-2\phi }={\frac {M}{\lambda }}-\lambda ^{2}uv}
,
where M is the ADM mass.
Singularities appear at uv = λ−3M.
The masslessness of the matter fields allow a black hole to completely evaporate away via Hawking radiation. In fact, this model was originally studied to shed light upon the black hole information paradox.
== See also ==
dilaton
general relativity
quantum gravity
RST model
Jackiw–Teitelboim gravity
Liouville gravity
== References == | Wikipedia/CGHS_model |
In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to string theory. In their setup, the vacuum has the usual 3 dimensions of space and one dimension of time but with another microscopic extra spatial dimension in the shape of a tiny circle. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions.
The five-dimensional (5D) theory developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Albert Einstein in 1919 and published them in 1921. Kaluza presented a purely classical extension of general relativity to 5D, with a metric tensor of 15 components. Ten components are identified with the 4D spacetime metric, four components with the electromagnetic vector potential, and one component with an unidentified scalar field sometimes called the "radion" or the "dilaton". Correspondingly, the 5D Einstein equations yield the 4D Einstein field equations, the Maxwell equations for the electromagnetic field, and an equation for the scalar field. Kaluza also introduced the "cylinder condition" hypothesis, that no component of the five-dimensional metric depends on the fifth dimension. Without this restriction, terms are introduced that involve derivatives of the fields with respect to the fifth coordinate, and this extra degree of freedom makes the mathematics of the fully variable 5D relativity enormously complex. Standard 4D physics seems to manifest this "cylinder condition" and, along with it, simpler mathematics.
In 1926, Oskar Klein gave Kaluza's classical five-dimensional theory a quantum interpretation, to accord with the then-recent discoveries of Werner Heisenberg and Erwin Schrödinger. Klein introduced the hypothesis that the fifth dimension was curled up and microscopic, to explain the cylinder condition. Klein suggested that the geometry of the extra fifth dimension could take the form of a circle, with the radius of 10−30 cm. More precisely, the radius of the circular dimension is 23 times the Planck length, which in turn is of the order of 10−33 cm. Klein also made a contribution to the classical theory by providing a properly normalized 5D metric. Work continued on the Kaluza field theory during the 1930s by Einstein and colleagues at Princeton University.
In the 1940s, the classical theory was completed, and the full field equations including the scalar field were obtained by three independent research groups: Yves Thiry, working in France on his dissertation under André Lichnerowicz; Pascual Jordan, Günther Ludwig, and Claus Müller in Germany, with critical input from Wolfgang Pauli and Markus Fierz; and Paul Scherrer working alone in Switzerland. Jordan's work led to the scalar–tensor theory of Brans–Dicke; Carl H. Brans and Robert H. Dicke were apparently unaware of Thiry or Scherrer. The full Kaluza equations under the cylinder condition are quite complex, and most English-language reviews, as well as the English translations of Thiry, contain some errors. The curvature tensors for the complete Kaluza equations were evaluated using tensor-algebra software in 2015, verifying results of J. A. Ferrari and R. Coquereaux & G. Esposito-Farese. The 5D covariant form of the energy–momentum source terms is treated by L. L. Williams.
== Kaluza hypothesis ==
In his 1921 article, Kaluza established all the elements of the classical five-dimensional theory: the Kaluza–Klein metric, the Kaluza–Klein–Einstein field equations, the equations of motion, the stress–energy tensor, and the cylinder condition. With no free parameters, it merely extends general relativity to five dimensions. One starts by hypothesizing a form of the five-dimensional Kaluza–Klein metric
g
~
a
b
{\displaystyle {\widetilde {g}}_{ab}}
, where Latin indices span five dimensions. Let one also introduce the four-dimensional spacetime metric
g
μ
ν
{\displaystyle {g}_{\mu \nu }}
, where Greek indices span the usual four dimensions of space and time; a 4-vector
A
μ
{\displaystyle A^{\mu }}
identified with the electromagnetic vector potential; and a scalar field
ϕ
{\displaystyle \phi }
. Then decompose the 5D metric so that the 4D metric is framed by the electromagnetic vector potential, with the scalar field at the fifth diagonal. This can be visualized as
g
~
a
b
≡
[
g
μ
ν
+
ϕ
2
A
μ
A
ν
ϕ
2
A
μ
ϕ
2
A
ν
ϕ
2
]
.
{\displaystyle {\widetilde {g}}_{ab}\equiv {\begin{bmatrix}g_{\mu \nu }+\phi ^{2}A_{\mu }A_{\nu }&\phi ^{2}A_{\mu }\\\phi ^{2}A_{\nu }&\phi ^{2}\end{bmatrix}}.}
One can write more precisely
g
~
μ
ν
≡
g
μ
ν
+
ϕ
2
A
μ
A
ν
,
g
~
5
ν
≡
g
~
ν
5
≡
ϕ
2
A
ν
,
g
~
55
≡
ϕ
2
,
{\displaystyle {\widetilde {g}}_{\mu \nu }\equiv g_{\mu \nu }+\phi ^{2}A_{\mu }A_{\nu },\qquad {\widetilde {g}}_{5\nu }\equiv {\widetilde {g}}_{\nu 5}\equiv \phi ^{2}A_{\nu },\qquad {\widetilde {g}}_{55}\equiv \phi ^{2},}
where the index
5
{\displaystyle 5}
indicates the fifth coordinate by convention, even though the first four coordinates are indexed with 0, 1, 2, and 3. The associated inverse metric is
g
~
a
b
≡
[
g
μ
ν
−
A
μ
−
A
ν
g
α
β
A
α
A
β
+
1
ϕ
2
]
.
{\displaystyle {\widetilde {g}}^{ab}\equiv {\begin{bmatrix}g^{\mu \nu }&-A^{\mu }\\-A^{\nu }&g_{\alpha \beta }A^{\alpha }A^{\beta }+{\frac {1}{\phi ^{2}}}\end{bmatrix}}.}
This decomposition is quite general, and all terms are dimensionless. Kaluza then applies the machinery of standard general relativity to this metric. The field equations are obtained from five-dimensional Einstein equations, and the equations of motion from the five-dimensional geodesic hypothesis. The resulting field equations provide both the equations of general relativity and of electrodynamics; the equations of motion provide the four-dimensional geodesic equation and the Lorentz force law, and one finds that electric charge is identified with motion in the fifth dimension.
The hypothesis for the metric implies an invariant five-dimensional length element
d
s
{\displaystyle ds}
:
d
s
2
≡
g
~
a
b
d
x
a
d
x
b
=
g
μ
ν
d
x
μ
d
x
ν
+
ϕ
2
(
A
ν
d
x
ν
+
d
x
5
)
2
.
{\displaystyle ds^{2}\equiv {\widetilde {g}}_{ab}\,dx^{a}\,dx^{b}=g_{\mu \nu }\,dx^{\mu }\,dx^{\nu }+\phi ^{2}(A_{\nu }\,dx^{\nu }+dx^{5})^{2}.}
== Field equations from the Kaluza hypothesis ==
The Kaluza–Klein–Einstein field equations of the five-dimensional theory were never adequately provided by Kaluza or Klein because they ignored the scalar field. The full Kaluza field equations are generally attributed to Thiry, who obtained vacuum field equations, although Kaluza originally provided a stress–energy tensor for his theory, and Thiry included a stress–energy tensor in his thesis. But as described by Gonner, several independent groups worked on the field equations in the 1940s and earlier. Thiry is perhaps best known only because an English translation was provided by Applequist, Chodos, & Freund in their review book. Applequist et al. also provided an English translation of Kaluza's article. Translations of the three (1946, 1947, 1948) Jordan articles can be found on the ResearchGate and Academia.edu archives. The first correct English-language Kaluza field equations, including the scalar field, were provided by Williams.
To obtain the 5D Kaluza–Klein–Einstein field equations, the 5D Kaluza–Klein–Christoffel symbols
Γ
~
b
c
a
{\displaystyle {\widetilde {\Gamma }}_{bc}^{a}}
are calculated from the 5D Kaluza–Klein metric
g
~
a
b
{\displaystyle {\widetilde {g}}_{ab}}
, and the 5D Kaluza–Klein–Ricci tensor
R
~
a
b
{\displaystyle {\widetilde {R}}_{ab}}
is calculated from the 5D connections.
The classic results of Thiry and other authors presume the cylinder condition:
∂
g
~
a
b
∂
x
5
=
0.
{\displaystyle {\frac {\partial {\widetilde {g}}_{ab}}{\partial x^{5}}}=0.}
Without this assumption, the field equations become much more complex, providing many more degrees of freedom that can be identified with various new fields. Paul Wesson and colleagues have pursued relaxation of the cylinder condition to gain extra terms that can be identified with the matter fields, for which Kaluza otherwise inserted a stress–energy tensor by hand.
It has been an objection to the original Kaluza hypothesis to invoke the fifth dimension only to negate its dynamics. But Thiry argued that the interpretation of the Lorentz force law in terms of a five-dimensional geodesic militates strongly for a fifth dimension irrespective of the cylinder condition. Most authors have therefore employed the cylinder condition in deriving the field equations. Furthermore, vacuum equations are typically assumed for which
R
~
a
b
=
0
,
{\displaystyle {\widetilde {R}}_{ab}=0,}
where
R
~
a
b
≡
∂
c
Γ
~
a
b
c
−
∂
b
Γ
~
c
a
c
+
Γ
~
c
d
c
Γ
~
a
b
d
−
Γ
~
b
d
c
Γ
~
a
c
d
{\displaystyle {\widetilde {R}}_{ab}\equiv \partial _{c}{\widetilde {\Gamma }}_{ab}^{c}-\partial _{b}{\widetilde {\Gamma }}_{ca}^{c}+{\widetilde {\Gamma }}_{cd}^{c}{\widetilde {\Gamma }}_{ab}^{d}-{\widetilde {\Gamma }}_{bd}^{c}{\widetilde {\Gamma }}_{ac}^{d}}
and
Γ
~
b
c
a
≡
1
2
g
~
a
d
(
∂
b
g
~
d
c
+
∂
c
g
~
d
b
−
∂
d
g
~
b
c
)
.
{\displaystyle {\widetilde {\Gamma }}_{bc}^{a}\equiv {\frac {1}{2}}{\widetilde {g}}^{ad}(\partial _{b}{\widetilde {g}}_{dc}+\partial _{c}{\widetilde {g}}_{db}-\partial _{d}{\widetilde {g}}_{bc}).}
The vacuum field equations obtained in this way by Thiry and Jordan's group are as follows.
The field equation for
ϕ
{\displaystyle \phi }
is obtained from
R
~
55
=
0
⇒
◻
ϕ
=
1
4
ϕ
3
F
α
β
F
α
β
,
{\displaystyle {\widetilde {R}}_{55}=0\Rightarrow \Box \phi ={\frac {1}{4}}\phi ^{3}F^{\alpha \beta }F_{\alpha \beta },}
where
F
α
β
≡
∂
α
A
β
−
∂
β
A
α
,
{\displaystyle F_{\alpha \beta }\equiv \partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha },}
◻
≡
g
μ
ν
∇
μ
∇
ν
,
{\displaystyle \Box \equiv g^{\mu \nu }\nabla _{\mu }\nabla _{\nu },}
and
∇
μ
{\displaystyle \nabla _{\mu }}
is a standard, 4D covariant derivative. It shows that the electromagnetic field is a source for the scalar field. Note that the scalar field cannot be set to a constant without constraining the electromagnetic field. The earlier treatments by Kaluza and Klein did not have an adequate description of the scalar field and did not realize the implied constraint on the electromagnetic field by assuming the scalar field to be constant.
The field equation for
A
ν
{\displaystyle A^{\nu }}
is obtained from
R
~
5
α
=
0
=
1
2
ϕ
g
β
μ
∇
μ
(
ϕ
3
F
α
β
)
−
A
α
ϕ
◻
ϕ
.
{\displaystyle {\widetilde {R}}_{5\alpha }=0={\frac {1}{2\phi }}g^{\beta \mu }\nabla _{\mu }(\phi ^{3}F_{\alpha \beta })-A_{\alpha }\phi \Box \phi .}
It has the form of the vacuum Maxwell equations if the scalar field is constant.
The field equation for the 4D Ricci tensor
R
μ
ν
{\displaystyle R_{\mu \nu }}
is obtained from
R
~
μ
ν
−
1
2
g
~
μ
ν
R
~
=
0
⇒
R
μ
ν
−
1
2
g
μ
ν
R
=
1
2
ϕ
2
(
g
α
β
F
μ
α
F
ν
β
−
1
4
g
μ
ν
F
α
β
F
α
β
)
+
1
ϕ
(
∇
μ
∇
ν
ϕ
−
g
μ
ν
◻
ϕ
)
,
{\displaystyle {\begin{aligned}{\widetilde {R}}_{\mu \nu }-{\frac {1}{2}}{\widetilde {g}}_{\mu \nu }{\widetilde {R}}&=0\Rightarrow \\R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }R&={\frac {1}{2}}\phi ^{2}\left(g^{\alpha \beta }F_{\mu \alpha }F_{\nu \beta }-{\frac {1}{4}}g_{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right)+{\frac {1}{\phi }}(\nabla _{\mu }\nabla _{\nu }\phi -g_{\mu \nu }\Box \phi ),\end{aligned}}}
where
R
{\displaystyle R}
is the standard 4D Ricci scalar.
This equation shows the remarkable result, called the "Kaluza miracle", that the precise form for the electromagnetic stress–energy tensor emerges from the 5D vacuum equations as a source in the 4D equations: field from the vacuum. This relation allows the definitive identification of
A
μ
{\displaystyle A^{\mu }}
with the electromagnetic vector potential. Therefore, the field needs to be rescaled with a conversion constant
k
{\displaystyle k}
such that
A
μ
→
k
A
μ
{\displaystyle A^{\mu }\to kA^{\mu }}
.
The relation above shows that we must have
k
2
2
=
8
π
G
c
4
1
μ
0
=
2
G
c
2
4
π
ϵ
0
,
{\displaystyle {\frac {k^{2}}{2}}={\frac {8\pi G}{c^{4}}}{\frac {1}{\mu _{0}}}={\frac {2G}{c^{2}}}4\pi \epsilon _{0},}
where
G
{\displaystyle G}
is the gravitational constant, and
μ
0
{\displaystyle \mu _{0}}
is the permeability of free space. In the Kaluza theory, the gravitational constant can be understood as an electromagnetic coupling constant in the metric. There is also a stress–energy tensor for the scalar field. The scalar field behaves like a variable gravitational constant, in terms of modulating the coupling of electromagnetic stress–energy to spacetime curvature. The sign of
ϕ
2
{\displaystyle \phi ^{2}}
in the metric is fixed by correspondence with 4D theory so that electromagnetic energy densities are positive. It is often assumed that the fifth coordinate is spacelike in its signature in the metric.
In the presence of matter, the 5D vacuum condition cannot be assumed. Indeed, Kaluza did not assume it. The full field equations require evaluation of the 5D Kaluza–Klein–Einstein tensor
G
~
a
b
≡
R
~
a
b
−
1
2
g
~
a
b
R
~
,
{\displaystyle {\widetilde {G}}_{ab}\equiv {\widetilde {R}}_{ab}-{\frac {1}{2}}{\widetilde {g}}_{ab}{\widetilde {R}},}
as seen in the recovery of the electromagnetic stress–energy tensor above. The 5D curvature tensors are complex, and most English-language reviews contain errors in either
G
~
a
b
{\displaystyle {\widetilde {G}}_{ab}}
or
R
~
a
b
{\displaystyle {\widetilde {R}}_{ab}}
, as does the English translation of Thiry. In 2015, a complete set of 5D curvature tensors under the cylinder condition, evaluated using tensor-algebra software, was produced.
== Equations of motion from the Kaluza hypothesis ==
The equations of motion are obtained from the five-dimensional geodesic hypothesis in terms of a 5-velocity
U
~
a
≡
d
x
a
/
d
s
{\displaystyle {\widetilde {U}}^{a}\equiv dx^{a}/ds}
:
U
~
b
∇
~
b
U
~
a
=
d
U
~
a
d
s
+
Γ
~
b
c
a
U
~
b
U
~
c
=
0.
{\displaystyle {\widetilde {U}}^{b}{\widetilde {\nabla }}_{b}{\widetilde {U}}^{a}={\frac {d{\widetilde {U}}^{a}}{ds}}+{\widetilde {\Gamma }}_{bc}^{a}{\widetilde {U}}^{b}{\widetilde {U}}^{c}=0.}
This equation can be recast in several ways, and it has been studied in various forms by authors including Kaluza, Pauli, Gross & Perry, Gegenberg & Kunstatter, and Wesson & Ponce de Leon, but it is instructive to convert it back to the usual 4-dimensional length element
c
2
d
τ
2
≡
g
μ
ν
d
x
μ
d
x
ν
{\displaystyle c^{2}\,d\tau ^{2}\equiv g_{\mu \nu }\,dx^{\mu }\,dx^{\nu }}
, which is related to the 5-dimensional length element
d
s
{\displaystyle ds}
as given above:
d
s
2
=
c
2
d
τ
2
+
ϕ
2
(
k
A
ν
d
x
ν
+
d
x
5
)
2
.
{\displaystyle ds^{2}=c^{2}\,d\tau ^{2}+\phi ^{2}(kA_{\nu }\,dx^{\nu }+dx^{5})^{2}.}
Then the 5D geodesic equation can be written for the spacetime components of the 4-velocity:
U
ν
≡
d
x
ν
d
τ
,
{\displaystyle U^{\nu }\equiv {\frac {dx^{\nu }}{d\tau }},}
d
U
ν
d
τ
+
Γ
~
α
β
μ
U
α
U
β
+
2
Γ
~
5
α
μ
U
α
U
5
+
Γ
~
55
μ
(
U
5
)
2
+
U
μ
d
d
τ
ln
c
d
τ
d
s
=
0.
{\displaystyle {\frac {dU^{\nu }}{d\tau }}+{\widetilde {\Gamma }}_{\alpha \beta }^{\mu }U^{\alpha }U^{\beta }+2{\widetilde {\Gamma }}_{5\alpha }^{\mu }U^{\alpha }U^{5}+{\widetilde {\Gamma }}_{55}^{\mu }(U^{5})^{2}+U^{\mu }{\frac {d}{d\tau }}\ln {\frac {c\,d\tau }{ds}}=0.}
The term quadratic in
U
ν
{\displaystyle U^{\nu }}
provides the 4D geodesic equation plus some electromagnetic terms:
Γ
~
α
β
μ
=
Γ
α
β
μ
+
1
2
g
μ
ν
k
2
ϕ
2
(
A
α
F
β
ν
+
A
β
F
α
ν
−
A
α
A
β
∂
ν
ln
ϕ
2
)
.
{\displaystyle {\widetilde {\Gamma }}_{\alpha \beta }^{\mu }=\Gamma _{\alpha \beta }^{\mu }+{\frac {1}{2}}g^{\mu \nu }k^{2}\phi ^{2}(A_{\alpha }F_{\beta \nu }+A_{\beta }F_{\alpha \nu }-A_{\alpha }A_{\beta }\partial _{\nu }\ln \phi ^{2}).}
The term linear in
U
ν
{\displaystyle U^{\nu }}
provides the Lorentz force law:
Γ
~
5
α
μ
=
1
2
g
μ
ν
k
ϕ
2
(
F
α
ν
−
A
α
∂
ν
ln
ϕ
2
)
.
{\displaystyle {\widetilde {\Gamma }}_{5\alpha }^{\mu }={\frac {1}{2}}g^{\mu \nu }k\phi ^{2}(F_{\alpha \nu }-A_{\alpha }\partial _{\nu }\ln \phi ^{2}).}
This is another expression of the "Kaluza miracle". The same hypothesis for the 5D metric that provides electromagnetic stress–energy in the Einstein equations, also provides the Lorentz force law in the equation of motions along with the 4D geodesic equation. Yet correspondence with the Lorentz force law requires that we identify the component of 5-velocity along the fifth dimension with electric charge:
k
U
5
=
k
d
x
5
d
τ
→
q
m
c
,
{\displaystyle kU^{5}=k{\frac {dx^{5}}{d\tau }}\to {\frac {q}{mc}},}
where
m
{\displaystyle m}
is particle mass, and
q
{\displaystyle q}
is particle electric charge. Thus electric charge is understood as motion along the fifth dimension. The fact that the Lorentz force law could be understood as a geodesic in five dimensions was to Kaluza a primary motivation for considering the five-dimensional hypothesis, even in the presence of the aesthetically unpleasing cylinder condition.
Yet there is a problem: the term quadratic in
U
5
{\displaystyle U^{5}}
,
Γ
~
55
μ
=
−
1
2
g
μ
α
∂
α
ϕ
2
.
{\displaystyle {\widetilde {\Gamma }}_{55}^{\mu }=-{\frac {1}{2}}g^{\mu \alpha }\partial _{\alpha }\phi ^{2}.}
If there is no gradient in the scalar field, the term quadratic in
U
5
{\displaystyle U^{5}}
vanishes. But otherwise the expression above implies
U
5
∼
c
q
/
m
G
1
/
2
.
{\displaystyle U^{5}\sim c{\frac {q/m}{G^{1/2}}}.}
For elementary particles,
U
5
>
10
20
c
{\displaystyle U^{5}>10^{20}c}
. The term quadratic in
U
5
{\displaystyle U^{5}}
should dominate the equation, perhaps in contradiction to experience. This was the main shortfall of the five-dimensional theory as Kaluza saw it, and he gives it some discussion in his original article.
The equation of motion for
U
5
{\displaystyle U^{5}}
is particularly simple under the cylinder condition. Start with the alternate form of the geodesic equation, written for the covariant 5-velocity:
d
U
~
a
d
s
=
1
2
U
~
b
U
~
c
∂
g
~
b
c
∂
x
a
.
{\displaystyle {\frac {d{\widetilde {U}}_{a}}{ds}}={\frac {1}{2}}{\widetilde {U}}^{b}{\widetilde {U}}^{c}{\frac {\partial {\widetilde {g}}_{bc}}{\partial x^{a}}}.}
This means that under the cylinder condition,
U
~
5
{\displaystyle {\widetilde {U}}_{5}}
is a constant of the five-dimensional motion:
U
~
5
=
g
~
5
a
U
~
a
=
ϕ
2
c
d
τ
d
s
(
k
A
ν
U
ν
+
U
5
)
=
constant
.
{\displaystyle {\widetilde {U}}_{5}={\widetilde {g}}_{5a}{\widetilde {U}}^{a}=\phi ^{2}{\frac {c\,d\tau }{ds}}(kA_{\nu }U^{\nu }+U^{5})={\text{constant}}.}
== Kaluza's hypothesis for the matter stress–energy tensor ==
Kaluza proposed a five-dimensional matter stress tensor
T
~
M
a
b
{\displaystyle {\widetilde {T}}_{M}^{ab}}
of the form
T
~
M
a
b
=
ρ
d
x
a
d
s
d
x
b
d
s
,
{\displaystyle {\widetilde {T}}_{M}^{ab}=\rho {\frac {dx^{a}}{ds}}{\frac {dx^{b}}{ds}},}
where
ρ
{\displaystyle \rho }
is a density, and the length element
d
s
{\displaystyle ds}
is as defined above.
Then the spacetime component gives a typical "dust" stress–energy tensor:
T
~
M
μ
ν
=
ρ
d
x
μ
d
s
d
x
ν
d
s
.
{\displaystyle {\widetilde {T}}_{M}^{\mu \nu }=\rho {\frac {dx^{\mu }}{ds}}{\frac {dx^{\nu }}{ds}}.}
The mixed component provides a 4-current source for the Maxwell equations:
T
~
M
5
μ
=
ρ
d
x
μ
d
s
d
x
5
d
s
=
ρ
U
μ
q
k
m
c
.
{\displaystyle {\widetilde {T}}_{M}^{5\mu }=\rho {\frac {dx^{\mu }}{ds}}{\frac {dx^{5}}{ds}}=\rho U^{\mu }{\frac {q}{kmc}}.}
Just as the five-dimensional metric comprises the four-dimensional metric framed by the electromagnetic vector potential, the five-dimensional stress–energy tensor comprises the four-dimensional stress–energy tensor framed by the vector 4-current.
== Quantum interpretation of Klein ==
Kaluza's original hypothesis was purely classical and extended discoveries of general relativity. By the time of Klein's contribution, the discoveries of Heisenberg, Schrödinger, and Louis de Broglie were receiving a lot of attention. Klein's Nature article suggested that the fifth dimension is closed and periodic, and that the identification of electric charge with motion in the fifth dimension can be interpreted as standing waves of wavelength
λ
5
{\displaystyle \lambda ^{5}}
, much like the electrons around a nucleus in the Bohr model of the atom. The quantization of electric charge could then be nicely understood in terms of integer multiples of fifth-dimensional momentum. Combining the previous Kaluza result for
U
5
{\displaystyle U^{5}}
in terms of electric charge, and a de Broglie relation for momentum
p
5
=
h
/
λ
5
{\displaystyle p^{5}=h/\lambda ^{5}}
, Klein obtained an expression for the 0th mode of such waves:
m
U
5
=
c
q
G
1
/
2
=
h
λ
5
⇒
λ
5
∼
h
G
1
/
2
c
q
,
{\displaystyle mU^{5}={\frac {cq}{G^{1/2}}}={\frac {h}{\lambda ^{5}}}\quad \Rightarrow \quad \lambda ^{5}\sim {\frac {hG^{1/2}}{cq}},}
where
h
{\displaystyle h}
is the Planck constant. Klein found that
λ
5
∼
10
−
30
{\displaystyle \lambda ^{5}\sim 10^{-30}}
cm, and thereby an explanation for the cylinder condition in this small value.
Klein's Zeitschrift für Physik article of the same year, gave a more detailed treatment that explicitly invoked the techniques of Schrödinger and de Broglie. It recapitulated much of the classical theory of Kaluza described above, and then departed into Klein's quantum interpretation. Klein solved a Schrödinger-like wave equation using an expansion in terms of fifth-dimensional waves resonating in the closed, compact fifth dimension.
== Quantum field theory interpretation ==
== Group theory interpretation ==
In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set, and construction of this compact dimension is referred to as compactification.
In modern geometry, the extra fifth dimension can be understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U(1). In Kaluza–Klein theory this group suggests that gauge symmetry is the symmetry of circular compact dimensions. Once this geometrical interpretation is understood, it is relatively straightforward to replace U(1) by a general Lie group. Such generalizations are often called Yang–Mills theories. If a distinction is drawn, then it is that Yang–Mills theories occur on a flat spacetime, whereas Kaluza–Klein treats the more general case of curved spacetime. The base space of Kaluza–Klein theory need not be four-dimensional spacetime; it can be any (pseudo-)Riemannian manifold, or even a supersymmetric manifold or orbifold or even a noncommutative space.
The construction can be outlined, roughly, as follows. One starts by considering a principal fiber bundle P with gauge group G over a manifold M. Given a connection on the bundle, and a metric on the base manifold, and a gauge invariant metric on the tangent of each fiber, one can construct a bundle metric defined on the entire bundle. Computing the scalar curvature of this bundle metric, one finds that it is constant on each fiber: this is the "Kaluza miracle". One did not have to explicitly impose a cylinder condition, or to compactify: by assumption, the gauge group is already compact. Next, one takes this scalar curvature as the Lagrangian density, and, from this, constructs the Einstein–Hilbert action for the bundle, as a whole. The equations of motion, the Euler–Lagrange equations, can be then obtained by considering where the action is stationary with respect to variations of either the metric on the base manifold, or of the gauge connection. Variations with respect to the base metric gives the Einstein field equations on the base manifold, with the energy–momentum tensor given by the curvature (field strength) of the gauge connection. On the flip side, the action is stationary against variations of the gauge connection precisely when the gauge connection solves the Yang–Mills equations. Thus, by applying a single idea: the principle of least action, to a single quantity: the scalar curvature on the bundle (as a whole), one obtains simultaneously all of the needed field equations, for both the spacetime and the gauge field.
As an approach to the unification of the forces, it is straightforward to apply the Kaluza–Klein theory in an attempt to unify gravity with the strong and electroweak forces by using the symmetry group of the Standard Model, SU(3) × SU(2) × U(1). However, an attempt to convert this interesting geometrical construction into a bona-fide model of reality flounders on a number of issues, including the fact that the fermions must be introduced in an artificial way (in nonsupersymmetric models). Nonetheless, KK remains an important touchstone in theoretical physics and is often embedded in more sophisticated theories. It is studied in its own right as an object of geometric interest in K-theory.
Even in the absence of a completely satisfying theoretical physics framework, the idea of exploring extra, compactified, dimensions is of considerable interest in the experimental physics and astrophysics communities. A variety of predictions, with real experimental consequences, can be made (in the case of large extra dimensions and warped models). For example, on the simplest of principles, one might expect to have standing waves in the extra compactified dimension(s). If a spatial extra dimension is of radius R, the invariant mass of such standing waves would be Mn = nh/Rc with n an integer, h being the Planck constant and c the speed of light. This set of possible mass values is often called the Kaluza–Klein tower. Similarly, in Thermal quantum field theory a compactification of the euclidean time dimension leads to the Matsubara frequencies and thus to a discretized thermal energy spectrum.
However, Klein's approach to a quantum theory is flawed and, for example, leads to a calculated electron mass in the order of magnitude of the Planck mass.
Examples of experimental pursuits include work by the CDF collaboration, which has re-analyzed particle collider data for the signature of effects associated with large extra dimensions/warped models.
Robert Brandenberger and Cumrun Vafa have speculated that in the early universe, cosmic inflation causes three of the space dimensions to expand to cosmological size while the remaining dimensions of space remained microscopic.
== Space–time–matter theory ==
One particular variant of Kaluza–Klein theory is space–time–matter theory or induced matter theory, chiefly promulgated by Paul Wesson and other members of the Space–Time–Matter Consortium. In this version of the theory, it is noted that solutions to the equation
R
~
a
b
=
0
{\displaystyle {\widetilde {R}}_{ab}=0}
may be re-expressed so that in four dimensions, these solutions satisfy Einstein's equations
G
μ
ν
=
8
π
T
μ
ν
{\displaystyle G_{\mu \nu }=8\pi T_{\mu \nu }\,}
with the precise form of the Tμν following from the Ricci-flat condition on the five-dimensional space. In other words, the cylinder condition of the previous development is dropped, and the stress–energy now comes from the derivatives of the 5D metric with respect to the fifth coordinate. Because the energy–momentum tensor is normally understood to be due to concentrations of matter in four-dimensional space, the above result is interpreted as saying that four-dimensional matter is induced from geometry in five-dimensional space.
In particular, the soliton solutions of
R
~
a
b
=
0
{\displaystyle {\widetilde {R}}_{ab}=0}
can be shown to contain the Friedmann–Lemaître–Robertson–Walker metric in both radiation-dominated (early universe) and matter-dominated (later universe) forms. The general equations can be shown to be sufficiently consistent with classical tests of general relativity to be acceptable on physical principles, while still leaving considerable freedom to also provide interesting cosmological models.
== Geometric interpretation ==
The Kaluza–Klein theory has a particularly elegant presentation in terms of geometry. In a certain sense, it looks just like ordinary gravity in free space, except that it is phrased in five dimensions instead of four.
=== Einstein equations ===
The equations governing ordinary gravity in free space can be obtained from an action, by applying the variational principle to a certain action. Let M be a (pseudo-)Riemannian manifold, which may be taken as the spacetime of general relativity. If g is the metric on this manifold, one defines the action S(g) as
S
(
g
)
=
∫
M
R
(
g
)
vol
(
g
)
,
{\displaystyle S(g)=\int _{M}R(g)\operatorname {vol} (g),}
where R(g) is the scalar curvature, and vol(g) is the volume element. By applying the variational principle to the action
δ
S
(
g
)
δ
g
=
0
,
{\displaystyle {\frac {\delta S(g)}{\delta g}}=0,}
one obtains precisely the Einstein equations for free space:
R
i
j
−
1
2
g
i
j
R
=
0
,
{\displaystyle R_{ij}-{\frac {1}{2}}g_{ij}R=0,}
where Rij is the Ricci tensor.
=== Maxwell equations ===
By contrast, the Maxwell equations describing electromagnetism can be understood to be the Hodge equations of a principal U(1)-bundle or circle bundle
π
:
P
→
M
{\displaystyle \pi :P\to M}
with fiber U(1). That is, the electromagnetic field
F
{\displaystyle F}
is a harmonic 2-form in the space
Ω
2
(
M
)
{\displaystyle \Omega ^{2}(M)}
of differentiable 2-forms on the manifold
M
{\displaystyle M}
. In the absence of charges and currents, the free-field Maxwell equations are
d
F
=
0
and
d
⋆
F
=
0
,
{\displaystyle \mathrm {d} F=0\quad {\text{and}}\quad \mathrm {d} {\star }F=0,}
where
⋆
{\displaystyle \star }
is the Hodge star operator.
=== Kaluza–Klein geometry ===
To build the Kaluza–Klein theory, one picks an invariant metric on the circle
S
1
{\displaystyle S^{1}}
that is the fiber of the U(1)-bundle of electromagnetism. In this discussion, an invariant metric is simply one that is invariant under rotations of the circle. Suppose that this metric gives the circle a total length
Λ
{\displaystyle \Lambda }
. One then considers metrics
g
^
{\displaystyle {\widehat {g}}}
on the bundle
P
{\displaystyle P}
that are consistent with both the fiber metric, and the metric on the underlying manifold
M
{\displaystyle M}
. The consistency conditions are:
The projection of
g
^
{\displaystyle {\widehat {g}}}
to the vertical subspace
Vert
p
P
⊂
T
p
P
{\displaystyle \operatorname {Vert} _{p}P\subset T_{p}P}
needs to agree with metric on the fiber over a point in the manifold
M
{\displaystyle M}
.
The projection of
g
^
{\displaystyle {\widehat {g}}}
to the horizontal subspace
Hor
p
P
⊂
T
p
P
{\displaystyle \operatorname {Hor} _{p}P\subset T_{p}P}
of the tangent space at point
p
∈
P
{\displaystyle p\in P}
must be isomorphic to the metric
g
{\displaystyle g}
on
M
{\displaystyle M}
at
π
(
P
)
{\displaystyle \pi (P)}
.
The Kaluza–Klein action for such a metric is given by
S
(
g
^
)
=
∫
P
R
(
g
^
)
vol
(
g
^
)
.
{\displaystyle S({\widehat {g}})=\int _{P}R({\widehat {g}})\operatorname {vol} ({\widehat {g}}).}
The scalar curvature, written in components, then expands to
R
(
g
^
)
=
π
∗
(
R
(
g
)
−
Λ
2
2
|
F
|
2
)
,
{\displaystyle R({\widehat {g}})=\pi ^{*}\left(R(g)-{\frac {\Lambda ^{2}}{2}}|F|^{2}\right),}
where
π
∗
{\displaystyle \pi ^{*}}
is the pullback of the fiber bundle projection
π
:
P
→
M
{\displaystyle \pi :P\to M}
. The connection
A
{\displaystyle A}
on the fiber bundle is related to the electromagnetic field strength as
π
∗
F
=
d
A
.
{\displaystyle \pi ^{*}F=dA.}
That there always exists such a connection, even for fiber bundles of arbitrarily complex topology, is a result from homology and specifically, K-theory. Applying Fubini's theorem and integrating on the fiber, one gets
S
(
g
^
)
=
Λ
∫
M
(
R
(
g
)
−
1
Λ
2
|
F
|
2
)
vol
(
g
)
.
{\displaystyle S({\widehat {g}})=\Lambda \int _{M}\left(R(g)-{\frac {1}{\Lambda ^{2}}}|F|^{2}\right)\operatorname {vol} (g).}
Varying the action with respect to the component
A
{\displaystyle A}
, one regains the Maxwell equations. Applying the variational principle to the base metric
g
{\displaystyle g}
, one gets the Einstein equations
R
i
j
−
1
2
g
i
j
R
=
1
Λ
2
T
i
j
{\displaystyle R_{ij}-{\frac {1}{2}}g_{ij}R={\frac {1}{\Lambda ^{2}}}T_{ij}}
with the electromagnetic stress–energy tensor being given by
T
i
j
=
F
i
k
F
j
l
g
k
l
−
1
4
g
i
j
|
F
|
2
.
{\displaystyle T^{ij}=F^{ik}F^{jl}g_{kl}-{\frac {1}{4}}g^{ij}|F|^{2}.}
The original theory identifies
Λ
{\displaystyle \Lambda }
with the fiber metric
g
55
{\displaystyle g_{55}}
and allows
Λ
{\displaystyle \Lambda }
to vary from fiber to fiber. In this case, the coupling between gravity and the electromagnetic field is not constant, but has its own dynamical field, the radion.
=== Generalizations ===
In the above, the size of the loop
Λ
{\displaystyle \Lambda }
acts as a coupling constant between the gravitational field and the electromagnetic field. If the base manifold is four-dimensional, the Kaluza–Klein manifold P is five-dimensional. The fifth dimension is a compact space and is called the compact dimension. The technique of introducing compact dimensions to obtain a higher-dimensional manifold is referred to as compactification. Compactification does not produce group actions on chiral fermions except in very specific cases: the dimension of the total space must be 2 mod 8, and the G-index of the Dirac operator of the compact space must be nonzero.
The above development generalizes in a more-or-less straightforward fashion to general principal G-bundles for some arbitrary Lie group G taking the place of U(1). In such a case, the theory is often referred to as a Yang–Mills theory and is sometimes taken to be synonymous. If the underlying manifold is supersymmetric, the resulting theory is a super-symmetric Yang–Mills theory.
== Empirical tests ==
No experimental or observational signs of extra dimensions have been officially reported. Many theoretical search techniques for detecting Kaluza–Klein resonances have been proposed using the mass couplings of such resonances with the top quark. An analysis of results from the LHC in December 2010 severely constrains theories with large extra dimensions.
The observation of a Higgs-like boson at the LHC establishes a new empirical test which can be applied to the search for Kaluza–Klein resonances and supersymmetric particles.
The loop Feynman diagrams that exist in the Higgs interactions allow any particle with electric charge and mass to run in such a loop. Standard Model particles besides the top quark and W boson do not make big contributions to the cross-section observed in the H → γγ decay, but if there are new particles beyond the Standard Model, they could potentially change the ratio of the predicted Standard Model H → γγ cross-section to the experimentally observed cross-section. Hence a measurement of any dramatic change to the H → γγ cross-section predicted by the Standard Model is crucial in probing the physics beyond it.
An article from July 2018 gives some hope for this theory; in the article they dispute that gravity is leaking into higher dimensions as in brane theory. However, the article does demonstrate that electromagnetism and gravity share the same number of dimensions, and this fact lends support to Kaluza–Klein theory; whether the number of dimensions is really 3 + 1 or in fact 4 + 1 is the subject of further debate.
== See also ==
== Notes ==
== References ==
Kaluza, Theodor (1921). "Zum Unitätsproblem in der Physik". Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.): 966–972. Bibcode:1921SPAW.......966K. https://archive.org/details/sitzungsberichte1921preussi
Klein, Oskar (1926). "Quantentheorie und fünfdimensionale Relativitätstheorie". Zeitschrift für Physik A. 37 (12): 895–906. Bibcode:1926ZPhy...37..895K. doi:10.1007/BF01397481.
Witten, Edward (1981). "Search for a realistic Kaluza–Klein theory". Nuclear Physics B. 186 (3): 412–428. Bibcode:1981NuPhB.186..412W. doi:10.1016/0550-3213(81)90021-3.
Appelquist, Thomas; Chodos, Alan; Freund, Peter G. O. (1987). Modern Kaluza–Klein Theories. Menlo Park, Cal.: Addison–Wesley. ISBN 978-0-201-09829-7. (Includes reprints of the above articles as well as those of other important papers relating to Kaluza–Klein theory.)
Duff, M. J. (1994). "Kaluza–Klein Theory in Perspective". In Lindström, Ulf (ed.). Proceedings of the Symposium 'The Oskar Klein Centenary'. Singapore: World Scientific. pp. 22–35. ISBN 978-981-02-2332-8.
Overduin, J. M.; Wesson, P. S. (1997). "Kaluza–Klein Gravity". Physics Reports. 283 (5): 303–378. arXiv:gr-qc/9805018. Bibcode:1997PhR...283..303O. doi:10.1016/S0370-1573(96)00046-4. S2CID 119087814.
Wesson, Paul S. (2006). Five-Dimensional Physics: Classical and Quantum Consequences of Kaluza–Klein Cosmology. Singapore: World Scientific. Bibcode:2006fdpc.book.....W. ISBN 978-981-256-661-4.
== Further reading ==
The CDF Collaboration, Search for Extra Dimensions using Missing Energy at CDF, (2004) (A simplified presentation of the search made for extra dimensions at the Collider Detector at Fermilab (CDF) particle physics facility.)
John M. Pierre, SUPERSTRINGS! Extra Dimensions, (2003).
Chris Pope, Lectures on Kaluza–Klein Theory.
Edward Witten (2014). "A Note On Einstein, Bergmann, and the Fifth Dimension", arXiv:1401.8048 | Wikipedia/Kaluza–Klein_theory |
In theoretical physics, scalar electrodynamics is a theory of a U(1) gauge field coupled to a charged spin 0 scalar field that takes the place of the Dirac fermions in "ordinary" quantum electrodynamics. The scalar field is charged, and with an appropriate potential, it has the capacity to break the gauge symmetry via the Abelian Higgs mechanism.
== Matter content and Lagrangian ==
=== Matter content ===
The model consists of a complex scalar field
ϕ
(
x
)
{\displaystyle \phi (x)}
minimally coupled to a gauge field
A
μ
(
x
)
{\displaystyle A_{\mu }(x)}
.
This article discusses the theory on flat spacetime
R
1
,
3
{\displaystyle \mathbb {R} ^{1,3}}
(Minkowski space) so these fields can be treated (naïvely) as functions
ϕ
:
R
1
,
3
→
C
{\displaystyle \phi :\mathbb {R} ^{1,3}\rightarrow \mathbb {C} }
, and
A
μ
:
R
1
,
3
→
(
R
1
,
3
)
∗
{\displaystyle A_{\mu }:\mathbb {R} ^{1,3}\rightarrow (\mathbb {R} ^{1,3})^{*}}
. The theory can also be defined for curved spacetime but these definitions must be replaced with a more subtle one. The gauge field is also known as a principal connection, specifically a principal
U
(
1
)
{\displaystyle {\text{U}}(1)}
connection.
=== Lagrangian ===
The dynamics is given by the Lagrangian density
L
=
(
D
μ
ϕ
)
∗
D
μ
ϕ
−
V
(
ϕ
∗
ϕ
)
−
1
4
F
μ
ν
F
μ
ν
=
(
∂
μ
ϕ
)
∗
(
∂
μ
ϕ
)
−
i
e
(
(
∂
μ
ϕ
)
∗
ϕ
−
ϕ
∗
(
∂
μ
ϕ
)
)
A
μ
+
e
2
A
μ
A
μ
ϕ
∗
ϕ
−
V
(
ϕ
∗
ϕ
)
−
1
4
F
μ
ν
F
μ
ν
{\displaystyle {\begin{array}{lcl}{\mathcal {L}}&=&(D_{\mu }\phi )^{*}D^{\mu }\phi -V(\phi ^{*}\phi )-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\\&=&(\partial _{\mu }\phi )^{*}(\partial ^{\mu }\phi )-ie((\partial _{\mu }\phi )^{*}\phi -\phi ^{*}(\partial _{\mu }\phi ))A^{\mu }+e^{2}A_{\mu }A^{\mu }\phi ^{*}\phi -V(\phi ^{*}\phi )-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\end{array}}}
where
F
μ
ν
=
(
∂
μ
A
ν
−
∂
ν
A
μ
)
{\displaystyle F_{\mu \nu }=(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu })}
is the electromagnetic field strength, or curvature of the connection.
D
μ
ϕ
=
(
∂
μ
ϕ
−
i
e
A
μ
ϕ
)
{\displaystyle D_{\mu }\phi =(\partial _{\mu }\phi -ieA_{\mu }\phi )}
is the covariant derivative of the field
ϕ
{\displaystyle \phi }
e
=
−
|
e
|
<
0
{\displaystyle e=-|e|<0}
is the electric charge
V
(
ϕ
∗
ϕ
)
{\displaystyle V(\phi ^{*}\phi )}
is the potential for the complex scalar field.
=== Gauge-invariance ===
This model is invariant under gauge transformations parameterized by
λ
(
x
)
{\displaystyle \lambda (x)}
. This is a real-valued function
λ
:
R
1
,
3
→
R
.
{\displaystyle \lambda :\mathbb {R} ^{1,3}\rightarrow \mathbb {R} .}
ϕ
′
(
x
)
=
e
i
e
λ
(
x
)
ϕ
(
x
)
and
A
μ
′
(
x
)
=
A
μ
(
x
)
+
∂
μ
λ
(
x
)
.
{\displaystyle \phi '(x)=e^{ie\lambda (x)}\phi (x)\quad {\textrm {and}}\quad A_{\mu }'(x)=A_{\mu }(x)+\partial _{\mu }\lambda (x).}
==== Differential-geometric view ====
From the geometric viewpoint,
λ
{\displaystyle \lambda }
is an infinitesimal change of trivialization, which generates the finite change of trivialization
e
i
e
λ
:
R
1
,
3
→
U
(
1
)
.
{\displaystyle e^{ie\lambda }:\mathbb {R} ^{1,3}\rightarrow {\text{U}}(1).}
In physics, it is customary to work under an implicit choice of trivialization, hence a gauge transformation really can be viewed as a change of trivialization.
== Higgs mechanism ==
If the potential is such that its minimum occurs at non-zero value of
|
ϕ
|
{\displaystyle |\phi |}
, this model exhibits the Higgs mechanism. This can be seen by studying fluctuations about the lowest energy configuration: one sees that the gauge field behaves as a massive field with its mass proportional to
e
{\displaystyle e}
times the minimum value of
|
ϕ
|
{\displaystyle |\phi |}
. As shown in 1973 by Nielsen and Olesen, this model, in
2
+
1
{\displaystyle 2+1}
dimensions, admits time-independent finite energy configurations corresponding to vortices carrying magnetic flux. The magnetic flux carried by these vortices are quantized (in units of
2
π
e
{\displaystyle {\tfrac {2\pi }{e}}}
) and appears as a topological charge associated with the topological current
J
t
o
p
μ
=
ϵ
μ
ν
ρ
F
ν
ρ
.
{\displaystyle J_{top}^{\mu }=\epsilon ^{\mu \nu \rho }F_{\nu \rho }\ .}
These vortices are similar to the vortices appearing in type-II superconductors. This analogy was used by Nielsen and Olesen in obtaining their solutions.
=== Example ===
A simple choice of potential for demonstrating the Higgs mechanism is
V
(
|
ϕ
|
2
)
=
λ
(
|
ϕ
|
2
−
Φ
2
)
2
.
{\displaystyle V(|\phi |^{2})=\lambda (|\phi |^{2}-\Phi ^{2})^{2}.}
The potential is minimized at
|
ϕ
|
=
Φ
{\displaystyle |\phi |=\Phi }
, which is chosen to be greater than zero. This produces a circle of minima, with values
Φ
e
i
θ
{\displaystyle \Phi e^{i\theta }}
, for
θ
{\displaystyle \theta }
a real number.
== Scalar chromodynamics ==
This theory can be generalized from a theory with
U
(
1
)
{\displaystyle U(1)}
gauge symmetry containing a scalar field
ϕ
{\displaystyle \phi }
valued in
C
{\displaystyle \mathbb {C} }
coupled to a gauge field
A
μ
{\displaystyle A_{\mu }}
to a theory with gauge symmetry under the gauge group
G
{\displaystyle G}
, a Lie group.
The scalar field
ϕ
{\displaystyle \phi }
is valued in a representation space of the gauge group
G
{\displaystyle G}
, making it a vector; the label of "scalar" field refers only to the transformation of
ϕ
{\displaystyle \phi }
under the action of the Lorentz group, so it is still referred to as a scalar field, in the sense of a Lorentz scalar. The gauge-field is a
g
{\displaystyle {\mathfrak {g}}}
-valued 1-form, where
g
{\displaystyle {\mathfrak {g}}}
is the Lie algebra of G.
== References ==
H. B. Nielsen and P. Olesen (1973). "Vortex-line models for dual strings". Nuclear Physics B. 61: 45–61. Bibcode:1973NuPhB..61...45N. doi:10.1016/0550-3213(73)90350-7.
Peskin, M and Schroeder, D.; An Introduction to Quantum Field Theory (Westview Press, 1995) ISBN 0-201-50397-2 | Wikipedia/Scalar_electrodynamics |
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. The equation is validated by its rigorous accounting of the observed fine structure of the hydrogen spectrum and has become vital in the building of the Standard Model.
The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation, which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation.
In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-1/2 particles.
Dirac did not fully appreciate the importance of his results; however, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. This accomplishment has been described as fully on par with the works of Newton, Maxwell, and Einstein before him. The equation has been deemed by some physicists to be the "real seed of modern physics". The equation has also been described as the "centerpiece of relativistic quantum mechanics", with it also stated that "the equation is perhaps the most important one in all of quantum mechanics".
The Dirac equation is inscribed upon a plaque on the floor of Westminster Abbey. Unveiled on 13 November 1995, the plaque commemorates Dirac's life.
The equation, in its natural units formulation, is also prominently displayed in the auditorium at the ‘Paul A.M. Dirac’ Lecture Hall at the Patrick M.S. Blackett Institute (formerly The San Domenico Monastery) of the Ettore Majorana Foundation and Centre for Scientific Culture in Erice, Sicily.
== History ==
The Dirac equation in the form originally proposed by Dirac is:: 291
(
β
m
c
2
+
c
∑
n
=
1
3
α
n
p
n
)
ψ
(
x
,
t
)
=
i
ℏ
∂
ψ
(
x
,
t
)
∂
t
{\displaystyle \left(\beta mc^{2}+c\sum _{n=1}^{3}\alpha _{n}p_{n}\right)\psi (x,t)=i\hbar {\frac {\partial \psi (x,t)}{\partial t}}}
where ψ(x, t) is the wave function for an electron of rest mass m with spacetime coordinates x, t. p1, p2, p3 are the components of the momentum, understood to be the momentum operator in the Schrödinger equation. c is the speed of light, and ħ is the reduced Planck constant; these fundamental physical constants reflect special relativity and quantum mechanics, respectively. αn and β are 4 × 4 gamma matrices.
Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, thus allowing the atom to be treated in a manner consistent with relativity. He hoped that the corrections introduced this way might have a bearing on the problem of atomic spectra.
Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity—which were based on discretizing the angular momentum stored in the electron's possibly non-circular orbit of the atomic nucleus—had failed, and the new quantum mechanics of Heisenberg, Pauli, Jordan, Schrödinger, and Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter and introduced new mathematical classes of objects that are now essential elements of fundamental physics.
The new elements in this equation are the four 4 × 4 matrices α1, α2, α3 and β, and the four-component wave function ψ. There are four components in ψ because the evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron.
The 4 × 4 matrices αk and β are all Hermitian and are involutory:
α
i
2
=
β
2
=
I
4
{\displaystyle \alpha _{i}^{2}=\beta ^{2}=I_{4}}
and they all mutually anti-commute:
α
i
α
j
+
α
j
α
i
=
0
(
i
≠
j
)
α
i
β
+
β
α
i
=
0
{\displaystyle {\begin{aligned}\alpha _{i}\alpha _{j}+\alpha _{j}\alpha _{i}&=0\quad (i\neq j)\\\alpha _{i}\beta +\beta \alpha _{i}&=0\end{aligned}}}
These matrices and the form of the wave function have a deep mathematical significance. The algebraic structure represented by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th-century work of German mathematician Hermann Grassmann in his Lineare Ausdehnungslehre (Theory of Linear Expansion).
=== Making the Schrödinger equation relativistic ===
The Dirac equation is superficially similar to the Schrödinger equation for a massive free particle:
−
ℏ
2
2
m
∇
2
ϕ
=
i
ℏ
∂
∂
t
ϕ
.
{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\phi =i\hbar {\frac {\partial }{\partial t}}\phi ~.}
The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically as they do in the Maxwell equations that govern the behavior of light – the equations must be differentially of the same order in space and time. In relativity, the momentum and the energies are the space and time parts of a spacetime vector, the four-momentum, and they are related by the relativistically invariant relation
E
2
=
m
2
c
4
+
p
2
c
2
,
{\displaystyle E^{2}=m^{2}c^{4}+p^{2}c^{2},}
which says that the length of this four-vector is proportional to the rest mass m. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory produces the Klein–Gordon equation describing the propagation of waves, constructed from relativistically invariant objects,
(
−
1
c
2
∂
2
∂
t
2
+
∇
2
)
ϕ
=
m
2
c
2
ℏ
2
ϕ
,
{\displaystyle \left(-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}+\nabla ^{2}\right)\phi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\phi ,}
with the wave function
ϕ
{\displaystyle \phi }
being a relativistic scalar: a complex number that has the same numerical value in all frames of reference. Space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time-derivative in order to solve definite problems. Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression
ρ
=
ϕ
∗
ϕ
{\displaystyle \rho =\phi ^{*}\phi }
and this density is convected according to the probability current vector
J
=
−
i
ℏ
2
m
(
ϕ
∗
∇
ϕ
−
ϕ
∇
ϕ
∗
)
{\displaystyle J=-{\frac {i\hbar }{2m}}(\phi ^{*}\nabla \phi -\phi \nabla \phi ^{*})}
with the conservation of probability current and density following from the continuity equation:
∇
⋅
J
+
∂
ρ
∂
t
=
0
.
{\displaystyle \nabla \cdot J+{\frac {\partial \rho }{\partial t}}=0~.}
The fact that the density is positive definite and convected according to this continuity equation implies that one may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. To maintain the notion of a convected density, one must generalize the Schrödinger expression of the density and current so that space and time derivatives again enter symmetrically in relation to the scalar wave function. The Schrödinger expression can be kept for the current, but the probability density must be replaced by the symmetrically formed expression
ρ
=
i
ℏ
2
m
c
2
(
ψ
∗
∂
t
ψ
−
ψ
∂
t
ψ
∗
)
,
{\displaystyle \rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}\partial _{t}\psi -\psi \partial _{t}\psi ^{*}\right),}
which now becomes the 4th component of a spacetime vector, and the entire probability 4-current density has the relativistically covariant expression
J
μ
=
i
ℏ
2
m
(
ψ
∗
∂
μ
ψ
−
ψ
∂
μ
ψ
∗
)
.
{\displaystyle J^{\mu }={\frac {i\hbar }{2m}}\left(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*}\right).}
The continuity equation is as before. Everything is compatible with relativity now, but the expression for the density is no longer positive definite; the initial values of both ψ and ∂tψ may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus, one cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.
Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein–Gordon equation, and describes a spinless particle field (e.g. pi meson or Higgs boson). Historically, Schrödinger himself arrived at this equation before the one that bears his name but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the charge density, which can be positive or negative, and not the probability density.
=== Dirac's coup ===
Dirac thus thought to try an equation that was first order in both space and time. He postulated an equation of the form
E
ψ
=
(
α
→
⋅
p
→
+
β
m
)
ψ
{\displaystyle E\psi =({\vec {\alpha }}\cdot {\vec {p}}+\beta m)\psi }
where the operators
(
α
→
,
β
)
{\displaystyle ({\vec {\alpha }},\beta )}
must be independent of
(
p
→
,
t
)
{\displaystyle ({\vec {p}},t)}
for linearity and independent of
(
x
→
,
t
)
{\displaystyle ({\vec {x}},t)}
for space-time homogeneity. These constraints implied additional dynamical variables that the
(
α
→
,
β
)
{\displaystyle ({\vec {\alpha }},\beta )}
operators will depend upon; from this requirement Dirac concluded that the operators would depend upon 4 × 4 matrices, related to the Pauli matrices.: 205
One could, for example, formally (i.e. by abuse of notation, since it is not straightforward to take a functional square root of the sum of two differential operators) take the relativistic expression for the energy
E
=
c
p
2
+
m
2
c
2
,
{\displaystyle E=c{\sqrt {p^{2}+m^{2}c^{2}}}~,}
replace p by its operator equivalent, expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible.
As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator (see also half derivative) thus:
∇
2
−
1
c
2
∂
2
∂
t
2
=
(
A
∂
x
+
B
∂
y
+
C
∂
z
+
i
c
D
∂
t
)
(
A
∂
x
+
B
∂
y
+
C
∂
z
+
i
c
D
∂
t
)
.
{\displaystyle \nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}=\left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)\left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)~.}
On multiplying out the right side it is apparent that, in order to get all the cross-terms such as ∂x∂y to vanish, one must assume
A
B
+
B
A
=
0
,
…
{\displaystyle AB+BA=0,~\ldots ~}
with
A
2
=
B
2
=
⋯
=
1
.
{\displaystyle A^{2}=B^{2}=\dots =1~.}
Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's matrix mechanics, immediately understood that these conditions could be met if A, B, C and D are matrices, with the implication that the wave function has multiple components. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of spin, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least 4 × 4 matrices to set up a system with the properties required – so the wave function had four components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here.
Given the factorization in terms of these matrices, one can now write down immediately an equation
(
A
∂
x
+
B
∂
y
+
C
∂
z
+
i
c
D
∂
t
)
ψ
=
κ
ψ
{\displaystyle \left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)\psi =\kappa \psi }
with
κ
{\displaystyle \kappa }
to be determined. Applying again the matrix operator on both sides yields
(
∇
2
−
1
c
2
∂
t
2
)
ψ
=
κ
2
ψ
.
{\displaystyle \left(\nabla ^{2}-{\frac {1}{c^{2}}}\partial _{t}^{2}\right)\psi =\kappa ^{2}\psi ~.}
Taking
κ
=
m
c
ℏ
{\displaystyle \kappa ={\tfrac {mc}{\hbar }}}
shows that all the components of the wave function individually satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is
(
A
∂
x
+
B
∂
y
+
C
∂
z
+
i
c
D
∂
t
−
m
c
ℏ
)
ψ
=
0
.
{\displaystyle \left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}-{\frac {mc}{\hbar }}\right)\psi =0~.}
Setting
A
=
i
β
α
1
,
B
=
i
β
α
2
,
C
=
i
β
α
3
,
D
=
β
,
{\displaystyle A=i\beta \alpha _{1}\,,\,B=i\beta \alpha _{2}\,,\,C=i\beta \alpha _{3}\,,\,D=\beta ~,}
and because
D
2
=
β
2
=
I
4
{\displaystyle D^{2}=\beta ^{2}=I_{4}}
, the Dirac equation is produced as written above.
=== Covariant form and relativistic invariance ===
To demonstrate the relativistic invariance of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows:
D
=
γ
0
,
A
=
i
γ
1
,
B
=
i
γ
2
,
C
=
i
γ
3
,
{\displaystyle {\begin{aligned}D&=\gamma ^{0},\\A&=i\gamma ^{1},\quad B=i\gamma ^{2},\quad C=i\gamma ^{3},\end{aligned}}}
and the equation takes the form (remembering the definition of the covariant components of the 4-gradient and especially that ∂0 = 1/c∂t)
where there is an implied summation over the values of the twice-repeated index μ = 0, 1, 2, 3, and ∂μ is the 4-gradient. In practice one often writes the gamma matrices in terms of 2 × 2 sub-matrices taken from the Pauli matrices and the 2 × 2 identity matrix. Explicitly the standard representation is
γ
0
=
(
I
2
0
0
−
I
2
)
,
γ
1
=
(
0
σ
x
−
σ
x
0
)
,
γ
2
=
(
0
σ
y
−
σ
y
0
)
,
γ
3
=
(
0
σ
z
−
σ
z
0
)
.
{\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\quad \gamma ^{1}={\begin{pmatrix}0&\sigma _{x}\\-\sigma _{x}&0\end{pmatrix}},\quad \gamma ^{2}={\begin{pmatrix}0&\sigma _{y}\\-\sigma _{y}&0\end{pmatrix}},\quad \gamma ^{3}={\begin{pmatrix}0&\sigma _{z}\\-\sigma _{z}&0\end{pmatrix}}.}
The complete system is summarized using the Minkowski metric on spacetime in the form
{
γ
μ
,
γ
ν
}
=
2
η
μ
ν
I
4
{\displaystyle \left\{\gamma ^{\mu },\gamma ^{\nu }\right\}=2\eta ^{\mu \nu }I_{4}}
where the bracket expression
{
a
,
b
}
=
a
b
+
b
a
{\displaystyle \{a,b\}=ab+ba}
denotes the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-dimensional space with metric signature (+ − − −). The specific Clifford algebra employed in the Dirac equation is known today as the Dirac algebra. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this geometric algebra represents an enormous stride forward in the development of quantum theory.
The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light:
P
o
p
ψ
=
m
c
ψ
.
{\displaystyle \operatorname {P} _{\mathsf {op}}\psi =mc\psi .}
Using
∂
/
=
d
e
f
γ
μ
∂
μ
{\displaystyle {\partial \!\!\!/}\mathrel {\stackrel {\mathrm {def} }{=}} \gamma ^{\mu }\partial _{\mu }}
(
∂
/
{\displaystyle {\partial \!\!\!{\big /}}}
is pronounced "d-slash"), according to Feynman slash notation, the Dirac equation becomes:
i
ℏ
∂
/
ψ
−
m
c
ψ
=
0.
{\displaystyle i\hbar {\partial \!\!\!{\big /}}\psi -mc\psi =0.}
In practice, physicists often use units of measure such that ħ = c = 1, known as natural units. The equation then takes the simple form
A foundational theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transform:
γ
μ
′
=
S
−
1
γ
μ
S
.
{\displaystyle \gamma ^{\mu \prime }=S^{-1}\gamma ^{\mu }S~.}
If in addition the matrices are all unitary, as are the Dirac set, then S itself is unitary;
γ
μ
′
=
U
†
γ
μ
U
.
{\displaystyle \gamma ^{\mu \prime }=U^{\dagger }\gamma ^{\mu }U~.}
The transformation U is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator γμ∂μ to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the previously mentioned foundational theorem, one may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form
(
i
U
†
γ
μ
U
∂
μ
′
−
m
)
ψ
(
x
′
,
t
′
)
=
0
U
†
(
i
γ
μ
∂
μ
′
−
m
)
U
ψ
(
x
′
,
t
′
)
=
0
.
{\displaystyle {\begin{aligned}\left(iU^{\dagger }\gamma ^{\mu }U\partial _{\mu }^{\prime }-m\right)\psi \left(x^{\prime },t^{\prime }\right)&=0\\U^{\dagger }(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m)U\psi \left(x^{\prime },t^{\prime }\right)&=0~.\end{aligned}}}
If the transformed spinor is defined as
ψ
′
=
U
ψ
{\displaystyle \psi ^{\prime }=U\psi }
then the transformed Dirac equation is produced in a way that demonstrates manifest relativistic invariance:
(
i
γ
μ
∂
μ
′
−
m
)
ψ
′
(
x
′
,
t
′
)
=
0
.
{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m\right)\psi ^{\prime }\left(x^{\prime },t^{\prime }\right)=0~.}
Thus, settling on any unitary representation of the gammas is final, provided the spinor is transformed according to the unitary transformation that corresponds to the given Lorentz transformation.
The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function. The representation shown here is known as the standard representation – in it, the wave function's upper two components go over into Pauli's 2 spinor wave function in the limit of low energies and small velocities in comparison to light.
The considerations above reveal the origin of the gammas in geometry, hearkening back to Grassmann's original motivation; they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as γμγν represent oriented surface elements, and so on. With this in mind, one can find the form of the unit volume element on spacetime in terms of the gammas as follows. By definition, it is
V
=
1
4
!
ϵ
μ
ν
α
β
γ
μ
γ
ν
γ
α
γ
β
.
{\displaystyle V={\frac {1}{4!}}\epsilon _{\mu \nu \alpha \beta }\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\alpha }\gamma ^{\beta }.}
For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of √g, where g is the determinant of the metric tensor. Since this is negative, that factor is imaginary. Thus
V
=
i
γ
0
γ
1
γ
2
γ
3
.
{\displaystyle V=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}.}
This matrix is given the special symbol γ5, owing to its importance when one is considering improper transformations of space-time, that is, those that change the orientation of the basis vectors. In the standard representation, it is
γ
5
=
(
0
I
2
I
2
0
)
.
{\displaystyle \gamma _{5}={\begin{pmatrix}0&I_{2}\\I_{2}&0\end{pmatrix}}.}
This matrix will also be found to anticommute with the other four Dirac matrices:
γ
5
γ
μ
+
γ
μ
γ
5
=
0
{\displaystyle \gamma ^{5}\gamma ^{\mu }+\gamma ^{\mu }\gamma ^{5}=0}
It takes a leading role when questions of parity arise because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime.
== Comparison with related theories ==
=== Pauli theory ===
The necessity of introducing half-integer spin goes back experimentally to the results of the Stern–Gerlach experiment. A beam of atoms is run through a strong inhomogeneous magnetic field, which then splits into N parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two; the ground state therefore could not be integer, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into three parts, corresponding to atoms with Lz = −1, 0, +1. The conclusion is that silver atoms have net intrinsic angular momentum of 1/2. Pauli set up a theory that explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so in SI units: (Note that bold faced characters imply Euclidean vectors in 3 dimensions, whereas the Minkowski four-vector Aμ can be defined as
A
μ
=
(
ϕ
/
c
,
−
A
)
{\displaystyle A_{\mu }=\left(\phi /c,-\mathbf {A} \right)}
.)
H
=
1
2
m
(
σ
⋅
(
p
−
e
A
)
)
2
+
e
ϕ
.
{\displaystyle H={\frac {1}{\ 2\ m\ }}\ {\Bigl (}{\boldsymbol {\sigma }}\cdot {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}{\Bigr )}^{2}+e\ \phi .}
Here A and
ϕ
{\displaystyle \phi }
represent the components of the electromagnetic four-potential in their standard SI units, and the three sigmas are the Pauli matrices. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field in SI units:
H
=
1
2
m
(
p
−
e
A
)
2
+
e
ϕ
−
e
ℏ
2
m
σ
⋅
B
.
{\displaystyle H={\frac {1}{\ 2\ m\ }}\ {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}^{2}+e\ \phi -{\frac {e\ \hbar }{\ 2\ m\ }}\ {\boldsymbol {\sigma }}\cdot \mathbf {B} ~.}
This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form:
(
γ
μ
(
i
ℏ
∂
μ
−
e
A
μ
)
−
m
c
)
ψ
=
0
.
{\displaystyle {\Bigl (}\gamma ^{\mu }\ {\bigl (}i\ \hbar \ \partial _{\mu }-e\ A_{\mu }{\bigr )}-m\ c{\Bigr )}\ \psi =0~.}
A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by i, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. There is more however. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the SI units restored:
(
m
c
2
−
E
+
e
ϕ
+
c
σ
⋅
(
p
−
e
A
)
−
c
σ
⋅
(
p
−
e
A
)
m
c
2
+
E
−
e
ϕ
)
(
ψ
+
ψ
−
)
=
(
0
0
)
.
{\displaystyle {\begin{pmatrix}mc^{2}-E+e\phi \quad &+c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)\\-c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)&mc^{2}+E-e\phi \end{pmatrix}}{\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}~.}
so
(
E
−
e
ϕ
)
ψ
+
−
c
σ
⋅
(
p
−
e
A
)
ψ
−
=
m
c
2
ψ
+
c
σ
⋅
(
p
−
e
A
)
ψ
+
−
(
E
−
e
ϕ
)
ψ
−
=
m
c
2
ψ
−
.
{\displaystyle {\begin{aligned}(E-e\phi )\ \psi _{+}-c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)\ \psi _{-}&=mc^{2}\ \psi _{+}\\c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)\ \psi _{+}-\left(E-e\phi \right)\ \psi _{-}&=mc^{2}\ \psi _{-}\end{aligned}}~.}
Assuming the field is weak and the motion of the electron non-relativistic, the total energy of the electron is approximately equal to its rest energy, and the momentum going over to the classical value,
E
−
e
ϕ
≈
m
c
2
p
≈
m
v
{\displaystyle {\begin{aligned}E-e\phi &\approx mc^{2}\\\mathbf {p} &\approx m\mathbf {v} \end{aligned}}}
and so the second equation may be written
ψ
−
≈
1
2
m
c
σ
⋅
(
p
−
e
A
)
ψ
+
,
{\displaystyle \psi _{-}\approx {\frac {1}{\ 2\ mc\ }}\ {\boldsymbol {\sigma }}\cdot {\Bigl (}\mathbf {p} -e\ \mathbf {A} {\Bigr )}\ \psi _{+},}
which is of order
v
c
.
{\displaystyle \ {\tfrac {\ v\ }{c}}~.}
Thus, at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement
(
E
−
m
c
2
)
ψ
+
=
1
2
m
[
σ
⋅
(
p
−
e
A
)
]
2
ψ
+
+
e
ϕ
ψ
+
{\displaystyle {\bigl (}E-mc^{2}{\bigr )}\ \psi _{+}={\frac {1}{\ 2m\ }}\ {\Bigl [}{\boldsymbol {\sigma }}\cdot {\bigl (}\mathbf {p} -e\mathbf {A} {\bigr )}{\Bigr ]}^{2}\ \psi _{+}+e\ \phi \ \psi _{+}\ }
The operator on the left represents the particle's total energy reduced by its rest energy, which is just its classical kinetic energy, so one can recover Pauli's theory upon identifying his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus, the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious i that appears in it, and the necessity of a complex wave function, back to the geometry of spacetime through the Dirac algebra. It also highlights why the Schrödinger equation, although ostensibly in the form of a diffusion equation, actually represents wave propagation.
It should be strongly emphasized that the entire Dirac spinor represents an irreducible whole. The separation, done here, of the Dirac spinor into large and small components depends on the low-energy approximation being valid. The components that were neglected above, to show that the Pauli theory can be recovered by a low-velocity approximation of Dirac's equation, are necessary to produce new phenomena observed in the relativistic regime – among them antimatter, and the creation and annihilation of particles.
=== Weyl theory ===
In the massless case
m
=
0
{\displaystyle m=0}
, the Dirac equation reduces to the Weyl equation, which describes relativistic massless spin-1/2 particles.
The theory acquires a second
U
(
1
)
{\displaystyle {\text{U}}(1)}
symmetry: see below.
== Physical interpretation ==
=== Identification of observables ===
The critical physical question in a quantum theory is this: what are the physically observable quantities defined by the theory? According to the postulates of quantum mechanics, such quantities are defined by self-adjoint operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. To maintain this interpretation on passing to the Dirac theory, the Hamiltonian must be taken to be
H
=
γ
0
[
m
c
2
+
c
γ
k
(
p
k
−
q
A
k
)
]
+
c
q
A
0
.
{\displaystyle H=\gamma ^{0}\left[mc^{2}+c\gamma ^{k}\left(p_{k}-qA_{k}\right)\right]+cqA^{0}.}
where, as always, there is an implied summation over the twice-repeated index k = 1, 2, 3. This looks promising, because one can see by inspection the rest energy of the particle and, in the case of A = 0, the energy of a charge placed in an electric potential cqA0. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is
H
=
c
(
p
−
q
A
)
2
+
m
2
c
2
+
q
A
0
.
{\displaystyle H=c{\sqrt {\left(\mathbf {p} -q\mathbf {A} \right)^{2}+m^{2}c^{2}}}+qA^{0}.}
Thus, the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and one must take great care to correctly identify what is observable in this theory. Much of the apparently paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables.
=== Hole theory ===
The negative E solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, they cannot simply be ignored, for once the interaction between the electron and the electromagnetic field is included, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy. Real electrons obviously do not behave in this way, or they would disappear by emitting energy in the form of photons.
To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the Dirac sea. Since the Pauli exclusion principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates.
Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle. The hole possesses a positive energy because energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but Hermann Weyl pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the positron, experimentally discovered by Carl Anderson in 1932.
It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons have to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "jellium" background so that the net electric charge density of the vacuum is zero. In quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it.
In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, and although it too is referred to as an "electron hole", it is distinct from a positron. The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.
=== In quantum field theory ===
In quantum field theories such as quantum electrodynamics, the Dirac field is subject to a process of second quantization, which resolves some of the paradoxical features of the equation.
== Mathematical formulation ==
In its modern formulation for field theory, the Dirac equation is written in terms of a Dirac spinor field
ψ
{\displaystyle \psi }
taking values in a complex vector space described concretely as
C
4
{\displaystyle \mathbb {C} ^{4}}
, defined on flat spacetime (Minkowski space)
R
1
,
3
{\displaystyle \mathbb {R} ^{1,3}}
. Its expression also contains gamma matrices and a parameter
m
>
0
{\displaystyle m>0}
interpreted as the mass, as well as other physical constants. Dirac first obtained his equation through a factorization of Einstein's energy-momentum-mass equivalence relation assuming a scalar product of momentum vectors determined by the metric tensor and quantized the resulting relation by associating momenta to their respective operators.
In terms of a field
ψ
:
R
1
,
3
→
C
4
{\displaystyle \psi :\mathbb {R} ^{1,3}\rightarrow \mathbb {C} ^{4}}
, the Dirac equation is then
(
i
ℏ
γ
μ
∂
μ
−
m
c
)
ψ
(
x
)
=
0
{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi (x)=0}
and in natural units, with Feynman slash notation,
(
i
∂
/
−
m
)
ψ
(
x
)
=
0
{\displaystyle (i\partial \!\!\!/-m)\psi (x)=0}
The gamma matrices are a set of four
complex matrices (elements of
Mat
4
×
4
(
C
)
{\displaystyle {\text{Mat}}_{4\times 4}(\mathbb {C} )}
) that satisfy the defining anti-commutation relations:
{
γ
μ
,
γ
ν
}
=
2
η
μ
ν
I
4
{\displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=2\eta ^{\mu \nu }I_{4}}
where
η
μ
ν
{\displaystyle \eta ^{\mu \nu }}
is the Minkowski metric element, and the indices
μ
,
ν
{\displaystyle \mu ,\nu }
run over 0,1,2 and 3. These matrices can be realized explicitly under a choice of representation. Two common choices are the Dirac representation and the chiral representation. The Dirac representation is
γ
0
=
(
I
2
0
0
−
I
2
)
,
γ
i
=
(
0
σ
i
−
σ
i
0
)
,
{\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\quad \gamma ^{i}={\begin{pmatrix}0&\sigma ^{i}\\-\sigma ^{i}&0\end{pmatrix}},}
where
σ
i
{\displaystyle \sigma ^{i}}
are the Pauli matrices.
For the chiral representation the
γ
i
{\displaystyle \gamma ^{i}}
are the same, but
γ
0
=
(
0
I
2
I
2
0
)
.
{\displaystyle \gamma ^{0}={\begin{pmatrix}0&I_{2}\\I_{2}&0\end{pmatrix}}~.}
The slash notation is a compact notation for
A
/
:=
γ
μ
A
μ
{\displaystyle A\!\!\!/:=\gamma ^{\mu }A_{\mu }}
where
A
{\displaystyle A}
is a four-vector (often it is the four-vector differential operator
∂
μ
{\displaystyle \partial _{\mu }}
). The summation over the index
μ
{\displaystyle \mu }
is implied.
Alternatively the four coupled linear first-order partial differential equations for the four quantities that make up the wave function can be written as a vector. In Planck units this becomes:: 6
i
∂
x
[
+
ψ
4
+
ψ
3
−
ψ
2
−
ψ
1
]
+
∂
y
[
+
ψ
4
−
ψ
3
−
ψ
2
+
ψ
1
]
+
i
∂
z
[
+
ψ
3
−
ψ
4
−
ψ
1
+
ψ
2
]
−
m
[
+
ψ
1
+
ψ
2
+
ψ
3
+
ψ
4
]
=
i
∂
t
[
−
ψ
1
−
ψ
2
+
ψ
3
+
ψ
4
]
{\displaystyle i\partial _{x}{\begin{bmatrix}+\psi _{4}\\+\psi _{3}\\-\psi _{2}\\-\psi _{1}\end{bmatrix}}+\partial _{y}{\begin{bmatrix}+\psi _{4}\\-\psi _{3}\\-\psi _{2}\\+\psi _{1}\end{bmatrix}}+i\partial _{z}{\begin{bmatrix}+\psi _{3}\\-\psi _{4}\\-\psi _{1}\\+\psi _{2}\end{bmatrix}}-m{\begin{bmatrix}+\psi _{1}\\+\psi _{2}\\+\psi _{3}\\+\psi _{4}\end{bmatrix}}=i\partial _{t}{\begin{bmatrix}-\psi _{1}\\-\psi _{2}\\+\psi _{3}\\+\psi _{4}\end{bmatrix}}}
which makes it clearer that it is a set of four partial differential equations with four unknown functions.
(Note that the
∂
y
{\displaystyle \partial _{y}}
term is not preceded by i because σy is imaginary.)
=== Dirac adjoint and the adjoint equation ===
The Dirac adjoint of the spinor field
ψ
(
x
)
{\displaystyle \psi (x)}
is defined as
ψ
¯
(
x
)
=
ψ
(
x
)
†
γ
0
.
{\displaystyle {\bar {\psi }}(x)=\psi (x)^{\dagger }\gamma ^{0}.}
Using the property of gamma matrices (which follows straightforwardly from Hermicity properties of the
γ
μ
{\displaystyle \gamma ^{\mu }}
) that
(
γ
μ
)
†
=
γ
0
γ
μ
γ
0
,
{\displaystyle (\gamma ^{\mu })^{\dagger }=\gamma ^{0}\gamma ^{\mu }\gamma ^{0},}
one can derive the adjoint Dirac equation by taking the Hermitian conjugate of the Dirac equation and multiplying on the right by
γ
0
{\displaystyle \gamma ^{0}}
:
ψ
¯
(
x
)
(
−
i
γ
μ
∂
←
μ
−
m
)
=
0
{\displaystyle {\bar {\psi }}(x)(-i\gamma ^{\mu }{\overleftarrow {\partial }}_{\mu }-m)=0}
where the partial derivative
∂
←
μ
{\displaystyle {\overleftarrow {\partial }}_{\mu }}
acts from the right on
ψ
¯
(
x
)
{\displaystyle {\bar {\psi }}(x)}
: written in the usual way in terms of a left action of the derivative, we have
−
i
∂
μ
ψ
¯
(
x
)
γ
μ
−
m
ψ
¯
(
x
)
=
0.
{\displaystyle -i\partial _{\mu }{\bar {\psi }}(x)\gamma ^{\mu }-m{\bar {\psi }}(x)=0.}
=== Klein–Gordon equation ===
Applying
i
∂
/
+
m
{\displaystyle i\partial \!\!\!/+m}
to the Dirac equation gives
(
∂
μ
∂
μ
+
m
2
)
ψ
(
x
)
=
0.
{\displaystyle (\partial _{\mu }\partial ^{\mu }+m^{2})\psi (x)=0.}
That is, each component of the Dirac spinor field satisfies the Klein–Gordon equation.
=== Conserved current ===
A conserved current of the theory is
J
μ
=
ψ
¯
γ
μ
ψ
.
{\displaystyle J^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .}
Another approach to derive this expression is by variational methods, applying Noether's theorem for the global
U
(
1
)
{\displaystyle {\text{U}}(1)}
symmetry to derive the conserved current
J
μ
.
{\displaystyle J^{\mu }.}
=== Solutions ===
Since the Dirac operator acts on 4-tuples of square-integrable functions, its solutions should be members of the same Hilbert space. The fact that the energies of the solutions do not have a lower bound is unexpected.
==== Plane-wave solutions ====
Plane-wave solutions are those arising from an ansatz
ψ
(
x
)
=
u
(
p
)
e
−
i
p
⋅
x
{\displaystyle \psi (x)=u(\mathbf {p} )e^{-ip\cdot x}}
which models a particle with definite 4-momentum
p
=
(
E
p
,
p
)
{\displaystyle p=(E_{\mathbf {p} },\mathbf {p} )}
where
E
p
=
m
2
+
|
p
|
2
.
{\textstyle E_{\mathbf {p} }={\sqrt {m^{2}+|\mathbf {p} |^{2}}}.}
For this ansatz, the Dirac equation becomes an equation for
u
(
p
)
{\displaystyle u(\mathbf {p} )}
:
(
γ
μ
p
μ
−
m
)
u
(
p
)
=
0.
{\displaystyle \left(\gamma ^{\mu }p_{\mu }-m\right)u(\mathbf {p} )=0.}
After picking a representation for the gamma matrices
γ
μ
{\displaystyle \gamma ^{\mu }}
, solving this is a matter of solving a system of linear equations. It is a representation-free property of gamma matrices that the solution space is two-dimensional (see here).
For example, in the chiral representation for
γ
μ
{\displaystyle \gamma ^{\mu }}
, the solution space is parametrised by a
C
2
{\displaystyle \mathbb {C} ^{2}}
vector
ξ
{\displaystyle \xi }
, with
u
(
p
)
=
(
σ
μ
p
μ
ξ
σ
¯
μ
p
μ
ξ
)
{\displaystyle u(\mathbf {p} )={\begin{pmatrix}{\sqrt {\sigma ^{\mu }p_{\mu }}}\xi \\{\sqrt {{\bar {\sigma }}^{\mu }p_{\mu }}}\xi \end{pmatrix}}}
where
σ
μ
=
(
I
2
,
σ
i
)
,
σ
¯
μ
=
(
I
2
,
−
σ
i
)
{\displaystyle \sigma ^{\mu }=(I_{2},\sigma ^{i}),{\bar {\sigma }}^{\mu }=(I_{2},-\sigma ^{i})}
and
⋅
{\displaystyle {\sqrt {\cdot }}}
is the Hermitian matrix square-root.
These plane-wave solutions provide a starting point for canonical quantization.
=== Lagrangian formulation ===
Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by:
L
=
i
ℏ
c
ψ
¯
γ
μ
∂
μ
ψ
−
m
c
2
ψ
¯
ψ
{\displaystyle {\mathcal {L}}=i\hbar c{\overline {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -mc^{2}{\overline {\psi }}\psi }
If one varies this with respect to
ψ
{\displaystyle \psi }
one gets the adjoint Dirac equation. Meanwhile, if one varies this with respect to
ψ
¯
{\displaystyle {\bar {\psi }}}
one gets the Dirac equation.
In natural units and with the slash notation, the action is then
For this action, the conserved current
J
μ
{\displaystyle J^{\mu }}
above arises as the conserved current corresponding to the global
U
(
1
)
{\displaystyle {\text{U}}(1)}
symmetry through Noether's theorem for field theory. Gauging this field theory by changing the symmetry to a local, spacetime point dependent one gives gauge symmetry (really, gauge redundancy). The resultant theory is quantum electrodynamics or QED. See below for a more detailed discussion.
=== Lorentz invariance ===
The Dirac equation is invariant under Lorentz transformations, that is, under the action of the Lorentz group
SO
(
1
,
3
)
{\displaystyle {\text{SO}}(1,3)}
or strictly
SO
(
1
,
3
)
+
{\displaystyle {\text{SO}}(1,3)^{+}}
, the component connected to the identity.
For a Dirac spinor viewed concretely as taking values in
C
4
{\displaystyle \mathbb {C} ^{4}}
, the transformation under a Lorentz transformation
Λ
{\displaystyle \Lambda }
is given by a
4
×
4
{\displaystyle 4\times 4}
complex matrix
S
[
Λ
]
{\displaystyle S[\Lambda ]}
. There are some subtleties in defining the corresponding
S
[
Λ
]
{\displaystyle S[\Lambda ]}
, as well as a standard abuse of notation.
Most treatments occur at the Lie algebra level. For a more detailed treatment see here. The Lorentz group of
4
×
4
{\displaystyle 4\times 4}
real matrices acting on
R
1
,
3
{\displaystyle \mathbb {R} ^{1,3}}
is generated by a set of six matrices
{
M
μ
ν
}
{\displaystyle \{M^{\mu \nu }\}}
with components
(
M
μ
ν
)
ρ
σ
=
η
μ
ρ
δ
ν
σ
−
η
ν
ρ
δ
μ
σ
.
{\displaystyle (M^{\mu \nu })^{\rho }{}_{\sigma }=\eta ^{\mu \rho }\delta ^{\nu }{}_{\sigma }-\eta ^{\nu \rho }\delta ^{\mu }{}_{\sigma }.}
When both the
ρ
,
σ
{\displaystyle \rho ,\sigma }
indices are raised or lowered, these are simply the 'standard basis' of antisymmetric matrices.
These satisfy the Lorentz algebra commutation relations
[
M
μ
ν
,
M
ρ
σ
]
=
M
μ
σ
η
ν
ρ
−
M
ν
σ
η
μ
ρ
+
M
ν
ρ
η
μ
σ
−
M
μ
ρ
η
ν
σ
.
{\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=M^{\mu \sigma }\eta ^{\nu \rho }-M^{\nu \sigma }\eta ^{\mu \rho }+M^{\nu \rho }\eta ^{\mu \sigma }-M^{\mu \rho }\eta ^{\nu \sigma }.}
In the article on the Dirac algebra, it is also found that the spin generators
S
μ
ν
=
1
4
[
γ
μ
,
γ
ν
]
{\displaystyle S^{\mu \nu }={\frac {1}{4}}[\gamma ^{\mu },\gamma ^{\nu }]}
satisfy the Lorentz algebra commutation relations.
A Lorentz transformation
Λ
{\displaystyle \Lambda }
can be written as
Λ
=
exp
(
1
2
ω
μ
ν
M
μ
ν
)
{\displaystyle \Lambda =\exp \left({\frac {1}{2}}\omega _{\mu \nu }M^{\mu \nu }\right)}
where the components
ω
μ
ν
{\displaystyle \omega _{\mu \nu }}
are antisymmetric in
μ
,
ν
{\displaystyle \mu ,\nu }
.
The corresponding transformation on spin space is
S
[
Λ
]
=
exp
(
1
2
ω
μ
ν
S
μ
ν
)
.
{\displaystyle S[\Lambda ]=\exp \left({\frac {1}{2}}\omega _{\mu \nu }S^{\mu \nu }\right).}
This is an abuse of notation, but a standard one. The reason is
S
[
Λ
]
{\displaystyle S[\Lambda ]}
is not a well-defined function of
Λ
{\displaystyle \Lambda }
, since there are two different sets of components
ω
μ
ν
{\displaystyle \omega _{\mu \nu }}
(up to equivalence) that give the same
Λ
{\displaystyle \Lambda }
but different
S
[
Λ
]
{\displaystyle S[\Lambda ]}
. In practice we implicitly pick one of these
ω
μ
ν
{\displaystyle \omega _{\mu \nu }}
and then
S
[
Λ
]
{\displaystyle S[\Lambda ]}
is well defined in terms of
ω
μ
ν
.
{\displaystyle \omega _{\mu \nu }.}
Under a Lorentz transformation, the Dirac equation
i
γ
μ
∂
μ
ψ
(
x
)
−
m
ψ
(
x
)
=
0
{\displaystyle i\gamma ^{\mu }\partial _{\mu }\psi (x)-m\psi (x)=0}
becomes
i
γ
μ
(
(
Λ
−
1
)
μ
ν
∂
ν
)
S
[
Λ
]
ψ
(
Λ
−
1
x
)
−
m
S
[
Λ
]
ψ
(
Λ
−
1
x
)
=
0.
{\displaystyle i\gamma ^{\mu }((\Lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })S[\Lambda ]\psi (\Lambda ^{-1}x)-mS[\Lambda ]\psi (\Lambda ^{-1}x)=0.}
Associated to Lorentz invariance is a conserved Noether current, or rather a tensor of conserved Noether currents
(
J
ρ
σ
)
μ
{\displaystyle ({\mathcal {J}}^{\rho \sigma })^{\mu }}
. Similarly, since the equation is invariant under translations, there is a tensor of conserved Noether currents
T
μ
ν
{\displaystyle T^{\mu \nu }}
, which can be identified as the stress-energy tensor of the theory. The Lorentz current
(
J
ρ
σ
)
μ
{\displaystyle ({\mathcal {J}}^{\rho \sigma })^{\mu }}
can be written in terms of the stress-energy tensor in addition to a tensor representing internal angular momentum.
==== Further discussion of Lorentz covariance of the Dirac equation ====
The Dirac equation is Lorentz covariant. Articulating this helps illuminate not only the Dirac equation, but also the Majorana spinor and Elko spinor, which although closely related, have subtle and important differences.
Understanding Lorentz covariance is simplified by keeping in mind the geometric character of the process. Let
a
{\displaystyle a}
be a single, fixed point in the spacetime manifold. Its location can be expressed in multiple coordinate systems. In the physics literature, these are written as
x
{\displaystyle x}
and
x
′
{\displaystyle x'}
, with the understanding that both
x
{\displaystyle x}
and
x
′
{\displaystyle x'}
describe the same point
a
{\displaystyle a}
, but in different local frames of reference (a frame of reference over a small extended patch of spacetime).
One can imagine
a
{\displaystyle a}
as having a fiber of different coordinate frames above it. In geometric terms, one says that spacetime can be characterized as a fiber bundle, and specifically, the frame bundle. The difference between two points
x
{\displaystyle x}
and
x
′
{\displaystyle x'}
in the same fiber is a combination of rotations and Lorentz boosts. A choice of coordinate frame is a (local) section through that bundle.
Coupled to the frame bundle is a second bundle, the spinor bundle. A section through the spinor bundle is just the particle field (the Dirac spinor, in the present case). Different points in the spinor fiber correspond to the same physical object (the fermion) but expressed in different Lorentz frames. Clearly, the frame bundle and the spinor bundle must be tied together in a consistent fashion to get consistent results; formally, one says that the spinor bundle is the associated bundle; it is associated to a principal bundle, which in the present case is the frame bundle. Differences between points on the fiber correspond to the symmetries of the system. The spinor bundle has two distinct generators of its symmetries: the total angular momentum and the intrinsic angular momentum. Both correspond to Lorentz transformations, but in different ways.
The presentation here follows that of Itzykson and Zuber. It is very nearly identical to that of Bjorken and Drell. A similar derivation in a general relativistic setting can be found in Weinberg. Here we fix our spacetime to be flat, that is, our spacetime is Minkowski space.
Under a Lorentz transformation
x
↦
x
′
,
{\displaystyle x\mapsto x',}
the Dirac spinor to transform as
ψ
′
(
x
′
)
=
S
ψ
(
x
)
{\displaystyle \psi '(x')=S\psi (x)}
It can be shown that an explicit expression for
S
{\displaystyle S}
is given by
S
=
exp
(
−
i
4
ω
μ
ν
σ
μ
ν
)
{\displaystyle S=\exp \left({\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)}
where
ω
μ
ν
{\displaystyle \omega ^{\mu \nu }}
parameterizes the Lorentz transformation, and
σ
μ
ν
{\displaystyle \sigma _{\mu \nu }}
are the six 4×4 matrices satisfying:
σ
μ
ν
=
i
2
[
γ
μ
,
γ
ν
]
.
{\displaystyle \sigma ^{\mu \nu }={\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]~.}
This matrix can be interpreted as the intrinsic angular momentum of the Dirac field. That it deserves this interpretation arises by contrasting it to the generator
J
μ
ν
{\displaystyle J_{\mu \nu }}
of Lorentz transformations, having the form
J
μ
ν
=
1
2
σ
μ
ν
+
i
(
x
μ
∂
ν
−
x
ν
∂
μ
)
{\displaystyle J_{\mu \nu }={\frac {1}{2}}\sigma _{\mu \nu }+i(x_{\mu }\partial _{\nu }-x_{\nu }\partial _{\mu })}
This can be interpreted as the total angular momentum. It acts on the spinor field as
ψ
′
(
x
)
=
exp
(
−
i
2
ω
μ
ν
J
μ
ν
)
ψ
(
x
)
{\displaystyle \psi ^{\prime }(x)=\exp \left({\frac {-i}{2}}\omega ^{\mu \nu }J_{\mu \nu }\right)\psi (x)}
Note the
x
{\displaystyle x}
above does not have a prime on it: the above is obtained by transforming
x
↦
x
′
{\displaystyle x\mapsto x'}
obtaining the change to
ψ
(
x
)
↦
ψ
′
(
x
′
)
{\displaystyle \psi (x)\mapsto \psi '(x')}
and then returning to the original coordinate system
x
′
↦
x
{\displaystyle x'\mapsto x}
.
The geometrical interpretation of the above is that the frame field is affine, having no preferred origin. The generator
J
μ
ν
{\displaystyle J_{\mu \nu }}
generates the symmetries of this space: it provides a relabelling of a fixed point
x
.
{\displaystyle x~.}
The generator
σ
μ
ν
{\displaystyle \sigma _{\mu \nu }}
generates a movement from one point in the fiber to another: a movement from
x
↦
x
′
{\displaystyle x\mapsto x'}
with both
x
{\displaystyle x}
and
x
′
{\displaystyle x'}
still corresponding to the same spacetime point
a
.
{\displaystyle a.}
These perhaps obtuse remarks can be elucidated with explicit algebra.
Let
x
′
=
Λ
x
{\displaystyle x'=\Lambda x}
be a Lorentz transformation. The Dirac equation is
i
γ
μ
∂
∂
x
μ
ψ
(
x
)
−
m
ψ
(
x
)
=
0
{\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi (x)-m\psi (x)=0}
If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames:
i
γ
μ
∂
∂
x
′
μ
ψ
′
(
x
′
)
−
m
ψ
′
(
x
′
)
=
0
{\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi ^{\prime }(x^{\prime })-m\psi ^{\prime }(x^{\prime })=0}
The two spinors
ψ
{\displaystyle \psi }
and
ψ
′
{\displaystyle \psi ^{\prime }}
should both describe the same physical field, and so should be related by a transformation that does not change any physical observables (charge, current, mass, etc.) The transformation should encode only the change of coordinate frame. It can be shown that such a transformation is a 4×4 unitary matrix. Thus, one may presume that the relation between the two frames can be written as
ψ
′
(
x
′
)
=
S
(
Λ
)
ψ
(
x
)
{\displaystyle \psi ^{\prime }(x^{\prime })=S(\Lambda )\psi (x)}
Inserting this into the transformed equation, the result is
i
γ
μ
∂
x
ν
∂
x
′
μ
∂
∂
x
ν
S
(
Λ
)
ψ
(
x
)
−
m
S
(
Λ
)
ψ
(
x
)
=
0
{\displaystyle i\gamma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}S(\Lambda )\psi (x)-mS(\Lambda )\psi (x)=0}
The coordinates related by Lorentz transformation satisfy:
∂
x
ν
∂
x
′
μ
=
(
Λ
−
1
)
ν
μ
{\displaystyle {\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }}
The original Dirac equation is then regained if
S
(
Λ
)
γ
μ
S
−
1
(
Λ
)
=
(
Λ
−
1
)
μ
ν
γ
ν
{\displaystyle S(\Lambda )\gamma ^{\mu }S^{-1}(\Lambda )={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu }\gamma ^{\nu }}
An explicit expression for
S
(
Λ
)
{\displaystyle S(\Lambda )}
(equal to the expression given above) can be obtained by considering a Lorentz transformation of infinitesimal rotation near the identity transformation:
Λ
μ
ν
=
g
μ
ν
+
ω
μ
ν
,
(
Λ
−
1
)
μ
ν
=
g
μ
ν
−
ω
μ
ν
{\displaystyle {\Lambda ^{\mu }}_{\nu }={g^{\mu }}_{\nu }+{\omega ^{\mu }}_{\nu }\ ,\ {(\Lambda ^{-1})^{\mu }}_{\nu }={g^{\mu }}_{\nu }-{\omega ^{\mu }}_{\nu }}
where
g
μ
ν
{\displaystyle {g^{\mu }}_{\nu }}
is the metric tensor :
g
μ
ν
=
g
μ
ν
′
g
ν
′
ν
=
δ
μ
ν
{\displaystyle {g^{\mu }}_{\nu }=g^{\mu \nu '}g_{\nu '\nu }={\delta ^{\mu }}_{\nu }}
and is symmetric while
ω
μ
ν
=
ω
α
ν
g
α
μ
{\displaystyle \omega _{\mu \nu }={\omega ^{\alpha }}_{\nu }g_{\alpha \mu }}
is antisymmetric. After plugging and chugging, one obtains
S
(
Λ
)
=
I
+
−
i
4
ω
μ
ν
σ
μ
ν
+
O
(
Λ
2
)
,
{\displaystyle S(\Lambda )=I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }+{\mathcal {O}}\left(\Lambda ^{2}\right),}
which is the (infinitesimal) form for
S
{\displaystyle S}
above and yields the relation
σ
μ
ν
=
i
2
[
γ
μ
,
γ
ν
]
{\displaystyle \sigma ^{\mu \nu }={\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]}
. To obtain the affine relabelling, write
ψ
′
(
x
′
)
=
(
I
+
−
i
4
ω
μ
ν
σ
μ
ν
)
ψ
(
x
)
=
(
I
+
−
i
4
ω
μ
ν
σ
μ
ν
)
ψ
(
x
′
+
ω
μ
ν
x
′
ν
)
=
(
I
+
−
i
4
ω
μ
ν
σ
μ
ν
−
x
μ
′
ω
μ
ν
∂
ν
)
ψ
(
x
′
)
=
(
I
+
−
i
2
ω
μ
ν
J
μ
ν
)
ψ
(
x
′
)
{\displaystyle {\begin{aligned}\psi '(x')&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x)\\&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x'+{\omega ^{\mu }}_{\nu }\,x^{\prime \,\nu })\\&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }-x_{\mu }^{\prime }\omega ^{\mu \nu }\partial _{\nu }\right)\psi (x')\\&=\left(I+{\frac {-i}{2}}\omega ^{\mu \nu }J_{\mu \nu }\right)\psi (x')\\\end{aligned}}}
After properly antisymmetrizing, one obtains the generator of symmetries
J
μ
ν
{\displaystyle J_{\mu \nu }}
given earlier. Thus, both
J
μ
ν
{\displaystyle J_{\mu \nu }}
and
σ
μ
ν
{\displaystyle \sigma _{\mu \nu }}
can be said to be the "generators of Lorentz transformations", but with a subtle distinction: the first corresponds to a relabelling of points on the affine frame bundle, which forces a translation along the fiber of the spinor on the spin bundle, while the second corresponds to translations along the fiber of the spin bundle (taken as a movement
x
↦
x
′
{\displaystyle x\mapsto x'}
along the frame bundle, as well as a movement
ψ
↦
ψ
′
{\displaystyle \psi \mapsto \psi '}
along the fiber of the spin bundle.) Weinberg provides additional arguments for the physical interpretation of these as total and intrinsic angular momentum.
== Other formulations ==
The Dirac equation can be formulated in a number of other ways.
=== Curved spacetime ===
This article has developed the Dirac equation in flat spacetime according to special relativity. It is possible to formulate the Dirac equation in curved spacetime.
=== The algebra of physical space ===
This article developed the Dirac equation using four-vectors and Schrödinger operators. The Dirac equation in the algebra of physical space uses a Clifford algebra over the real numbers, a type of geometric algebra.
=== Coupled Weyl Spinors ===
As mentioned above, the massless Dirac equation immediately reduces to the homogeneous Weyl equation. By using the chiral representation of the gamma matrices, the nonzero-mass equation can also be decomposed into a pair of coupled inhomogeneous Weyl equations acting on the first and last pairs of indices of the original four-component spinor, i.e.
ψ
=
(
ψ
L
ψ
R
)
{\displaystyle \psi ={\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}}
, where
ψ
L
{\displaystyle \psi _{L}}
and
ψ
R
{\displaystyle \psi _{R}}
are each two-component Weyl spinors. This is because the skew block form of the chiral gamma matrices means that they swap the
ψ
L
{\displaystyle \psi _{L}}
and
ψ
R
{\displaystyle \psi _{R}}
and apply the two-by-two Pauli matrices to each:
γ
μ
(
ψ
L
ψ
R
)
=
(
σ
μ
ψ
R
σ
¯
μ
ψ
L
)
.
{\displaystyle \gamma ^{\mu }{\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}={\begin{pmatrix}\sigma ^{\mu }\psi _{R}\\{\overline {\sigma }}^{\mu }\psi _{L}\end{pmatrix}}.}
So the Dirac equation
(
i
γ
μ
∂
μ
−
m
)
(
ψ
L
ψ
R
)
=
0
{\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m){\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}=0}
becomes
i
(
σ
μ
∂
μ
ψ
R
σ
¯
μ
∂
μ
ψ
L
)
=
m
(
ψ
L
ψ
R
)
,
{\displaystyle i{\begin{pmatrix}\sigma ^{\mu }\partial _{\mu }\psi _{R}\\{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{L}\end{pmatrix}}=m{\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}},}
which in turn is equivalent to a pair of inhomogeneous Weyl equations for massless left- and right-helicity spinors, where the coupling strength is proportional to the mass:
i
σ
μ
∂
μ
ψ
R
=
m
ψ
L
{\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{R}=m\psi _{L}}
i
σ
¯
μ
∂
μ
ψ
L
=
m
ψ
R
.
{\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=m\psi _{R}.}
This has been proposed as an intuitive explanation of Zitterbewegung, as these massless components would propagate at the speed of light and move in opposite directions, since the helicity is the projection of the spin onto the direction of motion. Here the role of the "mass"
m
{\displaystyle m}
is not to make the velocity less than the speed of light, but instead controls the average rate at which these reversals occur; specifically, the reversals can be modeled as a Poisson process.
== U(1) symmetry ==
Natural units are used in this section. The coupling constant is labelled by convention with
e
{\displaystyle e}
: this parameter can also be viewed as modelling the electron charge.
=== Vector symmetry ===
The Dirac equation and action admits a
U
(
1
)
{\displaystyle {\text{U}}(1)}
symmetry where the fields
ψ
,
ψ
¯
{\displaystyle \psi ,{\bar {\psi }}}
transform as
ψ
(
x
)
↦
e
i
α
ψ
(
x
)
,
ψ
¯
(
x
)
↦
e
−
i
α
ψ
¯
(
x
)
.
{\displaystyle {\begin{aligned}\psi (x)&\mapsto e^{i\alpha }\psi (x),\\{\bar {\psi }}(x)&\mapsto e^{-i\alpha }{\bar {\psi }}(x).\end{aligned}}}
This is a global symmetry, known as the
U
(
1
)
{\displaystyle {\text{U}}(1)}
vector symmetry (as opposed to the
U
(
1
)
{\displaystyle {\text{U}}(1)}
axial symmetry: see below). By Noether's theorem there is a corresponding conserved current: this has been mentioned previously as
J
μ
(
x
)
=
ψ
¯
(
x
)
γ
μ
ψ
(
x
)
.
{\displaystyle J^{\mu }(x)={\bar {\psi }}(x)\gamma ^{\mu }\psi (x).}
=== Gauging the symmetry ===
If we 'promote' the global symmetry, parametrised by the constant
α
{\displaystyle \alpha }
, to a local symmetry, parametrised by a function
α
:
R
1
,
3
→
R
{\displaystyle \alpha :\mathbb {R} ^{1,3}\to \mathbb {R} }
, or equivalently
e
i
α
:
R
1
,
3
→
U
(
1
)
,
{\displaystyle e^{i\alpha }:\mathbb {R} ^{1,3}\to {\text{U}}(1),}
the Dirac equation is no longer invariant: there is a residual derivative of
α
(
x
)
{\displaystyle \alpha (x)}
.
The fix proceeds as in scalar electrodynamics: the partial derivative is promoted to a covariant derivative
D
μ
{\displaystyle D_{\mu }}
D
μ
ψ
=
∂
μ
ψ
+
i
e
A
μ
ψ
,
{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +ieA_{\mu }\psi ,}
D
μ
ψ
¯
=
∂
μ
ψ
¯
−
i
e
A
μ
ψ
¯
.
{\displaystyle D_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ieA_{\mu }{\bar {\psi }}.}
The covariant derivative depends on the field being acted on. The newly introduced
A
μ
{\displaystyle A_{\mu }}
is the 4-vector potential from electrodynamics, but also can be viewed as a
U
(
1
)
{\displaystyle {\text{U}}(1)}
gauge field (which, mathematically, is defined as a
U
(
1
)
{\displaystyle {\text{U}}(1)}
connection).
The transformation law under gauge transformations for
A
μ
{\displaystyle A_{\mu }}
is then the usual
A
μ
(
x
)
↦
A
μ
(
x
)
+
1
e
∂
μ
α
(
x
)
{\displaystyle A_{\mu }(x)\mapsto A_{\mu }(x)+{\frac {1}{e}}\partial _{\mu }\alpha (x)}
but can also be derived by asking that covariant derivatives transform under a gauge transformation as
D
μ
ψ
(
x
)
↦
e
i
α
(
x
)
D
μ
ψ
(
x
)
,
{\displaystyle D_{\mu }\psi (x)\mapsto e^{i\alpha (x)}D_{\mu }\psi (x),}
D
μ
ψ
¯
(
x
)
↦
e
−
i
α
(
x
)
D
μ
ψ
¯
(
x
)
.
{\displaystyle D_{\mu }{\bar {\psi }}(x)\mapsto e^{-i\alpha (x)}D_{\mu }{\bar {\psi }}(x).}
We then obtain a gauge-invariant Dirac action by promoting the partial derivative to a covariant one:
S
=
∫
d
4
x
ψ
¯
(
i
D
/
−
m
)
ψ
=
∫
d
4
x
ψ
¯
(
i
γ
μ
D
μ
−
m
)
ψ
.
{\displaystyle S=\int d^{4}x\,{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi =\int d^{4}x\,{\bar {\psi }}\,(i\gamma ^{\mu }D_{\mu }-m)\,\psi .}
The final step needed to write down a gauge-invariant Lagrangian is to add a Maxwell Lagrangian term,
S
Maxwell
=
∫
d
4
x
[
−
1
4
F
μ
ν
F
μ
ν
]
.
{\displaystyle S_{\text{Maxwell}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }\right].}
Putting these together gives
Expanding out the covariant derivative allows the action to be written in a second useful form:
S
QED
=
∫
d
4
x
[
−
1
4
F
μ
ν
F
μ
ν
+
ψ
¯
(
i
∂
/
−
m
)
ψ
−
e
J
μ
A
μ
]
{\displaystyle S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi -eJ^{\mu }A_{\mu }\right]}
=== Axial symmetry ===
Massless Dirac fermions, that is, fields
ψ
(
x
)
{\displaystyle \psi (x)}
satisfying the Dirac equation with
m
=
0
{\displaystyle m=0}
, admit a second, inequivalent
U
(
1
)
{\displaystyle {\text{U}}(1)}
symmetry.
This is seen most easily by writing the four-component Dirac fermion
ψ
(
x
)
{\displaystyle \psi (x)}
as a pair of two-component vector fields,
ψ
(
x
)
=
(
ψ
1
(
x
)
ψ
2
(
x
)
)
,
{\displaystyle \psi (x)={\begin{pmatrix}\psi _{1}(x)\\\psi _{2}(x)\end{pmatrix}},}
and adopting the chiral representation for the gamma matrices, so that
i
γ
μ
∂
μ
{\displaystyle i\gamma ^{\mu }\partial _{\mu }}
may be written
i
γ
μ
∂
μ
=
(
0
i
σ
μ
∂
μ
i
σ
¯
μ
∂
μ
0
)
{\displaystyle i\gamma ^{\mu }\partial _{\mu }={\begin{pmatrix}0&i\sigma ^{\mu }\partial _{\mu }\\i{\bar {\sigma }}^{\mu }\partial _{\mu }\ &0\end{pmatrix}}}
where
σ
μ
{\displaystyle \sigma ^{\mu }}
has components
(
I
2
,
σ
i
)
{\displaystyle (I_{2},\sigma ^{i})}
and
σ
¯
μ
{\displaystyle {\bar {\sigma }}^{\mu }}
has components
(
I
2
,
−
σ
i
)
{\displaystyle (I_{2},-\sigma ^{i})}
.
The Dirac action then takes the form
S
=
∫
d
4
x
ψ
1
†
(
i
σ
μ
∂
μ
)
ψ
1
+
ψ
2
†
(
i
σ
¯
μ
∂
μ
)
ψ
2
.
{\displaystyle S=\int d^{4}x\,\psi _{1}^{\dagger }(i\sigma ^{\mu }\partial _{\mu })\psi _{1}+\psi _{2}^{\dagger }(i{\bar {\sigma }}^{\mu }\partial _{\mu })\psi _{2}.}
That is, it decouples into a theory of two Weyl spinors or Weyl fermions.
The earlier vector symmetry is still present, where
ψ
1
{\displaystyle \psi _{1}}
and
ψ
2
{\displaystyle \psi _{2}}
rotate identically. This form of the action makes the second inequivalent
U
(
1
)
{\displaystyle {\text{U}}(1)}
symmetry manifest:
ψ
1
(
x
)
↦
e
i
β
ψ
1
(
x
)
,
ψ
2
(
x
)
↦
e
−
i
β
ψ
2
(
x
)
.
{\displaystyle {\begin{aligned}\psi _{1}(x)&\mapsto e^{i\beta }\psi _{1}(x),\\\psi _{2}(x)&\mapsto e^{-i\beta }\psi _{2}(x).\end{aligned}}}
This can also be expressed at the level of the Dirac fermion as
ψ
(
x
)
↦
exp
(
i
β
γ
5
)
ψ
(
x
)
{\displaystyle \psi (x)\mapsto \exp(i\beta \gamma ^{5})\psi (x)}
where
exp
{\displaystyle \exp }
is the exponential map for matrices.
This isn't the only
U
(
1
)
{\displaystyle {\text{U}}(1)}
symmetry possible, but it is conventional. Any 'linear combination' of the vector and axial symmetries is also a
U
(
1
)
{\displaystyle {\text{U}}(1)}
symmetry.
Classically, the axial symmetry admits a well-formulated gauge theory. But at the quantum level, there is an anomaly, that is, an obstruction to gauging.
=== Extension to color symmetry ===
We can extend this discussion from an abelian
U
(
1
)
{\displaystyle {\text{U}}(1)}
symmetry to a general non-abelian symmetry under a gauge group
G
{\displaystyle G}
, the group of color symmetries for a theory.
For concreteness, we fix
G
=
SU
(
N
)
{\displaystyle G={\text{SU}}(N)}
, the special unitary group of matrices acting on
C
N
{\displaystyle \mathbb {C} ^{N}}
.
Before this section,
ψ
(
x
)
{\displaystyle \psi (x)}
could be viewed as a spinor field on Minkowski space, in other words a function
ψ
:
R
1
,
3
↦
C
4
{\displaystyle \psi :\mathbb {R} ^{1,3}\mapsto \mathbb {C} ^{4}}
, and its components in
C
4
{\displaystyle \mathbb {C} ^{4}}
are labelled by spin indices, conventionally Greek indices taken from the start of the alphabet
α
,
β
,
γ
,
⋯
{\displaystyle \alpha ,\beta ,\gamma ,\cdots }
.
Promoting the theory to a gauge theory, informally
ψ
{\displaystyle \psi }
acquires a part transforming like
C
N
{\displaystyle \mathbb {C} ^{N}}
, and these are labelled by color indices, conventionally Latin indices
i
,
j
,
k
,
⋯
{\displaystyle i,j,k,\cdots }
. In total,
ψ
(
x
)
{\displaystyle \psi (x)}
has
4
N
{\displaystyle 4N}
components, given in indices by
ψ
i
,
α
(
x
)
{\displaystyle \psi ^{i,\alpha }(x)}
. The 'spinor' labels only how the field transforms under spacetime transformations.
Formally,
ψ
(
x
)
{\displaystyle \psi (x)}
is valued in a tensor product, that is, it is a function
ψ
:
R
1
,
3
→
C
4
⊗
C
N
.
{\displaystyle \psi :\mathbb {R} ^{1,3}\to \mathbb {C} ^{4}\otimes \mathbb {C} ^{N}.}
Gauging proceeds similarly to the abelian
U
(
1
)
{\displaystyle {\text{U}}(1)}
case, with a few differences. Under a gauge transformation
U
:
R
1
,
3
→
SU
(
N
)
,
{\displaystyle U:\mathbb {R} ^{1,3}\rightarrow {\text{SU}}(N),}
the spinor fields transform as
ψ
(
x
)
↦
U
(
x
)
ψ
(
x
)
{\displaystyle \psi (x)\mapsto U(x)\psi (x)}
ψ
¯
(
x
)
↦
ψ
¯
(
x
)
U
†
(
x
)
.
{\displaystyle {\bar {\psi }}(x)\mapsto {\bar {\psi }}(x)U^{\dagger }(x).}
The matrix-valued gauge field
A
μ
{\displaystyle A_{\mu }}
or
SU
(
N
)
{\displaystyle {\text{SU}}(N)}
connection transforms as
A
μ
(
x
)
↦
U
(
x
)
A
μ
(
x
)
U
(
x
)
−
1
+
1
g
(
∂
μ
U
(
x
)
)
U
(
x
)
−
1
,
{\displaystyle A_{\mu }(x)\mapsto U(x)A_{\mu }(x)U(x)^{-1}+{\frac {1}{g}}(\partial _{\mu }U(x))U(x)^{-1},}
and the covariant derivatives defined
D
μ
ψ
=
∂
μ
ψ
+
i
g
A
μ
ψ
,
{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +igA_{\mu }\psi ,}
D
μ
ψ
¯
=
∂
μ
ψ
¯
−
i
g
ψ
¯
A
μ
†
{\displaystyle D_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ig{\bar {\psi }}A_{\mu }^{\dagger }}
transform as
D
μ
ψ
(
x
)
↦
U
(
x
)
D
μ
ψ
(
x
)
,
{\displaystyle D_{\mu }\psi (x)\mapsto U(x)D_{\mu }\psi (x),}
D
μ
ψ
¯
(
x
)
↦
(
D
μ
ψ
¯
(
x
)
)
U
(
x
)
†
.
{\displaystyle D_{\mu }{\bar {\psi }}(x)\mapsto (D_{\mu }{\bar {\psi }}(x))U(x)^{\dagger }.}
Writing down a gauge-invariant action proceeds exactly as with the
U
(
1
)
{\displaystyle {\text{U}}(1)}
case, replacing the Maxwell Lagrangian with the Yang–Mills Lagrangian
S
Y-M
=
∫
d
4
x
−
1
4
Tr
(
F
μ
ν
F
μ
ν
)
{\displaystyle S_{\text{Y-M}}=\int d^{4}x\,-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })}
where the Yang–Mills field strength or curvature is defined here as
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
−
i
g
[
A
μ
,
A
ν
]
{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }-ig\left[A_{\mu },A_{\nu }\right]}
and
[
⋅
,
⋅
]
{\displaystyle [\cdot ,\cdot ]}
is the matrix commutator.
The action is then
==== Physical applications ====
For physical applications, the case
N
=
3
{\displaystyle N=3}
describes the quark sector of the Standard Model, which models strong interactions. Quarks are modelled as Dirac spinors; the gauge field is the gluon field. The case
N
=
2
{\displaystyle N=2}
describes part of the electroweak sector of the Standard Model. Leptons such as electrons and neutrinos are the Dirac spinors; the gauge field is the
W
{\displaystyle W}
gauge boson.
==== Generalisations ====
This expression can be generalised to arbitrary Lie group
G
{\displaystyle G}
with connection
A
μ
{\displaystyle A_{\mu }}
and a representation
(
ρ
,
G
,
V
)
{\displaystyle (\rho ,G,V)}
, where the colour part of
ψ
{\displaystyle \psi }
is valued in
V
{\displaystyle V}
. Formally, the Dirac field is a function
ψ
:
R
1
,
3
→
C
4
⊗
V
.
{\displaystyle \psi :\mathbb {R} ^{1,3}\to \mathbb {C} ^{4}\otimes V.}
Then
ψ
{\displaystyle \psi }
transforms under a gauge transformation
g
:
R
1
,
3
→
G
{\displaystyle g:\mathbb {R} ^{1,3}\to G}
as
ψ
(
x
)
↦
ρ
(
g
(
x
)
)
ψ
(
x
)
{\displaystyle \psi (x)\mapsto \rho (g(x))\psi (x)}
and the covariant derivative is defined
D
μ
ψ
=
∂
μ
ψ
+
ρ
(
A
μ
)
ψ
{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +\rho (A_{\mu })\psi }
where here we view
ρ
{\displaystyle \rho }
as a Lie algebra representation of the Lie algebra
g
=
L
(
G
)
{\displaystyle {\mathfrak {g}}={\text{L}}(G)}
associated to
G
{\displaystyle G}
.
This theory can be generalised to curved spacetime, but there are subtleties that arise in gauge theory on a general spacetime (or more generally still, a manifold), which can be ignored on flat spacetime. This is ultimately due to the contractibility of flat spacetime that allows us to view a gauge field and gauge transformations as defined globally on
R
1
,
3
{\displaystyle \mathbb {R} ^{1,3}}
.
== See also ==
== References ==
=== Citations ===
=== Selected papers ===
Anderson, Carl (1933). "The Positive Electron". Physical Review. 43 (6): 491. Bibcode:1933PhRv...43..491A. doi:10.1103/PhysRev.43.491.
Arminjon, M.; F. Reifler (2013). "Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations". Brazilian Journal of Physics. 43 (1–2): 64–77. arXiv:1103.3201. Bibcode:2013BrJPh..43...64A. doi:10.1007/s13538-012-0111-0. S2CID 38235437.
Dirac, P. A. M. (1928). "The Quantum Theory of the Electron" (PDF). Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 117 (778): 610–624. Bibcode:1928RSPSA.117..610D. doi:10.1098/rspa.1928.0023. JSTOR 94981. Archived (PDF) from the original on 2 January 2015.
Dirac, P. A. M. (1930). "A Theory of Electrons and Protons". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 126 (801): 360–365. Bibcode:1930RSPSA.126..360D. doi:10.1098/rspa.1930.0013. JSTOR 95359.
Frisch, R.; Stern, O. (1933). "Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. I". Zeitschrift für Physik. 85 (1–2): 4. Bibcode:1933ZPhy...85....4F. doi:10.1007/BF01330773. S2CID 120793548.
=== Textbooks ===
Bjorken, J D; Drell, S (1964). Relativistic Quantum mechanics. New York, McGraw-Hill.
Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 9780471887416.
Griffiths, D.J. (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH. ISBN 978-3-527-40601-2.
Rae, Alastair I. M.; Jim Napolitano (2015). Quantum Mechanics (6th ed.). Routledge. ISBN 978-1482299182.
Schiff, L.I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill.
Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). Plenum.
Thaller, B. (1992). The Dirac Equation. Texts and Monographs in Physics. Springer.
== External links ==
The history of the positron Lecture given by Dirac in 1975
The Dirac Equation at MathPages
The Dirac equation for a spin 1⁄2 particle
The Dirac Equation in natural units at the Paul M. Dirac Lecture Hall, EMFCSC, Erice, Sicily | Wikipedia/Dirac's_equation |
In theoretical physics, thermal quantum field theory (thermal field theory for short) or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature.
In the Matsubara formalism, the basic idea (due to Felix Bloch) is that the expectation values of operators in a canonical ensemble
⟨
A
⟩
=
Tr
[
exp
(
−
β
H
)
A
]
Tr
[
exp
(
−
β
H
)
]
{\displaystyle \langle A\rangle ={\frac {{\mbox{Tr}}\,[\exp(-\beta H)A]}{{\mbox{Tr}}\,[\exp(-\beta H)]}}}
may be written as expectation values in ordinary quantum field theory where the configuration is evolved by an imaginary time
τ
=
i
t
(
0
≤
τ
≤
β
)
{\displaystyle \tau =it(0\leq \tau \leq \beta )}
. One can therefore switch to a spacetime with Euclidean signature, where the above trace (Tr) leads to the requirement that all bosonic and fermionic fields be periodic and antiperiodic, respectively, with respect to the Euclidean time direction with periodicity
β
=
1
/
(
k
T
)
{\displaystyle \beta =1/(kT)}
(we are assuming natural units
ℏ
=
1
{\displaystyle \hbar =1}
). This allows one to perform calculations with the same tools as in ordinary quantum field theory, such as functional integrals and Feynman diagrams, but with compact Euclidean time. Note that the definition of normal ordering has to be altered.
In momentum space, this leads to the replacement of continuous frequencies by discrete imaginary (Matsubara) frequencies
v
n
=
n
/
β
{\displaystyle v_{n}=n/\beta }
and, through the de Broglie relation, to a discretized thermal energy spectrum
E
n
=
2
n
π
k
T
{\displaystyle E_{n}=2n\pi kT}
. This has been shown to be a useful tool in studying the behavior of quantum field theories at finite temperature.
It has been generalized to theories with gauge invariance and was a central tool in the study of a conjectured deconfining phase transition of Yang–Mills theory. In this Euclidean field theory, real-time observables can be retrieved by analytic continuation. The Feynman rules for gauge theories in the Euclidean time formalism, were derived by C. W. Bernard.
The Matsubara formalism, also referred to as imaginary time formalism, can be extended to systems with thermal variations. In this approach, the variation in the temperature is recast as a variation in the Euclidean metric. Analysis of the partition function leads to an equivalence between thermal variations and the curvature of the Euclidean space.
The alternative to the use of fictitious imaginary times is to use a real-time formalism which come in two forms. A path-ordered approach to real-time formalisms includes the Schwinger–Keldysh formalism and more modern variants. The latter involves replacing a straight time contour from (large negative) real initial time
t
i
{\displaystyle t_{i}}
to
t
i
−
i
β
{\displaystyle t_{i}-i\beta }
by one that first runs to (large positive) real time
t
f
{\displaystyle t_{f}}
and then suitably back to
t
i
−
i
β
{\displaystyle t_{i}-i\beta }
. In fact all that is needed is one section running along the real time axis, as the route to the end point,
t
i
−
i
β
{\displaystyle t_{i}-i\beta }
, is less important. The piecewise composition of the resulting complex time contour leads to a doubling of fields and more complicated Feynman rules, but obviates the need of analytic continuations of the imaginary-time formalism. The alternative approach to real-time formalisms is an operator based approach using Bogoliubov transformations, known as thermo field dynamics.
As well as Feynman diagrams and perturbation theory, other techniques such as dispersion relations and the finite temperature analog of Cutkosky rules can also be used in the real time formulation.
An alternative approach which is of interest to mathematical physics is to work with KMS states.
== See also ==
Matsubara frequency
Polyakov loop
Quantum thermodynamics
Quantum statistical mechanics
== References == | Wikipedia/Thermal_quantum_field_theory |
The Bullough–Dodd model is an integrable model in 1+1-dimensional quantum field theory introduced by Robin Bullough and Roger Dodd. Its
Lagrangian density is
L
=
1
2
(
∂
μ
φ
)
2
−
m
0
2
6
g
2
(
2
e
g
φ
+
e
−
2
g
φ
)
{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\varphi )^{2}-{\frac {m_{0}^{2}}{6g^{2}}}(2e^{g\varphi }+e^{-2g\varphi })}
where
m
0
{\displaystyle m_{0}\,}
is a mass parameter,
g
{\displaystyle g\,}
is the coupling constant and
φ
{\displaystyle \varphi \,}
is a real scalar field.
The Bullough–Dodd model belongs to the class of affine Toda field theories.
The spectrum of the model consists of a single massive particle.
== See also ==
List of integrable models
== References ==
Dodd, R. K.; Bullough, R. K. (4 February 1977). "Polynomial Conserved Densities for the Sine-Gordon Equations". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 352 (1671). The Royal Society: 481–503. Bibcode:1977RSPSA.352..481D. doi:10.1098/rspa.1977.0012. ISSN 1364-5021. S2CID 123071322.
Fring, A.; Mussardo, G.; Simonetti, P. (1993). "Form factors of the elementary field in the Bullough-Dodd model". Physics Letters B. 307 (1–2). Elsevier BV: 83–90. arXiv:hep-th/9303108. Bibcode:1993PhLB..307...83F. doi:10.1016/0370-2693(93)90196-o. ISSN 0370-2693. S2CID 16396002. | Wikipedia/Bullough–Dodd_model |
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. Propagators may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions (called "causal" to distinguish it from the elliptic Laplacian Green's function).
== Non-relativistic propagators ==
In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a particle to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t).
The Green's function G for the Schrödinger equation is a function
G
(
x
,
t
;
x
′
,
t
′
)
=
1
i
ℏ
Θ
(
t
−
t
′
)
K
(
x
,
t
;
x
′
,
t
′
)
{\displaystyle G(x,t;x',t')={\frac {1}{i\hbar }}\Theta (t-t')K(x,t;x',t')}
satisfying
(
i
ℏ
∂
∂
t
−
H
x
)
G
(
x
,
t
;
x
′
,
t
′
)
=
δ
(
x
−
x
′
)
δ
(
t
−
t
′
)
,
{\displaystyle \left(i\hbar {\frac {\partial }{\partial t}}-H_{x}\right)G(x,t;x',t')=\delta (x-x')\delta (t-t'),}
where H denotes the Hamiltonian, δ(x) denotes the Dirac delta-function and Θ(t) is the Heaviside step function. The kernel of the above Schrödinger differential operator in the big parentheses is denoted by K(x, t ;x′, t′) and called the propagator.
This propagator may also be written as the transition amplitude
K
(
x
,
t
;
x
′
,
t
′
)
=
⟨
x
|
U
(
t
,
t
′
)
|
x
′
⟩
,
{\displaystyle K(x,t;x',t')={\big \langle }x{\big |}U(t,t'){\big |}x'{\big \rangle },}
where U(t, t′) is the unitary time-evolution operator for the system taking states at time t′ to states at time t. Note the initial condition enforced by
lim
t
→
t
′
K
(
x
,
t
;
x
′
,
t
′
)
=
δ
(
x
−
x
′
)
.
{\displaystyle \lim _{t\to t'}K(x,t;x',t')=\delta (x-x').}
The propagator may also be found by using a path integral:
K
(
x
,
t
;
x
′
,
t
′
)
=
∫
exp
[
i
ℏ
∫
t
′
t
L
(
q
˙
,
q
,
t
)
d
t
]
D
[
q
(
t
)
]
,
{\displaystyle K(x,t;x',t')=\int \exp \left[{\frac {i}{\hbar }}\int _{t'}^{t}L({\dot {q}},q,t)\,dt\right]D[q(t)],}
where L denotes the Lagrangian and the boundary conditions are given by q(t) = x, q(t′) = x′. The paths that are summed over move only forwards in time and are integrated with the differential
D
[
q
(
t
)
]
{\displaystyle D[q(t)]}
following the path in time.
The propagator lets one find the wave function of a system, given an initial wave function and a time interval. The new wave function is given by
ψ
(
x
,
t
)
=
∫
−
∞
∞
ψ
(
x
′
,
t
′
)
K
(
x
,
t
;
x
′
,
t
′
)
d
x
′
.
{\displaystyle \psi (x,t)=\int _{-\infty }^{\infty }\psi (x',t')K(x,t;x',t')\,dx'.}
If K(x, t; x′, t′) only depends on the difference x − x′, this is a convolution of the initial wave function and the propagator.
=== Examples ===
For a time-translationally invariant system, the propagator only depends on the time difference t − t′, so it may be rewritten as
K
(
x
,
t
;
x
′
,
t
′
)
=
K
(
x
,
x
′
;
t
−
t
′
)
.
{\displaystyle K(x,t;x',t')=K(x,x';t-t').}
The propagator of a one-dimensional free particle, obtainable from, e.g., the path integral, is then
Similarly, the propagator of a one-dimensional quantum harmonic oscillator is the Mehler kernel,
The latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity,
exp
(
−
i
t
ℏ
(
1
2
m
p
2
+
1
2
m
ω
2
x
2
)
)
=
exp
(
−
i
m
ω
2
ℏ
x
2
tan
ω
t
2
)
exp
(
−
i
2
m
ω
ℏ
p
2
sin
(
ω
t
)
)
exp
(
−
i
m
ω
2
ℏ
x
2
tan
ω
t
2
)
,
{\displaystyle {\begin{aligned}&\exp \left(-{\frac {it}{\hbar }}\left({\frac {1}{2m}}{\mathsf {p}}^{2}+{\frac {1}{2}}m\omega ^{2}{\mathsf {x}}^{2}\right)\right)\\&=\exp \left(-{\frac {im\omega }{2\hbar }}{\mathsf {x}}^{2}\tan {\frac {\omega t}{2}}\right)\exp \left(-{\frac {i}{2m\omega \hbar }}{\mathsf {p}}^{2}\sin(\omega t)\right)\exp \left(-{\frac {im\omega }{2\hbar }}{\mathsf {x}}^{2}\tan {\frac {\omega t}{2}}\right),\end{aligned}}}
valid for operators
x
{\displaystyle {\mathsf {x}}}
and
p
{\displaystyle {\mathsf {p}}}
satisfying the Heisenberg relation
[
x
,
p
]
=
i
ℏ
{\displaystyle [{\mathsf {x}},{\mathsf {p}}]=i\hbar }
.
For the N-dimensional case, the propagator can be simply obtained by the product
K
(
x
→
,
x
→
′
;
t
)
=
∏
q
=
1
N
K
(
x
q
,
x
q
′
;
t
)
.
{\displaystyle K({\vec {x}},{\vec {x}}';t)=\prod _{q=1}^{N}K(x_{q},x_{q}';t).}
== Relativistic propagators ==
In relativistic quantum mechanics and quantum field theory the propagators are Lorentz-invariant. They give the amplitude for a particle to travel between two spacetime events.
=== Scalar propagator ===
In quantum field theory, the theory of a free (or non-interacting) scalar field is a useful and simple example which serves to illustrate the concepts needed for more complicated theories. It describes spin-zero particles. There are a number of possible propagators for free scalar field theory. We now describe the most common ones.
=== Position space ===
The position space propagators are Green's functions for the Klein–Gordon equation. This means that they are functions G(x, y) satisfying
(
◻
x
+
m
2
)
G
(
x
,
y
)
=
−
δ
(
x
−
y
)
,
{\displaystyle \left(\square _{x}+m^{2}\right)G(x,y)=-\delta (x-y),}
where
x, y are two points in Minkowski spacetime,
◻
x
=
∂
2
∂
t
2
−
∇
2
{\displaystyle \square _{x}={\tfrac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}}
is the d'Alembertian operator acting on the x coordinates,
δ(x − y) is the Dirac delta function.
(As typical in relativistic quantum field theory calculations, we use units where the speed of light c and the reduced Planck constant ħ are set to unity.)
We shall restrict attention to 4-dimensional Minkowski spacetime. We can perform a Fourier transform of the equation for the propagator, obtaining
(
−
p
2
+
m
2
)
G
(
p
)
=
−
1.
{\displaystyle \left(-p^{2}+m^{2}\right)G(p)=-1.}
This equation can be inverted in the sense of distributions, noting that the equation xf(x) = 1 has the solution (see Sokhotski–Plemelj theorem)
f
(
x
)
=
1
x
±
i
ε
=
1
x
∓
i
π
δ
(
x
)
,
{\displaystyle f(x)={\frac {1}{x\pm i\varepsilon }}={\frac {1}{x}}\mp i\pi \delta (x),}
with ε implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements.
The solution is
where
p
(
x
−
y
)
:=
p
0
(
x
0
−
y
0
)
−
p
→
⋅
(
x
→
−
y
→
)
{\displaystyle p(x-y):=p_{0}(x^{0}-y^{0})-{\vec {p}}\cdot ({\vec {x}}-{\vec {y}})}
is the 4-vector inner product.
The different choices for how to deform the integration contour in the above expression lead to various forms for the propagator. The choice of contour is usually phrased in terms of the
p
0
{\displaystyle p_{0}}
integral.
The integrand then has two poles at
p
0
=
±
p
→
2
+
m
2
,
{\displaystyle p_{0}=\pm {\sqrt {{\vec {p}}^{2}+m^{2}}},}
so different choices of how to avoid these lead to different propagators.
=== Causal propagators ===
==== Retarded propagator ====
A contour going clockwise over both poles gives the causal retarded propagator. This is zero if x-y is spacelike or y is to the future of x, so it is zero if x ⁰< y ⁰.
This choice of contour is equivalent to calculating the limit,
G
ret
(
x
,
y
)
=
lim
ε
→
0
1
(
2
π
)
4
∫
d
4
p
e
−
i
p
(
x
−
y
)
(
p
0
+
i
ε
)
2
−
p
→
2
−
m
2
=
−
Θ
(
x
0
−
y
0
)
2
π
δ
(
τ
x
y
2
)
+
Θ
(
x
0
−
y
0
)
Θ
(
τ
x
y
2
)
m
J
1
(
m
τ
x
y
)
4
π
τ
x
y
.
{\displaystyle G_{\text{ret}}(x,y)=\lim _{\varepsilon \to 0}{\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{(p_{0}+i\varepsilon )^{2}-{\vec {p}}^{2}-m^{2}}}=-{\frac {\Theta (x^{0}-y^{0})}{2\pi }}\delta (\tau _{xy}^{2})+\Theta (x^{0}-y^{0})\Theta (\tau _{xy}^{2}){\frac {mJ_{1}(m\tau _{xy})}{4\pi \tau _{xy}}}.}
Here
Θ
(
x
)
:=
{
1
x
≥
0
0
x
<
0
{\displaystyle \Theta (x):={\begin{cases}1&x\geq 0\\0&x<0\end{cases}}}
is the Heaviside step function,
τ
x
y
:=
(
x
0
−
y
0
)
2
−
(
x
→
−
y
→
)
2
{\displaystyle \tau _{xy}:={\sqrt {(x^{0}-y^{0})^{2}-({\vec {x}}-{\vec {y}})^{2}}}}
is the proper time from x to y, and
J
1
{\displaystyle J_{1}}
is a Bessel function of the first kind. The propagator is non-zero only if
y
≺
x
{\displaystyle y\prec x}
, i.e., y causally precedes x, which, for Minkowski spacetime, means
y
0
≤
x
0
{\displaystyle y^{0}\leq x^{0}}
and
τ
x
y
2
≥
0
.
{\displaystyle \tau _{xy}^{2}\geq 0~.}
This expression can be related to the vacuum expectation value of the commutator of the free scalar field operator,
G
ret
(
x
,
y
)
=
−
i
⟨
0
|
[
Φ
(
x
)
,
Φ
(
y
)
]
|
0
⟩
Θ
(
x
0
−
y
0
)
,
{\displaystyle G_{\text{ret}}(x,y)=-i\langle 0|\left[\Phi (x),\Phi (y)\right]|0\rangle \Theta (x^{0}-y^{0}),}
where
[
Φ
(
x
)
,
Φ
(
y
)
]
:=
Φ
(
x
)
Φ
(
y
)
−
Φ
(
y
)
Φ
(
x
)
.
{\displaystyle \left[\Phi (x),\Phi (y)\right]:=\Phi (x)\Phi (y)-\Phi (y)\Phi (x).}
==== Advanced propagator ====
A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if x-y is spacelike or if y is to the past of x, so it is zero if x ⁰> y ⁰.
This choice of contour is equivalent to calculating the limit
G
adv
(
x
,
y
)
=
lim
ε
→
0
1
(
2
π
)
4
∫
d
4
p
e
−
i
p
(
x
−
y
)
(
p
0
−
i
ε
)
2
−
p
→
2
−
m
2
=
−
Θ
(
y
0
−
x
0
)
2
π
δ
(
τ
x
y
2
)
+
Θ
(
y
0
−
x
0
)
Θ
(
τ
x
y
2
)
m
J
1
(
m
τ
x
y
)
4
π
τ
x
y
.
{\displaystyle G_{\text{adv}}(x,y)=\lim _{\varepsilon \to 0}{\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{(p_{0}-i\varepsilon )^{2}-{\vec {p}}^{2}-m^{2}}}=-{\frac {\Theta (y^{0}-x^{0})}{2\pi }}\delta (\tau _{xy}^{2})+\Theta (y^{0}-x^{0})\Theta (\tau _{xy}^{2}){\frac {mJ_{1}(m\tau _{xy})}{4\pi \tau _{xy}}}.}
This expression can also be expressed in terms of the vacuum expectation value of the commutator of the free scalar field.
In this case,
G
adv
(
x
,
y
)
=
i
⟨
0
|
[
Φ
(
x
)
,
Φ
(
y
)
]
|
0
⟩
Θ
(
y
0
−
x
0
)
.
{\displaystyle G_{\text{adv}}(x,y)=i\langle 0|\left[\Phi (x),\Phi (y)\right]|0\rangle \Theta (y^{0}-x^{0})~.}
==== Feynman propagator ====
A contour going under the left pole and over the right pole gives the Feynman propagator, introduced by Richard Feynman in 1948.
This choice of contour is equivalent to calculating the limit
G
F
(
x
,
y
)
=
lim
ε
→
0
1
(
2
π
)
4
∫
d
4
p
e
−
i
p
(
x
−
y
)
p
2
−
m
2
+
i
ε
=
{
−
1
4
π
δ
(
τ
x
y
2
)
+
m
8
π
τ
x
y
H
1
(
1
)
(
m
τ
x
y
)
τ
x
y
2
≥
0
−
i
m
4
π
2
−
τ
x
y
2
K
1
(
m
−
τ
x
y
2
)
τ
x
y
2
<
0.
{\displaystyle G_{F}(x,y)=\lim _{\varepsilon \to 0}{\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{p^{2}-m^{2}+i\varepsilon }}={\begin{cases}-{\frac {1}{4\pi }}\delta (\tau _{xy}^{2})+{\frac {m}{8\pi \tau _{xy}}}H_{1}^{(1)}(m\tau _{xy})&\tau _{xy}^{2}\geq 0\\-{\frac {im}{4\pi ^{2}{\sqrt {-\tau _{xy}^{2}}}}}K_{1}(m{\sqrt {-\tau _{xy}^{2}}})&\tau _{xy}^{2}<0.\end{cases}}}
Here, H1(1) is a Hankel function and K1 is a modified Bessel function.
This expression can be derived directly from the field theory as the vacuum expectation value of the time-ordered product of the free scalar field, that is, the product always taken such that the time ordering of the spacetime points is the same,
G
F
(
x
−
y
)
=
−
i
⟨
0
|
T
(
Φ
(
x
)
Φ
(
y
)
)
|
0
⟩
=
−
i
⟨
0
|
[
Θ
(
x
0
−
y
0
)
Φ
(
x
)
Φ
(
y
)
+
Θ
(
y
0
−
x
0
)
Φ
(
y
)
Φ
(
x
)
]
|
0
⟩
.
{\displaystyle {\begin{aligned}G_{F}(x-y)&=-i\langle 0|T(\Phi (x)\Phi (y))|0\rangle \\[4pt]&=-i\left\langle 0|\left[\Theta (x^{0}-y^{0})\Phi (x)\Phi (y)+\Theta (y^{0}-x^{0})\Phi (y)\Phi (x)\right]|0\right\rangle .\end{aligned}}}
This expression is Lorentz invariant, as long as the field operators commute with one another when the points x and y are separated by a spacelike interval.
The usual derivation is to insert a complete set of single-particle momentum states between the fields with Lorentz covariant normalization, and then to show that the Θ functions providing the causal time ordering may be obtained by a contour integral along the energy axis, if the integrand is as above (hence the infinitesimal imaginary part), to move the pole off the real line.
The propagator may also be derived using the path integral formulation of quantum theory.
==== Dirac propagator ====
Introduced by Paul Dirac in 1938.
=== Momentum space propagator ===
The Fourier transform of the position space propagators can be thought of as propagators in momentum space. These take a much simpler form than the position space propagators.
They are often written with an explicit ε term although this is understood to be a reminder about which integration contour is appropriate (see above). This ε term is included to incorporate boundary conditions and causality (see below).
For a 4-momentum p the causal and Feynman propagators in momentum space are:
G
~
ret
(
p
)
=
1
(
p
0
+
i
ε
)
2
−
p
→
2
−
m
2
{\displaystyle {\tilde {G}}_{\text{ret}}(p)={\frac {1}{(p_{0}+i\varepsilon )^{2}-{\vec {p}}^{2}-m^{2}}}}
G
~
adv
(
p
)
=
1
(
p
0
−
i
ε
)
2
−
p
→
2
−
m
2
{\displaystyle {\tilde {G}}_{\text{adv}}(p)={\frac {1}{(p_{0}-i\varepsilon )^{2}-{\vec {p}}^{2}-m^{2}}}}
G
~
F
(
p
)
=
1
p
2
−
m
2
+
i
ε
.
{\displaystyle {\tilde {G}}_{F}(p)={\frac {1}{p^{2}-m^{2}+i\varepsilon }}.}
For purposes of Feynman diagram calculations, it is usually convenient to write these with an additional overall factor of i (conventions vary).
=== Faster than light? ===
The Feynman propagator has some properties that seem baffling at first. In particular, unlike the commutator, the propagator is nonzero outside of the light cone, though it falls off rapidly for spacelike intervals. Interpreted as an amplitude for particle motion, this translates to the virtual particle travelling faster than light. It is not immediately obvious how this can be reconciled with causality: can we use faster-than-light virtual particles to send faster-than-light messages?
The answer is no: while in classical mechanics the intervals along which particles and causal effects can travel are the same, this is no longer true in quantum field theory, where it is commutators that determine which operators can affect one another.
So what does the spacelike part of the propagator represent? In QFT the vacuum is an active participant, and particle numbers and field values are related by an uncertainty principle; field values are uncertain even for particle number zero. There is a nonzero probability amplitude to find a significant fluctuation in the vacuum value of the field Φ(x) if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially correlated fluctuations to some extent. The nonzero time-ordered product for spacelike-separated fields then just measures the amplitude for a nonlocal correlation in these vacuum fluctuations, analogous to an EPR correlation. Indeed, the propagator is often called a two-point correlation function for the free field.
Since, by the postulates of quantum field theory, all observable operators commute with each other at spacelike separation, messages can no more be sent through these correlations than they can through any other EPR correlations; the correlations are in random variables.
Regarding virtual particles, the propagator at spacelike separation can be thought of as a means of calculating the amplitude for creating a virtual particle-antiparticle pair that eventually disappears into the vacuum, or for detecting a virtual pair emerging from the vacuum. In Feynman's language, such creation and annihilation processes are equivalent to a virtual particle wandering backward and forward through time, which can take it outside of the light cone. However, no signaling back in time is allowed.
==== Explanation using limits ====
This can be made clearer by writing the propagator in the following form for a massless particle:
G
F
ε
(
x
,
y
)
=
ε
(
x
−
y
)
2
+
i
ε
2
.
{\displaystyle G_{F}^{\varepsilon }(x,y)={\frac {\varepsilon }{(x-y)^{2}+i\varepsilon ^{2}}}.}
This is the usual definition but normalised by a factor of
ε
{\displaystyle \varepsilon }
. Then the rule is that one only takes the limit
ε
→
0
{\displaystyle \varepsilon \to 0}
at the end of a calculation.
One sees that
G
F
ε
(
x
,
y
)
=
1
ε
if
(
x
−
y
)
2
=
0
,
{\displaystyle G_{F}^{\varepsilon }(x,y)={\frac {1}{\varepsilon }}\quad {\text{if}}~~~(x-y)^{2}=0,}
and
lim
ε
→
0
G
F
ε
(
x
,
y
)
=
0
if
(
x
−
y
)
2
≠
0.
{\displaystyle \lim _{\varepsilon \to 0}G_{F}^{\varepsilon }(x,y)=0\quad {\text{if}}~~~(x-y)^{2}\neq 0.}
Hence this means that a single massless particle will always stay on the light cone. It is also shown that the total probability for a photon at any time must be normalised by the reciprocal of the following factor:
lim
ε
→
0
∫
|
G
F
ε
(
0
,
x
)
|
2
d
x
3
=
lim
ε
→
0
∫
ε
2
(
x
2
−
t
2
)
2
+
ε
4
d
x
3
=
2
π
2
|
t
|
.
{\displaystyle \lim _{\varepsilon \to 0}\int |G_{F}^{\varepsilon }(0,x)|^{2}\,dx^{3}=\lim _{\varepsilon \to 0}\int {\frac {\varepsilon ^{2}}{(\mathbf {x} ^{2}-t^{2})^{2}+\varepsilon ^{4}}}\,dx^{3}=2\pi ^{2}|t|.}
We see that the parts outside the light cone usually are zero in the limit and only are important in Feynman diagrams.
=== Propagators in Feynman diagrams ===
The most common use of the propagator is in calculating probability amplitudes for particle interactions using Feynman diagrams. These calculations are usually carried out in momentum space. In general, the amplitude gets a factor of the propagator for every internal line, that is, every line that does not represent an incoming or outgoing particle in the initial or final state. It will also get a factor proportional to, and similar in form to, an interaction term in the theory's Lagrangian for every internal vertex where lines meet. These prescriptions are known as Feynman rules.
Internal lines correspond to virtual particles. Since the propagator does not vanish for combinations of energy and momentum disallowed by the classical equations of motion, we say that the virtual particles are allowed to be off shell. In fact, since the propagator is obtained by inverting the wave equation, in general, it will have singularities on shell.
The energy carried by the particle in the propagator can even be negative. This can be interpreted simply as the case in which, instead of a particle going one way, its antiparticle is going the other way, and therefore carrying an opposing flow of positive energy. The propagator encompasses both possibilities. It does mean that one has to be careful about minus signs for the case of fermions, whose propagators are not even functions in the energy and momentum (see below).
Virtual particles conserve energy and momentum. However, since they can be off shell, wherever the diagram contains a closed loop, the energies and momenta of the virtual particles participating in the loop will be partly unconstrained, since a change in a quantity for one particle in the loop can be balanced by an equal and opposite change in another. Therefore, every loop in a Feynman diagram requires an integral over a continuum of possible energies and momenta. In general, these integrals of products of propagators can diverge, a situation that must be handled by the process of renormalization.
=== Other theories ===
==== Spin 1⁄2 ====
If the particle possesses spin then its propagator is in general somewhat more complicated, as it will involve the particle's spin or polarization indices. The differential equation satisfied by the propagator for a spin 1⁄2 particle is given by
(
i
∇̸
′
−
m
)
S
F
(
x
′
,
x
)
=
I
4
δ
4
(
x
′
−
x
)
,
{\displaystyle (i\not \nabla '-m)S_{F}(x',x)=I_{4}\delta ^{4}(x'-x),}
where I4 is the unit matrix in four dimensions, and employing the Feynman slash notation. This is the Dirac equation for a delta function source in spacetime. Using the momentum representation,
S
F
(
x
′
,
x
)
=
∫
d
4
p
(
2
π
)
4
exp
[
−
i
p
⋅
(
x
′
−
x
)
]
S
~
F
(
p
)
,
{\displaystyle S_{F}(x',x)=\int {\frac {d^{4}p}{(2\pi )^{4}}}\exp {\left[-ip\cdot (x'-x)\right]}{\tilde {S}}_{F}(p),}
the equation becomes
(
i
∇̸
′
−
m
)
∫
d
4
p
(
2
π
)
4
S
~
F
(
p
)
exp
[
−
i
p
⋅
(
x
′
−
x
)
]
=
∫
d
4
p
(
2
π
)
4
(
p̸
−
m
)
S
~
F
(
p
)
exp
[
−
i
p
⋅
(
x
′
−
x
)
]
=
∫
d
4
p
(
2
π
)
4
I
4
exp
[
−
i
p
⋅
(
x
′
−
x
)
]
=
I
4
δ
4
(
x
′
−
x
)
,
{\displaystyle {\begin{aligned}&(i\not \nabla '-m)\int {\frac {d^{4}p}{(2\pi )^{4}}}{\tilde {S}}_{F}(p)\exp {\left[-ip\cdot (x'-x)\right]}\\[6pt]={}&\int {\frac {d^{4}p}{(2\pi )^{4}}}(\not p-m){\tilde {S}}_{F}(p)\exp {\left[-ip\cdot (x'-x)\right]}\\[6pt]={}&\int {\frac {d^{4}p}{(2\pi )^{4}}}I_{4}\exp {\left[-ip\cdot (x'-x)\right]}\\[6pt]={}&I_{4}\delta ^{4}(x'-x),\end{aligned}}}
where on the right-hand side an integral representation of the four-dimensional delta function is used. Thus
(
p̸
−
m
I
4
)
S
~
F
(
p
)
=
I
4
.
{\displaystyle (\not p-mI_{4}){\tilde {S}}_{F}(p)=I_{4}.}
By multiplying from the left with
(
p̸
+
m
)
{\displaystyle (\not p+m)}
(dropping unit matrices from the notation) and using properties of the gamma matrices,
p̸
p̸
=
1
2
(
p̸
p̸
+
p̸
p̸
)
=
1
2
(
γ
μ
p
μ
γ
ν
p
ν
+
γ
ν
p
ν
γ
μ
p
μ
)
=
1
2
(
γ
μ
γ
ν
+
γ
ν
γ
μ
)
p
μ
p
ν
=
g
μ
ν
p
μ
p
ν
=
p
ν
p
ν
=
p
2
,
{\displaystyle {\begin{aligned}\not p\not p&={\tfrac {1}{2}}(\not p\not p+\not p\not p)\\[6pt]&={\tfrac {1}{2}}(\gamma _{\mu }p^{\mu }\gamma _{\nu }p^{\nu }+\gamma _{\nu }p^{\nu }\gamma _{\mu }p^{\mu })\\[6pt]&={\tfrac {1}{2}}(\gamma _{\mu }\gamma _{\nu }+\gamma _{\nu }\gamma _{\mu })p^{\mu }p^{\nu }\\[6pt]&=g_{\mu \nu }p^{\mu }p^{\nu }=p_{\nu }p^{\nu }=p^{2},\end{aligned}}}
the momentum-space propagator used in Feynman diagrams for a Dirac field representing the electron in quantum electrodynamics is found to have form
S
~
F
(
p
)
=
(
p̸
+
m
)
p
2
−
m
2
+
i
ε
=
(
γ
μ
p
μ
+
m
)
p
2
−
m
2
+
i
ε
.
{\displaystyle {\tilde {S}}_{F}(p)={\frac {(\not p+m)}{p^{2}-m^{2}+i\varepsilon }}={\frac {(\gamma ^{\mu }p_{\mu }+m)}{p^{2}-m^{2}+i\varepsilon }}.}
The iε downstairs is a prescription for how to handle the poles in the complex p0-plane. It automatically yields the Feynman contour of integration by shifting the poles appropriately. It is sometimes written
S
~
F
(
p
)
=
1
γ
μ
p
μ
−
m
+
i
ε
=
1
p̸
−
m
+
i
ε
{\displaystyle {\tilde {S}}_{F}(p)={1 \over \gamma ^{\mu }p_{\mu }-m+i\varepsilon }={1 \over \not p-m+i\varepsilon }}
for short. It should be remembered that this expression is just shorthand notation for (γμpμ − m)−1. "One over matrix" is otherwise nonsensical. In position space one has
S
F
(
x
−
y
)
=
∫
d
4
p
(
2
π
)
4
e
−
i
p
⋅
(
x
−
y
)
γ
μ
p
μ
+
m
p
2
−
m
2
+
i
ε
=
(
γ
μ
(
x
−
y
)
μ
|
x
−
y
|
5
+
m
|
x
−
y
|
3
)
J
1
(
m
|
x
−
y
|
)
.
{\displaystyle S_{F}(x-y)=\int {\frac {d^{4}p}{(2\pi )^{4}}}\,e^{-ip\cdot (x-y)}{\frac {\gamma ^{\mu }p_{\mu }+m}{p^{2}-m^{2}+i\varepsilon }}=\left({\frac {\gamma ^{\mu }(x-y)_{\mu }}{|x-y|^{5}}}+{\frac {m}{|x-y|^{3}}}\right)J_{1}(m|x-y|).}
This is related to the Feynman propagator by
S
F
(
x
−
y
)
=
(
i
∂̸
+
m
)
G
F
(
x
−
y
)
{\displaystyle S_{F}(x-y)=(i\not \partial +m)G_{F}(x-y)}
where
∂̸
:=
γ
μ
∂
μ
{\displaystyle \not \partial :=\gamma ^{\mu }\partial _{\mu }}
.
==== Spin 1 ====
The propagator for a gauge boson in a gauge theory depends on the choice of convention to fix the gauge. For the gauge used by Feynman and Stueckelberg, the propagator for a photon is
−
i
g
μ
ν
p
2
+
i
ε
.
{\displaystyle {-ig^{\mu \nu } \over p^{2}+i\varepsilon }.}
The general form with gauge parameter λ, up to overall sign and the factor of
i
{\displaystyle i}
, reads
−
i
g
μ
ν
+
(
1
−
1
λ
)
p
μ
p
ν
p
2
p
2
+
i
ε
.
{\displaystyle -i{\frac {g^{\mu \nu }+\left(1-{\frac {1}{\lambda }}\right){\frac {p^{\mu }p^{\nu }}{p^{2}}}}{p^{2}+i\varepsilon }}.}
The propagator for a massive vector field can be derived from the Stueckelberg Lagrangian. The general form with gauge parameter λ, up to overall sign and the factor of
i
{\displaystyle i}
, reads
g
μ
ν
−
k
μ
k
ν
m
2
k
2
−
m
2
+
i
ε
+
k
μ
k
ν
m
2
k
2
−
m
2
λ
+
i
ε
.
{\displaystyle {\frac {g_{\mu \nu }-{\frac {k_{\mu }k_{\nu }}{m^{2}}}}{k^{2}-m^{2}+i\varepsilon }}+{\frac {\frac {k_{\mu }k_{\nu }}{m^{2}}}{k^{2}-{\frac {m^{2}}{\lambda }}+i\varepsilon }}.}
With these general forms one obtains the propagators in unitary gauge for λ = 0, the propagator in Feynman or 't Hooft gauge for λ = 1 and in Landau or Lorenz gauge for λ = ∞. There are also other notations where the gauge parameter is the inverse of λ, usually denoted ξ (see Rξ gauges). The name of the propagator, however, refers to its final form and not necessarily to the value of the gauge parameter.
Unitary gauge:
g
μ
ν
−
k
μ
k
ν
m
2
k
2
−
m
2
+
i
ε
.
{\displaystyle {\frac {g_{\mu \nu }-{\frac {k_{\mu }k_{\nu }}{m^{2}}}}{k^{2}-m^{2}+i\varepsilon }}.}
Feynman ('t Hooft) gauge:
g
μ
ν
k
2
−
m
2
+
i
ε
.
{\displaystyle {\frac {g_{\mu \nu }}{k^{2}-m^{2}+i\varepsilon }}.}
Landau (Lorenz) gauge:
g
μ
ν
−
k
μ
k
ν
k
2
k
2
−
m
2
+
i
ε
.
{\displaystyle {\frac {g_{\mu \nu }-{\frac {k_{\mu }k_{\nu }}{k^{2}}}}{k^{2}-m^{2}+i\varepsilon }}.}
=== Graviton propagator ===
The graviton propagator for Minkowski space in general relativity is
G
α
β
μ
ν
=
P
α
β
μ
ν
2
k
2
−
P
s
0
α
β
μ
ν
2
k
2
=
g
α
μ
g
β
ν
+
g
β
μ
g
α
ν
−
2
D
−
2
g
μ
ν
g
α
β
k
2
,
{\displaystyle G_{\alpha \beta ~\mu \nu }={\frac {{\mathcal {P}}_{\alpha \beta ~\mu \nu }^{2}}{k^{2}}}-{\frac {{\mathcal {P}}_{s}^{0}{}_{\alpha \beta ~\mu \nu }}{2k^{2}}}={\frac {g_{\alpha \mu }g_{\beta \nu }+g_{\beta \mu }g_{\alpha \nu }-{\frac {2}{D-2}}g_{\mu \nu }g_{\alpha \beta }}{k^{2}}},}
where
D
{\displaystyle D}
is the number of spacetime dimensions,
P
2
{\displaystyle {\mathcal {P}}^{2}}
is the transverse and traceless spin-2 projection operator and
P
s
0
{\displaystyle {\mathcal {P}}_{s}^{0}}
is a spin-0 scalar multiplet.
The graviton propagator for (Anti) de Sitter space is
G
=
P
2
2
H
2
−
◻
+
P
s
0
2
(
◻
+
4
H
2
)
,
{\displaystyle G={\frac {{\mathcal {P}}^{2}}{2H^{2}-\Box }}+{\frac {{\mathcal {P}}_{s}^{0}}{2(\Box +4H^{2})}},}
where
H
{\displaystyle H}
is the Hubble constant. Note that upon taking the limit
H
→
0
{\displaystyle H\to 0}
and
◻
→
−
k
2
{\displaystyle \Box \to -k^{2}}
, the AdS propagator reduces to the Minkowski propagator.
== Related singular functions ==
The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important in quantum field theory. These functions are most simply defined in terms of the vacuum expectation value of products of field operators.
=== Solutions to the Klein–Gordon equation ===
==== Pauli–Jordan function ====
The commutator of two scalar field operators defines the Pauli–Jordan function
Δ
(
x
−
y
)
{\displaystyle \Delta (x-y)}
by
⟨
0
|
[
Φ
(
x
)
,
Φ
(
y
)
]
|
0
⟩
=
i
Δ
(
x
−
y
)
{\displaystyle \langle 0|\left[\Phi (x),\Phi (y)\right]|0\rangle =i\,\Delta (x-y)}
with
Δ
(
x
−
y
)
=
G
ret
(
x
−
y
)
−
G
adv
(
x
−
y
)
{\displaystyle \,\Delta (x-y)=G_{\text{ret}}(x-y)-G_{\text{adv}}(x-y)}
This satisfies
Δ
(
x
−
y
)
=
−
Δ
(
y
−
x
)
{\displaystyle \Delta (x-y)=-\Delta (y-x)}
and is zero if
(
x
−
y
)
2
<
0
{\displaystyle (x-y)^{2}<0}
.
==== Positive and negative frequency parts (cut propagators) ====
We can define the positive and negative frequency parts of
Δ
(
x
−
y
)
{\displaystyle \Delta (x-y)}
, sometimes called cut propagators, in a relativistically invariant way.
This allows us to define the positive frequency part:
Δ
+
(
x
−
y
)
=
⟨
0
|
Φ
(
x
)
Φ
(
y
)
|
0
⟩
,
{\displaystyle \Delta _{+}(x-y)=\langle 0|\Phi (x)\Phi (y)|0\rangle ,}
and the negative frequency part:
Δ
−
(
x
−
y
)
=
⟨
0
|
Φ
(
y
)
Φ
(
x
)
|
0
⟩
.
{\displaystyle \Delta _{-}(x-y)=\langle 0|\Phi (y)\Phi (x)|0\rangle .}
These satisfy
i
Δ
=
Δ
+
−
Δ
−
{\displaystyle \,i\Delta =\Delta _{+}-\Delta _{-}}
and
(
◻
x
+
m
2
)
Δ
±
(
x
−
y
)
=
0.
{\displaystyle (\Box _{x}+m^{2})\Delta _{\pm }(x-y)=0.}
==== Auxiliary function ====
The anti-commutator of two scalar field operators defines
Δ
1
(
x
−
y
)
{\displaystyle \Delta _{1}(x-y)}
function by
⟨
0
|
{
Φ
(
x
)
,
Φ
(
y
)
}
|
0
⟩
=
Δ
1
(
x
−
y
)
{\displaystyle \langle 0|\left\{\Phi (x),\Phi (y)\right\}|0\rangle =\Delta _{1}(x-y)}
with
Δ
1
(
x
−
y
)
=
Δ
+
(
x
−
y
)
+
Δ
−
(
x
−
y
)
.
{\displaystyle \,\Delta _{1}(x-y)=\Delta _{+}(x-y)+\Delta _{-}(x-y).}
This satisfies
Δ
1
(
x
−
y
)
=
Δ
1
(
y
−
x
)
.
{\displaystyle \,\Delta _{1}(x-y)=\Delta _{1}(y-x).}
=== Green's functions for the Klein–Gordon equation ===
The retarded, advanced and Feynman propagators defined above are all Green's functions for the Klein–Gordon equation.
They are related to the singular functions by
G
ret
(
x
−
y
)
=
Δ
(
x
−
y
)
Θ
(
x
0
−
y
0
)
{\displaystyle G_{\text{ret}}(x-y)=\Delta (x-y)\Theta (x^{0}-y^{0})}
G
adv
(
x
−
y
)
=
−
Δ
(
x
−
y
)
Θ
(
y
0
−
x
0
)
{\displaystyle G_{\text{adv}}(x-y)=-\Delta (x-y)\Theta (y^{0}-x^{0})}
2
G
F
(
x
−
y
)
=
−
i
Δ
1
(
x
−
y
)
+
ε
(
x
0
−
y
0
)
Δ
(
x
−
y
)
{\displaystyle 2G_{F}(x-y)=-i\,\Delta _{1}(x-y)+\varepsilon (x^{0}-y^{0})\,\Delta (x-y)}
where
ε
(
x
0
−
y
0
)
{\displaystyle \varepsilon (x^{0}-y^{0})}
is the sign of
x
0
−
y
0
{\displaystyle x^{0}-y^{0}}
.
== See also ==
Source field
LSZ reduction formula
== Notes ==
== References ==
Bjorken, J.; Drell, S. (1965). Relativistic Quantum Fields. New York: McGraw-Hill. ISBN 0-07-005494-0. (Appendix C.)
Bogoliubov, N.; Shirkov, D. V. (1959). Introduction to the theory of quantized fields. Wiley-Interscience. ISBN 0-470-08613-0. {{cite book}}: ISBN / Date incompatibility (help) (Especially pp. 136–156 and Appendix A)
Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2019). Quantum Mechanics, Volume 1. Weinheim: John Wiley & Sons. ISBN 978-3-527-34553-3.
DeWitt-Morette, C.; DeWitt, B. (eds.). Relativity, Groups and Topology. Glasgow: Blackie and Son. ISBN 0-444-86858-5. (section Dynamical Theory of Groups & Fields, Especially pp. 615–624)
Greiner, W.; Reinhardt, J. (2008). Quantum Electrodynamics (4th ed.). Springer Verlag. ISBN 9783540875604.
Greiner, W.; Reinhardt, J. (1996). Field Quantization. Springer Verlag. ISBN 9783540591795.
Griffiths, D. J. (1987). Introduction to Elementary Particles. New York: John Wiley & Sons. ISBN 0-471-60386-4.
Griffiths, D. J. (2004). Introduction to Quantum Mechanics. Upper Saddle River: Prentice Hall. ISBN 0-131-11892-7.
Halliwell, J.J.; Orwitz, M. (1993), "Sum-over-histories origin of the composition laws of relativistic quantum mechanics and quantum cosmology", Physical Review D, 48 (2): 748–768, arXiv:gr-qc/9211004, Bibcode:1993PhRvD..48..748H, doi:10.1103/PhysRevD.48.748, PMID 10016304, S2CID 16381314
Huang, Kerson (1998). Quantum Field Theory: From Operators to Path Integrals. New York: John Wiley & Sons. ISBN 0-471-14120-8.
Itzykson, C.; Zuber, J-B. (1980). Quantum Field Theory. New York: McGraw-Hill. ISBN 0-07-032071-3.
Pokorski, S. (1987). Gauge Field Theories. Cambridge: Cambridge University Press. ISBN 0-521-36846-4. (Has useful appendices of Feynman diagram rules, including propagators, in the back.)
Schulman, L. S. (1981). Techniques & Applications of Path Integration. New York: John Wiley & Sons. ISBN 0-471-76450-7.
Scharf, G. (1995). Finite Quantum Electrodynamics, The Causal Approach. Springer. ISBN 978-3-642-63345-4.
== External links ==
Three Methods for Computing the Feynman Propagator | Wikipedia/Propagator_(Quantum_Theory) |
Axiomatic quantum field theory is a mathematical discipline which aims to describe quantum field theory in terms of rigorous axioms. It is strongly associated with functional analysis and operator algebras, but has also been studied in recent years from a more geometric and functorial perspective.
There are two main challenges in this discipline. First, one must propose a set of axioms which describe the general properties of any mathematical object that deserves to be called a "quantum field theory". Then, one gives rigorous mathematical constructions of examples satisfying these axioms.
== Analytic approaches ==
=== Wightman axioms ===
The first set of axioms for quantum field theories, known as the Wightman axioms, were proposed by Arthur Wightman in the early 1950s. These axioms attempt to describe QFTs on flat Minkowski spacetime by regarding quantum fields as operator-valued distributions acting on a Hilbert space. In practice, one often uses the Wightman reconstruction theorem, which guarantees that the operator-valued distributions and the Hilbert space can be recovered from the collection of correlation functions.
=== Osterwalder–Schrader axioms ===
The correlation functions of a QFT satisfying the Wightman axioms often can be analytically continued from Lorentz signature to Euclidean signature. (Crudely, one replaces the time variable
t
{\displaystyle \;t\;}
with imaginary time
τ
=
−
−
1
t
;
{\displaystyle \;\tau =-{\sqrt {-1\,}}\,t~;}
the factors of
−
1
{\displaystyle \;{\sqrt {-1\,}}\;}
change the sign of the time-time components of the metric tensor.) The resulting functions are called Schwinger functions. For the Schwinger functions there is a list of conditions — analyticity, permutation symmetry, Euclidean covariance, and reflection positivity — which a set of functions defined on various powers of Euclidean space-time must satisfy in order to be the analytic continuation of the set of correlation functions of a QFT satisfying the Wightman axioms.
=== Haag–Kastler axioms ===
The Haag–Kastler axioms axiomatize QFT in terms of nets of algebras.
=== Euclidean CFT axioms ===
These axioms (see e.g.) are used in the conformal bootstrap approach to conformal field theory in
R
d
{\displaystyle \mathbb {R} ^{d}}
. They are also referred to as Euclidean bootstrap axioms.
== See also ==
Dirac–von Neumann axioms
== References ==
Streater, R. F.; Wightman, A. S. (1964). PCT, Spin and Statistics, and All That. New York: W. A. Benjamin. OCLC 930068.
Bogoliubov, N.; Logunov, A.; Todorov, I. (1975). Introduction to Axiomatic Quantum Field Theory. Reading, Massachusetts: W. A. Benjamin. OCLC 1527225.
Araki, H. (1999). Mathematical Theory of Quantum Fields. Oxford University Press. ISBN 0-19-851773-4. | Wikipedia/Axiomatic_quantum_field_theory |
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