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If a rope is lying in equilibrium under tangential forces on a rough orthotropic surface then all three following conditions are satisfied: |
This generalization has been obtained by Konyukhov. |
In general relativity and tensor calculus, the Palatini identity is: |
where formula_2 denotes the variation of Christoffel symbols and formula_3 indicates covariant differentiation. |
A proof can be found in the entry Einstein–Hilbert action. |
The "same" identity holds for the Lie derivative formula_4. In fact, one has: |
where formula_6 denotes any vector field on the spacetime manifold formula_7. |
The Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) interacting electromagnetically to the first order in perturbation theory. It accounts for magnetic interactions and r... |
The Breit equation is not only an approximation in terms of quantum mechanics, but also in terms of relativity theory as it is not completely invariant with respect to the Lorentz transformation. Just as does the Dirac equation, it treats nuclei as point sources of an external field for the particles it describes. For ... |
is the Dirac Hamiltonian (see Dirac equation) for particle "i" at position r"i" and "φ"(r"i") is the scalar potential at that position; "qi" is the charge of the particle, thus for electrons "qi" = −"e". |
The one-electron Dirac Hamiltonians of the particles, along with their instantaneous Coulomb interactions 1/"rij", form the "Dirac-Coulomb" operator. To this, Breit added the operator (now known as the (frequency-independent) Breit operator): |
where the Dirac matrices for electron "i": a("i") = [αx("i"),αy("i"),αz("i")]. The two terms in the Breit operator account for retardation effects to the first order. |
The wave function Ψ in the Breit equation is a spinor with 4"N" elements, since each electron is described by a Dirac bispinor with 4 elements as in the Dirac equation, and the total wave function is the tensor product of these. |
The total Hamiltonian of the Breit equation, sometimes called the Dirac-Coulomb-Breit Hamiltonian ("HDCB") can be decomposed into the following practical energy operators for electrons in electric and magnetic fields (also called the Breit-Pauli Hamiltonian), which have well-defined meanings in the interaction of molec... |
in which the consecutive partial operators are: |
Following is a list of the frequently occurring equations in the theory of special relativity. |
To derive the equations of special relativity, one must start with two postulates: |
In this context, "speed of light" really refers to the speed supremum of information transmission or of the movement of ordinary (nonnegative mass) matter, locally, as in a classical vacuum. Thus, a more accurate description would refer to formula_1 rather than the speed of light per se. However, light and other massle... |
From these two postulates, all of special relativity follows. |
In the following, the relative velocity "v" between two inertial frames is restricted fully to the "x"-direction, of a Cartesian coordinate system. |
The following notations are used very often in special relativity: |
where β = formula_7 and "v" is the relative velocity between two inertial frames. |
For two frames at rest, γ = 1, and increases with relative velocity between the two inertial frames. As the relative velocity approaches the speed of light, γ → ∞. |
In this example the time measured in the frame on the vehicle, "t", is known as the proper time. The proper time between two events - such as the event of light being emitted on the vehicle and the event of light being received on the vehicle - is the time between the two events in a frame where the events occur at the... |
This is the formula for length contraction. As there existed a proper time for time dilation, there exists a proper length for length contraction, which in this case is "". The proper length of an object is the length of the object in the frame in which the object is at rest. Also, this contraction only affects the dim... |
In what follows, bold sans serif is used for 4-vectors while normal bold roman is used for ordinary 3-vectors. |
where formula_18 is known as the metric tensor. In special relativity, the metric tensor is the Minkowski metric: |
In the above, "ds"2 is known as the spacetime interval. This inner product is invariant under the Lorentz transformation, that is, |
The sign of the metric and the placement of the "ct", "ct"', "cdt", and "cdt′" time-based terms can vary depending on the author's choice. For instance, many times the time-based terms are placed first in the four-vectors, with the spatial terms following. Also, sometimes "η" is replaced with −"η", making the spatial t... |
It is possible to express the above coordinate transformation via a matrix. To simplify things, it can be best to replace "t", "t′", "dt", and "dt′" with "ct", "ct"', "cdt", and "cdt′", which has the dimensions of distance. So: |
The vectors in the above transformation equation are known as four-vectors, in this case they are specifically the position four-vectors. In general, in special relativity, four-vectors can be transformed from one reference frame to another as follows: |
In the above, formula_28 and formula_29 are the four-vector and the transformed four-vector, respectively, and Λ is the transformation matrix, which, for a given transformation is the same for all four-vectors one might want to transform. So formula_28 can be a four-vector representing position, velocity, or momentum, ... |
Invariance and unification of physical quantities both arise from four-vectors. The inner product of a 4-vector with itself is equal to a scalar (by definition of the inner product), and since the 4-vectors are physical quantities their magnitudes correspond to physical quantities also. |
Doppler shift for emitter and observer moving right towards each other (or directly away): |
Doppler shift for emitter and observer moving in a direction perpendicular to the line connecting them: |
The Elliott formula describes analytically, or with few adjustable parameters such as the dephasing constant, the light absorption or emission spectra of solids. It was originally derived by Roger James Elliott to describe linear absorption based on properties of a single electron–hole pair. The analysis can be extende... |
One of the most accurate theories of semiconductor absorption and photoluminescence is provided by the SBEs and SLEs, respectively. Both of them are systematically derived starting from the many-body/quantum-optical system Hamiltonian and fully describe the resulting quantum dynamics of optical and quantum-optical obse... |
These exciton eigenstates provide valuable insight to SBEs and SLEs, especially, when one analyses the linear semiconductor absorption spectrum or photoluminescence at steady-state conditions. One simply uses the constructed eigenstates to diagonalize the homogeneous parts of the SBEs and SLEs. Under the steady-state c... |
The same approach can be applied to compute absorption spectrum for fields that are in the terahertz (abbreviated as THz) range of electromagnetic radiation. Since the THz-photon energy lies within the meV range, it is mostly resonant with the many-body states, not the interband transitions that are typically in the eV... |
All of these derivations rely on the steady-state conditions and the analytic knowledge of the exciton states. Furthermore, the effect of further many-body contributions, such as the excitation-induced dephasing, can be included microscopically to the Wannier solver, which removes the need to introduce phenomenological... |
Linear absorption of broadband weak optical probe can then be expressed as |
where formula_5 is the probe-photon energy, formula_6 is the oscillator strength of the exciton state formula_2, and formula_8 is the dephasing constant associated with the exciton state formula_2. For a phenomenological description, formula_8 can be used as a single fit parameter, i.e., formula_11. However, a full mic... |
Each of the exciton resonances can produce a peak to the absorption spectrum when the photon energy matches with formula_3. For direct-gap semiconductors, the oscillator strength is proportional to the product of dipole-matrix element squared and formula_18 that vanishes for all states except for those that are spheric... |
In general, the exciton eigen energies consist of a series of bound states that emerge energetically well below the fundamental bandgap energy and a continuum of unbound states that appear for energies above the bandgap. Therefore, a typical semiconductor's low-density absorption spectrum shows a series of exciton reso... |
The concentration of charge carriers influence the shape of the absorption spectrum considerably. For high enough densities, all formula_3 energies correspond to continuum states and some of the oscillators strengths may become negative-valued due to the Pauli-blocking effect. Physically, this can be understood as the ... |
Even though one can understand the principal behavior of semiconductor absorption on the basis of the Elliott formula, detailed predictions of the exact formula_3, formula_6, and formula_12 requires a full many-body computation already for moderate carrier densities. |
After the semiconductor becomes electronically excited, the carrier system relaxes into a quasiequilibrium. At the same time, vacuum-field fluctuations trigger spontaneous recombination of electrons and holes (electronic vacancies) via spontaneous emission of photons. At quasiequilibrium, this yields a steady-state pho... |
that is very similar to the Elliott formula for the optical absorption. As a major difference, the numerator has a new contribution – the spontaneous-emission source |
that contains electron and hole distributions formula_28 and formula_29, respectively, where formula_30 is the carrier momentum. Additionally, formula_31 contains also a direct contribution from exciton populations formula_32 that describes truly bound electron–hole pairs. |
The formula_33 term defines the probability to find an electron and a hole with same formula_34. Such a form is expected for a probability of two uncorrelated events to occur simultaneously at a desired formula_34 value. Therefore, formula_33 is the spontaneous-emission source originating from uncorrelated electron–hol... |
Like for the absorption, a direct-gap semiconductor emits light only at the resonances corresponding to the formula_20-like states. As a typical trend, a quasiequilibrium emission is strongly peaked around the 1"s" resonance because formula_31 is usually largest for the formula_40 ground state. This emission peak often... |
As discussed above, it is often meaningful to tune the electromagnetic field to be resonant with the transitions between two many-body states. For example, one can follow how a bound exciton is excited from its 1"s" ground state to a 2"p" state. In several semiconductor systems, one needs THz fields to induce such tran... |
(\omega) = \mathrm{Im}\left[ \frac{\sum_{\nu, \lambda} S^{\nu, \lambda} (\omega) \Delta N_{\nu,\lambda} - \left[ S^{\nu, \lambda}(-\omega) \Delta N_{\nu,\lambda}\right]^{\star} }{ \omega (\hbar \omega + \mathrm{i} \gamma(\omega))} \right]\;. |
In this notation, the diagonal contributions formula_41 determine the population of formula_2 excitons. The off-diagonal formula_43 elements formally determine transition amplitudes between two exciton states formula_44 and formula_45. For elevated densities, formula_43 build up spontaneously and they describe correlat... |
In contrast to optical absorption and photoluminescence, THz absorption may involve all exciton states. This can be seen from the spectral response function |
that contains the current-matrix elements formula_48 between two exciton states. The unit vector formula_49 is determined by the direction of the THz field. This leads to dipole selection rules among exciton states, in full analog to the atomic dipole selection rules. Each allowed transition produces a resonance in for... |
In contrast to optical and photoluminescence spectroscopy, THz absorption can directly measure the presence of exciton populations in full analogy to atomic spectroscopy. For example, the presence of a pronounced 1"s"-to-2"p" resonance in THz absorption uniquely identifies the presence of excitons as detected experimen... |
The sigma model was introduced by ; the name σ-model comes from a field in their model corresponding to a spinless meson called , 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... |
In conventional particle physics settings, the field is generally taken to be SU(N), or the vector subspace of quotient formula_1 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;... |
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. |
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 formula_2 a field over a spacetim... |
where the formula_5 is the metric tensor on the field space formula_6, and the formula_7 are the derivatives on the underlying spacetime manifold. |
This expression can be unpacked a bit. The field space formula_8 can be chosen to be any Riemannian manifold. Historically, this is the "sigma" of the sigma model; the historically-appropriate symbol formula_9 is avoided here to prevent clashes with many other common usages of formula_9 in geometry. Riemannian manifold... |
The base manifold formula_3 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 formula_18 is just the plain-old covariant deriv... |
Taking formula_26 the Kronecker delta, "i.e." the scalar dot product in Euclidean space, one gets the formula_27 non-linear sigma model. That is, write formula_28 to be the unit vector in formula_29, so that formula_30, with formula_31 the ordinary Euclidean dot product. Then formula_32 the formula_33-sphere, the isome... |
For formula_36, this is the continuum limit of the isotropic ferromagnet on a lattice, i.e. of the classical Heisenberg model. For formula_37, 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... |
as the finite difference on neighboring lattice locations formula_39 Then formula_40 in the limit formula_41, and formula_42 after dropping the constant terms formula_43 (the "bulk magnetization"). |
The sigma model can also be written in a more fully geometric notation, as a fiber bundle with fibers formula_8 over a differentiable manifold formula_3. Given a section formula_2, fix a point formula_47 The pushforward at formula_48 is a map of tangent bundles |
where formula_51 is taken to be an orthonormal vector space basis on formula_52 and formula_53 the vector space basis on formula_54. The formula_55 is a differential form. The sigma model action is then just the conventional inner product on vector-valued "k"-forms |
where the formula_57 is the wedge product, and the formula_58 is the Hodge star. This is an inner product in two different ways. In the first way, given "any" two differentiable forms formula_59 in formula_3, the Hodge dual defines an invariant inner product on the space of differential forms, commonly written as |
The above is an inner product on the space of square-integrable forms, conventionally taken to be the Sobolev space formula_62 In this way, one may write |
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 formula_3, the field formula_22 is a scalar, and so formula_55 can be recognized just the ordinary gradient of a scalar function. The Hodge star is merely a fancy device... |
which is the Dirichlet energy of formula_22. Classical extrema of the action (the solutions to the Lagrange equations) are then those field configurations that minimize the Dirichlet energy of formula_22. Another way to convert this expression into a more easily-recognizable form is to observe that, for a scalar functi... |
where formula_74 is the Laplace–Beltrami operator, "i.e." the ordinary Laplacian when formula_3 is flat. |
That there is "another", second inner product in play simply requires not forgetting that formula_55 is a vector from the point of view of formula_8 itself. That is, given "any" two vectors formula_78, the Riemannian metric formula_79 defines an inner product |
Since formula_55 is vector-valued formula_82 on local charts, one also takes the inner product there as well. More verbosely, |
The tension between these two inner products can be made even more explicit by noting that |
is a bilinear form; it is a pullback of the Riemann metric formula_79. The individual formula_86 can be taken as vielbeins. The Lagrangian density of the sigma model is then |
for formula_88 the metric on formula_19 Given this gluing-together, the formula_55 can be interpreted as a solder form; this is articulated more fully, below. |
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. |
given above. Taking formula_92, the function formula_93 can be interpreted as a wave function, and its Laplacian the kinetic energy of that wave function. The formula_94 is just some geometric machinery reminding one to integrate over all space. The corresponding quantum mechanical notation is formula_95 In flat space,... |
which is just the grand-total kinetic energy of the wave-function formula_98, up to a factor of formula_99. To conclude, the classical sigma model on formula_100 can be interpreted as the quantum mechanics of a free, non-interacting quantum particle. Obviously, adding a term of formula_101 to the Lagrangian results in ... |
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" formula_3 and formula_8 are Riemannian manifolds; the below is written for formula_8, the same can be done for formula_3. The cotangent bund... |
The trivialization supplies canonical coordinates formula_113 on the cotangent bundle. Given the metric tensor formula_79 on formula_8, define the Hamiltonian function |
where, as always, one is careful to note that the inverse of the mertric is used in this definition: formula_117 Famously, the geodesic flow on formula_8 is given by the Hamilton–Jacobi equations |
The geodesic flow is the Hamiltonian flow; the solutions to the above are the geodesics of the manifold. Note, incidentally, that formula_121 along geodesics; the time parameter formula_122 is the distance along the geodesic. |
The sigma model takes the momenta in the two manifolds formula_111 and formula_124 and solders them together, in that formula_55 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 p... |
The space formula_8 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 formula_55 lives... |
For the special case of formula_8 being a Lie group, the formula_79 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 corres... |
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 |
of the "left" and "right" chiral fields, and then constructs the sigma model on the "diagonal" |
Such a quotient space is a symmetric space, and so one can generically take formula_135 where formula_136 is the maximal subgroup of formula_137 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 formula_137 to a metric o... |
In physics, the most common and conventional statement of the sigma model begins with the definition |
Here, the formula_141 is the pullback of the Maurer-Cartan form, for formula_142, onto the spacetime manifold. The formula_143 is a projection onto the odd-parity piece of the Cartan involution. That is, given the Lie algebra formula_144 of formula_137, the involution decompses the space into odd and even parity compon... |
This is instantly recognizable as the first term of the Skyrme model. |
The equivalent metric form of this is to write a group element formula_142 as the geodesic formula_149 of an element formula_150 of the Lie algebra formula_144. The formula_152 are the basis elements for the Lie algebra; the formula_153 are the structure constants of formula_144. |
Plugging this directly into the above and applying the infinitesimal form of the Baker–Campbell–Hausdorff formula promptly leads to the equivalent expression |
where formula_156 is now obviously (proportional to) the Killing form, and the formula_157 are the vielbeins that express the "curved" metric formula_79 in terms of the "flat" metric formula_156. The article on the Baker–Campbell–Hausdorff formula provides an explicit expression for the vielbeins. They can be written a... |
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