Method of quantum characteristics

A one-sided association rule ${\displaystyle f(\xi )\to {\hat {f}}}$ was formulated by Weyl initially with the help of Taylor expansion of functions of operators of the canonical variables

${\displaystyle {\hat {f}}=f({\hat {\xi }})\equiv \sum _{s=0}^{\infty }{\frac {1}{s!}}{\frac {\partial ^{s}f(0)}{\partial \xi ^{i_{1}}\ldots \partial \xi ^{i_{s}}}}{\hat {\xi }}^{i_{1}}\ldots {\hat {\xi }}^{i_{s}}.}$

The operators ${\displaystyle {\hat {\xi }}}$ do not commute, so the Taylor expansion is not defined uniquely. The above prescription uses the symmetrized products of the operators. The real functions correspond to the Hermitian operators. The function ${\displaystyle f(\xi )}$ is called Weyl's symbol of operator ${\displaystyle {\hat {f}}}$.

Under the reverse association ${\displaystyle f(\xi )\leftarrow {\hat {f}}}$, the density matrix turns to the Wigner function.[2] Wigner functions have numerous applications in quantum many-body physics, kinetic theory, collision theory, quantum chemistry.

A refined version of the Weyl–Wigner association rule was proposed by Groenewold[3] and Stratonovich.[4]

The set of operators acting in the Hilbert space is closed under multiplication of operators by ${\displaystyle c}$-numbers and summation. Such a set constitutes a vector space ${\displaystyle \mathbb {V} }$. The association rule formulated with the use of the Taylor expansion preserves operations on the operators. The correspondence can be illustrated with the following diagram:

${\displaystyle \left.{\begin{array}{c}{\begin{array}{c}\left.{\begin{array}{ccc}f(\xi )&\longleftrightarrow &{\hat {f}}\\g(\xi )&\longleftrightarrow &{\hat {g}}\\c\times f(\xi )&\longleftrightarrow &c\times {\hat {f}}\\f(\xi )+g(\xi )&\longleftrightarrow &{\hat {f}}+{\hat {g}}\end{array}}\right\}\;{\text{vector space}}\;\;\mathbb {V} \end{array}}\\{\begin{array}{ccc}{f(\xi )\star g(\xi )}&{\longleftrightarrow }&\;\;{{\hat {f}}{\hat {g}}}\end{array}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{array}}\right\}{\text{algebra}}}$

Here, ${\displaystyle f(\xi )}$ and ${\displaystyle g(\xi )}$ are functions and ${\displaystyle {\hat {f}}}$ and ${\displaystyle {\hat {g}}}$ are the associated operators.

The elements of basis of ${\displaystyle \mathbb {V} }$ are labelled by canonical variables ${\displaystyle \xi ^{i}\in (-\infty ,+\infty )}$. The commonly used Groenewold-Stratonovich basis looks like

${\displaystyle {\hat {B}}(\xi )=\int {\frac {d^{2n}\eta }{(2\pi \hbar )^{n}}}\exp(-{\frac {i}{\hbar }}\eta _{k}(\xi -{\hat {\xi }})^{k})\in \mathbb {V} .}$

The Weyl–Wigner two-sided association rule for function ${\displaystyle f(\xi )}$ and operator ${\displaystyle {\hat {f}}}$ has the form

${\displaystyle f(\xi )=\operatorname {Tr} [{\hat {B}}(\xi ){\hat {f}}],}$

${\displaystyle {\hat {f}}=\int {\frac {d^{2n}\xi }{(2\pi \hbar )^{n}}}f(\xi ){\hat {B}}(\xi ).}$

The function ${\displaystyle f(\xi )}$ provides coordinates of the operator ${\displaystyle {\hat {f}}}$ in the basis ${\displaystyle {\hat {B}}(\xi )}$. The basis is complete and orthogonal:

${\displaystyle \int {\frac {d^{2n}\xi }{(2\pi \hbar )^{n}}}{\hat {B}}(\xi )\operatorname {Tr} [{\hat {B}}(\xi ){\hat {f}}]={\hat {f}},}$

${\displaystyle \operatorname {Tr} [{\hat {B}}(\xi ){\hat {B}}(\xi ^{\prime })]=(2\pi \hbar )^{n}\delta ^{2n}(\xi -\xi ^{\prime }).}$

Alternative operator bases are discussed also. [5] The freedom in choice of the operator basis is better known as the operator ordering problem. The coordinates of particle trajectories in phase space depend on the operator basis.

The set of operators of canonical variables is complete in the sense that any operator can be represented as a function of operators ${\displaystyle {\hat {\xi }}}$. Transformations

${\displaystyle {\hat {f}}\rightarrow {\acute {\hat {f}}}={\hat {U}}^{\dagger }{\hat {f}}{\hat {U}}}$

induce, under the Wigner association rule, transformations of phase-space functions,

${\displaystyle {\begin{aligned}&{}f(\xi ){\stackrel {q}{\longrightarrow }}{\acute {f}}(\xi )=\mathrm {Tr} [{\hat {B}}(\xi ){\hat {U}}^{\dagger }{\hat {f}}{\hat {U}}]\\&{}\updownarrow \;\;\;\;\;\;\;\;\;\;\,\updownarrow \\&{}{\hat {f}}\;\;\;\;{\stackrel {\hat {U}}{\longrightarrow }}\,{\acute {\hat {f}}}\;\;\;\;\;={\hat {U}}^{\dagger }{\hat {f}}{\hat {U}}\end{aligned}}}$

Using the Taylor expansion, the transformation of function ${\displaystyle f(\xi )}$ under evolution can be found to be

${\displaystyle f(\xi )\rightarrow {\acute {f}}(\xi )\equiv \mathrm {Tr} [{\hat {B}}(\xi ){\hat {U^{\dagger }}}f({\hat {\xi }}){\hat {U}}]=\sum _{s=0}^{\infty }{\frac {1}{s!}}{\frac {\partial ^{s}f(0)}{\partial \xi ^{i_{1}}\ldots \partial \xi ^{i_{s}}}}q^{i_{1}}(\xi ,\tau )\star \ldots \star q^{i_{s}}(\xi ,\tau )\equiv f(\star q(\xi ,\tau )).}$

The composite function defined in such a way is called ${\displaystyle \star }$-function.

The composition law differs from the classical one. However, the semiclassical expansion of ${\displaystyle f(\star q(\xi ,\tau ))}$ around ${\displaystyle f(q(\xi ,\tau ))}$ is formally well defined and involves even powers of ${\displaystyle \hbar }$ only. This equation shows that, given how quantum characteristics are constructed, the physical observables can be found without further reference to the Hamiltonian. The functions ${\displaystyle q^{i}(\xi ,\tau )}$ play the role of characteristics,[8] similarly to the classical characteristics used to solve the classical Liouville equation.

The Wigner transform of the evolution equation for the density matrix in the Schrödinger representation leads to a quantum Liouville equation for the Wigner function. The Wigner transform of the evolution equation for operators in the Heisenberg representation,

${\displaystyle {\frac {\partial }{\partial \tau }}{\hat {f}}=-{\frac {i}{\hbar }}[{\hat {f}},{\hat {H}}],}$

leads to the same equation with the opposite (plus) sign in the right-hand side:

${\displaystyle {\frac {\partial }{\partial \tau }}f(\xi ,\tau )=f(\xi ,\tau )\wedge H(\xi ).}$

${\displaystyle \star }$-function solves this equation in terms of quantum characteristics:

${\displaystyle f(\xi ,\tau )=f(\star q(\xi ,\tau ),0).}$

Similarly, the evolution of the Wigner function in the Schrödinger representation is given by

${\displaystyle W(\xi ,\tau )=W(\star q(\xi ,-\tau ),0).}$

The Liouville theorem of classical mechanics fails, to the extent that, locally, the phase space volume is not preserved in time. In fact, the quantum phase flow does not preserve all differential forms ${\displaystyle \omega ^{2s}}$ defined by exterior powers of ${\displaystyle \omega ^{2}=I^{kl}d\xi _{k}\curlywedge d\xi _{l}}$.

The Wigner function represents a quantum system in a more general form than the wave function. Wave functions describe pure states, while the Wigner function characterizes ensembles of quantum states. Any Hermitian operator can be diagonalized:

${\displaystyle {\hat {f}}=\sum _{s}\lambda _{s}|s\rangle \langle s|}$.

Those operators whose eigenvalues ${\displaystyle \lambda _{s}}$ are non-negative and sum to a finite number can be mapped to density matrices, i.e., to some physical states. The Wigner function is an image of the density matrix, so the Wigner functions admit a similar decomposition:

${\displaystyle W(\xi )=\sum _{s}\lambda _{s}W_{s}(\xi ),}$

with ${\displaystyle \lambda _{s}\geq 0}$ and

${\displaystyle W_{s}(\xi )\star W_{r}(\xi )=\delta _{sr}W_{s}(\xi )}$.

The Quantum Hamilton's equations can be obtained applying the Wigner transform to the evolution equations for Heisenberg operators of canonical coordinates and momenta,

${\displaystyle {\frac {\partial }{\partial \tau }}q^{i}(\xi ,\tau )=\{\zeta ^{i},H(\zeta )\}|_{\zeta =\star q(\xi ,\tau )}.}$

The right-hand side is calculated like in the classical mechanics. The composite function is, however, ${\displaystyle \star }$-function. The ${\displaystyle \star }$-product violates canonicity of the phase flow beyond the first order in ${\displaystyle \tau }$.

The antisymmetrized products of even number of operators of canonical variables are c-numbers as a consequence of the commutation relations. These products are left invariant by unitary transformations, which leads, in particular, to the relation

${\displaystyle q^{i}(\xi ,\tau )\wedge q^{j}(\xi ,\tau )=\xi ^{i}\wedge \xi ^{j}=-I^{ij}.}$

In general, the antisymmetrized product

${\displaystyle q^{[i_{1}}(\xi ,\tau )\star q^{i_{2}}(\xi ,\tau )\star \ldots \star q^{i_{2s}]}(\xi ,\tau )}$

is also invariant, that is, it does not depend on time, and moreover does not depend on the coordinate.

Phase-space transformations induced by the evolution operator preserve the Moyal bracket and do not preserve the Poisson bracket, so the evolution map

${\displaystyle \xi \rightarrow {\acute {\xi }}=q(\xi ,\tau ),}$

is not canonical beyond O(τ).[8] The first order in τ defines the algebra of the transformation group. As previously noted, the algebra of canonical transformations of classical mechanics coincides with the algebra of unitary transformations of quantum mechanics. These two groups, however, are different because the multiplication operations in classical and quantum mechanics are different.

Transformation properties of canonical variables and phase-space functions under unitary transformations in the Hilbert space have important distinctions from the case of canonical transformations in the phase space.

Quantum characteristics can hardly be treated visually as trajectories along which physical particles move. The reason lies in the star-composition law

${\displaystyle q(\xi ,\tau _{1}+\tau _{2})=q(\star q(\xi ,\tau _{1}),\tau _{2}),}$

which is non-local and is distinct from the dot-composition law of classical mechanics.

The energy conservation implies

${\displaystyle H(\xi )=H(\star q(\xi ,\tau )),}$

where

${\displaystyle H(\xi )=\mathrm {Tr} [{\hat {B}}(\xi ){\hat {H}}]}$

is Hamilton's function. In the usual geometric sense, ${\displaystyle H(\xi )}$ is not conserved along quantum characteristics.