Method of quantum characteristics

Quantum characteristics are phase-space trajectories that arise in the phase space formulation of quantum mechanics through the Wigner transform of Heisenberg operators of canonical coordinates and momenta. These trajectories obey the Hamilton equations in quantum form and play the role of characteristics in terms of which time-dependent Weyl's symbols of quantum operators can be expressed. In the classical limit, quantum characteristics reduce to classical trajectories. The knowledge of quantum characteristics is equivalent to the knowledge of quantum dynamics.

Weyl-Wigner association rule

In Hamiltonian dynamics, classical systems with ${\displaystyle n}$ degrees of freedom are described by ${\displaystyle 2n}$ canonical coordinates and momenta

${\displaystyle ^{\;}\xi ^{i}=(x^{1},...,x^{n},p_{1},...,p_{n})\in \mathbb {R} ^{2n},}$

that form a coordinate system in the phase space. These variables satisfy the Poisson bracket relations

${\displaystyle ^{\;}\{\xi ^{k},\xi ^{l}\}=-I^{kl}.}$

The skew-symmetric matrix ${\displaystyle ^{\;}I^{kl}}$,

${\displaystyle \left\|I\right\|=\left\|{\begin{array}{ll}0&-E_{n}\\E_{n}&0\end{array}}\right\|,}$

where ${\displaystyle ^{\;}E_{n}}$ is the ${\displaystyle n\times n}$ identity matrix, defines nondegenerate 2-form in the phase space. The phase space acquires thereby the structure of a symplectic manifold. The phase space is not metric space, so distance between two points is not defined. The Poisson bracket of two functions can be interpreted as the oriented area of a parallelogram whose adjacent sides are gradients of these functions. Rotations in Euclidean space leave the distance between two points invariant. Canonical transformations in symplectic manifold leave the areas invariant.

In quantum mechanics, the canonical variables ${\displaystyle ^{\;}\xi }$ are associated to operators of canonical coordinates and momenta

${\displaystyle {\hat {\xi }}^{i}=({\hat {x}}^{1},...,{\hat {x}}^{n},{\hat {p}}_{1},...,{\hat {p}}_{n})\in \operatorname {Op} (L^{2}(\mathbb {R} ^{n})).}$

These operators act in Hilbert space and obey commutation relations

${\displaystyle [{\hat {\xi }}^{k},{\hat {\xi }}^{l}]=-i\hbar I^{kl}.}$

Weyl’s association rule [1] extends the correspondence ${\displaystyle \xi ^{i}\rightarrow {\hat {\xi }}^{i}}$ to arbitrary phase-space functions and operators.

Taylor expansion

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

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

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 is proposed by Groenewold [3] and Stratonovich.[4]

Operator basis

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 ${\displaystyle ^{\;}\xi _{i}}$. 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.

Star-product

The set of operators ${\displaystyle Op(L^{2}(\mathbb {R} ^{n}))}$ is closed under the multiplication of operators. The vector space ${\displaystyle \mathbb {V} }$ is endowed thereby with an associative algebra structure. Given two functions

${\displaystyle f(\xi )=Tr[{\hat {B}}(\xi ){\hat {f}}]~~\mathrm {and} ~~g(\xi )=Tr[{\hat {B}}(\xi ){\hat {g}}],}$

one can construct a third function

${\displaystyle f(\xi )\star g(\xi )=Tr[{\hat {B}}(\xi ){\hat {f}}{\hat {g}}]}$

called ${\displaystyle \star }$-product [3]. It is given explicitly by

${\displaystyle f(\xi )\star g(\xi )=f(\xi )\exp({\frac {i\hbar }{2}}{\mathcal {P}})g(\xi ).}$

where

${\displaystyle {\mathcal {P}}=-{I}^{kl}{\overleftarrow {\frac {\partial }{\partial \xi ^{k}}}}{\overrightarrow {\frac {\partial }{\partial \xi ^{l}}}}}$

is the Poisson operator. The ${\displaystyle \star }$-product splits into symmetric and skew-symmetric parts

${\displaystyle f\star g=f\circ g+{\frac {i\hbar }{2}}f\wedge g.}$

The ${\displaystyle \circ }$-product is not associative. In the classical limit ${\displaystyle \circ }$-product becomes the dot-product. The skew-symmetric part ${\displaystyle f\wedge g}$ is known under the name of Moyal bracket. [6] This is the Weyl's symbol of commutator. In the classical limit Moyal bracket becomes Poisson bracket. Moyal bracket is quantum deformation of Poisson bracket.

Quantum characteristics

The correspondence ${\displaystyle \xi \leftrightarrow {\hat {\xi }}}$ shows that coordinate transformations in the phase space are accompanied by transformations of operators of the canonical coordinates and momenta and vice versa. Let ${\displaystyle \mathbf {\hat {U}} }$ be the evolution operator,

${\displaystyle {\hat {U}}=\exp {\Bigl (}-{\frac {i}{\hbar }}{\hat {H}}\tau {\Bigr )},}$

and ${\displaystyle {\hat {H}}}$ is Hamiltonian. Consider the following scheme:

${\displaystyle \xi {\stackrel {q}{\longrightarrow }}{\acute {\xi }}}$

${\displaystyle \updownarrow \;\;\;\;\;\;\updownarrow }$

${\displaystyle {\hat {\xi }}{\stackrel {\hat {U}}{\longrightarrow }}{\acute {\hat {\xi }}},}$

Quantum evolution transforms vectors in the Hilbert space and, upon the Wigner association rule, coordinates in the phase space. In Heisenberg representation, the operators of the canonical variables are transformed as

${\displaystyle {\hat {\xi }}^{i}\rightarrow {\acute {{\hat {\xi }}^{i}}}={\hat {U}}^{+}{\hat {\xi }}^{i}{\hat {U}}.}$

The phase-space coordinates ${\displaystyle {\acute {\xi }}^{i}}$ that correspond to new operators ${\displaystyle {\acute {{\hat {\xi }}^{i}}}}$ in the old basis ${\displaystyle {\hat {B}}(\xi )}$ are given by

${\displaystyle \xi ^{i}\rightarrow {\acute {\xi }}^{i}=q^{i}(\xi ,\tau )=Tr[{\hat {B}}(\xi ){\hat {U}}^{+}{\hat {\xi }}^{i}{\hat {U}}],}$

with the initial conditions

${\displaystyle ^{\;}q^{i}(\xi ,0)=\xi ^{i}.}$

The functions ${\displaystyle ^{\;}q^{i}(\xi ,\tau )}$ define quantum phase flow. In the general case, it is canonical to first order in ${\displaystyle \tau }$.[7]

Star-function

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

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

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

${\displaystyle f(\xi ){\stackrel {q}{\longrightarrow }}{\acute {f}}(\xi )=Tr[{\hat {B}}(\xi ){\hat {U}}^{+}{\hat {f}}{\hat {U}}]}$

${\displaystyle \updownarrow \;\;\;\;\;\;\;\;\;\;\,\updownarrow }$

${\displaystyle {\hat {f}}\;\;\;\;{\stackrel {\hat {U}}{\longrightarrow }}\,{\acute {\hat {f}}}\;\;\;\;\;={\hat {U}}^{+}{\hat {f}}{\hat {U}}}$

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

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

Composite function defined in such a way is called ${\displaystyle \star }$-function. The composition law differs from the classical one. However, 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 quantum characteristics are constructed, physical observables can be found without further addressing to Hamiltonian. The functions ${\displaystyle ^{\;}q^{i}(\xi ,\tau )}$ play the role of characteristics[8] similarly to classical characteristics used to solve classical Liouville equation.

Quantum 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 "probability" density in phase space is not preserved in time.

Quantum Hamilton's equations

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 ${\displaystyle \tau }$.

Conservation of Moyal bracket

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 and, in particular,

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

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.[8] 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:

Composition law

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.

Energy conservation

The energy conservation implies

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

where

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

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

Summary

The origin of the method of characteristics can be traced back to Heisenberg’s matrix mechanics. Suppose that we have solved in the matrix mechanics the evolution equations for the operators of the canonical coordinates and momenta in the Heisenberg representation. These operators evolve according to

${\displaystyle {\hat {\xi }}^{i}\rightarrow {\hat {\xi }}^{i}(\tau )={\hat {U}}^{+}{\hat {\xi }}^{i}{\hat {U}}.}$

It is known that for any operator ${\displaystyle {\hat {f}}}$ one can find a function f(ξ) through which ${\displaystyle {\hat {f}}}$ is represented in the form ${\displaystyle f({\hat {\xi }})}$. The same operator ${\displaystyle {\hat {f}}}$ at time τ is equal to

${\displaystyle {\hat {f}}(\tau )=U^{+}{\hat {f}}U=U^{+}f({\hat {\xi }})U=f(U^{+}{\hat {\xi }}U)=f({\hat {\xi }}(\tau )).}$

This equation shows that ${\displaystyle {\hat {\xi }}(\tau )}$ are characteristics that determine the evolution for all of the operators in Op(L²(Rⁿ)). This property is fully transferred to the phase space upon deformation quantization and, in the limit of ħ → 0, to the classical mechanics.

CLASSICAL DYNAMICS vs. QUANTUM DYNAMICS

Liouville equation
First-order PDE	Infinite-order PDE
${\displaystyle {\frac {\partial }{\partial \tau }}\rho (\xi ,\tau )=-\{\rho (\xi ,\tau ),{\mathcal {H}}(\xi )\}}$	${\displaystyle {\frac {\partial }{\partial \tau }}W(\xi ,\tau )=-W(\xi ,\tau )\wedge H(\xi )}$
Hamilton's equations
Finite-order ODE	Infinite-order PDE
${\displaystyle {\frac {\partial }{\partial \tau }}c^{i}(\xi ,\tau )=\{\zeta ^{i},{\mathcal {H}}(\zeta )\}|_{\zeta =c(\xi ,\tau )}}$	${\displaystyle {\frac {\partial }{\partial \tau }}q^{i}(\xi ,\tau )=\{\zeta ^{i},H(\zeta )\}|_{\zeta =\star q(\xi ,\tau )}}$
Initial conditions	Initial conditions
${\displaystyle ^{\;}c^{i}(\xi ,0)=\xi ^{i}}$	${\displaystyle ^{\;}q^{i}(\xi ,0)=\xi ^{i}}$
Composition law
Dot-composition	${\displaystyle \star }$-composition
${\displaystyle ^{\;}c(\xi ,\tau _{1}+\tau _{2})=c(c(\xi ,\tau _{1}),\tau _{2})}$	${\displaystyle q(\xi ,\tau _{1}+\tau _{2})=q(\star q(\xi ,\tau _{1}),\tau _{2})}$
Invariance
Poisson bracket	Moyal bracket
${\displaystyle ^{\;}\{c^{i}(\xi ,\tau ),c^{j}(\xi ,\tau )\}=\{\xi ^{i},\xi ^{j}\}}$	${\displaystyle q^{i}(\xi ,\tau )\wedge q^{j}(\xi ,\tau )=\xi ^{i}\wedge \xi ^{j}}$
Energy conservation
Dot-composition	${\displaystyle \star }$-composition
${\displaystyle ^{\;}H(\xi )=H(c(\xi ,\tau ))}$	${\displaystyle ^{\;}H(\xi )=H(\star q(\xi ,\tau ))}$
Solution to Liouville equation
Dot-composition	${\displaystyle \star }$-composition
${\displaystyle ^{\;}\rho (\xi ,\tau )=\rho (c(\xi ,-\tau ),0)}$	${\displaystyle ^{\;}W(\xi ,\tau )=W(\star q(\xi ,-\tau ),0)}$

Table compares properties of characteristics in classical and quantum mechanics. PDE and ODE indicate partial differential equations and ordinary differential equations, respectively. The quantum Liouville equation is the Weyl-Wigner transform of the von Neumann evolution equation for the density matrix in the Schrödinger representation. The quantum Hamilton equations are the Weyl-Wigner transforms of the evolution equations for operators of the canonical coordinates and momenta in the Heisenberg representation.

In classical systems, characteristics ${\displaystyle ^{\;}c^{i}(\xi ,\tau )}$ usually satisfy first-order ODEs, e.g., classical Hamilton's equations, and solve first-order PDEs, e.g., the classical Liouville equation. Functions ${\displaystyle ^{\;}q^{i}(\xi ,\tau )}$ are also characteristics, despite both ${\displaystyle ^{\;}q^{i}(\xi ,\tau )}$ and ${\displaystyle ^{\;}f(\xi ,\tau )}$ obeying infinite-order PDEs.

The quantum phase flow contains all of the information about the quantum evolution. Semiclassical expansion of quantum characteristics and ${\displaystyle \star }$-functions of quantum characteristics in a power series in ${\displaystyle \hbar }$ allows calculation of the average values of time-dependent physical observables by solving a finite-order coupled system of ODEs for phase space trajectories and Jacobi fields.[9][10] The order of the system of ODEs depends on the truncation of the power series. The tunneling effect is nonperturbative in ${\displaystyle \hbar }$ and is not captured by the expansion. The density of the quantum probability fluid is not preserved in phase-space, as the quantum fluid diffuses. [6] Quantum characteristics must therefore be distinguished from both the trajectories of the de Broglie - Bohm theory [11] and the trajectories of the path-integral method in phase space for the amplitudes [12] and the Wigner function. [13][14] Thus far, only several quantum systems have been explicitly solved using the method of quantum characteristics [15][16].

See also

• Method of characteristics
• Wigner–Weyl transform
• Deformation theory
• Wigner distribution function
• Modified Wigner distribution function
• Wigner quasiprobability distribution
• Negative probability

References

1. Weyl, H. (1927). "Quantenmechanik und gruppentheorie". Zeitschrift für Physik. 46 (1–2): 1–46. Bibcode:1927ZPhy...46....1W. doi:10.1007/BF02055756.
2. Wigner, E. P. (1932). "On the quantum correction for thermodynamic equilibrium". Physical Review. 40 (5): 749–759. Bibcode:1932PhRv...40..749W. doi:10.1103/PhysRev.40.749. hdl:10338.dmlcz/141466.
3. Groenewold, H. J. (1946). "On the principles of elementary quantum mechanics". Physica. 12 (7): 405–460. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4.
4. R. L. Stratonovich, Sov. Phys. JETP 4, 891 (1957).
5. Mehta, C. L. (1964). "Phase-Space Formulation of the Dynamics of Canonical Variables". Journal of Mathematical Physics. 5 (5): 677–686. Bibcode:1964JMP.....5..677M. doi:10.1063/1.1704163.
6. Moyal, J. E. (1949). "Quantum mechanics as a statistical theory". Mathematical Proceedings of the Cambridge Philosophical Society. 45 (1): 99–124. Bibcode:1949PCPS...45...99M. doi:10.1017/S0305004100000487.
7. P. A. M. Dirac, The Principles of Quantum Mechanics, First Edition (Oxford: Clarendon Press, 1930).
8. Krivoruchenko, M. I.; Faessler, A. (2007). "Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics". Journal of Mathematical Physics. 48 (5): 052107. arXiv:quant-ph/0604075. Bibcode:2007JMP....48e2107K. doi:10.1063/1.2735816.
9. Krivoruchenko, M. I.; Fuchs, C.; Faessler, A. (2007). "Semiclassical expansion of quantum characteristics for many-body potential scattering problem". Annalen der Physik. 519 (9): 587–614. arXiv:nucl-th/0605015. Bibcode:2007AnP...519..587K. doi:10.1002/andp.200610251.
10. Maximov, S. (2009). "On a special picture of dynamical evolution of nonlinear quantum systems in the phase-space representation". Physica D. 238 (18): 1937–1950. Bibcode:2009PhyD..238.1937M. doi:10.1016/j.physd.2009.07.001.
11. P. R. Holland, The Quantum Theory of Motion: An Account of the De Broglie-Bohm Causal Interpretation of Quantum Mechanics, (Cambridge University Press, 1993), ISBN 0-521-35404-8.
12. Berezin, F. A. (1980). "Feynman path integrals in a phase space". Soviet Physics Uspekhi. 23 (11): 763–788. Bibcode:1980SvPhU..23..763B. doi:10.1070/PU1980v023n11ABEH005062.
13. Marinov, M. S. (1991). "A new type of phase-space path integral". Physics Letters A. 153 (1): 5–11. Bibcode:1991PhLA..153....5M. doi:10.1016/0375-9601(91)90352-9.
14. Wong, C. Y. (2003). "Explicit solution of the time evolution of the Wigner function". Journal of Optics B: Quantum and Semiclassical Optics. 5 (3): S420–S428. arXiv:quant-ph/0210112. Bibcode:2003JOptB...5S.420W. doi:10.1088/1464-4266/5/3/381.
15. Braunss, G. (2013). "Quantum dynamics in phase space: Moyal trajectories 2". Journal of Mathematical Physics. 54 (1): 012105. Bibcode:2013JMP....54a2105B. doi:10.1063/1.4773229.
16. Braunss, G. (2017). "Quantum dynamics in phase space: Moyal trajectories 3". Journal of Mathematical Physics. 58 (6): 062104. Bibcode:2017JMP....58f2104B. doi:10.1063/1.4984592.

Textbooks

• H. Weyl, The Theory of Groups and Quantum Mechanics, (Dover Publications, New York Inc., 1931).
• V. I. Arnold, Mathematical Methods of Classical Mechanics, (2-nd ed. Springer-Verlag, New York Inc., 1989).
• M. V. Karasev and V. P. Maslov, Nonlinear Poisson brackets. Geometry and quantization. Translations of Mathematical Monographs, 119. (American Mathematical Society, Providence, RI, 1993).