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| '''Quantum characteristics'''<ref>{{cite doi|10.1088/1464-4266/5/3/381|noedit}}</ref> are phase-space trajectories that arise in the [[phase space formulation]] of [[quantum mechanics]], related in [[deformation theory]] through the [[Weyl-Wigner transform]] of Heisenberg operators of canonical coordinates and momenta. These trajectories obey the Hamilton’s equations in quantum form and play the role of [[Method of characteristics|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. However, the density of the quantum probability fluid is ''not'' preserved in phase-space, and there is no well-defined notion of unambiguous trajectories for quantum systems, as the quantum fluid "diffuses".<ref>[[José Enrique Moyal|J. E. Moyal]], ''Quantum mechanics as a statistical theory'',
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| ''Proceedings of the Cambridge Philosophical Society'', 45, 99 (1949).</ref> The knowledge of quantum characteristics is equivalent to the knowledge of quantum dynamics, in principle, although few quantum systems have been explicitly solved in this way so far, in practice.
| |
| | |
| == Weyl-Wigner association rule ==
| |
| | |
| In [[Hamiltonian mechanics|Hamiltonian dynamics]] classical systems with <math>n</math> degrees of freedom are described by <math>2n</math> canonical coordinates and momenta
| |
| :<math>^{\;}\xi^{i} = (x^1, . . . , x^n, p_1, . . . , p_n) \in \mathbb{R}^{2n},</math>
| |
| that form a coordinate system in the phase space. These variables satisfy the [[Poisson bracket]] relations
| |
| :<math>^{\;}\{\xi^{k},\xi^{l}\}=-I^{kl}.</math>
| |
| The skew-symmetric matrix <math>^{\;}I^{kl}</math>,
| |
| | |
| :<math>\left\| I\right\| =\left\|
| |
| \begin{array}{ll}
| |
| 0 & -E_{n} \\
| |
| E_{n} & 0
| |
| \end{array}
| |
| \right\|,</math>
| |
| | |
| where <math>^{\;}E_n</math> is the <math>n \times n</math> 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 <math>^{\;}\xi</math> are associated to operators of canonical coordinates and momenta
| |
| | |
| :<math>\hat{\xi}^{i} = (\hat{x}^1, . . . , \hat{x}^n, \hat{p}_1, . . . , \hat{p}_n) \in Op(L^2(\mathbb{R}^n)).</math>
| |
| | |
| These operators act in [[Hilbert space]] and obey commutation relations
| |
| | |
| :<math>[\hat{\xi}^{k},\hat{\xi}^{l}]=-i\hbar I^{kl}.</math>
| |
| | |
| Weyl’s [[Wigner-Weyl transform|association rule]] <ref>[[Hermann Weyl|H. Weyl]], Z. Phys. 46, 1 (1927).</ref> extends the correspondence <math>\xi^i \rightarrow \hat{\xi}^i</math> to arbitrary phase-space functions and operators.
| |
| | |
| === Taylor expansion ===
| |
| | |
| A one-sided association rule <math>f(\xi) \to \hat{f}</math> was formulated by Weyl initially with the help of [[Taylor series|Taylor expansion]] of functions of operators of the canonical variables
| |
| | |
| :<math>\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}}.</math>
| |
| | |
| The operators <math>\hat{\xi}</math> 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 <math>^{\;}f(\xi)</math> is called Weyl's symbol of operator <math>\hat{f}</math>.
| |
| | |
| Under the reverse association <math>f(\xi) \leftarrow \hat{f}</math>, the [[density matrix]] turns to the [[Wigner quasi-probability distribution|Wigner function]].<ref>[[Eugene Wigner|E. P. Wigner]], ''On the quantum correction for thermodynamic equilibrium'', Phys. Rev. 40, 749 (1932).</ref> 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 Stratonovich.<ref>[[Ruslan L. Stratonovich|R. L. Stratonovich]], Sov. Phys. JETP 4, 891 (1957).</ref>
| |
| | |
| === Stratonovich basis ===
| |
| | |
| The set of operators acting in the Hilbert space is closed under multiplication of operators by <math>c</math>-numbers and summation. Such a set constitutes a vector space <math>\mathbb{V}</math>. 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:
| |
| :::::::::::<math>
| |
| \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\} \;\mathrm{vector\;space}\;\; \mathbb{V}
| |
| \end{array}
| |
| \\
| |
| \begin{array}{ccc}
| |
| { f(\xi )\star g(\xi )} & {\longleftrightarrow} & \;\; { \hat{f}\hat{g} }
| |
| \end{array}
| |
| \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
| |
| \end{array}
| |
| \right\} {\mathrm{algebra}}
| |
| </math>
| |
| Here, <math>^{\;}f(\xi)</math> and <math>^{\;}g(\xi)</math> are functions and <math>\hat{f}</math> and <math>\hat{g}</math> are the associated operators.
| |
| | |
| The elements of basis of <math>^{\;}\mathbb{V}</math> are labelled by canonical variables <math>^{\;}\xi_i</math>. The commonly used Stratonovich basis looks like
| |
| | |
| :<math>\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}.</math>
| |
| | |
| The Weyl-Wigner two-sided association rule for function <math>^{\;}f(\xi)</math> and operator <math>\hat{f}</math> has the form
| |
| | |
| :<math>f(\xi )=Tr[\hat{B}(\xi )\hat{f}],</math>
| |
| :<math>\hat{f} =\int \frac{d^{2n}\xi }{(2\pi \hbar )^{n}}f(\xi )\hat{B}(\xi ).</math>
| |
| | |
| The function <math>^{\;}f(\xi)</math> provides coordinates of the operator <math>\hat{f}</math> in the basis <math>\hat{B}(\xi )</math>. The basis is complete and orthogonal:
| |
| :<math>\int \frac{d^{2n}\xi }{(2\pi \hbar )^{n}}\hat{B}(\xi )Tr[\hat{B}(\xi )\hat{f}] =\hat{f},</math>
| |
| :<math>Tr[\hat{B}(\xi )\hat{B}(\xi ^{\prime })] = (2\pi \hbar )^{n}\delta^{2n}(\xi -\xi ^{\prime }).</math>
| |
| | |
| Alternative operator bases are discussed also.<ref>C. L. Mehta, J. Math. Phys. 5, 677 (1964).</ref> The freedom in
| |
| choice of the operator basis is better known as operator ordering problem.
| |
| | |
| == Star-product ==
| |
| | |
| The set of operators <math>Op(L^2(\mathbb{R}^n))</math> is closed under the multiplication of operators. The vector space <math>\mathbb{V}</math> is endowed thereby with an associative algebra structure. Given two functions
| |
| :<math>f(\xi ) = Tr[\hat{B}(\xi )\hat{f}]~~\mathrm{and}~~g(\xi ) = Tr[\hat{B}(\xi )\hat{g}],</math>
| |
| one can construct a third function | |
| | |
| :<math>f(\xi )\star g(\xi )=Tr[\hat{B}(\xi )\hat{f}\hat{g}]</math>
| |
| | |
| called <math>\star</math>-product
| |
| <ref>[[Hilbrand J. Groenewold|H. J. Groenewold]], ''On the Principles of elementary quantum mechanics'', Physica, 12, 405 (1946).</ref>
| |
| or [[Moyal product]]. It is given explicitly by
| |
| | |
| :<math>f(\xi )\star g(\xi )=f(\xi )\exp (\frac{i\hbar }{2}\mathcal{P})g(\xi ).</math>
| |
| | |
| where
| |
| :<math>\mathcal{P} = -{I}^{kl}
| |
| \overleftarrow{
| |
| \frac{\partial} {\partial \xi^{k}}
| |
| }
| |
| \overrightarrow{
| |
| \frac{\partial} {\partial \xi^{l}}}</math>
| |
| is the Poisson operator. The <math>\star</math>-product splits into symmetric and skew-symmetric parts
| |
| | |
| :<math>f\star g=f\circ g+\frac{i\hbar}{2} f\wedge g.</math>
| |
| | |
| The <math>\circ</math>-product is not associative. In the classical limit <math>\circ</math>-product becomes the dot-product. The skew-symmetric part <math>f \wedge g</math> is known under the name of [[Moyal bracket]]. This is the Weyl's symbol of commutator. In the classical limit Moyal bracket becomes Poisson bracket. Moyal bracket is [[Deformation theory|quantum deformation]] of Poisson bracket.
| |
| | |
| == Quantum characteristics ==
| |
| | |
| The correspondence <math>\xi \leftrightarrow \hat{\xi}</math> shows that coordinate transformations in the phase space are accompanied by transformations of operators of the canonical coordinates and momenta and ''vice versa''. Let <math>\mathbf{\hat{U}}</math> be the evolution operator,
| |
| :<math>\hat{U} = \exp(-\frac{i}{\hbar} \hat{H}\tau),</math>
| |
| and <math>\hat{H}</math> is Hamiltonian. Consider the following scheme:
| |
| :::::::::::::::::::<math>\xi \stackrel{q} \longrightarrow \acute{\xi}</math>
| |
| :::::::::::::::::::<math>\updownarrow \;\;\;\;\;\; \updownarrow</math>
| |
| :::::::::::::::::::<math>\hat{\xi} \stackrel{\hat{U}}\longrightarrow \acute{\hat{\xi}},</math>
| |
| | |
| Quantum evolution transforms vectors in the Hilbert space and, upon the Wigner association rule, coordinates in the phase space. In [[Heisenberg picture|Heisenberg representation]], the operators of the canonical variables are transformed as
| |
| :<math>\hat{\xi}^{i} \rightarrow \acute{\hat{\xi}^{i}}=\hat{U}^{+}\hat{\xi}^{i}\hat{U}.</math> | |
| The phase-space coordinates <math>\acute{\xi}^{i}</math> that correspond to new operators <math>\acute{\hat{\xi}^{i}}</math> in the old basis <math>\hat{B}(\xi)</math> are given by
| |
| :<math>\xi^{i} \rightarrow \acute{\xi}^{i} = q^{i}(\xi,\tau) = Tr[\hat{B}(\xi ) \hat{U}^{+} \hat{\xi}^{i} \hat{U}],</math>
| |
| with the initial conditions
| |
| :<math>^{\;}q^{i}(\xi,0)=\xi^{i}.</math>
| |
| The functions <math>^{\;}q^{i}(\xi,\tau)</math> define '''quantum phase flow'''. In the general case, it is canonical to first order in <math>\tau</math>.<ref>[[Paul Dirac|P. A. M. Dirac]], ''The Principles of Quantum Mechanics'', First Edition (Oxford: Clarendon Press, 1930).</ref>
| |
| | |
| === 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 <math>\hat{\xi}</math>. Transformations
| |
| :<math>\hat{f} \rightarrow \acute{\hat{f}}=\hat{U}^{+}\hat{f}\hat{U}</math>
| |
| induce under the Wigner association rule transformations of phase-space functions:
| |
| | |
| ::::::::::::::::<math>f(\xi) \stackrel{q}\longrightarrow \acute{f}(\xi) = Tr[\hat{B}(\xi )\hat{U}^{+}\hat{f}\hat{U}]</math>
| |
| ::::::::::::::::<math>\updownarrow \;\;\;\;\;\;\;\;\;\;\, \updownarrow</math>
| |
| ::::::::::::::::<math>\hat{f} \;\;\;\; \stackrel{\hat{U}}\longrightarrow \,\acute{\hat{f}} \;\;\;\;\; =\hat{U}^{+}\hat{f}\hat{U}</math>
| |
| | |
| Using the Taylor expansion, the transformation of function <math>^{\;}f(\xi )</math> under the evolution can be found to be
| |
| | |
| :<math>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)).</math>
| |
| | |
| Composite function defined in such a way is called <math>\star</math>-function.
| |
| The composition law differs from the classical one. However, semiclassical expansion of <math>f(\star q(\xi,\tau ))</math> around <math>f(q(\xi ,\tau))^{\;}</math> is formally well defined and involves even powers of <math>\hbar</math> only. | |
| This equation shows that, given quantum characteristics are constructed, physical observables can be found without further addressing to Hamiltonian.
| |
| The functions <math>^{\;}q(\xi ,\tau)^{i}</math> play the role of characteristics <ref name = mikaf /> similarly to [[Method of characteristics|classical characteristics]] used to solve classical [[Liouville's theorem (Hamiltonian)|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,
| |
| :<math>\frac{\partial }{\partial \tau} \hat{f} = -\frac{i}{\hbar}[\hat{f},\hat{H}],</math>
| |
| leads to the same equation with the '''opposite (plus) sign''' in the right-hand side:
| |
| :<math>\frac{\partial }{\partial \tau} f(\xi,\tau) = f(\xi,\tau) \wedge H(\xi ).</math>
| |
| <math>\star</math>-function solves this equation in terms of quantum characteristics:
| |
| :<math>f(\xi ,\tau)=f(\star q(\xi ,\tau),0).</math>
| |
| Similarly, the evolution of the Wigner function in the Schrödinger representation is given by
| |
| :<math>W(\xi ,\tau)=W(\star q(\xi ,- \tau),0).</math>
| |
| Note, however, that [[Liouville's theorem (Hamiltonian)]] 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
| |
| | |
| :<math>\frac{\partial }{\partial \tau }q^{i}(\xi ,\tau ) = \{\zeta ^{i},H(\zeta )\}|_{\zeta =\star q(\xi ,\tau )}.</math>
| |
| | |
| The right-hand side is calculated like in the classical mechanics. The composite function is, however, <math>\star</math>-function. The <math>\star</math>-product violates canonicity of the phase flow beyond the first order in <math>\tau</math>.
| |
| | |
| === 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,
| |
| :<math>q^{i}(\xi,\tau)\wedge q^{j}(\xi,\tau)=\xi ^{i}\wedge \xi ^{j}=- {I}^{ij}.</math>
| |
| | |
| Phase-space transformations induced by the evolution operator preserve the Moyal bracket and do not preserve the Poisson bracket, so the evolution map
| |
| :<math>\xi \rightarrow \acute{\xi} = q(\xi,\tau),</math>
| |
| is not canonical.<ref name = mikaf>M. I. Krivoruchenko, A. Faessler, [http://arxiv.org/abs/quant-ph/0604075 '' Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics''], J. Math. Phys. 48, 052107 (2007).</ref> 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
| |
| :<math>q(\xi ,\tau_1 + \tau_2 ) = q(\star q(\xi ,\tau_1 ),\tau_2),</math>
| |
| which is non-local and is distinct from the dot-composition law of classical mechanics.
| |
| | |
| === Energy conservation ===
| |
| | |
| The energy conservation implies
| |
| :<math>H(\xi )=H(\star q(\xi ,\tau ))</math>,
| |
| where
| |
| :<math>H(\xi )=Tr[\hat{B}(\xi )\hat{H}]</math>
| |
| is Hamilton's function. In the usual geometric sense, <math>^{\;}H(\xi )</math> is not conserved along quantum characteristics.
| |
| | |
| == Summary ==
| |
| | |
| Table compares properties of characteristics in classical and quantum mechanics. PDE and ODE are [[partial differential equation]]s and [[ordinary differential equation]]s, respectively. The quantum Liouville equation is the Weyl-Wigner transform of the von Neumann evolution equation for the density matrix in [[Schrödinger picture|Schrödinger representation]]. The quantum Hamilton's equations are the Weyl-Wigner transforms of the evolution equations for operators of the canonical coordinates and momenta in [[Heisenberg picture|Heisenberg representation]].
| |
| | |
| In classical systems, characteristics <math>^{\;}c^{i}(\xi,\tau)</math> satisfy usually first-order ODE, e.g., classical Hamilton's equations, and solve first-order PDE, e.g., classical Liouville equation. Functions <math>^{\;}q^{i}(\xi,\tau)</math> are characteristics also, despite both <math>^{\;}q^{i}(\xi,\tau)</math> and <math>^{\;}f(\xi,\tau)</math> obey infinite-order PDE.
| |
| | |
| ::::::::::{| class="wikitable"
| |
| |-
| |
| ! <math>\mathrm{CLASSICAL \;DYNAMICS}</math>
| |
| ! <math>\mathrm{QUANTUM \;DYNAMICS}</math> | |
| |-
| |
| | '''Liouville equation'''
| |
| |
| |
| |-
| |
| | ''Finite-order PDE''
| |
| | ''Infinite-order PDE''
| |
| |-
| |
| |<math>\frac{\partial}{\partial \tau} \rho(\xi,\tau) = - \{ \rho(\xi,\tau), \mathcal{H}(\xi) \}</math>
| |
| |<math>\frac{\partial }{\partial \tau }W(\xi ,\tau ) = - W(\xi ,\tau ) \wedge H(\xi )</math>
| |
| |-
| |
| | '''Hamilton's equations'''
| |
| |
| |
| |-
| |
| | ''Finite-order ODE''
| |
| | ''Infinite-order PDE''
| |
| |-
| |
| | <math>\frac{\partial}{\partial \tau} c^{i}(\xi,\tau) = \{\zeta^{i}, \mathcal{H}(\zeta)\}|_{\zeta = c(\xi,\tau)}</math>
| |
| | <math>\frac{\partial }{\partial \tau }q^{i}(\xi ,\tau ) = \{\zeta ^{i},H(\zeta )\}|_{\zeta =\star q(\xi ,\tau )}</math>
| |
| |-
| |
| | ''Initial conditions''
| |
| | ''Initial conditions''
| |
| |-
| |
| | <math>^{\;}c^{i}(\xi,0) = \xi^{i}</math>
| |
| | <math>^{\;}q^{i}(\xi,0) = \xi^{i}</math>
| |
| |-
| |
| | ''Composition law''
| |
| | ''<math>\star</math>-composition law''
| |
| |-
| |
| | <math>^{\;}c(\xi ,\tau_1 + \tau_2 ) = c( c(\xi ,\tau_1 ),\tau_2)</math>
| |
| | <math>q(\xi ,\tau_1 + \tau_2 ) = q(\star q(\xi ,\tau_1 ),\tau_2)</math>
| |
| |-
| |
| | ''Conservation of Poisson bracket''
| |
| | ''Conservation of Moyal bracket''
| |
| |-
| |
| | <math>^{\;}\{c^{i}(\xi,\tau), c^{j}(\xi,\tau)\}=\{\xi ^{i}, \xi ^{j}\} </math>
| |
| | <math>q^{i}(\xi,\tau)\wedge q^{j}(\xi,\tau)=\xi ^{i}\wedge \xi ^{j} </math>
| |
| |-
| |
| | ''Energy conservation''
| |
| | ''Energy conservation''
| |
| |-
| |
| | <math>^{\;}H(\xi )=H( c(\xi ,\tau ))</math>
| |
| | <math>^{\;}H(\xi )=H(\star q(\xi ,\tau ))</math>
| |
| |-
| |
| | '''Solutions to Liouville equation'''
| |
| |
| |
| |-
| |
| | <math>^{\;}\rho(\xi,\tau) = \rho(c(\xi ,- \tau ),0)</math>
| |
| | <math>^{\;}W(\xi,\tau) = W(\star q(\xi ,- \tau ),0)</math>
| |
| |}
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| The quantum phase flow contains entire information on the quantum evolution. Semiclassical expansion of quantum characteristics and <math>\star</math>-functions of quantum characteristics in power series in <math>\hbar</math> allows calculation of the average values of time-dependent physical observables by solving a finite-order coupled system of ODE for phase space trajectories and Jacobi fields.<ref>M. I. Krivoruchenko, C. Fuchs, A. Faessler, [http://arxiv.org/abs/nucl-th/0605015 ''Semiclassical expansion of quantum characteristics for many-body potential scattering problem''], Annalen der Physik 16, 587 (2007).</ref><ref>S. Maximov, [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVK-4WPJ5XM-1&_user=10&_coverDate=09%2F30%2F2009&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=2750e50b886dfe8d63ad0cac6839e683 ''On a special picture of dynamical evolution of nonlinear quantum systems in the phase-space representation''], Physica D238, 1937 (2009).
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| </ref> The order of the system of ODE depends on truncation of the power series. The tunneling effect is nonperturbative in <math>\hbar</math> and is not captured by the expansion. Quantum characteristics are distinct from trajectories of the de Broglie - Bohm theory.<ref>P. R. Holland, ''The Quantum Theory of Motion'', (Cambridge Uni. Press, Cambridge, 1993).</ref>
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| ==See also==
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| * [[Weyl quantization]]
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| * [[Wigner distribution function]]
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| * [[Modified Wigner distribution function]]
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| * [[Negative probability]]
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| * [[Method of characteristics]]
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| == References ==
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| <references/>
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| == Textbooks ==
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| * V. I. Arnold, ''Mathematical Methods of Classical Mechanics'', (2-nd ed. Springer-Verlag, New York Inc., 1989).
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| * M. V. Karasev and V. P Maslov, ''Nonlinear Poisson Brackets'', (Nauka, Moscow, 1991).
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| [[Category:Partial differential equations]]
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