Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Infinite Dimensional Analysis Quantum Probability and Related Topics

On the generalization of moyal equation for an arbitrary linear quantization

Leonid Andreevich Borisov, Yuriy N. Orlov

Published Mar 1, 2021• 0 citations
Publisher

Abstract

For an arbitrary linear combination of quantizations, the kernel of the inverse operator is constructed. An equation for the evolution of the Wigner function for an arbitrary linear quantization is derived and it is shown that only for Weyl quantization this equation does not contain a source of quasi-probability. Stationary solutions for the Wigner function of a harmonic oscillator are constructed, depending on the characteristic function of the quantization rule. In the general case of Hermitian linear quantization these solutions are real but not positive. We found the representation of Weyl quantization in the form of the limit of a sequence of linear Hermitian quantizations, such that for each element of this sequence the stationary solution of the Moyal equation is positive.

DOI
10.1142/s021902572150003x

Related papers

Generalized Evolution Equation of Wigner Function for an Arbitrary Linear Quantization

Leonid Andreevich Borisov, Yu. N. Orlov

Abstract The arbitrary linear quantization is considered for definition of Wigner function. The evolution equation of this function is derived with the use of inversion of linear quantization kernel. In general case the Moyal equation depends on the characteristic function of quantization. It is shown that only for Weyl quantization this equation does not consist the source of quasi-probability. The stationary solutions of Moyal equation are constructed for an arbitrary linear quantization of a harmonic oscillator. The case of anharmonic oscillator is presented as a practical example of quantum system model in the so-called post-newtonian approximation. This model exhibits the intersection effect between quantization dependences of Hamiltonian and Wigner function.

Lobachevskii Journal of Mathematics•Jan 1, 2021•2 citations

On the Inversion Formula of Linear Quantization and the Evolution Equation for the Wigner Function

Leonid Andreevich Borisov, Yu. N. Orlov

Abstract We consider the inversion problem for linear quantization defined by an integral transformation relating the matrix of a quantum operator to its classical symbol. For an arbitrary linear quantization, we construct evolution equations for the density matrix and the Wigner function. It is shown that the Weyl quantization is the only one for which the evolution equation of the Wigner function is free of a quasi-probability source, which distinguishes this quantization as the only physically adequate one in the class under consideration. As an example, we give an exact stationary solution for the Wigner function of a harmonic oscillator with an arbitrary linear quantization, and construct a sequence of quantizations that approximate the Weyl quantization and tend to it in the weak sense so that the Wigner function remains positive definite.

Proceedings of the Steklov Institute of Mathematics•Jul 1, 2021•1 citation

On the Statistical Properties of Linear Quantization and Inverse Operator

Yu. N. Orlov, V. Zh. Sakbaev

Abstract The paper uses an approach to the formal description of linear quantization as a statistical mixture of operators of the so-called tau quantization. This allows us to consider arbitrary Hermitian linear quantization as a linear integral operator acting on functions from the space of canonical variables and translating them into matrix elements of the quantum operator. In such a scheme, various quantization rules are uniformly studied: Weyl, Born, Jordan, and a number of others. The main result relates to the proof of the uniqueness of Weyl quantization as an operator, which is answered by the inverse operator generating a positive definite probability measure. For any other quantization, the inverse operator (dequantization) generates a measure that is not positive definite.

Lobachevskii Journal of Mathematics•Sep 1, 2025•0 citations

Wigner function of a quantum system with polynomial potential

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +1 more

Abstract The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.

Journal of Statistical Mechanics Theory and Experiment•May 1, 2020•15 citations

Extended Wigner function for the harmonic oscillator in the phase space

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +1 more

New time dependent extended Wigner functions for the quantum harmonic oscillator have been obtained in this work. The Moyal equation for the harmonic oscillator has been associated with the wave equation of a 2D membrane in the phase plane. The values of the extended Wigner function are equal to the deviation values of the points on the surface of the membrane from the equilibrium state. The positive and negative values of the extended Wigner function correspond to the direction of the deviation from the equilibrium state. As an example, a time dependent extended Wigner function corresponding to the standing wave of quasi-probability density arising in the phase plane is considered.

arXiv:2003.11737•Results in Physics•Nov 5, 2020•6 citations

Is the Moyal equation for the Wigner function a quantum analogue of the Liouville equation?

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +2 more

Abstract The Moyal equation describes the evolution of the Wigner function of a quantum system in the phase space. The right-hand side of the equation contains an infinite series with coefficients proportional to powers of the Planck constant. There is an interpretation of the Moyal equation as a quantum analogue of the classical Liouville equation. Indeed, if one uses the notion of the classical passage to the limit as the Planck constant tends to zero, then formally the right-hand side of the Moyal equation tends to zero. As a result, the Moyal equation becomes the classical Liouville equation for the distribution function. In this paper, we show that the right side of the Moyal equation does not explicitly depend on the Planck constant, and all terms of the series can make a significant contribution. The transition between the classical and quantum descriptions is related not to the Planck constant, but to the spatial scale. For a model quantum system with a potential in the form of a «quadratic funnel», an exact 3D solution of the Schrödinger equation is found and the corresponding Wigner function is constructed in the paper. Using trajectory analysis in the phase space, based on the representation of the right-hand side of the Moyal equation, it is shown that on the spatial microscale there is an infinite number of «trajectories» of the particle motion (thereby the concept of a trajectory is indefinite), and when passing to the macroscale, all «trajectories» concentrate around the classical trajectory.

arXiv:2307.16316•Journal of Statistical Mechanics Theory and Experiment•Sep 1, 2023•8 citations
For an arbitrary linear combination of quantizations, the kernel of the inverse operator is constructed. An equation for the evolution of the Wigner function for an arbitrary linear quantization is derived and it is shown that only for Weyl quantization this equation does not contain a source of quasi-probability. Stationary solutions for the Wigner function of a harmonic oscillator are constructed, depending on the characteristic function of the quantization rule. In the general case of Hermitian linear quantization these solutions are real but not positive. We found the representation of Weyl quantization in the form of the limit of a sequence of linear Hermitian quantizations, such that for each element of this sequence the stationary solution of the Moyal equation is positive.
Abstract The arbitrary linear quantization is considered for definition of Wigner function. The evolution equation of this function is derived with the use of inversion of linear quantization kernel. In general case the Moyal equation depends on the characteristic function of quantization. It is shown that only for Weyl quantization this equation does not consist the source of quasi-probability. The stationary solutions of Moyal equation are constructed for an arbitrary linear quantization of a harmonic oscillator. The case of anharmonic oscillator is presented as a practical example of quantum system model in the so-called post-newtonian approximation. This model exhibits the intersection effect between quantization dependences of Hamiltonian and Wigner function.
Abstract We consider the inversion problem for linear quantization defined by an integral transformation relating the matrix of a quantum operator to its classical symbol. For an arbitrary linear quantization, we construct evolution equations for the density matrix and the Wigner function. It is shown that the Weyl quantization is the only one for which the evolution equation of the Wigner function is free of a quasi-probability source, which distinguishes this quantization as the only physically adequate one in the class under consideration. As an example, we give an exact stationary solution for the Wigner function of a harmonic oscillator with an arbitrary linear quantization, and construct a sequence of quantizations that approximate the Weyl quantization and tend to it in the weak sense so that the Wigner function remains positive definite.
Abstract The paper uses an approach to the formal description of linear quantization as a statistical mixture of operators of the so-called tau quantization. This allows us to consider arbitrary Hermitian linear quantization as a linear integral operator acting on functions from the space of canonical variables and translating them into matrix elements of the quantum operator. In such a scheme, various quantization rules are uniformly studied: Weyl, Born, Jordan, and a number of others. The main result relates to the proof of the uniqueness of Weyl quantization as an operator, which is answered by the inverse operator generating a positive definite probability measure. For any other quantization, the inverse operator (dequantization) generates a measure that is not positive definite.
Abstract The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.
New time dependent extended Wigner functions for the quantum harmonic oscillator have been obtained in this work. The Moyal equation for the harmonic oscillator has been associated with the wave equation of a 2D membrane in the phase plane. The values of the extended Wigner function are equal to the deviation values of the points on the surface of the membrane from the equilibrium state. The positive and negative values of the extended Wigner function correspond to the direction of the deviation from the equilibrium state. As an example, a time dependent extended Wigner function corresponding to the standing wave of quasi-probability density arising in the phase plane is considered.
Abstract The Moyal equation describes the evolution of the Wigner function of a quantum system in the phase space. The right-hand side of the equation contains an infinite series with coefficients proportional to powers of the Planck constant. There is an interpretation of the Moyal equation as a quantum analogue of the classical Liouville equation. Indeed, if one uses the notion of the classical passage to the limit as the Planck constant tends to zero, then formally the right-hand side of the Moyal equation tends to zero. As a result, the Moyal equation becomes the classical Liouville equation for the distribution function. In this paper, we show that the right side of the Moyal equation does not explicitly depend on the Planck constant, and all terms of the series can make a significant contribution. The transition between the classical and quantum descriptions is related not to the Planck constant, but to the spatial scale. For a model quantum system with a potential in the form of a «quadratic funnel», an exact 3D solution of the Schrödinger equation is found and the corresponding Wigner function is constructed in the paper. Using trajectory analysis in the phase space, based on the representation of the right-hand side of the Moyal equation, it is shown that on the spatial microscale there is an infinite number of «trajectories» of the particle motion (thereby the concept of a trajectory is indefinite), and when passing to the macroscale, all «trajectories» concentrate around the classical trajectory.