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Journal of Physics A Mathematical and Theoretical

Dynamics of the τ-Wigner distribution function

Bartłomiej J. Spisak, D. Woźniak, Damian Kołaczek

Published Dec 8, 2022• 2 citations
Publisher

Abstract

Abstract Some of the non-classical distribution functions defined on the phase space can be unified owing to specific parameterization. The latter allows one to think about the general form of the equation of motion (EOM) for such parameterized distribution functions. Motivated by this idea, we derive the EOM for so-called the τ-Wigner distribution function (WDF). This parameterization directly results from a modification of the linear transformation of the spatial variables which is used to derive the original WDF and its EOM. The derived equation is analytically solved for the case of linear potential. The point symmetries of this last equation are also analyzed.

DOI
10.1088/1751-8121/acaa15

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Abstract Some of the non-classical distribution functions defined on the phase space can be unified owing to specific parameterization. The latter allows one to think about the general form of the equation of motion (EOM) for such parameterized distribution functions. Motivated by this idea, we derive the EOM for so-called the τ-Wigner distribution function (WDF). This parameterization directly results from a modification of the linear transformation of the spatial variables which is used to derive the original WDF and its EOM. The derived equation is analytically solved for the case of linear potential. The point symmetries of this last equation are also analyzed.
We present a general procedure for obtaining equations of motion for the Wigner distribution of functions that are governed by ordinary and partial differential equations. For the case of fields we show that in general one must consider Wigner distribution of the four variables, position, momentum, time and frequency. We also show that in general one cannot write an equation of motion for position and momentum however it can be done in some cases, the Schrodinger equation being one such case. Our method leads to an equation of motion for the Schrodinger equation with time dependent potentials in contrast to the result obtained by Wigner and Moyal which was for time independent potentials.
Wigner distribution function has much importance in quantum statistical mechanics. It finds applications in various disciplines of physics including condense matter, quantum optics, to name but a few. Wigner distribution function is introduced by E. Wigner in 1932. However, there is no analytical derivation of Wigner distribution function in the literatures, to date. In this paper, a simple analytical derivation of Wigner distribution function is presented. Our derivation is based on two assumptions, these are A) by taking the integral of Wigner distribution function, with respect to configuration space, the momentum space distribution function is obtained B) WDF is real. Similarly, and in addition to Wigner distribution function, the distribution function of Sobouti-Nasiri, which is imaginary, is also derived.
A general canonical transformation of mechanical operators of position and momentum is considered. It is shown that it automatically generates a parameter s which leads to a generalized (or s-parameterized) Wigner function. This allows one to derive a generalized (s-parameterized) Moyal brackets for any dimensions. In the classical limit the s-parameterized Wigner averages of the momentum and its square yield the respective classical values. Interestingly enough,in the latter case the classical Hamilton-Jacobi equation emerges as a consequence of such a transition only if there is a non-zero parameter s.
The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.
We discuss two sets of conditions that are necessary and sufficient for a function defined on phase space to be a Wigner distribution function (WDF). The first set is well known and involves the function itself; the second set is less familiar and involves the function's ``symplectic'' Fourier transform. After explaining why these two sets are equivalent, we explore some properties and applications of the second one. Among other things, we show that that set includes the position-momentum uncertainty relations as a special case, and in doing so we give a new derivation of them. This derivation itself serves as the starting point for the discussion of a quantum-mechanical moment problem. It also enables us to exhibit a real-valued phase-space function that obeys the uncertainty relations but that is not a WDF.