PSI-Moyal equation

TL;DR

This paper introduces a generalized phase space framework for quantum and classical systems with radiation, deriving a new PSI-Moyal equation for a fourth-rank Wigner function, expanding the mathematical tools for high kinematical value systems.

Contribution

It develops a novel generalized von Neumann equation and a fourth-rank Wigner function, extending the Moyal equation to accommodate high kinematical values in phase space.

Findings

Derived the PSI-Moyal equation for fourth-rank Wigner functions.

Proved theorems on properties and solutions of the PSI-Moyal equation.

Analyzed a model quantum system using the new framework.

Abstract

A full consideration of classical and quantum systems with radiation (electromagnetic/gravitational) requires the involvement of a mathematical description in the generalized phase space of high kinematical values. Based on the dispersion chain of equations of quantum mechanics, we construct a generalization of the von Neumann equation for the density matrix in the phase space of fourth-order kinematical values. The paper introduces a new extended definition of the fourth rank Wigner function, which is constructed from the wave functions of the second rank. A new extended Moyal equation (PSI-Moyal equation) for the Wigner function of the fourth rank is obtained. Theorems on the properties of the new PSI-Moyal equation and its solutions are proved. An example of a model quantum system is considered in detail.