Directly computing Wigner functions for open quantum systems

TL;DR

This paper presents a method to directly compute the time-dependent Wigner function for open quantum systems, enhancing its applicability without additional approximations.

Contribution

We derive an explicit expression for the time-dependent Wigner function in open quantum systems, applicable to non-relativistic particles interacting with relativistic environments.

Findings

Derived a direct computation formula for the Wigner function.

Applied the method to a scalar particle with Yukawa interaction.

Enhanced the practical use of Wigner functions in open quantum systems.

Abstract

The Wigner function is a well-known phase space distribution function with many applications in quantum mechanics. In this article, we consider an open quantum system consisting of a non-relativistic single particle interacting with a general, possibly relativistic environment. For this system, we derive an expression for directly computing the time-dependent Wigner function from its initial values. This result renders time-dependent Wigner functions more applicable without having to make additional approximations that would otherwise be required in order to solve the corresponding equation of motion. As an illustration of our findings, we discuss the example of a non-relativistic single scalar particle interacting via a Yukawa interaction with an environment comprising another type of scalar field that is treated relativistically.