Wigner distribution functions for complex dynamical systems: a path integral approach

TL;DR

This paper develops a path integral method to explicitly construct the propagator for the Wigner distribution function in quantum systems, including coupled systems like the Caldeira-Leggett model, extending quantum phase space analysis.

Contribution

It introduces a path integral approach to derive the propagator for the Wigner function, including for systems coupled to an environment, with explicit formulas for harmonic oscillator baths.

Findings

Explicit propagator expressions for quadratic Lagrangians.

Extension of the method to coupled systems using influence functional.

Application to the Caldeira-Leggett model.

Abstract

Starting from Feynman's Lagrangian description of quantum mechanics, we propose a method to construct explicitly the propagator for the Wigner distribution function of a single system. For general quadratic Lagrangians, only the classical phase space trajectory is found to contribute to the propagator. Inspired by Feynman's and Vernon's influence functional theory we extend the method to calculate the propagator for the reduced Wigner function of a system of interest coupled to an external system. Explicit expressions are obtained when the external system consists of a set of independent harmonic oscillators. As an example we calculate the propagator for the reduced Wigner function associated with the Caldeira-Legett model.