On Simulating Liouvillian Flow From Quantum Mechanics Via Wigner Functions

TL;DR

This paper explores how quantum mechanics and classical probabilistic mechanics relate for relativistic particles using Wigner functions, providing a Lorentz-covariant framework and comparing with recent methods.

Contribution

It introduces a Lorentz-covariant construction of Wigner functions for relativistic particles using bilocal currents and Fourier transforms, extending to Dirac, Klein-Gordon, and Proca equations.

Findings

Wigner functions are constructed in a Lorentz-covariant manner.

The approach is parallel for Klein-Gordon and Proca cases using Kemmer-Duffin formalism.

Results are compared with recent derivations by Bosanac.

Abstract

The interconnection between quantum mechanics and probabilistic classical mechanics for a free relativistic particle is derived in terms of Wigner functions (WF) for both Dirac and Klein-Gordon (K-G) equations. Construction of WF is achieved by first defining a bilocal 4-current and then taking its Fourier transform w.r.t. the relative 4-coordinate. The K-G and Proca cases also lend themselves to a closely parallel treatment provided the Kemmer- Duffin beta-matrix formalism is employed for the former. Calculation of WF is carried out in a Lorentz-covariant fashion by standard `trace' techniques. The results are compared with a recent derivation due to Bosanac.