Wigner's representation of quantum mechanics in integral form and its applications

TL;DR

This paper explores quantum phase-space dynamics using Wigner's integral form, demonstrating its advantages and proving that only quadratic potentials preserve phase-space volume in quantum mechanics.

Contribution

It introduces an integral form approach for Wigner's representation, providing a simple proof that only quadratic potentials satisfy Liouville's theorem in quantum mechanics.

Findings

Integral form enhances numerical stability and symmetry understanding.

Only quadratic potentials preserve phase-space volume in quantum mechanics.

Non-Liouvillian behavior cannot be eliminated through transformations.

Abstract

We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current~$\bm J$ as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold star products. Our central result is an explicit, elementary proof which shows that only systems up to quadratic in their potential fulfill Liouville's theorem of volume preservation in quantum mechanics. Contrary to a recent suggestion, our proof shows that the non-Liouvillian character of quantum phase-space dynamics cannot be transformed away.