On the propagation of semiclassical Wigner functions

TL;DR

This paper clarifies the differences between semiclassical Wigner function propagation and classical Liouville dynamics, providing a geometrical prescription for the former based on classical trajectories of initial phase space features.

Contribution

It introduces a new geometrical method to accurately describe semiclassical Wigner function propagation, contrasting it with classical Liouville evolution.

Findings

Semiclassical Wigner functions propagate according to trajectories of chord tips.

Classical Liouville propagation follows trajectories of phase space points.

The paper provides a geometrical prescription for semiclassical propagation.

Abstract

We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we re-discuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centered on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves.