Energy transitions driven by phase space reflection operators

TL;DR

This paper explores how phase space reflection operators influence quantum state transitions, providing a classical approximation framework and addressing singularities with Airy functions, enhancing understanding of quantum-classical correspondence.

Contribution

It introduces a classical approximation of transition probabilities via phase space reflection operators and resolves singularities in spectral Wigner functions using Airy functions.

Findings

Transition probabilities are given by integrals over phase space intersections.

Singularities at caustics are integrable except for one degree of freedom.

Airy functions effectively resolve spectral Wigner function singularities.

Abstract

Phase space reflection operators lie at the core of the Wigner-Weyl representation of density operators and observables. The role of the corresponding classical reflections is known in the construction of semiclassical approximations to Wigner functions of pure eigenstates and their coarsegrained microcanonical superpositions, which are not restricted to classically integrable systems. In their active role as unitary operators, they generate transitions between pairs of eigenstates specified by transition Wigner functions (or cross-Wigner functions): The square modulus of the transition Wigner function at each point in phase space is the transition probability for the reflection through that point. Coarsegraining the initial and final energies provides a transition probability density as a phase space path integral. It is here investigated in the simplest classical approximation involving microcanonical Wigner functions. A reflection operator generates a transition between a pair of energy shells with a probability density given by the integral of the inverse modulus of a Poisson bracket over the intersection of a shell with the reflection of its pair. The singularity of the pair of Wigner functions at their dominant caustics is nicely integrable over their intersection, except for a single degree of freedom. Even though this case is not directly relevant for future investigations of chaotic systems, it is shown here how the improved approximation of the spectral Wigner functions in terms of Airy functions resolves the singularity.