Husimi, Wigner, T{\"o}plitz, quantum statistics and anticanonical transformations

TL;DR

This paper investigates how Husimi, Wigner, and Toeplitz symbols of quantum states transform under quantum statistics, revealing their connection to canonical transformations on phase space and complex canonical relations.

Contribution

It establishes a link between quantum statistical operations and canonical transformations on phase space, including complexified versions, providing new insights into quantum-classical correspondence.

Findings

Quantum statistics induce canonical transformations on phase space.

These transformations correspond to complex canonical relations in Toeplitz representations.

The work connects quantum symmetries with classical geometric structures.

Abstract

We study the behaviour of Husimi, Wigner and T{\"o}plitz symbols of quantum density matrices when quantum statistics are tested on them, that is when on exchange two coordinates in one of the two variables of their integral kernel. We show that to each of these actions is associated a canonical transform on the cotangent bundle of the underlying classical phase space. Equivalently can one associate a complex canonical transform on the complexification of the phase-space. In the off-diagonal T{\"o}plitz representation introduced in [P], the action considered is associated to a complex aanticanonical relation.