Maximal Covariance Group of Wigner Transforms and Pseudo-Differential Operators

TL;DR

This paper identifies the maximal covariance group for Wigner transforms and Weyl operators as linear symplectic and anti-symplectic transformations, using new symplectic geometry results to refine classical capacity preservation theorems.

Contribution

It establishes the maximal covariance group for Wigner and Weyl operators and introduces a new symplectic geometry result to refine classical capacity preservation theorems.

Findings

Linear symplectic and anti-symplectic transformations form the maximal covariance group.

A new symplectic geometry result characterizes symplectic and anti-symplectic matrices.

Refinement of classical results on symplectic capacities of ellipsoids.

Abstract

We show that the linear symplectic and anti-symplectic transformations form the maximal covariance group for both the Wigner transform and Weyl operators. The proof is based on a new result from symplectic geometry which characterizes symplectic and anti-symplectic matrices, and which allows us, in addition, to refine a classical result on the preservation of symplectic capacities of ellipsoids.