Quantum mappings acting by coordinate transformations on Wigner distributions

TL;DR

This paper characterizes the unique properties of Wigner distributions, showing they are the only quasidistributions invariant under certain transformations, and identifies their maximal symmetry group.

Contribution

It proves the uniqueness of the Wigner transform under specific covariance conditions and identifies the full group of transformations preserving Wigner distributions.

Findings

Wigner transform is the only sesquilinear map with boundedness and covariance properties.

Wigner distributions are uniquely invariant under linear symplectic transformations.

The maximal group preserving Wigner distributions includes translations and linear symplectic/antisymplectic transformations.

Abstract

We prove two results about Wigner distributions. Firstly, that the Wigner transform is the only sesquilinear map ${\mathcal S}(\mathbb{R}^n) \times {\mathcal S}(\mathbb{R}^n) \to {\mathcal S}(\mathbb{R}^{2n})$ which is bounded and covariant under phase-space translations and linear symplectomorphisms. Consequently, the Wigner distributions form the only set of quasidistributions which is invariant under linear symplectic transformations. Secondly, we prove that the maximal group of (linear or non-linear) coordinate transformations that preserves the set of (pure or mixed) Wigner distributions consists of the translations and the linear symplectic and antisymplectic transformations.