Wigner functions on non-standard symplectic vector spaces

TL;DR

This paper explores the properties of Wigner functions in non-standard symplectic vector spaces, extending classical results to more general settings and analyzing their behavior under linear transformations.

Contribution

It generalizes the theory of Wigner functions and related concepts to arbitrary non-standard symplectic spaces, including new results on symplectic spectrum and transformations.

Findings

Wigner functions on different symplectic spaces are distinct but intersect.

Extended Williamson's theorem and symplectic spectrum concepts to non-standard spaces.

Analyzed how Wigner functions transform under linear coordinate changes.

Abstract

We consider the Weyl quantization on a flat non-standard symplectic vector space. We focus mainly on the properties of the Wigner functions defined therein. In particular we show that the sets of Wigner functions on distinct symplectic spaces are different but have non-empty intersections. This extends previous results to arbitrary dimension and arbitrary (constant) symplectic structure. As a by-product we introduce and prove several concepts and results on non-standard symplectic spaces which generalize those on the standard symplectic space, namely the symplectic spectrum, Williamson's theorem and Narcowich-Wigner spectra. We also show how Wigner functions on non-standard symplectic spaces behave under the action of an arbitrary linear coordinate transformation.