Wigner functions, Fresnel optics, and symplectic connections on phase space

TL;DR

This paper reveals that Wigner functions inherently include a symplectic connection, which covariantizes the symplectic exterior derivative, and explores its implications for local symplectic covariance in quantum mechanics.

Contribution

It introduces the concept of a symplectic connection within Wigner functions and analyzes its role in local symplectic covariance of quantum states.

Findings

Wigner functions contain a symplectic connection.

The symplectic connection covariantizes the exterior derivative on phase space.

Local symplectic covariance is relevant in the Schrödinger equation on phase space.

Abstract

We prove that Wigner functions contain a symplectic connection. The latter covariantises the symplectic exterior derivative on phase space. We analyse the role played by this connection and introduce the notion of local symplectic covariance of quantum-mechanical states. This latter symmetry is at work in the Schroedinger equation on phase space.