Quantum mechanics in phase space: The Schr\"odinger and the Moyal representations

TL;DR

This paper develops a phase space formulation of quantum mechanics, establishing its equivalence to the standard approach, and explores the Moyal representation where wave functions are cross-Wigner functions, connecting to deformation quantization.

Contribution

It introduces a phase space Schrödinger representation, derives the associated calculus, and relates it to the Moyal representation and deformation quantization.

Findings

The phase space formulation is unitarily equivalent to the standard quantum mechanics.

The Moyal representation uses cross-Wigner functions as wave functions.

The paper details the pseudo-differential calculus and spectral properties in these representations.

Abstract

We present a phase space formulation of quantum mechanics in the Schr\"odinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schr\"odinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.