Quantum Mechanics in Phase Space

TL;DR

This paper discusses the development and understanding of quantum mechanics in phase space, highlighting historical hesitations and the eventual acceptance of this formalism since the 20th century.

Contribution

It reviews the historical evolution and conceptual foundations of quantum mechanics in phase space, clarifying its formalism and acceptance over time.

Findings

Formalism for quantum mechanics in phase space was established in the 1940s.

Widespread acceptance of phase space formalism occurred in the late 20th century.

The approach addresses the joint consideration of positions and momenta in quantum mechanics.

Abstract

Ever since Werner Heisenberg's 1927 paper on uncertainty, there has been considerable hesitancy in simultaneously considering positions and momenta in quantum contexts, since these are incompatible observables. But this persistent discomfort with addressing positions and momenta jointly in the quantum world is not really warranted, as was first fully appreciated by Hilbrand Groenewold and Jos\'e Moyal in the 1940s. While the formalism for quantum mechanics in phase space was wholly cast at that time, it was not completely understood nor widely known --- much less generally accepted --- until the late 20th century.