Quantum Mechanics in Phase Space

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 José 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.