Symplectically Covariant Schr\"{o}dinger Equation in Phase Space

TL;DR

This paper develops a phase-space formulation of quantum mechanics by constructing an irreducible representation of the Heisenberg group on phase space, extending Weyl calculus, and discussing the probabilistic interpretation of the phase-space Schrödinger equation.

Contribution

It introduces a symplectically covariant Schrödinger equation in phase space using the Wigner-Moyal transform, extending the Weyl calculus and providing a new perspective on quantum mechanics.

Findings

Constructed an irreducible Heisenberg group representation on phase space.

Extended Weyl calculus into a phase-space framework.

Established the equivalence of phase-space quantum mechanics to standard formulations.

Abstract

A classical theorem of Stone and von Neumann says that the Schr\"{o}dinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schr\"{o}dinger equation