A Phase Space Representation of the Metaplectic Group

TL;DR

This paper develops a phase space extension of the metaplectic group, enabling it to act on functions directly in phase space, using advanced operator representations and pseudodifferential techniques.

Contribution

It introduces a novel extension of the metaplectic group acting on phase space functions, expanding the mathematical framework of symplectic and metaplectic representations.

Findings

Constructed an explicit phase space representation of Mp(n)

Connected twisted Weyl symbols with Bopp pseudodifferential operators

Extended the action of the metaplectic group to phase space functions

Abstract

The symplectic group Sp(n) acts on phase space while the unitary representation of its double cover, Mp(n), the metaplectic group, acts on functions defined on configuration space. We will construct an extension Mp(n) of Mp(n) acting on square integrable functions on phase space. This is performed using previous results of ours involving explicit expressions of the twisted Weyl symbols of metaplectic operators and Bopp pseudodifferential operators, which are phase space extensions of the usual Weyl operators.