A Phase Space Representation of the Metaplectic Group
Maurice de Gosson
TL;DR
The paper develops a phase-space extension $\widetilde{\mathrm{Mp}}(n)$ of the metaplectic group by leveraging twisted Weyl symbols and the Bopp pseudodifferential calculus, thereby allowing unitary action on $L^2(\mathbb{R}^{2n})$ and aligning the domains of Sp(n) and Mp(n). Core results include explicit twisted Weyl symbols for metaplectic operators, a factorization framework, and a concrete phase-space representation obtained via Bopp operators; these tools enable a phase-space formulation of metaplectic operators and their relation to Wigner transforms. A key application is the Feichtinger algebra $S_0(\mathbb{R}^n)$, with a phase-space metaplectic characterization that yields robust, invariant criteria under both Mp(n) and phase-space displacements, along with asymptotic analyses as $\hbar \to 0$. Overall, the work strengthens the interplay between time-frequency analysis and phase-space quantization, providing new avenues for modulation-space methods in quantum mechanics.
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.