Algebras of distributions suitable for phase-space quantum mechanics. I
José M. Gracia-Bondía, Joseph C. Várilly
TL;DR
The paper constructs a rigorous phase-space formulation of quantum mechanics by extending the twisted product $\times$ and twisted convolution $\diamond$ from the Schwartz space $\mathcal{S}_2$ to tempered distributions, yielding the Moyal $*$-algebra $\mathcal{M} = \mathcal{M}_L \cap \mathcal{M}_R$. It proves $\mathcal{M}$ is a Fourier-invariant $*$-algebra that contains $\mathcal{S}_2$, as well as distributions with compact support, and equips $\mathcal{S}_2'$ with a bimodule structure under $\mathcal{M}$. A pivotal innovation is the matricial representation based on the twisted Hermite basis $f_{mn}$, which diagonalizes the oscillator Hamiltonian and turns $f\times g$ into a matrix product with $(ab)_{mn} = \sum_k a_{mk} b_{kn}$. The work also introduces Sobolev-like scales $\mathcal{G}_{s,t}$ that form Banach algebras under $\times$, providing a framework for analyzing regularity and growth, and sets the stage for a spectral-theoretic treatment of phase-space quantum mechanics in a subsequent paper.
Abstract
The twisted product of functions on $R^{2N}$ is extended to a $*$-algebra of tempered distributions which contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation. The regularity properties of the twisted product are investigated. A matrix presentation of the twisted product is given, with respect to an appropriate orthonormal basis, which is used to construct a family of Banach algebras under this product.