Resolvent algebra in Fock-Bargmann representation

Abstract

The resolvent algebra $\mathcal{R}(X,\sigma )$ associated to a symplectic space $(X,\sigma )$ was introduced by D. Buchholz and H. Grundling as a convenient model of the canonical commutation relation (CCR) in quantum mechanics. We first study a representation of $\mathcal{R}({\mathbb{C}}^n,\sigma )$ with the standard symplectic form $\sigma$ inside the full Toeplitz algebra over the Fock-Bargmann space. We prove that $\mathcal{R}({\mathbb{C}}^n,\sigma )$ itself is a Toeplitz algebra. In the sense of R. Werner's correspondence theory we determine its corresponding shift-invariant and closed space of symbols. Finally, we discuss a representation of the resolvent algebra $\mathcal{R}(\mathcal{H},\tilde{\sigma })$ for an infinite dimensional symplectic separable Hilbert space $(\mathcal{H},\tilde{\sigma })$. More precisely, we find a representation of $\mathcal{R}(\mathcal{H},\tilde{\sigma })$ inside the full Toeplitz algebra over the Fock-Bargmann space in infinitely many variables.

29 pages; suggestions and questions are welcome