Resolvent algebra in Fock-Bargmann representation

The resolvent algebra $\mathcal{R}(X, σ)$ associated to a symplectic space $(X, σ)$ 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 $\mathcal{R}(\mathbb{C}^n, σ)$ with the standard symplectic form $σ$ inside the full Toeplitz algebra over the Fock-Bargmann space. We prove that $\mathcal{R}(\mathbb{C}^n, σ)$ 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 $\mathcal{R}(\mathcal{H}, \tildeσ)$ for an infinite dimensional symplectic separable Hilbert space $(\mathcal{H}, \tildeσ)$. More precisely, we find a representation of $\mathcal{R}(\mathcal{H}, \tildeσ)$ inside the full Toeplitz algebra over the Fock-Bargmann space in infinitely many variables.

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