Resolvent algebra in Fock-Bargmann representation

TL;DR

This paper explores the resolvent algebra related to quantum mechanics within the Fock-Bargmann space, proving its structure as a Toeplitz algebra and extending representations to infinite-dimensional cases.

Contribution

It demonstrates that the resolvent algebra can be realized as a Toeplitz algebra in the Fock-Bargmann representation and extends this to infinite-dimensional symplectic Hilbert spaces.

Findings

Resolvant algebra is a Toeplitz algebra in the Fock-Bargmann space.

Identifies the symbol space for the algebra.

Provides a representation in infinite dimensions.

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.