# Quantization and the Resolvent Algebra

## TL;DR

This paper introduces a new commutative C*-algebra related to symplectic vector spaces, and develops deformation and Berezin-type quantizations that connect it to the resolvent algebra, with detailed finite-dimensional analysis.

## Contribution

It constructs a novel commutative algebra and establishes compatible quantization maps linking it to the resolvent algebra, expanding the mathematical framework of quantum field theory.

## Key findings

- Defined a new commutative C*-algebra $C_
{R}(X)$ on symplectic spaces.
- Developed a strict deformation quantization mapping to the resolvent algebra.
- Analyzed the algebra's structure and spectrum in finite dimensions.

## Abstract

We introduce a novel commutative C*-algebra $C_\mathcal{R}(X)$ of functions on a symplectic vector space $(X,\sigma)$ admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra of $C_\mathcal{R}(X)$ to the resolvent algebra introduced by Buchholz and Grundling [JFA, 2008]. The associated quantization map is a field-theoretical Weyl quantization compatible with the work of Binz, Honegger and Rieckers [AHPO, 2004]. We also define a Berezin-type quantization map on all of $C_\mathcal{R}(X)$, which continuously and injectively maps it onto a dense subset of the resolvent algebra. The commutative resolvent algebra $C_\mathcal{R}(X)$, generally defined on a real inner product space $X$, intimately depends on the finite dimensional subspaces of $X$. We thoroughly analyze the structure of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its Gelfand spectrum.