Category of Quantizations and Inverse Problem

TL;DR

This paper develops a categorical framework for all quantizations of Poisson algebras, enabling a unified approach to classical limits and inverse problems, with applications to Lie algebras and matrix regularization.

Contribution

It introduces a new categorical structure for quantizations of Poisson algebras and proposes a novel classical limit formulation for inverse problems.

Findings

Defined equivalence of quantizations via the category.

Established two types of classical limits within the categorical framework.

Presented a method to derive quantization sequences from the principle of least action.

Abstract

We introduce a category composed of all quantizations of all Poisson algebras. By the category, we can treat in a unified way the various quantizations for all Poisson algebras and develop a new classical limit formulation. This formulation proposes a new method for the inverse problem, that is, the problem of finding the classical limit from a quantized space. Equivalence of quantizations is defined by using this category, and the conditions under which the two quantizations are equivalent are investigated. Two types of classical limits are defined as the limits in the context of category theory, and they are determined by giving a sequence of objects. Using these classical limits, we discuss the inverse problem of determining the classical limit from some noncommutative Lie algebra. From a Lie algebra, we construct a sequence of quantized spaces, from which we determine a Poisson algebra. We also present a method to obtain this sequence of quantizations from the principle of least action by using matrix regularization. Apart from the above category of quantizations of all Poisson algebras, a category of quantizations of a fixed single Poisson algebra is also introduced. In this category, the other classical limit is defined, and it is automatically determined for the category.