Category of Quantizations and Inverse Problem

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.