Abstract
We propose a generalization of quantization using a categorical approach. For a fixed Poisson algebra, quantization categories are defined as subcategories of the R-module category equipped with the structure of classical limits. We then construct the generalized quantization categories including matrix regularization, strict deformation quantization, prequantization, and Poisson enveloping algebra. It is shown that the categories of strict deformation quantization, prequantization, and matrix regularization with certain conditions are equivalent categories. On the other hand, the categories of Poisson enveloping algebra are not equivalent to the other categories.
Highlights
Noncommutative geometry is regarded as one of the key concepts for formulating the quantum gravity theory or non-perturbative string theory
We define a generalized quantization of a Poisson algebra as a subcategory of the category of modules over a commutative algebra
It is shown that matrix regularization, strict deformation quantization, and prequantization are included in this generalized quantization
Summary
Noncommutative geometry is regarded as one of the key concepts for formulating the quantum gravity theory or non-perturbative string theory. When we regard quantization as a mapfrom functions to operators acting linearly on a Hilbert space, it is generally considered that the axioms for quantization mapssatisfy the following conditions: (1) (Ĥ 1 + H2) = H 1 + H 2. Let us consider the definition of matrix regularization of a symplectic manifold (M, ω). If the following conditions are satisfied, we call the pair (Tk, h) a C1-convergent matrix regularization of (M, ω): 1. When we consider the case of Berezin–Toeplitz quantization, the naïve k → ∞ limit does not exist Such a kind of approximation of Poisson algebras via matrix algebras is studied in Ref. 3 under the name of Lα approximation. Formal deformation quantization is defined as follows: Definition 1.2.
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