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Quantization as a Categorical Equivalence

Benjamin H. Feintzeig

Abstract

We demonstrate that, in certain cases, quantization and the classical limit provide functors that are "almost inverse" to each other. These functors map between categories of algebraic structures for classical and quantum physics, establishing a categorical equivalence.

Quantization as a Categorical Equivalence

Abstract

We demonstrate that, in certain cases, quantization and the classical limit provide functors that are "almost inverse" to each other. These functors map between categories of algebraic structures for classical and quantum physics, establishing a categorical equivalence.
Paper Structure (5 sections, 6 theorems, 53 equations)

This paper contains 5 sections, 6 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Quantization and the Classical Limit
  3. The C*-Weyl Algebra for Linear Phase Spaces
  4. Rieffel's Quantization for Actions of $\mathbb{R}^d$
  5. Conclusion

Key Result

Proposition 1

If $\alpha_0: AP(V')\to AP(U')$ is a morphism in $\mathbf{LinClass}$ for symplectic topological vector spaces $V$ and $U$, then there is a character $\chi: V\to \mathbb{C}$ on the additive group $V$ and an additive, symplectic, origin-preserving transformation $T: V\to U$ such that for each $f\in V$.

Theorems & Definitions (21)

  • Definition 1: continuous bundle of C*-algebras
  • Definition 2: strict deformation quantization
  • Definition 3: morphisms
  • Definition 4
  • Definition 5
  • Proposition 1
  • proof
  • Corollary 1
  • Definition 6
  • Definition 7
  • ...and 11 more