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Quantization as a Categorical Equivalence for Hilbert Bimodules and Lagrangian Relations

Benjamin H. Feintzeig

Abstract

It is well known that classical and quantum theories carry distinct types of representations, each type of representation corresponding to possible values of generalized charges in the classical or quantum context. This paper demonstrates a sense in the structure of these representation theories is preserved from classical to quantum physics. To show this, I discuss distinct representation-theory preserving morphisms in the classical and quantum contexts. Specifically, I consider categories whose morphisms are Lagrangian relations in the classical context and Hilbert bimodules in the quantum context. These morphisms are significant because they give rise to induced representations of classical and quantum theories, respectively. I consider quantization and the classical limit as determining functors between these categories. I treat quantization via the strict deformation quantization of a Poisson algebra and the classical limit via the extension of a uniformly continuous bundle of C*-algebras. With these tools, I prove that the quantization and classical limit functors are "almost-inverse" to each other, thus establishing a categorical equivalence.

Quantization as a Categorical Equivalence for Hilbert Bimodules and Lagrangian Relations

Abstract

It is well known that classical and quantum theories carry distinct types of representations, each type of representation corresponding to possible values of generalized charges in the classical or quantum context. This paper demonstrates a sense in the structure of these representation theories is preserved from classical to quantum physics. To show this, I discuss distinct representation-theory preserving morphisms in the classical and quantum contexts. Specifically, I consider categories whose morphisms are Lagrangian relations in the classical context and Hilbert bimodules in the quantum context. These morphisms are significant because they give rise to induced representations of classical and quantum theories, respectively. I consider quantization and the classical limit as determining functors between these categories. I treat quantization via the strict deformation quantization of a Poisson algebra and the classical limit via the extension of a uniformly continuous bundle of C*-algebras. With these tools, I prove that the quantization and classical limit functors are "almost-inverse" to each other, thus establishing a categorical equivalence.
Paper Structure (12 sections, 15 theorems, 101 equations)

This paper contains 12 sections, 15 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Classical and Quantum Models
  3. Systems associated to Lie groupoids
  4. Quantization and the Classical Limit for Lie Groupoids
  5. Classical and Quantum Morphisms
  6. Dual Pairs and Hilbert Bimodules
  7. Quantization and the Classical Limit of Morphisms
  8. Categorical Equivalence
  9. Bimodules between Continuous Bundles
  10. Local Structure of Classical Dual Pairs
  11. Natural Isomorphism for the Classical Limit
  12. Conclusion

Key Result

Lemma 1

For any $\underline{\varphi},\underline{\psi}\in\widetilde{\underline{\mathcal{E}}}_M$, the map $\hbar\mapsto \langle\underline{\varphi}_\hbar,\underline{\psi}_\hbar\rangle_{\mathcal{E}_M}$ is uniformly continuous with respect to the norm on $C^*(H)$.

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 24 more