Quantization as a Categorical Equivalence for Hilbert Bimodules and Lagrangian Relations

TL;DR

This paper establishes a categorical equivalence between classical and quantum theories by demonstrating that quantization and classical limit functors are almost-inverses, preserving the structure of their respective representation theories.

Contribution

It introduces a categorical framework linking classical and quantum representations via Lagrangian relations and Hilbert bimodules, showing their structural preservation through quantization and classical limit.

Findings

Quantization and classical limit functors are almost-inverses.

Categories of Lagrangian relations and Hilbert bimodules are equivalent.

Representation theories are structurally preserved across classical and quantum contexts.

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.