Quantization as a Categorical Equivalence for Hilbert Bimodules and Lagrangian Relations

• URL: http://arxiv.org/abs/2602.15188v1
• Date: Mon, 16 Feb 2026 20:56:24 GMT
• Title: Quantization as a Categorical Equivalence for Hilbert Bimodules and Lagrangian Relations
• Authors: Benjamin H. Feintzeig,
• Abstract summary: I discuss distinct representation-theory preserving morphisms in the classical and quantum contexts.<n>I consider categories whose morphisms are Lagrangian relations in the classical context and Hilbert bimodules in the quantum context.<n>I treat quantization and the classical limit as determining functors between these categories.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• The 2-Category of Topological Quantum Computation [0.0]
  It has been widely assumed that the same category formalizes a topological quantum computing model.<n>In this paper, we argue that a categorical formalism that captures and unifies both anyonic theories and a model of topological quantum computing is a braided (fusion) 2-category.
  arXiv  Detail & Related papers  (2025-05-28T09:41:17Z)
• Kochen-Specker for many qubits and the classical limit [55.2480439325792]
  It is shown that quantum and classical predictions converge as the number of qubits is increases to the macroscopic scale.<n>This way to explain the classical limit concurs with, and improves, a result previously reported for GHZ states.
  arXiv  Detail & Related papers  (2024-11-26T22:30:58Z)
• A Universal Kinematical Group for Quantum Mechanics [0.0]
  In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms.
  arXiv  Detail & Related papers  (2024-04-28T18:46:24Z)
• Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity [0.0]
  We construct a theory of quantum matter fields and Nordstrom gravity in which the space-time metric is treated classically. The dynamics is constructed via the classical-quantum path integral and is completely positive, trace preserving (CPTP)
  arXiv  Detail & Related papers  (2024-01-10T19:23:00Z)
• Quantum Foundations as a Guide for Refining Particle Theories [0.0]
  All quantum field theories describe interacting bosonic elementary particles. We indicate that introducing interactions still leads to classical theories that can be compared with the quantum theories.
  arXiv  Detail & Related papers  (2023-12-14T23:17:54Z)
• Matter relative to quantum hypersurfaces [44.99833362998488]
  We extend the Page-Wootters formalism to quantum field theory. By treating hypersurfaces as quantum reference frames, we extend quantum frame transformations to changes between classical and nonclassical hypersurfaces.
  arXiv  Detail & Related papers  (2023-08-24T16:39:00Z)
• Connecting classical finite exchangeability to quantum theory [45.76759085727843]
  Exchangeability is a fundamental concept in probability theory and statistics.<n>It allows to model situations where the order of observations does not matter.<n>It is well known that both theorems do not hold for finitely exchangeable sequences.
  arXiv  Detail & Related papers  (2023-06-06T17:15:19Z)
• Correspondence Between the Energy Equipartition Theorem in Classical Mechanics and its Phase-Space Formulation in Quantum Mechanics [62.997667081978825]
  In quantum mechanics, the energy per degree of freedom is not equally distributed. We show that in the high-temperature regime, the classical result is recovered.
  arXiv  Detail & Related papers  (2022-05-24T20:51:03Z)
• Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality [69.62715388742298]
  We introduce a formulation of quantum theory (QT) as a general probabilistic theory but expressed via quasi-expectation operators (QEOs) We will show that QT for both distinguishable and indistinguishable particles can be formulated in this way. We will show that finitely exchangeable probabilities for a classical dice are as weird as QT.
  arXiv  Detail & Related papers  (2022-03-08T14:47:39Z)
• The classical limit of Schr\"{o}dinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as emergent phenomenon [0.0]
  A strict deformation quantization is analysed on the classical phase space $bR2n$. The existence of this classical limit is in particular proved for ground states of a wide class of Schr"odinger operators. The support of the classical state is included in certain orbits in $bR2n$ depending on the symmetry of the potential.
  arXiv  Detail & Related papers  (2021-03-22T14:55:57Z)
• Emergence of classical behavior in the early universe [68.8204255655161]
  Three notions are often assumed to be essentially equivalent, representing different facets of the same phenomenon. We analyze them in general Friedmann-Lemaitre- Robertson-Walker space-times through the lens of geometric structures on the classical phase space. The analysis shows that: (i) inflation does not play an essential role; classical behavior can emerge much more generally; (ii) the three notions are conceptually distinct; classicality can emerge in one sense but not in another.
  arXiv  Detail & Related papers  (2020-04-22T16:38:25Z)
• From a quantum theory to a classical one [117.44028458220427]
  We present and discuss a formal approach for describing the quantum to classical crossover. The method was originally introduced by L. Yaffe in 1982 for tackling large-$N$ quantum field theories.
  arXiv  Detail & Related papers  (2020-04-01T09:16:38Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.