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Quantization as a categorical equivalence

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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.

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Notes

  1. Others [4, 14, 28] have also provided results concerning senses in which quantization is a functor.

  2. For more details on continuous bundles of C*-algebras, see Dixmier [7, Ch.10], Kirchberg and Wasserman [15], or Landsman [19, §II.1.2].

  3. See Steeger and Feintzeig [31, Appendix B] for more details on different continuity conditions for bundles of C*-algebras. All bundles considered in this paper are uniformly continuous.

  4. It is possible to include more general locally compact metric spaces as base spaces. See Steeger and Feintzeig [31].

  5. In fact, we leave it as an open question whether there even exist morphisms between fiber C*-algebras of a strict deformation quantization that do not satisfy the scaling condition in cases of interest. We have not been able to find morphisms between the fiber C*-algebras used in Sects. 3–4 that fail the scaling condition.

  6. This is a slight generalization of the results in [6, §D], which treats the case where T is bijective and linear. One can easily check that these conditions are not necessary for \(\alpha _\hbar \) as given in Eq. (27) to define a *-homomorphism.

  7. See [3] for closely related results on automorphisms of the polynomial Weyl algebras.

  8. For more discussion of issues with the Weyl algebra, see [8, 10,11,12,13].

  9. The methods developed by Rieffel [27] for quantization apply much more broadly, even to deforming products on non-commutative C*-algebras carrying actions of \({\mathbb {R}}^d\). The methods have been further generalized by Landsman [17,18,19,20] to cases where the construction is employed locally, including Riemannian manifolds, principal bundles, and Lie groupoids. Also, Bieliavsky and Gayral [4] have provided a generalization of the quantization prescription for a much wider class of group actions.

  10. This is a small extension of the well-known fact known as “Milnor’s exercise” [16, Cor. 35.10]. We have simply extended the correspondence between algebra homomorphisms and smooth maps from algebras of the type \(C^\infty (M)\) to algebras of the type \(C_c^\infty (M)\) for a manifold M.

  11. The fact that \(\alpha _1\), so defined, is compatible with the group action follows from [28, Thm. 7.1].

  12. The fact that \(\alpha _0\), so defined, is compatible with the group action follows from [28, Thm. 7.1].

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Acknowledgements

Funding was provided by the National Science Foundation (Grant No. 2043089).

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  1. Department of Philosophy, University of Washington, Seattle, USA

    Benjamin H. Feintzeig

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Feintzeig, B.H. Quantization as a categorical equivalence. Lett Math Phys 114, 19 (2024). https://doi.org/10.1007/s11005-023-01765-w

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