Quantization as a functor

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dc.creatorLandsman, N. P.
dc.date2001-07-23
dc.date2002-02-13
dc.date.accessioned2026-07-07T04:28:34Z
dc.date.available2026-07-07T04:28:34Z
dc.descriptionNotwithstanding known obstructions to this idea, we formulate an attempt to turn quantization into a functorial procedure. We define a category PO of Poisson manifolds, whose objects are integrable Poisson manifolds and whose arrows are isomorphism classes of regular Weinstein dual pairs; it follows that identity arrows are symplectic groupoids, and that two objects are isomorphic in PO iff they are Morita equivalent in the sense of P. Xu. It has a subcategory LPO that has duals of integrable Lie algebroids as objects and cotangent bundles as arrows. We argue that naive C*-algebraic quantization should be functorial from LPO to the well-known category KK, whose objects are separable C*-algebras and whose arrows are Kasparov's KK-groups. This limited functoriality of quantization would already imply the Atiyah-Singer index theorem, as well as its far-reaching generalizations developed by Connes and others. In the category KK, isomorphism of objects implies isomorphism of K-theory groups, so that the functoriality of quantization on all of PO would imply that Morita equivalent Poisson algebras are quantized by C*-algebras with isomorphic K-theories. Finally, we argue that the correct codomain for the possible functoriality of quantization is the category RKK(I), which takes the deformation aspect of quantization into account.
dc.description20 pages LaTeX. Major revision
dc.identifierhttps://arxiv.org/abs/math-ph/0107023
dc.identifierhttp://arxiv.org/abs/math-ph/0107023
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/56837
dc.subjectMathematical Physics
dc.subjectOperator Algebras
dc.subjectSymplectic Geometry
dc.subject81S10; 46L65; 53D20; 53D55
dc.titleQuantization as a functor
dc.typetext

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