ar X iv : m at h - ph / 0 10 70 23 v 2 1 3 Fe b 20 02 Quantization as a functor ∗

Notwithstanding known obstructions to this idea, we formulate an attempt to turn quantization into a functorial procedure. We define a category Poisson 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 Poisson iff they are Morita equivalent in the sense of P. Xu. It has a subcategory LPoisson 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 LPoisson 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 Poisson 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. 1 " First quantization is a mystery, but second quantization is a func-tor " (E. Nelson) Comme l'on sait la " quantification géometrique " consistè a rechercher un certain foncteur de la catégorie des variétes symplectiques et sym-plectomorphismes dans celle des espaces de Hilbert complexes et des transformations unitaires (.. .) Il est bien connu qu'un tel foncteur n'existe pas. (from A. Crumeyrolle's review MR81g:58016 of [19])