Quantization as a Functor

" First quantization is a mystery, but second quantization is a func-tor " (E. Nelson) Comme l'on sait la " quantification geometrique " consiste a rechercher un certain foncteur de la categorie des varietes 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. Abstract We define a category Poisson of Poisson manifolds and a category IC * of operator algebras, such that (strict) quantization should be a functor Q : Poisson → IC *. More precisely, Poisson consists of integrable Poisson manifolds as objects and isomorphism classes of regular Weinstein dual pairs as arrows, whereas IC * has so-called C(I) C *-algebras (or, equivalently, upper semicontinuous fields of C *-algebras over the interval I), as objects, and unitary equivalence classes of C(I) Hilbert bimodules as arrows. We construct Q on the subcategory of Poisson whose objects are duals of integrable Lie algebroids, and whose arrows are cotangent bundles. Here quantization is indeed functorial. In general, the functoriality of quantization incorporates the " quantization commutes with reduction " principle, and in addition implies that quantization preserves Morita equivalence.