作为函子的量子化
Quantization as a functor
摘要：
尽管已知的障碍，这一想法，我们制定了一个试图把量化成一个函子程序。我们定义了一类Poisson流形PO，其对象是可积Poisson流形，其箭头是正则Weinstein对偶对的同构类;由此得出单位箭头是辛群胚，并且两个对象在PO中同构当且仅当它们在P. Xu意义下是Morita等价的.它有一个子范畴LPO，它以可积的李代数胚为对象，以余切丛为箭头。我们认为朴素C*-代数量子化应该是从LPO到著名范畴KK的函子，其对象是可分C*-代数，其箭头是Kasparov的KK-群。量子化的这种有限的功能性已经暗示了阿蒂亚-辛格指数定理，以及由康纳斯和其他人发展的意义深远的推广。在范畴KK中，对象的同构蕴涵着K-理论群的同构，因此在PO的全体上量子化的函子性意味着Morita等价的Poisson代数被具有同构K-理论的C*-代数量子化。最后，我们论证了量子化的可能函性的正确余域是范畴RKK（I），它考虑了量子化的变形方面。
英文摘要：
Notwithstanding 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.
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