Quantum States Localized on Lagrangian Submanifolds

Let X be a symplectic manifold and Aut(L) the automorphism group of a Kostant-Souriau line bundle on X. *Quantum states for X*, as defined by J.-M. Souriau in the 1990s, are certain positive-definite functions on Aut(L) or, less ambitiously, on any “large enough” subgroup G of Aut(L). This definition has two major drawbacks: when G = Aut(L) there are no known examples; and when G is a Lie subgroup the notion is far from selective enough. In this talk I’ll introduce the concept of a quantum state *localized at Y *, where Y is a coadjoint orbit of a subgroup H of G. I’ll explain how such states often exist and are unique when Y has lagrangian preimage in X, and how this can be regarded as a solving, in a number of cases, A. Weinstein’s “fundamental quantization problem” of attaching state vectors to lagrangian submanifolds.