Complex time evolution in geometric quantization and generalized coherent state transforms

摘要

For the cotangent bundle T~*K of a compact Lie group K, we study the complex-time evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space L~2(K) under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between L~2(K) and a certain weighted L~2-space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time -τ) within L~2(K), followed by a polarization-changing geometric-quantization evolution (for complex time +τ). In this case, our construction yields the usual generalized Segal-Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemann's "complexifier" method (which generalizes the construction of adapted complex structures).