Complex time evolution in geometric quantization and generalized coherent state transforms

TL;DR

This paper explores complex-time evolution in geometric quantization of cotangent bundles of compact Lie groups, introducing generalized coherent state transforms that unify several known transforms and relate to Thiemann's complexifier method.

Contribution

It constructs a family of generalized coherent state transforms for geometric quantization, connecting complex-time Hamiltonian flows with known transforms like the Segal--Bargmann transform.

Findings

Constructed a family of unitary isomorphisms between $L^{2}(K)$ and holomorphic function spaces.

Decomposed the isomorphism into Heisenberg-type evolution and polarization change.

Linked the construction to Thiemann's complexifier method and Mackey's theorem.

Abstract

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 $-\tau$) within $L^{2}(K)$, followed by a polarization--changing geometric quantization evolution (for complex time $+\tau$). 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). We will also investigate some properties of the generalized CSTs, and discuss how their existence can be understood in terms of Mackey's generalization of the Stone-von Neumann theorem.