Complex time evolution in geometric quantization and generalized coherent state transforms

Abstract

For the cotangent bundle T * K of a compact Lie group K, we study the complextime 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). 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.

Quantization in singular real polarizations: Kähler regularization, Maslov correction and pairings

“…In the examples in [KMN1, KMN2,KMN4], the space of P-polarized quantum states was already known and the limit actually recovers the correct Hilbert space. Conjecturally, however, one could possibly start with a badly behaved (and hence difficult to quantize directly) polarization P and define H P as a limit of well-behaved quantizations in Kähler polarizations.…”

“…, n C , there are many Thiemann complexifiers or regulators of the first type. It follows from [KMN2] that any strongly convex function of the momenta is a P Sch regulator of the first type. Functions of both p and q can also be used as, for example, the Hamiltonians of harmonic oscillators which will be studied in [Es].…”

“…Denoting the corresponding real polarization by P µ , the proposal of [MN2] corresponds then to obtaining the Hilbert space H µ , of quantum states in the polarization P µ , as the infinite imaginary time limit √ −1s with s → +∞, of the family defined by applying the imaginary time flow of the Hamiltonian vector field of the norm square of the moment map, X ||µ|| 2 , to the Hilbert space corresponding to a starting Kähler quantization. In order to relate the Schrödinger representation to this Kähler polarization, we consider the Thiemann complexifier method [Th1,Th2] adapted to geometric quantization in [HK1,HK2,KMN2,KMN3,KMN4]. In the toric case, these families of polarizations were first introduced in [BFMN].…”

“…These families were also used in [KMN1] to derive the Maslov shift of levels of the Bohr-Sommerfeld leaves. The momentum space quantization T * K, for compact Lie groups K, was also defined in [KW2] through the infinite imaginary time limit of the flow of the Hamiltonian vector field of the norm squared of the (in general non-abelian) moment map of the action of K on T * K (see also [KMN2]). …”

Partial coherent state transforms, G × T-invariant Kähler structures and geometric quantization of cotangent bundles of compact Lie groups

“…In fact, these two polarizations can be connected by a continuous family of G × G-invariant Kähler polarizations, which are related among themselves by compositions of Hall's coherent state transforms (CST) [FMMN1;FMMN2;KW]. On the other hand, these, as well as other more general [N;KMN1;KMN2], natural families of G × G-invariant Kähler structures are also very interesting from the point of view of Kähler geometry. Indeed, they are generated by the analytic continuation to complex time of Hamiltonian flows on T * G, of a so-called complexifier Hamiltonian function [Th; HK], and correspond to geodesics for the Mabuchi affine connection on the space of Kähler metrics on T * G [KMN1;MN].…”