Complex time evolution in geometric quantization and generalized coherent state transforms
| Autor: | João P. Nunes, José Mourão, William D. Kirwin |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2012 |
| Předmět: |
Geometric quantization
Mathematics - Differential Geometry High Energy Physics - Theory Pure mathematics 81S10 53D50 22E30 Holomorphic function FOS: Physical sciences 01 natural sciences symbols.namesake 0103 physical sciences FOS: Mathematics 0101 mathematics Mathematical Physics Mathematics 010102 general mathematics Hilbert space Time evolution Lie group Mathematical Physics (math-ph) 16. Peace & justice Differential Geometry (math.DG) High Energy Physics - Theory (hep-th) Mathematics - Symplectic Geometry Vertical tangent symbols Symplectic Geometry (math.SG) Cotangent bundle 010307 mathematical physics Hamiltonian (quantum mechanics) Analysis |
| Popis: | 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. Comment: 28 pages |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|