2012/03/21 by William D. Kirwin, Kirwin, William D., José M. Mourão +3
Mathematics · Physics and Astronomy · #22E30 #53D50 #81S10 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Laser-Matter Interactions and Applications #Mathematical Physics (math-ph) #Quantum Mechanics and Applications #Symplectic Geometry (math.SG) #hep-th #math-ph #math.DG #math.MP #math.SG #msc:22E30 #msc:53D50 #msc:81S10
paper · pdf · doi:10.48550/arxiv.1203.4767
28 pages
arxiv created 2012/03/21 · openalex publication_date 2012/03/21 · arxiv updated 2012/03/22 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
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 L2(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 L2(K) and a certain weighted L2-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 L2(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.