José Mourão - Non-uniqueness of quantization, complex time evolution and generalized coherent ...
Jurekfest 2019 - day 2, lecture 5.
Speaker: José Mourão (introduction by Fernando Barbero).
Title: Non-uniqueness of quantization, complex time evolution and generalized coherent state transforms.
Abstract: Space of quantizations/quantization ambiguity: From the rather messy quantization ambiguity of a phase space M with a real polarization (or,equivalently, a maximal set of independent local preferred real observables in involution) one gets a very nice, infinite dimensional, symmetric space of Kähler quantizations. This is done, in geometric quantization, by allowing the preferred local observables (defining a polarization) to be complex-valued. If the associated polarization is appropriately nondegenerate, i.e. Kähler, than there is no problem of reality conditions.
Geometry on the space of quantizations and coherent state transforms: Also, it becomes easier to study the quantization ambiguity as different Kähler quantizations can be linked with a geodesic (with respect to the natural Mabuchi metric on the space of Kähler quantizations) or a imaginary time canonical transformation. By appropriately lifting these geodesics to the quantum bundle one gets integral transforms relating different quantizations, the generalized coherent state transforms or Kostant–Souriau–Heisenberg (KSH) transforms. The usual coherent state transforms correspond to M = T*G, where G is a Lie group of compact type, and to geodesics starting from the real vertical (or Schrödinger) polarization (which can be considered a point in the boundary of the space of Kähler polarizations) to the standard, bi-invariant, Kähler polarization.
Slides: http://jurekfest.fuw.edu.pl/slides/mo...
Link to conference's webpage: http://jurekfest.fuw.edu.pl