Differential Equations and Numerical Analysis Seminar - 18/01/2012

Wednesday, 18 January 2012, 12:00 p.m.

Lecturer: Carlos Florentino, CAMGSD - Instituto Superior Técnico/UTL

Title: "Geometric quantization - from classical phase space to quantum Hilbert space”.

Local: Room 1.5, Edifício VII
Faculdade de Ciências e Tecnologia, Quinta da Torre, Caparica

 

Abstract: Given a classical mechanical system, described by observables defined in a certain phase space, geometric quantization is one attempt to produce the most appropriate quantum version of it, where phase space is replaced by a quantum Hilbert space and observables are replaced by operators in that Hilbert space. We shall indicate some of the desired properties of this passage (which cannot be simultaneously satisfied), and present the method of geometric pre-quantization, where the Hilbert space is produced out of geometric data: a line bundle with connection whose curvature coincides with the symplectic form. In the special case where phase space is Kähler, geometric pre-quantization has a satisfactory answer allowing for a essentially unique solution, and some examples can be described explicitly. Finally, we present joint work w/ T. Baier, J. Mourao and J. P. Nunes (J. Funct. Anal. *192* (2002) 410-424; arXiv:0806.0606) where the coherent state transform and the heat kernel for the circle are used to show uniqueness of pre-quantization in the cases of abelian varieties and smooth toric varieties.