Coherent states and geometric quantization.

摘要

A system of coherent states associated to a symplectic manifold M is a map phi : M → H for some Hilbert space H. In this thesis, we define and study overcomplete systems of coherent states that arise from the program of geometric quantization, and their relations to the Fourier transform in quantum mechanics, the Bargmann transform, traditional, Perelomov-type and Rawnsley-type coherent states, and deformation quantization.; In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier and Bargmann transforms arise in the context of geometric quantization. We consider a Hilbert space bundle H over the space J of all choices of complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection that was first described by Axelrod-Della Pietra-Witten. We construct the kernel for the integral parallel transport operator for this connection. The kernel is a Heisenberg-Weyl coherent state as well as the Bergman reproducing kernel. By extending geodesics to the boundary (for which the metaplectic correction is essential) we recover the Fourier transform rule from quantum mechanics, and also the Bargmann transform.; Next, we define coherent states associated to an arbitrary symplectic manifold using reproducing kernels introduced by Pasternak-Winiarski-Monastyrski-Wojcieszynski. These coherent states satisfy all of the traditional properties of coherent states, including overcompleteness. In the case that M is Kahler, they are related to heat kernels of Dirac operators. This relation gives a simple proof that the integral of the coherent density (relative to which the coherent states are overcomplete) is a topological invariant The coherent density is fundamental in the study of deformation quantization, and coincides with an object constructed by different means by Rawnsley, as does the coherent state 2-point function.; Finally, we explore the relation between coherent states and deformation quantization. By interpreting the limit h → 0 as the limit k → infinity for k-th tensor powers of a line bundle over M, coherent states provide an alternative quantization of M, which we show to be a pure state quantization.