On Kählerian Coherent States

Abstract

A reformulation of Rawnsley’s Kählerian coherent states (in the framework of geometric quantization) is used in order to investigate the interplay between their local and global properties (projective embeddings) and the relationship with Klauder quantization (via path integrals and the introduction of a metric on the classical phase space). A Klauder type formula is established for the projection operator onto the quantum Hilbert space (the kernel of a Bochner Laplacian) in terms of a phase space path integral. As a further application, a Riemann surface diastatic identity is derived, yielding, via Green function theory, a short proof of the Abel-Jacobi theorem (and conversely), together with some coherent state induced theta function identities.