Berezin-Toeplitz quantization and Berezin transform

Abstract

In this lecture results on the Berezin-Toeplitz quantization of arbitrary compact quantizable Kaehler manifolds are presented. These results are obtained in joint work with M. Bordemann and E. Meinrenken. The existence of the Berezin-Toeplitz deformation quantization is also covered. Recent results obtained in joint work with A. Karabegov on the asymptotic expansion of the Berezin transform for arbitrary quantizable compact Kaehler manifolds are explained. As an application the asymptotic expansion of the Fubini-Study fundamental form under the coherent state embedding is considered. Some comments on the dynamics of the quantum operators are given. (This is an extended write-up of an invited lecture presented at the workshop ``Asymptotic properties of time evolutions in classical and quantum systems'', September 13-17, 1999, Bologna, Italy.)