Geometric Quantization

Abstract

Geometric quantization is a geometric way to quantize symplectic manifolds. Its denition and properties are presented. Symplectic manifolds are called quantizable if there exists a hermitian line bundle with a compatible connection such that the curvature is essentially equal to the symplectic form of the manifold. In a first step the prequantum operators acting on the sections of the hermitian line bundle are introduced. In a second step a polarization is introduced and the quantum operators are defined by restricting the prequantum operators to the space of polarized sections. Different polarizations are discussed. In the compact Kähler manifold case with Kähler polarization the geometric quantum operator are related to the Berezin—Toeplitz quantum operators. Some other concepts discussed are asymptotic expansions by considering higher tensor powers of the quantum line bundle, half-form corrections, and deformation quantization.