Geometric Quantization

Geometric quantization is a geometric way to quantize symplectic manifolds. Its definition 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 correc-
tions, and deformation quantization.