Prequantization, geometric quantization, corrected geometric quantization

Abstract

A comparison on some facts concerning the geometric quantization of symplectic manifolds is presented here. Criticism, facts and improvements on the sophisticated theory of geometric quantization are presented touching briefly, all the "salient points of the theory". The unfamiliar reader can consider this as a "soft" introduction to the topic. Contents 1. Introduction 2. Kähler manifolds, classical mechanics and symmetries 2.1. Kähler manifolds 2.2. Kähler manifolds and classical mechanics 2.3. Kähler manifolds and classical mechanics with symmetries 3. Prequantization 3.1. Prequatization conditions 3.2. Examples 4. Geometric quantization 4.1. The Dirac axioms and quantum operators 4.2. Kähler polarizations 4.3. Kähler quantization, holomorphic sections and Szegö kernels 5.The corrected geometric quantization: the case of complex cotangent bundle 5.1. The square-root bundle, the half form hilbert space and the harmonic oscillator 5.2. The BKS pairing 5.3. The BKS pairing on the cotangent bundle, the time evolution and the Schrödinger equation 5.4. Reconstruction of Schrödinger equation on the cotangent bundle 5.5. The Klein-Gordon equation and the cotangent bundle 5.6. Considerations on the BKS pairing 5.7. The BKS pairing for one complex and one real polarization: the Segal-Bargmann transform 5.8. The BKS pairing between two complex polarizations: the Bogoliubov transformation 6. The Bohr-Sommerfeld