Abstract: To quantize a classical system we have to consider the kinematic relation between the classical and quantum case: In the quantum case the states of a system are represented by the rays in a Hilbert space H and the observables by a collection of symmetric operators on H. In the classical case the state space is a symplectic manifold M and the observables are the algebra of smooth functions on M. The kinematic problem is: given M and its symplectic form is it possible to reconstruct the Hilbert space H and the symmetric operators?

Geometric quantization gives a well defined procedure to construct the Hilbert space H and the operators corresponding to the classical observables. This procedure also satisfies the Dirac's quantum conditions. In this talk I will discuss these constructions in detail and the three stages of geometric quantization: pre-quantization, polarization and metaplectic correction.