Applications of Geometric Quantization

Two applications of geometric quantization are described here. In the first, the classical system whose phase space is a two-sphere is quantized with an anti-holomorphic Kahler polarization. The result is an isolated quantum spin degree of freedom, with integral or half-integral total spin. A convenient formalism for handling geometric quantization with a Kahler polarization is exhibited. The second application is a discussion of several approaches to the quantization of classical systems with first class constraints. It is shown that there is a quantization procedure, called here the constrained polarization method, which is equivalent to quantizing (with a real polarization) the true classical degrees of freedom. This new method is often more convenient than directly quantizing the reduced phase space. In an example, it is shown that the time -honored procedure of imposing constraints as quantum operator equations can result in a mathematically non-viable quantum theory, if the chosen polarization is not compatible with the constraints. For this particular example, which is somewhat reminiscent of the canonical description of general relativity, there is a polarization compatible with constraints.