Geometric quantization and constrained systems

The problem of obtaining the quantum theory of systems with first class constraints is discussed in the context of geometric quantization. The precise structure needed on the constraint surface of the full phase space to obtain a polarization on the reduced phase space is displayed in a form that is particularly convenient for applications. For unconstrained systems, any polarization on the phase space leads to a mathematically consistent quantum description, although not all of these descriptions may be viable from a physical standpoint. It is pointed out that the situation is worse in the presence of constraints: a general polarization on the full phase space need not lead to even a mathematically consistent quantum theory. Examples are given to illustrate the general constructions as well as the subtle difficulties.