A dynamical mechanism for the selection of physical states in `geometric quantization schemes' (Q1387182)

The author discusses Kostant's pre-quantization scheme. The problem arising in this approach is that the pre-quantum Hilbert space is too large for physics and it is therefore necessary to introduce a mechanism capable of selecting the subspace of physical states. The general viewpoint is that of picking out a real or complex polarization on the classical phase space and requiring that the physical states are the ones preserving the polarization. Such a prescription appears more and more problematic as soon as the dynamics of systems with less symmetry or no symmetry at all is considered. It is the aim of this paper to present a slightly different approach to the problem, focusing more on the dynamical aspects rather than on the kinematical ones. Working in a coordinate free manner, the author faces the problem of directly defining the quantum dynamics without going through the quantization of the whole algebra of classical observables. This yields a dynamical mechanism that produces the selection of the right set of physical quantum states.