Equivalence of dirac and intrinsic quantization for non-free group actions

Abstract

In this article we generalize some results on the equivalence of Dirac quantization and intrinsic quantization proven in [3]: We consider systems with first class constraints that may be considered as the vanishing of the momentum map to a lifted group action, but drop the assumption that the group action is free as well as the assumption that the group is compact. Using a generalized Weyl ordering prescription applicable to arbitrary cotangent bundles we derive necessary and sufficient conditions for the equivalence of the two approaches for different classes of functions analogous to those for the free case, although the proofs given in [3] must be considerably modified and refined due to the noncompactness of the orbits and the lack of sufficiently many invariant vector fields. The same strong obstruction as in the free case is found if one requires equivalence for all invariant functions, essentially only admitting trivial bundles.