Dirac versus reduced quantization: a geometrical approach

Abstract

Two particular aspects of the quantization of systems with first class linear momentum constraints are discussed. First, the inequivalence between Dirac and reduced phase space quantization is explained in terms of an ambiguity in the ordering of gauge invariant operators in the reduced Hamiltonian. It is shown that this operator ordering ambiguity arises due to a potential contribution to the reduced Hilbert space measure from the volume of the gauge orbit over each point in the physical configuration space. Secondly, the relationship between the gauge fixed covariant Faddeev-Popov functional integral and the functional integral for the completely reduced theory is elucidated. In particular, it is shown explicitly how the Faddeev-Popov functional integral reduces to the functional integral obtained via reduced phase space quantization, despite the fact that the measure of the former contains a contribution from the volume of the covariant gauge orbits, while the latter retains no information about the gauge structure.