Is reduced phase space quantisation equivalent to Dirac quantisation?

Abstract

Two methods of quantising systems with first class constraints are discussed in the context of a simple model: a free particle on the 2-sphere subject to an additional constraint. It is found that reduced phase space quantisation can lead to different energy spectra to those given by Dirac quantisation; however, it can also lead to the same spectrum by a careful choice of the domain of the Hamiltonian. Even though the spectra can be made to agree, the eigenfunctions of the respective Hamiltonians are not the same. Therefore the two methods potentially lead to different predictions for physical measurements of quantities other than the energy. Thus the two methods are not guaranteed to be completely equivalent; moreover, equivalence or inequivalence cannot be established from the spectra alone.