Comparing Formulations of Generalized Quantum Mechanics for Reparametrization-Invariant Systems

A class of decoherence schemes is described for implementing the principles of generalized quantum theory in reparametrization-invariant ‘hyperbolic’ models such as minisuperspace quantum cosmology. The connection with sumover-histories constructions is exhibited and the physical equivalence or inequivalence of different such schemes is analyzed. The discussion focuses on comparing constructions based on the Klein-Gordon product with those based on the induced (a.k.a. Rieffel, Refined Algebraic, Group Averaging, or Spectral Analysis) inner product. It is shown that the Klein-Gordon and induced products can be simply related for the models of interest. This fact is then used to establish isomorphisms between certain decoherence schemes based