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 sum-over-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 (also known as 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 on these products.