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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.
Hartle et al. (Sat,) studied this question.