  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 (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 on these
products.
