Comparing Formulations of Generalized Quantum Mechanics for Reparametrization-Invariant Systems

TL;DR

This work analyzes decoherence functionals for reparametrization-invariant systems by contrasting Klein-Gordon and induced inner products within product space decoherence schemes. It shows that, for AFH models with asymptotically free regions, these two inner products are related by a linear operator, enabling isomorphisms between KG-based and induced-based decoherence schemes. The authors connect sum-over-histories constructions to product space formalisms and demonstrate how appropriate state-space restrictions yield equivalent physical predictions across approaches. The results clarify the canonical versus sum-over-histories relationship in quantum cosmology and provide practical guidance for choosing inner products and state spaces in constrained quantum systems.

Abstract

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.