Comparing Formulations of Generalized Quantum Mechanics for Reparametrization-Invariant Systems

TL;DR

This paper compares different decoherence schemes in generalized quantum mechanics for reparametrization-invariant models, analyzing their physical equivalence and establishing connections between various inner product constructions.

Contribution

It provides a detailed comparison of Klein-Gordon and induced inner product-based decoherence schemes, revealing their relations and equivalences in hyperbolic quantum models.

Findings

Klein-Gordon and induced products can be related for specific models.

Certain decoherence schemes based on these products are isomorphic.

The connection with sum-over-histories formulations is clarified.

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.