Conditional probabilities with Dirac observables and the problem of time in quantum gravity

TL;DR

This paper integrates the evolving constants approach with the Page--Wootters formalism in quantum gravity, using Dirac observables to address the problem of time and demonstrating a decoherence mechanism.

Contribution

It combines two approaches to relational time in quantum gravity, overcoming previous objections and explicitly working out the parameterized particle example.

Findings

Explicit construction of observables avoiding non-physical quantities

Calculation of the propagator for the parameterized particle

Prediction of a fundamental decoherence mechanism

Abstract

We combine the "evolving constants" approach to the construction of observables in canonical quantum gravity with the Page--Wootters formulation of quantum mechanics with a relational time for generally covariant systems. This overcomes the objections levied by Kucha\v{r} against the latter formalism. The construction is formulated entirely in terms of Dirac observables, avoiding in all cases the physical observation of quantities that do not belong in the physical Hilbert space. We work out explicitly the example of the parameterized particle, including the calculation of the propagator. The resulting theory also predicts a fundamental mechanism of decoherence.