Relational observables, reference frames, and conditional probabilities

TL;DR

This paper develops a framework for constructing and interpreting relational observables in quantum mechanics using conditional probabilities, with applications to quantum gravity and cosmology.

Contribution

It introduces a generalized formalism for relational observables in time-reparametrization invariant quantum systems, linking them to conditional probabilities and quantum reference frames.

Findings

Relational observables can be understood via conditional probabilities.

Constructed unitarily evolving quantum relational observables in a cosmological model.

Insights into diffeomorphism-invariant operators in quantum gravity.

Abstract

We discuss the construction of relational observables in time-reparametrization invariant quantum mechanics and we argue that their physical interpretation can be understood in terms of conditional probabilities, which are defined from the solutions of the quantum constraint equation in a generalization of the Page-Wootters formalism. In this regard, we show how conditional expectation values of worldline tensor fields are related to quantum averages of suitably defined relational observables. We also comment on how the dynamics of these observables can be related to a notion of quantum reference frames. After presenting the general formalism, we analyze a recollapsing cosmological model, for which we construct unitarily evolving quantum relational observables. We conclude with some remarks about the relevance of these results for the construction and interpretation of diffeomorphism-invariant operators in quantum gravity.