Relational observables, reference frames, and conditional probabilities

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.

arXiv commentv2: references updated, minor typos corrected, conceptual clarifications added, published in Physical Review D

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• Equivalence of approaches to relational quantum dynamics in relativistic settings
• Quantum Relativity of Subsystems
• Unitarity of quantum-gravitational corrections to primordial fluctuations in the Born-Oppenheimer approach
• Observations in Quantum Cosmology
• Internal quantum reference frames for finite Abelian groups
• A relational perspective on the Wigner-Araki-Yanase theorem
• Robustness of the Page-Wootters construction across different pictures, states of the universe and system-clock interactions
• Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy
• Beyond semiclassical time
• Perspective-neutral approach to quantum frame covariance for general symmetry groups
• Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system
• A magnetic clock for a harmonic oscillator
• Relational Dynamics with Periodic Clocks