The Trinity of Relational Quantum Dynamics

TL;DR

This paper establishes a fundamental equivalence between three approaches to relational quantum dynamics, providing a unified framework that clarifies conceptual issues and resolves criticisms without relying on approximations or idealized assumptions.

Contribution

It introduces a novel equivalence called the trinity among relational quantum dynamics approaches, develops a covariant quantization procedure, and extends the framework to changing temporal reference frames.

Findings

Proves the equivalence of three relational quantum dynamics approaches.

Develops a gauge-invariant quantization method for relational observables.

Resolves previous criticisms regarding the PW formalism and clarifies the role of entanglement.

Abstract

The problem of time in quantum gravity calls for a relational solution. Using quantum reduction maps, we establish a previously unknown equivalence between three approaches to relational quantum dynamics: 1) relational observables in the clock-neutral picture of Dirac quantization, 2) Page and Wootters' (PW) Schr\"odinger picture formalism, and 3) the relational Heisenberg picture obtained via symmetry reduction. Constituting three faces of the same dynamics, we call this equivalence the trinity. We develop a quantization procedure for relational Dirac observables using covariant POVMs which encompass non-ideal clocks. The quantum reduction maps reveal this procedure as the quantum analog of gauge-invariantly extending gauge-fixed quantities. We establish algebraic properties of these relational observables. We extend a recent clock-neutral approach to changing temporal reference frames, transforming relational observables and states, and demonstrate a clock dependent temporal nonlocality effect. We show that Kucha\v{r}'s criticism, alleging that the conditional probabilities of the PW formalism violate the constraint, is incorrect. They are a quantum analog of a gauge-fixed description of a gauge-invariant quantity and equivalent to the manifestly gauge-invariant evaluation of relational observables in the physical inner product. The trinity furthermore resolves a previously reported normalization ambiguity and clarifies the role of entanglement in the PW formalism. The trinity finally permits us to resolve Kucha\v{r}'s criticism that the PW formalism yields wrong propagators by showing how conditional probabilities of relational observables give the correct transition probabilities. Unlike previous proposals, our resolution does not invoke approximations, ideal clocks or ancilla systems, is manifestly gauge-invariant, and easily extends to an arbitrary number of conditionings.