Measurement events relative to temporal quantum reference frames

Abstract

The Page-Wootters formalism is a proposal for reconciling the background-dependent, quantum-mechanical notion of time with the background independence of general relativity. However, the physical meaning of this framework remains debated. In this work, we compare two consistent approaches to the Page-Wootters formalism to clarify the operational meaning of evolution and measurements with respect to a temporal quantum reference frame. The so-called "twirled observable" approach implements measurements as operators that are invariant with respect to the Hamiltonian constraint. The "purified measurement" approach instead models measurements dynamically by modifying the constraint itself. While both approaches agree in the limit of ideal clocks, a natural generalization of the purified measurement approach to the case of non-ideal, finite-resource clocks yields a radically different picture. We discuss the physical origin of this discrepancy and argue that these approaches describe operationally distinct situations. Moreover, we show that, for non-ideal clocks, the purified measurement approach yields a time non-local evolution equation, which can lead to non-unitary evolution. Moreover, it implies a fundamental limitation to the operational definition of the temporal order of events. Nevertheless, unitarity and definite temporal order can be restored if we assume that time is discrete.

References in corpus (13)

• Reference frames, superselection rules, and quantum information
• Conditional probabilities with Dirac observables and the problem of time in quantum gravity
• An effective approach to the problem of time
• Effective approach to the problem of time: general features and examples
• The WAY theorem and the quantum resource theory of asymmetry
• Spacetime Quantum Reference Frames and superpositions of proper times
• Multiple-event probability in general-relativistic quantum mechanics
• Noncausal Page-Wootters circuits
• Relative Quantum Time
• Non-inertial quantum clock frames lead to non-Hermitian dynamics
• Flow of time during energy measurements and the resulting time-energy uncertainty relations
• Multiple-event probability in general-relativistic quantum mechanics: a discrete model
• Robustness of the Page-Wootters construction across different pictures, states of the universe and system-clock interactions

Cited by in corpus (3)

• Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy
• Measuring time in a timeless universe
• Relational Dynamics with Periodic Clocks