Observers and Timekeepers: From the Page-Wootters Mechanism to the Gravitational Path Integral

TL;DR

This paper explores how the problem of time and Hilbert space dimension in quantum gravity can be addressed by relative physics, using a gravitational path integral model and revisiting the Page-Wootters mechanism with new insights on observers and timekeepers.

Contribution

It introduces a hierarchy between observers and timekeepers, and proposes a method to incorporate them into the gravitational path integral, leading to an observer-dependent form of holography.

Findings

Distinct non-perturbative effects cause the problem of time and Hilbert space dimension.

Revisiting the Page-Wootters mechanism clarifies its application to quantum gravity.

Incorporating timekeepers yields a new observer-dependent holographic framework.

Abstract

Quantum gravity in a closed universe faces two a priori distinct yet seemingly related issues: the problem of time and the fact that its Hilbert space dimension is one. Both have been argued to be resolvable by formulating physics relative to an observer. Using a simple gravitational path integral model, we explain that the two issues arise from two distinct non-perturbative effects: the former from summing over metrics and the latter from summing over topologies. We then revisit the Page-Wootters mechanism, one of the earliest frameworks for formulating quantum mechanics relative to an observer, see how it applies to both issues, and introduce some new ingredients. In particular, we emphasize a hierarchy between an observer and a timekeeper. An observer is a subsystem of the universe whose specification results in a nontrivial Hilbert space, while a timekeeper is an observer with a specified history that can be used as a reference for the time of the environment and experiences a nontrivial time evolution. Finally, we propose a method for incorporating observers and timekeepers into the gravitational path integral and show that implementing a timekeeper in this way furnishes an observer-dependent generalization of holography.