The time of arrival problem in the Page-Wootters formalism

TL;DR

This paper constructs a relational time-of-arrival distribution within the Page-Wootters framework, linking it to existing approaches and highlighting interpretational challenges.

Contribution

It introduces a novel method to derive the time-of-arrival distribution using the Page-Wootters formalism, connecting relational quantum time with a concrete physical problem.

Findings

The derived distribution matches common approaches to the time-of-arrival problem.

The approach offers a relational interpretation of the time-of-arrival.

It reveals complications in interpreting the formalism as a theory of conditional probabilities.

Abstract

The time-of-arrival problem asks for the probability distribution for when a quantum particle reaches a specified location. It has been the subject of decades of debate, exemplifying the lack of a self-adjoint time observable in quantum theory. In the Page-Wootters framework, time is a relational quantity, emerging from correlations between a system and a clock induced by a global Hamiltonian constraint. We construct a time-of-arrival distribution by inverting the Page-Wootters approach, asking what time a clock reads given that the particle arrives at some fixed position. The result coincides with a common approach to the time-of-arrival problem, suggesting a potential relational interpretation of the latter. In addition to providing a relational description of the time-of-arrival problem, this gives an application of the Page-Wootters formalism to a concrete physical problem, and reveals some complications with its canonical interpretation as a theory of conditional probabilities.