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arXiv · 2604.00092

The time of arrival problem in the Page-Wootters formalism

Niyusha Hosseini·Maximilian P. E. Lock
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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.

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Niyusha Hosseini, Maximilian P. E. Lock. 2026-03-31. The time of arrival problem in the Page-Wootters formalism. https://arxiv.org/abs/2604.00092

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