Time of arrival on a ring and relativistic quantum clocks

TL;DR

This paper develops a quantum field theory framework for the time-of-arrival problem of particles on a ring, revealing how topology, rotation, and entanglement influence detection and quantum clock behavior.

Contribution

It introduces a QFT-based formalism for relativistic particles on a ring, including POVMs for arrival times, and explores effects of rotation and entanglement on detection probabilities.

Findings

Detection probabilities are periodic for particles on a ring.

Rotation introduces additional noise in detection, related to the Unruh effect.

Multi-time measurements reveal non-classical temporal correlations.

Abstract

We study the time-of-arrival problem for relativistic particles constrained to move on a ring, formulating the problem entirely within Quantum Field Theory (QFT). In contrast to its counterpart for motion in a line, the circle topology implies that particles may encounter the detector multiple times before detection, making a field-theoretic treatment of the measurement interaction essential. We employ the Quantum Temporal Probabilities (QTP) method to derive a class of Positive-Operator-Valued Measures (POVMs) for time-of-arrival observables directly from QFT. We analyze the resulting detection probabilities in both semiclassical and fully quantum regimes, identifying the relevant timescales and their dependence on the field-theoretic parameters. For ensembles of particles, the detection signal is a periodic function, providing a realization of a quantum clock whose operation reflects the local spacetime structure. We also extend the formalism to rotating rings and show that rotation induces additional noise in detection probabilities, interpretable as a manifestation of the rotational Unruh effect. Finally, we investigate multi-time measurements and demonstrate the emergence of non-classical temporal correlations due to entanglement.