Time-Ordering in the Dyson-Unruh Problem: Accelerated Observers and Quantum Fields

TL;DR

This paper develops a formalism for time-ordered perturbation theory in quantum field theory with accelerated detectors in curved spacetime, clarifying ordering ambiguities and systematically constructing interaction contributions.

Contribution

It introduces a method to express detector proper times via a global time coordinate, enabling consistent time-ordering in perturbative expansions for accelerated observers.

Findings

Constructed all interaction terms up to third order in perturbation theory.

Resolved ordering ambiguities caused by acceleration and causal structure.

Provided a systematic classification of S-matrix contributions.

Abstract

We investigate the structure of time-ordered perturbative expansions in quantum field theory (QFT) in curved spacetime, focusing on the interaction between a scalar field and multiple Unruh-DeWitt detectors undergoing uniform acceleration. Assuming the existence of a timelike Killing vector field, which serves as a global time coordinate, we express each detector's proper time in terms of this global time and consistently implement time-ordering in the Dyson series. Using light-cone coordinates and detector worldlines, we construct all possible interaction terms for two detectors at first, second, and third orders in perturbation theory. Our formalism enables a systematic classification of contributions to the S-matrix, resolving ordering ambiguities that arise due to differing accelerations and causal structure.