How often does the Unruh-DeWitt detector click? Regularisation by a spatial profile

TL;DR

This paper investigates the transition rate of an accelerated Unruh-DeWitt detector with spatial regularisation, deriving explicit finite formulas and analyzing various trajectories to understand its response spectrum.

Contribution

It provides explicit finite integral formulas for the detector's transition rate with spatial regularisation, including the zero size limit, and extends analysis to arbitrary profiles and trajectories.

Findings

Explicit finite formula for transition rate with Lorentzian profile

Recovery of Planckian spectrum for uniform acceleration

Analysis of stationary and nonstationary trajectories

Abstract

We analyse within first-order perturbation theory the instantaneous transition rate of an accelerated Unruh-DeWitt particle detector whose coupling to a massless scalar field on four-dimensional Minkowski space is regularised by a spatial profile. For the Lorentzian profile introduced by Schlicht, the zero size limit is computed explicitly and expressed as a manifestly finite integral formula that no longer involves regulators or limits. The same transition rate is obtained for an arbitrary profile of compact support under a modified definition of spatial smearing. Consequences for the asymptotic behaviour of the transition rate are discussed. A number of stationary and nonstationary trajectories are analysed, recovering in particular the Planckian spectrum for uniform acceleration.