Then again, how often does the Unruh-DeWitt detector click if we switch it carefully?

TL;DR

This paper investigates how the transition probability of an Unruh-DeWitt detector depends on the switching function, showing that smooth switching restores Lorentz invariance and finiteness to the results, unlike sudden switching.

Contribution

It demonstrates that non-Lorentz invariance in the detector's transition rate is due to sudden switching, and that smooth switching yields Lorentz invariant, finite results with conventional regularisation.

Findings

Smooth switching restores Lorentz invariance.

Conventional regularisation yields finite, Lorentz invariant results with smooth switching.

Sharp switching limit and spectral falloff properties are analyzed.

Abstract

The transition probability in first-order perturbation theory for an Unruh-DeWitt detector coupled to a massless scalar field in Minkowski space is calculated. It has been shown recently that the conventional $i\epsilon$ regularisation prescription for the correlation function leads to non-Lorentz invariant results for the transition rate, and a different regularisation, involving spatial smearing of the field, has been advocated to replace it. We show that the non-Lorentz invariance arises solely from the assumption of sudden switch-on and switch-off of the detector, and that when the model includes a smooth switching function the results from the conventional regularisation are both finite and Lorentz invariant. The sharp switching limit of the model is also discussed, as well as the falloff properties of the spectrum for large frequencies.