Considerations on the Unruh effect: causality and regularization

This paper is motivated by the observation that calculations of the Unruh effect based on idealized particle detectors are usually made in a way that involves integrations along the entire detector trajectory up to the infinitely remote future. We derive an expression which allows time-dependence of the detector response in the case of a non-stationary trajectory and conforms more explicitly to the principle of causality, namely that the response at a given instant of time depends only on the detectors past movements. On trying to reproduce the thermal Unruh spectrum we are led to an unphysical result, which we trace down to the use of the standard regularization t --> t - iepsilon of the correlation function. By consistently employing a rigid detector of finite extension, we are led to a different regularization which works fine with our causal response function.