  In this paper, we extend the Unruh-DeWitt (UDW) model to include a
relativistic quantized center of mass (c.m.) for the detector, which
traditionally has a classical c.m. and follows a classical trajectory. We
develop a relativistic model of an inertial detector following two different
approaches, starting from either a first- or second-quantized treatment, which
enables us to compare the fundamental differences between the two schemes. In
particular, we find that the notion of localization is different between the
two models, and leads to distinct predictions, which we study by comparing the
spontaneous emission rates for the UDW detector interacting with a massless
scalar field. Furthermore, we consider the UDW system in both a vacuum and
medium, and compare our results to existing models describing a classical or
quantized c.m. at low energies. We find that the predictions of each model,
including the two relativistic cases, can in principle be empirically
distinguished, and our results can be further extended to find optimal detector
states and processes to perform such experiments. This would clarify both the
role of a quantized c.m. for interactions with an external field, and the
differing localizations between the first- and second-quantized treatments.
