  We show that the predictions of spatially smeared particle detectors coupled
to quantum fields are not generally covariant outside the pointlike limit. This
lack of covariance manifests itself as an ambiguity in the time-ordering
operation. We analyze how the breakdown of covariance affects typical detector
models in quantum field theory such as the Unruh-DeWitt model. Specifically, we
show how the violations of covariance depend on the state of the
detectors-field system, the shape and state of motion of the detectors, and the
spacetime geometry. Furthermore, we provide the tools to explicitly evaluate
the magnitude of the violation, and identify the regimes where the predictions
of smeared detectors are either exactly or approximately covariant in
perturbative analyses.
