  If an Unruh-DeWitt detector moves with a uniform acceleration in Fock-space
vacuum, then the transition rate of the detector is proportional to the thermal
spectrum. It is well known that the transition rate of the detector crucially
depends on the two-point function along the detectors trajectory and in order
to compute it the standard "$i \epsilon$" regularization is used for Fock
space. Numerically, we show here that the regulator $\epsilon$ is generic in
polymer quantization, the quantization method used in \emph{loop quantum
gravity} with a finite value $\epsilon \approx 2.16$, which leads to
non-thermal spectrum for the uniformly accelerated detector. We also discuss
the response of a spatially smeared detector.
