Low energy Lorentz violation from modified dispersion at high energies

Many quantum theories of gravity propose Lorentz violating dispersion relations of the form $\omega = |k|\, f(|k|/M)$, with recovery of approximate Lorentz invariance at energy scales much below $M$. We show that a quantum field with this dispersion predicts drastic low energy Lorentz violation in atoms modelled as Unruh-DeWitt detectors, for any $f$ that dips below unity somewhere. As an example, we show that polymer quantization motivated by loop quantum gravity predicts such Lorentz violation below current ion collider rapidities.