On the non-existence of the micro-causal sector of Lorentz non-invariant quantum field theory on a continuum

We study a model for a Lorentz non-invariant quantum field theory. In this model Lorentz violation is due to a non-linear dispersion relation admitting signals in form of wave packets propagating with super-luminal (faster than the low-energy speed of light) group velocities. We characterize the causal sector of the theory by a cutoff condition protecting the theory against such signals and study the question as to whether the causal sector exhibits micro-causality (locality in all frames), in which case we call the causal sector the micro-causal sector. It is argued that in the causal sector the canonical commutation relation are changed in such a way that the imposition of locality postulate becomes incompatible with the continuity of space. The conclusion therefore is that there is no micro-causal sector of the theory on the continuum space underlying the model and in this respect it suggests the sensitivity of an effective Lorentz non-invariant quantum field theory to a short-distance cutoff.