  We study the consistency of having Lorentz invariance as a low energy
approximation within the quantum field theory framework. A model with a scalar
and a fermion field is used to show how a Lorentz invariance violating high
momentum scale, a physical cutoff rendering the quantum field theory finite,
can be made compatible with a suppression of Lorentz invariance violations at
low momenta. The fine tuning required to get this suppression and to have a
light scalar particle in the spectrum is determined at one loop.
