Lorentz invariance without trans-Planckian physics?

We explore the possibility that, in a quantum field theory with Planck scale cutoff Λ≃mp, observable quantities for low-energy processes respect the Lorentz symmetry. In particular, we compute the one-loop radiative correction Π to the self-energy of a scalar field with λϕ4 interaction, using a modified (non-invariant) propagator which vanishes in the trans-Planckian regime, as expected in the “classicalisation” scenario. We then show that, by imposing the result does not depend on Λ (in the limit Λ→mp), an explicit (albeit not unique) expression for Π can be derived, which is similar to the one simply obtained with the standard Feynman propagator and a cutoff Λ=mp.