Matter in Discrete Space-Times

The unitary representations of the Poincare group of a discrete space-time are constructed, following the Wigner method in continuum relativity. They can be interpreted as elementary particles with one significant new feature: the momentum space being the 4-torus is identified as the Brillouin zone of space-time where all physical phenomena occur. Consequently 4-momentum is defined and conserved only modulo a reciprocal lattice vector of the order of the Planck mass, implying that there is no notion of an invariant mass except when it vanishes. In particular the propagation of massive particles have superluminal phases under very large (trans-Planckian) boosts. This behaviour leads to new features in early cosmology which are not in conflict with current knowledge.