  We show how the Dirac equation in three space-dimensions emerges from the
large-scale dynamics of the minimal nontrivial quantum cellular automaton
satisfying unitariety, locality, homogeneity, and discrete isotropy, without
using the relativity principle. The Dirac equation is recovered for small
wave-vector and inertial mass, whereas Lorentz covariance is distorted in the
ultra-relativistic limit. The automaton can thus be regarded as a theory
unifying scales from Planck to Fermi. A simple asymptotic approach leads to a
dispersive Schroedinger equation describing the evolution of narrow-band states
at all scales.
