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We show how the Dirac equation in three space-dimensions emerges from thelarge-scale dynamics of the minimal nontrivial quantum cellular automatonsatisfying unitariety, locality, homogeneity, and discrete isotropy, withoutusing the relativity principle. The Dirac equation is recovered for smallwave-vector and inertial mass, whereas Lorentz covariance is distorted in theultra-relativistic limit. The automaton can thus be regarded as a theoryunifying scales from Planck to Fermi. A simple asymptotic approach leads to adispersive Schroedinger equation describing the evolution of narrow-band statesat all scales.

Derivation of the Dirac Equation from Principles of Information Processing