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Doubly-Special Relativity from Quantum Cellular Automata

It is shown how a Doubly-Special Relativity model can emerge from a quantum cellular automaton description of the evolution of countably many interacting quantum systems. We consider a one-dimensional automaton that spawns the Dirac evolution in the relativistic limit of small wave-vectors and masses (in Planck units). The assumption of invariance of dispersion relations for boosted observers leads to a non-linear representation of the Lorentz group on the $(ω,k)$ space, with an additional invariant given by the wave-vector $k=π/2$. The space-time reconstructed from the $(ω,k)$ space is intrinsically quantum, and exhibits the phenomenon of relative locality.