Quantum cellular automata and quantum walks provide a framework for the
foundations of quantum field theory, since the equations of motion of free
relativistic quantum fields can be derived as the small wave-vector limit of
quantum automata and walks starting from very general principles. The intrinsic
discreteness of this framework is reconciled with the continuous Lorentz
symmetry by reformulating the notion of inertial reference frame in terms of
the constants of motion of the quantum walk dynamics. In particular, among the
symmetries of the quantum walk which recovers the Weyl equation--the so called
Weyl walk--one finds a non linear realisation of the Poincar\'e group, which
recovers the usual linear representation in the small wave-vector limit. In
this paper we characterise the full symmetry group of the Weyl walk which is
shown to be a non linear realization of a group which is the semidirect product
of the Poincar\'e group and the group of dilations.Comment: 9 pages, 2 figure
