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Symmetries of the Dirac quantum walk and emergence of the de Sitter group

A quantum walk describes the discrete unitary evolution of a quantum particle on a discrete graph. Some quantum walks, referred to as the Weyl and Dirac quantum walks, provide a description of the free evolution of relativistic quantum fields in a regime where the wave-vectors involved in the particle state are small. The clash between the intrinsic discreteness of quantum walks and the symmetries of special relativity can be resolved by rethinking the notion of a change of inertial reference frame. We give here a definition of the latter that avoids a pre-defined space-time geometry, in terms of a change of values of the constants of motion that leaves the walk operator unchanged. Starting from the family of 1+1 dimensional Dirac quantum walks with all possible values of the mass parameter, we introduce a unique walk encompassing the latter as an extra degree of freedom, and we derive its group of changes of inertial frames. This symmetry group contains a non linear realization of $SO^+(2,1) \ltimes \mathbb{R}^3$; since one of the two space-like dimensions does not correspond to an actual spatial degree of freedom but rather the mass, we interpret it as a 2+1 dimensional de-Sitter group. This group group contains also a non-linear realisation of the proper orthochronous Poincaré group $SO^+(1,1) \ltimes \mathbb{R}^2$ in 1+1 dimension, as the ones considered within the framework of doubly special relativity, which recovers the usual relativistic symmetry of the Dirac Equation in the limit of small wave-vectors and masses. Surprisingly, if one considers the Dirac walk with a fixed value of the mass parameter, the group of allowed changes of reference frame does not have a consistent interpretation in the relativistic limit of small wave-vectors.