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 walks, provide a description of the free evolution of relativistic quantum fields in the small wave-vector regime. The clash between the intrinsic discreteness of quantum walks and the continuous symmetries of special relativity is resolved by giving a definition of change of inertial frame in terms of a change of values of the constants of motion, which leaves the walk operator unchanged. Starting from the family of 1 + 1 dimensional Dirac 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)⋉R3; 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 also contains a non-linear realization of the proper orthochronous Poincaré group SO+(1,1)⋉R2 in 1 + 1 dimension, as the ones considered within the framework of doubly special relativity, which recovers the usual relativistic symmetry in the limit of small wave-vectors and masses. Surprisingly, for 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 limit of small wave-vectors.