  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 Poincar\'e 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.
