  In this paper we consider quantum walks whose evolution converges to the
Dirac equation one in the limit of small wave-vectors. We show exact Fast
Fourier implementation of the Dirac quantum walks in one, two and three space
dimensions. The behaviour of particle states, defined as states smoothly peaked
in some wave-vector eigenstate of the walk, is described by an approximated
dispersive differential equation that for small wave-vectors gives the usual
Dirac particle and antiparticle kinematics. The accuracy of the approximation
is provided in terms of a lower bound on the fidelity between the exactly
evolved state and the approximated one. The jittering of the position operator
expectation value for states having both a particle and an antiparticle
component is analytically derived and observed in the numerical
implementations.
