  It has been observed that quantum walks on regular lattices can give rise to
wave equations for relativistic particles in the continuum limit. In this paper
we define the 3D walk as a product of three coined one-dimensional walks. The
factor corresponding to each one-dimensional walk involves two projection
operators that act on an internal coin space, each projector is associated with
either the "forward" or "backward" direction in that physical dimension. We
show that the simple requirement that there is no preferred axis or direction
along an axis---that is, that the walk be symmetric under parity
transformations and rotations that swap the axes of the cubic lattice---leads
to the requirement that the continuum limit of the walk is fully Lorentz
invariant. We show further that, in the case of a massive particle, this simple
symmetry requirement necessitates that inclusion of antimatter---the use of a
four-dimensional internal space---and that the "coin flip" operation is
generated by the parity transformation on the internal coin space, while the
differences of the projection operators associated to each dimension must all
anticommute. Finally, we discuss the leading correction to the continuum limit,
and the possibility of distinguishing through experiment between the discrete
random walk and the continuum-based Dirac equation as a description of fermion
dynamics.
