  The weak convergence theorems of the one- and two-dimensional simple quantum
walks, SQW$^{(d)}, d=1,2$, show a striking contrast to the classical
counterparts, the simple random walks, SRW$^{(d)}$. The limit distributions
have novel structures such that they are inverted-bell shaped and the supports
of them are bounded. In the present paper we claim that these properties of the
SQW$^{(d)}$ can be explained by the theory of relativistic quantum mechanics.
We show that the Dirac equation with a proper ultraviolet cutoff can provide a
quantum walk model in three dimensions, where the walker has a four-component
qubit. We clarify that the pseudovelocity $\V(t)$ of the quantum walker, which
solves the Dirac equation, is identified with the relativistic velocity. Since
the quantum walker should be a tardyon, not a tachyon, $|\V(t)| < c$, where $c$
is the speed of light, and this restriction (the causality) is the origin of
the finiteness of supports of the limit distributions universally found in
quantum walk models. By reducing the number of components of momentum in the
Dirac equation, we obtain the limit distributions of pseudovelocities for the
lower dimensional quantum walks. We show that the obtained limit distributions
for the one- and two-dimensional systems have the common features with those of
SQW$^{(1)}$ and SQW$^{(2)}$.
