Dirac equation with an ultraviolet cutoff and a quantum walk

Abstract

The weak convergence theorems of the one- and two-dimensional simple quantum walks, ${\mathrm{SQW}}^{(d)},d=1,2$, show a striking contrast to the classical counterparts, the simple random walks, ${\mathrm{SRW}}^{(d)}$. In the ${\mathrm{SRW}}^{(d)}$, the distribution of position $\mathbf{X}(t)$ of the particle starting from the origin converges to the Gaussian distribution in the diffusion scaling limit, in which the time scale $T$ and spatial scale $L$ both go to infinity as the ratio $L/\sqrt{T}$ is kept finite. On the other hand, in the ${\mathrm{SQW}}^{(d)}$, the ratio $L/T$ is kept to define the pseudovelocity $\mathbf{V}(t)=\mathbf{X}(t)/t$, and then all joint moments of the components ${V}_{j}(t),1\ensuremath{\leqslant}j\ensuremath{\leqslant}d$, of $\mathbf{V}(t)$ converge in the $T=L\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. The limit distributions have novel structures such that they are inverted-bell shaped and their supports are bounded. In the present paper we claim that these properties of the ${\mathrm{SQW}}^{(d)}$ 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 $\mathbf{V}(t)$ 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, $|\mathbf{V}(t)|<c$, 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 common features with those of ${\mathrm{SQW}}^{(1)}$ and ${\mathrm{SQW}}^{(2)}$.