Dirac equation with an ultraviolet cutoff and a quantum walk
Fumihito Sato, Makoto Katori · Physical Review A · 2010
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)|