Quantum diffusion of a relativistic particle in a time-dependent random potential

We present a rigorous study of quantum diffusion of a relativistic particle subjected to a time-dependent random potential with $\delta$ correlation in time. We find that in the asymptotic time limit the particle wave packet spreads ballistically in contrast with the nonrelativistic case, which in the same situation exhibits superballistic diffusion. The relativistic suppression of wave packet diffusion is discussed in connection with statistical conservation laws that follow from relativistic dynamics.

Figure 1

Numerical solution of Eq. (28) for $r=1, K=0, s=0.001, \alpha =\sqrt{\frac{4}{5}}, \beta =\sqrt{\frac{1}{5}}$, and other parameters equal to 1. The real part is odd and therefore vanishes at $Y=0$, and the solution is independent of $s$ at small $s$.

Figure 2

$\overline{\widetilde{G^{\prime }}}(Y,s)$ for $r=1, s=0.01$, and other parameters equal to 1. The inset is the plot of $-\overline{\widetilde{G^{\prime }}}(0,s)$ in log-log scale which shows the $\frac{1}{s}$ divergence in $s\to 0$.