Relaxation to quantum equilibrium for Dirac fermions in the de Broglie-Bohm pilot-wave theory

Numerical simulations indicate that the Born rule does not need to be postulated in the de Broglie-Bohm pilot-wave theory, but arises dynamically (relaxation to quantum equilibrium). These simulations were done for a particle in a two-dimensional box whose wave-function obeys the non-relativistic Schroedinger equation and is therefore scalar. The chaotic nature of the de Broglie-Bohm trajectories, thanks to the nodes of the wave-function which act as vortices, is crucial for a fast relaxation to quantum equilibrium. For spinors, we typically do not expect any node. However, in the case of the Dirac equation, the de Broglie-Bohm velocity field has vorticity even in the absence of nodes. This observation raises the question of the origin of relaxation to quantum equilibrium for fermions. In this article, we provide numerical evidence to show that Dirac particles also undergo relaxation, by simulating the evolution of various non-equilibrium distributions for two-dimensional systems (the 2D Dirac oscillator and the Dirac particle in a spherical 2D box).Comment: 11 pages, 9 figure