Radiative Transport Limit of Dirac Equations with Random Electromagnetic Field

Abstract

This paper concerns the kinetic limit of the Dirac equation with random electromagnetic field. We give a detailed mathematical analysis of the radiative transport limit for the phase space energy density of solutions to the Dirac equation. Our derivation is based on a martingale method and a perturbed test function expansion. This requires the electromagnetic field to be a space-time random field. The main mathematical tool in the derivation of the kinetic limit is the matrix-valued Wigner transform of the vector-valued Dirac solution. The major novelty compared to the scalar (Schr\"odinger) case is the proof of convergence of cross-modes to 0 weakly in space and almost surely in probability. The propagating modes are shown to converge in an appropriate strong sense to their deterministic limit.