Self-interactions as predicted by the Dirac-Maxwell equations

We solve the Maxwell-Dirac equations to study the dynamics of a spatially localized charged particle in one spatial dimension. While the coupling of the Maxwell equations to the Dirac equation predicts correctly the attractive or repulsive interaction between different particles, it also reveals an unphysical interaction of a single electron or positron with itself leading to an enhanced spatial spreading of the wave packet. Using a comparison with a relativistic ensemble of mutually interacting classical quasiparticles, we suggest that this quantum mechanical self-repulsion can be understood in terms of relativistic classical mechanics. We show that due to the simple form of the Coulomb law in one spatial dimension it is possible to find analytical expressions of the time-dependent spatial width for the interacting classical ensemble. A better understanding of the dynamical impact of this unavoidable self-repulsion effect is relevant for recent studies of the field-induced pair creation process from the vacuum.

Figure 1

The spatial probability density at various instants of time ($n$ − 1) 0.0013 a.u. for the quantum (solid) and classical mechanical (open circles) system. (Parameters are $L$ = 6.4 a.u., $N_z$ = 2048 spatial grid points, initial width Δ$z_0$ = 0.05 a.u. The number of particles in the classical ensemble was 10 000, $Q$ = −100, $M$ = 1.)

Figure 2

The spatial width Δ$z$($t$) of the charged particle as a function of time. The lowest curve/circles are the quantum/classical data for $Q$ = 0. For $Q$ = −100, the quantum calculation is compared with the analytical formula. (Parameters are $L$ = 6.4 a.u., $N_z$ = 2048 spatial grid points, initial width Δ$z_0$ = 0.05 a.u. 10 000 particles in the classical ensemble.)