Charge renormalization in time-dependent, relativistic quantum mechanics for electromagnetically self-interacting fermions

The time-dependent electromagnetically self-coupled Dirac equation is solved numerically by means of the MSD2 algorithm with special attention to stability. The expectation values of several dynamic operators are evaluated as functions of time and the asymptotic, i.e., physical values are obtained. It is observed that the positive and negative energy projections are separated from each other in space and time due to self-interactions from the Klein Effect. This is also observed in the gauge fields. A statistical method, employing a canonical ensemble whose temperature is the inverse of the spatial-grid size, is used to remove the momentum-dependence. Finite expectation values are obtained in the continuum limit. The charge renormalization is attributed to the contribution of the negative-energy components, enhanced by self-interactions. The physical charge is about 30% smaller than the bare value.