Klein-effect in electromagnetically self-interacting fermions and charge renormalization in time-dependent, relativistic quantum mechanics

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. This also affects 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 5% smaller than the bare value.