Currents of created pairs in strong electric fields

In this paper, we calculate tree-level currents of created particles in strong background electric fields in 4D QED for various initial states. Namely, we do that in pulse background for initial vacuum and thermal states at past infinity. In both cases, we find that the current grows linearly with the length of the pulse with coefficients of proportionality containing the characteristic Schwinger's factor. For the constant electric field background, we calculate the current for several different initial states. We observe that in such a case the current is either zero or linearly divergent. We explain the reason for such a behavior and compare the situation in ordinary and scalar QED. Finally, we calculate the current in two-dimensional situation in the presence of such settings when the so-called Klein paradox can be observed.