  We investigate the Schwinger pair production phenomena in spatially
homogeneous strong electric fields. We first consider scalar QED in
four-dimensions and discuss the potential ambiguity in the adiabatic order
assignment for the electromagnetic potential required to fix the
renormalization subtractions. We argue that this ambiguity can be solved by
invoking the conformal anomaly when both electric and gravitational backgrounds
are present. We also extend the adiabatic regularization method for spinor QED
in two-dimensions and find consistency with the chiral anomaly. We focus on the
issue of the renormalization of the electric current $\langle j^\mu \rangle$
generated by the created pairs. We illustrate how to implement the
renormalization of the electric current for the Sauter pulse.
