  The adiabatic regularization method was originally proposed by Parker and
Fulling to renormalize the energy-momentum tensor of scalar fields in expanding
universes. It can be extended to renormalize the electric current induced by
quantized scalar fields in a time-varying electric background. This can be done
in a way consistent with gravity if the vector potential is considered as a
variable of adiabatic order one. Assuming this, we further extend the method to
deal with Dirac fields in four spacetime dimensions. This requires a
self-consistent ansatz for the adiabatic expansion, in presence of a prescribed
time-dependent electric field, which is different from the conventional
expansion used for scalar fields. Our proposal is consistent, in the massless
limit, with the conformal anomaly. We also provide evidence that our proposed
adiabatic expansion for the fermionic modes parallels the Schwinger-DeWitt
adiabatic expansion of the two-point function. We give the renormalized
expression of the electric current and analyze, using numerical and analytical
tools, the pair production induced by a Sauter-type electric pulse. We also
analyze the scaling properties of the current for a large field strength.
