Semiclassical Study of the Quantum Back-Reaction Problem in Scalar QED

The quantum back-reaction problem in scalar QED is studied, in the semiclassical approximation, for nonstationary external electric fields. Using a model proposed in [Cooper, F., Mottola, E. (1989). Quantum back reaction in scalar QED as an initial-value problem. Physical Review D40(2), 456–464], based in a Hartree-type equation, we calculate, until order ħ, the induced current density and the induced electromagnetic field. Once we have computed the induced electromagnetic field, we obtain, until order ħ, the effective Lagrangian and the average density of produced pairs taking into account the back-reaction.