Electric current from Schwinger's time-ordered propagator

The quasistatic electric current density of fermions in the presence of an external electric field is determined through the utilization of a time-ordered Schwinger propagator. The study encompasses the necessary conditions for establishing a well-defined time-ordered propagator within the Schwinger formalism, specifically concentrating on constant and uniform electromagnetic fields. Within this theoretical framework, a chemical potential is introduced, resulting in non-zero values for the electric current and charge number density. Consequently, the dependence of electric conductivity on the strength of the electric field and the charge number density is investigated.