Schwinger production of scalar particles during and after inflation from
  the first principles
  By using the first-principles approach, we derive a system of three quantum
kinetic equations governing the production and evolution of charged scalar
particles by an electric field in an expanding universe. Analyzing the
ultraviolet asymptotic behavior of the kinetic functions, we found the
divergent parts of the electric current and the energy-momentum tensor of the
produced particles and determined the corresponding counterterms. The
renormalized system of equations is used to study the generation of
electromagnetic fields during and after inflation in the kinetic coupling model
$\mathcal{L}_{\rm EM}=-(1/4)f^{2}(\phi)F_{\mu\nu}F^{\mu\nu}$ with the Ratra
coupling function $f=\exp(\beta\phi/M_{p})$. It is found that the electric
current of created particles is retarded with respect to the electric field.
This leads to an oscillatory behavior of both quantities in agreement with the
results obtained previously in phenomenological kinetic and hydrodynamical
approaches.
