  This paper concerns the generation and evolution of the cosmological
(large-scale $\sim Mpc$) magnetic fields in an inflationary universe. The
universe during inflation is represented by de Sitter space-time. We started
with the Maxwell equations in spatially flat Friedmann-Robertson-Walker (FRW)
Cosmologies. Then we calculated the wave equations of the magnetic field and
electric field for the evolution. We consider the input current that was
produced from a massless charged scalar complex field. This field minimally
coupled to both gravity and the electromagnetic fields. The Lagrangian for
massless scalar electrodynamics is then
$L=\sqrt{-g}(D_\mu\phi(D^\mu\phi)^*-{1/4}F_{\mu\nu}F^{\mu\nu})$ . The complex
scalar field couples to electromagnetism through the usual gauge covariant
derivative $D_\mu=\partial_\mu-ieA_\mu$ . After the quantum field theoretical
deduction for the current, we put it back into the wave equation of the
magnetic field.
  After solving this wave equation, our result is $a^2B\sim
\frac{eH}{\sqrt{2}k^2}\mid\sin\sqrt{2}k\eta\mid$ . At the time $\eta_{RH}$ we
have $B_{RH}=\frac{e}{k_{phys}}$. This may imply that the breaking of the
conformal invariance due to the minimal coupling of a massless charged scalar
complex field to both gravitational and electromagnetic fields is not
sufficient for the production of seed galactic magnetic fields during
inflation. But since we are interested in the large-scale cosmological magnetic
field, this could be still a candidate, because of the $1/k$ factor.
