  We study the abelian Higgs model out-of-equilibrium in two different
approaches, a gauge invariant formulation, proposed by Boyanovsky et al.
\cite{Boyanovsky:1996dc} and in the Coulomb gauge. We show that both approaches
become equivalent in a consistent one loop approximation. Furthermore, we carry
out a proper renormalization for the model in order to prepare the equations
for a numerical implementation. The additional degrees of freedom, which arise
in gauge theories, influence the behavior of the system dramatically. A
comparison with results in the 't Hooft-Feynman background gauge found by us
recently, shows very good agreement.
