  With the improvement of the experimental techniques, many new phenomena where
the photon degrees of freedom are involved have been discovered. Typical
examples are the exciton-polariton quasi-particles, excitons strongly coupled
to photons. A correct description of these systems requires a full account of
the photon's dynamics, including the vector gauge degrees of freedom. In order
to include this contribution in \emph{ab initio} many-body perturbation
theories, here we present a generalization of the Hedin equations, originally
derived for a many electron system with Coulomb interaction. These equations
are now derived in order to take into account the strong coupling physics of
the electrons and the \textit{transverse} photon's degrees of freedom.
