  We provide the formal extension of Hedin's GW equations for single-particle
Green's functions with electron-electron interaction onto the Keldysh time-loop
contour. We show an application of our formalism to the plasmon model of a core
electron within the plasmon-pole approximation. We study in detail the
diagrammatic perturbation expansion of the core-electron/plasmon coupling on
the spectral functions of the so-called S-model which provides an exact
solution, concentrating especially on the effects of self-consistency and
vertex corrections on the GW self-energy. For the S-model, self-consistency is
essential for GW-like calculations to obtain the full spectral information. The
second- order exchange diagram (i.e. a vertex correction) is crucial to obtain
a better spectral description of the plasmon peak and side-band peaks in
comparison to GW-like calculations. However, the vertex corrections are well
reproduced within a non-self-consistent calculation. We also consider
conventional equilibrium GW calculations for the pure jellium model. We find
that with no second-order vertex correction, we cannot obtain the full set of
plasmon side-band peaks. Finally, we address the issues of formal connection
for the Dyson equations of the time-ordered Green's function and the Keldysh
Green's functions at equilibrium in the cases of zero and finite temperature.
