  Using many-body techniques we obtain the time-dependent Gaussian
approximation for interacting fermion-scalar field models. This method is
applied to an uniform system of relativistic spin-1/2 fermion field coupled,
through a Yukawa term, to a scalar field in 3+1 dimensions, the so-called
quantum scalar plasma model. The renormalization for the resulting Gaussian
mean-field equations, both static and dynamical, are examined and initial
conditions discussed. We also investigate solutions for the gap equation and
show that the energy density has a single minimum.
