  We investigate here the BCS BEC crossover in relativistic systems using a
variational construct for the ground state and the minimization of the
thermodynamic potential. This is first studied in a four fermion point
interaction model and with a BCS type ansatz for the ground state with fermion
pairs. It is shown that the antiparticle degrees of freedom play an important
role in the BCS BEC crossover physics, even when the ratio of fermi momentum to
the mass of the fermion is small. We also consider the phase structure for the
case of fermion pairing with imbalanced populations. Within the ansatz,
thermodynamically stable gapless modes for both fermions and anti fermions are
seen for strong coupling in the BEC regime. We further investigate the effect
of fluctuations of the condensate field by treating it as a dynamical field and
generalize the BCS ansatz to include quanta of the condensate field also in a
boson fermion model with quartic self interaction of the condensate field. It
is seen that the critical temperature decreases with inclusion of fluctuations.
