  In this work, we investigate the dynamical breakdown of Lorentz symmetry in 4
dimensions by the condensation of a fermionic field described by a Dirac
Lagrangean with a four-fermion interaction. Using the Keldysh formalism we show
that the Lorentz symmetry breaking modifies the Dyson equations of the
fermionic propagator. We analyze the non-perturbative solutions for the Dyson
equations using the combination of the rainbow and quenched approximations and
show that the Lorentz symmetry breakdown can occur in the strong coupling
regime and new features arise from this approach. Finally, we analyze the
contributions of temperature and chemical potential.
