  Based on the real-time formalism, especially, on Thermo Field Dynamics, we
derive the Schwinger-Dyson gap equation for the fermion propagator in QED and
Four-Fermion model at finite-temperature and -density. We discuss some
advantage of the real-time formalism in solving the self-consistent gap
equation, in comparison with the ordinary imaginary-time formalism. Once we
specify the vertex function, we can write down the SD equation with only
continuous variables without performing the discrete sum over Matsubara
frequencies which cannot be performed in advance without further approximation
in the imaginary-time formalism. By solving the SD equation obtained in this
way, we find the chiral-symmetry restoring transition at finite-temperature and
present the associated phase diagram of strong coupling QED.
  In solving the SD equation, we consider two approximations:
instantaneous-exchange and $p_0$-independent ones. The former has a direct
correspondence in the imaginary time formalism, while the latter is a new
approximation beyond the former, since the latter is able to incorporate new
thermal effects which has been overlooked in the ordinary imaginary-time
solution. However both approximations are shown to give qualitatively the same
results on the finite-temperature phase transition.
