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Finite-Temperature and -Density QED: Schwinger-Dyson Equation in the Real-Time Formalism

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.