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We construct the thermodynamic potential for quantum electrodynamics and quantum chromodynamics at finite temperature and density. We find the thermodynamic potential as a functional of the full propagators and vertices. This functional is stationary under variations of the full propagators and vertices by virtue of the Schwinger-Dyson equations, and may be interpreted as a classical action. We also discuss the renormalization of the propagators and vertices.
Freedman et al. (Mon,) studied this question.