Thermodynamic potential in quantum electrodynamics

The thermodynamic potential, $\Omega$, in quantum electrodynamics (QED) is derived using the path-integral formalism. Renormalization of $\Omega$ is shown by proving the following theorem: ${\Omega }_B(e_B,m_B,T,\mu )-{\Omega }_B(e_B,m_B,T=0,\mu =0)={\Omega }_R(e_R,m_R,T,\mu ,S)$, where $B$ and $R$ refer to bare and renormalized quantities, respectively, and $S$ is the Euclidean subtraction momentum squared. This theorem is proved explicitly to $e_R^{}{}_{}{}^4$ order and could be analogously extended to any higher order. Renormalization-group equations are derived for ${\Omega }_R$, and it is shown that perturbation theory in a medium is governed by effective coupling constants which are functions of the density. The behavior of the theory at high densities is governed by the Euclidean ultraviolet behavior of the theory in the vacuum.