A} = T \sum_{\omega,\alpha} A_{\alpha\alpha}(i\omega)$. In general, however, the functional $\Phi[{\bf G}]$ is not known explicitly which prevents an evaluation of ${\Omega} [{\bf G}]$ for a given ${\bf G}$. So-called conserving approximations [@BK61] provide an explicit but approximate functional $\Phi_{\rm pert.}[{\bf G}] \approx \Phi[{\bf G}]$. However, these are weak-coupling approaches where a certain subclass of $\Phi$ diagrams is summed up.

Here a different but still rigorous variational principle is proposed which is based on a functional ${\bf G} = {\bf G}[{\bf \Sigma}]$ defined as the inverse of ${\bf \Sigma} = {\bf \Sigma}[{\bf G}]$. We can assume the latter to be invertible (locally) provided that the system is not at a critical point for a phase transition (see Appendix \[sec:gofs\]). Consider then: $$\Omega_{\bf t}[{\bf \Sigma}] \equiv 
  {\rm Tr} \ln (- ({\bf G}_0^{-1} - {\bf \Sigma})^{-1}) + 
  F[{\bf \Sigma}]
\label{eq:var}$$ where $F[{\bf \Sigma}] \equiv \Phi[{\bf G}[{\bf \Sigma}]] - 
\mbox{Tr} ({\bf \Sigma} \, {\bf G}[{\bf \Sigma}])$ is the Legendre transform of $\Phi[{\bf G}]$. The subscript ${\bf t}$ indicates the explicit ${\bf t}$ dependence of $\Omega$ due to the free Green’s function ${\bf G}_0$. Using $T^{-1} \delta F[{\bf \Sigma}] / \delta {\bf \Sigma} = 
{\bf G}[{\bf \Sigma}]$, one finds that $$\delta \Omega_{\bf t} [{\bf 