These Feynman diagrams can be summed by solving the Dyson-Schwinger equation
$$
G = G_0 + G_0\Sigma G
$$
This is a self-consistency equation for $G$. Now write $G_0$ and $G$ in terms of single particle wave functions,
$$ G(x,x';\omega)=\sum_j \phi_j(x)\phi^*_j(x')\left[ \frac{\Theta(E_j-E_F)}{\omega-E_j+i\epsilon} +\frac{\Theta(E_F-E_j)}{\omega-E_j-i\epsilon} \right].
$$
Then the Dyson-Schwinger equation becomes a coupled set of equations for the eigenfunctions $\phi_j$ and the eigenvalues $E_j$. These are the standard Hartee-Fock equations. This is explained in some detail in many text books,
for example Negele and Orland, or Fetter and Walecka.

This post imported from StackExchange Physics at 2015-06-15 19:34 (UTC), posted by SE-user Thomas