Quantum Mechanics – Understanding Feynman Diagrams and Hartree-Fock

Question

I am puzzled by some lines I read in Mattuck's book on Feynman diagrams in many-body problems ( http://www.amazon.com/Feynman-Diagrams-Many-Body-Problem-Physics/dp/0486670473 )
Page 21 (1.14) for those who have the book.
Basically after representing the full propagator of an electron in an electron gas by expansion of the electron-electron interaction (not specified but QED I guess), it says: "this is the 'Hartree-Fock' approximation for the electron gas", which I still don't understand.
The Hartree-Fock method for me is just an iterative tool to calculate the collective wave-function of self-interacting fermions satisfying the correct anti-symmetrized form. This statement is evasive to me, and I'd like to understand in which way it makes sense.

Answer 1

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.