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I am puzzled by some lines I read in Mattuck's book on Feynman diagrams in many-body problems ( ) 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.

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I don't have this book but I think "Hartree-Fock approximation" means that you consider only states that are slater determinants of one-electron states, i.e. electron-electron correlations are neglected. – jjcale Nov 2 '12 at 18:10
Ok, and on what grounds do we neglect those correlations, it has always perturbed me that the use of a Slater determinant means complete separability between the electrons wave-functions without it being clearly justified. – Learning is a mess Nov 4 '12 at 10:45
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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.

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