- #1

pines-demon

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- TL;DR Summary
- What is the relation between perturbation theory and many-body diagrams?

I am trying to understand Green's functions in many-body theory for condensed matter. After much struggle, I managed to calculate my first diagrammatic expansion. However I am perplexed by getting more of the usual results.

The Hartree–Fock energy result I know from second quantization can be derived either by doing some mean-field approximation or by just calculating the first order perturbation energy of an electron gas. This is straightforward using ordinary quantum mechanics.

However now that I can write it using diagrams, I was expecting that doing a sum over a infinite number of diagrams would be equivalent to going over many orders of perturbation theory (even if some diagrams are not included). However the final result is the same as the one from the mean-field/1st order perturbation theory (hence it being called again Hartree–Fock).

What is going on here? Am I wrong at thinking that diagrammatic expansion is different from first order perturbation theory (when excluding some diagrams)? Or is there some sort of cancelation going on? Or are the results subtlely different?

The Hartree–Fock energy result I know from second quantization can be derived either by doing some mean-field approximation or by just calculating the first order perturbation energy of an electron gas. This is straightforward using ordinary quantum mechanics.

However now that I can write it using diagrams, I was expecting that doing a sum over a infinite number of diagrams would be equivalent to going over many orders of perturbation theory (even if some diagrams are not included). However the final result is the same as the one from the mean-field/1st order perturbation theory (hence it being called again Hartree–Fock).

What is going on here? Am I wrong at thinking that diagrammatic expansion is different from first order perturbation theory (when excluding some diagrams)? Or is there some sort of cancelation going on? Or are the results subtlely different?