In quantum field theories (to be relativistic, (non-)relativistic statistical or whatever), we have the powerful diagrammatic approach at our disposal. Most of the time we can not sum up all the diagrams, only a sub-class of them. I was thinking of the Hartree-Fock ones to give an example, but they are several classes we can re-sum exactly (skeleton, ladder, ...). Somehow this re-summation of diagrams is exact. I will keep italics to refer to this exactness.

Clearly the solution we find that way is not the full solution of the problem we considered at the beginning. By "problem" I obviously mean the eigenstates/eigenvalues problem of a secular equation given by a Hamiltonian/Lagrangian.

My question is the following: do we know the problem we exactly solved summing only partial class(es) of diagram ? I mean: can we write out an effective Hamiltonian/Lagrangian that our exact summation is the exact solution of ? Can we at least know the exact sub-space (eigenstates and/or eigenvalues), the solutions we found exactly belong to ?

  • $\begingroup$ Many of these mean-field type theories are exact limits of some $N\rightarrow\infty$ field theory. Is that the kind of thing you are looking for? Or are you looking for a fixed field theory at a fixed value of the couplings that reproduces the resummation. If you just want a Hamiltonian then a mean-field theory can be written as a self-consistent single site Hamiltonain. $\endgroup$ – BebopButUnsteady Jul 19 '13 at 15:44
  • $\begingroup$ @BebopButUnsteady Sorry, I do not clearly understand what you're talking about. My question was much more naive I believe: you start from a Hamiltonian (say interacting electrons with quartic term). We don't know the exact solution of it. But we can sum up partial class of diagram in a self-energy term. This is kind of an exact solution, isn't it ? My question is: it's an exact solution, ok, but which problem of ? Because clearly the full problem has as an exact solution the full summation of all classes of diagram, isn't it ? I'm not really looking for explicit examples. $\endgroup$ – FraSchelle Jul 19 '13 at 15:52
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    $\begingroup$ Yes - what I was saying is that many, if not all, of these resummations are known to be exact solutions of some Hamiltonian with some parameter taken to infinity. For example we can get a mean field theory of electrons by considering the theory where there a $N$ symmetric flavors of electrons and taking $N$ to infinity. $\endgroup$ – BebopButUnsteady Jul 19 '13 at 16:03
  • $\begingroup$ @BebopButUnsteady Ok then, that's really interesting :-) Sorry for my previous misunderstanding. I never heard about that. Do you know good references for some generic example in condensed matter for instance ? Maybe posting them as an answer would be great. Thanks in advance. $\endgroup$ – FraSchelle Jul 20 '13 at 6:59

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