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I'm studying the diagrammatics for a Bose system (in the superfluid phase) developed by Gavoret and Nozieres (Annals of Physics 28 349 (1964)).

In this paper, they show how to solve the problem using skeleton diagrams. In other words, they give equations for the two-point, three-point and four-point functions, involving the full Green's functions and irreducible diagrams (that they don't discuss, but I think they mean 2-PI irreducible here). In particular, they discuss in great length how these different functions are linked together, and what kind of Ward identities they have to fulfill in order to respect conservation laws.

Of course, this approach is useless unless one does approximations in order to compute correlation functions. What I don't get is how to approximate this skeleton diagrams in a consistent way to recover (for example) standard perturbation theory, which in this context is Bogoliubov theory, and at the same time be sure that the conservation laws are consistently recovered.

I can't find a nice reference that would tell me how to start from these diagrams and what I should do to them. All the textbooks I've looked at only have few pages on skeleton diagrams, and just show how to express the self energies and vertices with them, without discussing anything more.

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