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I'am currently learning the 2 particle irreducible (2PI) or Cornwall-Jackiw-Tomboulis (CJT) formalism. Does anybody know a textbook or a review that treats this subject? As far I only found the original papers like http://dx.doi.org/10.1103/PhysRevD.10.2428 but no modern review. Of course there are a lot of papers that use this formalism, but I haven't found a general introduction yet.

As far I know how to compute the generating functional, but not why this is a useful approximation, where it can be used (limits of the framework) and so on. At the end I am interested in renormalization in the Hartree approximation (I have resources for that, but not for the "beginning")

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There is an extensive, survey-like introduction by Juergen Berges, dating from 2004. For the renormalization of 2PI effective actions see also a very recent paper by Carrington et al..

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