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In Zagoskin's Quantum Theory of Many-Body Systems book, the Bethe-Salpeter equation is given as

$\Gamma(12;1'2')=\widetilde{\Gamma}_{PP}(12;1'2')+i\int d3\int d3'\int d4\int d4'\widetilde{\Gamma}_{PP}(12;3'4')G(33')G(44')\Gamma(3'4';1'2')$

Such an equation relates the vertex function $\Gamma$ with the Green's function--basically, it's a two-particle version of Dyson's equation.

My question concerns the above in the finite-temperature limit. For a finite-temperature Bethe-Salpeter equation, would I just replace the Green's function in the integral with a finite-temperature Green's function, or is it more subtle then that? I have tried searching for a finite-temperature Bethe-Salpeter equation, but I am only getting papers on QCD. I really need more of a general introduction geared more towards condensed matter physics (e.g., Fermi liquid systems). Any explanation or references at the level of Mahan's Many Body Physics would be greatly appreciated.

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This equation is valid at finite temperature using temperature Green's functions. A proper reference for this would be for instance "Many-Body Theory Exposed!" by Willem H. Dickhoff, Dimitri Van Neck, ISBN-13: 978-9812562944, chapter 9.

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