Expressing the decoupled equations of motion for the Green's function as a partial sum of Feynman diagrams

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Abstract
It is shown that any given decoupling of the equations of motion for the propagators of an interacting Fermi system is equivalent to a particular partial sum of Feynman diagrams. This result is established for many-time propagators in a straightforward way by iterating in the diagrammatic form of the equations of motion. The iteration method does not work for two-time propagators, but we are able to present an alternative argument which makes it plausible that the result holds true in this case as well. The decouplings leading to the Hartree-Fock, ladder, and ring diagram (RPA) partial sums are discussed in detail. It is conjectured that the reverse correspondence is not true, i.e. to any given partial sum there does not necessarily correspond a decoupling.