Resolution of the Hierarchy of Green's Functions for Fermions

The rigorous evolution equations for the one-body Green's functions ($g^{>},g^{<}$) describing a system of interacting fermions are not closed (an analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for many-body distribution functions). The analysis of these equations is made in terms of proper connected diagrams. It is established that in the bulk limit, $N$ (particle number) $\to \infty$, $\Omega$ (volume) $\to \infty$ while $\frac{N}{\Omega }$ remains finite, these equations can rigorously be transformed into closed equations with respect to ($g^{>},g^{<}$).