8 G c Λ V acuum diagram 3 λ 6 2 λ 4 2 λ 2 λ 2 2 λ 4 2 FIG. 8: Three possibilities to organize the vacuum diagram of sixth order of perturbation theory. The correct coefficient 3 2 is obtained by summing all the possibilities. them. Luttinger and Ward have proved the possibility to transform this expression into much more convenient for- mula without such integration. For that they used a special functional constricted from skeleton diagrams the lines of which are the renormalized electron Green’s func- tions. We shall use the skeleton diagrams of strongly correlated system which differ essential from Luttinger and Ward [24] case and transform equation (18) to more convenient form. In our strong correlated case we introduce the following functional which is the generalization of the Luttinger-Ward [24] equation just for the strongly correlated systems. Here operation T r use the summation by α, σ, i and integra- tion by τ . dence of full Green’s function G $^{c}$( V ). The contributions of the diagrams a), b) and c) are following The quantity Y ′ contains all peculiarities of the strongly correlated systems and is presented itself as a sum of closed linked skeleton diagrams, constructed from irreducible Green’s functions of correlated electrons and full Green’s functions of uncorrelated electrons. On the Fig. 10 are depicted some of simplest skeleton diagrams for functional Y $^{′}$. These diagrams depend of the interaction strength V not only through the factors in the front of each diagram but also through the depen- Y = − 1 2 β T r { ln( ˆ G c (0) ˆ Σ − 1) + ˆ G c ˆ Σ } + Y $^{′}$, (19) V 2 2 ∑ α$_{1}$α$_{2}$ ∑ 12 α$_{1}$α$_{2}$G f (0) α$_{1}$$_{α$_{2}$}$(1 , 2 | V ) G c α$_{2}$$_{α$_{1}$}$(2 , 1 | V ) , − V 4 8 ∑ α$_{1}$...α$_{4}$ ∑ 1 ... 4 α$_{1}$α$_{2}$α$_{3}$α$_{4}$G c α$_{1}$$_{α$_{2}$}$(1 , 2 | V ) G c α$_{3}$$_{α$_{4}$}$(3 , 4 | V ) × × 〈 T f − α$_{1}$ 1 f α$_{2}$ 2 f − α$_{3}$ 3 f α$_{4}$ 4 〉 ir 0 , V 6 48 ∑ α$_{1}$...α$_{6}$ ∑ 1 ... 6 α$_{1}$α$_{2}$α$_{3}$α$_{4}$α$_{5}$α$_{6}$G c α$_{1}$$_{α$_{2}$}$(1 , 2 | V ) G c α$_{3}$$_{α$_{4}$}$(3 , 4 | V ) ×