The linked-cluster Goldstones theorem

Many of the possible diagrams that can be drawn using the rules in Section 9.4 are of no importance, since on summation the corresponding terms cancel exactly in all orders. In order to classify and eliminate them, and to proceed to the simplest expressions for the wavefunction and energy, we need the following definitions.it 

It should be noted that, because it is only their topology that is significant a diagram may be distorted to display its type more openly. Thus (b) and (c) are respectively equivalent to 

We now introduce a basic factorization theorem (Frantz and Mills, 1960), which refers to the sum of all contributions associated with a set of diagrams that differ among themselves only through the displacement (with its vertices) of a subdiagram. An example is the set 

The upper and lower subdiagrams (A and B) show all possible orderings of the B vertex relative to the vertices in A all three diagrams correspond to the same final state 

The final state arising by summation of all the ( a + ne) / A B contributions whose diagrams differ only in the relative order of the vertices in A and B subdiagrams (containing a and vertices, respectively, in given internal order), is obtained by letting the operators for the A and B subdiagrams work separately on 0). (9.5.1) 

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