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If any vertex of a diagram can be reached from any other vertex by moving along the lines of the graph, the diagram is said to be connected. 5. Connected diagrams contributing to the second-order contributions to the Green-function generating functional in the ϕ 4 -model. graphs by cutting just one internal line. A similar terminology is applied to the actual Green functions: • • Parts of Green functions represented by connected and OPI diagrams are called connected Green functions Wn and OPI Green functions n , respectively.

X n ) (by equating the factors with equal powers of the source function J (x)). 6. 7. 8. 8 by K KG from the right and by W2−1 from the left, we obtain the equality for the so-called mass operator (or proper energy) of a particle. 9. 133)). In the same way we can derive the Schwinger equation for any field theoretical model with a polynomial Lagrangian. 141) for all fields enters the model. 34 Quantum field theory: the path-integral approach Ö Ö .. Ö n +1 = δn1 Ö 1 + 2 Ö + k+m=n Ö Ö Ö n! m!

2. Let us consider as an example the simplest scalar ϕ 3 -model with the action S[ϕ] = d4x 1 g ϕ(x)(£ + m 2 )ϕ(x) + ϕ 3 (x) . 2 3! 144) The renormalization of this model requires the addition of a ϕ 4 -vertex and hence, rigorously speaking, it is not self-consistent. However, this fact is not important for our formal functional manipulations and we choose this example as the simplest one to illustrate the general functional techniques. e. for any function f (x) from the domain of definition of the latter, K KG (x, x ) satisfies the equality (£ + m 2 ) f (x) ≡ d 4 x K KG (x, x ) f (x ).