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I have found a note about Generating functionals that seems to be very direct.

Since I faced a difficulty many times without solve it, I would like if anyone can explain me my troubles.

You can fine the note here (link).

In the following figure, the author describes the 3-point green function in the way it's defined

I don't understand what does it mean exactly "(more vertices)" and the 2 factor in front of the three-vertex. Does "more vertices" stands for other possibles interaction vertices, like ##\phi^4## etc…?

And the 2 factor? I should expect a factor of ##3!##.

Starting from this 2-factor, he prefers to write the above equation using this decomposition

Can you explain it in a more detailed way? What does the combinatoric factors comes from? I would like to understand it once for all.

There is also this diagram, where ##Z[J]## is the generating functional of green functions.

If I use the definition of ##Z[J]## that is

##

Z[J] = \Sigma_{n = 0}^{+ \infty} \frac{i^n}{n!}\int dx_1 … dx_n G(x_1,…,x_n)J(x_1)…J(x_n)

##

and if I perform the functional derivative

##

\frac{\delta Z[J]}{\delta J(y_1)} = \Sigma_{n=1}^{+ \infty}\frac{i^n}{(n-1)!}\int dx_1…dx_{n-1}G(x_1,…,x_{n-1},y_1)J(x_1)…J(x_n)

##

and I would like to obtain the expression he writes below, but i don't manage to do it.

Any help?? Thank you

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# Dyson equation and Feynman diagrams: a brief explain

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