# Dyson equation and Feynman diagrams: a brief explain

1. Jan 28, 2016

### FrancescoS

Hi!
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

2. Feb 2, 2016