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

Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?