How to prove useful property of logarithm of generating functional in QFT?

Question

How to prove that $\ln(Z(J))$ generates only connected Feynman diagrams? I can't find the proof of this statement, and have only met its demonstrations for case of 2- and 4-point.

Answer 1

Assume that the generating functional is given by a sum of all possible diagrams, i.e.

$$Z(J)=\sum_{n_i} D_{n_i}.$$

Furthermore, assume that each diagram D is given by a product of connected diagrams $C_i$, i.e. a diagram D can be disconnected. We will write this as

$$D_{n_i}=\prod_i\frac{1}{n_i!}C_i^{n_i},$$

where dividing by $n_i!$ amounts for a symmetry factor coming from exchanges of propagators and vertices between different diagrams. Combining this with our first expression, we get

$$Z(J)=\sum_{n_i}\prod_i\frac{1}{n_i!}C_i^{n_i}.$$

With some manipulation, this can be shown to be equivalent to

$$Z(J)=\exp\left(\sum_i C_i\right).$$

Taking the logarithm on both sides gives you the desired expression.

Answer 2

An intuitive interpretation from Timo Weigand's lecture notes:

Suppose $iW[J]$ contains all connected diagrams, then all possible connected and disconnected diagrams can be showed as products of $iW[J]$:

$$ \frac{Z[J]}{Z[0]} = 1 + iW[J] + \frac{1}{2!} {(iW[J])}^2 + \frac{1}{3!} {(iW[J])}^3 + ... = e^{iW[J]} $$

So

$$ iW[J] = ln \frac{Z[J]}{Z[0]} $$

This interpretation is just the same as Frederic's answer, but expressed in reverse order.