How to prove that the generator of proper vertices is the Legendre transform of $W(j) = \log \frac{Z[j]}{Z[0]}$

Question

I'm studying QFT from Le Bellac's book, but I can't understand very well his proof for the generator of proper vertices. Can someone give a more readable and/or understandable proof?

Answer 1

Very rougly, the logarithm has the function of transforming a product into a sum, therefore if the partition function is factorizable (i.e. disconnected), it will give a sum on connected parts.

For a slighly more rigorous discussion, consider the Hamiltonian of two non interacting systems, the partition function $z(j)=Z(j)/Z(0)$ (without vacuum fluctuactions) is obviously the product of the single partition functions.

$z(j)=z_1(j)z_2(j)$

Consider now the partition function generated by the connected diagrams $W(j)$, it is simply the sum of the the two connected partition functions, because if you want to create a connected diagram you can't select diagrams from both systems, since they are not interacting the diagram you created is disconnected.

$W(j)=W_1(j)+W_2(j)$

Then, we have the fact that the partition function is function only of the connected diagrams, since any non-connected diagram can be formed by joining two connected diagrams, then $Z(j)$ is a function of $W(j)$, that we will note as $Z(W(j))$.

From the previous discussion we have:

$z(W(j)) = z(W_1(j)+W_2(j)) = z(W_1(j))z(W_2(j))$

the only function that satisfies this identity is the exponential, as you can easily check, so

$z(W(j))=exp(a W(j))$

expanding around $j=0$ gives $a=1$.

I couldn't find a proof simpler that the one given in Lebellac for the question you asked, you might try for example this Zinn-Justin

Answer 2

try the lecture notes by van Hees from Frankfurt. Check out p. 114 http://faculty.ksu.edu.sa/djdou/Lectures%20Notes%20PHY556/Introduction%20to%20Q.F.T.pdf