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This question may be a little strange, but I'm currently going through my lecture notes and in the construction of the 1PI effective action there is a constant reference to 1PI correlation functions. I have no idea what these functions are, and the notes do not provide an explanation. Searching the internet is also not bringing me any answers. So my question is how are the 1PI correlation functions defined?

Some context on how the text is treating them:

The notes claim that the 1PI effective action is the generator of these correlation functions, and to prove it they construct relations between the n-point connected correlation functions (with sources) and the 1PI correlation functions and 2-point connected functions. For n=3 the relations are for example:

$$\tau_3(x_1,x_2,x_3) = \int dy_1 dy_2 dy_3 \prod_i \tau_2(x_i,y_i) \Gamma_3(y_1,y_2,y_3)$$

Where $\tau_n$ is the n-point connected correlation function with sources and $\Gamma_n$ the n-point 1PI correlation functions with sources. The higher relations are too involved to right down easily in a non-diagrammatic fashion.

The problem is I've never heard of these 1PI correlation functions before. They do not appear anywhere in the lecture notes before this chapter. We had discussed 1PI effective diagrams in Dyson resummation, but I do not see directly how it relates.

Qmechanic
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s.harp
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1 Answers1

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From your question, it looks like you know about the 1PI effective action. In a nutshell, it is the Legendre transform of the Wigner functional, $$ W[J] = \ln\left(Z[J]\right) \quad \Gamma[\phi] = \text{Sup}_J\left\{-W[J]+J \phi\right\} \, .$$ $Z[J]$ is the generating functional of correlation functions $Z[J] = \langle \text{e}^{J \phi}\rangle$.

The so-called 1PI correlation functions are the derivatives of $\Gamma[\phi]$ with respect to $\phi$ evaluated on the equations of motion, $$ \Gamma^{(n)} = \left. \frac{\delta^n \Gamma}{\delta^n \phi} \right|_{\phi=\phi_0} \, .$$ The field expectation value $\phi_0$ is defined through $$ \left. \frac{\delta \Gamma}{\delta \phi} \right|_{\phi = \phi_0} = 0 \, .$$

These can be used to compute actual correlation functions as you describe.

They are called 1 Particle Irreducible (1PI) because only loop diagrams enter their perturbative calculation. If you cut one of these diagrams once, it stays connected.

Steven Mathey
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  • Maybe I'm being dense, because this is the construction that was done in the lecture notes (I mean $\Gamma$ is the Legendre transform of W) and from it we deduced that it is the generating functional of the 1PI correlation functions. But this implies that these functions already had some form of meaning. – s.harp Sep 23 '15 at 20:31
  • Also in my lecture notes another formula for the 1PI correlation functions is given ($\Gamma_n (x_1,..,x_n)=\left.\frac{\delta}{\delta \phi(x_1)}...\frac{\delta}{\delta \phi(x_n)} \Gamma[\phi]\right\rvert_{\phi=0}$) than the one you give ($\Gamma_n=\left.\frac{\delta^n \Gamma}{\delta \phi^n}\right\rvert_{\phi=\phi_0}$). Although I imagine part of this discrepancy is from notation. – s.harp Sep 23 '15 at 20:35
  • Indeed, I use a short notation. If I were less lazy, I would write $\Gamma^{(n)}(x_1,...,x_n)$ etc. It seems that in your lecture notes, it is assumed that the field expectation value $\langle \phi(x) \rangle = \phi_0$ is zero. This is often the case. I'm used to system with a broken symmetry and did not set $\phi_0 = 0$. – Steven Mathey Sep 23 '15 at 21:47
  • If you want to give a meaning to $\Gamma^{(n)}$, you can think of them as effective vertices. If you take derivatives of the action of the system, you recover it's coupling constants, $\delta S^4/\delta^4 \phi \sim \lambda$. $\Gamma[\phi]$ is the effective action that you get once all the quantum fluctuations are integrated out. $\Gamma^{(n)}$ are the effective couplings. – Steven Mathey Sep 23 '15 at 21:48