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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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