What is the simplest way to describe the difference between these two concepts, that often go by the same name?


The Wilsonian effective action is an action with a given scale, where all short wavelength fluctuations (up to the scale) are integrated out. Thus the theory describes the effective dynamics of the long wavelength physics, but it is still a quantum theory and you still have an path integral to perform. So separating the fields into long and short wavelength parts $\phi = \phi_L + \phi_S$, the partition function will take the form (N.B. I'm using Euclidean path integral)

$$ Z = \int\mathcal D\phi e^{-S[\phi]} =\int\mathcal D\phi_{L}\left(\int D\phi_{S}e^{-S[\phi_L+\phi_S]}\right)=\int\mathcal D\phi_{L}e^{-S_{eff}[\phi_L]}$$ where $S_{eff}[\phi_L]$ is the Wilsonian effective action.

The 1PI effective action doesn't have a length scale cut-off, and is effectively looking like a classical action (but all quantum contribution are taken into account). Putting in a current term $J\cdot \phi$ we can define $Z[J] = e^{-W[J]}$ where $W[J]$ is the generating functional for connected correlation functions (analogous to the free energy in statistical physics). Define the "classical" field as $$\Phi[J] = \langle 0|\hat{\phi}|0\rangle_J/\langle 0| 0 \rangle_J = \frac 1{Z[J]}\frac{\delta}{\delta J}Z[J] = \frac{\delta}{\delta J}\left(-W[J]\right).$$

The 1PI effective action is given by a Legendre transformation $\Gamma[\Phi] = W[J] + J\cdot\Phi$ and thus the partition function takes the form

$$Z=\int\mathcal D\phi e^{-S[\phi] + J\cdot \phi} = e^{-\Gamma[\Phi] + J\cdot \Phi}.$$ As you can see, there is no path integral left to do.

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    $\begingroup$ Thank you for elegant and beautiful answer. As I understood, Wilsonian eff action has less information in it than full theory, but 1PI has even more information than needed. $\endgroup$ – Newman Dec 1 '11 at 17:24
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    $\begingroup$ You need to be a little careful about saying that Wilsonian eff. theory contain less information than the full theory, since performing the path integral over $\phi_L$ you get the full result. The high energy (low-distance) fluctuations are integrated out, but the information about them is still there, but hidden in $S_{eff}$. Wilsonian eff action is describing how the theory effectively behaves at low energy (long-distances), but you can't just "remove" short-distance physics. You need to integrate them about since for interacting theories they couple and contribute to the low-energy physics. $\endgroup$ – Heidar Dec 2 '11 at 13:31
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    $\begingroup$ I just found these note (arxiv.org/abs/hep-th/0701053v2) which I think contain more detailed discussion about differences and connections between 1PI and Wilsonian effective actions. $\endgroup$ – Heidar Dec 2 '11 at 13:43
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    $\begingroup$ As a follow up on the comment of Heidar: to obtain the Wilsonian action you usually have to perform some type of coarse graining. For instance, you might need to truncate the higher order terms in a perturbative approach to integrating out the fast modes. It's this coarse graining which causes the effective action to contain 'less information'. But this is somewhat of an artefact related to fact that in most theories you cannot integrate out the fast modes exactly. $\endgroup$ – Olaf Dec 7 '11 at 23:17
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    $\begingroup$ Why is often said that 1PI effectice action is nonlocal? $\endgroup$ – Newman Dec 16 '11 at 14:37

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