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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. $\endgroup$ Commented Dec 1, 2011 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. $\endgroup$ Commented Dec 2, 2011 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. $\endgroup$ Commented Dec 2, 2011 at 13:43
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    $\begingroup$ In fact, Wilsonian effective action is nonlocal. If you integrate out high ebergy degree of freedom, you just replace the short distance physics with a propagator. $\endgroup$ Commented Jun 23, 2012 at 12:23
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    $\begingroup$ Could we say that the 1PI action is obtained when one integrates over all energies/momenta? Which is the same as saying there is no cut-off. $\endgroup$ Commented Aug 9, 2015 at 14:13