Difference between 1PI effective action and Wilsonian effective action? What is the simplest way to describe the difference between these two concepts, that often go by the same name?
 A: 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.
