Difference between 1PI effective action and Wilsonian effective action?

Question

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

Answer 1

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.