What is the simplest ay to describe the difference between these two concepts, that often go by the same name?
Answer
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 ϕ=ϕL+ϕS, the partition function will take the form (N.B. I'm using euclidean path integral)
Z=∫Dϕe−S[ϕ]=∫DϕL(∫DϕSe−S[ϕL+ϕS])=∫DϕLe−Seff[ϕL]
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⋅ϕ 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 Φ[J]=⟨0|ˆϕ|0⟩J/⟨0|0⟩J=1Z[J]δδJZ[J]=δδJ(−W[J]).
The 1PI effective action is given by a Legendre transformation Γ[Φ]=W[J]+J⋅Φ and thus the partition function takes the form
Z=∫De−S[ϕ]+J⋅ϕ=e−Γ[Φ]+J⋅Φ.
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