I am struggling with the concept of quantum effective action, but first recall the definition : given a Wilsonian effective action $W[J]$ of our theory, the quantum effective action is just $$\Gamma[\phi_{J}]\equiv W[J]-\phi_{J}\cdot J\tag{1}$$ where the dot notation is used to indicate an integral over $d^{4}x$, and I have denoted with
$$\phi_{J}(x)\equiv<\phi>_{J}\equiv Z[J]^{-1}∫ \mathcal{D}\phi e^{-\frac{S[\phi]+J\cdot\phi}{\hbar}}\phi=\frac{δ W[J]}{δ J(x)}\tag{2}$$
the quantum average in presence of the external source $J$. If I understand correctly, $\Gamma[\phi_{J}]$ is just an object that, if somehow known, would let us to compute the partition function $\mathcal{Z}$ simply solving its equation of motion:
$$\frac{δ\Gamma}{δ\phi_{J}}\bigg|_{\phi_{J}=\phi_{c}}=0\iff\phi_{c}=\phi_{J=0}=<\phi>.\tag{3}$$
So plugging this "classical" solution into the quantum action gives
$$\mathcal{Z}[J]=e^{-\frac{1}{\hbar}W[J]}=e^{-\frac{1}{\hbar}(\Gamma[\phi_{J}]+J\cdot\phi_{J})}\Rightarrow \mathcal{Z}=∫ \mathcal{D}\phi e^{-\frac{S[\phi]}{\hbar}}=e^{-\frac{1}{\hbar}\Gamma[\phi_{c}]}.\tag{4}$$
which encodes the information about transition amplitudes. This shows how the quantum effective action "contains" all the quantum effects associated to the measure $\mathcal{D}\phi$. Here my questions:
- Why is this relevant if we are not able to compute fully $\Gamma$? I mean, it's usually said that it helps because it enable us to correct the potential with loop contributions $$V_{eff}=V+V_{1-loop}+...,\tag{5}$$ but then what do we do with that? To me this seems not easier than just do the loop calculations in perturbation theory using just $S[\phi]$, could you please explain how to put this things down to Earth? I don't think I really understand what does it mean to correct a potential with loops.
