Wilsonian renormalization The answer by Heider in this link points out that when we integrate out high momentum Fourier modes, we end up with Wilsonian effective action (not the 1PI action). This is the modern way of understanding renormalization.
I want to relate it to the old way of understanding renormalization i.e., not from a path-integral calculation but as perturbative renomalization (in the Hamiltonian formulation of field theory).
Counterterm renormalization In this technique, (for example, in scalar $\phi^4-$theory) separates $\mathcal{L}$ into a renormalized and a counterterm part as $$\mathcal{L}=\mathcal{L}_{renorm}+\mathcal{L}_{ct}.$$ Now $\mathcal{L}_{renorm}$ contains $V_r(\phi_r)=\frac{1}{2}m_r^2\phi_r^2+\frac{\lambda_r}{4!}\phi_r^4$ and the counterterm contains $$V_{ct}=\frac{1}{2}\delta_m\phi_r^2+\frac{\delta_\lambda}{4!}\phi_r^4$$ which cancels certain divergences of one-loop diagrams. The potential $V(\phi_r)$ is now in terms of measured parameters $m_r^2$ and $\lambda_r$.
Question 1: When you integrate high frequency modes, you generate terms proportional to $\phi^2$ and $\phi^4$ which looks like counterterms. My question is, can we regard the $\mathcal{L}_{ct}$ (in this approach) to be same as the contributions coming from integrating out high momentum modes (in the path integral)?
Question 2 Moreover, counterterm renormalization is a loopwise renormalization process. Is it also the same in the Wilsonian picture? Moreover, these two ways of understanding renormalization should be consistent. Can someone explain the connection between Wilsonian idea of renormalization and counterterm renormalization.
