Idrinehart, very interesting answer. Does the situation change if the perturbation is quartic in the fields? I ask because I've seen some literature where one construct the Fock space and in this Fock space the Hamiltonian is of the form quadractic term + quartic term which is considered as a perturbation.

@IgorKhavkine but even in the case $m^{2} = 0$, $\chi_{\kappa}$ is not really zero for $p^{2}$ large, right? I was expecting that the statement meant exactly what you wrote about the $\chi_{\kappa}$ having support in some shell of finite momenta, but it does never seem to be the case.

Also, what is the role of the estimates I mentioned when one does perturbative RG? Are this attempts to prove that higher order powers of $\phi$ are small enough so they do not contribute, as you mentioned?

Carlo, thanks for your answer. The lecture notes will certainly be helpful. But could you clarify a little more your answer? I mean, the renormalization group $\mathscr{R}$ I mentioned in my post changes the effective action, so your answer seem to indicate that one is actually trying to use this map $\mathscr{R}$ so the flow $G \to G' \to \cdots$ end up having a nicer/simpler form. Is this correct? If so, the idea is to get (hopefully) get something one can actually integrate? Or a fixed point, as I mentioned?

@MikaeldelaSalle thanks for the link and the comments. To answer your question, if $\psi \in \mathcal{F}^{\pm}(\mathscr{H})$ is an arbitrary element, the time-evolved element is $e^{-itH}\psi$. I edited the post to clarify this second question!