@PedroLauridsenRibeiro thanks for the comment. The point is that when Reed & Simon introduce $\varphi$ and $\pi$ in terms of the Segal quantization operator, they introduce for elements of $f \in \mathscr{H}_{C}$. That is my problem with the definition of $\varphi$ and $\pi$. Because $Ef$ is not necessarily $\mathscr{H}_{C}$, then these maps seems ill-defined.

Very nice answer as well! Thanks for the details! Really interesting points you raised!

Great answer! Right to the point! Thanks!

@FZaldivar right! Because you're taking $V$ to be $TV$. Thank you, your comment was really helpful to me!

@FZaldivar thanks for the comment! What you said is really enlightening! I think we shall demand $V$ to be finite-dimensional too in order to have Berezin's description, right? Because there we have a finite number of generators. Also, according to Berezin, $k_{n}$ are generators because every element of the Clifford algebra $f \in K_{n}$ can be written as $f = \sum_{l=0}^{n}\alpha_{i_{1},...,i_{l}}(k_{i_{1}}\cdots k_{i_{l}})$, where $\alpha_{i_{1},...,i_{l}}$ are numbers. Does this representation follow from your construction? I think I'd need a bit more work there, right?

@AbdelmalekAbdesselam amazing. I saved the link to read it later because I've been a little busy these past days and I want to read it carefully, and it demands time. I will read it as soon as I can and if something is not clear I reach you back. Anyway, thanks for the comments again, I've been learning so much from you!

For completeness, the forementioned answer was this one mathoverflow.net/questions/62770/…

I've always thought that the difference was explained because you were addressing the continuum limit, whereas the toy model in my post is trying to sketch a more general idea. But now that you mentioned the work of BK, this difference might be explained by the fact that BK were not trying to explain RG, but using an idea. As you mentioned, there is no reescaling at each step and I believe this is another way to say what I mentioned that after each interaction one end up with a different object. I'm getting it right?

@AbdelmalekAbdesselam your first comment maybe clarifies something it is not very clear to me yet. As you know, I study your answers frequently and in one of them you discuss a more concrete (yet, rather general) model and you proceed to reescale to a unit lattice. There, the initial integral ($Z$ in my post) becomes $Z$ itself but with $V_{0}$ replaced by $RG(V_{0})$. This is a very nice approach, and the ideas are clear to me. In my post, however, after each iteration, one ends up with different object, since at each step the covariance matrix changes $C_{n}\to C_{n+1}$. (Continues)