I must think more about all this. I'll possibly post another question on applications of these ideas. It might be better.

Let me see if I understand the application of your explanations. Suposse a field is $\phi: \mathbb{Z}^{d}\to \mathbb{R}^{d}$ so that the space of all fields is $\mathbb{R}^{\mathbb{Z}^{d}}$. For each $x \in \mathbb{Z}^{d}$ I can define random variables $f_{x}: \Omega=\mathbb{R}^{\mathbb{Z}^{d}}\to \mathbb{R}$ by $f_{x}(\phi) :=\phi(x)$. Thus, I can calculate, e.g. correlations $\mathbb{E}_{\nu_{C}}[f_{x}f_{y}] = \mathbb{E}_{\mu_{C}}[\phi(x)\phi(y)]$ both ways. That's why these are "basically the same"?

Thank you SO much for this amazing answer! This was exactly what I was searching for!

I see. I think this confusion of mine is worth a new post, to put it some context. Anyway, you've been really helpful! Thanks a lot!

I'm still having trouble understanding the origin of such differences. My main interest is statistical mechanics. I've been reading the section Gaussian Free Field (GFF) of two different books and the Hamiltonian on both books is the same. But when the time comes to discuss the thermodynamic limit, one book goes in the $l^{2}(\mathbb{Z}^{d})$ direction and the other goes in the $s'$ direction. But it seems that the target here is the same, but the constructions are differente and the results are different. Do you know why?

Thanks for the comment! Does this fact depend on the dimension $d$?

$\frac{d}{dt}(S+I+R) = 0$. But this is just a consequence of $N$ being a constant, right?

@MattF. thanks for your answer! If we include mortalities in the recovered population, then the formula $S+I+R = N$ totally makes sense but I wonder why not to consider $N = N(t)$ instead. I imagine it is because the equations would be much more difficult to solve, but it is just a guess. Anyway, woudn't we expect the model to be, idk, a bit more accurate if we consider $N$ as a time-dependent parameter?

@CarloBeenakker but that is the point. The book has an appendix on Grassmann variables and its associated calculus, but the exposure in the first few chapters states the results in real/complex variables. It really seems like a general approach since the discussion considers spin systems.

@AbdelmalekAbdesselam Ops! It's your answer. Well, all the other answers are useful too but your answer was really detailed and enlightening.

Thanks for the answer. So, if I understood it correctely, the partition function is my effective action (\ref{2}) at $\psi=0$?