And, also, it would be very clarifying to know why people use it, to what kind of problems, the differences between these approaches and so on.

@AaronBergman I think I am unexperienced enough to say that I don't know yet. My research area is statistical mechanics but QFT ideas end up being important at some level. What level? Still don't know for sure. I think this is one of the points that motivated my question in the first place. I'm having enough trouble trying to learn QFT on my own, and I know some people deal with it by using $C^{*}$-algebra and other tools I've never studied either.... But, on the other hand, I have a background on functional analysis and distribution theory. I wonder if this is enough to some extent.

At some point, Dimock states: "If $\psi \in L^{2}_{\pm}(\mathbb{R}^{3n})$ is a continuous function, then in $a(h)\psi$ we can take $h$ to be a $\delta$-function and define an operator $a(x)$ by $(a(x)\psi)(x_{1},...,x_{n-1}) = \sqrt{n}\psi(x,x_{1},...,x_{n-1})$." Question: according to you answer, this is precisely what my $\varphi^{\dagger}(x,\sigma)$ are, right? The only difference is that, in this case, the spins are not being taking into account, so that $\sigma$ is omitted.

@NikWeaver thank you so much for the amazing answer. This clarifies the things to me. Let me ask you something. You said that my second representation arises when the state $h$ is taken to be a delta. Dimock's book discusses many particle systems where $\mathcal{H}=L^{2}(\mathbb{R}^{3})$. In this case, $\mathcal{F}^{\pm}=\bigoplus_{n=0}^{\infty}\mathcal{H}^{\pm}$ , where $\mathcal{H}^{\pm} = L^{2}_{\pm}(\mathbb{R}^{3n})$ are, respectivelly, symmetric and anti-symmetric subspaces of $L^{2}(\mathbb{R}^{3n})$. continues...

@MircoA.Mannucci I think to recognize one's mistake is as important as learning from others. In my opinion, this is a very nice humble behavior of yours! This is well done science for sure!

@KonstantinosKanakoglou thank you so much for the comments! I'm going to take a look at this book right now.

@IgorKhavkine this is something that came to my mind, too. But this is just a different way to represent these operators, right? I mean, is it all? And, besides, why doing that? (Sorry if my questions are really basic, I'm still getting acquainted with all this).

Hello again Mirco! And thanks for the answer! You said that the Fock space formulation is already QFT, right? This is confusing to me. See, I know that, rigorously, a QFT theory is defined by Wightman axioms, right? But intuitively (since I'm not an expert), this seems really like a QM problem. What is you understanding on this?

Nice! I'll keep that in mind!

Mirco, that's really nice. I took a course many years ago with him as well. Very nice professor, indeed.

Great! I love these collection and I pretty much like everything Reed and Simon write. This sounds perfect!

Great answer! Thank you so much! I'll definitely read Edson's book. Btw, what's your opinion on Dimock's book "Quantum Mechanics and Quantum Field Theory"? It seems very good too.

perfectly clear! Thank you so much!