As I said before, I'm not a QFT expert but I'm trying to understand the basics of its rigorous formulation.

Let's take Dimock's book, where the foundation of QM and QFT is discussed. If we consider, say, two particles, one living in a Hilbert space $\mathcal{H}_{1}$ and the other in another Hilbert space $\mathcal{H}_{2}$, the description of the state of the two-particle system is given in terms of the tensor product $\mathcal{H}^{(2)}=\mathcal{H}_{1}\otimes \mathcal{H}_{2}$. Of course, we could go furhter and study a system $\mathcal{H}^{(N)}=\mathcal{H}_{1}\otimes \cdots \otimes \mathcal{H}_{n}$. If all the particles are identical, then $\mathcal{H}_{1}=\cdots = \mathcal{H}_{n} \equiv \mathcal{H}$ and we must take into account symmetric and anti-symmetric subspaces of $\mathcal{H}^{(N)}$, which correspond to the fact that particles may be either bosons or fermions, respectivelly. At this point, one defines symmetrization and anti-symetrization operators. The next step is to consider a system of an arbitrary number of particles. At this point, one defines Fock spaces $\mathcal{F}^{\pm}(\mathcal{H}) = \bigoplus_{n=0}^{\infty}\mathcal{H}_{n}^{\pm}$ for bosons and fermions. Also, one defines creation and annihilation operators $a(h)$ and $a^{\dagger}(h)$ on $\mathcal{F}^{\pm}(\mathcal{H})$.

Now, as far as I understand, this is all **quantum mechanics**, not QFT. However, these ideas seem to find analogues in QFT, and this is the point where I get confused.

On section I.5 of Feldman, Trubowitz and Knörrer's book there is a quick discussion on (fermionic) QFT and it is stated that, in this context, creation and annihilation operators are special families $\{\varphi^{\dagger}(x,\sigma):\hspace{0.1cm} x \in \mathbb{R}^{d}, \hspace{0.1cm} \sigma \in \mathcal{S}\}$ and $\{\varphi(x,\sigma):\hspace{0.1cm} x \in \mathbb{R}^{d}, \hspace{0.1cm} \sigma \in \mathcal{S}\}$ on a Hilbert space $\mathcal{H}$. This is very different than the creation and annihilation operators mentioned above. For instance, these are now families of operators indexed by $x$ and $\sigma$. I believe this is a reflection of the fact that we passed from QM to QFT. But I'm really lost here and I don't know what's the difference between these two constructions and definitions. Can anyone help me, please? I'm mainly interested in understanding the *second* approach, since the first one I believe I understand (at least sufficiently well). If, in addition, you could suggest some reference where these ideas of Feldman, Trubowitz and Knörrer are discussed in more details and with rigor, I'd appreciate!

**ADD:** Based on Feldman, Trubowitz and Knörrer's book, it seems to me that the understanding of these objects (to be more precise, the objects they briefly describe in the first 2 pages of section I.5) is fundamental to understand the formulation of a bunch of QFT models (at least for fermions). Thus, I'd appreciate if someone could elaborate a little more on the structure behind these creation and annihilation operators and its connections to the quantum case that is needed to understand the rest of the discussion on FTK's book. In other words, I think I just need to better understand these first definitions (and how are they connected with the usual quantum case I (seem) to know) to be able to understand the rest of the text.

Morallyspeaking, a linear functional on a function space is an integral. So $a(h) = \int a(x) h(x) dx$. Replace $a$ by $\varphi$ and make $h(x)$ multicomponent-valued to account for spin to get the FTK notation. If you shift the focus from rigor to motivation and historical context, you can't go wrong with Dirac's classicPrinciples of Quantum Mechanics(1930), §59–65. $\endgroup$ – Igor Khavkine Aug 15 '20 at 18:287more comments