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This is somehow a more mature version of an old question of mine. I'd like to have a more clear picture of the difference between QFT and QM from a mathematical point of view.

Okay, so we begin with a Hamiltonian $H$ on a Hilbert space $\mathscr{H}$, which is our single-particle space. We can extend the description to a system of $n$ particles by taking tensor products and, finally, consider the case where an arbitrary number of particles is allowed, so we get to our Fock space $\mathscr{F}(\mathscr{H})$. The Hamiltonian is extended by means of the second quantization $d\Gamma(H)$ and we can also define creation and annihilation operators $a^{\dagger}(f)$ and $a(f)$ on $\mathscr{F}(\mathscr{H})$ (I'm skipping the technical details since this is not the focus here).

This is where things get intriguing to me. The above describes a quantum mechanical system with an arbitrary number of particles. When Reed & Simon describe the quantization of the free field, they notice that the creation and annihilation operators have formal pontwise representations $a^{\dagger}(x)$ and $a(x)$ and, formally: $$a(f) = \int f(x) a(x)dx \quad \mbox{and} \quad a^{\dagger}(f) = \int \overline{f(x)}a^{\dagger}(x)dx$$ where the identities make mathematical sense in terms of quadratic forms. In the case of the free field, an analogous treatment is given to the free field operator and the associated Hamiltonian, so that the expressions obtained in the physical literature are actually the formal "integrands" of such representations, which have mathematical meaning as quadratic forms on $\mathscr{F}(\mathscr{H})$.

This led me the impression that, from a mathematical perspective, the bridge between (the general scheme of) QM and QFT is simply in the pointwise representation of the involved operators, as in the case of the creation and annihilation operators displayed above. However, this strikes a little bit odd to me since, well, these are formal objects which are actually quadratic forms in our usual Fock space; in other words, it seems more a matter of notation than anything else.

Question: Apart from more deep discussions on Gårding-Wightman axioms and what QFT really is in rigorous terms, are these formal expressions the real bridge between a QM and a QFT? If so, what is so special about these notations which provide such a connection between two theories (or, at least, two regimes of the same theory)? Is it really just a matter of notation?

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    $\begingroup$ There is really nothing fundamental about making a connection between quantum mechanics and free quantum field theory. The relationship between the two is not subtle, and it can be expressed rigorously in many different fashions. It is only when there are nontrivial interactions that quantum field theory becomes a fundamentally different "kind" of thing from quantum mechanics, so you are never going to get the key insights into what makes quantum field theory interesting in its own right by looking at free theories. And how to treat the interacting cases with rigor is, generally, unknown. $\endgroup$
    – Buzz
    Commented Jun 16, 2021 at 3:17
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    $\begingroup$ To formulate a QFT, we don't begin with a Fock space. Constructing a Fock space implies that you already know what the vacuum is. In that sense, the Fock space is a way of presenting the (at least partial) solution of a QFT rather than formulating the QFT. $\endgroup$
    – Michael Engelhardt
    Commented Jun 16, 2021 at 7:41
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    $\begingroup$ I think there are two separate things going on here. One is that QM and QFT are both quantum theories. The other is the idea that one can organize them both in terms of a Fock space. As far as the former is concerned, sure, QFT is just QM of infinitely many degrees of freedom (not that that's a trivial transition, cf. multivariable calculus vs. functional analysis). But a Fock space construction is something more specific that implies you've at least partially solved your theory. The interesting part of QFT is the solving, not the organizing into a Fock space at the end. $\endgroup$
    – Michael Engelhardt
    Commented Jun 16, 2021 at 15:23
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    $\begingroup$ Just to throw in an example that may help in sorting things: In light-front field theory, one argues that the vacuum is trivial (although this is a very subtle point). One then casts the theory in terms of a Fock space. It is still an interacting theory, the Hamiltonian is not quadratic yet! But one does have a Fock space representation, on the basis of which one can develop truncation schemes to approximately solve further, etc. This is why I write about "partially" solving. In light-front field theory, one has "partially solved" by making the argument that the vacuum is trivial. $\endgroup$
    – Michael Engelhardt
    Commented Jun 16, 2021 at 18:30
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    $\begingroup$ You can see the issue with the Fock state by looking at the Lee Model, which a toy model that has an exactly solvable spectrum, yet which still requires renormalization. If you try to set up the Fock space at the beginning, you will find that it does not really correspond to the state space of the theory, because the renormalization changes this fundamentally. $\endgroup$
    – Buzz
    Commented Jun 17, 2021 at 0:03

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