I have been trying to read a physics textbook on Quantum Field theory. There seems to me to be a bit of a disconnect in most texts I have looked at between quantum mechanics and quantum field theory, in the passage from multiparticle wave functions to fields. I'm curious if there is a physical reason for this. I'm asking this here because the question has a math physics flavor.

First, let me sketch out my simplistic understanding of field theory from the Hamiltonian point of view. I will ignore relativistic invariance (though if I understand correctly, with a bit of work it can be recovered in this picture). Let $\alpha$ be a bosonic particle and let $V = V_\alpha$ be a space of wavefunctions of a single $\alpha$ particle. For concreteness, let's assume $V = L^2_\mathbb{C}(\mathbb{R}^3),$ corresponding to a scalar field. Note that I don't really care about the details of $V$: any space will do (including a finite-dimensional one for bound particles). The one assumption I will make on $V$ is that we have fixed a real subspace $V_\mathbb{R}\subset V$ compatible with the Hermitian structure.

Then single-particle quantum mechanics says that a wavefunction $\psi\in V$ evolves according to the Schroedinger equation, $\dot{\psi} = -i H_\alpha\psi,$ for $H_\alpha$ the single-particle Hamiltonian. Similarly, for any $n$, there is a non-interacting hamiltonian $H_{\alpha, n} : = \text{Symmetrize}(H\otimes 1\otimes \dots \otimes 1)$ on the bosonic $n$-particle space $S^n(V)$.

Now my understanding is that field theory arises as soon as we perturb the collection of $n$-particle Hamiltonians $$\oplus H_{\alpha, n}\in \prod_n\operatorname{End}(S^n(V))$$ by an interacting term $H_{mix}$ that mixes particle numbers. The new Hamiltonian will now "create" and "annihilate" particles, and its time evolution will now be a time-dependent unitary automorphism $U_t$ of the space of power series $$\mathfrak{F}_{formal} = \widehat{S}^*(V) : = \prod_n S^n(V).$$ (I don't want to be too particular about the analysis here: in particular, perhaps I need to assume that $U_t$ has some decent convergence properties.)

Now $\mathfrak{F}_{formal}$ is of course the space of power series in a neighborhood of $0$ on the affine space $V_\mathbb{R}$ of fields. (Well, technically on its dual, but it has a Hilbert metric.) So the passage from a single bosonic particle state to a superposition of all its multiparticle states moves the quantum state space from $V$ to power series on $V$. (BTW, I'm surprised to have never encountered this point of view written down in a textbook: instead I sort of pieced it together from how physicists talk. Is this a standard, or even a correct point of view?)

Now my question is why field theory doesn't stop at power series. When mathematicians or physicists talk about the Hamiltonian formulation of field theory, the manifold of fields includes all $C^\infty$ (or something) *global* functions on $V_\mathbb{R}.$ Is this distinction important, and does it come from some specific physical context where one can measure the difference, or is it just an artifact of playing fast and loose with the analysis as physicists are wont to do?