Is there a physical reason that fields in QFT are globally defined?

Question

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?

Answer 1

You are describing what is commonly known as second quantization (as you probably already realize). In a nutshell, the main mathematical statement behind second quantization is the following: the algebra $\mathcal{A}_{particles}$ generated by creation/annihilation operators acting on the Fock space $\mathfrak{F}_{particles}$ is isomorphic to the algebra $\mathcal{P}_{field}$ of quantized polynomial observables on the classical phase space of some free field (roughly, a collection of infinitely many harmonic oscillators), and more over the representation of $\mathcal{P}_{field}$ on the Hilbert space $\mathfrak{H}_{field}$ of oscillator states is isomorphic to the representation of $\mathcal{A}_{particles}$ on $\mathfrak{F}_{particles}$. Now, what this "free field" is is determined by the structure of the single particle state space $V$ and Hamiltonian $H$. This field may have a spacetime interpretation, depending on what that structure is, but it may also not. You did not get to the equivalence itself in your description, but it is implicit in the construction of the Fock space.

Historically, this is how the focus shifted from particles to fields. The dictionary could be expanded further and include interactions. When applied to the example of QED (quantum electrodynamics), where initially electrons were quantized as particles and photons as fields, second quantization shows that equally well both electrons and photons can be quantized on equal footing from the start (as fields). One can carry this dictionary quite far, without deciding that either particles or fields are a preferred description. But eventually, the balance shifted towards fields: only those particles and interactions that correspond to local relativistic fields seem to be observed in nature, non-unitarily equivalent representations of $\mathcal{P}_{field}$ require drastic changes to the particle picture, non-perturbative (in the fields) phenomena are not (at least not easily) covered by the dictionary and yet there are good reasons to consider them (phase transitions).

The point of the historical summary is that, given the shift in focus from particles to fields, the mathematical questions change. Namely, one is not concerned a priori with a particle Fock space, rather one is concerned with the quantization of a field theory as an infinite dimensional classical system (either via Hamiltonian or path integral methods). Now, one no longer has a reason to restrict the polynomial field observables $\mathcal{P}_{field}$, other than convenience or technical necessity. It becomes a mathematical challenge to describe as large an algebra of observables $\mathcal{A}_{field} \supset \mathcal{P}_{field}$ as is reasonable. One can think of $\mathcal{A}_{field}$ as a quantization of $C^\infty(V_\mathbb{R})$, which you were wondering about, for a corresponding reasonable interpretation of $C^\infty$ on an infinite dimensional space. If such a quantization of a field system succeeds, and a particle description is possible and desired, then on can simply restrict this quantization to the polynomial observables $\mathcal{P}_{field}$ and use the second quantization dictionary.

P.S.: If you insist on describing Fock space as $\mathfrak{F}_{formal}$ as formal power series on $V$, then it is not a Hilbert space. If you would like Fock space to be a Hilbert space, you should restrict to those series that have finite norm, thus defining $\mathfrak{F}_{particles} = \bigoplus_n S^n(V)$ by the usual direct sum of Hilbert spaces. The connection of this description of Fock space to power series (or at least to polynomials) on $V$ is well-known, though it need not be mentioned in textbooks unless it's needed for some specific remarks. But the connection is mentioned already on Wikipedia.