Is an interacting QFT Hilbert space a physical particles Fock space?

Question

1. There are "Lectures on Quantum Field Theory" by P.A.M. Dirac, in which he claims that QFT state space is not a separable Hilbert space.
2. Also, I have seen some research papers (in axiomatic QFT), which claim that there is separable Hilbert space, describing field state at any fixed time $t$, and even there is separable Hilbert space for any finite time interval $[t_1, t_2]$, but there is no such thing (separable Hilbert space) for infinite time interval $(-\infty, \infty)$.

3. I have seen some research papers about "clothed particles representation". They claim, that field state (at finite time) cannot be described in terms of on-shell particles only. So I conclude, the "S-matrix"' Hilbert state is not "full" enough, despite it is full as Hilbert space in mathematical sense.

4. But there are books and papers, about QFT and S-matrix, for example, "Against particle/field duality: asymptotic particle states and interpolating fields in interacting QFT" by J. Bain, which talks about interacting QFT Hilbert space, and which even has a proof that such space is equal to asymptotic particles Hilbert space (S-matrix theory Hilbert space).

Thus, I cannot understand, if the S-matrix Hilbert space is only a part of/an approximation to "real" QFT space, but good enough to practical reasons (especially for scattering experiments, where only asymptotic states are physically observable), or it is "real" ("fundamental") space, and Dirac was wrong, when he said there is no such one.

Answer 1

Haag's theorem says that the Hilbert space on which interacting relativistic quantum fields can be defined cannot be a standard Fock space. The finite time dynamics happens in this space, hence not in a Fock space.

Haag-Ruelle theory on the other hand says that the space of asymptotic particles of a relativistic quantum field theory is a Fock space. The latter space is the space on which the S-matrix acts. Thus the S-matrix is a unitary operator on Fock space.

The Hilbert space of an interacting theory and the Hilbert space of the space of its asymptotic particles are therefore two different things.

Clothed particle theories work on the perturbative level only, where the structural differences between the free (Fock) Hilbert space and the interacting (non-Fock) Hilbert space is not visible.