Quote by Dickfore
I must admit this is the first time I hear of a Hilbert space being interacting or not. You may claim that the basis vectors constructed as a direct product of singleparticle kets of arbitrary power are eigenkets of the Hamiltonian of the system only when the theory is noninteracting, but the space spanned by them is independent of the basis, and, at least in principle, one should be able to diagonalize even the interacting Hamiltonian acting on kets in this Fock space.
In my opinion, the most important sentence in your post is the bolded one. If there are products of more than 2 field operators in the Lagrangian, then this is necessarily an interction.

What? Let's go to the context used by M.Y. Han book "A Story Of Light: A Short Introduction To Quantum Field Theory Of Quarks And Leptons".
I'll quote only the relevant passages and omit the math and other detailing part:
"The quantization of fields and the emergence of particles as quanta of the quantized fields discussed in Chapter 9 represent the very essence of quantum field theory. The fields mentioned so far  KleinGorden, electromagnetic as well as Dirac fields  are, however, only for the noninteracting cases, that is, for free fields devoid of any interactions, the forces. The theory of free fields by itself is devoid of any physical content: there is no such thing in the real world as a free, noninteracting electron that exerts no force on an adjacnet electron. The theory of free fields provides the foundations upon which one can build the framwork for introducing real physics, namely, the interaction among particles."
[omitting 2 pages of calculations and details]
"Quantum field theory for interacting particles would have been completely solved, and we could have moved beyond it. Well, not exactly. Not exactly, because no one can solve the highly nonlinear copuled equations for interacting fields that result from the interacting Lagrangian density obtained by the subtitution rule. Exact and analytical solutions for interacting fields have never been obtained: we ended up with the Lagrangian that we could not solve!"
[omitting a page]
"At this point, the quantum field theory of interacting particles proceeded towards the only other alternative left: when so justified, treat the interaction part of the Lagrangian as a small perturbatoin to the free part of the Lagrangian"
[I won't quote other paragraphs anymore. Just see it in amazon free page preview if necessary]
Do you know the part about "subtitution rule" he was talking about? Any relation to it that you are talking about? He basically said the subtitution rule couldn't be solved. And we are left only with perturbation, and we know it is seems ad hoc. Therefore Quantum Field Theory seems to be flawed. How then could they arrive at the right theory of Quantum Gravity with such a flawed foundation?!