By existence of Fock space, I refer to existence of creation and annihilation operators. And by vacuum, as I am restricting to flat spacetime, I am referring to Poincare-invariant vacuum state. (Different vacua exist even under flat spacetime, but I am restricting to this particular vacuum state.)
Does existence of vacuum state $|0\rangle$ in QFT imply existence of Fock space? My understanding is that the answer is no, but I could not find a definite source or think of a proof.
If the answer is no, then technically we should be able to do QFT calculations without relying on Fock space. In interacting theories for example, Fock space does not exist. And if vacuum state exists for such theories, then we should be able to rely on Green's function to get full pictures of QFT. (My understanding is that we can get propagator, correlation function, etc. out of Green's function) And solving for Green's function does not require Fock space picture. Thus, does this mean that we can do QFT without Fock space? Why then is absence of Fock space considered so problematic?
Adding to the original post:
Here is my attempt to understand things. It is likely to be wrong, but I shall try. So suppose that we have an interacting QFT. For non-linear interacting QFT, Green's function as we say in mathematics, no longer makes sense. So that means that we cannot use (mathematical) Green's function techniques to compute correlation function. That means resorting back to creation and annihilation operators, which we do not know, plus Fock space of creation and annihilation operators does not exist, at least the one that is Poincare-invariant. Thus, even if in free theory we seem to be able to use Green's function to define correlation function and use it to complete descriptions of a QFT, one cannot do that for interacting theory.
Is this the right way of understanding things? Technically, it seems that as long as we get correlation functions, we should be able to compute anything. I think this underlies many of axiomatic approaches as well.