# Does vacuum imply Fock space? Can we really do QFT without Fock space?

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.)

1. 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.

2. 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?