In this question, when I write Fock space, I mean "the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H", as it is described by Wikipedia.
When I began with many-particle-QM in my college-course, we constructed the many-particle space as a tensorproduct of single-particle-states, and then we constructed operators that act on this many-particle-space. From that point of view, it was natural to use the fock space as the space that constains the many-particle-states.
As I understand Quantum Field Theory, everything I do is, I take a classical field theory, and try to quantize it, which means I replace the field variables (like $\phi(x,t)$) with an Operator, then I set the Operator's commutation-relations in a way that makes sense. Afterwards I write the Hamiltonian down, and express it as a sum of some operators, which I afterwards interpret as "Creation and Annihilation"-operators, the create excitation of the quantum field. We then call this excitation "one particle".
My Question: Is the space that the operators act uppon a Fock-state by definition? Or is it just "any Hilbertspace"? If it's the latter: Can we interpret the Hilbertspace to be a Fock-space, after we work with the annihilation and creation operators? For example, If I found some strange EM-Field-space in QED, can I express it as the sum of tensorproducts of "single-particle-photon"-states?
More generaly (and that is the core of my question): Is there something like a "single-particle" in QFT? While in many-particle-QM, the single-particle-QM is the basis, QFT doesn't rely on this concept, that's why I'm asking.