What is the 'state space' of a quantum field theory called?

Question

This is just a terminological question, not a question about reality or mathematics.

I often want to talk about state spaces in quantum field theory. For example the space of [all possible vector states in] a free scalar quantum field.

I have been told in a comment on my other question this this object is not called a "quantum field", because a "quantum field" is an operator field (or a space of operator fields). I know an operator is a kind of mapping, and takes an input. The entity I want to be able to talk about is not a mapping, it is like a vector (or it is a vector), it just exists and does not act on something else. What is the standard name for it?

Edits:

I hope this is quite a clear example: I may want to talk about the 'state of photons in the universe'. I have been told this cannot be called a quantum field, because the quantum field is an operator not a state. So I presume this cannot be called the photon field or similar? Obviously it is not a quantum field theory either because it is not a theory, it is physical. So I don't know what to call it. I have never seen a phrase like "state of photons" or "space of photon states" in use.

I think it is fair to say what I am looking for is a term that means "Hilbert space equipped with a quantum field theory interpretation" (or physical entity represented by it) based on the helpful comments and answers.

Answer 1

The space of states of quantum field theory is a Hilbert space (or, if you want a space where every element is really a different state, the corresponding projective Hilbert space, since vectors that just differ by scalar multiplication represent the same state) just like in ordinary quantum mechanics.

Just like the classical observables of position and momentum get promoted to operators in quantum mechanics that act on such a space of states, quantum field theory promotes the classical fields of a field theory (e.g. electromagnetism) to operators acting on a space of states. The field is not the state, just like the position operator is not the state.

There is no such thing as a "space of states of a quantum field". The quantum field is the operator, not the state. There is a space of states of a quantum field theory, it is determined by all the fields of the theory and their interactions, not just by a single field, exactly how in non-field quantum mechanics there is no "space of states of position", just the single space of states both position and momentum act on.

Answer 2

I think this is a good question because it goes to the heart of something rather confusing in the terminology: the fact that the words 'quantum field' are attached to an operator-valued quantity such as $\hat{\phi}$ and yet people also say that the vacuum state such as $|0\rangle$ or $|\Omega\rangle$ is a state 'of the field(s)'. But as answer from ACuriousMind points out, an operator is not itself a state, nor does it 'have a state', but rather it acts on states in some space (here, Hilbert space).

But now the question arises, what is a state such as $|\Omega\rangle$ a state of? The standard terminology is to say it is a state of a field, or of a collection of fields. So the standard terminology is inconsistent and confusing.

I think a good way around this is to refer to an operator such as $\hat{\phi}$ or $\hat{\bf E}$ not as 'the field' but as 'the field amplitude'. Then you can say things like 'the field amplitude acting on the field state' and this form of words maps unambiguously onto the mathematical quantity $$ \hat{\phi} | \psi \rangle. $$ (This could be compared to a quantity such as $\hat{p} |\psi\rangle$ in ordinary quantum theory, which can be put into words as 'the particle momentum acting on the particle state', only of course we normally say 'the momentum operator acting on the state').