In which space the fields act?

Question

In Quantum Field Theory, as I understood, Quantum Fields are observable valued fields. In that sense, if $M$ is spacetime a Quantum Field is a function $\varphi : M \to \mathcal{L}(\mathcal{E})$ from spacetime into the set of operators on some Hilbert space $\mathcal{E}$, like the ones we use in Quantum Mechanics to describe states of systems. Furthermore, $\varphi(x)$ is an observable for each $x\in M$, so that it is a Hermitian operator.

The one thing I still can't grasp is: what is the space $\mathcal{E}$ on which $\varphi(x)$ acts for each $x\in M$? In Quantum Mechanics, to describe a system we pick a certain $\mathcal{E}$, and describe the system with kets belonging to $\mathcal{E}$.

Here the system is the field, but we are describing the system itself as one operator valued function. It is not clear the space where it acts.

So what is the space on which Quantum Fields acts and furthermore, how this relates to the state space of kets from Quantum Mechanics?

Answer 1

Well, a quantum field operator is actually a map $D(M) \ni f \mapsto \phi(f)$ where $M$ is a spacetime, $D(M)$ is some space of smooth of complex valued functions, usualy $C_0^\infty(M)$, and $\phi(f)$ is an operator defined on the Hilbert space of the theory $\cal H$, more precisely on a dense invariant domain ${\cal D} \subset \cal H$ in common with all choices of $f$. The map $D(M) \ni f \mapsto \phi(f)$ is also supposed to be linear (and sometime continuous with respect to some relevant topology), in view if the heuristic interpretation $$\phi(f) =\int_M \phi(x) f(x) d^nx\:.$$ Further properties depend on the type of field (it could be vector or spinor valued). In particular Hermiticity $$\phi(\overline{f}) = \phi(f)^\dagger|_{\cal D}\:,$$ and further properties related to the equation of motion and causality are usually assumed. When $\phi$ describes a free field, ${\cal H}$ turns out to be a Fock space.

A more abstract formulation eliminates the presence of the Hilbert space and think of $\phi(f)$ as the generator of an abstract $^*$-algebra. The structure of Hilbert space can be reintroduced later after having fixed an algebraic state by means of the GNS reconstruction procedure.