In the traditional presentation, Quantum Fields are usually presented as operator valued fields defined on spacetime, in the sense that $\varphi : M\to \mathcal{L}(\mathcal{H})$ for some Hilbert space $\mathcal{H}$.
On the other hand, recently I've read that Quantum Fields should be properly defined as operator valued distributions, and I believe that if I understood correctly this has to do with a way to deal with the problem of infinities that is usually associated to QFT. The only difference is that now we have a space of functions $\mathcal{D}(M)$, so that $\varphi : \mathcal{D}(M)\to \mathcal{L}(\mathcal{H})$.
Anyway, independently if we define fields as functions of events $\varphi(x)$ or functionals on functions $\varphi(f)$, the point is that $\varphi$ returns one operator.
Now, in Quantum Mechanics, there are two main kinds of operators which have a very clear meaning. The observables are hermitian operators, and those represent physical quantities by one postulate of the theory.
So whenever we have one observable $A$ is because we know of a physical quantity $A$ and the operator represents the quantity so that its spectrum is the set of allowed values and its eigenkets are the states with definite values of the said quantity. Everything is perfectly clear.
The other kind of operator I speak of are the unitary operators that usually represent transformations acting on the system, like translations, rotations and so forth. The meaning is also clear.
Now, in QFT, provided we have a field $\varphi: \mathcal{D}(M)\to \mathcal{L}(\mathcal{H})$, $\varphi(f)$ is one operator. What is the meaning of said operator?
I've heard it is one observable. But if it is one observable, what physical quantity it represents?
It is not at all clear for me what the operators $\varphi(f)$ represent, nor what is their action. And finally, how all this connects to the traditional Quantum Mechanics?
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