Consider a QFT consisting of a single, hermitian scalar field $\Phi$ on spacetime (say $\mathbb R^{3,1}$ for simplicity). At each point $x$ in spacetime, $\Phi(x)$ is an observable in the sense that it is a hermitian operator (operator distribution) on the Hilbert space of the theory, but is each such operator observable in a stronger, more physical sense? Is there an experiment one could hypothetically perform to measure the value of such a field at a given spacetime point?
This is one of those questions I glossed over while learning QFT, but now it's bugging me. In particular, I think this point is central in preventing me from understanding certain basic assumptions in QFT such as microcausality which I also never really think about anymore.
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