Thanks for the responses everyone, much appreciated. Anna, thanks for the detailed response, I have come to some of the conclusions you have stated, still, it leaves me feeling a bit uneasy as it's not a very satisfying conclusion.
Here is a quote of what is stated in the text I refer to:
"In QFT there are two kinds of operators. One kind is the usual one from NRQM and RQM representing dynamical variables of classical theory, such as the Hamiltonian (energy), the 3-momentum operator, charge, etc. The other kind comprises creation and destruction operators.
The first kind, when operating on an eigenstate, reproduces the original eigenstate, multiplied by an eigenvalue. The second kind changes the state to another state ... note that operators of this kind do not have eigenvalues, since their operation on a state changes that state, rather than reproducing it .. and hence they are generally not observable."
From another chapter:
"Recall ... fields such as phi, psi, A_mu are themselves not observable. They cannot be measured directly (we prove they have zero expectation value in Chapter 7). But properties of fields like energy, momentum and charge are measurable. Our dynamical variable operators, which include number operators, reflect this. They typically have non-zero expectation value
(phi is the scalar field of the Klein-Gordon equation, psi the spinor field of electrons and A_u the vector field of QED)
I guess we get into the whole, "what is real" debate. But, are fields "real", or are they as real as absolute space-time was in Newtonian Physics, in that they're just an approximation, or partial appearance, of some deeper underlying reality? Absolute space-time doesn't exist anywhere in nature, any more than a perfect circle does? Can the same be said for fields?