Fields: Fundamental and Physical, yet Unobservable?

Also, I got caught up on the notion of how something can be un-observable, yet also physical. And, how we can call fields fundamental, while at the same time fields are undetectable, even in principle?

Quantum field theory is a mathematical model, with creation and annnihilation operators operating on quantum fields . These do not have to be the fields of fundamental particles but cover many problems and are called effective field theories. In this sense the "field" on which the creation and annihilation operators operate can be various ,( starting from the harmonic oscillator ground state).

In particle physics the ground state is the wave function of the free particle solution for the respective particle fields from the table of elementary particles.

Wave functions are not directly observable. It is only their complex conjugate squared which gives probability density distributions.

A concrete example is the harmonic oscillator , using its solutions one can define a field theory for creation and annihilation operators:

This is astonishingly convenient. It presages a form of operator algebra that proceeds without ever looking at the form of ψ(x) and does not require direct evaluation of integrals

The fields existence is implied by the creation and annihilation operators on these ground states, which will describe the particles, and particles are observable.

To sum up, mathematical models fitting observations have components which are not directly observable, but are necessary for calculating measurable distributions. In particle physics calculations are made with Feynman diagrams which describe the perturbative expansion of crossections etc.

Fundamental but not physical: The (x,y,z) coordinates mapping a terrain are unobservable, a mathematical concept, nevertheless the map is described by the functions of (x,y,z), and the (x,y,z) concept is the context for having a map. Particle fields provide the fundamental context on which the physical observables of particles can be modeled.

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?