I'm just starting to dip my toes into quantum field theory using Klauber's "Student friendly quantum field theory." I just got through the section on free spin-0 waves and it mostly makes sense: we solve the Klein-Gordon equation to get an operator which is a function of space and time, i.e., a quantum field. The math steps make sense to me. What's confusing me is, why should I care about this quantum field we just worked so hard to find? How is it connected to reality? In non-relativistic QM, the wavefunction we solve for has a very definite meaning in terms of experiment: it gives the probability of different outcomes. But this quantum field is made up of raising and lowering operators, and it doesn't even need to be Hermitian. Is it observable, or is it some sort of handy mathematical tool, or what?