My understanding is that in general, operators -- corresponding to observables -- act on a state (itself a member of an infinite-dimensional Hilbert space), and the eigenvalue is the value in that state (at least, if it's a pure state). To get an expectation value, you take the dot product of the final and the operator of the initial state. That much at least is put forward in pretty much every text on QM of any level, and is basically given as received gospel, and it's pretty clear when working with "first quantized" operators, like position, energy or linear/angular momentum. Then, with second quantization, the state turns into a Fock space, and the operator (or state, depending on the representation) is defined over all of space-time. Even there, the meaning of the Number operator (destruction operator followed by creation operator) seems pretty clear: N(p, x, y, z, t) [ state > has an eigenvalue telling how many particles of momentum p there are at xyzt. But then when it comes to creation/annihilation operators, it seems like every text switches from an operator being an artifice for getting a measurement into something that actually "operates" on the state -- e.g. Create(p, x, t, z) [0> is described as "adding a particle of momentum p at position xyzt to the vacuum state"...so it's not clear at all what it "measures", if anything. My gut says that the expectation value should be the probability that a particle of momentum p is created/removed at xyzt, but I can't find anyone saying that...and even of that were the case, it seems like the operator would be redundant, since the final state would have the added(removed) particle wrt the initial one, so shouldn't just the dot product of the two give the amplitude of going from one to the other? Now, I did see a note in the most recent text I've flipped through (Duncan's Conceptual Foundations -- great stuff btw) that the creation/annihilation operators were non-hermitian...but even if that meant they were useless for measuring, that doesn't explain the entirely new interpretation.