I'm a grad student in math, and I've been trying to learn some physics on the side by taking some classes and reading books. I took a class on quantum field theory last semester that was taught out of Srednicki; the class was very good, but I found myself at the end with a conceptual question that I haven't been able to find an answer to. I've had a little trouble explaining what my question is in the past, but I'm going to do my best. In quantum mechanics, there are states, which are elements of some pre-specified Hilbert space, and observables, which are operators on that Hilbert space. Most treatments I've seen have used this framework, with some adjustments, to set up QFT. The example everyone likes to start with is the (free) Klein-Gordon field. You find that an operator that satisfies the Klein-Gordon equation has to have a Fourier expansion that looks just so, and you call the operators that appear in it creation and annihilation operators for the particles represented by your field. You can get expressions for these operators in terms of your field just by pushing symbols around. You then take your state space to be the smallest Hilbert space on which these operators can act nontrivially and which contains something that's killed by the annihilation operators. This is, as I understand it, called a "Fock space." The states are all given physical interpretations that I feel pretty good about. This all seems to go off without a hitch if you have more than one free field in your theory. Just tensor the Fock spaces together and there you go. But the impression I've been given is that once you introduce interactions, this all goes to hell. Certainly the analysis that allowed us to come up with the forms of the creation and annihilation operators doesn't work anymore. So my questions are: (1) Did we change the state space when we changed the Lagrangian? This isn't how it worked in QM, is it? What's the new state space? (2) Do we still have creation and annihilation operators? Do they act the same as in the free picture? Should I assign them the same physical interpretation? (3) Somewhat relatedly, what is the physical interpretation of the operator that has the same name as the field? It's not an observable, is it? It's not even Hermitian in general. I suspect the problem is that I'm just thinking of this whole thing wrong somehow. Anything anyone could say to set me straight would be greatly appreciated.