I'm not sure if this post should go here or into the quantum physics forum, but I figure this can't be a bad place to put it. I have a few questions about canonical quantization and quantum field theories with interactions that I hope someone can answer. 1. I've been told that in Klein-Gordon theory, the set of solutions of the classical field equation can be taken to be the Hilbert space of the quantum theory. Is that true, and in that case, does it contain just the one-particle states or the set of all possible states including the vaccuum? 2. Suppose we add a phi^4 term to the KG Lagrangian to get a theory with interactions. What's the Hilbert space of the quantum theory in this case? Is it the set of solutions of the classical field equation (with the phi^4 term)? 3. When we quantize the field in phi^4 theory, we do it by replacing Fourier coefficients of the solution of a classical field equation with creation and annihilation operators, but which field equation is that? The phi^4 equation or the KG equation? (I think I know the answer to this one: It's the KG equation, right?) 4. The canonical quantization procedure is a way to construct an irreducible representation of (the covering group of) the Poincaré group. Noether's theorem tells us how to express the generators in terms of creation and annihilation operators, and the theory of Lie groups and Lie algebras tells us how to use the generators to construct the symmetry operators (e.g. the translation operator). But what Lagrangian do we apply Noether's theorem to when we do this? Is it the phi^4 Lagrangian or the KG Lagrangian? I think I should stop here because the questions I want to ask depend on the answers to these questions.