Hey everybody, since the previous thread got locked I thought I would open this thread as a place to discuss rigorous issues in quantum field theory, be it on the constructive or axiomatic side of things. I apologize if one is not supposed to start a discussion with posts from old threads, but I thought it would be a nice way to get the discussion going. In the last thread: Yes, that is essentially my understanding. The upshot is that at the end you've constructed the perturbative expansion for [tex]S(g)[/tex] (the S-matrix in finite volume) in a completely rigorous way. Modern work on the Epstein-Glaser approach tries to take the limit [tex]g \rightarrow 1[/tex], to go to infinite volume, although it has proven extremely difficult. Yes, the Hamiltonian is not a well-defined finite operator on Fock space, but in most QFTs it can be shown to be a well-defined finite operator on another Hilbert space. Do you view this as a problem? For the two-dimensional models the best reference is: Segal, I. Notes towards the construction of nonlinear relativistic quantum elds I: The Hamiltonian in two spacetime dimensions as the generator of a C* - automorphism group", Proc. Nat. Acad. Sci. U.S.A. 57, 1178-1183. For the proof that general field theories have signals propogating at the speed of light a good read might be the treatise on algebraic quantum field theory: Araki, H. Mathematical Theory of Quantum Fields, Oxford, 2000. For several nonperturbatively constructed field theories there are actual calculations showing this to be the case. If you want references I can provide them. I understand you are skeptical because there isn't a well-defined finite Hamiltonian on Fock space, however there is one on the correct Hilbert space.