My understanding of Haag's theorem (see link below) is that there is a mismatch between the Hilbert spaces of free and interacting particles. The argument seems to be that we require both the free and interacting vacuum states to be invariant under Poincare transformations. Now since the entire Hilbert space of the free particle is built by acting on the vacuum with creation operators which are not invariant under Poincare transformations, then our only choice for the interacting vacuum is to have it be proportional to the free vacuum. Here we get a contradiction, since the free vacuum cannot be an eigenstate of both the free Hamiltonian and the full Hamiltonian. http://philsci-archive.pitt.edu/archive/00002673/01/earmanfraserfinalrevd.pdf" [Broken]) All that's good and well, except for one thing. Why do we require the free vacuum to be invariant under Poincare transformations? The free fields aren't the physical ones, the interacting ones are. I see no reason why the non-physical free vacuum can't transform non-trivially under a translation or a rotation. If, instead, we require only that the interacting vacuum be invariant under Poincare transformations, the mismatch between the Hilbert spaces disappears.