Suppose there are two interacting fields $\phi _1 $ and $\phi_2 $. Let $\psi [\phi_1, \phi_2]$ be a functional with the two fields as the input functions and complex numbers as the output, such that the functional integral $\int |\psi|^2 d[\phi_1]d[\phi_2]$ is normalizable to 1.
This means that $\psi$ can serve as the initial state of the system. The time evolution will be given by the functional Schrodinger equation.
The initial state is a vector in a Hilbert space, the Schrodinger equation is bound to map vectors to vectors. Given all this, I see no way the time evolution isn't part of the Hilbert space. But Haag's theorem seems to say this exact thing: The states of an interacting field theory aren't part of the Hilbert space.
Can someone please explain this? Or if I've got Haag's theorem wrong, please explain what exactly does Haag's theorem say in the Schrodinger picture.
Best Answer
Haag's theorem says that the representation of the CCR of a free field cannot be isomorphic to the representation of the CCR of an interacting field.
So if you start with an interacting field, there's no issue: All the states and operators live in the representation of the interacting field, and everything works normally. We're not saying "time evolution doesn't exist".
What Haag's theorem does is draw into question the rigorous meaning of the usual physical treatment of interacting quantum field theories, where we talk about the theory being the free theory in the asymptotic past/future and write formulae that seem to connect the free and the interacting vacuum (e.g. stuff that looks schematically like $\lvert 0\rangle = \lim_{T\to\infty} \mathrm{e}^{\mathrm{i}Ht} \lvert \Omega\rangle$): These formulae that involve both free and interacting states, connected by the action of an operator on "the Hilbert space", make no sense in light of Haag's theorem: The free and interacting states live in different Hilbert spaces, and the field operators that act on the free states are not the same field operators that act on the interacting states - the free and interacting space are different, non-isomorphic representations.
This, in particular, says that "the interaction picture" doesn't exist: In the interaction picture we must assume that the free $H_0$ and the interaction part $H_I$ are operators on the same space - otherwise nothing we write down in the interaction picture makes any sense. Therefore, this is a problem for the standard derivations of the LSZ formula in canonical quantization, which makes heavy use of the interaction picture.
Haag's theorem doesn't really have anything to say about the Schrödinger or Heisenberg pictures.