In view of Haag's Theorem, it seems the Hilbert spaces of a free theory and an interacting theory are not the same. Though it seems very believable, I could not find a result that states that this is the case for all theories.
Intuitively, it seems obvious that the Hilbert space of an interacting theory involving distinct fields is not the same as any of the Hilbert spaces of the individual free theories. However, what if we consider the scalar field with $\phi^4$ interactions? In this case I was unable to convince myself that the Hilbert space of the interacting theory must be different than the Hilbert space of the free $\phi$ field.
Is it always the case that Hilbert space of an interacting theory is different than any free theory?
Best Answer
Yes, the Hilbert space for a free theory and an interacting theory are different. There can not exist a unitary isomorphism that preserves the algebra of operators. (You need the caveat about preserving the algebra of operators, because all separable Hilbert spaces are unitarily isomorphic in many ways.)
In fact, Haag's theorem implies something stronger: any change in any one of the coupling constants gives rise to a different Hilbert space. So, not only do free & interacting theories get different Hilbert spaces, but also you get different Hilbert spaces for different values of the coupling constants $g \neq g'$. Even crazier, this is true of the quadratic coupling constants, aka, the mass terms. Different masses lead to different state spaces, even in free theories.