In article #34 of a recent thread about Haag's theorem, i.e., https://www.physicsforums.com/showthread.php?t=334424&page=3 a point of view was mentioned which I'd like to discuss further. Here's the context: I think I see a flaw in the argument above. Suppose I want to know the accelerations of the two particles (or maybe just their relative acceleration wrt each other). In the case of 2 free particles I can certainly ask the question, but the answer is always 0. But for two interacting particles, the answer is nonzero in general. Expressing this in the language of quantum logic and yes-no experiments, the question "is the acceleration 0" always yields "yes" in the free case, but can yield "no" in the interacting case. Similarly, the question "is the acceleration nonzero" always yields "no" in the free case but might yield "yes" in the interacting case. Denote the 2-free-particle Hilbert space as [itex]H_0[/itex] and the 2-interacting-particle Hilbert space as [itex]H[/itex]. Proceeding by contradiction, let's assume that [itex]H_0[/itex] and [itex]H[/itex] are unitarily equivalent, i.e., that any basis of [itex]H_0[/itex] also spans [itex]H[/itex]. Assume as well that the dynamical variable known as "acceleration" corresponds to (densely defined) self-adjoint operators in [itex]H_0[/itex] and [itex]H[/itex], and that the two operators are equivalent to each other up to a unitary transformation. Every state in [itex]H_0[/itex] is a trivial eigenstate of the acceleration operator, with eigenvalue 0. Expressed differently, the acceleration operator annihilates every state in [itex]H_0[/itex]. However, we expect to find states in [itex]H[/itex] corresponding to nonzero accelerations, i.e., states which the acceleration operator does not annihilate. This implies that the basis states of [itex]H_0[/itex] cannot span H, contradicting the initial assumption. Conclusion: [itex]H_0[/itex] and [itex]H[/itex] are not unitarily equivalent. So it's not enough that the same logical propositions (questions) can be asked in both spaces. The spectrum of the corresponding operator must also be considered, and whether both spaces accommodate the full spectrum. Or am I missing something?