Do the Gell-Mann and Low Theorem and Haag's Theorem contradict one another? I was wondering if these two theorems do not conflict with one another in some sense. It sees as if one allow us to move from the free to the interacting theory, while the other forbids it.
 A: Haag's theorem was originally formulated for QFT in continuous space, with a small number of axioms to define what "QFT" means. Within that axiomatic system, it's a theorem. It says that if you change the value of a coupling constant in the Lagrangian, say from zero to a finite value, then every state in the vacuum representation of one of those theories is orthogonal to every state in the vacuum representation of the other theory. Therefore, within that strict axiomatic system, "the interaction picture does not exist" (but see below).
The Gell-Mann and Low theorem is a valid theorem, but the version of QFT for which Haag's theorem was originally formulated doesn't satisfy the assumptions of the Gell-Mann and Low theorem. They don't contradict each other, because they make different assumptions. (I don't know much about the history. Maybe the original formulation/application of the Gell-Mann and Low theorem wasn't really valid. My point below is that it can be made valid by changing how it's formulated/applied, without changing the basic idea.)
Maybe the real question here is why the Gell-Mann and Low theorem can be applied effectively in QFT, where it shouldn't apply and where Haag's theorem says it can't work. The answer is that for applications to experimental high-energy physics, we don't need to define QFT in continuous space. Even if we can (which is rare), we don't need to. We can define it in discrete space instead, with a finite number of points. We can make the discretization so fine that it might as well be continuous, except that the strict version of Haag's theorem no longer applies, and the Gell-Mann and Low theorem does apply. That's one way to understand why it works.
Even though Haag's theorem as originally formulated doesn't apply to such discretized QFTs, the spirit of Haag's theorem still applies. It says that if you change the value of a coupling constant in the Lagrangian, then every state in the vacuum representation of one of those theories is very nearly orthogonal to every state in the vacuum representation of the other theory, approaching complete orthogonality as the continuum limit is approached. One implication of this is that even though small-coupling perturbation theory (as enabled by the Gell-Mann and Low theorem) works great for correlation functions (scattering amplitudes, etc), it doesn't work well at all for states. We can calculate QED scattering amplitudes very precisely using small-coupling perturbation theory, but we can't use that same approach to express the theory's true single-particle states in terms of the field operators when the coupling is nonzero, even if the coupling is tiny, because the perturbative approximation produces states that are nearly orthogonal to the true single-particle states. People have proposed using light-front quantization as a more tractable approach to analyzing states, but this approach comes with its own difficulties, as explained in Collins (2018), "The non-triviality of the vacuum in light-front quantization: An elementary treatment" (arXiv:1801.03960), section IV.
