LSZ formula V.S. Long-range force This site may already have a similar question, but I haven't been able to get a convincing answer, so let me ask you again.
My question is “ Why does LSZ formula apply to theories that include long-distance forces, and give good results?”
LSZ assumes “non-interacting particles” (of course without self interacting) at the far past and future, so long-range forces like EM interaction (which is mediated by photons) are originally out of theory because we cannot neglect interactions between particles regardless distance of them. However, LSZ is applied to QED and it succeed greatly.
Such a situation around LSZ looks strange to my eyes. If anyone has a good interpretation to justify or resolve this, would you please let me know?
 A: EM may be "long-range", but the asymptotic states in the LSZ formalism live in the asymptotic future/past where the particles are infinitely far apart, so it doesn't care about the range of the involved forces as long as the forces go to zero as the distance goes to infinity.
Now, the question might shift to why it is a viable assumption that this might ever be a good approximation for the finitely-far apart particles in reality. Note that the LSZ formalism when used to derive scattering amplitudes usually deals with pure momentum states, which are infinitely delocalized - but real particles are not infinitely delocalized, even when we accelerate them to high momentum, they are still confined to being at least inside the collider. Real particles are at least somewhat localized, i.e. they are wavepackets that are some superposition of pure momentum states that are localized both in momentum and position space and when the position wave functions of these particles after collision no longer overlap to a significant degree, then they do not behave noticably different from the "asymptotic" initial and final states of LSZ that don't interact at all.
