Independence of $S$-matrix of $\xi$-gauge in QED

On page 298 in Peskin and Schroeder, the authors attempt to argue that the $$S$$-matrix should be independent of the $$\xi$$-gauge in QED. However, I don't understand their argument, in particular the statement

"Since the $$S$$-matrix is defined between asymptotic states, we can compute $$S$$-matrix elements in a formalism in which the coupling constant is turned off adiabatically in the far past and far future".

I don't see what this statement has to do with the independence of the $$S$$-matrix on $$\xi$$. But more importantly, I don't understand why their statement is even correct. Fermions never stop interacting with the photon field, by virtue of their electric charge, as the single-particle four-momentum eigenstates for a fermion in the interacting theory is different than in the free theory. Roughly the fermions are always surrounded by a "photon cloud" (appropriately given by the Feynman diagrams contributing to the exact two-point correlator in the interacting theory, which, by the way, is not gauge-invariant). Hence, a fermion doesn't only interact with an incoming real photon, but it also continually interacts with virtual photons. If you remove the coupling constant (by which I assume they mean the electric charge), then even the single-particle fermions states (and so also the asymptotic states) would be fundamentally altered.

Could somebody provide a formal proof/argument that the $$S$$-matrix in QED is independent of choice of $$\xi$$-gauge? Ideally, I would love to see an argument that uses the formal definition of $$S$$-matrices from the LSZ formula, but I would be open to other approaches as well.

Edit: Some have commented in favor of the claim that we can turn off the coupling and turn the theory into the free theory in the far past and future (as claimed in Peskin and Schroeder). But I miss an argument why this implies the $$\xi$$-independence of the S-matrix. If somebody knows of such an argument, I would be happy about that approach.

• I only comment on the adiabatic switching on/off of the interaction. OP is correct in saying that the real electron is dressed by all the virtual whatever. The problem is, we do not know what explicitly that physical state is. The knowledge amounts to exactly solving QED. The best we can do is to start from the single-particle state of the free theory, adiabatically turn on the interaction, and hope that the free eigenstate adiabatically evolves into the exact physical state that we desired. Commented Jul 14 at 8:36

As far as I know, the rigorous proof of the LSZ formula fails in QED because of the photon cloud. Nevertheless, it still seems to work! We just have to physically assume that at $$t\to\pm\infty$$ the theory is free. I fear that this is also the case here.