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From the discussion in some posts example1, example2, we know that the S-matrix is the residue of the corresponding Green's function. On the other hand, S-matix is a physical observable in QFT, but Green's function is not gauge invariant in a gauge theory. How to fix this inconsistency?

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    $\begingroup$ Why do you say the S matrix is a physical observable? In experiments we measure cross sections, decay rates, momenta, energies, etc., but the S matrix is not something one measures directly, right? It is not even a hermitian operator. $\endgroup$
    – Wakabaloola
    Dec 27, 2018 at 17:34
  • $\begingroup$ This is what I often heard. See for example the discussion here physics.stackexchange.com/questions/4331/… $\endgroup$
    – thone
    Dec 28, 2018 at 0:47

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