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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1Why 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. – Wakabaloola Dec 27 '18 at 17:34
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This is what I often heard. See for example the discussion here https://physics.stackexchange.com/questions/4331/is-the-s-matrix-the-only-quantum-field-observable – thone Dec 28 '18 at 00:47