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The S-matrix (scattering matrix) relates the initial state and the final state of a physical system undergoing a scattering process in quantum mechanics and quantum field theory. It is the unitary matrix connecting asymptotic particle states in the Hilbert space of physical states (scattering channels).

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Optical theorem Peskin and Schroeder

From your comments it looks like you're mainly confused about the $E_1 E_2 | v_1 - v_2 |$ prefactor, so I'll try to be very explicit about that part. Start from Eq. (7.49) and take the initial and fin …
soundofthepolice's user avatar