How exactly analyticity of S-matrix comes from causality principle?

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Recently I've read that analyticity of S-matrix ($S(k)$, where $k$ corresponds to momentum, may be analytically extended into complex values of momentum) comes from causality principle. How to prove this statement?

This post imported from StackExchange Physics at 2016-09-27 17:01 (UTC), posted by SE-user Name YYY
Not my field, but from the theory of scattering matrices in general linear systems theory (as opposed to particle physics): the Paley Wiener theorem infers from the analyticity of a Fourier transform (which would here correspond to momentum space) certain bounds on the rates of decay of the untransformed distribution. The related Paley Wiener causality criterion infers constraints on the decay rates at infinity of the Fourier transform if the untransformed signal is to be causal (i.e. zero for $t<0$). Also, the S-matrix must be analytic for all complex frequencies with positive real part.
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