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There is a statement in optical theorem (From Peskin & Schroeder):

It is easily checked (in QED, for example) that each diagram contributing to an S-matrix element $\mathcal{M}$ is purely real unless some denominators vanish, so that $i\epsilon$ prescription for the treating the poles becomes relevant.

This statement comes from Peskin & Schroeder, pp.232. I guess we can verify this statement simply by looking into a general Feynman diagram and count how many propagators and how many vertices such that we know how many $i$ will be there. But I wonder whether there is a more general proof for this statement. Here is a hint:

S-matrix is defined as $$\langle\vec{p}_1\vec{p}_2...|S|\vec{k}_1\vec{k}_2...\rangle\equiv {}_\mbox{out}\langle\vec{p}_1\vec{p}_2...|\vec{k}_1\vec{k}_2...\rangle_\mbox{in}=\lim_{T\rightarrow\infty}\langle\vec{p}_1\vec{p}_2...|e^{-iH(2T)}|\vec{k}_1\vec{k}_2...\rangle.$$

In path-integral, S-matrix will be $$S\sim\lim_{T\rightarrow\infty(1-i\epsilon)}\int\mathcal{D}\phi e^{i\int d^4x\mathcal{L}}=\mbox{series of expansion}.\tag{1}$$ And according to $S=1+iT\sim 1+i\mathcal{M}$. The hope is to prove that $\mathcal{M}$ is purely real unless some denominators vanish in the series of expansion.

Would somebody give a complete proof? Thank you very much.

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  • $\begingroup$ While reading P&S I assumed that they simply made a qualitative observation, since residues always appear multiplied by $2\pi i$. $\endgroup$ – Styg Feb 12 at 10:28

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