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Often in QFT you calculate the Feynman amplitude $\mathscr{M}$ from Feynman diagrams. In David Griffiths book, he adds up Feynman diagrams and get $-i\mathscr{M}$. However, in other texts I have seen that the authors add up Feynman diagrams to get $i\mathscr{M}$ without the minus sign. My question: does this sign really matter? Because at the end of the day the physical observable is $|\mathscr{M}|^2 = i\mathscr{M}*-i\mathscr{M}$ so the $i$'s cancel out.

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As long as you are consistent, the overall phase of $\mathscr M$ is irrelevant. Not even the $i$ is really important.

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The overall phase of $\mathcal{M}$ is immaterial because you take the square modulus of it. Just make sure that the reference you study on use the same conventions, or you risk to mess up with signs and $i$ factors in case you have to do some algebraic manipulation before taking the square modulus.

Pay attention in the intermediate passages though: when you deal with independent diagrams, if you compute the two $\mathcal{M}$'s separately and only then you sum them up, you have to be careful on the relative signs and factors. A typical example is Compton scattering where you have a minus sign between the two diagrams!

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