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This question is in reference to eq. (2.52) on the bottom of page 27 in Peskin and Schroeder.

To evaluate the Klein-Gordon field propagator along a spacelike interval we wrap the contour around the branch cut at $p=+im$. My question is (possibly a mathematical one) why it is necessary to move the contour in the first place? The function is analytic along the real axis, which is what we are originally integrating along.

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All Peskin and Schroeder want to show is that the integral is non-zero. You could certainly try to exactly compute the integral over the original contour. However, I imagine that it's probably a lot easier to use the new contour wrapped around $p=im$. In any case, along the new contour, it is immediately clear that the integral is non-zero because the integrand is always positive. This is not clear along the original contour.

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  • $\begingroup$ Ah I see, it is just more instructive to use that contour. Thank you :) $\endgroup$
    – Charlie
    Sep 25 '20 at 14:36

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