I'm trying to show that the propagator for spacelike separation decays like [itex]e^{-m r}[/itex] and I'm stuck. At some point I hit the integral(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\int_{-\infty}^{\infty} \frac{dk}{\sqrt{k^2 + m^2}} e^{i k r}.

[/tex]

Integration of complex functions not being my forte, I only managed to get to this point using provided answers, but I don't understand the next step. Apparently the above integral is equal to

[tex]

2 \int_0^{\infty} dy e^{-(y+m) r} \frac{1}{\sqrt{(y+m)^2 - m^2}},

[/tex]

where the substitution [itex]k = i(m+y)[/itex] has been made. Now there are several things I don't understand. Firstly, I see how the integrand has changed, but since the integrand is not an even function of [itex]k[/itex] or [itex]y[/itex], how is it possible to change the integration limits to [itex]0[/itex] and [itex]\infty[/itex]? Secondly, shouldn't the integration now be along the complex axis, i.e. shouldn't the upper limit be [itex]i \infty[/itex], due to the [itex]i[/itex] in the substitution? Finally, there is also a bit of text where the author says the integrand has a branch cut going from [itex]i m[/itex] to [itex]i \infty[/itex], and one has to fold the contour of the integral around this cut. I don't see what that cut has to do with anything whatsoever, or exactly what contour he is using.

Could someone help me along with this? I think I understand the rest of the derivation, this is the last link I need.

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# Proving that a propagator decays exponentially for spacelike separations

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