The propagator in 4-dimensions for a free scalar field has the form:(adsbygoogle = window.adsbygoogle || []).push({});

Δ(x,0)=Θ(t)A(x,t)+Θ(-t)B(x,t)

where Θ is the step function (eq 23 of Zee's QFT book, 2nd edition). He then makes the claim that for spacelike x, one can set t=0, and define Θ(0)=1/2.

The going through all the math, he derives:

[tex]\Delta(x,0)|_{t=0}=-\frac{1}{8 \pi^2 r} \int^{\infty}_{-\infty} \frac{dk k}{\sqrt{k^2+m^2}}e^{ikr} [/tex]

This looks to me like it fails to converge as at large k is just oscillates without decreasing amplitude or period.

But then he does this:

[tex]\Delta(x,0)|_{t=0}=-\frac{1}{8 \pi^2 r} \int^{\infty}_{-\infty} \frac{dk k}{\sqrt{k^2+m^2}}e^{ikr}=

\frac{i}{8 \pi^2 r}\frac{\partial}{\partial r} \int^{\infty}_{-\infty} \frac{dk }{\sqrt{k^2+m^2}}e^{ikr}[/tex]

and now the integral converges by Jordan's lemma.

Is this to be viewed as an analytic continuation? The final results seems to be correct as he gets a Bessel function. But this doesn't seem to be correct mathematically unless he's claiming a continuation.

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# Zee's QFT book: equal time propagator

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