The Green's function for a scalar field in Euclidean space is(adsbygoogle = window.adsbygoogle || []).push({});

$$(2\pi)^4\delta^4(p+k) \frac{1}{p^2+m^2}$$

however when I continue to Minkowski space via G_{Min}(p_{Min})=G_{E}(-i(p_{Min})) there's seems to be a sign error:

$$(2\pi)^4\delta^4(-i (p+k)) \frac{1}{-p^2+m^2}=(2\pi)^4\delta^4(p+k) \frac{i}{-p^2+m^2}=-(2\pi)^4\delta^4(p+k) \frac{i}{p^2-m^2}$$

where I used δ(-ix)=(1/(-i))δ(x).

The error seems to be that the scaling of the delta function should instead be δ(-ix)=δ(ix)=(1/(i))δ(x).

But how do we know this? For real 'a' it can be argued δ(ax)=(1/|a|)δ(x) on the grounds that δ is postive, but δ(-ix) is not positive as it's not even a real number.

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# A Delta function in continuation back to Minkowski space

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