I'm trying to evaluate the energy shift in a scalar field described by the Klein-Gordon equation caused by adding two time-independent point sources. In Zee's Quantum Field Theory in a Nutshell, he shows the derivation for a (3+1)-dimensional universe, and I'm trying to do the same for an (N+1)-dimensional universe.(adsbygoogle = window.adsbygoogle || []).push({});

At some point I obtain the integral

[tex]

E = -\int \frac{d^N \vec{k}}{(2 \pi)^N} \frac{e^{i \vec{k} \cdot (\vec{x}_1 - \vec{x}_2)}}{\vec{k}^2 + m^2}

[/tex]

where the two [itex]x[/itex] vectors are the locations of the sources. This integral isn't hard to evaluate for 3+1 dimensions, it can be done by going to spherical coordinates, where you get an integration over [itex]d(\cos{\theta})[/itex], making the exponential doable. However, in N+1 dimensions, going to hyperspherical coordinates, I can't perform any such simplification because the volume element contains not a simple sine, but a [itex]sin^{N-2}[/itex]. I tried this road and it gives me some hypergeometric function. Can anyone see how I might proceed with this integral?

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# Scalar field energy for two delta function sources

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