## Coulomb’s Law in (n+1) dimensions

The pairwise interaction energy in spatial dimensions is given by

To evaluate this -dimensional Fourier integral, we must employ hyperspherical coordinates. The volume element is

The angles range from to and ranges from to . Writing , the integral can be written as

The constant is equal to the product of integrals of the form . The number of such integrals depends on the dimension . In two and three dimensions, . In more than 3 dimensions, there are such terms, for (the integral over produces a , which we’ve already factored out). The value of is for 2 dimensions, and is for 3 and higher dimensions.

The integral over produces a regularized confluent hypergeometric function . Specifically, for , the integral over produces

(As a check, for , this becomes , which is what we obtained for 2 spatial dimensions in a previous post.)

The resulting integration over is rather tricky. At this point, I don’t know if its possible to do it by hand, so I am using Mathematica to perform it.

**Edit: **a few hours later..

So, it seems this integral does not have a closed form representation, or at least not one that Mathematica can find.

I’ve found a closed form solution for this problem, though I needed to use Google to find an integration “trick” that I had never seen before (seems somewhat specific to this problem):

Using this and completing squares in the exponent I got

,

matching the results that were previously obtained for D = 2 and D = 3.

mchouzaDecember 21, 2010 at 17:57

Okay, sounds interesting. I would like to see your full working, as I’m not too smart with the math 😛

VivekDecember 22, 2010 at 08:38

It’s at http://mchouza.wordpress.com/zee-qft-part-i/#I.4.1.ii

mchouzaDecember 22, 2010 at 08:49