Coordinates and casimir effect

[epislon58]

Hello, I am reading this paper on the casimir effect and I am failing to understand where the 1/(2âˆ)^2 comes in and how the polar coordinates are converted to Cartesian. The equations are (3.23) and (3.24).

http://aphyr.com/data/journals/113/comps.pdf

Thank you!

 

[ChrisVer]

it's the momentum space integration, and in general the momentums are normalized as such. In fact it's the momentum that corresponds to some box in your phase diagram.
look then at page 7

 

[epislon58]

for the same equations as well, he goes from w.w^2s to (n(pie)/a)^3-2s. how does he come up with the 3-2s? Thank you