Checking integration over 4-momentum

[nikol]

Homework Statement

I would like to make sure I am performing the integration correctly. It is a loop integral in QFT:
[itex]\int \frac{d^{4}p}{(2\pi)^{4}}\frac{1}{p^{2}}[/itex]
where p is the 4-momentum, Minkowski space.

Homework Equations

The Attempt at a Solution

I think you must change to spherical coordinates in which:
[itex]d^{4}p=|p|^{3}sin^{2}\theta sin\phi_{1}d|p|d\theta d\phi_{1}d\phi[/itex]
where [itex]|p|=\sqrt{p_{1}^{2}+p_{2}^{2}+p_{3}^{2}-p_{0}^{2}}[/itex]
The integration over the 3 angles will give [itex]2\pi^{2}[/itex] and you have to solve now:
[itex]\int \frac{2\pi^{2}p^{3}dp}{(2\pi)^{4}}\frac{1}{p^{2}}=\frac{1}{8\pi^{2}} \int p dp=\frac{p^{2}}{16\pi^{2}}[/itex]

 

[Asim Riaz]

Now you can solve the last integration, with the limits from 0 to infinity (which is the range of integration for a loop integral). The result is\frac{p^{2}}{16\pi^{2}}=\frac{\infty^{2}}{16\pi^{2}}=\inftySo the solution of the integral is infinite.