# Integral in Commutator of Scalar fields

1. Jun 16, 2012

### praharmitra

So, in the calculation of $D(t,r) = \left[ \phi(x) , \phi(y) \right]$, where $t= x^0 - y^0,~ \vec{r} = \vec{x} - \vec{y}$ you need to calculate the following integral
$$D(t,r) = \frac{1}{2\pi^2 r} \int\limits_0^\infty dp \frac{ p \sin(p r) \sin \left[(p^2 + m^2)^{1/2} t \right]} { (p^2 + m^2 )^{1/2}}$$
For $m=0$, the integral is simple. We get
$$D(t,r) = \frac{1}{4\pi r} \left[ \delta(t - r) - \delta(t + r) \right]$$
I even know what the answer for $m \neq 0$. I have no idea how to calculate it though. Any help?