Klein Gordon ret. Greens function on closed form

[center o bass]

In this note (http://sgovindarajan.wdfiles.com/local--files/serc2009/greenfunction.pdf) the Klein-Gordon retarded green function is derived on the form $$G_{ret}(x − x′) = \theta(t − t') \int \frac{d^3 \vec k}{(2\pi)^3 \omega_k} \sin \omega_k (t − t′) e^{i \vec{k}\cdot (\vec x - \vec x')}$$

where ##\omega_k = \sqrt{\vec{k}^2 + m^2}##. The author then gives an exercise to carry out the rest of the integration and express the Greens function on closed form. But I do not see how to carry the integration out due to the dependence of ##\omega_k## on ##\vec k^2##. Does anyone have any suggestions on how this might be solved, or alternatively know where I can find a full derivation?

 

[dextercioby]

You need to convert to spherical coordinates in momentum space. Then you can do it. It ends up with Bessel functions.