Integrating a dot product inside an exponential

[naele]

Homework Statement

This is from Peskin & Schroeder p. 14 in case anybody's interested. The function is
$$U(t)=\frac{1}{(2\pi)^3}\int d^3p\, e^{-it \sqrt{p^2+m^2}}e^{i\vec p\cdot(\vec x-\vec x_0)}$$

Homework Equations

The Attempt at a Solution

Essentially you write out the dot product as $p\cdot x'=px'\cos\theta$ and then change to spherical coordinates and then effect a u-sub letting u=cos(theta). What I'm not sure on is why the angle is written with theta (the inclination angle, physicist convention) and not phi. I understand that the angle between two vectors is the same when projected onto a plane, but is that what's going on here? As in, the choice of theta is simply to make it easier for the integration?

 

[vela]

When you switch to spherical coordinates, the axes are set up so that the z axis is aligned with x-x₀. You could, in principle, orient the axes differently. In that case, the angle between the two vectors will be a function of both θ and Φ, but why make things complicated?

 

[naele]

Thank you, that clears everything up.