# Integrating a dot product inside an exponential

1. Aug 4, 2011

### naele

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

2. Relevant equations

3. 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?

2. Aug 4, 2011

### vela

Staff Emeritus
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?

3. Aug 4, 2011

### naele

Thank you, that clears everything up.

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