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Integrating a dot product inside an exponential

  1. Aug 4, 2011 #1
    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
    [tex]
    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)}
    [/tex]

    2. Relevant equations



    3. The attempt at a solution

    Essentially you write out the dot product as [itex]p\cdot x'=px'\cos\theta[/itex] 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. jcsd
  3. Aug 4, 2011 #2

    vela

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    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?
     
  4. Aug 4, 2011 #3
    Thank you, that clears everything up.
     
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