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Integral in Commutator of Scalar fields

  1. Jun 16, 2012 #1
    So, in the calculation of [itex] D(t,r) = \left[ \phi(x) , \phi(y) \right] [/itex], where [itex] t= x^0 - y^0,~ \vec{r} = \vec{x} - \vec{y} [/itex] 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 [itex]m=0[/itex], 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 [itex] m \neq 0 [/itex]. I have no idea how to calculate it though. Any help?
  2. jcsd
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