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Fourier Transform of propagators

  1. Oct 1, 2013 #1
    I've been assigned the following homework:
    I have to compute the spectral density of a QFT and in order to do so I have to compute Fourier tranform of the following quantity (in Minkowsky signature, mostly minus)

    [itex]\rho\left(p\right) = \int \frac{1}{\left(-x^2 + i \epsilon x_0\right)^{\Delta}} e^{i p \cdot x} d^4 x [/itex]

    Using residual theorem I've been able to compute exactly the case for [itex]\Delta = 1[/itex] and it turns out to be [itex] \rho(p) \propto \delta(p^2) \theta(p_0) [/itex] .
    For the case [itex]\Delta \neq 1[/itex] it's a bit tricky but I managed to perform the integration over [itex]x_0[/itex] using the residual once again and I found ([itex] r^2 = x_i x^i [/itex])

    [itex]\sum_{k=0}^{\Delta -1} {\Delta-1 \choose k} (i p_0)^k \frac{\Gamma(\Delta -1 - k)}{\Gamma(\Delta)^2} \frac{1}{(2 r)^{2 \Delta+1-k}} \left(e^{i p_0 r} + (-1)^{2 \Delta -1 - k} e^{-i p_0 r}\right)[/itex]

    Probably something is wrong since when I perform the remaining integrations (in spherical coordinates) I do not recover the following result:

    [itex] \rho\left(p\right) = \frac{\Delta -1}{4^{\Delta} \Gamma(\Delta)^2} \theta(p_0) \delta(p^2) (p^2)^{\Delta -2} [/itex]
    Any help would be great.
  2. jcsd
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