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Schwartz QFT book, Problem 14.3

  1. Mar 25, 2015 #1
    I am working on Schwartz QFT book problem 14.3, particularly part (c).
    Basically, it asks us to evaluate the following integration.
    [tex] \int \cfrac{d^3 p}{2\pi^3} \omega_p e^{i \vec{p} \cdot (\vec{x}-\vec{y})}[/tex] where
    [tex] \omega_p = \sqrt{p^2 + m^2} [/tex]
    I could perform the angular integration, and the result is
    [tex] \cfrac{2}{(2\pi)^2 |x-y|} \int_0^{\infty} dp \,\,p \sqrt{p^2+m^2} \sin{(p|x-y|)} = \cfrac{2}{(2\pi)^2 |x-y|} \textrm{Im}\left( \int_0^{\infty} dp \,\,p \sqrt{p^2+m^2} e^{i p |x-y|} \right) [/tex]
    The textbook says it should be expressed as some sort of Hankel functions, but I am not sure how I get to that point. Do you guys have any suggestion? Thank you.
     
  2. jcsd
  3. Mar 25, 2015 #2

    mfb

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    I don't know if it works, but a substitution p2+m2 = x could give interesting results. It makes the exponential more complicated but simplifies everything else.
     
  4. Mar 25, 2015 #3
    Could you tell me in more detail? In fact, I could not find any integral representations of Hankel functions that matches with my integration.
     
  5. Mar 27, 2015 #4

    mfb

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    Did you try it? What was the result?
     
  6. Feb 12, 2017 #5
    The result is the derivative of the modified Bessel function of the second kind, whose integral representation is
    upload_2017-2-12_12-27-45.png
    http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html

    And the solution for the massive case is given in Weinberg's QFT I page 387, which can be derived from the above integral representation. For the massless case, it seems to me that the result is just the limit ##m\rightarrow 0## of the massive case, is this right? In taking the limit that ##m\rightarrow 0##, I didn't encounter any singular behavior.
     
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