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Dimensional regularization

  1. Aug 14, 2008 #1
    how does dimensional regularization work ?

    i see , how can you calculate integrals in d-dimensions of the form

    [tex] \int d^{d} k F( \vec k ) [/tex] ??

    and for other cases , let us suppose we have the integral

    [tex] \lim_{\varepsilon\rightarrow 0^+}\int \frac{dp}{(2\pi)^{4-\varepsilon}} \frac{2\pi^{(4-\varepsilon)/2}}{\Gamma\left(\frac{4-\varepsilon}{2}\right)} \frac{p^{3-\varepsilon}}{\left(p^2+m^2\right)^1} [/tex]

    there is no way this integral can be calculated
     
  2. jcsd
  3. Aug 15, 2008 #2

    Avodyne

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    Science Advisor

    It works by analytic continuation. Consider the integral

    [tex]\int_0^\infty dx\,{x^n\over x^2+1}.[/tex]

    For [itex]-1<{\rm Re}\,n<1[/itex], the integral converges, and the result is

    [tex]{\pi/2\over\cos(n\pi/2)}.[/tex]

    We now define this to be the value of the integral for all [itex]n[/itex].
     
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