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Srednicki CH26

  1. May 11, 2010 #1

    I was wondering if anyone could explain how Srednicki gets to his eqn 26.7:

    [tex] \tilde{dk_1}\tilde{dk_2} \sim (\omega^{d-3}_{1}d\omega_1) (\omega^{d-3}_{2}d\omega_2)(sin^{d-3}\theta d\theta) [/tex]

    I thought this would be to do with transforming into some kind of d-dimensional polar coords so I start as:

    [tex] \tilde{dk_1}\tilde{dk_2}=\frac{d^{d-1}k_1}{(2\pi)^{d-1}2\omega_{1}}\frac{d^{d-1}k_2}{(2\pi)^{d-1}2\omega_{2}}=\frac{\vec{k_1}^{d-2}d\vec{k_1}d\Omega_{d-2}\vec{k_2}^{d-2}d\vec{k_2}d\Omega_{d-2}}{(2\pi)^{d-1}2\omega_{1}(2\pi)^{d-1}2\omega_{2} } [/tex]

    Now since he's working in the massless limit [tex] \omega_{1,2}=\vec{k}_{1,2} [/tex]

    [tex] \tilde{dk_1}\tilde{dk_2}=\frac{\omega^{d-3}_{1}d\omega_{1}d\Omega_{d-2}\omega^{d-3}_{2}d\omega_{2}\Omega_{d-2}}{4(2\pi)^{d-1}(2\pi)^{d-1} } [/tex]

    [tex] \tilde{dk_1}\tilde{dk_2}=(\omega^{d-3}_{1}d\omega_{1})(\omega^{d-3}_{2}d\omega_{2}) \frac{d\Omega_{d-2}d\Omega_{d-2}}{4(2\pi)^{d-1}(2\pi)^{d-1}} [/tex]

    Which looks quite similar to what he has, but not there yet. I'm guessing that the solid angle must go something like

    [tex] d\Omega_{d-2}=sin^{d-3}d\theta \times d\phi_{1}d\phi_{2}... [/tex]

    Which probably cancels out a few [tex]\pi[/tex]'s but then why doesnt he have two lot's of the sin term?

    Thanks for any help on this
  2. jcsd
  3. May 11, 2010 #2
    Some of my vectors should have modulus bars around them by the way, but I couldn't figure out the Latex command, hopefully it will be obvious from context anyway...
  4. May 11, 2010 #3
    Spherical coordinates in N dimensions are treated in Hassani "Mathematical Physics" p 593. If you open it in Google books you can find the relevant page. It has the volume element etc.
  5. May 11, 2010 #4
    Thanks, I can't seem to find the page you refer to, when I look at Hassani on google books I either seem to get his mathematica book or I get math methods but with not enough pages, could you possibly link me to the one you're looking at?

    I found the volume on wiki anyway I believe under http://en.wikipedia.org/wiki/N-sphere, suggesting to me if I'm in d-1 spatial dimensions:

    [tex] d\Omega=sin^{d-3}\theta_{1}sin^{d-4}\theta_{2}........[/tex]

    but given that I have two lots of [tex] d\Omega[/tex] I would still expect Srednicki to have his sine term squared? even if he's neglecting the lower power sines for whatever reason...
  6. May 12, 2010 #5
    I can't link directly to the page in question. The problem with the preview is that you can only look at a limited number of pages before you get blocked. Of course you can delete your cookie and try again until you get to the right page !

    Anyway it only contained the same info as the wiki page that you found. Hopefully you managed to sort out the problem now.

    Incidentally, I assume in eq 26.7, the tilde just means "is proportional to" - there are other angles in the volume elements, but they can all be integrated out when computing cross sections. However, the amplitude T will depend upon the angle [itex]\theta[/itex] between the spatial momenta, so the only bits we're interested in are the 2 [itex]d\omega[/itex]s and [itex]d\theta[/itex]
    Last edited: May 12, 2010
  7. May 12, 2010 #6


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    Zwiebach's book on String Theory also has a thorough treatment on this subject in one of the first chapters.
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