Srednicki CH26 Explained: Solving Eqn 26.7

[LAHLH]

Hi,

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

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

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


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

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


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

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

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

$$d\Omega_{d-2}=sin^{d-3}d\theta \times d\phi_{1}d\phi_{2}...$$

Which probably cancels out a few $$\pi$$'s but then why doesn't he have two lot's of the sin term?

Thanks for any help on this

 

[LAHLH]

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...

 

[sheaf]

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.

 

[LAHLH]

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:

$$d\Omega=sin^{d-3}\theta_{1}sin^{d-4}\theta_{2}...$$

but given that I have two lots of $$d\Omega$$ I would still expect Srednicki to have his sine term squared? even if he's neglecting the lower power sines for whatever reason...

 

[sheaf]

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 $\theta$ between the spatial momenta, so the only bits we're interested in are the 2 $d\omega$s and $d\theta$

 

[haushofer]

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