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    Hi - first off, i'm posting this in the maths forum because although it's at it's heart a physics problem, I think my difficulties are of a mathematical nature.

    I've attached the part of my lecture notes that i'm struggling with. I can understand everything up to eq.113, but i'm not sure what is going on with d^3p_1 for it to turn into p^2_1 d\Omega d|\textb{p_1}| in eq 113.1, and even what \Omega is meant to represent, as it's not mentioned in the text at any point.

    Similarly, i'm not really sure how eq 113.2 integrates out to give \frac{k}{M}, or really even where eq 113.2 comes from. I haven't spent very long looking at this step yet, so it might be something obvious that I've just missed.


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    Looking at particle physics before I sleep will give me nightmares
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    (Original post by Rubgish)
    Hi - first off, i'm posting this in the maths forum because although it's at it's heart a physics problem, I think my difficulties are of a mathematical nature.

    I've attached the part of my lecture notes that i'm struggling with. I can understand everything up to eq.113, but i'm not sure what is going on with d^3p_1 for it to turn into p^2_1 d\Omega d|\textb{p_1}| in eq 113.1, and even what \Omega is meant to represent, as it's not mentioned in the text at any point.
    d^3p_1 respresents an element of momentum space for the p_1 variable. I think that they are essentially decomposing this into spherical polars, with d\Omega representing an angular element (in fact, in scattering theory d\Omega usually represents a solid angle by convention) so they are writing

    d^3p_1 = p^2_1 dp_1 d\Omega = p^2_1 d|\bold{p_1}| d\Omega = p^2_1 d|\bold{p_1}| \sin \theta d\theta d\phi

    if you want to write it out in full. Here I think that they are equating the magnitude of the momentum with the spherical polar variable r.

    Similarly, i'm not really sure how eq 113.2 integrates out to give \frac{k}{M}, or really even where eq 113.2 comes from. I haven't spent very long looking at this step yet, so it might be something obvious that I've just missed.
    The delta function bit looks tricky. I can't see how they transform it into the quoted version in 113.2, but assuming this is correct with a typo in the denominator (p \rightarrow p_1), and that we can write E_1+E_2 = Mc^2 = M with c=1, then:

    \frac{1}{32\pi^2M}\int |\mathcal{M}^2|\frac{\delta(p_1-k)E_1E_2}{p_1M} \frac{p_1^2}{E_1E_2} dp_1 d\Omega = \\ \\ \frac{1}{32\pi^2M}\int |\mathcal{M}^2|\frac{\delta(p_1-k)}{M} p_1 dp_1 d\Omega = \\ \\ \frac{1}{32\pi^2M^2}\int |\mathcal{M}^2| d\Omega \int \delta(p_1-k) p_1 dp_1 =

     \frac{k}{32\pi^2M^2}\int |\mathcal{M}^2| d\Omega

    by the sifting property of the delta function.
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    (Original post by atsruser)
    d^3p_1 respresents an element of momentum space for the p_1 variable. I think that they are essentially decomposing this into spherical polars, with d\Omega representing an angular element (in fact, in scattering theory d\Omega usually represents a solid angle by convention) so they are writing

    d^3p_1 = p^2_1 dp_1 d\Omega = p^2_1 d|\bold{p_1}| d\Omega = p^2_1 d|\bold{p_1}| \sin \theta d\theta d\phi

    if you want to write it out in full. Here I think that they are equating the magnitude of the momentum with the spherical polar variable r.



    The delta function bit looks tricky. I can't see how they transform it into the quoted version in 113.2, but assuming this is correct with a typo in the denominator (p \rightarrow p_1), and that we can write E_1+E_2 = Mc^2 = M with c=1, then:

    \frac{1}{32\pi^2M}\int |\mathcal{M}^2|\frac{\delta(p_1-k)E_1E_2}{p_1M} \frac{p_1^2}{E_1E_2} dp_1 d\Omega = \\ \\ \frac{1}{32\pi^2M}\int |\mathcal{M}^2|\frac{\delta(p_1-k)}{M} p_1 dp_1 d\Omega = \\ \\ \frac{1}{32\pi^2M^2}\int |\mathcal{M}^2| d\Omega \int \delta(p_1-k) p_1 dp_1 =

     \frac{k}{32\pi^2M^2}\int |\mathcal{M}^2| d\Omega

    by the sifting property of the delta function.
    Hadn't considered that it could be a rewrite into polar coords, that makes a lot of sense cheers!
 
 
 
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