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Schwartz QFT Eqn 5.26

  1. Aug 21, 2014 #1
    Hello.
    From Schwartz QFT BOOK,
    How could Eqn 5.26 can be Eqn 5.27?

    [itex]d \Pi_{LIPS}=(2 \pi) ^{4} \delta^{4}(\Sigma p) \frac{d^{3} p_{3}}{(2 \pi) ^{3}} \frac{1}{2 E_{3}} \frac{d^{3} p_{4}}{(2 \pi)^{3}} \frac{1}{2 E_{4}} [/itex] Eqn(5.26)


    [itex]d \Pi_{LIPS}=\frac{1}{16 \pi ^{2}} dΩ ∫ d p_{f} \frac{{p_{f}}^2}{E_{3}} \frac{1}{E_{4}} \delta ( E_{3} + E_{4} - E_{CM}) [/itex] Eqn(5.27)
     
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  3. Aug 21, 2014 #2

    WannabeNewton

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    Just integrate over the ##\delta##-function and then switch to spherical coordinates in momentum space. Keep in mind ##\delta^4 (\Sigma p) = \delta^4 (p^{\mu}_1 + p^{\mu}_2 - p^{\mu}_3 - p^{\mu}_4)## so separate the ##\delta##-function into products over the 3-vectors and the energies.
     
  4. Aug 21, 2014 #3
    Integrate over? In (5.26), there is no integration?
     
  5. Aug 21, 2014 #4

    WannabeNewton

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    It's implicit.
     
  6. Aug 21, 2014 #5

    Avodyne

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    Yeah, there really shouldn't be an integral sign in 5.27 if there isn't one in 5.26. Also, p_3 has changed its name to p_f. Also, while 5.27 is Lorentz invariant, he's adopted a specific frame (the CM frame) in 5.27.
     
  7. Aug 21, 2014 #6
    OK. as [itex]\vec{p_{3}}=-\vec{p_{4}}[/itex] , we can insert integration (over [itex]\vec{p_{4}}[/itex] ) in Eqn(5.26) without altering the original. got it.
     
    Last edited: Aug 21, 2014
  8. Aug 21, 2014 #7
    thanks. I would ask some more questions about Shwartz QFT text book. I hope to solve all the exercises of this book within 2 years, though had solved only up to ch.4
     
  9. Aug 21, 2014 #8

    WannabeNewton

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    Cool, well good luck! I'm working through the book as well actually. I'm on ch.7 problems. So it looks like we have the same goals :)
     
  10. Aug 21, 2014 #9
    Good. Good luck!! This book is quite well written.
     
  11. Aug 22, 2014 #10

    WannabeNewton

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    Haha yes, it is the first QFT book I've personally come across that actually feels like a true physics book. It almost feels like cheating having this book in possession when my class's assigned text is (unfortunately) Peskin and Schroeder since the former provides all the intuition that the latter completely lacks, at least in Part I (I haven't even looked Parts II and beyond).

    EDIT: actually Aitchison and Hey is a really awesome physics book as well, George Jones told me about it.
     
    Last edited: Aug 22, 2014
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