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@ Peskin eqn 2.54

  1. Dec 7, 2009 #1
    1. The problem statement, all variables and given/known data
    I am facing problem to derive the 2nd expression from the first one. My problem is the 2nd term of the 2nd expression.


    2. Relevant equations
    [tex]
    \int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}[\exp(-ip\cdot(x - y)) - \exp(ip\cdot (x - y))]=\int\ \frac{d^3p} {(2\pi)^3}\ \{ \frac {1}{2E_p}\ e^{-ip.(x-y)}\left|_{p^0 = E_p}\ +\ \frac {1}{-2E_p}\ e^{-ip.(x-y)}\left|_{p^0 = -E_p}\ \}
    [/tex]

    3. The attempt at a solution
    [tex]
    p\cdot (x - y)= p^0(x^0 - y^0) - \textbf p\cdot(x-y)
    [/tex]

    For Po = - Ep we can take
    [tex]
    p\cdot (x - y)= - p^0(x^0 - y^0) - \textbf p\cdot(x-y)
    [/tex]
    If i am not wrong yet, then what now?
    should i change the dummy variable as p = - p? But if do it then i think another change comes d3p becomes -d3p for the 2nd term and i loose the minus sign before the 2nd term.

    i don't know how much wrong i am but i am expecting good solution from you guys.
     
    Last edited: Dec 7, 2009
  2. jcsd
  3. Dec 7, 2009 #2

    Ben Niehoff

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    Science Advisor
    Gold Member

    The trick is that the measure d3p actually does not change sign under p -> -p.
     
  4. Dec 7, 2009 #3
    Thanks Ben . But will you plz explain that why the sign of d3 does not change. Is it because of spherical co-ordinates : d3p = p2sin(theta)d(theta)d(phi)dp , where p = |p| ?
     
    Last edited: Dec 7, 2009
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