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Dirac Principle Value Identity applied to Propagators

  1. Dec 4, 2011 #1
    Hi,

    How is

    [tex]\frac{1}{\displaystyle{\not}{P}-m+i\epsilon}-\frac{1}{\displaystyle{\not}{P}-m-i\epsilon} = \frac{2\pi}{i}(\displaystyle{\not}{P}+m)\delta(P^2-m^2)[/tex]

    ? This is equation (4-91) of Itzykson and Zuber (page 189). I know that

    [tex]\frac{1}{x\mp i\epsilon} = \mathcal{P}\left(\frac{1}{x}\right) \pm i\pi\delta(x)[/tex]

    But this doesn't seem to give the right hand side of the first equation above. What am I missing?

    Thanks in advance!
     
  2. jcsd
  3. Dec 5, 2011 #2
    How does the [itex](\displaystyle{\not}{P} + m)[/itex] appear?
     
  4. Dec 5, 2011 #3

    Bill_K

    User Avatar
    Science Advisor

    It's because P/ - m is a matrix, and so first you have to write 1/(P/ - m) as (P/ + m)/(P2 - m2).

    So in detail,

    1/(P/ - m + iε) - 1/(P/ - m + iε) = (P/ + m)[1/(P2 - m2 + iε) - 1/(P2 - m2 - iε)]
    = (P/ + m)[-iπ δ(P2 - m2) -iπ δ(P2 - m2)] = (P/ + m)(-2iπ δ(P2 - m2))
     
  5. Dec 18, 2011 #4
    Thanks BillK, that cleared it up!
     
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