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Gamma matrices

  1. Apr 8, 2015 #1
    Hi all,
    I make some
    exercises in particle physics but I'm stuck in two problems related to Gamma matrices identities,
    First: the Fermion propagator ## \frac {i } { /\!\!\!p - m} = i \frac { /\!\!\!p + m } { p^2 - m^2} ## So how ##/ \!\!\!\!p ^2 = p^2 ## ? Where ## /\!\!\!p = \gamma_\mu p^\mu ##.

    I think ##/ \!\!\!\!p ^2 = \gamma_\mu p^\mu \gamma_\nu p^\nu =
    \gamma_\mu p^\mu \gamma_\nu g^{\mu\nu} p_\mu = \gamma_\mu \gamma^\mu p^2 = 4 p^2 ##, so I got a factor 4 ! What's wrong here?

    Second: It's related to the helicity operator, ## h= S . \bf{p} ## ( where S is 2 by 2 matrix, with ##\sigma^i ## on the diagonal ), that as mentioned in a reference as [arXiv:1006.1718], it commutes with the Dirac Hamiltonian ## H = \gamma^0 ( \gamma^i p^i + m ) ## equ. (3.3), due to Gamma matrices anticommutation relation, but this isn't clear for me at all..

    Thanx
     
    Last edited: Apr 8, 2015
  2. jcsd
  3. Apr 8, 2015 #2

    phyzguy

    User Avatar
    Science Advisor

    I'm not sure about the second one, but on the first one, you've just been sloppy with your summation indices, and you've used μ as a summation index twice. If you do what you were trying to do correctly, you get:
    [tex] \not p \not p = \gamma_\mu p^\mu \gamma_\nu p^\nu = \gamma_\mu p^\mu \gamma_\nu g^{\nu \lambda}p_\lambda = \gamma_\mu p^\mu \gamma^\lambda p_\lambda= ?[/tex]

    An easier way to see it is as follows:

    [tex]\not p \not p= \frac{1}{2}(\gamma^\mu p_\mu \gamma^\nu p_\nu + \gamma^\nu p_\nu \gamma^\mu p_\mu)= \frac{1}{2}p_\mu p_\nu(\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu) = \frac{1}{2}p_\mu p_\nu 2 g^{\mu \nu} = p_\mu p^\mu = p^2[/tex]
     
  4. Apr 8, 2015 #3
    For the first question:
    ##/ \!\!\!\!p ^2 = / \!\!\!\!p ^2## so by simple relabling
    ## \gamma_\mu p^\mu \gamma_\nu p^\nu = \gamma_\nu p^\nu \gamma_\mu p^\mu ##
    now ##/ \!\!\!\!p ^2 = \frac{1}{2}( / \!\!\!\!p ^2 + / \!\!\!\!p ^2) ##
    ## \gamma_\mu p^\mu \gamma_\nu p^\nu = \frac{1}{2}( \gamma_\mu p^\mu \gamma_\nu p^\nu + \gamma_\nu p^\nu \gamma_\mu p^\mu) = \frac{1}{2}\left\{ \gamma_\mu,\, \gamma_\nu\right\}p^\nu p^\mu##
    since ##[p^\nu, p^\mu ] = 0 ##
    You should be able to finish the last one or two steps from here.

    Edit: I didn't refresh my browser to see someone beat me to it before posting. Oh well.
     
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