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Notation relating to gamma matrices

  1. Feb 13, 2010 #1
    Hi

    My QFT course assumes the following notation for gamma matrices:

    [tex]\gamma ^{\mu_1 \mu_2 \mu_3 \mu_4} = {\gamma ^ {[\mu_1}}{\gamma ^ {\mu_2}}{\gamma ^ {\mu_3}}{\gamma ^ {\mu_4 ]}}[/tex]

    what does the thing on the right hand side actually mean? Its seems to be a commutator of some sort.
     
  2. jcsd
  3. Feb 13, 2010 #2

    tiny-tim

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    Hi vertices! :smile:

    (have a gamma: γ and a mu: µ :wink:)

    It means you add every possible permutation (just as in γµ[1µ2] you'd add every possible permutation, but there'd only be two of them! :wink:).
     
  4. Feb 13, 2010 #3

    Thanks tiny-tim:)

    So for example, would I be right in saying that:

    γµ1µ2µ1γµ2µ2γµ1?
     
  5. Feb 13, 2010 #4

    Ben Niehoff

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    Tiny-Tim forgot to tell you that you also must multiply each term by the sign of its permutation. That is, you must take a totally-antisymmetrized sum. You might also have to divide by N afterward, depending on the convention you're using (where N is the number of gamma matrices being antisymmetrized).
     
  6. Feb 13, 2010 #5
    Actually, the usual definition for the square bracket, like [tex]\gamma^{\mu\nu} \equiv \gamma^{[\mu} \gamma^{\nu]} [/tex] is the anti-symmetrization of the indices. For this case I just mentioned,
    [tex] \gamma^{\mu\nu} = \gamma^\mu\gamma^\nu - \gamma^\nu\gamma^\mu [/tex]
    And for example,
    [tex]\gamma^{\mu\nu\rho}
    \equiv \gamma^{[\mu}\gamma^{\nu}\gamma^{\rho]}
    = \frac{1}{3!}\left( \gamma^\mu\gamma^\nu\gamma^\rho - \gamma^\mu\gamma^\rho\gamma^\nu + \gamma^\nu\gamma^\rho\gamma^\mu -\gamma^\nu\gamma^\mu\gamma^\rho +\gamma^\rho\gamma^\mu\gamma^\nu -\gamma^\rho\gamma^\nu\gamma^\mu \right)
    [/tex]
     
  7. Feb 13, 2010 #6
    sorry can I ask a stupid question: what is the 'sign' of a permutation?
     
  8. Feb 13, 2010 #7
    can i ask you how the the sign of each term is determined? How do you decide which term is positive and which term is negative?

    thanks.
     
  9. Feb 13, 2010 #8
    Aahh I've figured it out i think - cyclic permutations are positive; anticyclic permutations are negative. A bit like the levi-civita thingy..
     
  10. Feb 13, 2010 #9

    Ben Niehoff

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    This works for 3 indices or less. For N indices, you check whether it is an even or odd permutation of {1,2,3,4,...,N}. That is, you count the number of pairwise interchanges required to bring the indices back to numerical order. If it takes an even number of switches, the term gets a plus sign; otherwise, a minus sign.
     
  11. Feb 15, 2010 #10
    interesting. thanks for pointing this out Ben.
     
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