Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Notation relating to gamma matrices

  1. Feb 13, 2010 #1

    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


    User Avatar
    Science Advisor
    Homework Helper

    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:

  5. Feb 13, 2010 #4

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    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,
    \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)
  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?

  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

    User Avatar
    Science Advisor
    Gold Member

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook