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

A Covariant gamma matrices

  1. Dec 5, 2016 #1
    Covariant gamma matrices are defined by



    The gamma matrix ##\gamma^{5}## is defined by

    $$\gamma^{5}\equiv i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}.$$


    Is the covariant matrix ##\gamma_{5}## then defined by

    $$\gamma_{5} = i\gamma_{0}(-\gamma_{1})(-\gamma_{2})(-\gamma_{3})?$$
  2. jcsd
  3. Dec 5, 2016 #2


    User Avatar
    Science Advisor
    Homework Helper

    No, typically the chirality matrix has the index "downstairs" and is defined in terms of the "downstair" gammas. So the three minuses in your last equality should be omitted.k
  4. Dec 5, 2016 #3
    But, in Peskin and Schroeder, page 50, ##\gamma^{5}## is defined as

    $$\gamma^{5}\equiv i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$$

    and the downstairs index is not used on the chirality matrix.
  5. Dec 5, 2016 #4


    User Avatar
    Science Advisor
    Homework Helper

    Iirc, there's only one matrix being used (either with the index "down" or "up"), not both in a book. I don't have a statistics in my head, but the lower 5 is prevalent.
  6. Dec 5, 2016 #5
    So, you mean

    $$\gamma_{5} \equiv \gamma^{5} \equiv i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}?$$
  7. Dec 5, 2016 #6


    User Avatar
    Science Advisor
    Homework Helper

    No, to have a consistent definition, you have gamma_5 = i gamma_0 * gamma_1 *...
    And separately gamma^5 = i gamma^0 * gamma^1 *...
    Because of the sign ambiguity (the metric has either 1 or 3 minuses), books will choose to use only one type of 5.
    Last edited: Dec 6, 2016
  8. Dec 5, 2016 #7
    But equation (36.46) in Srenicki has

    $$\gamma_{5} = i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$$
  9. Dec 6, 2016 #8


    User Avatar
    Science Advisor

    That's because the '5' is just a dummy name, not a legitimate index. The 2nd part of Srednicki's (36.46) is actually $$\gamma_5 ~=~ -\,\frac{i}{24}\, \epsilon_{\mu\nu\rho\sigma} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma ~.$$Peskin & Schroeder do something similar on p49 where they write $$\gamma^{\mu\nu\rho\sigma} ~=~ \gamma^{[\mu} \gamma^\nu \gamma^\rho \gamma^{\sigma]} ~,$$but then introduce a ##\gamma^5## in eq(3.68). Whichever place you put the "5" index, the 5th gamma is a pseudo-scalar. It should probably be called something else not involving the index "5".
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Covariant gamma matrices
  1. Gamma matrices (Replies: 2)

  2. Expanding gamma matrices (Replies: 12)