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Help with Dirac trace algebra

  1. Mar 20, 2016 #1
    1. The problem statement, all variables and given/known data
    This isn't a homework problem; it's just something I'm working on and I'm a little confused as to how to go about dealing with what I have. I have several traces of Dirac's gamma matrices, and I know that the trace of an odd number of gamma matrices is zero. So my first question is: does it matter *which* gamma matrices? For example, one of my traces has
    $$ \gamma_{\nu}\gamma^5\gamma^{\rho}\gamma^{\alpha}\gamma^{\sigma}\gamma^0\gamma^5\gamma^0\gamma^{\lambda}\gamma_{\mu}\gamma^{\beta} $$ Since each $$\gamma^5$$ is a product of 4 gamma matrices, altogether this would be a product of 17, which is odd. But some are $$\gamma^{\mu}$$, some $$\gamma^0$$ and some $$\gamma^5$$ and I'm just not sure what the rules are for this type of thing.

    2. Relevant equations
    $$Tr (ABC) = Tr (CAB) = Tr (BCA)$$
    $$Tr (A +B) = Tr(A)+Tr(B)$$
    $$Tr(aA) = aTr(A)$$

    3. The attempt at a solution
    I would think that this would, in fact, still be zero because there's an odd number of them. Otherwise I would probably use commutation/anticommutation relations to bring the $$gamma^0$$ and $$\gamma^5$$ together (respectively) to get unity, then just deal with the rest.
     
  2. jcsd
  3. Mar 21, 2016 #2

    Orodruin

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    And there is your answer.
     
  4. Mar 21, 2016 #3
    So does that mean that all the standard trace identies for gamma matrices (e.g. in Griffiths or Peskin Schroeder) hold for *any* gamma matrices, regardless of their index?
     
  5. Mar 21, 2016 #4

    Orodruin

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    As long as that index is 0, 1, 2, or 3. You will need to treat ##\gamma^5## differently.
     
  6. Mar 21, 2016 #5
    Isn't the last one just a product of the first 4?
     
  7. Mar 21, 2016 #6

    Orodruin

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    Yes (up to a factor), so you need to treat it differently.
     
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