I am attempting to prove a set of results for the products of gamma matrices and traces of products of gamma matrices, but got stuck on this particular one.


My previous strategy to get the metric in expressions has been to exploit the anti-commutation relation, writing $\gamma^{\mu\nu}$ as $$\gamma^{\mu\nu}+\gamma^{\nu\mu}-\gamma^{\nu\mu}=\{\gamma^{\mu},\gamma^{\nu}\}-\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}-\gamma^{\nu}\gamma^{\mu}.$$ I then feel that I will have to use the cyclical property of the trace to get the desired expression: I have tried with the case for three gamma matrices to see if it can be extended, but not sure how to do this. For example, if I have three gamma matrices in the trace we have


From linearity of the trace, I can write this as two traces. The second one is 0 because its the trace of a product of an odd number of gamma matrices.


The metric is symmetric so we can re-write:


This looks partly right but not sure how to get the metrics out of the trace and some of it is not in the right order anyway (also for even $n$ we can't use the trick where part of the trace went to 0).

  • 1
    $\begingroup$ Trace of an odd number of gamma's is zero, just use $\gamma^1 \gamma^i = 2g^{1i} - \gamma^i \gamma^1$ continually for the even case. $\endgroup$ – bolbteppa Apr 8 '18 at 11:58
  • 2
    $\begingroup$ Yes of course, what am I talking about, I was analysing the case for 3 matrices when I said myself that the trace for an odd number is 0. $\endgroup$ – Tom Apr 8 '18 at 15:08

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