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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.

$$Tr(\gamma^{\mu_1}...\gamma^{\mu_n})=g^{\mu_1\mu_2}Tr(\gamma^{\mu_3}...\gamma^{\mu_n})-g^{\mu_1\mu_3}Tr(\gamma^{\mu_2}\gamma^{\mu_4}...\gamma^{\mu_n})+...+g^{\mu_1\mu_n}Tr(\gamma^{\mu_2}...\gamma^{\mu_{n-1}}).$$

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

$Tr(\gamma^{\mu_1}\gamma^{\mu_2}\gamma^{\mu_3})=Tr(\{\gamma^{\mu_1},\gamma^{\mu_2}\}-\gamma^{\mu_2}\gamma^{\mu_1})\gamma^{\mu_3})=Tr(2g^{\mu_{1}\mu_{2}}\gamma^{\mu_3}-\gamma^{\mu_2}\gamma^{\mu_1}\gamma^{\mu_3})$

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.

$Tr(2g^{\mu_{1}\mu_{2}}\gamma^{\mu_3})$

The metric is symmetric so we can re-write:

$Tr((g^{\mu_{1}\mu_{2}}+g^{\mu_{2}\mu_{1}})\gamma^{\mu_3})=Tr(g^{\mu_{1}\mu_{2}}\gamma^{\mu_3})+Tr(g^{\mu_{2}\mu_{1}}\gamma^{\mu_3})$

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).

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    $\begingroup$ 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
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    $\begingroup$ 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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