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The question is simple, what is the result of the following multiplication of traces by the metric?

$Tr\left[\gamma^\alpha\gamma^\nu\gamma^\beta\gamma^\tau\gamma^5\right]Tr\left[\gamma^\delta\gamma^\mu\gamma^\epsilon\gamma^\kappa\gamma^5\right]g^{\mu\nu}g^{\kappa\tau}=-8\epsilon^{\alpha\nu\beta\tau}\epsilon^{\delta\mu\epsilon\kappa}g^{\mu\nu}g^{\kappa\tau}$

It seems confusing to me because I don't see clearly if it's equal to zero or not.

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    $\begingroup$ The formula is malformed, in relativity the sum convention is to sum over an index occurring twice once as co- and once as contravariant index. (I guess the indices on the $g$ should be down?) $\endgroup$ – Sebastian Riese May 10 '18 at 18:00
  • $\begingroup$ @SebastianRiese that's not the point actually, I edited the question. The problem is how to compute the product between Levi-Civita tensor and the metric tensors. $\endgroup$ – Enri May 10 '18 at 18:54
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    $\begingroup$ The Wikipedia page for the Levi-Civita has some contraction identities. $\endgroup$ – knzhou May 10 '18 at 18:55

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