I'm reading through some lecture notes and there is a proof that the gamma matrices are traceless that I've never seen before (I've seen the "identity 0" on wikipedia proof) and I can't work out some of the steps:(adsbygoogle = window.adsbygoogle || []).push({});

\begin{align*}

2\eta_{\mu\nu}Tr(\gamma_\lambda) &= Tr(\{\gamma_{\mu},\gamma_{\nu}\}\gamma_\lambda)

\\ &= Tr(\gamma_\mu\gamma_\nu\gamma_\lambda + \gamma_\nu\gamma_\mu\gamma_\lambda)

\\ &= Tr(\gamma_\mu\gamma_\nu\gamma_\lambda + \gamma_\mu\gamma_\lambda\gamma_\nu)

\\ &= Tr(\gamma_\mu\{\gamma_{\nu},\gamma_{\lambda}\})

\\ &= 2\eta_{\nu\lambda}Tr(\gamma_\mu)

\\ \mu = \nu \neq \lambda \implies Tr(\gamma_\lambda) = 0

\end{align*}

In particular I don't understand the very first equality and the very last (I assume the same method is being used in both) but I understand the rest using the trace and anti commutator properties. I understand that the 2 eta factor is equal to the anti commutator but I don't see how this allows you to pull it inside of the trace, I tried to work it out explicitly using a generic 4x4 matrix for the gammas but I can't get it.

Thank you in advance for any help

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# Gamma matrix traceless proof

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