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I need your help in order to prove this relation:

$$\gamma^{\mu} \not p \gamma_{\mu}$$ that has to give me $$-2 \not p$$

I tried this: $$ \begin{array}{rll} \gamma^{\mu} \not p \gamma_{\mu} & = \gamma^{\mu}\big(\gamma^{\nu}p_{\nu}\big)\gamma_{\mu} \\ & = \gamma^{\mu}\big(\gamma^{\nu}\gamma_{\mu}\big)p_{\nu} \\ & = \gamma^{\mu}\big(\gamma^{\nu}\gamma_{\mu} + \gamma^{\nu}\gamma_{\mu} - \gamma^{\nu}\gamma_{\mu}\big)p_{\nu} \\ & = \gamma^{\mu}\big(2\eta^{\nu}_{\mu} - \gamma^{\nu}\gamma_{\mu}\big)p_{\nu} \\ & = 2\gamma^{\mu}\eta^{\nu}_{\mu} - \gamma^{\mu}\gamma^{\nu}\gamma_{\mu}p_{\nu} \\ & = 2\gamma^{\nu}p_{\nu} - 4\gamma^{\nu}p_{\nu} \\ & = -2\not p \end{array} $$

My question is: is this valid? And in the second line: is the passage valid? Can I exchange the order of $\gamma$ and $p$?

Thank you everybody!!

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1 Answer 1

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Everything looks fine to me. If in doubt, the wikipedia article has the proof of all the common identities.

Also, the step from the first line to the second is valid because $p_\nu$ is a scalar quantity, so it commutes with the matrix product.

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