# Dirac Gamma matrices question

## Homework Statement

Given that $\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}*1$ where $1$ is the identity matrix and the $\gamma$ are the gamma matrices from the Dirac equation, prove that:

$\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2g_{\mu\nu}*1$

## Homework Equations

$g^{\mu\nu}\gamma_{\nu}=\gamma^{\mu}$ and $g_{\mu\nu}\gamma^{\nu}=\gamma_{\mu}$

## The Attempt at a Solution

I'm not sure what to start with. I tried expressing the terms of the relation to be proved as follows

$\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=g_{\mu\alpha}\gamma^{\alpha}g_{\nu \beta}\gamma^{\beta}+ g_{\nu\beta}\gamma^{\beta}g_{\mu\alpha}\gamma^{ \alpha }$

but that isn't going anywhere. So how do I approach this?

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dextercioby
Homework Helper
Hmm, just replace mu and nu with their possible values and see what you get. Don't forget that the metric tensor is diagonal (probably diag(+,-,-,-)).

George Jones
Staff Emeritus
Given that $\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}*1$
Another way to do it is to multiply both sides of this equation by $g_{\alpha \mu} g _{\beta \nu}$.
Thank you George Jones! That did the trick nicely. Using $g_{\mu\alpha}g^{\alpha\nu}=\delta^{\mu}_{\nu}$ the result follows easily.