# Gamma matrix identity (QFT)

## Homework Statement

Prove that $\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}\gamma^{e}\gamma_{a}$ = $2\left(\gamma^{e}\gamma^{b}\gamma^{c}\gamma^{d}+\gamma^{d}\gamma^{c} \gamma^{b}\gamma^{e}\right)$

Each of the $\gamma^{i}$s are as used in the Dirac equation.

## Homework Equations

$\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}\gamma_{a}$ = $-2\gamma^{d}\gamma^{c}\gamma^{b}$

$\gamma^{a}\gamma^{b} + \gamma^{b}\gamma^{a} = 2g^{ab}$

## The Attempt at a Solution

$\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}\gamma^{e}\gamma_{a}$ = $2g^{ab}\gamma^{c}\gamma^{d}\gamma^{e}\gamma_{a}$ - $\gamma^{b}\gamma^{a}\gamma^{c}\gamma^{d}\gamma^{e}\gamma_{a}$

= $2\gamma^{c}\gamma^{d}\gamma^{e}\gamma^{b}$ + $2\gamma^{b}\gamma^{e}\gamma^{d}\gamma^{c}$

Perhaps I mixed up something or there is a typo...

Instead, start over and try commuting ##\gamma^{e}## with ##\gamma_{a}##. To do this, you will need to use the identity obtained by lowering ##a## in the identity $\gamma^{a}\gamma^{b} + \gamma^{b}\gamma^{a} = 2g^{ab}$