1. ## Trace theorems

I didn't quite know where to put this one.

I'll get the definitions in here first:
$\displaystyle \gamma ^0 = \left ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix} \right )$

$\displaystyle \gamma ^1 = \left ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{matrix} \right )$

$\displaystyle \gamma ^2 = \left ( \begin{matrix}0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{matrix} \right )$

$\displaystyle \gamma ^3 = \left ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} \right )$

$\displaystyle \eta = \left ( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix} \right )$

My question is this: How do I evaluate $\displaystyle Tr \left [ \gamma ^{\mu} ~ \gamma ^{\sigma} ~ \gamma ^{\lambda} ~ \gamma ^0 \right ]$

The only other information I have that might be useful is the anti-commutator

$\displaystyle \left \{ \gamma ^{\mu} ,~ \gamma ^{\sigma} \right \} = \gamma ^{\mu} ~ \gamma ^{\sigma} + \gamma ^{\sigma} ~ \gamma ^{\mu} = 2~ \eta ^{\mu \sigma}$

Thanks!

-Dan

2. ## Re: Trace theorems

Got it!

It can be solved by repeated application of $\displaystyle \left \{ \gamma ^{\mu} , ~ \gamma ^{\nu} \right \} = 2 \eta ^{\mu \nu}$.

Rewrite:
$\displaystyle Tr \left [ \gamma ^{\mu} \gamma ^{\nu} \gamma ^{\lambda} \gamma ^0 \right ] = Tr \left [ \gamma ^0 \gamma ^{\mu} \gamma ^{\nu} \gamma ^{\lambda} \right ]$

and bring back the $\displaystyle \gamma ^0$ back through to its original position by use of the anti-commutator.

The result is: $\displaystyle Tr \left [ \gamma ^{\mu} \gamma ^{\nu} \gamma ^{\lambda} \gamma ^0 \right ] = 4 \left ( \eta ^{\mu \nu} \eta ^{\lambda 0} - \eta ^{\mu \lambda} \eta ^{\nu 0} + \eta ^{\mu 0} \eta ^{\nu \lambda} \right )$

-Dan

Correction: I have changed the indices between this post and the last. As both posts are essentially independent of the choice of indices I'm not going to bother to fix this.