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Thread: Trace theorems

  1. #1
    Forum Admin topsquark's Avatar
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    Trace theorems

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

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

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

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

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

    \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 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

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

    Thanks!

    -Dan
    Last edited by topsquark; Jun 3rd 2017 at 12:04 AM. Reason: Fixed anti-commutator brackets
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: Trace theorems

    Got it!

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

    Rewrite:
    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 \gamma ^0 back through to its original position by use of the anti-commutator.

    The result is: 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.
    Last edited by topsquark; Jun 3rd 2017 at 12:07 AM.
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