I'm trying to show that the generators of the spinor representation:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]M^{\mu \nu}=\frac{1}{2}\sigma^{\mu \nu}=\frac{i}{4}[\gamma^\mu,\gamma^\nu] [/tex]

obey the Lorentz algebra:

[tex][M^{\mu \nu},M^{\rho \sigma}]=i(\delta^{\mu \rho}M^{\nu \sigma}-\delta^{\nu \rho}M^{\mu \sigma}+\delta^{\nu \sigma}M^{\mu \rho}-\delta^{\mu \sigma}M^{\nu \rho}) [/tex]

However, I'm not getting the right answer, so I was hoping someone could point out where I went wrong:

[tex][M^{\mu \nu},M^{\rho \sigma}]=

\frac{-1}{16}[[\gamma^{\mu},\gamma^{\nu}],[\gamma^{\rho},\gamma^{\sigma}]]

[/tex]

[tex]

=\frac{-1}{16}[2\gamma^{\mu}\gamma^{\nu}-2g^{\mu \nu},2\gamma^{\rho}\gamma^{\sigma}-2g^{\rho \sigma}]=\frac{-1}{8}[\gamma^{\mu}\gamma^{\nu},\gamma^{\rho}\gamma^{\sigma}]

[/tex]

Now using these relations:

[AB,CD]=[AB,C]D+C[AB,D]

[AB,C]=A{B,C}-{A,C}B

[tex]

\frac{-1}{8}[\gamma^{\mu}\gamma^{\nu},\gamma^{\rho}\gamma^{\sigma}]

=\frac{-1}{8}(

[\gamma^{\mu}\gamma^{\nu},\gamma^{\rho}]\gamma^{\sigma}+

\gamma^{\rho}[\gamma^{\mu}\gamma^{\nu},\gamma^{\sigma}]

)=\frac{-1}{8}(\gamma^{\mu} \{\gamma^{\nu},\gamma^{\rho} \}\gamma^{\sigma}

-\{\gamma^{\mu},\gamma^{\rho} \}\gamma^\nu \gamma^\sigma

+

\gamma^{\rho}\gamma^{\mu} \{\gamma^{\nu},\gamma^{\sigma} \}

- \gamma^\rho \{\gamma^{\mu},\gamma^\sigma \} \gamma^\nu

)[/tex]

[tex]=\frac{-1}{4}(g^{\nu \rho}\gamma^{\mu}\gamma^{\sigma}

-g^{\mu \rho}\gamma^{\nu}\gamma^{\sigma}

+g^{\nu \sigma}\gamma^{\rho}\gamma^{\mu}

-g^{\mu \sigma}\gamma^{\rho}\gamma^{\nu}

) [/tex]

This last expression almost looks like the Lorentz algebra, but it is missing the partner in the commutator.

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# Gamma matrices and lorentz algebra

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