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.

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Gamma matrices and lorentz algebra

Loading...

Similar Threads for Gamma matrices lorentz | Date |
---|---|

A Gamma matrices | Jan 13, 2018 |

A Covariant gamma matrices | Dec 5, 2016 |

I Why are the gamma-matrices invariant? | Feb 24, 2016 |

Question about Lorentz Invariance and Gamma Matrices | Sep 6, 2014 |

Reps of lorentz group and pauli and gamma matrices | Apr 2, 2009 |

**Physics Forums - The Fusion of Science and Community**