Paraphrasing Peskin and Schroeder:(adsbygoogle = window.adsbygoogle || []).push({});

By repeated use of

[itex]\left\{ \gamma^{\mu} , \gamma^{\nu} \right\}= 2 g^{\mu\nu} \times \textbf{1}_{n \times n} [/itex] (Clifford/Dirac algebra),

verify that the n-dimensional representation of the Lorentz algebra,

[itex]S^{\mu \nu}=\frac{i}{4}\left[\gamma^{\mu},\gamma^{\nu}\right][/itex],

satisfies the commutation relation

[itex]\left[J^{\mu \nu},J^{\rho \sigma}\right]=i\left(g^{\nu \rho}J^{\mu \sigma}-g^{\mu \rho}J^{\nu \sigma}-g^{\nu \sigma}J^{\mu \rho}+g^{\mu \sigma}J^{\nu \rho}\right)[/itex].

I've tried many lengthy computations and always seem to be missing something.

Most obvious thing to try is just

[itex]\left[S^{\mu \nu},S^{\rho \sigma}\right]=S^{\mu \nu}S^{\rho \sigma}-S^{\rho \sigma}S^{\mu \nu}=\frac{-1}{16}\left(\left[\gamma^{\mu},\gamma^{\nu}\right]\left[\gamma^{\rho},\gamma^{\sigma}\right]-\left[\gamma^{\rho},\gamma^{\sigma}\right]\left[\gamma^{\mu},\gamma^{\nu}\right]\right)[/itex]

[itex]=\frac{-1}{16}\left(\left(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right)\left(\gamma^{\rho}\gamma^{\sigma}-\gamma^{\sigma}\gamma^{\rho}\right)-\left(\gamma^{\rho}\gamma^{\sigma}-\gamma^{\sigma}\gamma^{\rho}\right)\left(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right)\right)[/itex]

[itex]=\frac{-1}{16}\left( \gamma^{\mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma} - \gamma^{\mu} \gamma^{\nu} \gamma^{\sigma} \gamma^{\rho} - \gamma^{\nu} \gamma^{\mu} \gamma^{\rho} \gamma^{\sigma} + \gamma^{\nu} \gamma^{\mu} \gamma^{\sigma} \gamma^{\rho} - \gamma^{\rho} \gamma^{\sigma} \gamma^{\mu} \gamma^{\nu} + \gamma^{\rho} \gamma^{\sigma} \gamma^{\mu} \gamma^{\nu} + \gamma^{\sigma} \gamma^{\rho} \gamma^{\mu} \gamma^{\nu} - \gamma^{\sigma} \gamma^{\rho} \gamma^{\nu} \gamma^{\mu} \right) [/itex]

and then I've tried a few different commutation relations but to no avail.

Would be very grateful for any help in finishing this off.

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# Homework Help: Proving a representation of the Lorentz Algebra from Clifford Algebra/Gamma matrices.

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