Sorry to be asking again so soon, the help yesterday was great. I'm now trying to reach 2.17 from the generator commutation relation:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] [M^{\mu\nu}, M^{\rho\sigma}]=i\hbar(g^{\mu\rho}M^{\nu\sigma}-g^{\nu\rho}M^{\mu\sigma})+........ 2.16[/tex]

J is defined by its components as [tex] J_i=\frac{1}{2}\epsilon_{ijk}M^{jk} [/tex]

I'm trying to work out the commutation relation [tex] [J_i, J_l] [/tex]

I've started this and plugged in the expression for the J components and used the generator commutation relation. I came to a point where I had:

[tex] [J_i, J_l]= \frac{1}{4} i\hbar \epsilon_{ijk}\epsilon_{lmn} ( g^{jm}M^{kn}-g^{km}M^{jn}-g^{jn}M^{km}+g^{kn}M^{jm} ) [/tex]

I paused at this point and wondered how to get rid of the metrics, can I just raise the the relevant index on each Levi Cevita symbol? (I'm not sure as I've heard this isnt a standard tensor but a tensor density). I blindly assumed I could act with the metrics in this way, and raised the Levi Cevita symbol indices, and I ended up with (after relabelling lots of dummy indices, using the antisymmetry of the M, and using the fact that single index swaps of the Levi Cevita pick up a minus sign (although not sure if this is legal between a raised and lowered index which I did do):

[tex] [J_i, J_l]=i \hbar \epsilon_i^{.m}_k \epsilon_{lmn}M^{kn}[/tex]

The expression I actually want (Equation 2.17) is [tex] [J_i, J_l]=i \hbar \epsilon_{ilk}J_k=\frac{1}{2}i \hbar \epsilon_{ilk}\epsilon_{kmn}M^{mn} [/tex]

Cheers for any help at all

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# Srednicki 2.17. How does metric act on Levi Cevita symbol?

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