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vputz
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Homework Statement
As part of a problem set, I'm trying to show that the [tex]S^{\mu\nu}[/tex] matrices defined by [tex]S^{\mu\nu}=\frac{i}{4}[\gamma^\mu,\gamma^\nu][/tex] constitute a representation of the Lorentz algebra.
Well, luckily P&S assure me that by repeated use of
{[tex]\gamma^\mu,\gamma^\nu[/tex]}[tex]=2g^{\mu\nu}\times 1_n[/tex]
"...it is easy to verify that these matrices satisfy the commutation relations"
[[tex]J^{\mu\nu},J^\rho\sigma] = i(g^{\nu\rho}J^{\mu\sigma} - g^{\mu\rho}J^{\nu\sigma} - g^{\nu\sigma}J^{\mu\rho} + g^{\mu\sigma}J^{\nu\rho})[/tex]
Homework Equations
The Attempt at a Solution
Well, it is just ducky that it is easy to verify (grin). But it is not turning out to be easy for me! If I approach it by expanding [[tex]S^{\mu\nu},S^{\rho\sigma}[/tex]], I wind up with a lot of gamma matrices; writing [tex]\gamma^\mu[/tex] just as [tex]\mu[/tex], it looks something like
[tex]\mu\nu\rho\sigma - \mu\nu\sigma\rho - \nu\mu\rho\sigma + \nu\mu\sigma\rho - \rho\sigma\mu\nu + \rho\sigma\nu\mu + \sigma\rho\mu\nu - \sigma\rho\nu\mu[/tex]
Now, if I'm moving toward the Lorentz algebra, I need to get terms of the form [tex]g^{\nu\rho}S^{\mu\sigma}[/tex] after making use of the anticommutator relation--but I only see those pairings of indices in the middle of the groups of four. I am certain that I'm missing some obvious commutation or anticommutation trick to break those groups up so that I can make use of the anticommutation identity to introduce the metric and thus get toward a solution, but I've been staring at it for hours and no luck. I tried approaching it the other way (expanding the right hand side of the Lorentz algebra equation above) and wound up with sixteen groups of four gammas with different indices and still no obvious way to reduce them and show equality.
Stated a little differently, in order to get a [tex]g^{\mu\rho}S^{\mu\sigma}[/tex] term, I need a {[tex]\nu,\rho[/tex]}[[tex]\mu,\sigma[/tex]] term, which if I expand it comes out to be something like
[tex]\nu\rho\mu\sigma - \nu\rho\sigma\mu + \rho\nu\mu\sigma - \rho\nu\sigma\mu[/tex]
My original expansion does contain some terms with the right first and last indices, but the middle terms reversed from what I want. It also does NOT have some terms I need (for example, terms beginning and ending with [tex]\mu[/tex] and [tex]\nu[/tex]).
I know it should be straightforward and I'm just not seeing the link. Any suggestions?