- #1
spaghetti3451
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Homework Statement
1. Show that the Lorentz algebra generator ##J^{\mu \nu} = i(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu})## lead to the commutation relation ##[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})##.
2. Show that one particular representation of the the Lorentz group given by ##(J^{\mu \nu})_{\alpha \beta} = i(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}-\delta^{\mu}_{\beta}\delta^{\nu}_{\alpha})##.
Homework Equations
The Attempt at a Solution
##[J^{\mu \nu}, J^{\rho \sigma}]##
##= J^{\mu \nu}J^{\rho \sigma}-J^{\rho \sigma}J^{\mu \nu}##
##= -(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu})(x^{\rho}\partial^{\sigma}-x^{\sigma}\partial^{\rho})+(x^{\rho}\partial^{\sigma}-x^{\sigma}\partial^{\rho})(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu})##.
How do I get the metric tensor from here?