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as this is a basic QFT question, I wasn't sure to put it in the relativity or quantum section. Since this question specifically is about manipulating tensor expressions, i figured here would be appropriate.

My question is about equating coefficients in tensor expressions, 2.4.10-11 in Weinberg's Quantum Theory of Fields (2005).

[itex] i[ 1/2 \omega_{\mu \nu} J^{\mu \nu} - \epsilon_{mu}P^{\mu}, J^{\rho \sigma} ] = \omega_{\mu}^{\rho} J^{\mu \sigma} - \omega_{\nu}^{\sigma} J^{\rho \nu} - \epsilon^{\rho}P^{\sigma}+ \epsilon^{\sigma}P^{\rho}[/itex]

[itex] i [ 1/2 \omega_{\mu \nu} J^{\mu \nu} - \epsilon_{\mu} P^{\mu} ,P^{\rho} ]=\omega_{\mu}^{\rho} P^{\mu} [/itex]

The task is to equate coefficients on the epsilon and omega terms to find the commutators of the poincare algebra. I'm a bit confused because, for instance, the omega term on the LHS of the first equation has dummy indices, but on the RHS has real indices. I'm not sure what manipulations i can do besides raising and lowering with the metric.

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# Deriving the poincare algebra

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