- #1
stormyweathers
- 7
- 0
Hey guys,
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.
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.