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- Homework Statement
- Using differential expressions for the generator to verify the commutator expression in Poincare group

- Relevant Equations
- Definition for the diffrential expressions of the generators are given below

Using differential expressions for the generator, verify the commutator expression for ##[J_{\mu\nu},P_{\rho}]=i(\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu})## in Poincare group

Generator of translation: ##P_{\rho}=-i\partial_{\rho}##

Generator of rotation: ##J_{\mu\nu}=i(x_{\mu}\partial_{\nu}-x_{\nu}\partial{_\mu})##

Here is my working, I operate the commutator on a vector ##x^j##:

##[J_{\mu\nu},P_{\rho}]x^j##

##=(J_{\mu\nu}P_{\rho}-P_{\rho}J_{\mu\nu})x^j##

##=0-iP_{\rho}(x_{\mu}\partial_{\nu}-x_{\nu}\partial{_\mu})x^j##

##=-[(\partial_{\rho}x_{\mu})(\partial_{\nu}x^j)-(\partial_{\rho}x_{\nu})(\partial_{\mu}x^j)]##

##=-[\eta_{\rho\mu}(\partial_{\nu}x^j)-\eta_{\rho\nu}(\partial_{\mu}x^j)]##

##=-i(\eta_{\rho\mu}P_{\nu}x^j-\eta_{\rho\nu}P_{\mu}x^j)##

##=-i(\eta_{\rho\mu}P_{\nu}-\eta_{\rho\nu}P_{\mu})x^j##

My answer had one extra negative sign on it. Where did it go wrong?

Generator of translation: ##P_{\rho}=-i\partial_{\rho}##

Generator of rotation: ##J_{\mu\nu}=i(x_{\mu}\partial_{\nu}-x_{\nu}\partial{_\mu})##

Here is my working, I operate the commutator on a vector ##x^j##:

##[J_{\mu\nu},P_{\rho}]x^j##

##=(J_{\mu\nu}P_{\rho}-P_{\rho}J_{\mu\nu})x^j##

##=0-iP_{\rho}(x_{\mu}\partial_{\nu}-x_{\nu}\partial{_\mu})x^j##

##=-[(\partial_{\rho}x_{\mu})(\partial_{\nu}x^j)-(\partial_{\rho}x_{\nu})(\partial_{\mu}x^j)]##

##=-[\eta_{\rho\mu}(\partial_{\nu}x^j)-\eta_{\rho\nu}(\partial_{\mu}x^j)]##

##=-i(\eta_{\rho\mu}P_{\nu}x^j-\eta_{\rho\nu}P_{\mu}x^j)##

##=-i(\eta_{\rho\mu}P_{\nu}-\eta_{\rho\nu}P_{\mu})x^j##

My answer had one extra negative sign on it. Where did it go wrong?

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