- #1
crime9894
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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?
Last edited: