- #1
RicardoMP
- 49
- 2
- Homework Statement
- I want to prove that the Casimir operator of the Poincaré algebra ## P^2 ## satisfies ## [P^2,P_\mu]=0 ##.
- Relevant Equations
- The most relevant equation is ## [P_\mu,P_\nu]=0##.
I want to make certain that my proof is correct:
Since ## P^2 = P_\nu P^\nu=P^\nu P_\nu ##, then ## [P^2,P_\mu]=[P^\nu P_\nu,P_\mu]=P^\nu[P_\nu,P_\mu]+[P^\nu,P_\mu]P_\nu=[P^\nu,P_\mu]P_\nu=g^{\nu\alpha}[P_\alpha,P_\mu]P_\nu=0 ##, since ## g^{\nu\alpha} ## is just a number, I can bring it out of the commutator, thus giving me the desired result. Is this last step correct?
Since ## P^2 = P_\nu P^\nu=P^\nu P_\nu ##, then ## [P^2,P_\mu]=[P^\nu P_\nu,P_\mu]=P^\nu[P_\nu,P_\mu]+[P^\nu,P_\mu]P_\nu=[P^\nu,P_\mu]P_\nu=g^{\nu\alpha}[P_\alpha,P_\mu]P_\nu=0 ##, since ## g^{\nu\alpha} ## is just a number, I can bring it out of the commutator, thus giving me the desired result. Is this last step correct?