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This is an exercise from Peskin&Schroeder's book. The exercise requires to show that $\textbf{J+}$ and $\textbf{J-}$ commute with each other. What is the exact meaning of commutation between operators that are the form of vectors? Do I have to just show that $[\textbf{J+}^i, \textbf{J-}^i]=0$? Or do I have to show that $[\textbf{J+}^i, \textbf{J-}^j]=0$? I am confused...

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  • $\begingroup$ Are you confusing 'vector operator' (i.e. an operator that acts on vectors) with 'operator that is a vector'? $\endgroup$ – Acccumulation Mar 15 '18 at 15:17
  • $\begingroup$ I mean the latter. $\endgroup$ – Keith Mar 15 '18 at 15:18
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    $\begingroup$ You have to show that each component of the first vector commutes with each component of the second. $\endgroup$ – lcv Mar 18 '20 at 21:08
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in this specific case you have to show that $$[J_+^i,J_-^j]=0$$ meaning that every component commutes with every component.

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  • $\begingroup$ Only in this specific case? How about general cases? $\endgroup$ – Keith Mar 15 '18 at 15:13
  • $\begingroup$ The above relation is more general than $[J_+^i,J_-^i]=0$. $\endgroup$ – Ozz Mar 15 '18 at 15:15

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