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Vector operators $\vec{V}$ in quantum mechanics are usually defined as those that commute in a particular way with the spatial Angular Momentum $\vec{L}$: $[L_i,V_j]=i\hbar\varepsilon_{ijk}V_k$. I am not aware of a more general definition, but can we define some sort of "vectoriality" with respect to spin angular momentum (or any set of operators such that $[J_i,J_j]=i\hbar\varepsilon_{ijk}J_k$)?

Is it correct to think that the "vectorial"(or even "tensorial") property of operators is valid for each one of the tensor spaces ($\mathcal{E}_{spatial}\otimes\mathcal{E}_{spin}\otimes \cdots$) available once "spin" degrees of freedom have been added? Maybe there is even some sort of "tensoriality" with respect to a more general algebra of commutation.

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  • $\begingroup$ In English, only capitalize proper nouns and the first word of each sentence. $\endgroup$
    – DanielSank
    Jan 20 '16 at 0:12
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    $\begingroup$ Related: physics.stackexchange.com/q/73532/2451 $\endgroup$
    – Qmechanic
    Jan 20 '16 at 8:25
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    $\begingroup$ @DanielSank Welcome to English SE! $\endgroup$
    – N.S.JOHN
    Jan 20 '16 at 12:53
  • $\begingroup$ @N.S.JOHN Something wrong with helping folks communicate their physics questions more clearly? $\endgroup$
    – DanielSank
    Jan 20 '16 at 17:32

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