# Vector operators in quantum mechanics

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.

• In English, only capitalize proper nouns and the first word of each sentence. Jan 20 '16 at 0:12
• Jan 20 '16 at 8:25
• @DanielSank Welcome to English SE! Jan 20 '16 at 12:53
• @N.S.JOHN Something wrong with helping folks communicate their physics questions more clearly? Jan 20 '16 at 17:32