# Can vector operators in quantum mechanics be viewed as the components of some object?

The title pretty much says it all. For example, we sometimes think of the three momentum operators as components of a "vector operator" $$\hat{ \vec{p}}=(\hat p_x,\hat p_y,\hat p_z)$$. The terminology is often justified via commutation relations and can made rigorous using representation theory [1] [2].

But does there exist (or can we construct) a space in which the vector operator $$\hat{\vec p}$$ lives, as an actual vector (or generalized vector) such that the standard operators $$\hat p_i$$ are genuine components? This would be nice because then, for instance, we could derive the transformation laws for how the $$\hat p_i$$ change under a change of basis, rather than imposing them by hand.

A first thought would be something like End$$(H)\times{}$$End$$(H)\times{}$$End$$(H)$$. But this space has infinite dimension [assuming $$H=L^2(\mathbb R^3)$$], and we want a three-dimensional space. We also want the components to be operators, which a standard vector space can't accommodate. But maybe a module would work?

• Related/possible duplicate (at least the final paragraph of my answer there would be my answer to this question, too): physics.stackexchange.com/q/525919/50583 Jul 12, 2023 at 16:28
• I'm not convinced this has anything to do with quantum mechanics per se. You're wondering whether functions from $A\to B^k$ can be seen as belonging to a $k$-dimensional space with components in $A\to B$. Jul 12, 2023 at 20:04