Checking if Three Operators Form a 3-Vector with SO(3) Commutation Relations

[jfy4]

Hello,

Lets say I have three operators

$$k_3=\partial_\phi$$, $$k_1=\sin\phi\partial_\theta+\cot\theta\cos\phi\partial_\phi$$, $$k_2=\cos\phi\partial_\theta-\cot\theta\sin\phi\partial_\phi$$.

These operators satisfy the SO(3) commutation relations:

$$[k_i,k_j]=\epsilon_{ijk}k_k$$

How can I check to see if these three operators together form a 3-vector?

 

[Bill_K]

In general you prove a set of operators V_i form a vector by showing how they transform under rotations, namely [L_i, V_j] = e_ijk V_k. But in this case, your operators V_i already are L_i!

I think what I'd do is to start out by defining L_i in Cartesian coordinates, L_z = x∂_y - y∂_x, etc, transform them to spherical coordinates and show they reduce to your V_i. That substantially proves it.

 

[jfy4]

In general you prove a set of operators V_i form a vector by showing how they transform under rotations, namely [L_i, V_j] = e_ijk V_k. But in this case, your operators V_i already are L_i!

I think what I'd do is to start out by defining L_i in Cartesian coordinates, L_z = x∂_y - y∂_x, etc, transform them to spherical coordinates and show they reduce to your V_i. That substantially proves it.

Ok, I have managed to show that they come from the Cartesian coordinates, however, It still does not seem to me that these 3 operators are specifically assigned to anyone component of a vector. That is,

$$L_x=z\partial_y-y\partial_z$$

but after the transformation

$$L_x=\sin\phi\partial_\theta+\cot\theta\cos\phi\partial_\phi$$

What component would this correspond to in $$(r,\theta,\phi)$$ coordinates? How can I determine what component these three operators correspond too?

 

[jfy4]

bump.