Boost generator transforms as vector under rotations

[LAHLH]

Hi,

I've read quite a few times now in group theory and QFT books that $$[X_i,Y_j]=i\epsilon_{ijk}Y_k$$ can be regarded as saying that $$\vec{Y}$$, the vector of boost generators transforms as a vector under rotations (where X are SO(3) generators).

I don't really understand why this implies this fact, perhaps some could enlighten me.

Thanks

 

[dextercioby]

What's the definition in quantum mechanics for a vector operator ?

 

[LAHLH]

Well I'm guessing this is it, but I don't really understand why.

 

[Petr Mugver]

A 3-vector is a set of three quantities that transform correctly under rotations. In a Hilbert space the unitary rotation operators are

$$U(R)=1+i\mathbf{a}\mathbf{J}$$

where J is the total angular momentum and R is a clockwise rotation of angle |a| around the a/|a| direction. A vector operator Y transforms like

$$U(R)\mathbf{Y}U^\dagger(R)=R\mathbf{Y}$$

If you expand this in first order taylor series (i.e. if you consider infinitesimal rotations), you'll find the commutation relations you mentioned.

 

[LAHLH]

Thanks a lot, that has cleared it up for me.