Relativistic QM though maybe more of a math question

[Baggio]

Hi,

I'm a bit befuddled about something my lecturer wrote:


<br /> <br /> S^{\dagger}\sigma_{\alpha}R_{{\alpha}\beta}B_{\beta}S=R_{{\alpha}\beta}B_{\beta}S^{\dagger}\sigma_{\alpha}S<br />

R is a 3x3 rotation matrix which transforms the magnetic field B between frames, sigma_alpha are the pauli matricies. S is a rotation matrix that acts on spin wave vectors

I don't understand wh one can simply move the RB term to the left. It seems to make sense since RB is a vector rotation and S sigma S is a spin rotation operator and so they should be written in this way but I just don't know why mathematically one can do that.

thanks

 

[Perturbation]

R_{\alpha\beta}B_{\beta} are components of a vector, so just numbers, not operators or vectors themselves. The spin and Pauli operators are not operators on a Hilbert space, they're operators on a discrete, finite dimensional vector space, so they're just matrices. Therefore the vector R_{\alpha\beta}B_{\beta} commutes with these matrices. In simple linear algebra, if A and B are matrices and a and b scalars, ABab=abAB. If one wanted to fiddle about with the ordering of the spin and Pauli matrices you'd have to use their commutation relations.