Relativistic QM though maybe more of a math question

[Baggio]

Hi,

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


[tex]

S^{\dagger}\sigma_{\alpha}R_{{\alpha}\beta}B_{\beta}S=R_{{\alpha}\beta}B_{\beta}S^{\dagger}\sigma_{\alpha}S
[/tex]

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]

[itex]R_{\alpha\beta}B_{\beta}[/itex] 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 [itex]R_{\alpha\beta}B_{\beta}[/itex] 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.

 

[denian]

Hi there,

It seems like your lecturer is discussing a concept from relativistic quantum mechanics known as the Wigner rotation. This involves the transformation of spin states between different reference frames. In this case, the equation you provided shows the transformation of the magnetic field B from one frame to another using the rotation matrix R and the spin rotation operator S. The dagger symbol indicates the Hermitian conjugate, which is the transpose and complex conjugate of the matrix.

To answer your question, the reason why one can move the RB term to the left is because of the commutative property of matrices. In other words, the order in which you multiply matrices does not affect the result. This is similar to how you can change the order of multiplication in regular algebraic equations.

I hope this helps clarify things for you. If you have further questions, I suggest discussing them with your lecturer or seeking out additional resources on the Wigner rotation in relativistic quantum mechanics. Keep up the good work in your studies!