Classical vs. Quantum interpretation of spin 4-vector

[Zoot]

I have a few basic questions about the Pauli-Lubanski spin 4-vector S.

1. I've used it in quantum mechanical calculations as an operator, that is to say each of the components of S is a matrix operator that operates on an eigenvector or eigenspinor. But my question is about the utility of S in a classical sense, that is to say it represents the physical spin angular momentum. For example, in an electron's rest frame, is the spin 4-vector for the case spin-up along the z-axis given by S = (0, 0, 0, h/2) and for spin-down along x we have S = (0, -h/2, 0, 0) etc?

2. I know that in the particle's rest frame S = (0, Sx, Sy, Sz) where the spatial components are the spin angular momentum 3-vector components. However, when we Lorentz boost S, the time component is no longer zero. In this boosted case, do the 3 spatial components still give the spin angular momentum 3-vector (analogous to the case for 4-momentum where the 3 spatial components always give the 3-momentum), or do the spatial components now mean something else? The reason I'm not sure is that some 4-vectors, e.g. 4-velocity, have spatial components that do not represent 3-velocity at all since they may be superluminal, etc.

Thanks for any help on this!

 

[DrDu]

You should try to express the relativistic total angular momentum tensor in it's rest frame in terms of the LS vector. Then look how this tensor transforms under boosts!