# Classical vs. Quantum interpretation of spin 4-vector

1. Dec 18, 2012

### 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!

2. Dec 19, 2012

### 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!

Last edited: Dec 19, 2012