I have read that the 3-momentum of a relativistic particle cannot be orthogonal to its spin 3-vector. When thinking about how the spin vector transforms when the particle approaches light speed, it seems clear that it cannot be orthogonal to the boost direction, but I'm wondering what is the exact mechanism that prevents it.
An example for clarification: Let's take an electron at rest in the lab frame, which has been put in a spin-up eigenstate ($z$-axis). Then, a potential is added, such that the electron gains some $x$-momentum.
Since the spin was pure $z$ axis and the momentum pure $x$, some relativistic process must change the spin vector to prevent it from being orthogonal to the momentum.
EDIT: I'm starting to think that this must be answered with Dirac spinors. The $x$ momentum must have some effect on the spin vector direction. But I cannot see the quantitative answer, so I'm still looking for an answer.