So, we can break down the Dirac equation into 4 "component" equations for the wave function. I was going to post a question here a few days ago asking if a fermion (electron) could possess a "spin" even if it were at rest, I.e., p=0. I did an internet scan, though, and found out that, indeed, you can have zero momentum and still be in a spin up or down state. Why is that? What's the purpose of the Pauli "spin" matrices if you don't need them to imbue a particle with spin? What's their purpose? From what I gather from Viascience, if you're in a rest state, you can be in "pure" up or down spin state, but once you start moving, you confound that pure state when you start moving and add momentum to the equation. But, my central question remains. The Dirac equation is a 4 component coupled equation with 4 solutions. The first two are positive energy solutions and the next two are negative energy solutions. However, if you set the momentum to zero, there doesn't seem to be anything in the math that would suggest a "spin." Having a spin would seem to be a 3-D property. Once you set the momentum to zero, it seems as if the contribution of the spin matrices are irrelevant. The only thing I can think of is that perhaps there is something intrinsic to the two (say positive energy) solutions that imbue a spin up or spin down character just by virtue of the equation itself?