[wyager]

Exactly. Didn't want to confuse it with the normal lowercase `m` that might show up in a wave equation derivation for masses-on-a-string or whatever

Should probably have mentioned - the final equation in my derivation is the Klein-Gordon equation, which is a relativistic equation for the behavior of spinless particles (and maybe bosons in general? I forget)

To get an equation that describes fermion behavior, you need to do another step, which I believe Dirac was the first to do; try very hard to take the square root of both sides of this equation, so you only have first-order derivatives. Dirac really dislike the idea of having a second-order equation, because it leaves an extra initial condition you have to specify. If you expand the p^u p_u term, you can see that it's impossible to take the square root of both sides using normal algebra, because you're trying to take the square root of the sum of multiple terms (d^2/dt^2 - d^2/dx^2 - d^2/dy^2 - d^2/dz^2) . You have to introduce gamma matrices or clifford algebras (IMO the better option) to do it, which seems like a weird and non-physically-motivated approach, but if you do it, spin up and spin down states miraculously fall out of the equation. Eigenchris on youtube has a video that helped me to figure out what was going on there