If space and time are equivalent, what's Spin in time dimension This troubles me: We are talking about time and space being equivalent, but still only consider Spin in the $x$, $y$ or $z$-direction. What's Spin in time dimension? Is it distinction between particles and antiparticles?
 A: It is not true that "we" consider spin only in spatial directions, except "we" means something like "undergraduate students", maybe. Instead relativistic physics is all controled by spacetime spinors, namely by representatins of the double cover of the Poincaré  Lie group of spacetime translations, rotations and boosts.
Maybe the best way to get an intuitive feeling for what a timelike spinor "means", physically, is to see how two such spinors combine into a twistor and how that endodes momentum and chirality of massless particles. 
A: If you want to do ordinary non-relativistic quantum mechanics with electrons without having to put spin in "by hand", you would start with the Dirac equation $(i\gamma^\mu \partial_\mu - m) \psi = 0$ and use this to derive the Schrödinger equation as the $|\mathbf{p}|^2 \ll m^2$ limit. The field $\psi$ is a 4-component spinor, although the number 4 is misleading (it would also have 4 components in 5 spacetime dimensions).
In any event, if you go to the rest frame of the electron, i.e., $\mathbf{p}=0$, then there are 4 solutions to the Dirac equation: $\uparrow, \downarrow$ with energy $E=m$ (electrons with spin up, down respectively) and $\uparrow, \downarrow$ with energy $E=-m$ (positrons).
For $\mathbf{p} \ne 0$, the 4 solutions become coupled, and instead of spin, it is better to talk about helicity or chirality.
A: The thing with spin is that it does not quite have a physical interpretation, it is rather a mathematical formulation for the intrinsic angular momentum of a particle. Spin corresponds to an orthogonal translation of a Lorentz Group (a lie group), which one can usually represent in terms of spinors. Spinors have very interesting, and complex, properties, which are the reason as to why their behavior is usually in time dimensions become so hard to understand.
Let me know if you want a further explanation on the subject.
Hope it helped!
