While I'm fairly familiar with non-relativistic quantum mechanics, I have just recently started delving into relativistic QM, my main reference being Sakurai's "Advanced Quantum Mechanics" (chapter 3 specifically, which deals with spin 1/2 particles). I think I understood the basic mathematical arguments as to why the state of such particles in a relativistic regime has to be described by a four-component object (the Dirac spinor). What I don't understand is how to "go back" to the Pauli-Schrödinger two-component spinor. More concretely: given a Dirac spinor, how can I find the corresponding Pauli spinor?
From Sakurai's words, I already know that in the nonrelativistic regime $E \approx mc^2$ the upper two components of the Dirac spinor coincide with the Pauli-Schrödinger spinor, and so if I undestand correctly, I could (in this regime) "throw away" the lower half of the spinor and get the two-component state I want. But in the general case I see no hints as to how I can find such correspondence, and even IF there is such a correspondence.
Now, my first guess would be that, as velocities become relativistic, the other two components become more promiment and relevant, and you would have to abandon the 2 component description altogether. My problem with that is that, in this article by A. Peres, P. F. Scudo, and D. R. Terno: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.88.230402 they do a relativistic treatment of a spin-1/2 particle using two-component wavefunctions, which seem to coincide with the usual non-relativistic Pauli spinor. In fact, the article's text-book reference (F. Halpern's "Special Relativity and Quantum Mechanics") gives "two by two representations" of the Lorentz transformations, which seem to be SU(2) matrices which act on a two-component wavefunction. Peres et al. seem to use it in this manner at least.
So, what am I missing here?