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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?

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I think your last paragraph really hits on the key point here; in a formal sense, you absolutely cannot go down from the four component spinor wave function to a two component function. But, as you also correctly pointed out, the negative energy components of the spinor wavefunction are suppressed in the non-relativistic limit $c\rightarrow \infty$. Thus we just throw away these terms since they are not doing anything for us anymore, and we get the Pauli-Schrodinger equation exactly as you said. So in so far as we are leaving the relativistic limit, the Pauli spinor IS the Dirac spinor, or at least the pieces that correspond to positive energy eigenvalues and are not being suppressed. I've been working out a lot of these details myself lately, so I hope this helps.

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