referframe said:
Thank you for the very complete reply. It confirms what I have read so far, that the answer to my question is NOT simple.
haha, ouch, that means I was not clear at all!
It's a nice topic, but indeed full of twists, and it depends on what are you interested. One way of looking at spinors is that they 'generalize' in some sense the kind of 'fields' you can have in space or spacetime. For example, in pre-relativity physics you have a real 3-dimensional vector space with the usual euclidean metric. With this metric you have its special orthogonal group SO(3), the usual 3-d 'rotations'. In this vector space, you have vectors and you can build tensors, and they all transform under the rotations in the usual way. SU(2) spinors are more 'general' entities, and they allow you to recover all this usual 3-d vector algebra. These spinors are the usual ones in non-relativistic QM.
But in relativity, the (real) vector space is 4-dimensional and with a metric of Lorentzian signature, the group is the (for now, restricted; i.e., without space or time inversions) Lorentz group, SO'(1,3) (the ' indicates that we are taking the connected component of the identity). One may naturally ask, is there a similar spinor construction for this vector space? And the answer is yes, SL(2,C) spinors, i.e., you just take a bigger subgroup of GL(2, C). The underlying space on which these matrices act is still a 2-dimensional complex vector space, i.e., these spinors are always two-component entities. This approach can be found in, e.g., http://home.uchicago.edu/~geroch/Links_to_Notes.html on spinors or in chapter 13 of Wald's GR. So, in this approach, they are pretty analogous entities, you only change the underlying groups, real vector space and metric.
Now, in relativistic QM, you want a state space equipped with the (for now, restricted) Poincaré group acting as the relativity symmetry group of the system. This implies that you need a true unitary irrepresentation of its universal cover. You can build explicitly all these possible representations by using the 'Mackey machine' method. The inequivalent classes are labeled by mass and spin (helicity, if m=0). For m>0, spin-1/2 and positive energy (the ones with negative energy give you the antimatter when you consider the wavefunctions as classical fields to be quantized), the representation space is the Hilbert space of certain (tempered-distributional) solutions of the following equation:
##\left(\square+m^{2}\right)\varphi^{A}=0##
where $$\varphi^{A}$$ is a two-component SL(2,C) spinor field in spacetime. The representation is given by the natural action of the Poincaré group on SL(2,C) spinor fields. The structure of the equation simply comes from the usual relativistic formula $$E^{2}-p^{2}=m^{2}$$ (in momentum space, we are taking our functions on the positive energy mass shell, and that's how this formula enters here).
But this is a second order equation. If we define the following auxiliary variable, $$
\sigma_{A'}=\frac{\sqrt{2}}{m}\partial_{A'A}\varphi^{A}$$ (the derivative is just the usual derivative operator generalized to spinors), the second order equation implies $$
\partial^{A'A}\sigma_{A'}=-\frac{m}{\sqrt{2}}\varphi^{A}$$, i.e., the initial second order equation leads to a coupled system of two first order equtions. The implication also works in the other direction, i.e., the system implies the initial equation. If you define the Dirac spinor as $$\psi=(\psi_{0},\psi_{1},\psi_{2},\psi_{3})=(\varphi^{A},\sigma_{A'})$$, you can see that this system is simply the Dirac equation (in the Weyl representation). Also, the transformations laws of SL(2,C) spinors imply the correct transformation law for Dirac spinors. In this sense, SL(2,C) spinors can be seen as the 'atoms' of spinors, from which you can build other 'bigger' spinors in relativity. The 4-components in the Dirac spinor arise when one tries to express the 'wave equation' we got as a first order equation. Notice that the Clifford algebra arises when one tries to write a first order multicomponent relativistic-covariant wave equation (this is what Dirac did). We know that the minimum number of components for that is 4, so everything is self-consistent. Of course, with the two-components SL(2,C) spinors one can see more clearly the connection with the usual spin-1/2 representation of SU(2), which is obscured in the 4-components formalism of Dirac spinors. The usual ways in which you can get real spacetime 4-vectors (the conserved current vector, in particular) from Dirac spinors wavefunctions, can be seen as arising from the fundamental property of SL(2,C) spinors that motivated their introduction, and of course the usual formulas can be obtained in this way.
So, if we are only working with the restricted relativity groups, you can choose second order/two components or first order/four components. But, as
@king vitamin said, if you add parity to the picture, since this operation interchanges the pairs in the Dirac spinors, then one has to use the 4-components formalism.
