# What is the difference between a Pauli spinor, a Weyl spinor, and a Cartan spinor?

I know that a spinor is a complex two components "vector", which is acted on by the $SU(2)$ group under a rotation. In the physics litterature, I often read "Weyl spinors", "Pauli spinors", "Cartan spinors". What are the differences ? Aren't they all the same mathematical objects ?

Take note that I'm not talking about the Dirac bi-spinors (a pair of spinors) and all their variations (Majorana spinors and Weyl decomposition).

• I have never heard the terms "Pauli spinor" or "Cartan spinor", although I am aware that Cartan was one of the first to deal with the notion of spinors. Can you add a reference for the usage of these terms? Apr 27, 2017 at 13:37
• It would be a lot of work for me to find back all the books and docs on the internet that use these names. I suspect that it's all about the same mathematical objects (2 components spinors), but I need to be sure, in case there are some subtleties that I'm unaware of. It may simply be related to the historical context where the spinors where introduced in some calculations : "Pauli spinors" when it was about Pauli describing spin, "Cartan spinors" when Cartan introduced his spinors, etc. Or may it be related to the various ways of defining spinors ?
– Cham
Apr 27, 2017 at 13:42
• @ACuriousMind "Pauli spinor" is a term you can find in the literature e.g. in Frescura and Hiley's 1998 paper "Geometric interpretation of the Pauli spinor". dx.doi.org/10.1119/1.12548 Apr 30, 2017 at 19:23
• @ACuriousMind "Cartan Spinors" are discussed e.g. by Budinich, P. & Rigoli, M. Nuov Cim B (1988) 102: 609. doi:10.1007/BF02725619 (Too late to add it into the preceding comment; but as this question is close to my own interests, I'm tempted to try to work up a proper answer, as time allows, over the next few days) Apr 30, 2017 at 19:33
• @iSeeker, thanks ! Is there really a difference ?
– Cham
Apr 30, 2017 at 22:22

Perhaps the shortest answer (from my elementary understanding of representation theory) would be that the differences between the types of spinors you asked for lie primarily in terms of the representations of the rotation group under which they transform: If the spinors are symbolised by ψ, then the transformation rule:

            Ψ’ = M Ψ,  where M is one of the matrix reps of the rotation group.


will have:

M = SL(2,C) for 2-component Weyl (and other relativistic, e.g. Lorentz) spinors, which obey the Weyl equation (the massless form of the Dirac equation)

M = SU(2) for the non-relativistic 2-component (Pauli) spinors, which obey the Schrodinger-Pauli equation – the non-relativistic but massive limit of the Dirac equation (I suspect their components, for normalised wave functions are each restricted to the unit circle in the complex plane.)

M = something much more general than either of the above for Cartan spinors in their most general form (possibly via SO(p,q) for general p, q?) as Cartan claimed his spinors are the most general mathematical form of spinors, and they deal with rotations in spaces of any number of dimensions. They should therefore range beyond 2-component objects (as do even Weyl spinors for e.g. 6-D space).

Although you profess disinterest in Dirac and Majorana spinors you might also like to refer to a comparable type (but much more expert) answer comparing Weyl with Dirac and Majorana spinors.

Meanwhile, as others can probably provide a better answer, I regard this as an opportunity to learn more by inviting corrections; otherwise, I might post a more detailed answer, with references, later.

UPDATE: Abstract of the ref given as this paper in Physics StackExch 381625 states:
"...The physical observables in Schrödinger–Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit." (behind a paywall).