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schieghoven is offline
Apr28-08, 05:09 AM
P: 78
I found it easiest to construct the theory in terms of (Weyl) 2-spinors - these have 2 complex components and transform according to SL(2). SL(2) is the set of 2-by-2 complex matrices with unit determinant. There are two distinct ways to do this: if [tex] R \in SL(2) [/tex], then a weyl spinor \psi could transform as
[tex] \psi \rightarrow R \psi [/tex]
or as
[tex] \psi \rightarrow (R^\star)^{-1} \psi [/tex]
These are called Left-handed or Right-handed Weyl spinors, respectively. (This is what is meant by the Lorentz group having 2 distinct representations in SL(2)). A Dirac spinor is a pair of one LH Weyl spinor and one RH Weyl spinor.

SL(2) arises because its Lie algebra is isomorphic to that of the Lorentz group SO(1,3) - they both have 6 generators corresponding to a closed subalgebra of 3 infinitesimal rotations, and 3 infinitesimal boosts. This gives a geometric interpretation to spinors: infinitesimal lorentz transformations correspond one-to-one with infinitesimal SL(2) transformations of spinors.

This is possible on curved spacetime as well. In this case we assume a general metric g_ij with Minkowskian signature, and define the symmetry group in terms of this metric. There are 6 generators, which we denote J_ab with antisymmetry in the two indices. The Lie-algebra structure is characterised by the bracket
[J_{ab},{J_{ij}] = g_{ai} J_{bj} + g_{bj} J_{ai} - g_{aj} J_{bi} - g_{bi} J_{aj}
and it is in fact possible to choose 2-by-2 matrices from sl(2) that satisfy this bracket... if you like I can post some further details about how to do this. Anyway the short answer is that the symmetry group corresponding to the general metric g_ij is also SL(2), and has LH and RH representations in much the same way as in flat space. In this way it's possible to define spinors at each point of a curved manifold.

So, I think that spinors require more structure on the manifold than vectors... vectors can be defined independently of the metric, but spinors require a metric.

I think this is a fairly complicated topic, so I hope this condensed summary of results doesn't just cloud the water further.... :-)