Lorentz spinors appear as irreducible representations of the group SL(2,C). Elements of the group are 2x2 matrices with complex entries and unity determinant. A Lorentz spinor is a two component vector ψA,χA∈V2 with A=1,2. The Levi-Civita tensor ϵAB is an invariant tensor under SL(2,C). This means that if we have an irrep ψA of SL(2,C) we can transform it with the Levi-Civita tensor and this will give an equivalent irrep. So, the covariant vectors ψA∈˜V2 made as ψA=ψBϵBA are equivalent to the contravariant vectors ψA; it doesn't matter whether we use ψA or ψA because both quantities represent the same physical thing.
The situation is the same as in Minkowski spacetime when we use xμ or xμ=ημλxλ. Covariant and contravariant vectors in Minkowski spacetime are equivalent irreps of the Lorentz group O(1,3) because the metric ημλ is an invariant tensor.
The only problem with using the Levi-Civita tensor to lower a spinor index is that it is antisymmetric so that it matters which index is summed. I've decided to lower spinor indices as ψA=ψBϵBA and so, for consistency, I've got to stick to this convention and not be tempted to use ψA=ϵABψB.
Having picked a lowering convention, I'm forced to raise a spinor index with ψA=ϵABψB because then both operations are consistent.
ψA=ψBϵBA=ϵBCψCϵBA=δCAψC=ψA
Now we can make a SL(2,C) scalar ψAχA=χAψA. The order of the vectors does not matter because the components ψ1,ψ2 are just complex numbers. The following bit of index gymnastics recovers the property in the second equation in soliton's question.
ψAχA=ψBϵBAχA=−ψBχAϵAB=−ψAχA
Everything so far has been classical. When we go over to quantum theory, the spinors are promoted to operators ψA→ˆψA. These operators represent fermions: as operators, they have to obey anticommutation relations, in this case [ˆψA,ˆχB]+=0. So, repeating the last calculation with operators,
ˆψAˆχA=ˆψBϵBAˆχA=−ˆψBˆχAϵAB=−ˆψAˆχA=+ˆχAˆψA
recovers the first equation in soliton's question.
I have to own up to the fact that I've not yet studied supersymmetry, but I think this is what must be going on based on general principles.
This post imported from StackExchange Physics at 2014-12-08 12:15 (UTC), posted by SE-user Stephen Blake