I don't think that Sasha made a mistake. I'll use the dotted/undotted notation which may clarify the possible SL(2,C) invariants. Let ξA, θA, η˙A and ϕ˙A be Weyl spinors. The Levi-Civita tensors ϵAB and ϵ˙A˙B transform trivially under SL(2,C) so they can be used to lower indices. The consistent rules are,
ξA=ξBϵBA
and,
η˙A=ϵ˙A˙Bη˙B
The SL(2,C) invariant Levi-Civita tensors are just similarity transformations which connect equivalent SL(2,C) irreps.
Using upstairs and downstairs indices and complex conjugation
(∗), one can make four SL(2,C) invariants,
ξAθA,
η˙Aϕ˙A,
(ξA)∗η˙A=[ξ∗]˙Aη˙A
and
ξA(η˙A)∗=ξA[η∗]A
The first and second are invariant under parity. The third and fourth are not invariant under parity. By adding and subtracting the third and fourth SL(2,C) invariants, one can make Sasha's bilinear forms
D1 and
D2.
D1 transforms trivially under parity whilst
D2 changes sign under parity. Thus
D1 is an
O(3,1) scalar and
D2 is an
O(3,1) pseudoscalar. Sasha's invariant
χTLσ2ψL is my
η˙Aϕ˙A modulo a factor of
i so Sasha's O(3,1) invariant
D3 is made by summing my first and second invariants.
Edit:
My earlier draft said, "I don't see any contradiction with representation theory here because I don't see any reason for the expansion of Dirac spinors
[(1/2,0)⊕(0,1/2)]⊗[(1/2,0)⊕(0,1/2)]
to exhaust the SL(2,C) invariants of the Weyl spinors." On reflection, my words were wrong. All I've done here is to list the bilinear O(3,1) invariants . I guess Sasha wants to see the decomposition of a general rank 2 Dirac tensor into O(3,1) irreps, I haven't done this part.
This post imported from StackExchange Physics at 2014-04-13 14:36 (UCT), posted by SE-user Stephen Blake