You are correct that for a massive spinor, helicity is not Lorentz invariant. For a massless spinor, helicity is Lorentz invariant and coincides with chirality. Chirality is always Lorentz invariant.
Helicity defined
ˆh=→Σ⋅ˆp,
commutes with the Hamiltonian,
[ˆh,H]=0,
but is clearly not Lorentz invariant, because it contains a dot product of a three-momentum.
Chirality defined
γ5=iγ0…γ3,
is Lorentz invariant, but does not commute with the Hamiltonian,
[γ5,H]∝m
because a mass term mixes chirality, mˉψLψR. If m=0, you can show from the massless Dirac equation that γ5=ˆh when acting on a spinor.
Your second answer is closest to the truth:
The weak interaction couples only with left chiral spinors and is not frame/observer dependent.
A left chiral spinor can be written
ψL=12(1+γ5)ψ.
If m=0, the left and right chiral parts of a spinor are independent. They obey separate Dirac equations.
If m≠0, the mass states ψ,
m(ˉψRψL+ˉψLψR)=mˉψψψ=ψL+ψR
are not equal to the interaction states, ψL and ψR. There is a single Dirac equation for ψ that is not separable into two equations of motion (one for ψR and one for ψL).
If an electron, say, is propagating freely, it is a mass eigenstate, with both left and right chiral parts propagating.
This post imported from StackExchange Physics at 2014-04-13 14:39 (UCT), posted by SE-user innisfree