From what I'm understanding about Dirac spinors, using the Weyl basis for the γ matrices the first two components behave as a left handed Weyl spinor, while the third and the fourth form a right handed Weyl spinor. By boosting in a direction or in the opposite, I can "asymptotically kill" either the left or right handed part of the (massive) spinor. Since only the left-handed part interacts with the weak force, does that mean that when I see an electron travelling very fast in one direction (same as/opposite to spin) I see/don't see it weakly interacting? This sounds very odd indeed.
I have two hypotheses:
- Massive spinors don't have an intrinsic chirality (since they are not eigenstates of chirality operator), the only information I have is about helicity, and the odd thing I described before is actually observed (really odd to me).
- Massive particles have an intrinsic chirality, but I don't see how the chirality information gets encoded into the Dirac spinor / how the weak interaction couples to only half of it. To me it seems that only the helicity information is carried by a spinor.
Answer
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]∝mbecause 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 mass states ψ, m(ˉψRψL+ˉψLψR)=mˉψψψ=ψL+ψR
If an electron, say, is propagating freely, it is a mass eigenstate, with both left and right chiral parts propagating.
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