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I read this article about chirality and helicity. At some point it says

For massless particles, chirality is the same as helicity.

But as far as I know, helicity takes form in numbers, $(-1/2, +1/2)$, while chirality takes form in left or right (left chiral, right chiral). So how do these two different properties become the same?

Shiki Ryougi
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Jungwoon Song
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  • For massless particles, the eigenstates of helicity are *also* those of the two chirality projection operators. The "how" is explained in QFT texts describing Dirac spinors. Are you comfortable with those solutions of the Dirac equation? – Cosmas Zachos May 14 '22 at 13:56
  • I think the Wikipedia article means that for massless particles the chirality and helicity are always left, left or right, right and not that the chirality and helicity eigenvalues are the same. – John Rennie May 14 '22 at 14:16
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    related: https://physics.stackexchange.com/q/232591/50583 – ACuriousMind May 14 '22 at 14:32

1 Answers1

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I'm not quite sure this is what you want, but, for massless particles, so the 3-momentum cannot vanish, you can define, in any frame, $\hat p\equiv \vec p /|\vec p|$; hence, you have $$ h=\hat p \cdot \vec S= \frac{1}{2} \gamma^0 \gamma^5 \vec \gamma \cdot \hat p. $$ But this manifestly commutes with chirality, $\gamma^5$, so the two operators share eigenvectors.

Left-chiral ones have negative helicity, and right-chiral ones positive helicity. The numerical magnitude of eigenvalues is irrelevant here: The article focusses on the sign of the helicity.

For more details of how Dirac equation spinors realize the above for both particles and antiparticles, see here. Note how a fermion reverses chirality, spin and helicity upon charge conjugation.

Cosmas Zachos
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