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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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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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