# How do massless particles have the same chirality and helicity when they are different properties?

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?

• 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? Commented May 14, 2022 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. Commented May 14, 2022 at 14:16
• Commented May 14, 2022 at 14:32

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.