Difference between left- and right-handed, helicity and chirality

What is the difference? I know there is the (almost) same question What's the difference between helicity and chirality? but when a particle is given as left-handed. Is it helicity or chirality?

• Yeesh, this is a clear duplicate, but I don't blame you, because the answers to that old question are all terrible. – knzhou Aug 21 at 23:49
• – Chiral Anomaly Aug 22 at 3:35

When we consider spinors of the Lorentz group $$SO(3,1)$$, recall that the universal covering of $$SO(3,1)^+$$ (the component of the Lorentz group connected to the identity) is isomorphic to $$SL(2,\mathbb C)$$.

Two-component spinors are elements of two-dimensional irreducible modules of $$SL(2,\mathbb C)$$. However, noting that the complexification of the Lie algebra of $$SL(2,\mathbb C)$$ is $$A_1 \oplus A_1$$, there are two inequivalent such modules.

These modules have weights $$(1,0)$$ and $$(0,1)$$, or in physics language, spins $$(\frac12,0)$$ and $$(0,\frac12)$$ respectively. Objects with indices corresponding to each have different transformation properties, namely, for the former,

$$\psi_\alpha \to M^\beta_\alpha \psi_\beta$$

for some $$M\in SL(2,\mathbb C)$$ whereas for the latter,

$$\psi_{\dot\alpha} \to \overline M^{\dot\beta}_{\dot\alpha} \psi_{\dot\beta}.$$

Typically, we refer to the undotted indices as left-handed and the dotted indices as right-handed. Note that in some cases in lower dimension, they are not distinct (which is simpler as one does not need Van der Waerden notation to distinguish them.)

It should be noted they transform in the same way under rotations, but they transform oppositely under boosts, motivating the nomenclature. Normally you may have been introduced to spinors first through the Dirac spinor, which lies in the $$(\frac12,0) \oplus (0,\frac12)$$ representation, being comprised of two chiral spinors.

Helicity is the projection of spin onto momentum of a particle:

$$h = \frac{\vec s\cdot \vec p}{|\vec p|}$$

If a particle with spin-1 moves exactly in the same direction as its spin points (let's say the spin point in $$z$$-direction and it also moves in $$z$$-direction), then the helicity is $$h=+1$$. If it moves in the exact opposite direction, towards $$-z$$, the helicity is $$h=-1$$.

As for the terminology, a particle for which $$h=-|\vec s|$$ is called left-handed, and $$h=-|\vec s|$$ is called right-handed.

Chirality is a property of a particle. As Wikipedia puts it, "it is determined by whether the particle transforms in a right- or left-handed representation of the Poincaré group."

Though terms like "left-chiral" and "right-chiral" might be more suitable, people usually also use the terms left-handed and right-handed when talking about chirality. Another thing to watch out for is that for massless particles, its helicity is the same as its chirality.

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