Helicity (particle physics)

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This article is about helicity in particle physics. For other uses, see helicity.

In particle physics, helicity is the projection of the spin $\vec S$ onto the direction of momentum, $\hat p$ :

$h = \vec S\cdot \hat p,\qquad \hat p = \vec p / |\vec p|$

Because the spin with respect to an axis has discrete values, helicity is discrete, too. For spin-1/2 particles such as the electron, the helicity can either be positive ( $+\hbar/2$ ) - the particle is then "right-handed" - or negative ( $-\hbar/2$ ) - the particle is then "left-handed". Note that helicity can equivalently be written with the total angular momentum operator $\vec J$ , as the contribution from orbital angular momentum vanishes, $\vec L\cdot \vec p=0$ .

In 3+1 dimensions, the little group for a massless particle is the double cover of SE(2). This has unitary representations which are invariant under the SE(2) "translation"s and transform as e^ihθ under a SE(2) rotation by θ. This is the helicity h representation. We also have another unitary representation which transforms nontrivially under the SE(2) translations. This is the continuous spin representation.

In d+1 dimensions, the little group is the double cover of SE(d-1) (the case where d<=2 is more complicated because of anyons, etc). As before, we have unitary reps which don't transform under the SE(d-1) "translations" (the "standard" reps) and "continuous spin" reps.

For massless (or extremely light) spin-1/2 particles, helicity is equivalent to the operator of chirality multiplied by $\hbar/2$ .