Helicity (particle physics)

From Wikipedia, the free encyclopedia

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 eigenvalues of spin with respect to an axis has discrete values, the eigenvalues of helicity are also discrete. For a particle of spin S, the eigenvalues of helicity are S, (S-1), ..., -S. The measured helicity of a spin S particle will range from -S to + S. Note that helicity can equivalently be written with the total angular momentum operator $\vec J$ , instead of $\vec S$ , because the projection of orbital angular momentum along the linear 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 $\hbar/2$ .