Helicity (particle physics)

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:This article is about helicity in particle physics. For other uses of the term see helicity.

In particle physics, helicity is the projection of the angular momentum to the direction of motion:

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

Because the angular momentum 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".

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 $\hbar/2$ .