# Helicity (particle physics)

In particle physics, helicity is the projection of the spin onto the direction of momentum.

## Overview

The angular momentum J is the sum of an orbital angular momentum L and a spin S . The relationship between orbital angular momentum L , the position operator r and the linear momentum (orbit part) p is

${\vec {L}}={\vec {r}}\times {\vec {p}},$ so L 's component in the direction of p is zero. Thus, helicity is just the projection of the spin onto the direction of linear momentum. The helicity of a particle is right-handed if the direction of its spin is the same as the direction of its motion and left-handed if opposite. Helicity is conserved.

Because the eigenvalues of spin with respect to an axis have discrete values, the eigenvalues of helicity are also discrete. For a massive particle of spin S, the eigenvalues of helicity are S, S − 1, S − 2, ..., −S.:12 In massless particles, not all of these correspond to physical degrees of freedom: for example, the photon is a massless spin 1 particle with helicity eigenvalues −1 and +1, and the eigenvalue 0 is not physically present.

All known spin ​12 particles have non-zero mass; however, for hypothetical massless spin ​12 particles, helicity is equivalent to the chirality operator multiplied by ​12ħ. By contrast, for massive particles, distinct chirality states (e.g., as occur in the weak interaction charges) have both positive and negative helicity components, in ratios proportional to the mass of the particle.

## Little group

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) "translations" and transform as ei under a SE(2) rotation by θ. This is the helicity h representation. There is also another unitary representation which transforms non-trivially 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, there are unitary representations which don't transform under the SE(d − 1) "translations" (the "standard" representations) and "continuous spin" representations.