### Helicity (particle physics)

In particle physics, **helicity** is the projection of the angular momentum onto the direction of momentum. The angular momentum is the sum of an orbital momentum *L* → and a spin *S* →, and by definition its relation to linear momentum *p*→ is

- \vec L = \vec r \times \vec p,

so its component in the direction of *p*→ is zero. Thus, helicity is also the projection of the spin onto the direction of momentum. This quantity is conserved.^{[1]}

Because the eigenvalues of spin with respect to an axis have 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.^{[2]}^{:12}

For massless spin-^{1}⁄_{2} particles, helicity is equivalent to the chirality operator multiplied by ħ/2. 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 e^{ihθ} 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.

## See also

## References