# 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

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.[1]

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.[2]^{: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.[3]

All known spin ^{1}⁄_{2} particles have non-zero mass; however, for hypothetical massless spin ^{1}⁄_{2} particles, helicity is equivalent to the chirality operator multiplied by ^{1}⁄_{2}ħ. 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.

## References

- Landau, L.D.; Lifshitz, E.M. (2013).
*Quantum mechanics*. A shorter course of theoretical physics.**2**. Elsevier. pp. 273–274. ISBN 9781483187228. - Troshin, S.M.; Tyurin, N.E. (1994).
*Spin phenomena in particle interactions*. Singapore: World Scientific. ISBN 9789810216924. - Thomson (2011). "Handout 13" (PDF). High Energy Physics. Part III, Particles. U.K.: Cambridge U.

- Povh, Bogdan; Lavelle, Martin; Rith, Klaus; Scholz, Christoph; Zetsche, Frank (2008).
*Particles and nuclei an introduction to the physical concepts*(6th ed.). Berlin: Springer. ISBN 9783540793687. - Schwartz, Matthew D. (2014). "Chirality, helicity and spin".
*Quantum field theory and the standard model*. Cambridge: Cambridge University Press. pp. 185–187. ISBN 9781107034730. - Taylor, John (1992). "Gauge theories in particle physics". In Davies, Paul (ed.).
*The new physics*(1st pbk. ed.). Cambridge, [England]: Cambridge University Press. pp. 458–480. ISBN 9780521438315.