Helicity (particle physics)

In particle physics, helicity is the projection of the angular momentum onto the direction of momentum. The angular momentum J → is the sum of an orbital momentum L → and a spin S →; its relation to the position operator r→ and the linear momentum 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. 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

• Wigner's classification
• Pauli–Lubanski pseudovector

References

1. Landau, L D; Lifshitz, E M (2013). Quantum mechanics. A shorter course of theoretical physics. 2. Elsevier. pp. 273–274. ISBN 9781483187228.
2. Troshin, S. M.; Tyurin, N. E. (1994). Spin phenomena in particle interactions. Singapore: World Scientific. ISBN 9789810216924.

• 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. The new physics (1st pbk. ed.). Cambridge, [England]: Cambridge University Press. pp. 458–480. ISBN 9780521438315.