Helicity (particle physics)


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

Overview

The angular momentum is the sum of an orbital angular momentum and a spin. The relationship between orbital angular momentum, the position operator and the linear momentum is
so 's component in the direction of 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, the eigenvalues of helicity are,,,..., −. 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 particles have non-zero mass; however, for hypothetical massless spin particles, helicity is equivalent to the chirality operator multiplied by. By contrast, for massive particles, distinct chirality states have both positive and negative helicity components, in ratios proportional to the mass of the particle.

Little group

In dimensions, the little group for a massless particle is the double cover of SE. This has unitary representations which are invariant under the SE "translations" and transform as under a SE rotation by. This is the helicity representation. There is also another unitary representation which transforms non-trivially under the SE translations. This is the continuous spin representation.
In dimensions, the little group is the double cover of SE. As before, there are unitary representations which don't transform under the SE "translations" and "continuous spin" representations.