# Helicity (particle physics)

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

$h = \vec J\cdot\hat p = \vec L\cdot\hat p + \vec S\cdot \hat p = \vec S\cdot \hat p,\qquad \hat p = \frac{\vec p}{\left|\vec p\right|} ,$

as the projection of orbital angular momentum along the linear momentum is zero, L = 0.

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.

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.

For massless spin-12 particles, helicity is equivalent to the chirality operator multiplied by 12ħ.