Helicity (particle physics)
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.
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.: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.
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 1⁄2ħ. 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.
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 eihθ 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.
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