Helicity (particle physics)

In particle physics, helicity is the projection of the spin $vec\; S$ onto the direction of momentum, $hat\; p$::$h\; =\; vec\; Scdot\; hat\; p,qquad\; hat\; p\; =\; vec\; p\; /\; |vec\; p|$

Because the spin with respect to an axis has discrete values, helicity is discrete, too. For spin-1/2 particles such as the electron, the helicity can either be positive ($+hbar/2$) - the particle is then "right-handed" - or negative ($-hbar/2$) - the particle is then "left-handed". Note that helicity can equivalently be written with the total angular momentum operator $vec\; J$, as the contribution from orbital angular momentum vanishes, $vec\; Lcdot\; vec\; p=0$.

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) "translation"s and transform as e^ihθ under a SE(2) rotation by θ. This is the helicity h representation. We also have another unitary representation which transforms nontrivially 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, we have unitary reps which don't transform under the SE(d-1) "translations" (the "standard" reps) and "continuous spin" reps.

For massless (or extremely light) spin-1/2 particles, helicity is equivalent to the operator of chirality multiplied by $hbar/2$.

Etymology

Helicity derives from the Latin "helix", from Greek; akin to Greek eilyein to roll, wrap. [ [http://www.m-w.com/dictionary/helix helix - Definition from the Merriam-Webster Online Dictionary ] ]

ee also

* Wigner's classification

References