Helicity (particle physics)

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In particle physics, helicity is the projection of the spin \vec S onto the direction of momentum, \hat p :

h = \vec S\cdot \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 L\cdot \vec p=0.

Image:Right left helicity.jpg

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 eihθ 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.

Helicity derives from the Latin "helix", from Greek; akin to Greek eilyein to roll, wrap.[1]

  1. ^ http://www.m-w.com/dictionary/helix
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