# Helicity is different form Spin for massless particle(photon)?

1. May 26, 2014

### time601

As we know photon's helicity are $\pm$1. Helicity is the projection of the spin S onto the direction of momentum, p, which is considered as Sz.
What about Sx and Sy? They are both ZERO?

2. May 26, 2014

### time601

Is it reasonable to say that massless particles have no spin but just helicity?

3. May 26, 2014

### Bill_K

What we mean by the spin of a particle is the subgroup of the Lorentz group that commutes with its 4-momentum (the "little group"). For a particle with mass, go to its rest frame where the 4-momentum is Pμ = (0,0,0,1) and the spin operators are the rotations in 3-space, Sx, Sy and Sz. They form SO(3).

For a massless particle there is no rest frame, so take the 4-momentum in the z-direction, kμ = (0,0,1,1), and its spin operators are the three operators that preserve kμ.

The first one is a rotation in the (x,y) plane. This is the helicity. It acts on the components of the particle's 4-potential as Ax ± iAy → ±(Ax ± iAy).

The other two are null rotations, x → x + εk and y → y + εk. These operations just add a multiple of k to the 4-potential. But this is just a gauge transformation. So helicity is the only observable part.