Question

What is the 4-spin vector of a photon?

Answer 1

Electrons have spin 1/2, and as they are charged, they also have an associated magnetic moment, which can be measured by an electron beam splitting up in an inhomogeneous magnetic field or through the interaction of the electrons's magnetic moment with an external magnetic field in spectroscopic measurements.

Answer 2

Massless particles are a representation of a different little group than massive particles. That's the fancy group theory way of saying that photons have no rest frame. More formally, the little group is the subgroup of Lorentz transformations that leaves the 4-momentum invariant.

It is true that the little group of massive particles, SO(3)SO(3), has 2s+12s+1 states associated with a spin ss particle.

The little group of massless participles is ISO(2)ISO(2), which is different. Massless spin-0 particles have 1 degree of freedom. All higher spin-ss particles have 2 degrees of freedom. (There's a tricky topological argument for this given for example in Weinberg Vol 1, chapter 2). A massless spin s>0s>0 particle has helicity h=±sh=±s states. Helicity is defined by transformation under rotation about the 3-momentum. Under rotation by an angle θθ about the 3-momentum, a helicity hhstate transforms as |h⟩→eihθ|h⟩|h⟩→eihθ|h⟩.