How do photons evade the unavoidable consequence of angular momentum algebra?

Question

I think an inescapable consequence of the angular momentum algebra is that a particle with spin-$j$ must have $(2j+1)$ spin projections in any direction. However, photons seem to evade this conclusion. Why?

Answer 1

The technical answer is that spin is property associated with representation of the Poincare group P. The representations are are induced from the "little group" i.e. the sub-group of P that leaves the particle's four momentum fixed. For a massive particle we can go to the rest frame in which the four momentum is $p=(m,0,0,0)$. This vector this is left fixed by rotations SO(3), so for a massive particle spin is a property of rotations and their spin $j$ representations.

For a massless particle there is no rest frame and the reference momentum must be a null vector $p =(|{\bf p}_0|, {\bf p}_0)$. The little group now consists of space rotations SO(2) about the three-vector ${\bf p}$, together with operations that are generated by infinitesimal Lorentz boosts in directions perpendicular to ${\bf p}_0$ combined with compensating infinitesimal rotations. Remarkably the combined operations mutually commute, possess all the algebraic properties of Euclidean translations, and the resulting little group is isomorphic to the symmetry group SE(2) of the two-dimensional Euclidean plane. Wigner argued that the translations must do nothing and so the spin of a massless particle is asociated with the one-dimensional representations of SO(2).

A more intuitive explanation is that for something moving at the speed of light, any vector originally have a component perpendicular to the drection of travel will be Lorentz trasformed to one pointing in the direction of travel

Answer 2

The electromagnetic field is transversal because of charge conservation. The state of polarisation or spin that is perpendicular to the direction of propagation is therefore never excited.