# Why the little group of a massless spin-1 particle is ISO(2) rather than SO(2)?

As the title suggests, after going through a lot of Wikipedia, and references of books, I have learned that: The little group of a massive spin-1 particle is SO(3), while the little group of a massless spin-1 particle is ISO(2) or E(2), as far as the unitary representation of Poincare group is concerned.

My question is: Why the little group of a massless spin-1 particle is ISO(2) rather than SO(2)? I don't really see how the translations of the 2-dimensional Euclidean group play a role here (for example, take the momentum to be $$p^\mu=(E,0,0,E)$$ with polarizations $$\epsilon_1=(0,1,0,0)$$ and $$\epsilon_2=(0,0,1,0)$$. Or I can similarly ask why the little group of a massive spin-1 particle is SO(3) rather than ISO(3).

• You need to work with the full symmetry group to ensure your equations that emerge (in this case Maxwell’s equations) possess all the relevant symmetries. As an aside, have you read Wigner’s original article on the matter? He explains this quite clearly IMHO. – Max Lein Sep 7 '20 at 6:19

Here is a brief explanation: In the 3+1D massless case we can assume that the only non-zero component of the momentum $$P_{\mu}$$ is the lightcone coordinate $$P_+$$. Besides the homogeneous $$SO(2)$$ group such momentum also commute with the Lorentz generators $$M_{-i}$$ with index $$i\neq \pm$$, which make up the inhomogeneous part of $$ISO(2)$$.