Massless particle little group ($ISO(D-2)$)

The stabilizer of a nonzero null four-momentum has connected group $ISO(D-2)$. For ordinary finite-polarization particles its translation part acts trivially, leaving a representation of $SO(D-2)$.

For a future null standard momentum $k$, choose Lorentz transformations $L(p)k=p$ and a unitary irreducible representation $d$ of its little group. The Wigner rotation $R(\Lambda ,p)=L(\Lambda p{)}^{-1}\Lambda L(p)$ acts on the internal state. The resulting induced representation acts on wavefunctions with invariant measure $d^{3}p/(2p^0)$. Trivial little-group translations give fixed helicity; nontrivial translations give a continuous-spin representation with infinitely many polarization states.

In a finite-dimensional unitary representation of $ISO(2)$, its two commuting Hermitian translation generators have a finite joint spectrum. Conjugating by every planar rotation rotates that spectrum. A finite set invariant under all rotations contains only the origin, so both generators vanish. An irreducible representation is then a one-dimensional rotation character, giving a helicity state. Nontrivial translations instead require the infinite-dimensional continuous-spin representation. The same argument applies to the physical double cover of this little group.

For a standard null momentum $(E,0,0,E)$ in signature $(+---)$, define $K_i=M^{0i}$ and $J_i={ϵ}_{ijk}{M}^{jk}/2$. The stabilizing generators are $J_3,N_1,N_2$. The Poincare algebra gives $[J_3,N_1]=i{N}_2$, $[J_3,N_2]=-i{N}_1$ and $[N_1,N_2]=0$, the Lie algebra of planar Euclidean motions. Evaluating the Pauli-Lubanski pseudovector gives $W^2=-E^2(N_1^2+N_2^2)$.

An ordinary massless spin-one particle has transverse vector polarizations under $SO(D-2)$. A covariant polarization satisfies $p\cdot ϵ=0$ and is identified under $ϵ\mapsto ϵ+\lambda p$. The quotient has $D-2$ positive polarizations.