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The little group of a massive four-momentum is $SO(3)$, so a massive spin-one one-particle state has $2s+1=3$ polarizations. For $p^{\mu }=(E,0,0,p)$ and metric signature $(+---)$, one convenient basis isAll three obey $p_{\mu }{ϵ}^{\mu }=0$ and ${ϵ}^{\ast }\cdot ϵ=-1$. Their completeness relation isFor a massless momentum, the finite-helicity representations of the $ISO(2)$ little group carry only the two helicity states $\lambda =\pm 1$. Gauge equivalence removes the timelike and longitudinal polarizations; correspondingly, ${ϵ}_L$ has no finite $M_X\to 0$ limit.

The Wigner classification labels a massive one-particle state by $|p;j,m_s⟩$, where $p^2=m^2>0$, $p^0>0$, $j$ is the spin representation of the $SO(3)$ little group, and $m_s=-j,\dots ,j$. A finite-spin massless state is $|p;\lambda ⟩$, where $p^2=0$, $p^0>0$, and the helicity $\lambda$ labels a one-dimensional representation of the rotational part of the $ISO(2)$ little group. A parity-invariant theory pairs nonzero helicities $\lambda$ and $-\lambda$.

The physical polarization counts areA scalar obeys the Klein-Gordon equation. A spinor obeys the Dirac equation, and a massless irreducible spinor additionally has a fixed chirality. A massive vector obeys the Proca equation, whose divergence gives ${\partial }_{\mu }{A}^{\mu }=0$ and leaves three polarizations. A massless vector instead has the gauge redundancy $A_{\mu }\sim A_{\mu }+{\partial }_{\mu }\alpha$, leaving two transverse polarizations. A massive symmetric tensor obeys the Fierz-Pauli equations: symmetry, transversality, and tracelessness leave five components. A massless tensor has linearized diffeomorphism redundancy $h_{\mu \nu }\sim h_{\mu \nu }+{\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$, leaving helicities $\pm 2$. A redundancy identifies field configurations representing the same physical state; gauge invariance is invariance under that identification.

The helicity of a massless state is the component of angular momentum along its momentum,When the translation part of the massless little group is trivial, the Pauli-Lubanski pseudovector obeysA rotation through angle $\theta$ about $\mathbf{P}$ acts by the phase $e^{-i\lambda \theta }$. On the double cover, a $4\pi$ rotation is the identity, soHence $\lambda =0,\pm \frac{1}{2},\pm 1,\pm \frac{3}{2},\dots$. A parity-invariant theory pairs the $\lambda$ and $-\lambda$ representations, although one chiral massless representation need not contain both.