Little group

The little group of a fixed four-momentum $p$ is its stabilizer subgroup in the Lorentz group: $G_p=\{\Lambda :\Lambda p=p\}$. It changes internal states while leaving that momentum unchanged. In four spacetime dimensions, a massive rest momentum has spatial rotations as its little group; the corresponding Spin group is SU(2), permitting half-integer spin. A nonzero future-directed massless four-momentum instead has the planar Euclidean little-group structure. Its finite-helicity quantum states have trivial translation action. These group representations, combined with the momentum orbit, underlie Wigner's classification.^[ref]

A particle polarization is a physical internal spin state at fixed four-momentum. In $D$ spacetime dimensions, massive states use the rotation little group $SO(D-1)$ or its appropriate Spin group; half-integer spin requires the cover. Finite-spin massless states use the transverse rotation part $SO(D-2)$, with trivial little-group translations. Thus a massive vector has three polarizations in four dimensions, while a photon has two.^[ref] A massive two-form field has $\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$ states, versus $\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)$ for a two-form gauge field, after imposing the field equations and identifying states related by gauge symmetry.

For an on-shell massive vector with $k^{\mu }=(E,|\mathbf{k}|\hat{\mathbf{k}})$ in Minkowski metric signature $+---$, the displayed vector satisfies $k\cdot {ϵ}_L=0$ and ${ϵ}_L^2=-1$. It specifies the longitudinal one of the three physical polarization states. It is not a replacement for a final polarization sum when other states are unobserved.

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.

The stabilizer of a timelike four-momentum in $D$-dimensional Minkowski spacetime has connected group $SO(D-1)$. Massive particle polarizations form representations of this group.

A massive spin-one particle transforms as a vector under its massive particle little group $SO(D-1)$. Its $D-1$ polarizations are orthogonal to its timelike four-momentum. Under transverse $SO(D-2)$ rotations they split into a vector and a scalar.

The symmetry used to classify an open string must preserve its endpoint boundary background as well as its momentum. A Neumann-Dirichlet open-string boundary condition breaks target Lorentz transformations mixing ND and common directions with Neumann boundary conditions. For a D1–D25 pair with 24 ND directions, the common worldvolume has Lorentz group $SO(1,1)$, whose connected massive little group is trivial. The transverse $SO(24)$ remains as an internal rotation group. Thus a massive 24-component oscillator multiplet is not an $SO(25)$ vector of the full 26-dimensional massive little group; a vector of that full group would have 25 components.

A massive vector in $D$ spacetime dimensions has $D-1$ polarizations perpendicular to its timelike momentum. A massless vector has transverse polarizations modulo its null longitudinal polarization, leaving $D-2$ physical degrees of freedom.

Helicity is the projection of angular momentum onto the momentum direction. For a finite-helicity massless particle, the Pauli-Lubanski vector satisfies $W^{\mu }=\lambda P^{\mu }$.

A physical photon has two transverse helicity states. Circular polarization distinguishes their populations; a real photon has no longitudinal helicity-zero polarization.

For a relativistic neutrino, the normalized helicity is approximately $-1$ in ordinary weak production. For a massive particle, helicity and fermion chirality differ; suppressed opposite-helicity amplitudes and frame dependence prevent promoting the massless selection rule to an exact statement about arbitrary boosts.

For $\mathbf{k}=k\hat{z}$, these complex vectors form an orthonormal basis of the transverse plane using the Hermitian inner product. They have opposite phases under rotations about $\hat{z}$, so they describe the two helicity components. Since $\mathbf{e}\cdot m^{(\pm )}=\sin \theta e^{\pm i\phi }/\sqrt{2}$, transverse-vector sources carry only the angular sectors $m=\pm 1$ of spherical harmonics. Streaming by $ik\cos \theta$ preserves $m$ and couples neighboring angular degrees.

A continuous-spin representation is a massless unitary irreducible representation of the Poincare group in which the translation subgroup of the $ISO(2)$ little group acts nontrivially. It contains infinitely many helicities.

Wigner's classification obtains irreducible positive-energy representations of the Poincaré group by inducing from unitary irreducible representations of the little group of a standard momentum.

Fix $m>0$, $k=(m,0)$, and a rotationless Lorentz boost $L(p)$ taking $k$ to $p$ on the positive-energy mass shell. The little group on the Lorentz spinor double cover is SU(2). Its spin-$s$ irreducible representation induces a unitary representation on $L^2(d^{3}p/(2p^0))\otimes {\mathbb{C}}^{2s+1}$. Here $W(\Lambda ,p)=L(\Lambda p{)}^{-1}\Lambda L(p)$ is the Wigner rotation, lifted to SU(2) when necessary, and the translation convention is $U(I,a)|p⟩=e^{ip\cdot a}|p⟩$. The little-group cocycle proves the group law, and Wigner's classification gives irreducibility. Every nonnegative integer or half-integer spin is allowed on the double cover; descent to the ordinary proper orthochronous Poincaré group requires integer spin, because its $2\pi$ rotation must act as the identity.