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The little group of a momentum $p$ is its stabilizer in the Lorentz group. Its unitary irreducible representations label the internal degrees of freedom of one-particle states on the orbit of $p$.

These stabilizer representations are used in Wigner's classification of relativistic particle states.

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.

Consider positive-energy unitary irreducible representations of the proper orthochronous Poincaré group, allowing its double cover for physical half-integer spin. Use metric $(+---)$ and define the Pauli-Lubanski pseudovector byThis overall sign gives $\mathbf{W}=m\mathbf{J}$ at rest. The antisymmetric tensor and commuting translation generators give $P_{\mu }{W}^{\mu }=0$. The given commutators say $W$ commutes with translations and transforms as a Lorentz vector. Hence both $P^2$ and $W^2$ commute with the Poincare algebra; they are Casimir operators.

The commuting Hermitian translation generators admit generalized joint momentum eigenstates. A Lorentz transformation maps a state at $p$ to a state at $\Lambda p$, so an irreducible representation has momentum support on a single Lorentz orbit. Choose a standard momentum $k$. Its little group consists of the transformations fixing $k$ and acts on the internal states there. Transport those internal states over the orbit by chosen transformations $L(p)k=p$. This gives the construction underlying Wigner's classification.

For timelike future-directed momenta, $P^2=m^2$ with $m>0$ and $k=(m,0,0,0)$. The massive particle little group is SO(3), or SU(2) on the double cover. Write $J_i={ϵ}_{ijk}d(M^{jk})/2$. The rotation commutators give $[J_i,J_j]=i{ϵ}_{ijk}{J}_k$, so the irreducible little-group representation hasAt rest, $W^0=0$ and $\mathbf{W}=m\mathbf{J}$. Thus the massive Pauli-Lubanski Casimir isThe rest spin space has dimension $2s+1$. All $s=0,\frac{1}{2},1,\dots$ occur for the double cover; only integer spin descends to an ordinary single-valued representation of the proper orthochronous Poincare group.

For the detailed representation, choose a Lorentz boost $L(p)$ from $k$ to every point of the positive-energy mass shell. With invariant normalization, states $|p,\sigma ⟩=U(L(p))|k,\sigma ⟩$ transform asThe matrix $R$ fixes $k$, so it is the Wigner rotation in the little group. Its cocycle identityproves the representation law. Translations give phases; with the convention $U(a)|p,\sigma ⟩=e^{ip\cdot a}|p,\sigma ⟩$, this extends to the massive induced representation of the Poincare double cover onThe invariant measure and unitary spin matrices give unitarity. A closed invariant subspace restricts at fixed momentum to a little-group invariant spin subspace; irreducibility of that spin representation and transitivity over the mass shell force the full representation to be irreducible. Conversely the momentum-orbit construction recovers every massive positive-energy irreducible representation. The labels are therefore $(m,s)$.

For a nonzero future null momentum choose $k=(E,0,0,E)$. The massless particle little group is $ISO(2)$. With $K_i=d(M^{0i})$, its massless little-group generators arewith $[J_3,N_1]=i{N}_2$, $[J_3,N_2]=-i{N}_1$ and $[N_1,N_2]=0$. Evaluating the Pauli-Lubanski vector gives $W^2=-E^2(N_1^2+N_2^2)$.

In a finite-dimensional unitary little-group representation, the joint spectrum of the commuting Hermitian $N_1,N_2$ is finite and invariant under continuous planar rotations. Its only possible values are zero. Thus finite-dimensional unitary representations of the massless little group have trivial translations. Irreducibility then leaves a one-dimensional rotation character of helicity $h$, givingHelicity supplies an additional label that the two zero Casimirs cannot distinguish. It is integer for the ordinary group and may be half-integer on its physical double cover. Nontrivial little-group translations instead lead to an infinite-dimensional continuous-spin representation with $W^2<0$; the Poincare algebra by itself allows these too. This completes the timelike and nonzero null cases of physical interest.

Use units $c=\hslash =1$ and the Minkowski metric $g=\mathrm{diag}(1,-1,-1,-1)$. Composing two affine Poincaré group transformations, with the rightmost applied first, givesThe translations form a normal subgroup, so this is the semidirect product of the Lorentz group with ${\mathbb{R}}^{1,3}$. The transformations $(\mathrm{diag}(1,R),0)$, $R\in SO(3)$, fix time and the spatial origin. Their products and inverses correspond to $R{R}^{\prime }$ and $R^{-1}$, hence they form a subgroup isomorphic to SO(3).

A rotationless Lorentz boost changes the velocity of a frame without adding a spatial rotation. For $|\mathbf{v}|<1$ a convenient active convention iswith $B(0)=I$. It satisfies $B^{T}gB=g$, $\det B=1$, and $B(m,0)=(m\gamma ,m\gamma \mathbf{v})$. Collinear boosts form a one-parameter subgroup with additive rapidity, but general boosts do not. For nonzero boosts along $x$ and $y$, $(B_x{B}_y{)}_{02}={\gamma }_x{\gamma }_y{v}_y$ while $(B_x{B}_y{)}_{20}={\gamma }_y{v}_y$: their product is not symmetric, whereas every rotationless boost above is symmetric. The product contains a Wigner rotation. Infinitesimally, the Poincare algebra gives $[K_i,K_j]=-i{ϵ}_{ijk}{J}_k$ for Hermitian generators, so the boost generators do not close amongst themselves.

There is a group-versus-cover qualification in the spin range: an honest unitary representation of the proper orthochronous Poincaré group permits integer spin only. Half-integer spin requires its double cover $SL(2,\mathbb{C})⋉{\mathbb{R}}^{1,3}$, or a projective unitary representation of the ordinary group. The construction below gives the requested range on that double cover and descends for integer spin.

Fix $m>0$ and the rest momentum $k=(m,0)$. Its little group consists of rotations: if a proper orthochronous Lorentz transformation fixes $k$, its time column is $(1,0)$, the metric condition forces the corresponding time row, and the remaining block belongs to SO(3). On the Lorentz spinor double cover, this little group is SU(2). Choose its irreducible spin-$s$ representation, with rest-frame states $|k,s,\sigma ⟩$, $\sigma =-s,-s+1,\dots ,s$, and matrices $D^{(s)}$, the Wigner D-matrices.

For each $p$ on the forward mass shell, $p^2=m^2$, $p^0>0$, choose the standard rotationless Lorentz boost $L(p)=B(\mathbf{p}/p^0)$ and defineThe labels are canonical spin projections, defined by this choice of boosts. For any proper orthochronous $\Lambda$, factorThe last factor fixes $k$ and hence is a Wigner rotation. On the double cover, use its lift to SU(2). Taking the translation convention $U[I,a]|p,s,\sigma ⟩=e^{ip\cdot a}|p,s,\sigma ⟩$ yields the massive induced representation of the Poincare double cover:The argument of the translation phase is $\Lambda p$, since $U[\Lambda ,a]=U[I,a]U[\Lambda ,0]$. The Wigner rotation identityproves the Lorentz part of the representation law; the phases multiply according to the affine product already derived.

Normalize the generalized momentum states by $⟨p,s,\sigma |q,s,\tau ⟩=2p^0(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q}){\delta }_{\sigma \tau }$. The invariant mass shell measure $d^{3}p/(2p^0)$ and unitarity of $D^{(s)}$ make the induced representation unitary on $L^2(d^{3}p/(2p^0))\otimes {\mathbb{C}}^{2s+1}$. This covariant normalization accounts for the absence of an energy square-root factor in the displayed transformation law. To see irreducibility, translations force an operator in the commutant to act within momentum fibres; the little-group action and Schur lemma make its action scalar on each irreducible spin fibre. Lorentz transitivity on the forward mass shell makes this scalar constant. The commutant is therefore scalar, so the unitary representation is irreducible. Conversely, the simultaneous translation spectrum and its little-group representation give the massive positive-energy sector of Wigner's classification.

Thus every $m>0$ and $s\in \{0,\frac{1}{2},1,\dots \}$ gives such an irreducible representation on the double cover. Its central element over a $2\pi$ spatial rotation acts as $(-1{)}^{2s}$. Since this rotation is the identity in the ordinary Poincaré group,

For the parity operator, put $\Pi =\mathrm{diag}(1,-1,-1,-1)$ and $\bar{p}=\Pi p=(p^0,-\mathbf{p})$. The extension obeys $PU[\Lambda ,a]P^{-1}=U[\Pi \Lambda \Pi ,\Pi a]$. At rest, $P$ commutes with the rotations; Schur lemma therefore gives $P|k,s,\sigma ⟩={\eta }_P|k,s,\sigma ⟩$ with a common phase $|{\eta }_P|=1$. Since $\Pi L(p)\Pi =L(\bar{p})$, the parity action on canonical massive spin states isFor the usual reflection convention $P^2=I$, ${\eta }_P=\pm 1$ is the intrinsic parity. Canonical spin is an axial quantity and its label $\sigma$ is unchanged; a helicity label instead changes sign because momentum is reversed. Either intrinsic parity extends a given massive spin representation.

The connected $(1+2)$-dimensional Poincaré group isThe semidirect product records that Lorentz transformations act nontrivially on translations, which is what organizes momentum orbits. For a representative momentum $p$, its little group is the stabilizer $G_p=\{\Lambda \in S{O}^{+}(1,2):\Lambda p=p\}$. Wigner's classification constructs irreducible particle representations by inducing a unitary irreducible representation of $G_p$ along the Lorentz orbit of $p$.