= Solution
Use the metric $\eta=\operatorname{diag}(1,-1,-1,-1)$ and first consider the connected <Poincaré group>, $\mathbb R^{1,3}\rtimes SO^+(1,3)$. Physical half-integer spins require <unitary representations> of its double cover $\mathbb R^{1,3}\rtimes SL(2,\mathbb C)$; genuine representations of the group without this cover permit only integer spin. Write $P^\mu=d(P^\mu)$ and $M^{\mu\nu}=d(M^{\mu\nu})$. Strong continuity gives self-adjoint generators, with <commutators> understood on a common invariant dense domain.
The commuting translation generators have a joint spectral resolution. <Lorentz transformations> carry the spectral four-momentum $p$ to $\Lambda p$, so an <irreducible representation> has its momentum spectral measure concentrated on a single Lorentz orbit. For positive-energy particles the orbits of interest are the future <mass shells> $p^2=m^2>0$, or $p^2=0$ with $p\neq0$ and $p^0>0$. The zero-momentum orbit and spacelike or negative-energy orbits are outside these two particle cases.
Choose $\epsilon^{0123}=1$ and define the <Pauli-Lubanski pseudovector> with the overall sign convention
$$
W^\mu=-\frac12\epsilon^{\mu\nu\rho\sigma}P_\nu M_{\rho\sigma}.
$$
This sign makes the spatial rest-frame vector equal to $m\mathbf J$. The generators' Hermiticity and the antisymmetric epsilon contraction make $W^\mu$ Hermitian: the ordering correction from commuting $P$ through $M$ contracts metric tensors into epsilon and vanishes. The same antisymmetry proves <Pauli-Lubanski orthogonality>, $W^\mu P_\mu=0$.
The assumed <commutators> say that $P^\mu$ and $W^\mu$ transform as four-vectors and that $W$ commutes with all translations. Consequently
$$
C_1=P_\mu P^\mu,\qquad C_2=W_\mu W^\mu
$$
commute with the entire <Poincare algebra>. For instance the two terms in the Lorentz <commutator> with a contracted vector square cancel. The <Casimir operators> therefore have fixed spectral values in a <unitary irreducible representation>, by the <Schur lemma> applied to their spectral projections. A second ingredient, the <little group>, distinguishes the internal states.
Fix a standard momentum $k$ on the orbit. Its <little group> is the subgroup of <Lorentz transformations> leaving $k$ fixed. Choose $L(p)$ with $L(p)k=p$. A <Lorentz transformation> acts on the internal fibre by the <Wigner rotation>
$$
R(\Lambda,p)=L(\Lambda p)^{-1}\Lambda L(p),
\qquad R(\Lambda,p)k=k.
$$
If $d_K$ is a <unitary irreducible representation> of this stabilizer, the <induced representation> acts on wavefunctions as
$$
(U(a,\Lambda)\psi)(p)
=e^{-ia\cdot p}\,
d_K\!\left(R(\Lambda,\Lambda^{-1}p)\right)\psi(\Lambda^{-1}p).
$$
The measure $d^3p/(2p^0)$ on either future <mass shell> is Lorentz invariant, so this action is unitary. The identity
$$
R(\Lambda_1\Lambda_2,p)
=R(\Lambda_1,\Lambda_2p)R(\Lambda_2,p)
$$
proves the group multiplication law. Conversely, covariance of the translation spectral resolution supplies precisely these fibres and their stabilizer action. A commuting operator acts fibrewise; transitivity relates its fibre values, and irreducibility of $d_K$ makes it scalar. This explains why the construction gives, and classifies, the <unitary irreducible representations> on each orbit.
For a timelike momentum choose $k=(m,0,0,0)$. A <Lorentz transformation> fixing it acts only as a rotation of its spatial <orthogonal complement>, so the <massive particle little group> is $SO(3)$, or $SU(2)$ in the double cover. Put
$$
J_i=\frac12\epsilon_{ijk}M^{jk},\qquad K_i=M^{0i}.
$$
The <Poincare algebra> gives $[J_i,J_j]=i\epsilon_{ijk}J_k$. The <unitary irreducible representations> of $SU(2)$ are labelled by $j=0,\frac12,1,\ldots$, with $\mathbf J^2=j(j+1)$ and $J_3$ <eigenvalues> $-j,-j+1,\ldots,j$. Evaluation of the <Pauli-Lubanski pseudovector> at $k$ gives $W^0=0$, $\mathbf W=m\mathbf J$. Thus the <massive Pauli-Lubanski Casimir> and particle labels are
$$
\boxed{P^2=m^2,\qquad W^2=-m^2j(j+1),\qquad
(m,j),\quad 2j+1\text{ spin states}.}
$$
The full <massive induced representation of the Poincare double cover> is infinite-dimensional because momentum ranges continuously over the orbit; its finite internal multiplicity is $2j+1$.
For a null momentum choose $k=(E,0,0,E)$ with $E>0$. The infinitesimal stabilizer generators are
$$
R=J_3,\qquad N_1=K_1-J_2,\qquad N_2=K_2+J_1.
$$
Their <commutators> with momentum, evaluated at $k$, vanish. From the rotation and boost <commutators> of the <Poincare algebra> one obtains
$$
[R,N_1]=iN_2,\qquad [R,N_2]=-iN_1,\qquad[N_1,N_2]=0.
$$
These are the <massless little-group generators> of $ISO(2)$: a planar rotation and two commuting null rotations, which act like translations in the little-group algebra. Hence the <massless particle little group> is the Euclidean plane group, with the appropriate double cover for half-integer spin.
The definition of $W$ gives $W^0=\mathbf P\cdot\mathbf J$ and $\mathbf W=P^0\mathbf J-\mathbf P\times\mathbf K$. At the standard null momentum,
$$
W^\mu=(ER,EN_2,-EN_1,ER),\qquad
W^2=-E^2(N_1^2+N_2^2).
$$
Suppose the little-group representation has finite <dimension>, as for particles with finitely many polarizations. The commuting Hermitian $N_1,N_2$ have a finite joint spectrum. Conjugation by $e^{-i\phi R}$ rotates this spectrum through every planar angle. The only finite set invariant under all these rotations is the origin. Thus both $N_i$ vanish, proving that <finite-dimensional unitary representations of the massless little group have trivial translations>. Irreducibility then leaves a one-dimensional rotation character, $R=hI$. The <massless longitudinal Pauli-Lubanski eigenvalues> are
$$
\boxed{P^2=0,\qquad W^\mu=hP^\mu,\qquad W^2=0,\qquad
h\in\tfrac12\mathbb Z}
$$
for the double cover, with $h\in\mathbb Z$ for a genuine $SO(2)$ character. The number $h$ is <helicity>. Different helicities have the same two <Casimir eigenvalues>, so the <little group> is indispensable for the classification. A connected-group <irreducible representation> has one <helicity>; if parity is also implemented, it exchanges $h$ and $-h$ and normally requires their pair.
Unitarity alone also permits nontrivial null-rotation generators. The rotation-invariant operator $N_1^2+N_2^2$ is scalar in an irreducible little-group representation; if it has value $(\rho/E)^2>0$, one obtains a <continuous-spin representation>. A concrete realization on angular wavefunctions is
$$
N_1=\frac{\rho}{E}\cos\phi,\qquad
N_2=\frac{\rho}{E}\sin\phi,\qquad
R=-i\frac{d}{d\phi}.
$$
These operators satisfy the same <commutators>. Periodic or antiperiodic angular wavefunctions give integer or half-integer rotation modes, and the translation generators connect infinitely many such modes. The invariant labels are
$$
\boxed{P^2=0,\qquad W^2=-\rho^2,\qquad\rho>0.}
$$
These are the additional infinite-polarization massless <unitary irreducible representations>. Restricting to finitely many particle polarizations excludes them; they must not be ruled out merely by assuming unitarity. This completes the timelike and null cases of the <Wigner classification>.
Back to article page
