Wigner Classification Preview

Wigner’s particle construction combines a momentum orbit with a unitary representation of its little group. It explains why mass and spin or helicity label a free particle and why a boost can rotate its spin label in a momentum-dependent way. This page constructs the positive-energy massive and finite-helicity massless cases. It is a working introduction, not a classification of every orbit or a derivation of interactions and statistics.

Required background. Massive and Massless Representations supplies the little groups, and Relativistic Phase Space derives the invariant measure. Helpful background. Free Dirac Spinors separates spinor coefficients from energy-sector states.

Standard boosts define the spin labels

Use natural units and the connected Poincaré cover. First take $m>0$, a standard momentum $k=(m,0,0,0)$, and orthogonal rest-spin labels $\sigma=-s,\ldots,s$. Choose a Lorentz transformation $L(p)$ with

$$L(p)k=p,\qquad p^0=E_{\mathbf p}>0.$$

Define momentum-spin kets, with a consistent normalization, by transporting the rest basis with $U(L(p))$. The choice of $L(p)$ specifies what “the same spin label” means at different momenta.

A common choice is the rotationless standard boost:

$$\begin{aligned} L(p)^0{}_0&=E/m,& L(p)^0{}_j&=p_j/m,\\ L(p)^i{}_0&=p_i/m,& L(p)^i{}_j&=\delta_{ij} +\frac{p_ip_j}{m(E+m)}. \end{aligned}$$

Here spatial subscripts denote Euclidean components. The positive off-diagonal signs make $L(p)k=(E,\mathbf p)$. It is the inverse of the passive frame boost with velocity $\mathbf p/E$ used in the toolkit. It has the continuous value $I$ at $\mathbf p=0$.

A general transformation leaves a little-group remainder

Transform a state from $p$ to $\Lambda p$. Compare the route $\Lambda L(p)$ with the chosen standard route $L(\Lambda p)$. Their difference is

$$\mathcal W(\Lambda,p) =L(\Lambda p)^{-1}\Lambda L(p).$$

It fixes $k$, since applying the factors in order gives $k\to p\to\Lambda p\to k$. For the massive case it is a rest-frame rotation, represented on the spin labels by a unitary matrix $D^{(s)}(\mathcal W)$. We use $\mathcal W$ to avoid confusing this group element with the Pauli–Lubanski vector $W^\mu$.

The resulting ket law is

$$U(\Lambda)|p,\sigma\rangle =\sum_{\sigma'} |\Lambda p,\sigma'\rangle D^{(s)}_{\sigma'\sigma}(\mathcal W(\Lambda,p)).$$

The lift to the cover fixes the sign for half-integer spin. The rotation can depend on $p$, even though $\Lambda$ is the same spacetime transformation for the entire state.

Covariant normalization removes an energy square root

Choose

$$\langle p',\sigma'|p,\sigma\rangle =2E_{\mathbf p}(2\pi)^3 \delta^3(\mathbf p'-\mathbf p)\delta_{\sigma'\sigma}.$$

The corresponding measure and completeness relation are

$$d\mu(p)=\frac{d^3p}{(2\pi)^3\,2E_{\mathbf p}}, \qquad I=\sum_\sigma\int d\mu(p)\,|p,\sigma\rangle\langle p,\sigma|.$$

The measure is invariant under proper orthochronous Lorentz transformations. Equivalently, the on-shell Jacobian obeys $d^3(\Lambda p)=E_{\Lambda p}d^3p/E_p$, so the delta function and the factor $2E$ transform together. There is no additional square-root energy factor in the ket law with this normalization.

If instead $|p,\sigma\rangle_0=|p,\sigma\rangle/\sqrt{2E_p}$, then the norm is $(2\pi)^3\delta^3$ and the law becomes

$$U(\Lambda)|p,\sigma\rangle_0 =\sqrt{\frac{E_{\Lambda p}}{E_p}} \sum_{\sigma'} |\Lambda p,\sigma'\rangle_0D_{\sigma'\sigma}(\mathcal W).$$

The two formulas describe the same unitary representation in different normalizations. Combining the measure from one with the ket law from the other produces incorrect packet probabilities.

Wave packets and unitarity

A normalizable packet has the expansion and norm

$$|\psi\rangle =\sum_\sigma\int d\mu(p)\,f_\sigma(p)|p,\sigma\rangle, \qquad \|\psi\|^2=\int d\mu(p)\,f(p)^\dagger f(p).$$

Fix the translation convention $T(a)=e^{iP\cdot a}$ and $U(a,\Lambda)=T(a)U(\Lambda)$. Writing the transformed packet at momentum $q$ gives

$$f'(q) =e^{iq\cdot a} D\!\left(\mathcal W(\Lambda,\Lambda^{-1}q)\right) f(\Lambda^{-1}q).$$

The phase has unit modulus. Unitarity of $D$ and invariance of $d\mu$ then give $\|f'\|=\|f\|$. The momentum argument, measure, and spin matrix act together; this is not unitarity of a four-by-four spinor boost in a Euclidean component norm.

Translations obey $U(\Lambda)T(a)U(\Lambda)^{-1}=T(\Lambda a)$, so this convention realizes the same semidirect product as $x\mapsto\Lambda x+a$. The translation convention is stated separately from Schrödinger time evolution $e^{-iHt}$.

Composition and the choice of standard boosts

Inserting an identity between two transformations gives

$$\begin{aligned} \mathcal W(\Lambda_2\Lambda_1,p) ={}&\mathcal W(\Lambda_2,\Lambda_1p)\\ &\times\mathcal W(\Lambda_1,p). \end{aligned}$$

The momentum in the first factor on the right is $\Lambda_1p$, not $p$. This cocycle identity makes the induced state transformations compose correctly.

Changing the standard boosts to $L'(p)=L(p)R(p)$, with $R(p)$ in the little group, changes the remainder to

$$\mathcal W'(\Lambda,p) =R(\Lambda p)^{-1}\mathcal W(\Lambda,p)R(p).$$

This is a momentum-dependent change of spin basis. Canonical spin, helicity bases, and other conventions can give different spin matrices while describing the same complete state. Probabilities agree when states and observables are transformed together.

In particular, tracing over momentum and comparing only a spin density matrix requires a specified spin basis and measurement. A boost generally correlates momentum and the transformed spin label through $\mathcal W(\Lambda,p)$.

Massless finite-helicity states

For $m=0$, replace the rest reference by a nonzero null vector and choose $L(p)$ that maps it to $p$. The construction of $\mathcal W$ is unchanged, but its little group is the massless stabilizer. In a finite-helicity irrep its null rotations act trivially, and its rotation contributes the phase $e^{-ih\theta(\Lambda,p)}$.

The invariant measure remains $d^3p/[(2\pi)^3\,2|\mathbf p|]$ for $\mathbf p\ne0$. A single helicity has one internal label per momentum. Opposite helicities, antiparticle charges, or flavors may require additional representation sectors; they do not follow just from the value of $P^2$.

The mathematical classification includes other orbits and the continuous-spin massless branch. Restricting this discussion to positive-energy massive and finite-helicity states is an explicit physical scope choice. Neither this construction nor Lorentz covariance of a wave equation alone proves the spin–statistics theorem.

Exercises

1. With rotationless standard boosts, show that a pure spatial rotation $R$ has $\mathcal W(R,p)=R$.

Solution

The explicit boost satisfies $L(Rp)=RL(p)R^{-1}$. Thus $L(Rp)^{-1}RL(p)=RL(p)^{-1}R^{-1}RL(p)=R$. Canonical spin labels therefore rotate by the same ordinary rotation at every momentum.

2. Derive the cocycle identity by expanding its right-hand side.

Solution

The product contains $L(\Lambda_2\Lambda_1p)^{-1}\Lambda_2 L(\Lambda_1p)L(\Lambda_1p)^{-1}\Lambda_1L(p)$. The middle pair cancels, leaving the definition of $\mathcal W(\Lambda_2\Lambda_1,p)$.

3. Why is the massive spin space finite dimensional while its one-particle Hilbert space is infinite dimensional?

Solution

There are $2s+1$ internal spin labels at each momentum, but a normalizable state has an arbitrary square-integrable amplitude over a continuous mass shell. Its Hilbert space is $L^2(d\mu)\otimes\mathbb C^{2s+1}$ in the chosen spin trivialization, not merely $\mathbb C^{2s+1}$.

References

• W. N. Polyzou, W. Glöckle, and H. Witała, “Spin in Relativistic Quantum Theory,” Few-Body Systems 54, 1667–1704, 2013, doi:10.1007/s00601-012-0526-8, arXiv:1208.5840 — standard boosts and alternative spin bases.
• N. Straumann, “Unitary Representations of the inhomogeneous Lorentz Group and their Significance in Quantum Physics,” 2008, arXiv:0809.4942 — induced-representation construction.
• E. P. Wigner, “On Unitary Representations of the Inhomogeneous Lorentz Group,” Annals of Mathematics 40, 149–204, 1939, doi:10.2307/1968551 — the classification underlying particle labels.