Wigner's classification

Statement

Let $\tilde P = \R^{1,3} \rtimes \mathrm{SL}_2(\C)$ be the universal cover of the identity component of the Poincaré group. Here $\R^{1,3}$ is Minkowski space with the form $p \cdot p = p_0^2 - p_1^2 - p_2^2 - p_3^2$, and SL(2,C) acts on it through the covering map onto the restricted Lorentz group. The group $\mathrm{SL}_2(\C)$ also acts on the momentum space $\widehat{\R^{1,3}} \cong \R^{1,3}$ of characters $x \mapsto e^{i p \cdot x}$ of the translations. For a momentum $p$, let $H_p \subseteq \mathrm{SL}_2(\C)$ be its stabilizer, the little group.

1. The irreducible unitary representations of $\tilde P$, up to equivalence, are in bijection with certain pairs $(\mathcal{O}, \sigma)$. Here $\mathcal{O}$ is an orbit of $\mathrm{SL}_2(\C)$ in momentum space. And $\sigma$ is an irreducible unitary representation of $H_p$ for a chosen $p \in \mathcal{O}$, up to equivalence.
2. The representation attached to $(\mathcal{O}, \sigma)$ is the induced representation $\mathrm{Ind}_{\R^{1,3} \rtimes H_p}^{\tilde P} (e^{i p \cdot x} \sigma)$.
3. For the orbit $p \cdot p = m^2$ with $m > 0$ and $p_0 > 0$, the little group is isomorphic to SU(2), and $\sigma$ is labelled by a spin $s \in \set{0, \tfrac12, 1, \tfrac32, \dots}$.
4. For the orbit $p \cdot p = 0$ with $p_0 > 0$, the little group is isomorphic to a double cover of the Euclidean group of the plane. The representations $\sigma$ that are trivial on its translation part are characters labelled by a helicity $h \in \tfrac12 \Z$; the others form the continuous spin representations.