Positive-energy Wigner–Mackey classification for connected covers

By [F2], the dual of translations is the full momentum space, all its $K$-orbits are locally closed and countably separated, and the dual action is the stated Lorentz action because the cover is onto. For every momentum $p$, the stabilizer $K_p$ is closed by [F3]; hence $H_p={\mathbb{R}}^D⋊K_p$ is a closed subgroup of the second-countable locally compact group $G$. The group $G$ is second-countable locally compact because $K$ is, and ${\mathbb{R}}^D$ is. Therefore [F4] applies and classifies every irreducible strongly continuous representation of $G$ by one orbit and one irreducible little-group representation. For the inducing representation $\sigma (a,h)={\chi }_p(a)\theta (h)$, invariance of ${\chi }_p$ under $K_p$ gives multiplicativity. For every vector $v$, $\mathrm{\parallel }\sigma (a,h)v-\sigma (a_0,h_0)v\mathrm{\parallel }\le \mathrm{\parallel }\theta (h)v-\theta (h_0)v\mathrm{\parallel }+\mathrm{\mid }{\chi }_p(a)-{\chi }_p(a_0)\mathrm{\mid }\mathrm{\parallel }v\mathrm{\parallel }$, which tends to zero by strong continuity of $\theta$ and continuity of ${\chi }_p$ in [F1]. Thus $\sigma$ meets the continuity hypothesis of [F7], which then guarantees strong continuity of its induction.

In the induced model for ${\pi }_{p,\theta }$, translations act by multiplication by the dual character at each base point of the orbit, and [F4] says that the translation PVM has full support on that orbit. If $p\in C_{+}$, Lorentz transformations preserve $C_{+}$, so its entire orbit lies in $C_{+}$; the PVM of ${\pi }_{p,\theta }$ is consequently concentrated on $C_{+}$ and the representation is positive-energy by [F1].

Conversely, let $\pi$ be an irreducible positive-energy representation. By step 1.1 and [F4], it is equivalent to ${\pi }_{p,\theta }$ for some orbit $O=K\cdot p$ and an irreducible $\theta$ of $K_p$. The orbit list in [F2] shows that every Lorentz orbit either lies entirely in $C_{+}$ (future timelike, future null, or zero) or is disjoint from it (past timelike, spacelike, or past null; in $D=2$ the two future null rays are included in the first case and the past rays in the last). If $O$ were disjoint from $C_{+}$, [F4] would give $P_{\pi }(O)=I$, while positive energy gives $P_{\pi }(C_{+})=I$ by [F1]. The sets are disjoint, so [F8] implies $I=P_{\pi }(C_{+})P_{\pi }(O)=0$, impossible because an irreducible representation acts on a nonzero Hilbert space. Thus $O\subseteq C_{+}$ and its representative may be chosen there.

By [F2], distinct momentum orbits give distinct translation spectral supports. Within one orbit, [F4] says equivalence holds exactly for conjugate little-group representations at the chosen representative. This proves precisely the equivalence relation in the theorem.

The orbit list [F2] and stabilizer computation [F3] give the massive and zero-orbit cases in the statement. Applying [F5] to the nonzero null orbit gives the stated complete null representation branches in each dimension. Applying [F6] gives the separate descent condition for every central cover kernel. These deductions preserve the full zero-orbit Lorentz-cover representation and make no inference about disconnected symmetries. Steps 1.1 and 2.1–2.3 prove the asserted positive-energy classification and its stated boundaries.