Connected Poincaré covers and positive translation energy

Fix an integer spacetime dimension $D\ge 2$ and the mostly-plus form

$$\eta (p,a)=-p^0{a}^0+\sum _{j=1}^{D-1}{p}^j{a}^j\mathrm{.}$$

Let $e_0=(1,0,\dots ,0)$, and let

$$L=S{O}_0(D-1,1)=\{\mathrm{\Lambda }\in G{L}_D(\mathbb{R}):{\mathrm{\Lambda }}^T\eta \mathrm{\Lambda }=\eta ,\det \mathrm{\Lambda }=1,\mathrm{\Lambda }{e}_0\text{ is future-pointing}\}$$

be the proper orthochronous Lorentz group, the identity component of the indicated matrix group. A connected Lorentz cover is a connected Lie group $K$ with a covering homomorphism $c:K\to L$ in the sense of Covering homomorphisms of Lie groups. It acts on translations $N=({\mathbb{R}}^D,+)$ by ${\alpha }_k(a)=c(k)a$. The associated connected-cover Poincaré group is the external semidirect product $G={\mathbb{R}}^D{⋊}_{\alpha }K,(a,k)(b,\mathrm{\ell })=(a+c(k)b,k\mathrm{\ell })\mathrm{.}$ with its product topology, using The external semidirect product $N{⋊}_{\alpha }H$ and The semidirect-product multiplication makes $N\times H$ a group.

For $p\in {\mathbb{R}}^D$, write

$${\chi }_p(a)=\exp (i\eta (p,a))\in \mathbb{T}\mathrm{.}$$

This is a continuous character of $({\mathbb{R}}^D,+)$; the Pontryagin dual and its compact-open topology are those of The Pontryagin dual with the compact-open topology. Put ${\widehat{C}}_{+}=\{{\chi }_p:p^0\ge 0,\eta (p,p)\le 0\}$. The local orbit result below proves that $p\mapsto {\chi }_p$ is the full dual-coordinate homeomorphism. Under this convention ${\chi }_p(t{e}_0)=e^{-it{p}^0}$.

Assume AC. If $U:G\to U(\mathcal{H})$ is a strongly continuous unitary representation as in Strongly continuous unitary representations, invariant linear subspaces and intertwiners, the restriction $a\mapsto U(a,e)$ has its unique regular projection-valued spectral measure $P_U$ on $\widehat{{\mathbb{R}}^D}$ supplied by Spectral measure of a unitary representation of an abelian group, covariance, and ergodicity. The representation is positive-energy when

$$P_U({\widehat{C}}_{+})=I_{\mathcal{H}}\mathrm{.}$$

The cited spectral-measure lemma states its covariant result for separable carriers. For arbitrary $\mathcal{H}$, the zero space has the zero PVM; otherwise the same integrated-$C_0(\widehat{{\mathbb{R}}^D})$ construction in its proof, steps 2.1–7.1, gives the displayed PVM. These steps use the nondegenerate $C_0$-representation theorem (its supplier lem-nondegenerate-czero-representations-have-regular-pvms), which has no separability hypothesis. The separability use in the spectral-measure proof's step 1.1 is only to show that the dual is second-countable and standard Borel; neither property is needed for the regular PVM existence and uniqueness used here. Haar measure on ${\mathbb{R}}^D$ is $\sigma$-finite, and its dual is locally compact Hausdorff by The dual of a locally compact abelian group is locally compact abelian.

Equivalently, after the orbit lemma identifies $\widehat{{\mathbb{R}}^D}$ with momentum space, $P_U$ is concentrated on the closed future cone $C_{+}=\{p:p^0\ge 0,\eta (p,p)\le 0\}$ and the spectral coordinate $p^0$ is the nonnegative generator $P^0$ in $U(t{e}_0,e)=e^{-it{P}^0}$. A unitary representation is irreducible and strongly continuous with the meanings in Strongly continuous unitary representations, invariant linear subspaces and intertwiners.