What compactifications of the Poincaré group have been studied? As we know, the Poincaré group is non-compact. Poincaré invariance has been observed at velocities and energies up to $10^{20}$ eV in cosmic rays. The other day, I was thinking about how the $SU(2)$ homomorphism to $SO(3)$ imposes a double cover, and I keep wondering if something like that could exist in the Poincaré group, but of course the main problem is that the group is not compact.
I wonder if it is possible at all to make a compactification of the group that is consistent with low-energy physics, and still preserves some form of the isotropy of spacetime. For instance, I considered identifying the different connected components (either CP or PT inverted) of the group at some boundary consistent with energies of the order of $10^{28}$ eV, but with meaningful dimensional analysis, but have not succeeded in analyzing the symmetry properties and algebraic properties of the resulting manifold (is it still a Lie group after such an identification?)
The physical interpretation of such an identification is up for discussion, but I think the concrete example compactification I gave would basically amount to a duality that continuously maps between particles with energies above $E_p$ (some arbitrary boundary energy) and particles with energy below $E_p$ and $P$ or $CP$ reversed. This would for instance make electric charge conservation an approximate symmetry.
Has something like this been attempted? Or are there good reasons known why this could not work?
 A: You cannot embed the Poincare group or the Lorentz group into a compact Lie group $G$. Indeed, denote the Lie algebra of $G$ as $\mathfrak{g}$ and the Lorentz algebra as $\mathfrak{l}=\mathfrak{o}(1,3)\cong\mathfrak{sl}_2(\mathbf{C})$.
The Killing form on $\mathfrak{g}$ is non-positive-definite, but then so is its restriction to $\mathfrak{l}$. Restriction of the Killing form on $\mathfrak{g}$ to $\mathfrak{l}$ is $\mathfrak{l}$-invariant and is therefore proportional to the Killing form on $\mathfrak{l}$, since the latter is a simple real Lie algebra. Finally, the Killing form on $\mathfrak{l}$ has signature $(3,3)$, contradiction.
By the same reasoning, you cannot mod out by a discrete subgroup of the Lorentz group and get a compact group: the Lie algebra does not change, so the Killing form cannot become non-positive-definite.
On the other hand, there are well-known 'compactifications' of the translation group. You can either mod out $\mathbf{R}/\mathbf{Z}=S^1$ or immerse $\mathbf{R}\rightarrow T^2$ as an irrational winding depending on what kind of compactifications you are interested in.
