what compactifications of the Poincare group have been studied?

Question

as we know the Poincare group is non-compact. Poincare invariance have been observed in velocities and energies up to $10^{20}$ eV in cosmic rays. The other day i was thinking in how $SU(2)$ homeomorphism in $SO(3)$ imposes a double cover, and i keep wondering if something like that could exist in the Poincare 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 isotropy of space-time. For instance, i considered indentifying 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 analysising the symmetry properties of the resulting manifold and the algebraic properties of it (it is still a Lie group after such identification?)

The physical interpretation of such identification is up to discussion, but i think that it would basically stand for a duality that maps continuously (in the concrete example compactification i gave) particles with energies above $E_p$ (some abritrary boundary energy) with particles with energy below $E_p$ and $P$ or $CP$ reversed. This latter would make for instance, electric charge conservation an approximate symmetry.

Has something like this been attempted? or are there good reasons known why this could not work?

Answer 1

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.