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what compactifications of the Poincare group have been studied?

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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?

This post has been migrated from (A51.SE)
asked Nov 19, 2011 in Theoretical Physics by CharlesJQuarra (510 points) [ no revision ]
The double cover of the identity component of the Poincare group is a standard object. There is no problem with the group being non-compact

This post has been migrated from (A51.SE)
well, thats the part i'm not sure because i don't know if there is a well-defined compactification procedure for groups as there are for manifolds. What i was hoping is to take the Lie group as a manifold, apply the compactification (basically by identifying it with other stuff at a prescribed boundary), and see what "needs to happen" in the boundary so that the resulting manifold is still a Lie group

This post has been migrated from (A51.SE)
Could you explain what you mean by a group compactification? Do you have an example in mind? The connected component of the identity of the Lorentz group $O(1,3)$ is isomorphic to $PSL_2(\mathbf{C})$, which has an obvious double cover $SL_2(\mathbf{C})$. This also extends to the Poincare group.

This post has been migrated from (A51.SE)

1 Answer

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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.

This post has been migrated from (A51.SE)
answered Nov 19, 2011 by Pavel Safronov (1,115 points) [ no revision ]

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