for all practical purposes we can do most physics under the assumption that space-time is locally flat

but it isn’t.

Locally space-time has some curvature, but that doesn’t matter so much.

Globally, there is also an issue, since we seem to live in a universe with a positive cosmological constant, and going off to infinity is not going to get us to a nice flat Minkowski space.

Technically, this confounds a lot of assumptions in physics, where it is assumed that there is some asymptotic flat space, over there somewhere, where everything is well defined, cleanly separated.

In practise, this ought to not matter, because locally the asymptotic flat space is a very good approximation at the energies particles are at. Space is not interestingly non-flat for most purposes, except near relativistic singularities.

So our favourite recent topic – Lisi’s “exceptionally simple theory of everything” – an aspect of the theory is that it uses spacetime symmetries to absorb group symmetries from the large symmetry group, E8, which Lisi looks to for a unified theory. E8 is a very large group, with much higher symmetry than needed for just the Standard Model symmetries, so it is good to use those symmetries without having to add new physics.

But, the Coleman-Mandula theorem forbids this. It shows that you can’t mix Poincare symmetries and internal gauge symmetries except for trivial theories (with no interactions), or supersymmetric theories.

Lisi’s counterargument is that his spacetime symmetry is deSitter, not Poincare so the theorem does not apply.

The counter-argument is that this doesn’t matter, Coleman-Mandula must be at least approximately correct since the Poincare symmetry is a very good observational symmetry for local physics.

So in some formal sense, Coleman-Mandula can only be weakly violated.

So… what if it is in fact weakly violated?

To break Coleman-Mandula, the theory must have interacting fields and positive mass lowest state, a mass gap.

A weak violation implies the mass gap should be small, ie the lowest mass state should have a very low mass.

Which it does. The order unity mass is the Planck mass at 10^{19}GeV or so. The Standard Model masses are mostly down at ~ 1 GeV, or 10^{-19} in natural units, and the lowest non-zero masses are probably the neutrinos down at around 10^{-29} or so of the Planck mass.

That is about right for a weak violation of the Coleman-Mandula theorem, isn’t it?

Which would also nicely “explain” the hierarchy problem, particle masses are small because the Coleman-Mandula theorem is almost exact, and we get a bonus cosmological constant that is in fact tied to the mass scale – ie the Higgs mass.

Since the violation in zero mass, the presence of the mass gap, is induced by the cosmological constant making spacetime non-Poincare.

How is that for arm waving?