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 1019GeV 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?
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"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."
Hang on, what about the massless photons?
Perhaps what you really mean is that the mass spectrum shouldn't have a continuous part that includes zero? But I don't see how that constitues a breaking of Coleman-Mandula...
hm, you're allowed lots of massless states - photon and graviton for sure.
if I recall correctly, what you lose is a lowest non-zero mass state in the spectrum of what should be the massive particles - so the theory is made trivial
If the theorem is weakly violated, this ought to show as a mass scale close to zero, compared presumably to a planck mass.
Arguably that is in fact the case, and is the hierarchy problem.
So maybe that is the solution - or the post hoc rationalisation.
Or maybe this is completely off the wall.
Ok, I think I'm starting to get the picture now. My take on what you're saying is this: One way to escape the C-M theorem is to violate the assumption in the theorem that says that for every M > 0 there are only finitely many states (particle types) with mass less than M. The natural way for this assumption to be violated is if all the states are massless. But then if you break exact Poincare symmetry by turning on a very small cosmological constant Lambda, it allows you to deviate slightly from the previous situation such that particles may acquire small masses (although those that prefer to remain massless are welcome to do so) while still having nontrivial particle interactions (scatterings).
This is a cool idea! To my inexpert mind it doesn't sound off the wall at all -- kudos to you for coming up with it.
One question though: do the exicted states of, say, the proton count as seperate states in all this? I guess they must in order to violate the aforementioned assumption of C-M by having infinitely many massless states. But then there is a bit of a problem: turning on a small Lambda is supposed to only allow small masses, but in the real world the excited states of the proton etc can have arbitrarily high energies/masses (although they decay very quickly in practice).