Humour me for a minute here and let\'s imagine that all interesting and plausibl

Question

Humour me for a minute here and let's imagine that all interesting and plausible supersymmetry models have been "cornered" out by the experimental data;

what sort of alternatives are there for having quantum field theories with Poincare symmetry that are allowed to have nontrivial internal symmetries? i.e: that the Coleman-Mandula theorem does not apply?

What other assumptions of the theorem can be relaxed or dropped, and still leave us with workable QFT?

Are we forced to drop full Lorentz-Poincare symmetry or will the theorem still apply with slight violations of that symmetry?

Explanation / Answer

The Coleman-Mandula Theorem is a theorem about the infinitesimal symmetry generators of S-matrices.

1) It's only a theorem about Lie algebras. It doesn't see discrete symmetries like parity and it can't tell the difference between Spin(3,1) and SO(3,1). It also assumes that the symmetry generators form a lie algebra rather than a super Lie algebra.

2) It's a theorem about asymptotic momentum-state scattering. No asymptotics, no theorem. So it doesn't apply in deSitter space, AKA our world. And it doesn't necessarily apply to extended object scattering.

3) It assumes that the theory has a mass gap and a number operator. So doesn't apply to CFTs. Or to QED, which doesn't exist anyways. It should apply QED's electron scattering channels though.

4) It assumes that scattering maps 1-particle states to 1-particle states. False, if there is particle creation, from a current or gravitationally-driven.

5) It assumes that 2-particle scattering is analytic and non-trivial at all but finitely many energies and scattering angles. You can relax the mass gap assumption and keep this assumption, and you'll be allowed conformal symmetries instead of just Poincare.

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