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?


2 Answers 2


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.

  • $\begingroup$ in quantum gravity we are already forced to discard asymptotic scattering (seen as the "mythologic absence of true observables") so, i don't see that as much of a problem. It is a problem to make sense of calculations in general though $\endgroup$
    – lurscher
    Nov 15, 2012 at 4:37
  • $\begingroup$ "mythologic absence of true observables" There is not a single Google hit for this phrase. What do you mean by it? $\endgroup$
    – user1504
    Nov 15, 2012 at 4:51
  • $\begingroup$ motls.blogspot.com/2008/10/observables-in-quantum-gravity.html $\endgroup$
    – lurscher
    Nov 15, 2012 at 4:53
  • $\begingroup$ i'm not convinced of that argument (the absence of observables in Quantum Gravity) but the experts are, so it is important to understand why. I think within this problem lies the subtle distinction between symmetry and gauge transformations. When we say "diffeomorphism is a gauge symmetry" we are mixing different stuff together. But that's another topic, i just mentioned it because it raises precisely the point that you highlighted: the asymptotic scattering states are no longer physically interesting $\endgroup$
    – lurscher
    Nov 15, 2012 at 4:56
  • 1
    $\begingroup$ @lurscher: It's not at all clear to me that quantum gravity forces us to discard asymptotic scattering states. This is not the case in string theory, where the whole goal is to compute the S-matrix. $\endgroup$
    – user1504
    Nov 15, 2012 at 13:07

There is a construction (http://arxiv.org/abs/1512.03328), which essentially relaxes the condition that 1-particle states are mapped to 1-particle states. Actually, in an other view, it extends the traditionally compact gauge group with a so called solvable part. (Actually, SUSY does also similar trick, but in the traditional presentation it is not so obvious to see that.)


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