Coleman–Mandula theorem

In the early 1960s, the global ${\displaystyle {\text{SU}}(3)}$ symmetry associated with the eightfold way was shown to successfully describe the hadron spectrum for hadrons of the same spin. This led to efforts to expand the global ${\displaystyle {\text{SU}}(3)}$ symmetry to a larger ${\displaystyle {\text{SU}}(6)}$ symmetry mixing both flavour and spin, an idea similar to that previously considered in nuclear physics by Eugene Wigner in 1937 for an ${\displaystyle {\text{SU}}(4)}$ symmetry.[2] This non-relativistic ${\displaystyle {\text{SU}}(6)}$ model united vector and pseudoscalar mesons of different spin into a 35-dimensional multiplet and it also united the two baryon decuplets into a 56-dimensional multiplet.[3] While this was reasonably successful in describing various aspects of the hadron spectrum, from the perspective of quantum chromodynamics this success is merely a consequence of the flavour and spin independence of the force between quarks. There were many attempts to generalize this non-relativistic ${\displaystyle {\text{SU}}(6)}$ model into a fully relativistic one, but these all failed.

At the time it was also an open question whether there existed a symmetry for which particles of different masses could belong to the same multiplet. Such a symmetry could then account for the mass splitting found in mesons and baryons.[4] It was only later understood that this is instead a consequence of the breakdown of the ${\displaystyle {\text{SU}}(3)}$ internal symmetry.

These two motivations led to a series of no-go theorems to show that spacetime symmetries and internal symmetries could not be combined in any but a trivial way.[5] The first notable theorem was proved by William McGlinn in 1964,[6] with a subsequent generalization by Lochlainn O'Raifeartaigh in 1965.[7] These efforts culminated with the most general theorem by Sidney Coleman and Jeffrey Mandula in 1967.

Little notice was given to this theorem in subsequent years. As a result, the theorem played no role in the early development of supersymmetry, which instead emerged in the early 1970s from the study of dual resonance models, which are the precursor to string theory, rather than from any attempts to overcome the no-go theorem.[8] Similarly, the Haag–Łopuszański–Sohnius theorem, a supersymmetric generalization of the Coleman–Mandula theorem, was proved in 1975 after the study of supersymmetry was already underway.[9]

Consider a theory that can be described by an S-matrix and that satisfies the following conditions[1]

• The symmetry group is a Lie group which includes the Poincaré group as a subgroup,
• Below any mass, there are only a finite number of particle types,
• Any two-particle state undergoes some reaction at almost all energies,
• The amplitudes for elastic two-body scattering are analytic functions of the scattering angle at almost all energies and angles,
• A technical assumption that the group generators are distributions in momentum space.

The Coleman–Mandula theorem states that the symmetry group of this theory is necessarily a direct product of the Poincaré group and an internal symmetry group.[10] The last technical assumption is unnecessary if the theory is described by a quantum field theory and is only needed to apply the theorem in a wider context.

A kinematic argument for why the theorem should hold was provided by Edward Witten.[11] The argument is that Poincaré symmetry acts as a very strong constraint on elastic scattering, leaving only the scattering angle unknown. Any additional spacetime dependent symmetry would overdetermine the amplitudes, making them nonzero only at discrete scattering angles. Since this conflicts with the assumption of the analyticity of the scattering angles, such additional spacetime dependent symmetries are ruled out.

• Extended supersymmetry
• Supergroup
• Supersymmetry algebra

• Coleman–Mandula theorem on Scholarpedia
• Sascha Leonhardt on the Coleman–Mandula theorem