Coleman–Mandula theorem

In theoretical physics, the Coleman–Mandula theorem is a no-go theorem stating that spacetime and internal symmetries can only combine in a trivial way. This means that the charges associated with internal symmetries must always transform as Lorentz scalars. Some notable exceptions to the no-go theorem are conformal symmetry and supersymmetry. It is named after Sidney Coleman and Jeffrey Mandula who proved it in 1967 as the culmination of a series of increasingly generalized no-go theorems investigating how internal symmetries can be combined with spacetime symmetries. The supersymmetric generalization is known as the Haag–Łopuszański–Sohnius theorem. In the early 1960s, the global 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 symmetry to a larger symmetry mixing both flavour and spin, an idea similar to that previously considered in nuclear physics by Eugene Wigner in 1937 for an symmetry. This non-relativistic 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. 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 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. It was only later understood that this is instead a consequence of the breakdown of the 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.