Теорема Коулмана — Мандулы: о тривиальности симметрий пространства-времени и внутренних симметрий.

No go theorem pertaining the triviality of space time and internal symmetries
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 flavor 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 differing up , down , and strange quark masses which leads to a breakdown of the internal flavor 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. The first notable theorem was proved by William McGlinn in 1964, with a subsequent generalization by Lochlainn O'Raifeartaigh in 1965. 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. 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.

Consider a theory that can be described by an S matrix and that satisfies the following conditions 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. 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.

The theorem does not apply to a theory of massless particles, with these allowing for conformal symmetry as an additional spacetime dependent symmetry.

Models with nonlocal symmetries whose charges do not act on multiparticle states as if they were a tensor product of one particle states, evade the theorem. Such an evasion is found more generally for quantum group symmetries which avoid the theorem because the corresponding algebra is no longer a Lie algebra.

For other spacetime symmetries besides the Poincaré group, such as theories with a de Sitter background or non relativistic field theories with Galilean invariance, the theorem no longer applies. It also does not hold for discrete symmetries, since these are not Lie groups, or for spontaneously broken symmetries since these do not act on the S matrix level and thus do not commute with the S matrix.