Editing Coleman–Mandula theorem (section)

==History==

In the early 1960s, the [[Symmetry (physics)#Local and global|global]] <math>\text{SU}(3)</math> flavor symmetry associated with the [[eightfold way (physics)|eightfold way]] was shown to successfully describe the [[hadron spectroscopy|hadron spectrum]] for [[hadron]]s of the same [[Spin (physics)|spin]]. This led to efforts to expand the global <math>\text{SU}(3)</math> symmetry to a larger <math>\text{SU}(6)</math> symmetry mixing both [[flavour (particle physics)|flavour]] and spin, an idea similar to that previously considered in [[nuclear physics]] by [[Eugene Wigner]] in 1937 for an <math>\text{SU}(4)</math> symmetry.<ref>{{cite journal|last1=Wigner|first1=E.|authorlink1=Eugene Wigner|date=1937|title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei|url=https://link.aps.org/doi/10.1103/PhysRev.51.106|journal=Phys. Rev.|volume=51|issue=2|pages=106–119|doi=10.1103/PhysRev.51.106|pmid=|arxiv=|bibcode=1937PhRv...51..106W |s2cid=|access-date=|url-access=subscription}}</ref> This non-relativistic <math>\text{SU}(6)</math> model united [[vector meson|vector]] and [[pseudoscalar meson|pseudoscalar]] [[meson]]s of different spin into a 35-dimensional [[multiplet]] and it also united the two [[baryon]] decuplets into a 56-dimensional multiplet.<ref>{{cite journal|last1=Wess|first1=J.|authorlink1=Julius Wess|date=2009|title=From symmetry to supersymmetry|journal=The European Physical Journal C |volume=59 |issue=2 |pages=177–183 |doi=10.1140/epjc/s10052-008-0837-6 |arxiv=0902.2201|bibcode=2009EPJC...59..177W |s2cid=14917968 }}</ref> 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 [[quark]]s. There were many attempts to generalize this non-relativistic <math>\text{SU}(6)</math> model into a fully [[theory of relativity|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 [[mass]]es could belong to the same multiplet. Such a symmetry could then account for the mass splitting found in mesons and baryons.<ref>{{cite book|last=Duplij|first=S.|author-link=|date=2003|title=Concise Encyclopedia of Supersymmetry|url=|doi=|location=|publisher=Springer|chapter=|pages=265–266|isbn=978-1402013386}}</ref> 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 <math>\text{SU}(3)</math> 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.<ref>{{cite book|last1=Shifman|first1=M.|author-link1=Mikhail Shifman|last2=Kane|first2=G.|author-link2=Gordon L. Kane|date=2000|title=The Supersymmetric World:The Beginnings of the Theory|url=|doi=|location=|publisher=World Scientific Publishing|chapter=|pages=184–185|isbn=978-9810245221}}</ref> The first notable theorem was proved by William McGlinn in 1964,<ref>{{cite journal|last1=McGlinn|first1=W.D.|authorlink1=|date=1964|title=Problem of Combining Interaction Symmetries and Relativistic Invariance|url=https://link.aps.org/doi/10.1103/PhysRevLett.12.467|journal=Phys. Rev. Lett.|volume=12|issue=16|pages=467–469|doi=10.1103/PhysRevLett.12.467|pmid=|arxiv=|bibcode=1964PhRvL..12..467M |s2cid=|access-date=|url-access=subscription}}</ref> with a subsequent generalization by [[Lochlainn O'Raifeartaigh]] in 1965.<ref>{{cite journal|last1=O'Raifeartaigh|first1=L.|authorlink1=Lochlainn O'Raifeartaigh|date=1965|title=Lorentz Invariance and Internal Symmetry|url=https://link.aps.org/doi/10.1103/PhysRev.139.B1052|journal=Phys. Rev.|volume=139|issue=4B|pages=B1052–B1062|doi=10.1103/PhysRev.139.B1052|pmid=|arxiv=|bibcode=1965PhRv..139.1052O |s2cid=|access-date=|url-access=subscription}}</ref> 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 model]]s, which are the precursor to [[string theory]], rather than from any attempts to overcome the no-go theorem.<ref>{{cite book|last=Cao|first=T.Y.|author-link=|date=2004|title=Conceptual Foundations of Quantum Field Theory|url=|doi=|location=|publisher=Cambridge University Press|chapter=19|page=282|isbn=978-0521602723}}</ref> 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.<ref>{{cite journal|last1=Haag|first1=R.|authorlink1=Rudolf Haag|last2=Łopuszański|first2=J.T.|authorlink2=Jan Łopuszański (physicist)|last3=Sohnius|first3=M.|authorlink3=|date=1975|title=All possible generators of supersymmetries of the S-matrix|url=https://dx.doi.org/10.1016/0550-3213%2875%2990279-5|journal=Nuclear Physics B|volume=88|issue=2|pages=257–274|doi=10.1016/0550-3213(75)90279-5|pmid=|arxiv=|bibcode=1975NuPhB..88..257H |s2cid=|access-date=|url-access=subscription}}</ref>