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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.^[1] The supersymmetric generalization is known as the Haag–Łopuszański–Sohnius theorem.

History

In the early 1960s, the global ${\text{SU}}(3)$ 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 ${\text{SU}}(3)$ symmetry to a larger ${\text{SU}}(6)$ symmetry mixing both flavour and spin, an idea similar to that previously considered in nuclear physics by Eugene Wigner in 1937 for an ${\text{SU}}(4)$ symmetry.^[2] This non-relativistic ${\text{SU}}(6)$ 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 ${\text{SU}}(6)$ 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 differing up-, down-, and strange-quark masses which leads to a breakdown of the ${\text{SU}}(3)$ 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.^[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]

Theorem

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.

Limitations

Conformal symmetry

The theorem does not apply to a theory of massless particles, with these allowing for conformal symmetry as an additional spacetime dependent symmetry.^[10] In particular, the algebra of this group is the conformal algebra, which consists of the Poincaré algebra together with the commutation relations for the dilaton generator and the special conformal transformations generator.

Supersymmetry

The Coleman–Mandula theorem assumes that the only symmetry algebras are Lie algebras, but the theorem can be generalized by instead considering Lie superalgebras. Doing this allows for additional anticommutating generators known as supercharges which transform as spinors under Lorentz transformations. This extension gives rise to the super-Poincaré algebra, with the associated symmetry known as supersymmetry. The Haag–Łopuszański–Sohnius theorem is the generalization of the Coleman–Mandula theorem to Lie superalgebras, with it stating that supersymmetry is the only new spacetime dependent symmetry that is allowed. For a theory with massless particles, the theorem is again evaded by conformal symmetry which can be present in addition to supersymmetry giving a superconformal algebra.

Low dimensions

In a one or two dimensional theory the only possible scattering is forwards and backwards scattering so analyticity of the scattering angles is no longer possible and the theorem no longer holds. Spacetime dependent internal symmetries are then possible, such as in the massive Thirring model which can admit an infinite tower of conserved charges of ever higher tensorial rank.^[12]

Quantum groups

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.^[13] 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.

Other limitations

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.^[14] 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.^[15]

Notes

^ ^a ^b Coleman, S.R.; Mandula, J. (1967). "All Possible Symmetries of the S Matrix". Phys. Rev. 159 (5): 1251–1256. Bibcode:1967PhRv..159.1251C. doi:10.1103/PhysRev.159.1251.
^ Wigner, E. (1937). "On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei". Phys. Rev. 51 (2): 106–119. Bibcode:1937PhRv...51..106W. doi:10.1103/PhysRev.51.106.
^ Wess, J. (2009). "From symmetry to supersymmetry". The European Physical Journal C. 59 (2): 177–183. arXiv:0902.2201. Bibcode:2009EPJC...59..177W. doi:10.1140/epjc/s10052-008-0837-6. S2CID 14917968.
^ Duplij, S. (2003). Concise Encyclopedia of Supersymmetry. Springer. pp. 265–266. ISBN 978-1402013386.
^ Shifman, M.; Kane, G. (2000). The Supersymmetric World:The Beginnings of the Theory. World Scientific Publishing. pp. 184–185. ISBN 978-9810245221.
^ McGlinn, W.D. (1964). "Problem of Combining Interaction Symmetries and Relativistic Invariance". Phys. Rev. Lett. 12 (16): 467–469. Bibcode:1964PhRvL..12..467M. doi:10.1103/PhysRevLett.12.467.
^ O'Raifeartaigh, L. (1965). "Lorentz Invariance and Internal Symmetry". Phys. Rev. 139 (4B): B1052–B1062. Bibcode:1965PhRv..139.1052O. doi:10.1103/PhysRev.139.B1052.
^ Cao, T.Y. (2004). "19". Conceptual Foundations of Quantum Field Theory. Cambridge University Press. p. 282. ISBN 978-0521602723.
^ Haag, R.; Łopuszański, J.T.; Sohnius, M. (1975). "All possible generators of supersymmetries of the S-matrix". Nuclear Physics B. 88 (2): 257–274. Bibcode:1975NuPhB..88..257H. doi:10.1016/0550-3213(75)90279-5.
^ ^a ^b Weinberg, S. (2005). "24". The Quantum Theory of Fields: Supersymmetry. Vol. 3. Cambridge University Press. pp. 12–22. ISBN 978-0521670555.
^ Zichichi, A. (2012). The Unity of the Fundamental Interactions: 19. Springer. pp. 305–315. ISBN 978-1461336570.
^ Berg, B.; Karowski, M.; Thun, H.J. (1976). "Conserved currents in the massive thirring model". Physics Letters B. 64 (3): 286–288. Bibcode:1976PhLB...64..286B. doi:10.1016/0370-2693(76)90203-3.
^ Bernard, D.; LeClair, A. (1991). "Quantum group symmetries and non-local currents in 2D QFT". Communications in Mathematical Physics. 142 (1): 99–138. Bibcode:1991CMaPh.142...99B. doi:10.1007/BF02099173. S2CID 119026420.
^ Fotopoulos, A.; Tsulaia, M. (2010). "On the Tensionless Limit of String theory, Off - Shell Higher Spin Interaction Vertices and BCFW Recursion Relations". JHEP. 2010 (11): 086. arXiv:1009.0727. Bibcode:2010JHEP...11..086F. doi:10.1007/JHEP11(2010)086. S2CID 119287675.
^ Fabrizio, N.; Percacci, R. (2008). "Graviweak Unification". J. Phys. A. 41 (7): 075405. arXiv:0706.3307. Bibcode:2008JPhA...41g5405N. doi:10.1088/1751-8113/41/7/075405. S2CID 15045658.

@@ Line 1: / Line 1: @@
 {{Short description|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 symmetries|spacetime]] and internal [[symmetry (physics)|symmetries]] can only combine in a trivial way. This means that the charges associated with internal symmetries must always transform as [[Lorentz scalar]]s. 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.<ref name="CM">{{cite journal|last1=Coleman|first1=S.R.|authorlink1=Sidney Coleman|last2=Mandula|first2=J.|authorlink2=Jeffrey Mandula|date=1967|title=All Possible Symmetries of the S Matrix|url=|journal=Phys. Rev.|volume=159|issue=|pages=1251–1256|doi=10.1103/PhysRev.159.125|pmid=|arxiv=|s2cid=|access-date=}}</ref> The supersymmetric generalization is known as the [[Haag–Łopuszański–Sohnius theorem]].
+In [[theoretical physics]], the '''Coleman–Mandula theorem''' is a [[no-go theorem]] stating that [[spacetime symmetries|spacetime]] and internal [[symmetry (physics)|symmetries]] can only combine in a trivial way. This means that the charges associated with internal symmetries must always transform as [[Lorentz scalar]]s. 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.<ref name="CM">{{cite journal|last1=Coleman|first1=S.R.|authorlink1=Sidney Coleman|last2=Mandula|first2=J.|authorlink2=Jeffrey Mandula|date=1967|title=All Possible Symmetries of the S Matrix|url=|journal=Phys. Rev.|volume=159|issue=5|pages=1251–1256|doi=10.1103/PhysRev.159.1251 |pmid=|arxiv=|bibcode=1967PhRv..159.1251C |s2cid=|access-date=}}</ref> The supersymmetric generalization is known as the [[Haag–Łopuszański–Sohnius theorem]].
 ==History==
-In the early 1960s, the [[Symmetry (physics)#Local and global|global]] <math>\text{SU}(3)</math> 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=|s2cid=|access-date=}}</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 arXiv|last1=Wess|first1=J.|authorlink1=Julius Wess|date=2009|title=From Symmetry to Supersymmetry|eprint=0902.2201}}</ref> While this was reasonably successful in describing various aspects of the hadron spectrum, from the perspective of [[quantum chromodynamics]] this 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.
+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=}}</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]] could belong to the same multiplet. Such a symmetry could then possibly account for the mass splitting found in mesons and baryons at the time.<ref>{{cite book|last=Duplij|first=S.|author-link=|date=2003|title=Concise Encyclopedia of Supersymmetry|url=|doi=|location=|publisher=Springer|chapter=|page=265–266|isbn=978-1402013386}}</ref> It was only later understood that this is instead a consequence of the breakdown of the <math>\text{SU}(3)</math> internal symmetry.
+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=|page=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=|s2cid=|access-date=}}</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=|s2cid=|access-date=}}</ref> These efforts culminated with the most general theorem by Sidney Coleman and Jeffrey Mandula in 1967.
+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=}}</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=}}</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 a study of [[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 derived in 1975 after the study of supersymmetry was well 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://www.sciencedirect.com/science/article/pii/0550321375902795|journal=Nuclear Physics B|volume=88|issue=2|pages=257–274|doi=10.1016/0550-3213(75)90279-5|pmid=|arxiv=|s2cid=|access-date=}}</ref>
+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=}}</ref>
 ==Theorem==
@@ Line 20: / Line 20: @@
 * The [[scattering amplitude|amplitudes]] for [[elastic scattering|elastic]] two-body scattering are [[analytic function]]s of the scattering angle at almost all energies and angles,
 * A technical assumption that the group [[generator (mathematics)|generators]] are [[distribution (mathematics)|distributions]] in [[position and momentum space|momentum space]].
-The Coleman–Mandula theorem states that the symmetry of this theory is necessarily a [[direct product of groups|direct product]] of the Poincaré group and an internal symmetry group.<ref name="Weinberg">{{cite book|last=Weinberg|first=S.|author-link=Steven Weinberg|date=2005|title=The Quantum Theory of Fields: Supersymmetry|volume=3|url=|doi=|location=|publisher=Cambridge University Press|chapter=24|page=12–22|isbn=978-0521670555}}</ref> Note that the last technical assumption is unnecessary if the theory described by a [[quantum field theory]] and is only needed to apply the theorem in a wider context.
+The Coleman–Mandula theorem states that the symmetry group of this theory is necessarily a [[direct product of groups|direct product]] of the Poincaré group and an internal symmetry group.<ref name="Weinberg">{{cite book|last=Weinberg|first=S.|author-link=Steven Weinberg|date=2005|title=The Quantum Theory of Fields: Supersymmetry|volume=3|url=|doi=|location=|publisher=Cambridge University Press|chapter=24|pages=12–22|isbn=978-0521670555}}</ref> 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 [[kinematics|kinematic]] argument for why the theorem should hold was provided by [[Edward Witten]].<ref>{{cite book|last=Zichichi|first=A.|author-link=Antonino Zichichi|date=2012|title=The Unity of the Fundamental Interactions: 19|url=|doi=|location=|publisher=Springer|chapter=|page=305–315|isbn=978-1461336570}}</ref> The argument is that Poincaré symmetry is far too strong of a constraint for elastic scattering, leaving only the scattering angle unknown. Hence, any additional spacetime dependent symmetry would [[overdetermined system|overdetermines]] 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 symmetries are ruled out.
+A [[kinematics|kinematic]] argument for why the theorem should hold was provided by [[Edward Witten]].<ref>{{cite book|last=Zichichi|first=A.|author-link=Antonino Zichichi|date=2012|title=The Unity of the Fundamental Interactions: 19|url=|doi=|location=|publisher=Springer|chapter=|pages=305–315|isbn=978-1461336570}}</ref> 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 [[overdetermined system|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.
 ==Limitations==
@@ Line 28: / Line 28: @@
 ===Conformal symmetry===
-The theorem does not apply to a theory of [[massless particle]]s, with these possibly admitting an additional spacetime symmetry called conformal symmetry.<ref name="Weinberg"/> In particular, the allowed [[algebra over a field|algebra]] is the Poincaré algebra together with the commutation relations for the [[homothety|dilaton]] generator and the [[special conformal transformation]]s generator, giving the [[conformal symmetry#commutation relations|conformal algebra]].
+The theorem does not apply to a theory of [[massless particle]]s, with these allowing for conformal symmetry as an additional spacetime dependent symmetry.<ref name="Weinberg"/> In particular, the [[algebra over a field|algebra]] of this group is the [[conformal symmetry#commutation relations|conformal algebra]], which consists of the Poincaré algebra together with the commutation relations for the [[homothety|dilaton]] generator and the [[special conformal transformation]]s generator.
 ===Supersymmetry===
-The Coleman–Mandula theorem assumes that the only symmetry algebras are [[Lie algebra]]s, but this can be generalized to [[Lie superalgebra]]s. Doing this allows for additional [[commutator#ring theory|anticommutating]] generators known as [[supercharge]]s which transform as [[spinor]]s under [[Lorentz transformation]]s. The resulting algebra is known as a [[super-Poincaré algebra]], with the associated symmetry known as supersymmetry. The Haag–Łopuszański–Sohnius theorem is the generalization of the Coleman–Mandula theorem to Lie superalgebras, with it stating that supersymmetry is the only new spacetime dependent symmetry that is allowed. For a theory with massless particles, the theorem is again evaded by conformal symmetry which can be present in addition to supersymmetry giving a [[superconformal algebra]].
+The Coleman–Mandula theorem assumes that the only symmetry algebras are [[Lie algebra]]s, but the theorem can be generalized by instead considering [[Lie superalgebra]]s. Doing this allows for additional [[commutator#ring theory|anticommutating]] generators known as [[supercharge]]s which transform as [[spinor]]s under [[Lorentz transformation]]s. This extension gives rise to the [[super-Poincaré algebra]], with the associated symmetry known as supersymmetry. The Haag–Łopuszański–Sohnius theorem is the generalization of the Coleman–Mandula theorem to Lie superalgebras, with it stating that supersymmetry is the only new spacetime dependent symmetry that is allowed. For a theory with massless particles, the theorem is again evaded by conformal symmetry which can be present in addition to supersymmetry giving a [[superconformal algebra]].
 ===Low dimensions===
-In a one or two dimensional theory the only possible scattering is forwards and backwards scattering so analyticity of the scattering angles is no longer possible and the theorem no longer holds. Spacetime dependent internal symmetries are then possible, such as in the massive [[Thirring model]] which can admit an infinite tower of [[charge conservation|conserved charges]] of ever higher [[tensor|tensorial]] [[rank (linear algebra)|rank]].<ref>{{cite journal|last1=Berg|first1=B.|authorlink1=|last2=Karowski|first2=M.|authorlink2=|last3=Thun|first3=H.J.|authorlink3=|date=1976|title=Conserved currents in the massive thirring model|url=https://www.sciencedirect.com/science/article/pii/0370269376902033|journal=Physics Letters B|volume=64|issue=3|pages=286–288|doi=10.1016/0370-2693(76)90203-3|pmid=|arxiv=|s2cid=|access-date=}}</ref>
+In a one or two dimensional theory the only possible scattering is forwards and backwards scattering so analyticity of the scattering angles is no longer possible and the theorem no longer holds. Spacetime dependent internal symmetries are then possible, such as in the massive [[Thirring model]] which can admit an infinite tower of [[charge conservation|conserved charges]] of ever higher [[tensor]]ial [[rank (linear algebra)|rank]].<ref>{{cite journal|last1=Berg|first1=B.|authorlink1=|last2=Karowski|first2=M.|authorlink2=|last3=Thun|first3=H.J.|authorlink3=|date=1976|title=Conserved currents in the massive thirring model|url=https://dx.doi.org/10.1016/0370-2693%2876%2990203-3|journal=Physics Letters B|volume=64|issue=3|pages=286–288|doi=10.1016/0370-2693(76)90203-3|pmid=|arxiv=|bibcode=1976PhLB...64..286B |s2cid=|access-date=}}</ref>
 ===Quantum groups===
-Models with [[principle of locality|nonlocal]] symmetries whose charges do not act on multiparticle states as if they were a [[tensor product]] of one-particle states, evade the theorem.<ref>{{cite journal|last1=Bernard|first1=D.|authorlink1=|last2=LeClair|first2=A.|authorlink2=|date=1991|title=Quantum group symmetries and non-local currents in 2D QFT|url=https://doi.org/10.1007/BF02099173|journal=Communications in Mathematical Physics|volume=142|issue=1|pages=99–138|doi=10.1007/BF02099173|pmid=|arxiv=|s2cid=|access-date=}}</ref> 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.
+Models with [[principle of locality|nonlocal]] symmetries whose charges do not act on multiparticle states as if they were a [[tensor product]] of one-particle states, evade the theorem.<ref>{{cite journal|last1=Bernard|first1=D.|authorlink1=|last2=LeClair|first2=A.|authorlink2=|date=1991|title=Quantum group symmetries and non-local currents in 2D QFT|url=https://doi.org/10.1007/BF02099173|journal=Communications in Mathematical Physics|volume=142|issue=1|pages=99–138|doi=10.1007/BF02099173|pmid=|arxiv=|bibcode=1991CMaPh.142...99B |s2cid=119026420|access-date=}}</ref> 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.
 ===Other limitations===
-For other spacetime symmetries besides the Poincaré group, such as theories with a [[de Sitter space|de Sitter background]] or non-relativistic [[classical field theory|field theories]] with [[Galilean invariance]], the theorem no longer applies.<ref>{{cite journal|last1=Fotopoulos|first1=A.|authorlink1=|last2=Tsulaia|first2=M.|authorlink2=|date=2010|title=On the Tensionless Limit of String theory, Off - Shell Higher Spin Interaction Vertices and BCFW Recursion Relations|url=|journal=JHEP|volume=11|issue=|pages=086|doi=10.1007/JHEP11(2010)086|pmid=|arxiv=1009.0727|s2cid=|access-date=}}</ref> It also does not hold for [[discrete symmetry|discrete symmetries]], since these are not Lie groups, or for [[spontaneous symmetry breaking|spontaneously broken symmetries]] since these do not act on the S-matrix level and thus do not commute with the S-matrix.<ref>{{cite journal|last1=Fabrizio|first1=N.|authorlink1=|last2=Percacci|first2=R.|authorlink2=|date=2008|title=Graviweak Unification|url=|journal=J. Phys. A|volume=41|issue=|pages=075405|doi=10.1088/1751-8113/41/7/075405|pmid=|arxiv=0706.3307|s2cid=|access-date=}}</ref>
+For other spacetime symmetries besides the Poincaré group, such as theories with a [[de Sitter space|de Sitter background]] or non-relativistic [[classical field theory|field theories]] with [[Galilean invariance]], the theorem no longer applies.<ref>{{cite journal|last1=Fotopoulos|first1=A.|authorlink1=|last2=Tsulaia|first2=M.|authorlink2=|date=2010|title=On the Tensionless Limit of String theory, Off - Shell Higher Spin Interaction Vertices and BCFW Recursion Relations|url=|journal=JHEP|volume=2010|issue=11|pages=086|doi=10.1007/JHEP11(2010)086|pmid=|arxiv=1009.0727|bibcode=2010JHEP...11..086F |s2cid=119287675|access-date=}}</ref> It also does not hold for [[discrete symmetry|discrete symmetries]], since these are not Lie groups, or for [[spontaneous symmetry breaking|spontaneously broken symmetries]] since these do not act on the S-matrix level and thus do not commute with the S-matrix.<ref>{{cite journal|last1=Fabrizio|first1=N.|authorlink1=|last2=Percacci|first2=R.|authorlink2=|date=2008|title=Graviweak Unification|url=|journal=J. Phys. A|volume=41|issue=7|pages=075405|doi=10.1088/1751-8113/41/7/075405|pmid=|arxiv=0706.3307|bibcode=2008JPhA...41g5405N |s2cid=15045658|access-date=}}</ref>
 ==See also==
@@ Line 65: / Line 65: @@
 [[Category:Supersymmetry]]
 [[Category:Theorems in quantum mechanics]]
+[[Category:No-go theorems]]

v t e Supersymmetry
General topics	Supersymmetry Supersymmetric gauge theory Supersymmetric quantum mechanics Supergravity Superstring theory Super vector space Supergeometry
Supermathematics	Superalgebra Lie superalgebra Super-Poincaré algebra Superconformal algebra Supersymmetry algebra Supergroup Superspace Harmonic superspace Super Minkowski space Supermanifold
Concepts	Supercharge R-symmetry Supermultiplet Short supermultiplet BPS state Superpotential D-term FI D-term F-term Moduli space Supersymmetry breaking Konishi anomaly Seiberg duality Seiberg–Witten theory Witten index Wess–Zumino gauge Localization Mu problem Little hierarchy problem Electric–magnetic duality
Theorems	Coleman–Mandula Haag–Łopuszański–Sohnius Nonrenormalization
Field theories	Wess–Zumino N = 1 super Yang–Mills 4D N = 1 N = 4 super Yang–Mills Super QCD MSSM NMSSM 6D (2,0) superconformal ABJM superconformal
Supergravity	Pure 4D N = 1 supergravity 4D N = 1 supergravity N = 8 supergravity Higher dimensional 11D supergravity Type I supergravity Type IIA supergravity Type IIB supergravity Gauged supergravity
Superpartners	Axino Chargino Gaugino Goldstino Graviphoton Graviscalar Higgsino LSP Neutralino R-hadron Sfermion Sgoldstino Stop squark Superghost
Researchers	Affleck Bagger Batchelor Berezin Dine Fayet Gates Golfand Iliopoulos Montonen Olive Salam Seiberg Siegel Roček Rogers Wess Witten Zumino