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In theoretical physics, the Haag–Łopuszański–Sohnius theorem states that if both commutating and anticommutating generators are considered, then the only way to nontrivially mix spacetime and internal symmetries is through supersymmetry. The anticommutating generators must be spin-1/2 spinors which can additionally admit their own internal symmetry known as R-symmetry. The theorem is a generalization of the Coleman–Mandula theorem to Lie superalgebras. It was proved in 1975 by Rudolf Haag, Jan Łopuszański, and Martin Sohnius^[1] as a response to the development of the first supersymmetric field theories by Julius Wess and Bruno Zumino in 1974.

History

During the 1960s, a set of theorems investigating how internal symmetries can be combined with spacetime symmetries were proved, with the most general being the Coleman–Mandula theorem.^[2] It showed that the Lie group symmetry of an interacting theory must necessarily be a direct product of the Poincaré group with some compact internal group. Unaware of this theorem, during the early 1970s a number of authors independently came up with supersymmetry, seemingly in contradiction to the theorem since there some generators do transform non-trivially under spacetime transformations.

In 1974 Jan Łopuszański visited Karlsruhe from Wrocław shortly after Julius Wess and Bruno Zumino constructed the first supersymmetric quantum field theory, the Wess–Zumino model.^[3] Speaking to Wess, Łopuszański was interested in figuring out how these new theories managed to overcome the Coleman–Mandula theorem. While Wess was too busy to work with Łopuszański, his doctoral student Martin Sohnius was available. Over the next few weeks they devised a proof of their theorem after which Łopuszański went to CERN where he worked with Rudolf Haag to significantly refine the argument and also extend it to the massless case. Later, after Łopuszański went back to Wrocław, Sohnius went to CERN to finish the paper with Haag, which was published in 1975.

Theorem

The main assumptions of the Coleman–Mandula theorem are that the theory includes an S-matrix with analytic scattering amplitudes such that any two-particle state must undergo some reaction at almost all energies and scattering angles.^[4] Furthermore, there must only be a finite number of particle types below any mass, disqualifying massless particles. The theorem then restricts the Lie algebra of the theory to be a direct sum of the Poincare algebra with some internal symmetry algebra.

The Haag–Łopuszański–Sohnius theorem is based on the same assumptions, except for allowing additional anticommutating generators, elevating the Lie algebra to a Lie superalgebra. In four dimensions, the theorem states that the only nontrivial anticommutating generators that can be added are a set of ${\mathcal {N}}$ pairs of supercharges $Q_{\alpha }^{L}$ and ${\bar {Q}}_{\dot {\alpha }}^{R}$ , indexed by $\alpha$ , which commute with the momentum generator and transform as left-handed and right-handed Weyl spinors. The undotted and dotted index notation, known as Van der Waerden notation, distinguishes left-handed and right-handed Weyl spinors from each other. Generators of other spin, such spin-3/2 or higher, are disallowed by the theorem.^[5] In a basis where $({\bar {Q}}_{\dot {\alpha }}^{A})=(Q_{\alpha }^{A})^{\dagger }$ , these supercharges satisfy

\{Q_{\alpha }^{A},Q_{\beta }^{B}\}=\epsilon _{\alpha \beta }Z^{AB},\ \ \ \ \ \ \ \ \ \ \{Q_{\alpha }^{A},{\bar {Q}}_{\dot {\beta }}^{B}\}=\delta ^{AB}\sigma _{\alpha {\dot {\beta }}}^{\mu }P_{\mu },

where $Z^{AB}$ are known as central charges, which commute with all generators of the superalgebra. Together with the Poincaré algebra, this Lie superalgebra is known as the super-Poincaré algebra. Since four dimensional Minkowski spacetime also admits Majorana spinors as fundamental spinor representations, the algebra can equivalently be written in terms of four-component Majorana spinor supercharges, with the algebra expressed in terms of gamma matrices and the charge conjugation operator rather than Pauli matrices used for the two-component Weyl spinors.^[6]

The supercharges can also admit an additional Lie algebra symmetry known as R-symmetry, whose generators $B_{i}$ satisfy

[Q_{\alpha }^{A},B_{i}]=\sum _{B}s_{i}^{AB}Q_{\alpha }^{B},

where $s_{i}^{AB}$ are Hermitian representation matrices of the generators in the ${\mathcal {N}}$ -dimensional representation of the R-symmetry group.^[7] For ${\mathcal {N}}=1$ the central charge must vanish and the R-symmetry is given by a ${\text{U}}(1)$ group, while for extended supersymmetry ${\mathcal {N}}>1$ the central charges need not vanish, while the R-symmetry is a ${\text{U}}({\mathcal {N}})$ group.

If massless particles are allowed, then the algebra can additionally be extended using conformal generators: the dilaton generator $D$ and the special conformal transformations generator $K_{\mu }$ . For ${\mathcal {N}}$ supercharges, there must also be the same number of superconformal generators $S_{\alpha }$ which satisfy

\{S_{\alpha }^{A},{\bar {S}}_{\dot {\beta }}^{B}\}=\delta ^{AB}\sigma _{\alpha {\dot {\beta }}}^{\mu }K_{\mu },

with both the supercharges and the superconformal generators being charged under a ${\text{U}}({\mathcal {N}})$ R-symmetry.^[8] This algebra is an example of a superconformal algebra, which in this four dimensional case is denoted by ${\mathfrak {su}}(2,2|{\mathcal {N}})$ .^[9] Unlike for non-conformal supersymmetric algebras, R-symmetry is always present in superconformal algebras.^[10]

Limitations

The Haag–Łopuszański–Sohnius theorem was originally derived in four dimensions, however the result that supersymmetry is the only nontrivial extension to spacetime symmetries holds in all dimensions greater than two. The form of the supersymmetry algebra however changes. Depending on the dimension, the supercharges can be Weyl, Majorana, Weyl–Majorana, or symplectic Weyl–Majorana spinors. Furthermore, R-symmetry groups differ according to the dimensionality and the number of supercharges.^[11] This superalgebra also only applies in Minkowski spacetime, being modified in other spacetimes. For example, there exists an extension to anti-de Sitter space for one or more supercharges, while an extension to de Sitter space only works if multiple supercharges are present.^[12]

In two or fewer dimensions the theorem breaks down. The reason for this is that analyticity of the scattering amplitudes can no longer hold since for example in two dimensions the only scattering is forward and backward scattering. The theorem also does not apply to discrete symmetries or to spontaneously broken symmetries since these are not symmetries at the level of the S-matrix.

References

^ 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.
^ Weinberg, S. (2005). "24". The Quantum Theory of Fields: Supersymmetry. Vol. 3. Cambridge University Press. pp. 1–2. ISBN 978-0521670555.
^ Sohnius, M.F. (2001). "Recollections of a young contributor". Nucl. Phys. B Proc. Suppl. 101 (1–3): 129–132. Bibcode:2001NuPhS.101..129S. doi:10.1016/S0920-5632(01)01499-2.
^ 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.
^ Duplij, S. (2003). Concise Encyclopedia of Supersymmetry. Springer. pp. 181–182. ISBN 978-1402013386.
^ Năstase, H. (2015). "3". Introduction to the AdS/CFT Correspondance. Cambridge: Cambridge University Press. pp. 45–46. ISBN 978-1107085855.
^ Wess, J.; Bagger, B. (1992). "1". Supersymmetry and Supergravity. Princeton University Press. pp. 3–9. ISBN 978-0691025308.
^ Ammon, M.; Erdmenger, J. (2015). "3". Gauge/Gravity Duality: Foundations and Applications. Cambridge University Press. pp. 139–140. ISBN 978-1107010345.
^ Freund, P.G.O. (1988). "4". Introduction to Supersymmetry. Cambridge University Press. p. 26. ISBN 978-0521356756.
^ Akhond, M.; et al. (2021). "The Hitchhiker's Guide to 4d N=2 Superconformal Field Theories". SciPost Phys. Lect. Notes. arXiv:2112.14764. doi:10.21468/SciPostPhysLectNotes.64. S2CID 245537355.
^ Cecotti, S. (2015). "2". Supersymmetric Field Theories: Geometric Structures and Dualities. Cambridge: Cambridge University Press. p. 46. ISBN 978-1107053816.
^ Dall'Agata, G.; Zagermann, M. (2021). "4". Supergravity: From First Principles to Modern Applications. Springer. pp. 51–55. ISBN 978-3662639788.

[1] 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.

[2] Weinberg, S. (2005). "24". The Quantum Theory of Fields: Supersymmetry. Vol. 3. Cambridge University Press. pp. 1–2. ISBN 978-0521670555.

[3] Sohnius, M.F. (2001). "Recollections of a young contributor". Nucl. Phys. B Proc. Suppl. 101 (1–3): 129–132. Bibcode:2001NuPhS.101..129S. doi:10.1016/S0920-5632(01)01499-2.

[4] 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.

[5] Duplij, S. (2003). Concise Encyclopedia of Supersymmetry. Springer. pp. 181–182. ISBN 978-1402013386.

[6] Năstase, H. (2015). "3". Introduction to the AdS/CFT Correspondance. Cambridge: Cambridge University Press. pp. 45–46. ISBN 978-1107085855.

[7] Wess, J.; Bagger, B. (1992). "1". Supersymmetry and Supergravity. Princeton University Press. pp. 3–9. ISBN 978-0691025308.

[8] Ammon, M.; Erdmenger, J. (2015). "3". Gauge/Gravity Duality: Foundations and Applications. Cambridge University Press. pp. 139–140. ISBN 978-1107010345.

[9] Freund, P.G.O. (1988). "4". Introduction to Supersymmetry. Cambridge University Press. p. 26. ISBN 978-0521356756.

[10] Akhond, M.; et al. (2021). "The Hitchhiker's Guide to 4d N=2 Superconformal Field Theories". SciPost Phys. Lect. Notes. arXiv:2112.14764. doi:10.21468/SciPostPhysLectNotes.64. S2CID 245537355.

[11] Cecotti, S. (2015). "2". Supersymmetric Field Theories: Geometric Structures and Dualities. Cambridge: Cambridge University Press. p. 46. ISBN 978-1107053816.

[12] Dall'Agata, G.; Zagermann, M. (2021). "4". Supergravity: From First Principles to Modern Applications. Springer. pp. 51–55. ISBN 978-3662639788.

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 {{Short description|Theorem in theoretical physics}}
-In [[theoretical physics]], the '''Haag–Łopuszański–Sohnius theorem''' states that if both [[commutative property|commutating]] and [[anticommutative property|anticommutating]] [[generator (mathematics)|generators]] are considered, then the only way to nontrivially mix [[spacetime symmetries|spacetime]] and internal symmetries is through [[supersymmetry]]. The anticommutating generators are [[spin (physics)|spin]]-1/2 [[spinor]]s which can additionally admit their own internal symmetry known as [[R-symmetry]]. The theorem is a generalization of the [[Coleman-Mandula theorem]] to [[Lie superalgebras]]. It was proved in 1975 by [[Rudolf Haag]], [[Jan Łopuszański (physicist)|Jan Łopuszański]], and Martin Sohnius<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> as a response to the development of the first supersymmetric field theories by [[Julius Wess]] and [[Bruno Zumino]] in 1974.
+In [[theoretical physics]], the '''Haag–Łopuszański–Sohnius theorem''' states that if both [[commutative property|commutating]] and [[anticommutative property|anticommutating]] [[generator (mathematics)|generators]] are considered, then the only way to nontrivially mix [[spacetime symmetries|spacetime]] and internal symmetries is through [[supersymmetry]]. The anticommutating generators must be [[Spin (physics)|spin]]-1/2 [[spinor]]s which can additionally admit their own internal symmetry known as [[R-symmetry]]. The theorem is a generalization of the [[Coleman–Mandula theorem]] to [[Lie superalgebras]]. It was proved in 1975 by [[Rudolf Haag]], [[Jan Łopuszański (physicist)|Jan Łopuszański]], and Martin Sohnius<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> as a response to the development of the first supersymmetric field theories by [[Julius Wess]] and [[Bruno Zumino]] in 1974.
 ==History==
-Prior to the Haag–Łopuszański–Sohnius theorem, the [[Coleman–Mandula theorem]] was the strongest of a series of [[no-go theorem]]s, stating that the symmetry group of a [[consistent]] [[space-time|4-dimensional]] [[quantum field theory]] is the [[direct product of groups|direct product]] of the [[internal symmetry]] group and the [[Poincaré group]].
+During the 1960s, a set of theorems investigating how internal symmetries can be combined with spacetime symmetries were proved, with the most general being the Coleman–Mandula theorem.<ref>{{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=1–2|isbn=978-0521670555}}</ref> It showed that the [[Lie group]] [[symmetry (physics)|symmetry]] of an interacting theory must necessarily be a [[direct product]] of the [[Poincaré group]] with some [[compact group|compact]] internal group. Unaware of this theorem, during the early 1970s a number of authors independently came up with supersymmetry, seemingly in contradiction to the theorem since there some generators do transform non-trivially under spacetime transformations.
-In 1971 [[Yuri Golfand]] and [[E. P. Likhtman]] published the first paper on four-dimensional supersymmetry which presented (in modern notation) N=1 superalgebra and N=1 super-QED with charged matter and a mass term for the photon field. They proved that the conserved supercharges can exist in four dimensions by allowing both commuting and [[anticommutator|anticommuting]] symmetry generators, thus providing a nontrivial extension of the [[Poincaré algebra]], namely the [[supersymmetry algebra]]. In 1975, [[Rudolf Haag]], [[Jan Łopuszański (physicist)|Jan Łopuszański]], and [[Martin Sohnius]] further generalized superalgebras by analyzing extended supersymmetries (e.g. N=2) and introducing additional central charges.
+In 1974 Jan Łopuszański visited [[Karlsruhe]] from [[Wrocław]] shortly after Julius Wess and Bruno Zumino constructed the first supersymmetric [[quantum field theory]], the [[Wess–Zumino model]].<ref>{{cite journal|last1=Sohnius|first1=M.F.|authorlink1=|date=2001|title=Recollections of a young contributor|journal=Nucl. Phys. B Proc. Suppl.|volume=101|issue=1–3|pages=129–132|doi=10.1016/S0920-5632(01)01499-2|bibcode=2001NuPhS.101..129S }}</ref> Speaking to Wess, Łopuszański was interested in figuring out how these new theories managed to overcome the Coleman–Mandula theorem. While Wess was too busy to work with Łopuszański, his doctoral student Martin Sohnius was available. Over the next few weeks they devised a proof of their theorem after which Łopuszański went to [[CERN]] where he worked with Rudolf Haag to significantly refine the argument and also extend it to the [[massless particle|massless]] case. Later, after Łopuszański went back to Wrocław, Sohnius went to CERN to finish the paper with Haag, which was published in 1975.
-==Importance==
-What is most fundamental in this result (and thus in supersymmetry), is that there can be an interplay of spacetime symmetry with internal symmetry (in the sense of "mixing particles"): the supersymmetry generators transform [[boson]]ic particles into [[fermion]]ic ones and vice versa, but the [[anticommutator]] of two such transformations yields a [[translation]] in [[spacetime]]. Precisely such an interplay seemed excluded by the [[Coleman–Mandula theorem]], which stated that (bosonic) internal symmetries cannot interact non-trivially with spacetime symmetry.
+==Theorem==
-This theorem was also an important justification of the previously found [[Wess–Zumino model]], an interacting four-dimensional [[quantum field theory]] with [[supersymmetry]], leading to a [[renormalizable]] theory.
+The main assumptions of the Coleman–Mandula theorem are that the theory includes an [[S-matrix]] with [[analytic function|analytic]] [[scattering amplitude]]s such that any two-[[particle]] state must undergo some reaction at almost all [[energy|energies]] and scattering angles.<ref>{{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|journal=Phys. Rev.|volume=159|issue=5|pages=1251–1256|doi=10.1103/PhysRev.159.1251 |bibcode=1967PhRv..159.1251C }}</ref> Furthermore, there must only be a finite number of particle types below any [[mass]], disqualifying massless particles. The theorem then restricts the [[Lie algebra]] of the theory to be a direct sum of the Poincare algebra with some internal symmetry [[algebra over a field|algebra]].
+The Haag–Łopuszański–Sohnius theorem is based on the same assumptions, except for allowing additional anticommutating generators, elevating the Lie algebra to a Lie superalgebra. In four dimensions, the theorem states that the only nontrivial anticommutating generators that can be added are a set of <math>\mathcal N</math> pairs of [[supercharge]]s <math>Q^L_\alpha</math> and <math>\bar Q^R_{\dot \alpha}</math>, indexed by <math>\alpha</math>, which commute with the [[momentum operator|momentum generator]] and transform as [[chirality (physics)|left-handed and right-handed]] [[Weyl equation|Weyl spinors]]. The undotted and dotted index notation, known as [[Van der Waerden notation]], distinguishes left-handed and right-handed Weyl spinors from each other. Generators of other spin, such spin-3/2 or higher, are disallowed by the theorem.<ref>{{cite book|last=Duplij|first=S.|author-link=|date=2003|title=Concise Encyclopedia of Supersymmetry|url=|doi=|location=|publisher=Springer|chapter=|pages=181–182|isbn=978-1402013386}}</ref> In a basis where <math>(\bar Q^A_{\dot \alpha}) = (Q^A_\alpha)^\dagger</math>, these supercharges satisfy
+:<math>
+\{Q^A_\alpha, Q^B_\beta\} = \epsilon_{\alpha \beta} Z^{AB}, \ \ \ \ \ \ \ \ \ \ \{Q^A_\alpha, \bar Q^B_{\dot \beta}\} = \delta^{AB}\sigma^\mu_{\alpha \dot \beta}P_\mu,
+</math>
+where <math>Z^{AB}</math> are known as [[central charge]]s, which commute with all generators of the superalgebra. Together with the Poincaré algebra, this Lie superalgebra is known as the [[super-Poincaré algebra]]. Since four dimensional Minkowski spacetime also admits [[Majorana equation|Majorana spinors]] as fundamental spinor representations, the algebra can equivalently be written in terms of four-component Majorana spinor supercharges, with the algebra expressed in terms of [[gamma matrices]] and the [[C-symmetry|charge conjugation operator]] rather than Pauli matrices used for the two-component Weyl spinors.<ref>{{cite book|last=Năstase|first=H.|author-link=|date=2015|title=Introduction to the AdS/CFT Correspondance|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=3|pages=45–46|isbn=978-1107085855}}</ref>
+The supercharges can also admit an additional Lie algebra symmetry known as R-symmetry, whose generators <math>B_i</math> satisfy
+:<math>
+[Q^A_\alpha, B_i] =\sum_B s^{AB}_i Q^B_\alpha,
+</math>
+where <math>s_i^{AB}</math> are [[Hermitian matrix|Hermitian]] [[group representation|representation matrices]] of the generators in the <math>\mathcal N</math>-dimensional representation of the R-symmetry group.<ref>{{cite book|last1=Wess|first1=J.|author-link1=Julius Wess|last2=Bagger|first2=B.|author-link2=Jonathan Bagger|date=1992|title=Supersymmetry and Supergravity|url=|doi=|location=|publisher=Princeton University Press|chapter=1|pages=3–9|isbn=978-0691025308}}</ref> For <math>\mathcal N=1</math> the central charge must vanish and the R-symmetry is given by a <math>\text{U}(1)</math> group, while for [[extended supersymmetry]] <math>\mathcal N>1</math> the central charges need not vanish, while the R-symmetry is a <math>\text{U}(\mathcal N)</math> group.
+If massless particles are allowed, then the algebra can additionally be extended using [[conformal symmetry|conformal]] generators: the [[homothety|dilaton]] generator <math>D</math> and the [[special conformal transformation]]s generator <math>K_\mu</math>. For <math>\mathcal N</math> supercharges, there must also be the same number of superconformal generators <math>S_\alpha</math> which satisfy
+:<math>
+\{S^A_\alpha, \bar S^B_{\dot \beta}\} = \delta^{AB}\sigma^\mu_{\alpha \dot \beta}K_\mu,
+</math>
+with both the supercharges and the superconformal generators being charged under a <math>\text{U}(\mathcal N)</math> R-symmetry.<ref>{{cite book|last1=Ammon|first1=M.|author-link1=|last2=Erdmenger|first2=J.|author-link2=|date=2015|title=Gauge/Gravity Duality: Foundations and Applications|url=|doi=|location=|publisher=Cambridge University Press|chapter=3|pages=139–140|isbn=978-1107010345}}</ref> This algebra is an example of a [[superconformal algebra]], which in this four dimensional case is denoted by <math>\mathfrak{su}(2,2|\mathcal N)</math>.<ref>{{cite book|last=Freund|first=P.G.O.|author-link=Peter Freund|date=1988|title=Introduction to Supersymmetry|url=|doi=|location=|publisher=Cambridge University Press|chapter=4|page=26|isbn=978-0521356756}}</ref> Unlike for non-conformal supersymmetric algebras, R-symmetry is always present in superconformal algebras.<ref>{{cite journal|last1=Akhond|first1=M.|authorlink1=|display-authors=etal|date=2021|title=The Hitchhiker's Guide to 4d N=2 Superconformal Field Theories|url=|journal=SciPost Phys. Lect. Notes|volume=|issue=|pages=|doi=10.21468/SciPostPhysLectNotes.64|pmid=|arxiv=2112.14764|s2cid=245537355|access-date= |doi-access=free }}</ref>
 ==Limitations==
+The Haag–Łopuszański–Sohnius theorem was originally derived in [[four-dimensional space|four dimensions]], however the result that supersymmetry is the only nontrivial extension to spacetime symmetries holds in all dimensions greater than two. The form of the supersymmetry algebra however changes. Depending on the dimension, the supercharges can be Weyl, Majorana, Weyl–Majorana, or symplectic Weyl–Majorana spinors. Furthermore, R-symmetry groups differ according to the dimensionality and the number of supercharges.<ref>{{cite book|last1=Cecotti|first1=S.|author-link1=Sergio Cecotti|date=2015|title=Supersymmetric Field Theories: Geometric Structures and Dualities|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=2|page=46|isbn=978-1107053816}}</ref> This superalgebra also only applies in Minkowski spacetime, being modified in other spacetimes. For example, there exists an extension to [[anti-de Sitter space]] for one or more supercharges, while an extension to [[de Sitter space]] only works if multiple supercharges are present.<ref>{{cite book|last1=Dall'Agata|first1=G.|author-link1=|last2=Zagermann|first2=M.|author-link2=|date=2021|title=Supergravity: From First Principles to Modern Applications|url=|doi=|location=|publisher=Springer|chapter=4|pages=51–55|isbn=978-3662639788}}</ref>
-The theorem only deals with "visible symmetries, i.e., with symmetries of the S-matrix" and thus it is still possible that "the fundamental equations may have a higher symmetry". Expressed differently, this means the theorem does not restrict broken symmetry, but only unbroken symmetries.
+In two or fewer dimensions the theorem breaks down. The reason for this is that analyticity of the scattering amplitudes can no longer hold since for example in two dimensions the only scattering is forward and backward scattering. The theorem also does not apply to [[discrete symmetry|discrete symmetries]] or to [[spontaneous symmetry breaking|spontaneously broken symmetries]] since these are not symmetries at the level of the S-matrix.
 ==See also==
@@ Line 21: / Line 47: @@
 ==References==
+{{reflist}}
-*{{Citation | last1=Haag | first1=Rudolf | last2=Sohnius | first2=Martin | last3=Łopuszański | first3=Jan T. | title=All possible generators of supersymmetries of the S-matrix | doi=10.1016/0550-3213(75)90279-5 | mr=0411396  | year=1975 | journal=Nuclear Physics B | volume=88 | issue=2 | pages=257–274|bibcode = 1975NuPhB..88..257H }}
 {{Supersymmetry topics}}
 {{DEFAULTSORT:Haag-Łopuszański-Sohnius theorem}}
+[[Category:Supersymmetry]]
 [[Category:Quantum field theory]]
-[[Category:Supersymmetric quantum field theory]]
 [[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