Editing Coleman–Mandula theorem (section)

==Limitations==

===Conformal symmetry===

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 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]]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=|url-access=subscription}}</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=|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=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>