Editing Coleman–Mandula theorem (section)

==Theorem==

Consider a theory that can be described by an [[S-matrix]] and that satisfies the following conditions<ref name="CM"/>
* The symmetry [[group (mathematics)|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 [[energy|energies]],
* 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 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=1983|title=The Unity of the Fundamental Interactions|url=https://doi.org/10.1007/978-1-4613-3655-6|doi=|location=|publisher=Springer|chapter=|pages=305–315|isbn=978-0-306-41242-4}}</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.