Coleman–Mandula theorem

A theorem stating that, under specified assumptions, continuous symmetries of a relativistic particle-scattering matrix cannot nontrivially mix spacetime and internal symmetries.

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S-matrix: An operator encoding transition amplitudes between incoming and outgoing states in a scattering process. The theorem constrains symmetries of this scattering operator.
Lorentz group: The group of linear transformations preserving the spacetime interval of special relativity. Its transformations form the relativistic part of the Poincaré symmetry.
Sidney Coleman: An American theoretical physicist known for work in quantum field theory and particle physics. He co-authored the theorem's original formulation.
Haag–Łopuszański–Sohnius theorem: A classification theorem showing that supersymmetry extends the allowed spacetime and internal symmetry algebra. It adds graded symmetries while preserving the theorem's broader structural constraints.
Supersymmetric quantum field theory: A quantum field theory whose fields and interactions obey supersymmetry. The theorem's supersymmetric extension made such theories a viable framework for unified symmetries.
Poincaré group: The group of spacetime translations and Lorentz transformations in special relativity. Its generators supply the spacetime symmetries covered by the theorem.
Noether's theorem: A theorem relating continuous symmetries of an action to conserved quantities. It connects the continuous symmetries in the theorem to conserved generators.
Jeffrey Mandula: A theoretical physicist who co-authored the theorem on symmetries of the S-matrix. His work with Coleman established the no-mixing result.
Supersymmetry algebra: A graded algebra whose fermionic generators transform bosonic states into fermionic states and vice versa. Its anticommutators include spacetime translations, enabling a qualified symmetry mixing.
Grand unified theory: A theory that embeds the Standard Model gauge interactions in a larger gauge group. The theorem distinguishes internal gauge unification from forbidden mixing with spacetime symmetry.