nLab Coleman-Mandula theorem

Idea

The Coleman–Mandula theorem is a no-go theorem motivated by the possibilities of Lie group symmetries in quantum field theory in Minkowski space-time.

Statement

Any Lie group containing the Poincaré group $P$ (in 4d) as a subgroup and containing a maximal internal symmetry group $G$ must be a direct product of those. In addition, $G$ must be a semisimple Lie group with additional $U(1)$ (circle group) factors.

The generalization of this statement to super Lie algebras is known as the Haag–Łopuszański–Sohnius theorem.

Remarks

Gel’fand and Likhtman showed that with a slight extension of the concept of Lie group, one can get that $P$ and $G$ combine in a nontrivial way. This happens for example in the supersymmetric case.

References

• Sidney Coleman, Jeffrey Ellis Mandula, All Possible Symmetries of the S Matrix, Physical Review 159 (5): 1251–1256 (1967) (doi:10.1103/PhysRev.159.1251)

• I. M. Gel'fand, E. S. Likhtman, JETP Letters 13, 323 (1971)

Review includes

• Wikipedia (English)