Coleman–Mandula theorem

Statement

Let $G$ be a connected Lie group of unitary symmetries of the S-matrix of a relativistic quantum theory in four spacetime dimensions. Assume the following.

1. Lorentz invariance: $G$ contains a subgroup locally isomorphic to the Poincaré group.
2. Particle finiteness: for every $M > 0$ there are only finitely many types of particles of mass less than $M$.
3. Weak elastic analyticity: elastic two-body scattering amplitudes are analytic functions of the Mandelstam variables $s$ and $t$ near the physical region, except at normal thresholds.
4. Occurrence of scattering: let $T = S - 1$ be the transition part of the S-matrix. For one-particle momenta $p$ and $p'$, the vector $T \lvert p, p' \rangle$ is nonzero, except perhaps at isolated values of $s$.
5. Technical assumption: the generators of $G$ act on one-particle states as integral operators in momentum space with distribution kernels.

Then $G$ is locally isomorphic to the direct product of the Poincaré group and an internal symmetry group. The generators of the internal group commute with all Poincaré generators and act only on the particle type labels. The Lie algebra of the internal group is the direct sum of a compact semisimple Lie algebra and an abelian Lie algebra.