Coleman–Mandula theorem

Description

Every quantum field theory satisfying the assumptions,

1. Below any mass M, there are only finite number of particle types
2. Any two-particle state undergoes some reaction at almost all energies
3. The amplitude for elastic two body scattering are analytic functions of scattering angle at almost all energies,

and that has non-trivial interactions can only have a Lie group symmetry which is always a direct product of the Poincaré group and an internal group if there is a mass gap: no mixing between these two is possible. As the authors say in the introduction to the 1967 publication, "We prove a new theorem on the impossibility of combining space-time and internal symmetries in any but a trivial way."

Different spacetime symmetries

The first condition for the theorem is that the unified group "G contains a subgroup locally isomorphic to the Poincare group." Therefore the theorem only makes a statement about the unification of the Poincare group with an internal symmetry group. However, if the Poincare group is replaced with a different spacetime symmetry, for example, with the de Sitter group the theorem no longer holds. In addition, if all particles are massless the Coleman–Mandula theorem allows a combination of internal and spacetime symmetries, because the spacetime symmetry group is then the conformal group.

Spontaneous symmetry breaking

Note that this theorem only constrains the symmetries of the S-matrix itself. As such, it places no constraints on spontaneously broken symmetries which do not show up directly on the S-matrix level. In fact, it is easy to construct spontaneously broken symmetries (in interacting theories) which unify spatial and internal symmetries.

Discreteness

This theorem also only applies to discrete Lie algebras and not continuous Lie groups. As such, it does not apply to discrete symmetries or globally for Lie groups. As an example of the latter, we might have a model where a rotation by 2π (a discrete spacetime symmetry) is identified with an involutive internal symmetry which commutes with all the other internal symmetries.

If there is no mass gap, it could be a tensor product of the conformal algebra with an internal Lie algebra. But in the absence of a mass gap, there are also other possibilities. For example, quantum electrodynamics has vector and tensor conserved charges. See infraparticle for more details.

Supersymmetry

Supersymmetry may be considered a possible "loophole" of the theorem because it contains additional generators (supercharges) that are not scalars but rather spinors. This loophole is possible because supersymmetry is a Lie superalgebra, not a Lie algebra. The corresponding theorem for supersymmetric theories with a mass gap is the Haag–Łopuszański–Sohnius theorem.

Quantum group symmetry, present in some two-dimensional integrable quantum field theories like the sine-Gordon model, exploits a similar loophole.