Coleman–Mandula theorem

The Coleman–Mandula theorem, named after Sidney Coleman and Jeffrey Mandula, is a no-go theorem in theoretical physics. It states that "space-time and internal symmetries cannot be combined in any but a trivial way".^[1] The only conserved quantities in a "realistic" theory with a mass gap, apart from the generators of the Poincaré group, must be Lorentz scalars.

Description

Every quantum field theory satisfying certain technical assumptions about its S-matrix that has non-trivial interactions can only have a symmetry Lie algebra 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."^[2][3]

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.

This theorem also only applies to Lie algebras and not 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 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 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–Lopuszanski–Sohnius theorem.

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

Notes

References

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