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". Since "realistic" theories contain a mass gap, the only conserved quantities, apart from the generators of the Poincaré group, must be Lorentz scalars.

Every quantum field theory satisfying the assumptions,

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."

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.

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.

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