In theoretical physics, the Haag–Łopuszański–Sohnius theorem shows that the possible symmetries of a consistent 4-dimensional quantum field theory do not only consist of internal symmetries and Poincaré symmetry, but can also include supersymmetry as a nontrivial extension of the Poincaré algebra. This significantly generalized the Coleman–Mandula theorem.
One of the important results is that the fermionic part of the Lie superalgebra has to have spin-1/2 (spin 3/2 or higher are ruled out).
Prior to the Haag–Łopuszański–Sohnius theorem, the Coleman–Mandula theorem was the strongest of a series of no-go theorems, stating that the symmetry group of a consistent 4-dimensional quantum field theory is the direct product of the internal symmetry group and the Poincaré group.
In 1975, Rudolf Haag, Jan Łopuszański, and Martin Sohnius published their proof that weakening the assumptions of the Coleman–Mandula theorem by allowing both commuting and anticommuting symmetry generators, there is a nontrivial extension of the Poincaré algebra, namely the supersymmetry algebra.
What is most fundamental in this result (and thus in supersymmetry), is that there can be an interplay of spacetime symmetry with internal symmetry (in the sense of "mixing particles"): the supersymmetry generators transform bosonic particles into fermionic ones and vice versa, but the anticommutator of two such transformations yields a translation in spacetime. Precisely such an interplay seemed excluded by the Coleman–Mandula theorem, which stated that (bosonic) internal symmetries cannot interact non-trivially with spacetime symmetry.
This theorem was also an important justification of the previously found Wess–Zumino model, an interacting four-dimensional quantum field theory with supersymmetry, leading to a renormalizable theory.
The theorem only deals with "visible symmetries, i.e., with symmetries of the S-matrix" and thus it is still possible that "the fundamental equations may have a higher symmetry". Expressed differently, this means the theorem does not restrict broken symmetry, but only unbroken symmetries.