In theoretical physics, the Haag–Łopuszański–Sohnius theorem states that if both commutating and anticommutating generators are considered, then the only way to nontrivially mix spacetime and internal symmetries is through supersymmetry.
The anticommutating generators must be spin-1/2 spinors which can additionally admit their own internal symmetry known as R-symmetry.
It was proved in 1975 by Rudolf Haag, Jan Łopuszański, and Martin Sohnius[1] as a response to the development of the first supersymmetric field theories by Julius Wess and Bruno Zumino in 1974.
Unaware of this theorem, during the early 1970s a number of authors independently came up with supersymmetry, seemingly in contradiction to the theorem since there some generators do transform non-trivially under spacetime transformations.
In 1974 Jan Łopuszański visited Karlsruhe from Wrocław shortly after Julius Wess and Bruno Zumino constructed the first supersymmetric quantum field theory, the Wess–Zumino model.
[3] Speaking to Wess, Łopuszański was interested in figuring out how these new theories managed to overcome the Coleman–Mandula theorem.
While Wess was too busy to work with Łopuszański, his doctoral student Martin Sohnius was available.
Over the next few weeks they devised a proof of their theorem after which Łopuszański went to CERN where he worked with Rudolf Haag to significantly refine the argument and also extend it to the massless case.
Later, after Łopuszański went back to Wrocław, Sohnius went to CERN to finish the paper with Haag, which was published in 1975.
The main assumptions of the Coleman–Mandula theorem are that the theory includes an S-matrix with analytic scattering amplitudes such that any two-particle state must undergo some reaction at almost all energies and scattering angles.
In four dimensions, the theorem states that the only nontrivial anticommutating generators that can be added are a set of
, which commute with the momentum generator and transform as left-handed and right-handed Weyl spinors.
The undotted and dotted index notation, known as Van der Waerden notation, distinguishes left-handed and right-handed Weyl spinors from each other.
are known as central charges, which commute with all generators of the superalgebra.
Since four dimensional Minkowski spacetime also admits Majorana spinors as fundamental spinor representations, the algebra can equivalently be written in terms of four-component Majorana spinor supercharges, with the algebra expressed in terms of gamma matrices and the charge conjugation operator rather than Pauli matrices used for the two-component Weyl spinors.
[6] The supercharges can also admit an additional Lie algebra symmetry known as R-symmetry, whose generators
[10] The Haag–Łopuszański–Sohnius theorem was originally derived in four dimensions, however the result that supersymmetry is the only nontrivial extension to spacetime symmetries holds in all dimensions greater than two.
Depending on the dimension, the supercharges can be Weyl, Majorana, Weyl–Majorana, or symplectic Weyl–Majorana spinors.
Furthermore, R-symmetry groups differ according to the dimensionality and the number of supercharges.
For example, there exists an extension to anti-de Sitter space for one or more supercharges, while an extension to de Sitter space only works if multiple supercharges are present.