In theoretical physics, the HaagLopuszanskiSohnius theorem shows that the possible symmetries of a consistent 4dimensional 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 ColemanMandula theorem. One of the important results is that the fermionic part of the Lie superalgebra has to have spin1/2 (spin 3/2 or higher are ruled out)
History Prior to the HaagLopuszanskiSohnius theorem, the ColemanMandula theorem was the strongest of a series of nogo theorems, stating that the symmetry group of a consistent 4dimensional 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 ColemanMandula theorem by allowing both commuting and anticommuting symmetry generators, there is a nontrivial extension of the Poincaré algebra, namely the supersymmetry algebra. Importance 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 commutator of two such transformations yields a translation in spacetime. Precisely such an interplay seemed excluded by the ColemanMandula theorem, which stated that (bosonic) internal symmetries cannot interact nontrivially with spacetime symmetry. This theorem was also an important justification of the previously found WessZumino model, an interacting fourdimensional quantum field theory with supersymmetry, leading to a renormalizable theory. See also * Supergravity * Smatrix References * R. Haag, J. T. Lopuszanski and M. Sohnius, "All Possible Generators Of Supersymmetries Of The S Matrix", Nucl. Phys. B 88 (1975) 257. Retrieved from "http://en.wikipedia.org/"

