are the precursor to string theory, rather than from any attempts to overcome the no-go theorem. Similarly, the Haag–Łopuszański–Sohnius theorem, a supersymmetric generalization of the Coleman–Mandula theorem, was proved in 1975 after the study of supersymmetry was already underway. Theorem Consider a theory that can be described by an S-matrix and that satisfies the following conditions The symmetry group is a Lie group which includes the Poincaré group as a subgroup, Below any mass, there are only a finite number of particle types, Any two-particle state undergoes some reaction at almost all energies, The amplitudes for elastic two-body scattering are analytic functions of the scattering angle at almost all energies and angles, A technical assumption that the group generators are distributions in momentum space. The Coleman–Mandula theorem states that the symmetry of this theory is necessarily a direct product of the Poincaré group and an internal symmetry group. Note that the last technical assumption is unnecessary if the theory is described by a quantum field theory and is only needed to apply the theorem in a wider context. A kinematic argument for why the theorem should hold was provided by Edward Witten. The argument is that Poincaré symmetry is far too strong of a constraint for elastic scattering, leaving only the scattering angle unknown. Hence, any additional spacetime dependent symmetry would overdetermine the amplitudes, making them nonzero only at discrete scattering angles. Since this conflicts with the assumption of the analyticity of the scattering angles, such additional symmetries are ruled out. Limitations Conformal symmetry The theorem does not apply to a theory of massless particles, with these possibly admitting an additional spacetime symmetry called conformal symmetry. In particular, the allowed algebra is the Poincaré algebra together with the commutation relations for the dilaton generator and the special conformal transformations generator, giving the conformal algebra. Supersymmetry The Coleman–Mandula theorem assumes that the only symmetry algebras are Lie algebras, but this can be generalized to Lie superalgebras. Doing this allows for additional anticommutating generators known as supercharges which transform as spinors under Lorentz transformations. The resulting algebra is known as a super-Poincaré algebra, with the associated symmetry known as supersymmetry. The Haag–Łopuszański–Sohnius theorem is the generalization of the Coleman–Mandula theorem to Lie superalgebras, with it stating that supersymmetry is the only new spacetime dependent symmetry that is allowed. For a theory with massless particles, the theorem is again evaded by conformal symmetry