nLab Haag–Łopuszański–Sohnius theorem

Idea

The Coleman-Mandula theorem says that the only possible symmetry Lie algebra of the S-matrix of a 4-dimensional quantum field theory with mass gap is a direct sum of the Poincare Lie algebra and some other Lie algebra of internal gauge symmetries, hence that there cannot be a non-trivial mixing between the spacetime symmetry and the internal symmetry at the level of Lie algebras and in the presence of a mass gap.

(This applies to fundamental local symmetry of the S-matrix, not to global symmetry after spontaneous symmetry breaking.)

The theorem of Haag–Łopuszański–Sohnius 75 makes the analogous statement for super Lie algebras: the only possible super Lie algebra symmetry of an S-matrix is a direct sum of the super Poincare Lie algebra (“supersymmetry”) and another super Lie algebra.

Notice that the super Poincare Lie algebra itself does mix the plain bosonic Poincare Lie algebra symmetry with a kind of “internal” symmetry, see also at extended supersymmetry.

References

The original article:

• Rudolf Haag, Jan Łopuszański, Martin Sohnius, All possible generators of supersymmetries of the S-matrix, Nuclear Physics B, 88: 257–274 (1975) (doi:10.1016/0550-3213(75)90279-5, spire:90850)

On the history of the result:

• Jan Łopuszański, The story of central charges, p. 16-17 in: Steven Duplij, Warren Siegel, Jonathan Bagger (eds.), Concise encyclopedia of supersymmetry – and noncommutative structures in mathematics and physics, Springer 2005 (doi:10.1007/1-4020-4522-0 pdf)

See also:

• Wikipedia, Haag-Łopuszański-Sohnius theorem