Historical supersymmetry: bosons and fermions

Our first example of supersymmetry is the classical example of a symmetry relating fermions and bosons. Let us put this into its proper historical context. The name derives from supermultiplet, a word coined by Wigner to denote a representation of some higher symmetry. This supersymmetry would presumably unify the spacetime symmetries — e.g., the Poincaré or conformal symmetries — with the internal symmetries governing the spectrum of excitations in nuclear or particle physics. Such a symmetry would have underlied the so-called relativistic quark model.

A celebrated theorem of Coleman and Mandula put an end to the search for the relativistic quark model. They showed that the maximal Lie algebra of symmetries of the scattering matrix of a four-dimensional relativistic quantum field theory is the direct product of a spacetime Lie algebra — the Poincaré or conformal Lie algebra — with the Lie algebra of a compact Lie group. Being a direct product, the Casimir invariants of the spacetime Lie algebra are also invariant under the internal symmetries and hence irreducible representations of the direct product are spin- and mass-degenerate.

The insight of Haag, Łopuszański and Sohnius was to realise that one could escape the rather negative consequences of the Coleman-Mandula theorem by relaxing the notion of symmetry to allow for what is now known as a Lie superalgebra. These are algebras with a notion of parity which is preserved by the bracket in a natural way.

This parity (or ℤ2-grading), which corresponds physically to the parity associated to fermion number, has come to be synonymous with supersymmetry in some (more algebraic) quarters. There is some element of truth in this way of thinking, of course, but this is not where I want us to go.

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