In a standard advanced field theory course, one learns about a number of symmetries: Poincaré invariance, global continuous symmetries, discrete symmetries, gauge symmetries, approximate and exact symmetries. These latter symmetries all have the property that they commute with Lorentz transformations and in particular with rotations. So, the multiplets of the symmetries always contain particles of the same spin; in particular, they always consist of either bosons or fermions.

For a long time, it was believed that these were the only allowed types of symmetry; this statement was even embodied in a theorem, known as the Coleman–Mandula theorem. However, physicists studying theories based on strings stumbled on a symmetry which related fields of different spin. Others quickly worked out simple field theories with this new symmetry, called supersymmetry.

Supersymmetric field theories can be formulated in dimensions up to eleven. These higher-dimensional theories will be important when we consider string theory. In this chapter we consider theories in four dimensions. The supersymmetry charges, because they change spin, must themselves carry spin – they are spin-1/2 operators. They transform as doublets under the Lorentz group, just like the two-component spinors χ and χ*. (The theory of two-component spinors is reviewed in Appendix A, where our notation, which is essentially that of the text byWess and Bagger (1992), is explained.) There can be 1, 2, 4 or 8 such spinors; correspondingly, the symmetry is said to be N = 1, 2, 4 or 8 supersymmetry. Like the generators of an ordinary group, the supersymmetry generators obey an algebra; unlike an ordinary bosonic group, however, the algebra involves anticommutators as well as commutators (it is said to be “graded”).

There are at least four reasons to think that supersymmetry might have something to do with TeV-scale physics. The first is the hierarchy problem: as we will see, supersymmetry can both explain how hierarchies arise, and why there are no large radiative corrections. The second is the unification of couplings. We have seen that while the gauge group of the Standard Model can in a rather natural way be unified in a larger group, the couplings do not unify properly. In the minimal supersymmetric extension of the Standard Model (the minimal supersymmetric Standard Model, or MSSM) the couplings unify nicely if the scale of supersymmetry breaking is about 1 TeV.