9 - Supersymmetry

Summary

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 they commute 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: 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 X and X*. (The theory of two-component spinors is reviewed in Appendix A, where our notation, which is essentially that of the text by Wess 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 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”).