The history of supersymmetry is as peculiar as anything in the history of science. Suggested in the early 1970s, supersymmetry has been elaborated since then into a beautiful mathematical formalism that unites particles of different spin into symmetry multiplets and has profound implications for fundamental physics. Yet there is so far not a shred of direct experimental evidence and only a few bits of indirect evidence that supersymmetry has anything to do with the real world. If (as I expect) supersymmetry does turn out to be relevant to nature, it will represent a striking success of purely theoretical insight.

Chapter 25 will begin the construction of supersymmetry theories from first principles. In the present chapter we shall introduce supersymmetry along chronological rather than logical lines.

Unconventional Symmetries and ‘No-Go’ Theorems

In the early 1960s the symmetry SU(3) of Gell-Mann and Ne'eman (discussed in Section 19.7) successfully explained the relations between various strongly interacting particles of different charge and strangeness but of the same spin. The idea then grew up that perhaps SU(3) is part of a larger symmetry, which has the unconventional effect of uniting SU(3) multiplets of different spin. There is such an approximate symmetry in the non-relativistic quark model, under S (7(6) transformations on quark spins and flavors, analogous to an earlier SU(4) symmetry of nuclear physics that had been introduced in 1937 by Wigner.

This volume deals with quantum field theories that are governed by supersymmetry, a symmetry that unites particles of integer and half-integer spin in common symmetry multiplets. These theories offer a possible way of solving the ‘hierarchy problem,’ the mystery of the enormous ratio of the Planck mass to the 300 GeV energy scale of electroweak symmetry breaking. Supersymmetry also has the quality of uniqueness that we search for in fundamental physical theories. There is an infinite number of Lie groups that can be used to combine particles of the same spin in ordinary symmetry multiplets, but there are only eight kinds of supersymmetry in four spacetime dimensions, of which only one, the simplest, could be directly relevant to observed particles.

These are reasons enough to devote this third volume of The Quantum Theory of Fields to supersymmetry. In addition, the quantum field theories based on supersymmetry have remarkable properties that are not found among other field theories: some supersymmetric theories have couplings that are not renormalized in any order of perturbation theory; other theories are finite; and some even allow exact solutions. Indeed, much of the most interesting work in quantum field theory over the past decade has been in the context of supersymmetry.

Physical phenomena at energies accessible in today's accelerator laboratories are accurately described by the standard model, the renormalizable theory of quarks, leptons, and gauge bosons, governed by the gauge group SU(3) × SU(2) × U(1), described in Sections 18.7 and 21.3. The standard model is today usually understood as a low-energy approximation to some as-yet-unknown fundamental theory in which gravitation appears unified with the strong and electroweak forces at an energy somewhere in the range of 1016 to 1018 GeV. This raises the hierarchy problem: what accounts for the enormous ratio of this fundamental energy scale and the energy scale ≈ 300 GeV that characterizes the standard model?

The strongest theoretical motivation for supersymmetry is that it offers a hope of solving the hierarchy problem. Quarks, leptons, and gauge bosons are required by the SU(3) × SU(2) × U(1) gauge symmetry to appear with zero masses in the Lagrangian of the standard model, so that the physical masses of these particles are proportional to the electroweak breaking scale, which in turn is proportional to the mass of the scalar fields responsible for the electroweak symmetry breakdown. The crux of the hierarchy problem1a is that the scalar fields, unlike the fermion and gauge boson fields, are not protected from acquiring large bare masses by any symmetry of the standard model, so it is difficult to see why their masses, and hence all other masses, are not in the neighborhood of 1016 to 1018 GeV.

Now we know the structure of the most general supersymmetry algebras, and we have seen how to work out the implications of this symmetry for the particle spectrum. In order to learn what supersymmetry has to say about particle interactions, we need to see how to construct supersymmetric field theories.

Originally the construction of field supermultiplets was done directly, by a repeated use of the Jacobi identities, much as in the construction of supermultiplets of one-particle states in Sections 25.4 and 25.5. Section 26.1 presents one example of this technique, used here to construct supermultiplets containing only scalar and Dirac fields. Fortunately there is an easier technique, invented by Salam and Strathdee, in which supermultiplets of fields are gathered into ‘superfields,’ which depend on fermionic coordinates as well as on the usual four coordinates of spacetime. Superfields are introduced in Section 26.2, and used to construct supersymmetric field theories and to study some of their consequences in Sections 26.3-26.8. This chapter will be concerned only with N = 1 supersymmetry, where the superfield formalism has been chiefly useful. At the end of the next chapter we will construct theories with N-extended supersymmetry by imposing the U(N) /∧-symmetry on theories of N = 1 superfields.

Direct Construction of Field Supermultiplets

To illustrate the direct construction of a field supermultiplet, we will consider fields that can destroy the particles belonging to the simplest supermultiplet of arbitrary mass discussed in Section 25.5: two spinless particles and one particle of spin 1/2.

We have encountered a number of infinite-dimensional symmetry algebras on the world-sheet: conformal, superconformal, and current. While we have used these symmetries as needed to obtain specific physical results, in the present chapter we would like to take maximum advantage of them in determining the form of the world-sheet theory. An obvious goal, not yet reached, would be to construct the general conformal or superconformal field theory, corresponding to the general classical string background.

This subject is no longer as central as it once appeared to be, as spacetime rather than world-sheet symmetries have been the principal tools in recent times. However, it is a subject of some beauty in its own right, with various applications to string compactification and also to other areas of physics.

We first discuss the representations of the conformal algebra, and the constraints imposed by conformal invariance on correlation functions. We then study some examples, such as the minimal models, Sugawara and coset theories, where the symmetries do in fact determine the theory completely. We briefly summarize the representation theory of the N = 1 superconformal algebra. We then discuss a framework, rational conformal field theory, which incorporates all these CFTs. To conclude this chapter we present some important results about the relation between conformal field theories and nearby two-dimensional field theories that are not conformally invariant, and the application of CFT in statistical mechanics.