Introduction

So far, in our study of instanton physics, we have focused on theories with bosonic fields only. In this chapter we will analyze instanton effects in theories with fermionic fields. We will first consider an extension of the quantum-mechanical models studied in Chapter 1, and in particular we will look at supersymmetric QM. This is our only incursion into supersymmetry in this book, but it has some important lessons for instanton physics: first of all, it gives yet another example of a phenomenon which is purely non-perturbative, namely the breaking of supersymmetry in some quantum mechanical models. Second, our study of these simplified models will give us a first taste of one important aspect of instanton calculus in the presence of fermions: the existence of zero modes for the fermion fields.

After this detailed study of instantons and fermions in a simplified setting, we will move on to QCD. We will first study chiral symmetries due to the presence of quark flavors, and we will present a quick introduction to the effective Lagrangian of QCD at low energies. This is a subject which is both elegant and relevant to the phenomenology of QCD. Finally, we will discuss the U (1) problem in QCD and how it can be solved in principle by taking into account the anomaly in the axial current. In Chapter 7 we will make this more precise, at large N, through the Witten–Veneziano formula.

Instantons in supersymmetric Quantum Mechanics

We will now study a very interesting variant of the quantum mechanical models that were considered in Chapter 1: supersymmetric QM. We will restrict ourselves to one-dimensional models.

General aspects

In order to introduce supersymmetric QM, we have to add fermionic coordinates to the standard bosonic coordinate q. In terms of operators, this means that on top of the usual bosonic operators q, p, we have to introduce Grassmann operators ψ1, 2, obeying the anticommutation relations,