The Sigma-Model Limit

In this chapter we will discuss the infinite-coupling limit of the polynomial models, which is, in a way, the antithesis of the perturbative approach. We will see that in the $\lambda\rightarrow\infty$ limit there is an exact representation of the polynomial models, which is another class of non-linear models of scalar fields, known as the sigma models. As we will see in the simple case we will deal with here, in this limit the quantum $\lambda \varphi ^{4}$ models can be identified with a simple example of these sigma models. In the one-component models that we are examining here these sigma models reduce to the well-known Ising models. Note that we are dealing here with an equivalence between the quantum versions of the models, not between the corresponding classical field theories, which are quite different from one another.