This chapter is devoted to the study of the non-linear σ-model, a quantum field theory (QFT) where the (scalar) field is an N-component vector of fixed length, mostly in dimensions close to 2. The model possesses a global, non-linearly realized symmetry, O(N) symmetry: under a group transformation, the transformed field is a non-linear function of the field itself. The non-linear σ-model belongs to a class of models constructed on special homogeneous spaces, symmetric spaces that, as Riemannian manifolds, admit a unique metric. Unlike what happens in a (ϕ2)2 -like field theory with the same symmetry, in the non-linear σ-model, in the tree approximation, the O(N) symmetry is always spontaneously broken: the action describes the interactions of (N−1) massless fields, the Goldstone modes. Since the fields are massless, in two dimensions infrared divergences appear in the perturbative expansion and an infrared regulator is required. To understand the phase structure beyond leading order, a renormalization group (RG) analysis is necessary. This requires understanding how the model renormalizes. Power counting shows that the model is renormalizable in two dimensions. Since the field then is dimensionless, although the degree of divergence of Feynman diagrams is bounded, an infinite number of counterterms is generated, because all correlation functions are divergent. A quadratic master equation satisfied by the generating functional of vertex functions is derived, which makes it possible to prove that the coefficients of all counterterms are related, and that the renormalized theory depends only on two parameters.
Oscillator Representation and Generalized van der Waals Hamiltonians
