1/N

Abstract

There exist families of field theories with symmetry group SO(N) (or SU(N)) that become simpler as N becomes larger. More precisely, the solutions to these theories possess an expansion in powers of 1/N. This expansion is the subject of these lectures. The 1/N expansion can be used to analyze model field theories. The 1/N expansion is developed for phi/sup 4/ theory and applied to two two-dimensional models with similar combinatoric structures, the Gross-Neveu model and the CP/sup N-1/ model. These models display (in the leading 1/N approximation) such interesting phenomena as asymptotic freedom, dynamical symmetry breaking, dimensional transmutation, and non-perturbative confinement. It is possible that the 1/N expansion, with N the number of colors, might fruitfully be applied to quantum chromodynamics. Unfortunately, it is not possible to make a decisive test of the approximation, because no one knows how to compute even the first term in the expansion in closed form. However, it is possible to argue that this first term, whatever its detailed form, has many properties that are also shared by the real world, and which are otherwise underived from field theory. These include the saturation of scattering amplitudes by an infinite number of narrow resonances, the essential feature of dual-resonance models. (RWR)