Introduction

More variables usually means greater complexity, but not always. 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.

There are two reasons to study the 1/N expansion.

(1) It can be used to analyze model field theories. This is important. Most of us have a good intuition for the phenomena of classical mechanics. We were not born with this intuition; we developed it toiling over problems involving rigid spheres that roll without slipping and similar extreme but instructive simplifications of reality. One reason we have such a poor intuition for the phenomena of quantum field theory is that there are so few simple examples; essentially all we have to play with is perturbation theory and a handful of soluble models. The 1/N expansion enables us to enlarge this set.

In Section 2 I develop the 1/N expansion for φ4 theory and apply it to two-dimensional models with similar combinatoric structures, the Gross–Neveu model and the ℂPN−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; they are worth studying.

(2) It is possible that the 1/N expansion, with N the number of colors, might fruitfully be applied to quantum chromodynamics.