Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 An introduction to unitary symmetry
- 2 Soft pions
- 3 Dilatations
- 4 Renormalization and symmetry: a review for non-specialists
- 5 Secret symmetry: an introduction to spontaneous symmetry breakdown and gauge fields
- 6 Classical lumps and their quantum descendants
- 7 The uses of instantons
- 8 1/N
8 - 1/N
Published online by Cambridge University Press: 10 November 2010
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 An introduction to unitary symmetry
- 2 Soft pions
- 3 Dilatations
- 4 Renormalization and symmetry: a review for non-specialists
- 5 Secret symmetry: an introduction to spontaneous symmetry breakdown and gauge fields
- 6 Classical lumps and their quantum descendants
- 7 The uses of instantons
- 8 1/N
Summary
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.
- Type
- Chapter
- Information
- Aspects of SymmetrySelected Erice Lectures, pp. 351 - 402Publisher: Cambridge University PressPrint publication year: 1985
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