The renormalization group and the epsilon expansion

Kenneth G Wilson

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in e = 4 – d is explained [d is the dimension of space (statistical mechanics) or space- time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1 – 8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6-8 include the approximate rcnormalization group recursion formula and the Feynman graph method for calculating exponents. Sections 10-13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11-13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.