The Renormalization Group: Fundamental Aspects

Abstract

This chapter is of central importance. We introduce here the renormalization group, based on the observation that our choice of the elementary segment involves a large amount of arbitrariness. Thus there should exist a mapping from segments of size ℓ = ℓ₀ to segments of size ℓ₁ which does not change macroscopic observables. It also should not affect the basic structure of the model, viz. Gaussian segments interacting via some local two-body repulsion. Our previous arguments on the structure of the model never refered to any distinguished value of ℓ. In Sect. 8.1 we formulate these considerations more precisely in terms of the ‘Renormalization Group Hypothesis’. This is to be amended by the ‘Fixed Point Hypothesis’, assuming that with increasing size of the elementary segment the excluded volume constant tends to some finite limiting value. Universal scaling and power laws follow naturally.