23 - Relevant and irrelevant fields

Summary

In this chapter we continue to study the massless scalar field described by the action (22.1). We have seen that this model has a gapless excitation spectrum and the correlation functions of bosonic exponents followpower laws. This behaviour implies that the correlation length is infinite and the system is in its critical phase. It is certainly very important to knowhowstable this critical point is with respect to perturbations. Any model is only an idealization; when one derives it certain interactions are neglected. Howdoes one decide which interactions are important and which are not? Obviously, weak interactions are those whose influence on the correlation functions is small. The trouble is, however, that usually the correlation functions are affected differently on different scales, the long distance asymptotics being affected the most. Therefore it can happen that a certain perturbation causes only tiny changes at short distances, but changes the large distance behaviour profoundly. In the renormalization group picture this is observed as a growth of the coupling constant associated with the perturbing operator. Such growth is a frequent phenomenon in critical theories; a slow decay of their correlation functions gives rise to divergences in their diagram series.

Operators whose influence grows on large scales (small momenta) are called ‘relevant’. The problem of relevancy of perturbations can be formulated and solved in a general form. Suppose we have a system at criticality (i.e. all its correlation functions decay as a power law at large distances).