Renormalization Group - Relevant and Irrelevant Operators, Universality Classes

Relevant and Irrelevant Operators, Universality Classes

Let us consider a certain observable of a physical system undergoing an RG transformation. The magnitude of the observable as the length scale of the system goes from small to large may be (a) always increasing, (b) always decreasing or (c) other. In the first case, the observable is said to be a relevant observable; in the second, irrelevant and in the third, marginal.

A relevant operator is needed to describe the macroscopic behaviour of the system; an irrelevant observable is not. Marginal observables may or may not need to be taken into account. A remarkable fact is that most observables are irrelevant, i.e.: the macroscopic physics is dominated by only a few observables in most systems. In other terms: microscopic physics contains (Avogadro's number) variables, and macroscopic physics only a few.

Before the RG, there was an astonishing empirical fact to explain: the coincidence of the critical exponents (i.e.: the behaviour near a second order phase transition) in very different phenomena, such as magnetic systems, superfluid transition (Lambda transition), alloy physics, etc. This was called universality and is successfully explained by RG, just showing that the differences between all those phenomena are related to irrelevant observables.

Thus, many macroscopic phenomena may be grouped into a small set of universality classes, described by the set of relevant observables.