Editing Renormalization group (section)

==Relevant and irrelevant operators and universality classes==
<!-- This section is linked from [[Power law]] -->
{{See also|Universality class| Dangerously irrelevant operator|Phase transition#Critical exponents and universality classes|Scale invariance#CFT description}}

Consider a certain observable {{mvar|A}} 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 determines the importance of the observable(s) for the scaling law:
{| style="margin-left:3em;"
| <small>'''''If its magnitude''''' ... </small>
| <small>'''''then the observable is''''' ...</small>
|-
| always increases
| '''relevant'''
|-
| always decreases
| '''irrelevant'''
|-
| other
| '''marginal'''
|}
A ''relevant'' observable is needed to describe the macroscopic behaviour of the system; ''irrelevant'' observables are not needed. ''Marginal'' observables may or may not need to be taken into account. A remarkable broad fact is that ''most observables are irrelevant'', i.e., ''the macroscopic physics is dominated by only a few observables in most systems''.

As an example, in microscopic physics, to describe a system consisting of a [[Mole (unit)|mole]] of [[carbon-12]] atoms we need of the order of 10{{sup|23}} (the [[Avogadro constant|Avogadro number]]) variables, while to describe it as a macroscopic system (12&nbsp;grams of carbon-12) we only need a few.

Before Wilson's RG approach, there was an astonishing empirical fact to explain: The coincidence of the [[critical exponents]] (i.e., the exponents of the reduced-temperature dependence of several quantities near a [[second order phase transition]]) in very disparate phenomena, such as magnetic systems, superfluid transition ([[Lambda transition]]), alloy physics, etc. So in general, thermodynamic features of a system near a phase transition ''depend only on a small number of variables'', such as the dimensionality and symmetry, but are insensitive to details of the underlying microscopic properties of the system.

This coincidence of critical exponents for ostensibly quite different physical systems, called [[universality (dynamical systems)|universality]], is easily explained using the renormalization group, by demonstrating that the differences in phenomena among the individual fine-scale components are determined by ''irrelevant observables'', while the ''relevant observables'' are shared in common. Hence many macroscopic phenomena may be grouped into a small set of '''[[universality class]]es''', specified by the shared sets of relevant observables.{{efn|A superb technical exposition by [[:de:Jean Zinn-Justin|J. Zinn-Justin]] (2010) is the classic article {{cite journal |title=Critical Phenomena: Field theoretical approach |journal=Scholarpedia |year=2010 |doi=10.4249/scholarpedia.8346 |last1=Zinn-Justin |first1=Jean |volume=5 |issue=5 |pages=8346 |bibcode=2010SchpJ...5.8346Z |doi-access=free}}. For example, for Ising-like systems with a <math>\mathbb{Z}_2</math> symmetry or, more generally, for models with an O(N) symmetry, the Gaussian (free) fixed point is long-distance stable above space dimension four, marginally stable in dimension four, and unstable below dimension four. See [[Quantum triviality]].}}