Editing Renormalization group (section)

=== Deeper understanding ===
The renormalization group emerges from the [[renormalization]] of the quantum field variables, which normally has to address the problem of infinities in a quantum field theory.{{efn|Although note that the RG exists independently of the infinities.}} This problem of systematically handling the infinities of quantum field theory to obtain finite physical quantities was solved for QED by [[Richard Feynman]], [[Julian Schwinger]] and [[Shin'ichirō Tomonaga]], who received the 1965 Nobel prize for these contributions. They effectively devised the theory of mass and charge renormalization, in which the infinity in the momentum scale is [[Cutoff (physics)|cut off]] by an ultra-large [[Regularization (physics)|regulator]], Λ.{{efn|The regulator parameter Λ could ultimately be taken to be infinite – infinities reflect the pileup of contributions from an infinity of degrees of freedom at infinitely high energy scales.}}

The dependence of physical quantities, such as the electric charge or electron mass, on the scale Λ is hidden, effectively swapped for the longer-distance scales at which the physical quantities are measured, and, as a result, all observable quantities end up being finite instead, even for an infinite Λ. Gell-Mann and Low thus realized in these results that, infinitesimally, while a tiny change in '' g'' is provided by the above RG equation given ψ(''g''), the self-similarity is expressed by the fact that ψ(''g'') depends explicitly only upon the parameter(s) of the theory, and not upon the scale ''μ''. Consequently, the above renormalization group equation may be solved for (''G'' and thus) ''g''(''μ'').

A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilation group of conventional ''renormalizable'' theories, considers methods where widely different scales of lengths appear simultaneously. It came from [[condensed matter physics]]: [[Leo P. Kadanoff]]'s paper in 1966 proposed the "block-spin" renormalization group.<ref name="Kadanoff">{{cite journal |author-link=Leo P. Kadanoff |first=Leo P. |last=Kadanoff |year=1966 |title=Scaling laws for Ising models near <math>T_c</math> |journal=Physics Physique Fizika |volume=2 |issue=6 |page=263|doi=10.1103/PhysicsPhysiqueFizika.2.263 |doi-access=free }}</ref> The "blocking idea" is a way to define the components of the theory at large distances as aggregates of components at shorter distances.

This approach covered the conceptual point and was given full computational substance in the extensive important work of [[Kenneth G. Wilson|Kenneth Wilson]]. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the [[Kondo effect|Kondo problem]], in 1975,<ref>{{cite journal |author-link=Kenneth G. Wilson |first=K.G. |last=Wilson |year=1975 |title=The renormalization group: Critical phenomena and the Kondo problem |journal=Rev. Mod. Phys. |volume=47 |issue=4 |page=773|doi=10.1103/RevModPhys.47.773 |bibcode=1975RvMP...47..773W }}</ref> as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and [[critical phenomena]] in 1971.<ref>{{Cite journal |last=Wilson |first=K.G. |author-link=Kenneth G. Wilson |title=Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture |doi=10.1103/PhysRevB.4.3174 |journal=Physical Review B |volume=4 |issue=9 |pages=3174–3183 |year=1971 |bibcode=1971PhRvB...4.3174W|doi-access=free }}</ref><ref>{{Cite journal |last=Wilson |first=K. |author-link=Kenneth G. Wilson |title=Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior |doi=10.1103/PhysRevB.4.3184 |journal=Physical Review B |volume=4 |issue=9 |pages=3184–3205 |year=1971 |bibcode=1971PhRvB...4.3184W|doi-access=free }}</ref><ref>{{cite journal |last1=Wilson |first1=K.G. |author1-link=Kenneth G. Wilson |last2=Fisher |first2=M. |year=1972 |title=Critical exponents in 3.99 dimensions |journal=Physical Review Letters |volume=28 |issue=4 |page=240 |doi=10.1103/physrevlett.28.240 |bibcode=1972PhRvL..28..240W }}</ref> He was awarded the Nobel prize for these decisive contributions in 1982.<ref>{{cite web |url=https://www.nobelprize.org/uploads/2018/06/wilson-lecture-2.pdf |title=Wilson's Nobel Prize address |website=NobelPrize.org |first=Kenneth G. |last=Wilson |author-link=Kenneth G. Wilson}}</ref>