Editing Renormalization group (section)

==History==<!--'History of renormalization group theory' redirects here-->
The idea of scale transformations and scale invariance is old in physics: Scaling arguments were commonplace for the [[Pythagoreanism|Pythagorean school]], [[Euclid]], and up to [[Galileo]].<ref>{{cite web |url=http://www.av8n.com/physics/scaling.htm |title=Introduction to Scaling Laws |website=av8n.com}}</ref> They became popular again at the end of the 19th century, perhaps the first example being the idea of enhanced [[viscosity]] of [[Osborne Reynolds]], as a way to explain turbulence.

The renormalization group was initially devised in particle physics, but nowadays its applications extend to [[solid-state physics]], [[fluid mechanics]], [[physical cosmology]], and even [[nanotechnology]]. An early article<ref>{{cite journal |author1-link=Ernst Stueckelberg |last1=Stueckelberg |first1=E.C.G. |author2-link=André Petermann |first2=A. |last2=Petermann |year=1953 |url=https://www.e-periodica.ch/cntmng?pid=hpa-001:1953:26::894 |title=La renormalisation des constants dans la théorie de quanta |journal=Helv. Phys. Acta |volume=26 |pages=499–520 |language=FR}}</ref> by [[Ernst Stueckelberg]] and [[André Petermann]] in 1953 anticipates the idea in [[quantum field theory]]. Stueckelberg and Petermann opened the field conceptually. They noted that [[renormalization]] exhibits a [[Group (mathematics)|group]] of transformations which transfers quantities from the bare terms to the counter terms. They introduced a function ''h''(''e'') in [[Quantum electrodynamics| quantum electrodynamics (QED)]], which is now known as the [[Beta function (physics)|beta function]] (see below).

===Beginnings===
[[Murray Gell-Mann]] and [[Francis E. Low]] restricted the idea to scale transformations in QED in 1954,<ref>{{cite journal |last=Gell-Mann |first=M. |author-link=Murray Gell-Mann |author2=Low, F. E. |author-link2=Francis E. Low |year=1954 |title=Quantum Electrodynamics at Small Distances |journal=Physical Review |volume=95 |issue=5 |pages=1300–1312 |doi=10.1103/PhysRev.95.1300 |bibcode=1954PhRv...95.1300G |url=https://authors.library.caltech.edu/60469/1/PhysRev.95.1300.pdf}}</ref> which are the most physically significant, and focused on asymptotic forms of the photon propagator at high energies.  They determined the variation of the electromagnetic coupling in QED, by appreciating the simplicity of the scaling structure of that theory. They thus discovered that the coupling parameter ''g''(''μ'') at the energy scale ''μ'' is effectively given by the (one-dimensional translation) group equation
<math display="block">g(\mu)=G^{-1}\left(\left(\frac{\mu}{M}\right)^d G(g(M))\right)</math>
or equivalently, <math>G\left(g(\mu)\right)= G(g(M))\left({\mu}/{M}\right)^d</math>, for an arbitrary function ''G'' (known as [[Franz Wegner|Wegner]]'s scaling function) and a constant ''d'', in terms of the coupling ''g(M)'' at a reference scale ''M''.

Gell-Mann and Low realized in these results that the effective scale can be arbitrarily taken as ''μ'', and can vary to define the theory at any other scale:
<math display="block">g(\kappa)=G^{-1}\left(\left(\frac{\kappa}{\mu}\right)^d G(g(\mu))\right) = G^{-1}\left(\left(\frac{\kappa}{M}\right)^d G(g(M))\right)</math>

The gist of the RG is this group property: as the scale ''μ'' varies, the theory presents a self-similar replica of itself, and any scale can be accessed similarly from any other scale, by group action, a formal transitive conjugacy of couplings<ref>{{cite journal |last1=Curtright |first1=T.L. |author-link1=Thomas Curtright |last2=Zachos |first2=C.K. |date=March 2011 |title=Renormalization Group Functional Equations |journal=Physical Review D |volume=83 |issue=6 |pages=065019 |doi=10.1103/PhysRevD.83.065019 |bibcode=2011PhRvD..83f5019C |arxiv=1010.5174|s2cid=119302913 }}</ref> in the mathematical sense ([[Schröder's equation]]).

On the basis of this (finite) group equation and its scaling property, Gell-Mann and Low could then focus on infinitesimal transformations, and invented a computational method based on a mathematical flow function {{math|''ψ''(''g'') {{=}} ''G'' ''d''/(∂''G''/∂''g'')}} of the coupling parameter ''g'', which they introduced. Like the function ''h''(''e'') of Stueckelberg-Petermann, their function determines the differential change of the coupling ''g''(''μ'') with respect to a small change in energy scale ''μ'' through a differential equation, the ''renormalization group equation'':
<math display="block"> \displaystyle\frac{\partial g}{\partial \ln\mu} = \psi(g) = \beta(g) </math>
The modern name is also indicated, the [[Beta function (physics)|beta function]], introduced by [[Curtis Callan|C. Callan]] and [[Kurt Symanzik|K. Symanzik]] in 1970.<ref name=CS/> Since it is a mere function of ''g'', integration in ''g'' of a perturbative estimate of it permits specification of the renormalization trajectory of the coupling, that is, its variation with energy, effectively the function ''G'' in this perturbative approximation. The renormalization group prediction (cf. Stueckelberg–Petermann and Gell-Mann–Low works) was confirmed 40 years later at the [[LEP]] accelerator experiments: the [[Fine-structure constant|fine structure "constant"]] of QED was measured<ref>{{Cite journal|last=Fritzsch|first=Harald|date=2002|title=Fundamental Constants at High Energy|journal=Fortschritte der Physik|volume=50|issue=5–7|pages=518–524|doi=10.1002/1521-3978(200205)50:5/7<518::AID-PROP518>3.0.CO;2-F|arxiv=hep-ph/0201198|bibcode=2002ForPh..50..518F |s2cid=18481179 }}</ref> to be about {{frac|1|127}} at energies close to 200 GeV, as opposed to the standard low-energy physics value of {{frac|1|137}}.{{efn|Early applications to [[quantum electrodynamics]] are discussed in the influential 1959 book ''The Theory of Quantized Fields'' by [[Nikolay Bogolyubov]] and [[Dmitry Shirkov]].<ref>{{cite book |author1-link=Nikolay Bogolyubov |first1=N.N. |last1=Bogoliubov |author2-link=Dmitry Shirkov |first2=D.V. |last2=Shirkov |year=1959 |title=The Theory of Quantized Fields |place=New York, NY |publisher=Interscience}}</ref>}}

=== 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>

===Reformulation===
Meanwhile, the RG in particle physics had been reformulated in more practical terms by Callan and Symanzik in 1970.<ref name=CS>{{cite journal |last=Callan |first=C.G. |year=1970 |title=Broken scale invariance in scalar field theory |doi=10.1103/PhysRevD.2.1541 |journal=Physical Review D |volume=2 |issue=8 |pages=1541–1547 |bibcode=1970PhRvD...2.1541C}}</ref><ref>{{cite journal |last=Symanzik |first=K. |year=1970 |title=Small distance behaviour in field theory and power counting |doi=10.1007/BF01649434 |journal=Communications in Mathematical Physics |volume=18 |issue=3 |pages=227–246 |bibcode=1970CMaPh..18..227S|s2cid=76654566 |url=http://projecteuclid.org/euclid.cmp/1103842537 }}</ref> The above beta function, which describes the "running of the coupling" parameter with scale, was also found to amount to the "canonical trace anomaly", which represents the quantum-mechanical breaking of scale (dilation) symmetry in a field theory.{{efn|Remarkably, the trace anomaly and the running coupling quantum mechanical procedures can themselves induce mass.}} Applications of the RG to particle physics exploded in number in the 1970s with the establishment of the [[Standard Model]].

In 1973,<ref>{{cite journal |first1=D.J. |last1=Gross |first2=F. |last2=Wilczek |year=1973 |title=Ultraviolet behavior of non-Abelian gauge theories |journal=[[Physical Review Letters]] |volume=30 |issue=26 |pages=1343–1346 |doi=10.1103/PhysRevLett.30.1343 |doi-access=free |bibcode=1973PhRvL..30.1343G}}</ref><ref>{{cite journal |first=H.D. |last=Politzer |year=1973 |title=Reliable perturbative results for strong interactions |journal=[[Physical Review Letters]]  |volume=30 |issue=26 |pages=1346–1349 |bibcode=1973PhRvL..30.1346P |doi=10.1103/PhysRevLett.30.1346 |doi-access=free}}</ref> it was discovered that a theory of interacting colored quarks, called [[quantum chromodynamics]], had a negative beta function. This means that an initial high-energy value of the coupling will eventuate a special value of {{mvar|μ}} at which the coupling blows up (diverges). This special value is the [[Strong coupling constant#QCD and asymptotic freedom|scale of the strong interactions]], [[Coupling constant#QCD scale|{{mvar|μ}} = {{math|&Lambda;}}{{sub|QCD}}]] and occurs at about 200 MeV. Conversely, the coupling becomes weak at very high energies ([[asymptotic freedom]]), and the quarks become observable as point-like particles, in [[deep inelastic scattering]], as anticipated by Feynman–Bjorken scaling. QCD was thereby established as the quantum field theory controlling the strong interactions of particles.

Momentum space RG also became a highly developed tool in solid state physics, but was hindered by the extensive use of perturbation theory, which prevented the theory from succeeding in strongly correlated systems.{{efn|For strongly correlated systems, [[Calculus of variations|variational]] techniques are a better alternative.}}

=== Conformal symmetry ===
Conformal symmetry is associated with the vanishing of the beta function. This can occur naturally if a coupling constant is attracted, by running, toward a ''fixed point'' at which ''β''(''g'') = 0. In QCD, the fixed point occurs at short distances where ''g'' → 0 and is called a ([[Quantum triviality|trivial]]) [[ultraviolet fixed point]]. For heavy quarks, such as the [[top quark]], the coupling to the mass-giving [[Higgs boson]] runs toward a fixed non-zero (non-trivial) [[infrared fixed point]], first predicted by Pendleton and Ross (1981),<ref>{{cite journal |first1=Brian |last1=Pendleton |first2=Graham |last2=Ross |title=Mass and mixing angle predictions from infrared fixed points |journal=Physics Letters B |volume=98 |issue=4 |year=1981 |pages=291–294 |doi=10.1016/0370-2693(81)90017-4
|bibcode=1981PhLB...98..291P }}</ref> and [[C. T. Hill]].<ref>{{cite journal |first=Christopher T. |last=Hill |author-link=C. T. Hill |title=Quark and lepton masses from renormalization group fixed points |journal=Physical Review D |volume=24 |issue=3 |year=1981 |pages=691–703 |doi=10.1103/PhysRevD.24.691|bibcode=1981PhRvD..24..691H }}</ref> 
The top quark Yukawa coupling lies slightly below the infrared fixed point of the Standard Model suggesting the possibility of additional new physics, such as sequential heavy Higgs bosons.{{citation needed|date=December 2022}}

In [[string theory]], conformal invariance of the string world-sheet is a fundamental symmetry: ''β'' = 0 is a requirement. Here, ''β'' is a function of the geometry of the space-time in which the string moves. This determines the space-time dimensionality of the string theory and enforces Einstein's equations of [[general relativity]] on the geometry. The RG is of fundamental importance to string theory and theories of [[grand unification]].

It is also the modern key idea underlying [[critical phenomena]] in condensed matter physics.<ref>{{Cite journal |last=Shankar |first=R. |doi=10.1103/RevModPhys.66.129 |title=Renormalization-group approach to interacting fermions |journal=Reviews of Modern Physics |volume=66 |issue=1 |pages=129–192 |year=1994 |arxiv=cond-mat/9307009 |bibcode=1994RvMP...66..129S}} (For nonsubscribers see {{cite journal |title= Renormalization-group approach to interacting fermions|arxiv = cond-mat/9307009|doi = 10.1103/RevModPhys.66.129|last =  Shankar|first = R. |journal = Reviews of Modern Physics|year = 1993|volume = 66|issue = 1|pages = 129–192|bibcode = 1994RvMP...66..129S}}.)</ref> Indeed, the RG has become one of the most important tools of modern physics.<ref>{{cite journal |first1=L.Ts. |last1=Adzhemyan |first2=T.L. |last2=Kim |first3=M.V. |last3=Kompaniets |first4=V.K. |last4=Sazonov |title=Renormalization group in the infinite-dimensional turbulence: determination of the RG-functions without renormalization constants |journal=Nanosystems: Physics, Chemistry, Mathematics |date=August 2015 |volume=6 |issue=4 |page=461|doi=10.17586/2220-8054-2015-6-4-461-469 |doi-access=free }}</ref> It is often used in combination with the [[Monte Carlo method]].<ref name="CallawayPetronzio1984">{{cite journal |last1=Callaway |first1=David J.E. |last2=Petronzio |first2=Roberto |title=Determination of critical points and flow diagrams by Monte Carlo renormalization group methods |journal=Physics Letters B |volume=139 |issue=3 |year=1984 |pages=189–194 |issn=0370-2693 |doi=10.1016/0370-2693(84)91242-5 |bibcode=1984PhLB..139..189C |url=https://cds.cern.ch/record/149868}}</ref>