Editing Renormalization group (section)

===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>}}