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{{Quantum book backlink|Quantum field theory}}
The '''renormalization group''' ('''RG''') is a framework in theoretical physics for studying how the description of a physical system changes when the observation scale changes.<ref name="GellMannLow1954">{{cite journal |last=Gell-Mann |first=M. |last2=Low |first2=F. E. |year=1954 |title=Quantum Electrodynamics at Small Distances |journal=Physical Review |volume=95 |issue=5 |pages=1300–1312 |doi=10.1103/PhysRev.95.1300}}</ref><ref name="Wilson1975">{{cite journal |last=Wilson |first=K. G. |year=1975 |title=The renormalization group: Critical phenomena and the Kondo problem |journal=Reviews of Modern Physics |volume=47 |issue=4 |pages=773–840 |doi=10.1103/RevModPhys.47.773}}</ref> In quantum field theory, it describes how coupling constants, masses, and fields depend on the energy scale <math>\mu</math> at which a process is probed.<ref name="Callan1970">{{cite journal |last=Callan |first=C. G. |year=1970 |title=Broken scale invariance in scalar field theory |journal=Physical Review D |volume=2 |issue=8 |pages=1541–1547 |doi=10.1103/PhysRevD.2.1541}}</ref><ref name="Symanzik1970">{{cite journal |last=Symanzik |first=K. |year=1970 |title=Small distance behaviour in field theory and power counting |journal=Communications in Mathematical Physics |volume=18 |issue=3 |pages=227–246 |doi=10.1007/BF01649434}}</ref>
The same idea is equally important in statistical mechanics and condensed matter physics, where it explains critical phenomena, universality, and the emergence of effective large-scale laws from microscopic interactions.<ref name="Kadanoff1966">{{cite journal |last=Kadanoff |first=L. P. |year=1966 |title=Scaling laws for Ising models near <math>T_c</math> |journal=Physics Physique Fizika |volume=2 |issue=6 |pages=263–272 |doi=10.1103/PhysicsPhysiqueFizika.2.263}}</ref><ref name="WilsonFisher1972">{{cite journal |last=Wilson |first=K. G. |last2=Fisher |first2=M. E. |year=1972 |title=Critical exponents in 3.99 dimensions |journal=Physical Review Letters |volume=28 |issue=4 |pages=240–243 |doi=10.1103/PhysRevLett.28.240}}</ref>

[[File:Renormalization_group_flow.jpg|thumb|400px|Renormalization group flow showing how couplings evolve with scale toward fixed points, revealing scale dependence, universality, and critical behavior.]]

== Scale dependence and running couplings ==

A central RG idea is that the parameters of a theory are not fixed once and for all, but depend on the scale at which the system is examined.<ref name="GellMannLow1954" /> If <math>g(\mu)</math> denotes a coupling at scale <math>\mu</math>, its scale dependence is encoded by the beta function

<math>\beta(g)=\mu \frac{d g}{d\mu}.</math>

This equation tells us how the coupling changes under scale transformations. In quantum electrodynamics, for example, the effective electric charge depends on distance or momentum scale because vacuum polarization screens the bare charge.<ref name="GellMannLow1954" />

This scale dependence is often called the '''running''' of couplings. It is one of the most important physical consequences of renormalization theory.<ref name="SchwartzQFT">{{cite book |last=Schwartz |first=Matthew D. |title=Quantum Field Theory and the Standard Model |publisher=Cambridge University Press |year=2014 |page=314 |doi=10.1017/9781139540940}}</ref>

== Renormalization group equation ==

The RG equation expresses the fact that a theory should describe the same physics even when the arbitrary renormalization scale is changed.<ref name="Callan1970" /><ref name="Symanzik1970" /> In differential form this becomes

<math>\frac{\partial g}{\partial \ln \mu}=\beta(g).</math>

More generally, whole Green functions or effective actions satisfy RG equations. In particle physics this leads to the Callan–Symanzik formalism, which makes the scale dependence of quantum field theories explicit.<ref name="Callan1970" /><ref name="Symanzik1970" />

The RG therefore provides not just a calculational trick, but a systematic way to connect descriptions of the same system across different scales.

== Fixed points and scale invariance ==

A '''fixed point''' of the RG flow is a value <math>g^*</math> such that

<math>\beta(g^*)=0.</math>

At a fixed point, the couplings stop running, and the system becomes scale invariant.<ref name="Wilson1975" /><ref name="Goldenfeld">{{cite book |last=Goldenfeld |first=N. |title=Lectures on Phase Transitions and the Renormalization Group |publisher=Addison-Wesley |year=1992}}</ref> In many important cases this extends to conformal invariance, especially in critical systems and in quantum field theories with enhanced symmetry.

Fixed points organize the possible large-scale behaviors of a theory. They determine whether a theory becomes weakly coupled at high energies, strongly coupled in the infrared, or flows into a critical state.

== Quantum field theory applications ==

In quantum field theory, the RG explains how couplings evolve with energy and why different interactions behave differently at short and long distances.<ref name="SchwartzQFT" /> A classic example is quantum chromodynamics (QCD), where the beta function is negative, so the strong coupling decreases at high energy.<ref name="GrossWilczek1973">{{cite journal |last=Gross |first=D. J. |last2=Wilczek |first2=F. |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}}</ref><ref name="Politzer1973">{{cite journal |last=Politzer |first=H. D. |year=1973 |title=Reliable perturbative results for strong interactions |journal=Physical Review Letters |volume=30 |issue=26 |pages=1346–1349 |doi=10.1103/PhysRevLett.30.1346}}</ref>

This property is called '''asymptotic freedom'''. At very high energies, quarks behave almost as free particles, while at low energies the coupling grows and confinement becomes important.<ref name="GrossWilczek1973" /><ref name="Politzer1973" />

In quantum electrodynamics, by contrast, the effective electromagnetic coupling grows slowly with energy.<ref name="GellMannLow1954" /> The RG therefore gives a unified language for comparing the scale dependence of different quantum field theories.

== Critical phenomena and universality ==

One of the deepest achievements of the renormalization group is its explanation of critical phenomena.<ref name="Wilson1975" /><ref name="Kadanoff1966" /> Near a continuous phase transition, the correlation length becomes very large, and the system becomes insensitive to many microscopic details.

This is why very different physical systems can share the same critical exponents and scaling laws. They flow toward the same RG fixed point and therefore belong to the same '''universality class'''.<ref name="WilsonFisher1972" /><ref name="Goldenfeld" />

The RG explains why macroscopic behavior near criticality depends mainly on broad properties such as:

* dimensionality
* symmetry
* number of relevant operators

rather than on microscopic details of the underlying material.

== Block-spin picture ==

A particularly intuitive version of the RG is Kadanoff’s block-spin picture.<ref name="Kadanoff1966" /> In a lattice system such as the Ising model, one groups nearby microscopic degrees of freedom into blocks and replaces them by coarse-grained variables.

After one blocking step, the theory has fewer degrees of freedom but a similar form, with renormalized parameters such as temperature or interaction strength. Repeating this procedure shows how the theory flows under successive coarse-graining steps.

This makes the RG a natural tool for understanding how long-distance physics emerges from short-distance structure.

== Relevant, irrelevant, and marginal operators ==

In RG language, operators are classified by how their couplings behave under scale transformations.<ref name="Goldenfeld" /><ref name="Wilson1975" />

* '''Relevant operators''' grow under coarse-graining and strongly affect long-distance physics.
* '''Irrelevant operators''' decrease and become unimportant at large scales.
* '''Marginal operators''' remain roughly scale-neutral and require more detailed analysis.

This classification is central to effective field theory and universality. It explains why only a few parameters are often needed to describe large-scale behavior, even when the microscopic system is enormously complicated.

== Momentum-space renormalization ==

In field theory, RG transformations are often carried out in momentum space by integrating out high-momentum modes.<ref name="Shankar1994">{{cite journal |last=Shankar |first=R. |year=1994 |title=Renormalization-group approach to interacting fermions |journal=Reviews of Modern Physics |volume=66 |issue=1 |pages=129–192 |doi=10.1103/RevModPhys.66.129}}</ref> Since large momentum corresponds to short distance, removing these modes produces an effective description valid at lower energies.

This momentum-space approach is especially natural in perturbative quantum field theory, where one studies how the couplings of an action change as the cutoff or renormalization scale is varied.<ref name="SchwartzQFT" />

It is also widely used in condensed matter theory, although strongly correlated systems often require nonperturbative extensions.

== Exact and functional renormalization group ==

Beyond perturbative RG, there are exact or functional renormalization group equations that describe the flow of entire effective actions rather than a few couplings.<ref name="Polchinski1984">{{cite journal |last=Polchinski |first=J. |year=1984 |title=Renormalization and Effective Lagrangians |journal=Nuclear Physics B |volume=231 |issue=2 |pages=269–295 |doi=10.1016/0550-3213(84)90287-6}}</ref><ref name="Wetterich1993">{{cite journal |last=Wetterich |first=C. |year=1993 |title=Exact evolution equation for the effective potential |journal=Physics Letters B |volume=301 |issue=1 |pages=90–94 |doi=10.1016/0370-2693(93)90726-X}}</ref>

A major example is the Wetterich equation,

<math>\frac{d}{dk}\Gamma_k[\varphi]
=\tfrac{1}{2}\operatorname{Tr}\left[\left(\frac{\delta^2 \Gamma_k}{\delta \varphi \delta \varphi}+R_k\right)^{-1}\cdot\frac{d}{dk}R_k\right],</math>

where <math>\Gamma_k</math> is a scale-dependent effective action and <math>R_k</math> is an infrared regulator.<ref name="Wetterich1993" />

These approaches are particularly useful in nonperturbative problems, including critical phenomena, quantum gravity, and strongly interacting systems.

== Historical development ==

The modern RG emerged from several stages. Early scale ideas appeared long before quantum field theory, but the decisive conceptual beginnings in field theory came from Stueckelberg and Petermann in 1953, followed by Gell-Mann and Low in 1954 for quantum electrodynamics.<ref name="Stueckelberg1953">{{cite journal |last=Stueckelberg |first=E. C. G. |last2=Petermann |first2=A. |year=1953 |title=La renormalisation des constantes dans la théorie des quanta |journal=Helvetica Physica Acta |volume=26 |pages=499–520}}</ref><ref name="GellMannLow1954" />

A deeper physical interpretation came from Kadanoff’s scaling picture and Wilson’s full renormalization group framework in the late 1960s and early 1970s.<ref name="Kadanoff1966" /><ref name="Wilson1971I">{{cite journal |last=Wilson |first=K. G. |year=1971 |title=Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture |journal=Physical Review B |volume=4 |issue=9 |pages=3174–3183 |doi=10.1103/PhysRevB.4.3174}}</ref><ref name="Wilson1971II">{{cite journal |last=Wilson |first=K. G. |year=1971 |title=Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior |journal=Physical Review B |volume=4 |issue=9 |pages=3184–3205 |doi=10.1103/PhysRevB.4.3184}}</ref>

The discovery of asymptotic freedom in QCD then established the RG as a central tool of modern particle physics.<ref name="GrossWilczek1973" /><ref name="Politzer1973" />

== Summary ==

The renormalization group is one of the most powerful ideas in modern physics because it explains how theories change with scale.<ref name="Wilson1975" /><ref name="Goldenfeld" /> It provides the language for:

* running couplings in quantum field theory
* fixed points and scale invariance
* critical phenomena and universality
* effective field theories and coarse-graining
* nonperturbative flow equations

It therefore unifies key ideas across particle physics, statistical mechanics, condensed matter theory, and modern field theory.

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Renormalization group}}

Physics:Quantum Renormalization group - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

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