Physics:Quantum Renormalization group - Revision history

WikiHarold: Restore missing Quantum reference definitions

2026-05-24T00:31:28Z

Restore missing Quantum reference definitions

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	== Scale dependence and running couplings ==		== 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		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">{{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> 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>		<math>\beta(g)=\mu \frac{d g}{d\mu}.</math>
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	== Renormalization group equation ==		== 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		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">{{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> In differential form this becomes

	<math>\frac{\partial g}{\partial \ln \mu}=\beta(g).</math>		<math>\frac{\partial g}{\partial \ln \mu}=\beta(g).</math>
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	<math>\beta(g^*)=0.</math>		<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.		At a fixed point, the couplings stop running, and the system becomes scale invariant.<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><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.		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.
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	== Critical phenomena and universality ==		== 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.		One of the deepest achievements of the renormalization group is its explanation of critical phenomena.<ref name="Wilson1975" /><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> 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" />		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">{{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><ref name="Goldenfeld" />

	The RG explains why macroscopic behavior near criticality depends mainly on broad properties such as:		The RG explains why macroscopic behavior near criticality depends mainly on broad properties such as:
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	{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}

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

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}
	{{Sourceattribution\|Physics:Quantum Renormalization group\|1}}		{{Sourceattribution\|Physics:Quantum Renormalization group\|1}}

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	'''Renormalization group''' ~~is a Book I topic in~~ the ~~Quantum Collection. 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~~. In quantum field theory, it describes how coupling constants, masses, and fields depend on the energy scale \mu at which a process is probed~~. 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. 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. ~~In quantum field~~ theory, ~~it describes how coupling constants, masses, and fields~~ depend on the ~~energy~~ scale ~~\mu~~ at which ~~a process~~ is ~~probed~~.		'''Renormalization group''' 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. 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. 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. 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.
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	~~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>		'''Renormalization group''' is a Book I topic in the Quantum Collection. 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. In quantum field theory, it describes how coupling constants, masses, and fields depend on the energy scale \mu at which a process is probed. 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. 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. In quantum field theory, it describes how coupling constants, masses, and fields depend on the energy scale \mu at which a process is probed.
	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>~~
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	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}
	{{Sourceattribution\|Renormalization group}}		{{Sourceattribution\|Physics:Quantum Renormalization group\|1}}

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			{{Short description\|Quantum Collection topic on Quantum Renormalization group}}

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	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 '''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>		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>
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			[[File:Renormalization_group_flow.jpg\|thumb\|280px\|Quantum Renormalization group.]]
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	[[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.]]		</div>

	== Scale dependence and running couplings ==		== Scale dependence and running couplings ==

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	{{Short description\|Quantum Collection topic on Quantum Renormalization group}}		{{Short description\|Quantum Collection topic on Quantum Renormalization group}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
	{{Quantum article nav\|previous=Physics:Quantum Renormalization in field theory\|previous label=Renormalization in field theory\|next=Physics:Quantum Field Theory Gauge symmetry\|next label=Field Theory Gauge symmetry}}		{{Quantum article nav\|previous=Physics:Quantum Renormalization in field theory\|previous label=Renormalization in field theory\|next=Physics:Quantum Field Theory Gauge symmetry\|next label=Field Theory Gauge symmetry}}