Physics:Quantum triviality - Revision history

WikiHarold: Use Quantum See also index module

2026-05-23T22:26:18Z

Use Quantum See also index module

← Older revision		Revision as of 22:26, 23 May 2026
Line 129:		Line 129:
	</ref><ref>Urs Heller, Markus Klomfass, Herbert Neuberger, and Pavlos Vranas, (1993).		</ref><ref>Urs Heller, Markus Klomfass, Herbert Neuberger, and Pavlos Vranas, (1993).
	"Numerical analysis of the Higgs mass triviality bound", ''Nucl. Phys.'', '''B405''': 555-573.</ref>		"Numerical analysis of the Higgs mass triviality bound", ''Nucl. Phys.'', '''B405''': 555-573.</ref>

	~~==See also==~~
	* Hierarchy problem

	== See also ==		== See also ==

Maintenance script: Normalize quantum page header order

2026-05-22T11:33:42Z

Normalize quantum page header order

← Older revision		Revision as of 11:33, 22 May 2026
Line 1:		Line 1:
	{{~~short~~ description\|Possible outcome of renormalization in physics}}		{{Short description\|Possible outcome of renormalization in physics}}
	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
	{{Quantum article nav\|previous=Physics:Quantum Standard Model\|previous label=Standard Model\|next=Physics:Quantum confinement problem\|next label=Confinement problem}}		{{Quantum article nav\|previous=Physics:Quantum Standard Model\|previous label=Standard Model\|next=Physics:Quantum confinement problem\|next label=Confinement problem}}
	{{quantum field theory}}		{{quantum field theory}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Clean Book label remnants and backlink spacing

2026-05-22T11:12:00Z

Clean Book label remnants and backlink spacing

← Older revision		Revision as of 11:12, 22 May 2026
Line 1:		Line 1:
	{{short description\|Possible outcome of renormalization in physics}}		{{short description\|Possible outcome of renormalization in physics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
	{{Quantum article nav\|previous=Physics:Quantum Standard Model\|previous label=Standard Model\|next=Physics:Quantum confinement problem\|next label=Confinement problem}}		{{Quantum article nav\|previous=Physics:Quantum Standard Model\|previous label=Standard Model\|next=Physics:Quantum confinement problem\|next label=Confinement problem}}

Maintenance script: Remove hidden BOM characters and direct Book label after Short description

2026-05-22T11:04:13Z

Remove hidden BOM characters and direct Book label after Short description

← Older revision		Revision as of 11:04, 22 May 2026
Line 1:		Line 1:
	~~~~{{short description\|Possible outcome of renormalization in physics}}		{{short description\|Possible outcome of renormalization in physics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}

Maintenance script: Resize Quantum triviality lead image

2026-05-20T12:56:09Z

Resize Quantum triviality lead image

← Older revision		Revision as of 12:56, 20 May 2026
Line 1:		Line 1:
	~~~~{{short description\|Possible outcome of renormalization in physics}}		{{short description\|Possible outcome of renormalization in physics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
Line 29:		Line 29:

	<div style="width:300px;">		<div style="width:300px;">
	[[File:Quantum_book1_triviality_yellow.png]]		[[File:Quantum_book1_triviality_yellow.png\|thumb\|280px\|Quantum triviality.]]
	</div>		</div>

Maintenance script: Apply Quantum previous-next navigation

2026-05-20T12:26:24Z

Apply Quantum previous-next navigation

← Older revision		Revision as of 12:26, 20 May 2026
Line 1:		Line 1:
	{{short description\|Possible outcome of renormalization in physics}}		{{short description\|Possible outcome of renormalization in physics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
			{{Quantum article nav\|previous=Physics:Quantum Standard Model\|previous label=Standard Model\|next=Physics:Quantum confinement problem\|next label=Confinement problem}}
	{{quantum field theory}}		{{quantum field theory}}

Maintenance script: Clean Book I red links, intro, and image slots

2026-05-20T08:36:58Z

Clean Book I red links, intro, and image slots

← Older revision		Revision as of 08:36, 20 May 2026
Line 1:		Line 1:
	{{short description\|Possible outcome of renormalization in physics}}		{{short description\|Possible outcome of renormalization in physics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
Line 11:		Line 11:

	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	In a [[Physics:Quantum field theory\|quantum field theory]], ~~[[Physics:Asymptotic freedom#Screening and antiscreening\|~~charge screening]] can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as '''quantum triviality'''. Strong evidence supports the idea that a field theory involving only a scalar Higgs boson is trivial in four spacetime dimensions,<ref>{{cite book \| author1 = R. Fernandez \| author2 = J. Froehlich \| author3 = A. D. Sokal \| year = 1992 \| title = Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory \| publisher = Springer \| isbn = 0-387-54358-9 }}</ref><ref name="TrivPurs"/> but the situation for realistic models including other particles in addition to the Higgs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the Standard Model of particle physics, the question of triviality in Higgs models is of great importance.		In a [[Physics:Quantum field theory\|quantum field theory]], charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as '''quantum triviality'''. Strong evidence supports the idea that a field theory involving only a scalar Higgs boson is trivial in four spacetime dimensions,<ref>{{cite book \| author1 = R. Fernandez \| author2 = J. Froehlich \| author3 = A. D. Sokal \| year = 1992 \| title = Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory \| publisher = Springer \| isbn = 0-387-54358-9 }}</ref><ref name="TrivPurs"/> but the situation for realistic models including other particles in addition to the Higgs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the Standard Model of particle physics, the question of triviality in Higgs models is of great importance.

	This Higgs triviality is similar to the Landau pole problem in [[Physics:Quantum electrodynamics\|quantum electrodynamics]], where this quantum theory may be inconsistent at very high momentum scales unless the renormalized charge is set to zero, i.e., unless the field theory has no interactions. The Landau pole question is generally considered to be of minor academic interest for quantum electrodynamics because of the inaccessibly large momentum scale at which the inconsistency appears. This is not however the case in theories that involve the elementary scalar Higgs boson, as the momentum scale at which a "trivial" theory exhibits inconsistencies may be accessible to present experimental efforts such as at the Large Hadron Collider (LHC) at CERN. In these Higgs theories, the interactions of the Higgs particle with itself are posited to generate the masses of the W and Z bosons, as well as lepton masses like those of the electron and muon. If realistic models of particle physics such as the Standard Model suffer from triviality issues, the idea of an elementary scalar Higgs particle may have to be modified or abandoned.		This Higgs triviality is similar to the Landau pole problem in [[Physics:Quantum electrodynamics\|quantum electrodynamics]], where this quantum theory may be inconsistent at very high momentum scales unless the renormalized charge is set to zero, i.e., unless the field theory has no interactions. The Landau pole question is generally considered to be of minor academic interest for quantum electrodynamics because of the inaccessibly large momentum scale at which the inconsistency appears. This is not however the case in theories that involve the elementary scalar Higgs boson, as the momentum scale at which a "trivial" theory exhibits inconsistencies may be accessible to present experimental efforts such as at the Large Hadron Collider (LHC) at CERN. In these Higgs theories, the interactions of the Higgs particle with itself are posited to generate the masses of the W and Z bosons, as well as lepton masses like those of the electron and muon. If realistic models of particle physics such as the Standard Model suffer from triviality issues, the idea of an elementary scalar Higgs particle may have to be modified or abandoned.
Line 42:		Line 42:

	Now we consider a certain blocking transformation of the state variables <math>\{s_i\}\to \{\tilde s_i\}</math>,		Now we consider a certain blocking transformation of the state variables <math>\{s_i\}\to \{\tilde s_i\}</math>,
	the number of <math>\tilde s_i</math> must be lower than the number of <math>s_i</math>. Now let us try to rewrite the <math>Z</math> function ''only'' in terms of the <math>\tilde s_i</math>. If this is achievable by a certain change in the parameters, <math>\{J_k\} \to \{\tilde J_k\}</math>, then the theory is said to be '''renormalizable'''. The most important information in the RG flow are its '''fixed points'''. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be '''trivial'''. Numerous fixed points appear in the study of ~~[[Physics:Lattice gauge theory#Quantum triviality\|~~lattice Higgs theories]], but the nature of the quantum field theories associated with these remains an open question.<ref name="TrivPurs"/>		the number of <math>\tilde s_i</math> must be lower than the number of <math>s_i</math>. Now let us try to rewrite the <math>Z</math> function ''only'' in terms of the <math>\tilde s_i</math>. If this is achievable by a certain change in the parameters, <math>\{J_k\} \to \{\tilde J_k\}</math>, then the theory is said to be '''renormalizable'''. The most important information in the RG flow are its '''fixed points'''. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be '''trivial'''. Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question.<ref name="TrivPurs"/>

	==Historical background==		==Historical background==

Maintenance script: Clean Book I red links, intro, and image slots

2026-05-20T08:16:54Z

Clean Book I red links, intro, and image slots

← Older revision		Revision as of 08:16, 20 May 2026
Line 1:		Line 1:
	{{short description\|Possible outcome of renormalization in physics}}		{{short description\|Possible outcome of renormalization in physics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
Line 11:		Line 11:

	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	In a [[Physics:Quantum field theory\|quantum field theory]], [[Physics:Asymptotic freedom#Screening and antiscreening\|charge screening]] can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as '''quantum triviality'''. Strong evidence supports the idea that a field theory involving only a scalar ~~[[Physics:~~Higgs boson~~\|Higgs boson]]~~ is trivial in four spacetime dimensions,<ref>{{cite book \| author1 = R. Fernandez \| author2 = J. Froehlich \| author3 = A. D. Sokal \| year = 1992 \| title = Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory \| publisher = Springer \| isbn = 0-387-54358-9 }}</ref><ref name="TrivPurs"/> but the situation for realistic models including other particles in addition to the Higgs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the ~~[[Physics:Standard Model\|~~Standard Model]] of ~~[[Physics:Particle physics\|~~particle physics]], the question of triviality in Higgs models is of great importance.		In a [[Physics:Quantum field theory\|quantum field theory]], [[Physics:Asymptotic freedom#Screening and antiscreening\|charge screening]] can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as '''quantum triviality'''. Strong evidence supports the idea that a field theory involving only a scalar Higgs boson is trivial in four spacetime dimensions,<ref>{{cite book \| author1 = R. Fernandez \| author2 = J. Froehlich \| author3 = A. D. Sokal \| year = 1992 \| title = Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory \| publisher = Springer \| isbn = 0-387-54358-9 }}</ref><ref name="TrivPurs"/> but the situation for realistic models including other particles in addition to the Higgs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the Standard Model of particle physics, the question of triviality in Higgs models is of great importance.

	This Higgs triviality is similar to the ~~[[Physics:Landau pole\|~~Landau pole]] problem in [[Physics:Quantum electrodynamics\|quantum electrodynamics]], where this quantum theory may be inconsistent at very high momentum scales unless the renormalized charge is set to zero, i.e., unless the field theory has no interactions. The Landau pole question is generally considered to be of minor academic interest for quantum electrodynamics because of the inaccessibly large momentum scale at which the inconsistency appears. This is not however the case in theories that involve the elementary scalar Higgs boson, as the momentum scale at which a "trivial" theory exhibits inconsistencies may be accessible to present experimental efforts such as at the ~~[[Physics:~~Large Hadron Collider~~\|Large Hadron Collider]]~~ (LHC) at ~~[[Organization:CERN\|~~CERN]]. In these Higgs theories, the interactions of the Higgs particle with itself are posited to generate the masses of the ~~[[Physics:W and Z bosons\|~~W and Z bosons]], as well as ~~[[Physics:Lepton\|~~lepton]] masses like those of the ~~[[Physics:Electron\|~~electron]] and ~~[[Physics:Muon\|~~muon]]. If realistic models of particle physics such as the Standard Model suffer from triviality issues, the idea of an elementary scalar Higgs particle may have to be modified or abandoned.		This Higgs triviality is similar to the Landau pole problem in [[Physics:Quantum electrodynamics\|quantum electrodynamics]], where this quantum theory may be inconsistent at very high momentum scales unless the renormalized charge is set to zero, i.e., unless the field theory has no interactions. The Landau pole question is generally considered to be of minor academic interest for quantum electrodynamics because of the inaccessibly large momentum scale at which the inconsistency appears. This is not however the case in theories that involve the elementary scalar Higgs boson, as the momentum scale at which a "trivial" theory exhibits inconsistencies may be accessible to present experimental efforts such as at the Large Hadron Collider (LHC) at CERN. In these Higgs theories, the interactions of the Higgs particle with itself are posited to generate the masses of the W and Z bosons, as well as lepton masses like those of the electron and muon. If realistic models of particle physics such as the Standard Model suffer from triviality issues, the idea of an elementary scalar Higgs particle may have to be modified or abandoned.

	The situation becomes more complex in theories that involve other particles however. In fact, the addition of other particles can turn a trivial theory into a nontrivial one, at the cost of introducing constraints. Depending on the details of the theory, the Higgs mass can be bounded or even predictable.<ref name="TrivPurs">		The situation becomes more complex in theories that involve other particles however. In fact, the addition of other particles can turn a trivial theory into a nontrivial one, at the cost of introducing constraints. Depending on the details of the theory, the Higgs mass can be bounded or even predictable.<ref name="TrivPurs">
Line 20:		Line 20:
	\| year=1988		\| year=1988
	\| title=Triviality Pursuit: Can Elementary Scalar Particles Exist?		\| title=Triviality Pursuit: Can Elementary Scalar Particles Exist?
	\| journal=~~[[Physics:Physics Reports\|~~Physics Reports]]		\| journal=Physics Reports
	\| volume=167		\| volume=167
	\| issue=5 \| pages=241–320		\| issue=5 \| pages=241–320
Line 28:		Line 28:

	<div style="width:300px;">		<div style="width:300px;">
	[[File:~~not found~~]]		[[File:Quantum_book1_triviality_yellow.png]]
	</div>		</div>

Line 34:		Line 34:

	==Triviality and the renormalization group==		==Triviality and the renormalization group==
	Modern considerations of triviality are usually formulated in terms of the real-space ~~[[Physics:Renormalization group\|~~renormalization group]], largely developed by ~~[[Biography:Kenneth G. Wilson\|~~Kenneth Wilson]] and others. Investigations of triviality are usually performed in the context of ~~[[Physics:Lattice gauge theory\|~~lattice gauge theory]]. A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional ''renormalizable'' theories, came from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group.<ref>~~[[Biography:Leo Kadanoff\|~~L.P. Kadanoff]] (1966): "Scaling laws for Ising models near <math>T_c</math>", Physics (Long Island City, N.Y.) '''2''', 263.</ref> The ''blocking idea'' is a way to define the components of the theory at large distances as aggregates of components at shorter distances.		Modern considerations of triviality are usually formulated in terms of the real-space renormalization group, largely developed by Kenneth Wilson and others. Investigations of triviality are usually performed in the context of lattice gauge theory. A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional ''renormalizable'' theories, came from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group.<ref>L.P. Kadanoff (1966): "Scaling laws for Ising models near <math>T_c</math>", Physics (Long Island City, N.Y.) '''2''', 263.</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<ref>~~[[Biography:Kenneth G. Wilson\|~~K.G. Wilson]](1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. '''47''', 4, 773.</ref> in Wilson's extensive important contributions. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the ~~[[Physics:Kondo effect\|~~Kondo problem]], in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and ~~[[Critical phenomena\|~~critical phenomena]] in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.		This approach covered the conceptual point and was given full computational substance<ref>K.G. Wilson(1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. '''47''', 4, 773.</ref> in Wilson's extensive important contributions. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem, in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.

	In more technical terms, let us assume that we have a theory described by a certain function <math>Z</math> of the state variables <math>\{s_i\}</math> and a certain set of coupling constants <math>\{J_k\}</math>. This function may be a ~~[[Physics:Partition function (quantum field theory)\|~~partition function]], an ~~[[Physics:Action\|~~action]], a ~~[[Physics:Hamiltonian (quantum mechanics)\|~~Hamiltonian]], etc. It must contain the		In more technical terms, let us assume that we have a theory described by a certain function <math>Z</math> of the state variables <math>\{s_i\}</math> and a certain set of coupling constants <math>\{J_k\}</math>. This function may be a partition function, an action, a Hamiltonian, etc. It must contain the
	whole description of the physics of the system.		whole description of the physics of the system.

Line 73:		Line 73:
	The growth of {{math\|''g''<sub>0</sub>}} with {{math\|Λ}} invalidates Eqs. ({{EquationNote\|1}}) and ({{EquationNote\|2}}) in the region {{math\|''g''<sub>0</sub> ≈ 1}} (since they were obtained for {{math\|''g''<sub>0</sub> ≪ 1}}) and the existence of the "Landau pole" in Eq.2 has no physical meaning.		The growth of {{math\|''g''<sub>0</sub>}} with {{math\|Λ}} invalidates Eqs. ({{EquationNote\|1}}) and ({{EquationNote\|2}}) in the region {{math\|''g''<sub>0</sub> ≈ 1}} (since they were obtained for {{math\|''g''<sub>0</sub> ≪ 1}}) and the existence of the "Landau pole" in Eq.2 has no physical meaning.

	The actual behavior of the charge {{math\|''g''(''μ'')}} as a function of the momentum scale {{mvar\|μ}} is determined by the full ~~[[Physics:Renormalization group\|~~Gell-Mann–Low equation]]		The actual behavior of the charge {{math\|''g''(''μ'')}} as a function of the momentum scale {{mvar\|μ}} is determined by the full Gell-Mann–Low equation
	{{NumBlk\|:\|<math>\frac{dg}{d \ln \mu} =\beta(g)=\beta_2 g^2+\beta_3 g^3+\ldots ~,</math>\|{{EquationRef\|3}}}}		{{NumBlk\|:\|<math>\frac{dg}{d \ln \mu} =\beta(g)=\beta_2 g^2+\beta_3 g^3+\ldots ~,</math>\|{{EquationRef\|3}}}}
	which gives Eqs.({{EquationNote\|1}}),({{EquationNote\|2}}) if it is integrated under conditions {{math\|1=''g''(''μ'') = ''g''<sub>obs</sub>}} for {{math\|1=''μ'' = ''m''}} and {{math\|1=''g''(''μ'') = ''g''<sub>0</sub>}} for {{math\|1=''μ'' = Λ}}, when only the term with <math>\beta_2</math> is retained in the right hand side.		which gives Eqs.({{EquationNote\|1}}),({{EquationNote\|2}}) if it is integrated under conditions {{math\|1=''g''(''μ'') = ''g''<sub>obs</sub>}} for {{math\|1=''μ'' = ''m''}} and {{math\|1=''g''(''μ'') = ''g''<sub>0</sub>}} for {{math\|1=''μ'' = Λ}}, when only the term with <math>\beta_2</math> is retained in the right hand side.
Line 92:		Line 92:
	}}		}}

	The latter case corresponds to the quantum triviality in the full theory (beyond its perturbation context), as can be seen by ~~[[Philosophy:Reductio ad absurdum\|~~reductio ad absurdum]]. Indeed, if {{math\|''g''<sub>obs</sub>}} is finite, the theory is internally inconsistent. The only way to avoid it, is to tend <math>\mu_0</math> to infinity, which is possible only for {{math\|''g''<sub>obs</sub> → 0}}.		The latter case corresponds to the quantum triviality in the full theory (beyond its perturbation context), as can be seen by reductio ad absurdum. Indeed, if {{math\|''g''<sub>obs</sub>}} is finite, the theory is internally inconsistent. The only way to avoid it, is to tend <math>\mu_0</math> to infinity, which is possible only for {{math\|''g''<sub>obs</sub> → 0}}.

	==Conclusions==		==Conclusions==

	As a result, the question of whether the ~~[[Physics:~~Standard Model~~\|Standard Model]]~~ of ~~[[Physics:Particle physics\|~~particle physics]] is nontrivial remains a serious unresolved question. Theoretical proofs of triviality of the pure scalar field theory exist, but the situation for the full standard model is unknown. The implied constraints on the standard model have been discussed.<ref>{{Cite journal		As a result, the question of whether the Standard Model of particle physics is nontrivial remains a serious unresolved question. Theoretical proofs of triviality of the pure scalar field theory exist, but the situation for the full standard model is unknown. The implied constraints on the standard model have been discussed.<ref>{{Cite journal
	\| last1 = Callaway \| first1 = D.		\| last1 = Callaway \| first1 = D.
	\| last2 = Petronzio \| first2 = R.		\| last2 = Petronzio \| first2 = R.
Line 110:		Line 110:
	\|author=I. M. Suslov		\|author=I. M. Suslov
	\|year=2010		\|year=2010
	\|title=Asymptotic Behavior of the ''β'' Function in the ''φ''<sup>4</sup> Theory: A Scheme Without Complex Parameters \|journal=~~[[Physics:Journal of Experimental and Theoretical Physics\|~~Journal of Experimental and Theoretical Physics]]		\|title=Asymptotic Behavior of the ''β'' Function in the ''φ''<sup>4</sup> Theory: A Scheme Without Complex Parameters \|journal=Journal of Experimental and Theoretical Physics
	\|volume=111		\|volume=111
	\|issue=3 \|pages=450–465		\|issue=3 \|pages=450–465
Line 121:		Line 121:
	\| doi = 10.1007/BF01479540		\| doi = 10.1007/BF01479540
	\| title = Implications of triviality for the standard model		\| title = Implications of triviality for the standard model
	\| journal = ~~[[Physics:~~Zeitschrift für Physik C~~\|Zeitschrift für Physik C]]~~		\| journal = Zeitschrift für Physik C
	\| volume = 31		\| volume = 31
	\| issue = 2		\| issue = 2
Line 132:		Line 132:

	==See also==		==See also==
	* ~~[[Physics:~~Hierarchy problem~~\|Hierarchy problem]]~~		* Hierarchy problem

	== See also ==		== See also ==

Maintenance script: Normalize Book I Quantum page structure

2026-05-19T22:50:38Z

Normalize Book I Quantum page structure

← Older revision		Revision as of 22:50, 19 May 2026
Line 133:		Line 133:
	==See also==		==See also==
	* [[Physics:Hierarchy problem\|Hierarchy problem]]		* [[Physics:Hierarchy problem\|Hierarchy problem]]

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

	==References==		==References==
Line 142:		Line 145:
	[[Category:Physical phenomena]]		[[Category:Physical phenomena]]

	{{Sourceattribution\|Quantum triviality}}		{{Author\|Harold Foppele}}

			{{Sourceattribution\|Physics:Quantum triviality\|1}}

Harold: Arrange page top as TOC lead image columns

2026-05-17T14:02:21Z

Arrange page top as TOC lead image columns

@@ Line 3: / Line 3: @@
 {{Quantum book backlink|Quantum field theory}}
 {{quantum field theory}}
 In a [[Physics:Quantum field theory|quantum field theory]], [[Physics:Asymptotic freedom#Screening and antiscreening|charge screening]] can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as '''quantum triviality'''. Strong evidence supports the idea that a field theory involving only a scalar [[Physics:Higgs boson|Higgs boson]] is trivial in four spacetime dimensions,<ref>{{cite book | author1 = R. Fernandez |  author2 = J. Froehlich |  author3 = A. D. Sokal | year = 1992 | title = Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory | publisher = Springer | isbn = 0-387-54358-9 }}</ref><ref name="TrivPurs"/> but the situation for realistic models including other particles in addition to the Higgs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the [[Physics:Standard Model|Standard Model]] of [[Physics:Particle physics|particle physics]], the question of triviality in Higgs models is of great importance.
@@ Line 17: / Line 25: @@
   | doi=10.1016/0370-1573(88)90008-7
   }}</ref> These quantum triviality constraints are in sharp contrast to the picture one derives at the classical level, where the Higgs mass is a free parameter.
 ==Triviality and the renormalization group==