Physics:Quantum Yang-Mills mass gap - Revision history

WikiHarold: Restore missing Quantum reference definitions

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	The problem has two closely related parts.		The problem has two closely related parts.

	First, one must prove that quantum Yang–Mills theory exists as a mathematically well-defined quantum field theory. This means more than writing down the formal equations used in physics. The theory must satisfy rigorous axiomatic conditions of the kind used in constructive quantum field theory and axiomatic quantum field theory.<ref name="official" /><ref>{{Cite book \|last1=Streater \|first1=R. F. \|title=PCT, spin and statistics, and all that \|last2=Wightman \|first2=A. S. \|date=2000 \|publisher=Princeton University Press \|isbn=978-0-691-07062-9 \|edition=1st with rev \|series=Princeton landmarks in physics \|location=Princeton, N.J}}</ref><ref>{{Cite journal \|last1=Osterwalder \|first1=Konrad \|last2=Schrader \|first2=Robert \|year=1973 \|title=Axioms for Euclidean Green's functions \|journal=Communications in Mathematical Physics \|volume=31 \|issue=2 \|pages=83–112 \|bibcode=1973CMaPh..31...83O \|doi=10.1007/BF01645738 \|s2cid=189829853}}</ref><ref>{{Cite journal \|last1=Osterwalder \|first1=Konrad \|last2=Schrader \|first2=Robert \|year=1975 \|title=Axioms for Euclidean Green's functions II \|journal=Communications in Mathematical Physics \|volume=42 \|issue=3 \|pages=281–305 \|bibcode=1975CMaPh..42..281O \|doi=10.1007/BF01608978 \|s2cid=119389461}}</ref>		First, one must prove that quantum Yang–Mills theory exists as a mathematically well-defined quantum field theory. This means more than writing down the formal equations used in physics. The theory must satisfy rigorous axiomatic conditions of the kind used in constructive quantum field theory and axiomatic quantum field theory.<ref name="official">{{Cite web \|last1=Jaffe \|first1=Arthur \|last2=Witten \|first2=Edward \|title=Quantum Yang-Mills theory \|url=https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf \|url-status=live \|archive-url=https://web.archive.org/web/20230620074636/https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf \|archive-date=2023-06-20 \|website=Clay Mathematics Institute \|access-date=7 May 2026}}</ref><ref>{{Cite book \|last1=Streater \|first1=R. F. \|title=PCT, spin and statistics, and all that \|last2=Wightman \|first2=A. S. \|date=2000 \|publisher=Princeton University Press \|isbn=978-0-691-07062-9 \|edition=1st with rev \|series=Princeton landmarks in physics \|location=Princeton, N.J}}</ref><ref>{{Cite journal \|last1=Osterwalder \|first1=Konrad \|last2=Schrader \|first2=Robert \|year=1973 \|title=Axioms for Euclidean Green's functions \|journal=Communications in Mathematical Physics \|volume=31 \|issue=2 \|pages=83–112 \|bibcode=1973CMaPh..31...83O \|doi=10.1007/BF01645738 \|s2cid=189829853}}</ref><ref>{{Cite journal \|last1=Osterwalder \|first1=Konrad \|last2=Schrader \|first2=Robert \|year=1975 \|title=Axioms for Euclidean Green's functions II \|journal=Communications in Mathematical Physics \|volume=42 \|issue=3 \|pages=281–305 \|bibcode=1975CMaPh..42..281O \|doi=10.1007/BF01608978 \|s2cid=119389461}}</ref>

	Second, one must prove that the theory has a positive mass gap. This means that the vacuum is separated from the next possible energy state by a strictly positive amount. In particle language, the theory must not contain arbitrarily light physical particles.		Second, one must prove that the theory has a positive mass gap. This means that the vacuum is separated from the next possible energy state by a strictly positive amount. In particle language, the theory must not contain arbitrarily light physical particles.
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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}}

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	<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;">
	~~The~~ '''Yang–Mills existence and mass gap problem~~'''~~ is an open problem in mathematical physics and one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute. It asks for a mathematically rigorous construction of quantum Yang–Mills theory in four-dimensional spacetime and a proof that the theory has a positive mass gap.<ref name="official">{{Cite web \|last1=Jaffe \|first1=Arthur \|last2=Witten \|first2=Edward \|title=Quantum Yang-Mills theory \|url=https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf \|url-status=live \|archive-url=https://web.archive.org/web/20230620074636/https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf \|archive-date=2023-06-20 \|website=Clay Mathematics Institute \|access-date=7 May 2026}}</ref>		'''Yang-Mills mass gap''' the Yang–Mills existence and mass gap problem is an open problem in mathematical physics and one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute. It asks for a mathematically rigorous construction of quantum Yang–Mills theory in four-dimensional spacetime and a proof that the theory has a positive mass gap. In physical terms, the problem concerns the mathematical foundations of non-abelian quantum gauge theory. Yang–Mills theories form the basis of the gauge-field description used in the Standard Model of particle physics. In particular, quantum chromodynamics, the theory of the strong interaction, is based on a non-abelian gauge theory. The problem has two closely related parts.

	In physical terms, the problem concerns the mathematical foundations of non-abelian quantum gauge theory. Yang–Mills theories form the basis of the gauge-field description used in the Standard Model of particle physics. In particular, quantum chromodynamics, the theory of the strong interaction, is based on a non-abelian gauge theory.

	The ~~official~~ problem ~~statement asks one to prove that, for any compact simple gauge group <math>G</math>, a non-trivial quantum Yang–Mills theory exists on <math>\mathbb{R}^4</math> and~~ has a mass gap <math>\Delta > 0</math>.<ref name="official" /> Here <math>\mathbb{R}^4</math> denotes four-dimensional Euclidean space, and the mass gap is the positive energy difference between the vacuum and the lightest physical excitation of the theory.

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	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

	{{Sourceattribution\|~~Yang–Mills existence and~~ mass gap\|1}}		{{Sourceattribution\|Physics:Quantum Yang-Mills mass gap\|1}}

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	The official problem statement asks one to prove that, for any compact simple gauge group <math>G</math>, a non-trivial quantum Yang–Mills theory exists on <math>\mathbb{R}^4</math> and has a mass gap <math>\Delta > 0</math>.<ref name="official" /> Here <math>\mathbb{R}^4</math> denotes four-dimensional Euclidean space, and the mass gap is the positive energy difference between the vacuum and the lightest physical excitation of the theory.		The official problem statement asks one to prove that, for any compact simple gauge group <math>G</math>, a non-trivial quantum Yang–Mills theory exists on <math>\mathbb{R}^4</math> and has a mass gap <math>\Delta > 0</math>.<ref name="official" /> Here <math>\mathbb{R}^4</math> denotes four-dimensional Euclidean space, and the mass gap is the positive energy difference between the vacuum and the lightest physical excitation of the theory.
	~~<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:360px;">~~
	~~<div style="font-size:90%;">The Yang–Mills mass gap problem asks why a quantum gauge field theory can produce massive excitations even when the classical gauge fields are massless.</div>~~
	~~</div>~~
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	[[File:Quantum_Yang-Mills_mass_gap_yellow.png\|thumb\|280px\|~~Quantum Yang-Mills~~ mass gap.]]		[[File:Quantum_Yang-Mills_mass_gap_yellow.png\|thumb\|280px\|The Yang–Mills mass gap problem asks why a quantum gauge field theory can produce massive excitations even when the classical gauge fields are massless.]]
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	{{Short description\|Millennium Prize Problem in quantum field theory}}		{{Short description\|Millennium Prize Problem in quantum field theory}}

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	The '''Yang–Mills existence and mass gap problem''' is an open problem in mathematical physics and one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute. It asks for a mathematically rigorous construction of quantum Yang–Mills theory in four-dimensional spacetime and a proof that the theory has a positive mass gap.<ref name="official">{{Cite web \|last1=Jaffe \|first1=Arthur \|last2=Witten \|first2=Edward \|title=Quantum Yang-Mills theory \|url=https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf \|url-status=live \|archive-url=https://web.archive.org/web/20230620074636/https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf \|archive-date=2023-06-20 \|website=Clay Mathematics Institute \|access-date=7 May 2026}}</ref>		The '''Yang–Mills existence and mass gap problem''' is an open problem in mathematical physics and one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute. It asks for a mathematically rigorous construction of quantum Yang–Mills theory in four-dimensional spacetime and a proof that the theory has a positive mass gap.<ref name="official">{{Cite web \|last1=Jaffe \|first1=Arthur \|last2=Witten \|first2=Edward \|title=Quantum Yang-Mills theory \|url=https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf \|url-status=live \|archive-url=https://web.archive.org/web/20230620074636/https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf \|archive-date=2023-06-20 \|website=Clay Mathematics Institute \|access-date=7 May 2026}}</ref>

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	The official problem statement asks one to prove that, for any compact simple gauge group <math>G</math>, a non-trivial quantum Yang–Mills theory exists on <math>\mathbb{R}^4</math> and has a mass gap <math>\Delta > 0</math>.<ref name="official" /> Here <math>\mathbb{R}^4</math> denotes four-dimensional Euclidean space, and the mass gap is the positive energy difference between the vacuum and the lightest physical excitation of the theory.		The official problem statement asks one to prove that, for any compact simple gauge group <math>G</math>, a non-trivial quantum Yang–Mills theory exists on <math>\mathbb{R}^4</math> and has a mass gap <math>\Delta > 0</math>.<ref name="official" /> Here <math>\mathbb{R}^4</math> denotes four-dimensional Euclidean space, and the mass gap is the positive energy difference between the vacuum and the lightest physical excitation of the theory.
	<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:360px;">		<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:360px;">
	~~[[File:Quantum_Yang-Mills_mass_gap_yellow.png\|340px]]~~
	<div style="font-size:90%;">The Yang–Mills mass gap problem asks why a quantum gauge field theory can produce massive excitations even when the classical gauge fields are massless.</div>		<div style="font-size:90%;">The Yang–Mills mass gap problem asks why a quantum gauge field theory can produce massive excitations even when the classical gauge fields are massless.</div>
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			[[File:Quantum_Yang-Mills_mass_gap_yellow.png\|thumb\|280px\|Quantum Yang-Mills mass gap.]]
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	== Meaning of the problem ==		== Meaning of the problem ==

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	{{Quantum article nav\|previous=Physics:Quantum unsolved problems\|previous label=Unsolved problems\|next=Physics:Quantum gravity problem\|next label=Gravity problem}}		{{Quantum article nav\|previous=Physics:Quantum unsolved problems\|previous label=Unsolved problems\|next=Physics:Quantum gravity problem\|next label=Gravity problem}}