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{{Short description|Millennium Prize Problem in quantum field theory}}
{{Quantum book backlink|Advanced and frontier topics}}

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>

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.
<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>
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== Meaning of the problem ==

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>

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.

For the gauge group <math>SU(3)</math>, which is relevant to the strong interaction, the problem is closely connected with the expected existence of massive glueballs. A successful proof would show, in a mathematically rigorous way, why the quantum gauge theory produces massive color-neutral excitations rather than massless isolated gluons.

== Yang–Mills theory ==

Yang–Mills theory is a generalization of gauge theory in which the gauge group is non-abelian. Unlike electromagnetism, whose gauge group is abelian, non-abelian gauge theories contain gauge fields that interact with one another.

This self-interaction is one reason Yang–Mills theory is difficult. It is also why it is physically important. Non-abelian gauge theories describe the weak and strong nuclear interactions in the Standard Model. In the strong interaction, the gauge group is <math>SU(3)</math>, and the corresponding theory is quantum chromodynamics.

In the classical theory, the gauge fields can be massless. In the quantum theory, however, the physical spectrum is expected to contain a lowest nonzero mass. This is the mass gap.

== Mass gap ==

In quantum field theory, a '''mass gap''' is the difference between the vacuum energy and the energy of the lightest excited state. If the vacuum has energy zero, then the mass gap is the smallest positive energy that can appear in the physical spectrum.

A theory with no mass gap can have excitations of arbitrarily small energy. A theory with a positive mass gap has a lowest nonzero energy scale. For Yang–Mills theory, the Clay problem requires proving that such a positive gap exists.<ref name="official" />

One way of seeing the mass gap is through correlation functions. For a field <math>\phi</math>, the long-time behavior of a two-point function may contain exponential decay of the form

<math display="block">\langle\phi(0,t)\phi(0,0)\rangle\sim \sum_n A_n\exp(-\Delta_n t).</math>

The smallest positive value <math>\Delta_0</math> corresponds to the mass gap. Lattice gauge theory calculations support the existence of such a gap in Yang–Mills theory and have been used to estimate glueball spectra.<ref name="teper">{{Cite journal |last1=Lucini |first1=Biagio |last2=Teper |first2=Michael |last3=Wenger |first3=Urs |year=2004 |title=Glueballs and k-strings in SU(N) gauge theories: calculations with improved operators |journal=Journal of High Energy Physics |volume=2004 |issue=6 |page=012 |arxiv=hep-lat/0404008 |bibcode=2004JHEP...06..012L |doi=10.1088/1126-6708/2004/06/012 |s2cid=14807677}}</ref><ref name="morningstar">{{Cite journal |last1=Chen |first1=Y. |last2=Alexandru |first2=A. |last3=Dong |first3=S. J. |last4=Draper |first4=T. |last5=Horváth |first5=I. |last6=Lee |first6=F. X. |last7=Liu |first7=K. F. |last8=Mathur |first8=N. |last9=Morningstar |first9=C. |last10=Peardon |first10=M. |last11=Tamhankar |first11=S. |last12=Young |first12=B. L. |last13=Zhang |first13=J. B. |year=2006 |title=Glueball spectrum and matrix elements on anisotropic lattices |journal=Physical Review D |volume=73 |issue=1 |article-number=014516 |arxiv=hep-lat/0510074 |bibcode=2006PhRvD..73a4516C |doi=10.1103/PhysRevD.73.014516 |s2cid=15741174}}</ref>

== Relation to confinement ==

The mass gap problem is closely related to confinement in quantum chromodynamics. Confinement is the property that quarks and gluons are not observed as isolated particles under ordinary low-energy conditions. Instead, they appear inside color-neutral bound states such as hadrons.

In a pure Yang–Mills theory without quarks, the corresponding color-neutral bound states are glueballs. If glueballs exist as physical excitations, they should have positive mass. This is why the mass gap is physically connected with the absence of freely propagating massless gluons in the observed low-energy spectrum.

At the level of theoretical physics and lattice gauge theory, there is strong evidence for confinement and a mass gap. The Millennium problem, however, asks for a mathematically rigorous proof, not only a physical argument or numerical calculation.

== Wightman and Osterwalder–Schrader axioms ==

The official Clay problem requires the constructed quantum Yang–Mills theory to satisfy rigorous axiomatic standards. The problem statement refers to the Wightman axioms and the Osterwalder–Schrader axioms.<ref name="official" /><ref>{{Cite book |last1=Streater |first1=R. F. |url=https://archive.org/details/pctspinstatistic0000stre |title=PCT, spin and statistics, and all that |last2=Wightman |first2=A. |publisher=New York, W.A. Benjamin |year=1964 |url-access=registration}}</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>

The Wightman framework describes quantum fields as operator-valued distributions acting on a Hilbert space. It imposes conditions such as relativistic covariance, a unique vacuum, positivity of energy, and local commutativity for spacelike separated fields.

The Osterwalder–Schrader framework gives related conditions for Euclidean quantum field theory. These axioms are important because Yang–Mills theory is often studied using Euclidean methods and lattice approximations. A rigorous construction must show that the Euclidean theory corresponds to a valid relativistic quantum field theory.

== Importance ==

The problem is important because most interacting quantum field theories in four dimensions are difficult to define rigorously at all length scales. Many theories are best understood as effective field theories, valid below some cutoff scale.

Non-abelian Yang–Mills theory is different because it has asymptotic freedom. At high energies, the interaction becomes weaker, making it a natural candidate for a nontrivial four-dimensional quantum field theory that may exist at all scales.

A proof of existence and mass gap would therefore be a major result for both physics and mathematics. It would give a rigorous foundation to a central structure used in modern particle physics and would clarify the mathematical status of non-abelian gauge theory.

== Relation to undecidability ==

The general problem of determining whether a quantum many-body system has a spectral gap is known to be undecidable in certain settings. This means that no universal algorithm can decide the answer for every possible system.<ref>{{Cite journal |last1=Cubitt |first1=Toby S. |last2=Pérez-García |first2=David |last3=Wolf |first3=Michael |date=2018-10-01 |title=The Unsolvable Problem |url=https://www.scientificamerican.com/article/the-unsolvable-problem/ |journal=Scientific American |volume=319 |issue=4 |pages=28–37 |doi=10.1038/scientificamerican1018-28 |pmid=30273308 |bibcode=2018SciAm.319d..28C |url-access=subscription }}</ref><ref>{{Cite news |last=Castelvecchi |first=Davide |date=9 December 2015 |title=Paradox at the heart of mathematics makes physics problem unanswerable |url=https://www.nature.com/articles/nature.2015.18983 |journal=Nature |doi=10.1038/nature.2015.18983 }}</ref>

This undecidability result does not by itself solve or refute the Yang–Mills mass gap problem. The Clay problem concerns a specific class of quantum gauge theories, not arbitrary quantum systems. It does, however, illustrate why spectral-gap questions can be mathematically subtle.

== Status ==

The Yang–Mills existence and mass gap problem remains unsolved. Physicists have strong theoretical and numerical evidence that four-dimensional non-abelian Yang–Mills theory has a mass gap, but a complete proof satisfying the required axiomatic standards is not known.

The problem remains one of the most important open questions connecting quantum field theory, particle physics, geometry, analysis, and mathematical physics.

== See also ==

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

== References ==

{{reflist|3}}

== Further reading ==

* {{Cite book |last1=Bogoliubov |first1=N. |title=General Principles of Quantum Field Theory |last2=Logunov |first2=A. |last3=Oksak |last4=Todorov |first4=I. |publisher=Springer Netherlands |year=1990 |isbn=978-94-010-6707-2 |location=Dordrecht |language=en |doi=10.1007/978-94-009-0491-0}}
* {{Cite book |last=Strocchi |first=Franco |title=Selected topics on the general properties of quantum field theory: lecture notes |publisher=World Scientific |year=1993 |isbn=978-981-02-1149-3 |series=World Scientific lecture notes in physics |location=Singapore}}
* {{Cite journal |last=Dynin |first=A. |year=2014 |title=Quantum Yang-Mills-Weyl Dynamics in the Schrödinger paradigm |journal=Russian Journal of Mathematical Physics |volume=21 |issue=2 |pages=169–188 |bibcode=2014RJMP...21..169D |doi=10.1134/S1061920814020046 |s2cid=121878861}}
* {{Cite journal |last=Dynin |first=A. |year=2014 |title=On the Yang-Mills mass gap problem |journal=Russian Journal of Mathematical Physics |volume=21 |issue=3 |pages=326–328 |bibcode=2014RJMP...21..326D |doi=10.1134/S1061920814030042 |s2cid=120135592}}
* {{Cite book |last1=Bushhorn |first1=G. |title=Fundamental physics-- Heisenberg and beyond: Werner Heisenberg Centennial Symposium "Developments in Modern Physics" |last2=Wess |first2=J. |publisher=Springer |year=2004 |isbn=978-3-540-20201-1 |location=Berlin ; New York}}

{{Author|Harold Foppele}}

{{Sourceattribution|Yang–Mills existence and mass gap|1}}

Physics:Quantum Yang-Mills mass gap - Revision history

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