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The problem is phrased as follows:<ref name="official">{{Cite web |last1=Jaffe |first1=Arthur |author-link=Arthur Jaffe |last2=Witten |first2=Edward |author-link2=Edward Witten |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=Claymath.org}}</ref> :''Yang–Mills Existence and Mass Gap.'' Prove that for any compact simple [[Gauge group (mathematics)|gauge group]] G, a [[non-trivial]] quantum [[Yang–Mills theory]] exists on <math>\mathbb{R}^4</math> and has a [[mass gap]] Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in {{Harvtxt|Streater|Wightman|1964}}, {{Harvtxt|Osterwalder|Schrader|1973}} and {{Harvtxt|Osterwalder|Schrader|1975}}. In this statement, a quantum [[Yang–Mills theory]] is a [[Non-abelian group|non-abelian]] [[quantum field theory]] similar to that underlying the [[Standard Model]] of [[particle physics]]; <math>\mathbb{R}^4</math> is [[Euclidean space|Euclidean 4-space]]; the [[mass gap]] Δ is the mass of the least massive particle predicted by the theory. Therefore, the winner must prove that: * Yang–Mills theory exists and satisfies the standard of rigor that characterizes contemporary [[mathematical physics]], in particular [[constructive quantum field theory]],<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>{{Harvtxt|Osterwalder|Schrader|1973}}</ref> and * The mass of all particles of the force field predicted by the theory are strictly positive. For example, in the case of G=SU(3)—the strong nuclear interaction—the winner must prove that [[glueball]]s have a lower mass bound, and thus cannot be arbitrarily light. The general problem of determining the presence of a [[spectral gap (physics)|spectral gap]] in a system is known to be [[Undecidable problem|undecidable]].<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/ |access-date=2024-09-11 |journal=[[Scientific American]] |volume=319 |issue=4 |pages=28–37 |doi=10.1038/scientificamerican1018-28 |pmid=30273308 |language=en}}</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 |access-date=2024-09-11 |journal=[[Nature (journal)|Nature]] |language=en |doi=10.1038/nature.2015.18983 }}</ref> ==Background== {{Cquote|[...] one does not yet have a mathematically complete example of a [[quantum gauge theory]] in four-dimensional [[space-time]], nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so!|15px|15px|From the Clay Institute's official problem description by [[Arthur Jaffe]] and [[Edward Witten]].}} The problem requires the construction of a QFT satisfying the Wightman axioms and showing the existence of a mass gap. Both of these topics are described in sections below. ===The Wightman axioms=== {{main|Wightman axioms}} The Millennium problem requires the proposed Yang–Mills theory to satisfy the [[Wightman axioms]] or similarly stringent axioms.<ref name="official" /> There are four axioms: ;W0 (assumptions of relativistic quantum mechanics) [[Quantum mechanics]] is described according to [[John von Neumann|von Neumann]]; in particular, the [[pure state]]s are given by the rays, i.e. the one-dimensional subspaces, of some [[separable space|separable]] complex [[Hilbert space]]. The Wightman axioms require that the [[Poincaré group]] acts [[unitary representation|unitarily]] on the Hilbert space. In other words, they have position dependent operators called '''''quantum fields''''' which form covariant [[representations of the Poincaré group]]. The group of space-time translations is [[commutative]], and so the operators can be simultaneously diagonalised. The generators of these groups give us four [[self-adjoint operator]]s, <math>P_j,j=0,1,2,3</math>, which transform under the homogeneous group as a [[four-vector]], called the [[energy-momentum relation|energy-momentum]] four-vector. The second part of the zeroth axiom of Wightman is that the representation ''U''(''a'', ''A'') fulfills the spectral condition—that the simultaneous spectrum of energy-momentum is contained in the forward cone: :<math>P_0\geq 0,\;\;\;\;P_0^2 - P_jP_j\geq 0.</math> The third part of the axiom is that there is a unique state, represented by a ray in the Hilbert space, which is invariant under the action of the Poincaré group. It is called a vacuum. ;W1 (assumptions on the domain and continuity of the field) For each test function ''f'', there exists a set of operators <math>A_1(f),\ldots ,A_n(f)</math> which, together with their adjoints, are defined on a dense subset of the Hilbert state space, containing the vacuum. The fields ''A'' are operator-valued [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distributions]]. The Hilbert state space is spanned by the field polynomials acting on the vacuum (cyclicity condition). ;W2 (transformation law of the field) The fields are covariant under the action of [[Poincaré group]], and they transform according to some representation S of the [[Lorentz group]], or SL(2,'''C''') if the spin is not integer: :<math>U(a,L)^{\dagger}A(x)U(a,L)=S(L)A(L^{-1}(x-a)).</math> ;W3 (local commutativity or microscopic causality) If the supports of two fields are [[space-like]] separated, then the fields either commute or anticommute. Cyclicity of a vacuum, and uniqueness of a vacuum are sometimes considered separately. Also, there is the property of asymptotic completeness—that the Hilbert state space is spanned by the asymptotic spaces <math>H^{in}</math> and <math>H^{out}</math>, appearing in the collision [[S matrix]]. The other important property of field theory is the [[mass gap]] which is not required by the axioms—that the energy-momentum spectrum has a gap between zero and some positive number. ===Mass gap=== {{main|Mass gap}} In [[quantum field theory]], the '''mass gap''' is the difference in energy between the vacuum and the next lowest [[energy state]]. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle. For a given real field <math>\phi(x)</math>, we can say that the theory has a mass gap if the [[Green%27s_function_(many-body_theory)#Two-point_functions_2|two-point function]] has the property :<math>\langle\phi(0,t)\phi(0,0)\rangle\sim \sum_nA_n\exp\left(-\Delta_nt\right)</math> with <math>\Delta_0>0</math> being the lowest energy value in the spectrum of the Hamiltonian and thus the mass gap. This quantity, easy to generalize to other fields, is what is generally measured in lattice computations. It was proved in this way that [[Yang–Mills theory]] develops a mass gap on a lattice.<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 |pages=012 |arxiv=hep-lat/0404008 |bibcode=2004JHEP...06..012L |doi=10.1088/1126-6708/2004/06/012 |issn=1029-8479 |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 |language=en |volume=73 |issue=1 |pages=014516 |arxiv=hep-lat/0510074 |bibcode=2006PhRvD..73a4516C |doi=10.1103/PhysRevD.73.014516 |issn=1550-7998 |s2cid=15741174}}.</ref> ==Importance of Yang–Mills theory== Most known and nontrivial (i.e. interacting) [[quantum field theories]] in 4 dimensions are [[effective field theory|effective field theories]] with a [[cutoff (physics)|cutoff]] scale. Since the [[Beta function (physics)|beta function]] is positive for most models, it appears that most such models have a [[Landau pole]] as it is not at all clear whether or not they have nontrivial [[UV fixed point]]s. This means that if such a [[Quantum field theory|QFT]] is well-defined at all scales, as it has to be to satisfy the axioms of [[axiomatic quantum field theory]], it would have to be trivial (i.e. a [[free field theory]]). [[Yang–Mills theory|Quantum Yang–Mills theory]] with a [[Non-abelian gauge theory|non-abelian]] [[gauge group]] and no quarks is an exception, because [[asymptotic freedom]] characterizes this theory, meaning that it has a trivial [[UV fixed point]]. Hence it is the simplest nontrivial constructive QFT in 4 dimensions. ([[Quantum chromodynamics|QCD]] is a more complicated theory because it involves [[quark]]s.) ===Quark confinement=== {{main|Quantum chromodynamics|color confinement|lattice gauge theory}} At the level of rigor of [[theoretical physics]], it has been well established that the quantum Yang–Mills theory for a non-abelian [[Lie group]] exhibits a property known as [[color confinement|confinement]]; though proper [[mathematical physics]] has more demanding requirements on a proof. A consequence of this property is that above the [[Color confinement|confinement scale]], the color charges are connected by [[QCD string|chromodynamic flux tubes]] leading to a linear potential between the charges. Hence isolated color charge and isolated [[gluon]]s cannot exist. In the absence of confinement, we would expect to see massless gluons, but since they are confined, all we would see are color-neutral bound states of gluons, called [[glueball]]s. If glueballs exist, they are massive, which is why a mass gap is expected. ==References== {{Reflist}} ==Further reading== *{{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}} *{{Cite journal |last1=Osterwalder |first1=Konrad |last2=Schrader |first2=Robert |year=1973 |title=Axioms for Euclidean Green's functions |journal=Communications in Mathematical Physics |language=en |volume=31 |issue=2 |pages=83–112 |bibcode=1973CMaPh..31...83O |doi=10.1007/BF01645738 |issn=0010-3616 |s2cid=189829853}} *{{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 |language=en |volume=42 |issue=3 |pages=281–305 |bibcode=1975CMaPh..42..281O |doi=10.1007/BF01608978 |issn=0010-3616 |s2cid=119389461}} *{{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 |editor-last=Bogolubov |editor-first=N. N. |location=Dordrecht |language=en |doi=10.1007/978-94-009-0491-0 |editor-last2=Logunov |editor-first2=A. A. |editor-last3=Oksak |editor-first3=A. I. |editor-last4=Todorov |editor-first4=I. T.}} *{{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 |language=en |volume=21 |issue=2 |pages=169–188 |bibcode=2014RJMP...21..169D |doi=10.1134/S1061920814020046 |issn=1061-9208 |s2cid=121878861}} *{{Cite journal |last=Dynin |first=A. |year=2014 |title=On the Yang-Mills mass gap problem |journal=Russian Journal of Mathematical Physics |language=en |volume=21 |issue=3 |pages=326–328 |bibcode=2014RJMP...21..326D |doi=10.1134/S1061920814030042 |issn=1061-9208 |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 |editor-last=Heisenberg |editor-first=Werner |location=Berlin ; New York |editor-last2=Buschhorn |editor-first2=Gerd W. |editor-last3=Wess |editor-first3=Julius}} ==External links== * [https://www.claymath.org/millennium/yang-mills-the-maths-gap/ The Millennium Prize Problems: Yang–Mills and Mass Gap] {{DEFAULTSORT:Yang-Mills Existence And Mass Gap}} [[Category:Millennium Prize Problems]] [[Category:Gauge theories]] [[Category:Quantum chromodynamics]] [[Category:Unsolved problems in mathematics]] [[Category:Unsolved problems in physics]] </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p><span id="templatesused">Pages transcluded onto the current version of this page<span class="posteditwindowhelplinks"> (<a href="/wiki/Help:Transclusion" title="Help:Transclusion">help</a>)</span>:</span> </p></div><ul> <li><a href="/wiki/Template:Blockquote" 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