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<div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Yang-Mills-Existenz-_und_Massenl%C3%BCckeproblem" title="Yang-Mills-Existenz- und Massenlückeproblem – German" lang="de" hreflang="de" data-title="Yang-Mills-Existenz- und Massenlückeproblem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%96%91-%EB%B0%80%EC%8A%A4_%EC%A7%88%EB%9F%89_%EA%B0%84%EA%B7%B9_%EA%B0%80%EC%84%A4" title="양-밀스 질량 간극 가설 – Korean" lang="ko" hreflang="ko" data-title="양-밀스 질량 간극 가설" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Keberadaan_Yang%E2%80%93Mills_dan_celah_massa" title="Keberadaan Yang–Mills dan celah massa – Indonesian" lang="id" hreflang="id" data-title="Keberadaan Yang–Mills dan celah massa" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Esistenza_di_Yang-Mills_e_del_gap_di_massa" title="Esistenza di Yang-Mills e del gap di massa – Italian" lang="it" hreflang="it" data-title="Esistenza di Yang-Mills e del gap di massa" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A4%E3%83%B3%E2%80%93%E3%83%9F%E3%83%AB%E3%82%BA%E6%96%B9%E7%A8%8B%E5%BC%8F%E3%81%A8%E8%B3%AA%E9%87%8F%E3%82%AE%E3%83%A3%E3%83%83%E3%83%97%E5%95%8F%E9%A1%8C" title="ヤン–ミルズ方程式と質量ギャップ問題 – Japanese" lang="ja" hreflang="ja" data-title="ヤン–ミルズ方程式と質量ギャップ問題" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Exist%C3%AAncia_de_Yang-Mills_e_intervalo_de_massa" title="Existência de Yang-Mills e intervalo de massa – Portuguese" lang="pt" hreflang="pt" data-title="Existência de Yang-Mills e intervalo de massa" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Yangin%E2%80%93Millsin_olemassaolo" 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Birch_and_Swinnerton-Dyer_conjecture" title="Birch and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</a></li> <li><a href="/wiki/Hodge_conjecture" title="Hodge conjecture">Hodge conjecture</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_existence_and_smoothness" title="Navier–Stokes existence and smoothness">Navier–Stokes existence and smoothness</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li> <li><a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a> (solved)</li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a class="mw-selflink selflink">Yang–Mills existence and mass gap</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Millennium_Prize_Problems" title="Template:Millennium Prize Problems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Millennium_Prize_Problems" title="Template talk:Millennium Prize Problems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Millennium_Prize_Problems" title="Special:EditPage/Template:Millennium Prize Problems"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>Yang–Mills existence and mass gap problem</b> is an <a href="/wiki/Open_problem" title="Open problem">unsolved problem</a> in <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a> and <a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">mathematics</a>, and one of the seven <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a> defined by the <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a>, which has offered a prize of US$1,000,000 for its solution. </p><p>The problem is phrased as follows:<sup id="cite_ref-official_1-0" class="reference"><a href="#cite_note-official-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><i>Yang–Mills Existence and Mass Gap.</i> Prove that for any compact simple <a href="/wiki/Gauge_group_(mathematics)" title="Gauge group (mathematics)">gauge group</a> G, a <a href="/wiki/Non-trivial" class="mw-redirect" title="Non-trivial">non-trivial</a> quantum <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a> exists on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"></span> and has a <a href="/wiki/Mass_gap" title="Mass gap">mass gap</a> Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in <a href="#CITEREFStreaterWightman1964">Streater & Wightman (1964)</a>, <a href="#CITEREFOsterwalderSchrader1973">Osterwalder & Schrader (1973)</a> and <a href="#CITEREFOsterwalderSchrader1975">Osterwalder & Schrader (1975)</a>.</dd></dl> <p>In this statement, a quantum <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a> is a <a href="/wiki/Non-abelian_group" title="Non-abelian group">non-abelian</a> <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> similar to that underlying the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"></span> is <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean 4-space</a>; the <a href="/wiki/Mass_gap" title="Mass gap">mass gap</a> Δ is the mass of the least massive particle predicted by the theory. </p><p>Therefore, the winner must prove that: </p> <ul><li>Yang–Mills theory exists and satisfies the standard of rigor that characterizes contemporary <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a>, in particular <a href="/wiki/Constructive_quantum_field_theory" title="Constructive quantum field theory">constructive quantum field theory</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and</li> <li>The mass of all particles of the force field predicted by the theory are strictly positive.</li></ul> <p>For example, in the case of G=SU(3)—the strong nuclear interaction—the winner must prove that <a href="/wiki/Glueball" title="Glueball">glueballs</a> have a lower mass bound, and thus cannot be arbitrarily light. </p><p>The general problem of determining the presence of a <a href="/wiki/Spectral_gap_(physics)" title="Spectral gap (physics)">spectral gap</a> in a system is known to be <a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Yang%E2%80%93Mills_existence_and_mass_gap&action=edit&section=1" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>[...] one does not yet have a mathematically complete example of a <a href="/wiki/Quantum_gauge_theory" class="mw-redirect" title="Quantum gauge theory">quantum gauge theory</a> in four-dimensional <a href="/wiki/Space-time" class="mw-redirect" title="Space-time">space-time</a>, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so!</p><div class="templatequotecite">— <cite>From the Clay Institute's official problem description by <a href="/wiki/Arthur_Jaffe" title="Arthur Jaffe">Arthur Jaffe</a> and <a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a>.</cite></div></blockquote> <p>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. </p> <div class="mw-heading mw-heading3"><h3 id="The_Wightman_axioms">The Wightman axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Yang%E2%80%93Mills_existence_and_mass_gap&action=edit&section=2" title="Edit section: The Wightman axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Wightman_axioms" title="Wightman axioms">Wightman axioms</a></div> <p>The Millennium problem requires the proposed Yang–Mills theory to satisfy the <a href="/wiki/Wightman_axioms" title="Wightman axioms">Wightman axioms</a> or similarly stringent axioms.<sup id="cite_ref-official_1-1" class="reference"><a href="#cite_note-official-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> There are four axioms: </p> <dl><dt>W0 (assumptions of relativistic quantum mechanics)</dt></dl> <p><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> is described according to <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a>; in particular, the <a href="/wiki/Pure_state" class="mw-redirect" title="Pure state">pure states</a> are given by the rays, i.e. the one-dimensional subspaces, of some <a href="/wiki/Separable_space" title="Separable space">separable</a> complex <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>. </p><p>The Wightman axioms require that the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> acts <a href="/wiki/Unitary_representation" title="Unitary representation">unitarily</a> on the Hilbert space. In other words, they have position dependent operators called <i><b>quantum fields</b></i> which form covariant <a href="/wiki/Representations_of_the_Poincar%C3%A9_group" class="mw-redirect" title="Representations of the Poincaré group">representations of the Poincaré group</a>. </p><p>The group of space-time translations is <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, and so the operators can be simultaneously diagonalised. The generators of these groups give us four <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operators</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{j},j=0,1,2,3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{j},j=0,1,2,3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/782439e13da7ea2d14b7c73d661fda5fe5131c73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.244ex; height:2.843ex;" alt="{\displaystyle P_{j},j=0,1,2,3}"></span>, which transform under the homogeneous group as a <a href="/wiki/Four-vector" title="Four-vector">four-vector</a>, called the <a href="/wiki/Energy-momentum_relation" class="mw-redirect" title="Energy-momentum relation">energy-momentum</a> four-vector. </p><p>The second part of the zeroth axiom of Wightman is that the representation <i>U</i>(<i>a</i>, <i>A</i>) fulfills the spectral condition—that the simultaneous spectrum of energy-momentum is contained in the forward cone: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}\geq 0,\;\;\;\;P_{0}^{2}-P_{j}P_{j}\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}\geq 0,\;\;\;\;P_{0}^{2}-P_{j}P_{j}\geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e20bdbff9ee0b9eb6414dcb53610fad75561349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.85ex; height:3.176ex;" alt="{\displaystyle P_{0}\geq 0,\;\;\;\;P_{0}^{2}-P_{j}P_{j}\geq 0.}"></span></dd></dl> <p>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. </p> <dl><dt>W1 (assumptions on the domain and continuity of the field)</dt></dl> <p>For each test function <i>f</i>, there exists a set of operators <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}(f),\ldots ,A_{n}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}(f),\ldots ,A_{n}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f5c4007f99674c38b8179caa50732b06f9d3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.113ex; height:2.843ex;" alt="{\displaystyle A_{1}(f),\ldots ,A_{n}(f)}"></span> which, together with their adjoints, are defined on a dense subset of the Hilbert state space, containing the vacuum. The fields <i>A</i> are operator-valued <a href="/wiki/Distribution_(mathematics)#Tempered_distributions_and_Fourier_transform" title="Distribution (mathematics)">tempered distributions</a>. The Hilbert state space is spanned by the field polynomials acting on the vacuum (cyclicity condition). </p> <dl><dt>W2 (transformation law of the field)</dt></dl> <p>The fields are covariant under the action of <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>, and they transform according to some representation S of the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>, or SL(2,<b>C</b>) if the spin is not integer: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(a,L)^{\dagger }A(x)U(a,L)=S(L)A(L^{-1}(x-a)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>L</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mi>A</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(a,L)^{\dagger }A(x)U(a,L)=S(L)A(L^{-1}(x-a)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db1f07b6cc0676a69d139e6e5b1962688322fa08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.035ex; height:3.176ex;" alt="{\displaystyle U(a,L)^{\dagger }A(x)U(a,L)=S(L)A(L^{-1}(x-a)).}"></span></dd></dl> <dl><dt>W3 (local commutativity or microscopic causality)</dt></dl> <p>If the supports of two fields are <a href="/wiki/Space-like" class="mw-redirect" title="Space-like">space-like</a> separated, then the fields either commute or anticommute. </p><p>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 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{in}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{in}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab71d84aa6e9365eb0abfafa80188bd778142283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.889ex; height:2.676ex;" alt="{\displaystyle H^{in}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{out}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{out}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8e16c549b8a1f9d082cb858316f586bc463cc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.667ex; height:2.509ex;" alt="{\displaystyle H^{out}}"></span>, appearing in the collision <a href="/wiki/S_matrix" class="mw-redirect" title="S matrix">S matrix</a>. The other important property of field theory is the <a href="/wiki/Mass_gap" title="Mass gap">mass gap</a> which is not required by the axioms—that the energy-momentum spectrum has a gap between zero and some positive number. </p> <div class="mw-heading mw-heading3"><h3 id="Mass_gap">Mass gap</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Yang%E2%80%93Mills_existence_and_mass_gap&action=edit&section=3" title="Edit section: Mass gap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mass_gap" title="Mass gap">Mass gap</a></div> <p>In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, the <b>mass gap</b> is the difference in energy between the vacuum and the next lowest <a href="/wiki/Energy_state" class="mw-redirect" title="Energy state">energy state</a>. 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. </p><p>For a given real field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546b660b2f3cfb5f34be7b3ed8371d54f5c74227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.524ex; height:2.843ex;" alt="{\displaystyle \phi (x)}"></span>, we can say that the theory has a mass gap if the <a href="/wiki/Green%27s_function_(many-body_theory)#Two-point_functions_2" title="Green's function (many-body theory)">two-point function</a> has the property </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi (0,t)\phi (0,0)\rangle \sim \sum _{n}A_{n}\exp \left(-\Delta _{n}t\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>∼</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi (0,t)\phi (0,0)\rangle \sim \sum _{n}A_{n}\exp \left(-\Delta _{n}t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b30c3a13317e64c367221ea3860f26d8589f81f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.947ex; height:5.509ex;" alt="{\displaystyle \langle \phi (0,t)\phi (0,0)\rangle \sim \sum _{n}A_{n}\exp \left(-\Delta _{n}t\right)}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{0}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{0}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2060d33ac218f529c9b1f24eab707b22b4842a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.251ex; height:2.509ex;" alt="{\displaystyle \Delta _{0}>0}"></span> 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 <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a> develops a mass gap on a lattice.<sup id="cite_ref-teper_6-0" class="reference"><a href="#cite_note-teper-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-morningstar_7-0" class="reference"><a href="#cite_note-morningstar-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Importance_of_Yang–Mills_theory"><span id="Importance_of_Yang.E2.80.93Mills_theory"></span>Importance of Yang–Mills theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Yang%E2%80%93Mills_existence_and_mass_gap&action=edit&section=4" title="Edit section: Importance of Yang–Mills theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most known and nontrivial (i.e. interacting) <a href="/wiki/Quantum_field_theories" class="mw-redirect" title="Quantum field theories">quantum field theories</a> in 4 dimensions are <a href="/wiki/Effective_field_theory" title="Effective field theory">effective field theories</a> with a <a href="/wiki/Cutoff_(physics)" title="Cutoff (physics)">cutoff</a> scale. Since the <a href="/wiki/Beta_function_(physics)" title="Beta function (physics)">beta function</a> is positive for most models, it appears that most such models have a <a href="/wiki/Landau_pole" title="Landau pole">Landau pole</a> as it is not at all clear whether or not they have nontrivial <a href="/wiki/UV_fixed_point" class="mw-redirect" title="UV fixed point">UV fixed points</a>. This means that if such a <a href="/wiki/Quantum_field_theory" title="Quantum field theory">QFT</a> is well-defined at all scales, as it has to be to satisfy the axioms of <a href="/wiki/Axiomatic_quantum_field_theory" title="Axiomatic quantum field theory">axiomatic quantum field theory</a>, it would have to be trivial (i.e. a <a href="/wiki/Free_field_theory" class="mw-redirect" title="Free field theory">free field theory</a>). </p><p><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Quantum Yang–Mills theory</a> with a <a href="/wiki/Non-abelian_gauge_theory" class="mw-redirect" title="Non-abelian gauge theory">non-abelian</a> <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a> and no quarks is an exception, because <a href="/wiki/Asymptotic_freedom" title="Asymptotic freedom">asymptotic freedom</a> characterizes this theory, meaning that it has a trivial <a href="/wiki/UV_fixed_point" class="mw-redirect" title="UV fixed point">UV fixed point</a>. Hence it is the simplest nontrivial constructive QFT in 4 dimensions. (<a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">QCD</a> is a more complicated theory because it involves <a href="/wiki/Quark" title="Quark">quarks</a>.) </p> <div class="mw-heading mw-heading3"><h3 id="Quark_confinement">Quark confinement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Yang%E2%80%93Mills_existence_and_mass_gap&action=edit&section=5" title="Edit section: Quark confinement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a>, <a href="/wiki/Color_confinement" title="Color confinement">color confinement</a>, and <a href="/wiki/Lattice_gauge_theory" title="Lattice gauge theory">lattice gauge theory</a></div> <p>At the level of rigor of <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>, it has been well established that the quantum Yang–Mills theory for a non-abelian <a href="/wiki/Lie_group" title="Lie group">Lie group</a> exhibits a property known as <a href="/wiki/Color_confinement" title="Color confinement">confinement</a>; though proper <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a> has more demanding requirements on a proof. A consequence of this property is that above the <a href="/wiki/Color_confinement" title="Color confinement">confinement scale</a>, the color charges are connected by <a href="/wiki/QCD_string" class="mw-redirect" title="QCD string">chromodynamic flux tubes</a> leading to a linear potential between the charges. Hence isolated color charge and isolated <a href="/wiki/Gluon" title="Gluon">gluons</a> 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 <a href="/wiki/Glueball" title="Glueball">glueballs</a>. If glueballs exist, they are massive, which is why a mass gap is expected. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Yang%E2%80%93Mills_existence_and_mass_gap&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-official-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-official_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-official_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJaffeWitten" class="citation web cs1"><a href="/wiki/Arthur_Jaffe" title="Arthur Jaffe">Jaffe, Arthur</a>; 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Retrieved <span class="nowrap">2024-09-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=The+Unsolvable+Problem&rft.volume=319&rft.issue=4&rft.pages=28-37&rft.date=2018-10-01&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican1018-28&rft_id=info%3Apmid%2F30273308&rft.aulast=Cubitt&rft.aufirst=Toby+S.&rft.au=P%C3%A9rez-Garc%C3%ADa%2C+David&rft.au=Wolf%2C+Michael&rft_id=https%3A%2F%2Fwww.scientificamerican.com%2Farticle%2Fthe-unsolvable-problem%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCastelvecchi2015" class="citation news cs1">Castelvecchi, Davide (9 December 2015). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/nature.2015.18983">"Paradox at the heart of mathematics makes physics problem unanswerable"</a>. <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature.2015.18983">10.1038/nature.2015.18983</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-09-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Paradox+at+the+heart+of+mathematics+makes+physics+problem+unanswerable&rft.date=2015-12-09&rft_id=info%3Adoi%2F10.1038%2Fnature.2015.18983&rft.aulast=Castelvecchi&rft.aufirst=Davide&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2Fnature.2015.18983&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span></span> </li> <li id="cite_note-teper-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-teper_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuciniTeperWenger2004" class="citation journal cs1">Lucini, Biagio; Teper, Michael; Wenger, Urs (2004). "Glueballs and k -strings in SU( N ) gauge theories: calculations with improved operators". <i><a href="/wiki/Journal_of_High_Energy_Physics" title="Journal of High Energy Physics">Journal of High Energy Physics</a></i>. <b>2004</b> (6): 012. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-lat/0404008">hep-lat/0404008</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004JHEP...06..012L">2004JHEP...06..012L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1126-6708%2F2004%2F06%2F012">10.1088/1126-6708/2004/06/012</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1029-8479">1029-8479</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14807677">14807677</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+High+Energy+Physics&rft.atitle=Glueballs+and+k+-strings+in+SU%28+N+%29+gauge+theories%3A+calculations+with+improved+operators&rft.volume=2004&rft.issue=6&rft.pages=012&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14807677%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2004JHEP...06..012L&rft_id=info%3Aarxiv%2Fhep-lat%2F0404008&rft.issn=1029-8479&rft_id=info%3Adoi%2F10.1088%2F1126-6708%2F2004%2F06%2F012&rft.aulast=Lucini&rft.aufirst=Biagio&rft.au=Teper%2C+Michael&rft.au=Wenger%2C+Urs&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span>.</span> </li> <li id="cite_note-morningstar-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-morningstar_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChenAlexandruDongDraper2006" class="citation journal cs1">Chen, Y.; Alexandru, A.; Dong, S. J.; Draper, T.; Horváth, I.; Lee, F. X.; Liu, K. F.; Mathur, N.; Morningstar, C.; Peardon, M.; Tamhankar, S.; Young, B. L.; Zhang, J. B. (2006). "Glueball spectrum and matrix elements on anisotropic lattices". <i>Physical Review D</i>. <b>73</b> (1): 014516. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-lat/0510074">hep-lat/0510074</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006PhRvD..73a4516C">2006PhRvD..73a4516C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.73.014516">10.1103/PhysRevD.73.014516</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1550-7998">1550-7998</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15741174">15741174</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Glueball+spectrum+and+matrix+elements+on+anisotropic+lattices&rft.volume=73&rft.issue=1&rft.pages=014516&rft.date=2006&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15741174%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2006PhRvD..73a4516C&rft_id=info%3Aarxiv%2Fhep-lat%2F0510074&rft.issn=1550-7998&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.73.014516&rft.aulast=Chen&rft.aufirst=Y.&rft.au=Alexandru%2C+A.&rft.au=Dong%2C+S.+J.&rft.au=Draper%2C+T.&rft.au=Horv%C3%A1th%2C+I.&rft.au=Lee%2C+F.+X.&rft.au=Liu%2C+K.+F.&rft.au=Mathur%2C+N.&rft.au=Morningstar%2C+C.&rft.au=Peardon%2C+M.&rft.au=Tamhankar%2C+S.&rft.au=Young%2C+B.+L.&rft.au=Zhang%2C+J.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Yang%E2%80%93Mills_existence_and_mass_gap&action=edit&section=7" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStreaterWightman1964" class="citation book cs1">Streater, R. F.; Wightman, A. (1964). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/pctspinstatistic0000stre"><i>PCT, spin and statistics, and all that</i></a></span>. New York, W.A. Benjamin.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=PCT%2C+spin+and+statistics%2C+and+all+that&rft.pub=New+York%2C+W.A.+Benjamin&rft.date=1964&rft.aulast=Streater&rft.aufirst=R.+F.&rft.au=Wightman%2C+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpctspinstatistic0000stre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOsterwalderSchrader1973" class="citation journal cs1">Osterwalder, Konrad; Schrader, Robert (1973). "Axioms for Euclidean Green's functions". <i>Communications in Mathematical Physics</i>. <b>31</b> (2): 83–112. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1973CMaPh..31...83O">1973CMaPh..31...83O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01645738">10.1007/BF01645738</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0010-3616">0010-3616</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:189829853">189829853</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Mathematical+Physics&rft.atitle=Axioms+for+Euclidean+Green%27s+functions&rft.volume=31&rft.issue=2&rft.pages=83-112&rft.date=1973&rft_id=info%3Adoi%2F10.1007%2FBF01645738&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A189829853%23id-name%3DS2CID&rft.issn=0010-3616&rft_id=info%3Abibcode%2F1973CMaPh..31...83O&rft.aulast=Osterwalder&rft.aufirst=Konrad&rft.au=Schrader%2C+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOsterwalderSchrader1975" class="citation journal cs1">Osterwalder, Konrad; Schrader, Robert (1975). "Axioms for Euclidean Green's functions II". <i>Communications in Mathematical Physics</i>. <b>42</b> (3): 281–305. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1975CMaPh..42..281O">1975CMaPh..42..281O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01608978">10.1007/BF01608978</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0010-3616">0010-3616</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119389461">119389461</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Mathematical+Physics&rft.atitle=Axioms+for+Euclidean+Green%27s+functions+II&rft.volume=42&rft.issue=3&rft.pages=281-305&rft.date=1975&rft_id=info%3Adoi%2F10.1007%2FBF01608978&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119389461%23id-name%3DS2CID&rft.issn=0010-3616&rft_id=info%3Abibcode%2F1975CMaPh..42..281O&rft.aulast=Osterwalder&rft.aufirst=Konrad&rft.au=Schrader%2C+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBogoliubovLogunovOksakTodorov1990" class="citation book cs1">Bogoliubov, N.; Logunov, A.; Oksak; Todorov, I. (1990). Bogolubov, N. N.; Logunov, A. A.; Oksak, A. I.; Todorov, I. T. (eds.). <i>General Principles of Quantum Field Theory</i>. Dordrecht: Springer Netherlands. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-009-0491-0">10.1007/978-94-009-0491-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-010-6707-2" title="Special:BookSources/978-94-010-6707-2"><bdi>978-94-010-6707-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Principles+of+Quantum+Field+Theory&rft.place=Dordrecht&rft.pub=Springer+Netherlands&rft.date=1990&rft_id=info%3Adoi%2F10.1007%2F978-94-009-0491-0&rft.isbn=978-94-010-6707-2&rft.aulast=Bogoliubov&rft.aufirst=N.&rft.au=Logunov%2C+A.&rft.au=Oksak&rft.au=Todorov%2C+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrocchi1993" class="citation book cs1">Strocchi, Franco (1993). <i>Selected topics on the general properties of quantum field theory: lecture notes</i>. World Scientific lecture notes in physics. Singapore: World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-1149-3" title="Special:BookSources/978-981-02-1149-3"><bdi>978-981-02-1149-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Selected+topics+on+the+general+properties+of+quantum+field+theory%3A+lecture+notes&rft.place=Singapore&rft.series=World+Scientific+lecture+notes+in+physics&rft.pub=World+Scientific&rft.date=1993&rft.isbn=978-981-02-1149-3&rft.aulast=Strocchi&rft.aufirst=Franco&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDynin2014" class="citation journal cs1">Dynin, A. (2014). "Quantum Yang-Mills-Weyl Dynamics in the Schrödinger paradigm". <i>Russian Journal of Mathematical Physics</i>. <b>21</b> (2): 169–188. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014RJMP...21..169D">2014RJMP...21..169D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2FS1061920814020046">10.1134/S1061920814020046</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1061-9208">1061-9208</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121878861">121878861</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Russian+Journal+of+Mathematical+Physics&rft.atitle=Quantum+Yang-Mills-Weyl+Dynamics+in+the+Schr%C3%B6dinger+paradigm&rft.volume=21&rft.issue=2&rft.pages=169-188&rft.date=2014&rft_id=info%3Adoi%2F10.1134%2FS1061920814020046&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121878861%23id-name%3DS2CID&rft.issn=1061-9208&rft_id=info%3Abibcode%2F2014RJMP...21..169D&rft.aulast=Dynin&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDynin2014" class="citation journal cs1">Dynin, A. (2014). "On the Yang-Mills mass gap problem". <i>Russian Journal of Mathematical Physics</i>. <b>21</b> (3): 326–328. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014RJMP...21..326D">2014RJMP...21..326D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2FS1061920814030042">10.1134/S1061920814030042</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1061-9208">1061-9208</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120135592">120135592</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Russian+Journal+of+Mathematical+Physics&rft.atitle=On+the+Yang-Mills+mass+gap+problem&rft.volume=21&rft.issue=3&rft.pages=326-328&rft.date=2014&rft_id=info%3Adoi%2F10.1134%2FS1061920814030042&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120135592%23id-name%3DS2CID&rft.issn=1061-9208&rft_id=info%3Abibcode%2F2014RJMP...21..326D&rft.aulast=Dynin&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBushhornWess2004" class="citation book cs1">Bushhorn, G.; Wess, J. (2004). Heisenberg, Werner; Buschhorn, Gerd W.; Wess, Julius (eds.). <i>Fundamental physics-- Heisenberg and beyond: Werner Heisenberg Centennial Symposium "Developments in Modern Physics"</i>. Berlin ; New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-20201-1" title="Special:BookSources/978-3-540-20201-1"><bdi>978-3-540-20201-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+physics--+Heisenberg+and+beyond%3A+Werner+Heisenberg+Centennial+Symposium+%22Developments+in+Modern+Physics%22&rft.place=Berlin+%3B+New+York&rft.pub=Springer&rft.date=2004&rft.isbn=978-3-540-20201-1&rft.aulast=Bushhorn&rft.aufirst=G.&rft.au=Wess%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AYang%E2%80%93Mills+existence+and+mass+gap" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Yang%E2%80%93Mills_existence_and_mass_gap&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.claymath.org/millennium/yang-mills-the-maths-gap/">The Millennium Prize Problems: Yang–Mills and Mass Gap</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Yang–Mills_existence_and_mass_gap&oldid=1258691530">https://en.wikipedia.org/w/index.php?title=Yang–Mills_existence_and_mass_gap&oldid=1258691530</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Millennium_Prize_Problems" title="Category:Millennium Prize Problems">Millennium Prize Problems</a></li><li><a href="/wiki/Category:Gauge_theories" title="Category:Gauge theories">Gauge theories</a></li><li><a href="/wiki/Category:Quantum_chromodynamics" title="Category:Quantum chromodynamics">Quantum chromodynamics</a></li><li><a href="/wiki/Category:Unsolved_problems_in_mathematics" title="Category:Unsolved problems in mathematics">Unsolved problems in mathematics</a></li><li><a href="/wiki/Category:Unsolved_problems_in_physics" title="Category:Unsolved problems in physics">Unsolved problems in physics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Use_American_English_from_March_2019" title="Category:Use American English from March 2019">Use American English from March 2019</a></li><li><a href="/wiki/Category:All_Wikipedia_articles_written_in_American_English" title="Category:All Wikipedia articles written in American English">All Wikipedia articles written in American English</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 21 November 2024, at 01:58<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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