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<span>Description</span> </div> </a> <button aria-controls="toc-Description-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Description subsection</span> </button> <ul id="toc-Description-sublist" class="vector-toc-list"> <li id="toc-Global_and_local_symmetries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Global_and_local_symmetries"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Global and local symmetries</span> </div> </a> <ul id="toc-Global_and_local_symmetries-sublist" class="vector-toc-list"> <li id="toc-Global_symmetry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Global_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Global symmetry</span> </div> </a> <ul id="toc-Global_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_of_global_symmetry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_of_global_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Example of global symmetry</span> </div> </a> <ul id="toc-Example_of_global_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Local_symmetry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Local_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Local symmetry</span> </div> </a> <ul id="toc-Local_symmetry-sublist" class="vector-toc-list"> <li id="toc-Use_of_fiber_bundles_to_describe_local_symmetries" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Use_of_fiber_bundles_to_describe_local_symmetries"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3.1</span> <span>Use of fiber bundles to describe local symmetries</span> </div> </a> <ul id="toc-Use_of_fiber_bundles_to_describe_local_symmetries-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Gauge_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gauge_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Gauge fields</span> </div> </a> <ul id="toc-Gauge_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physical_experiments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physical_experiments"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Physical experiments</span> </div> </a> <ul id="toc-Physical_experiments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuum_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuum_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Continuum theories</span> </div> </a> <ul id="toc-Continuum_theories-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_field_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_field_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Quantum field theories</span> </div> </a> <ul id="toc-Quantum_field_theories-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Classical_gauge_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classical_gauge_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Classical gauge theory</span> </div> </a> <button aria-controls="toc-Classical_gauge_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Classical gauge theory subsection</span> </button> <ul id="toc-Classical_gauge_theory-sublist" class="vector-toc-list"> <li id="toc-Classical_electromagnetism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_electromagnetism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Classical electromagnetism</span> </div> </a> <ul id="toc-Classical_electromagnetism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-An_example:_Scalar_O(n)_gauge_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#An_example:_Scalar_O(n)_gauge_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>An example: Scalar O(<i>n</i>) gauge theory</span> </div> </a> <ul id="toc-An_example:_Scalar_O(n)_gauge_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Yang–Mills_Lagrangian_for_the_gauge_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Yang–Mills_Lagrangian_for_the_gauge_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>The Yang–Mills Lagrangian for the gauge field</span> </div> </a> <ul id="toc-The_Yang–Mills_Lagrangian_for_the_gauge_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-An_example:_Electrodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#An_example:_Electrodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>An example: Electrodynamics</span> </div> </a> <ul id="toc-An_example:_Electrodynamics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_formalism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_formalism"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Mathematical formalism</span> </div> </a> <ul id="toc-Mathematical_formalism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantization_of_gauge_theories" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quantization_of_gauge_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Quantization of gauge theories</span> </div> </a> <button aria-controls="toc-Quantization_of_gauge_theories-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Quantization of gauge theories subsection</span> </button> <ul id="toc-Quantization_of_gauge_theories-sublist" class="vector-toc-list"> <li id="toc-Methods_and_aims" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Methods_and_aims"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Methods and aims</span> </div> </a> <ul id="toc-Methods_and_aims-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Anomalies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Anomalies"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Anomalies</span> </div> </a> <ul id="toc-Anomalies-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pure_gauge" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pure_gauge"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Pure gauge</span> </div> </a> <ul id="toc-Pure_gauge-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Gauge theory</span></h1> <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. Available in 29 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-29" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">29 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D9%85%D9%82%D9%8A%D8%A7%D8%B3" title="نظرية المقياس – Arabic" lang="ar" hreflang="ar" data-title="نظرية المقياس" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_gauge" title="Teoria de gauge – Catalan" lang="ca" hreflang="ca" data-title="Teoria de gauge" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Eichtheorie" title="Eichtheorie – German" lang="de" hreflang="de" data-title="Eichtheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CE%B2%CE%B1%CE%B8%CE%BC%CE%AF%CE%B4%CE%B1%CF%82" title="Θεωρία βαθμίδας – Greek" lang="el" hreflang="el" data-title="Θεωρία βαθμίδας" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_campo_de_gauge" title="Teoría de campo de gauge – Spanish" lang="es" hreflang="es" data-title="Teoría de campo de gauge" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ga%C5%AD%C4%9Da_teorio" title="Gaŭĝa teorio – Esperanto" lang="eo" hreflang="eo" data-title="Gaŭĝa teorio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D9%BE%DB%8C%D9%85%D8%A7%D9%86%D9%87%E2%80%8C%D8%A7%DB%8C" title="نظریه پیمانه‌ای – Persian" lang="fa" hreflang="fa" data-title="نظریه پیمانه‌ای" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_de_jauge" title="Théorie de jauge – French" lang="fr" hreflang="fr" data-title="Théorie de jauge" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Tomhasteoiric" title="Tomhasteoiric – Irish" lang="ga" hreflang="ga" data-title="Tomhasteoiric" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B2%8C%EC%9D%B4%EC%A7%80_%EC%9D%B4%EB%A1%A0" title="게이지 이론 – Korean" lang="ko" hreflang="ko" data-title="게이지 이론" 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/Teori_ukuran_(fisika)" title="Teori ukuran (fisika) – Indonesian" lang="id" hreflang="id" data-title="Teori ukuran (fisika)" 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/Teoria_di_gauge" title="Teoria di gauge – Italian" lang="it" hreflang="it" data-title="Teoria di gauge" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%9B%D7%99%D7%95%D7%9C" title="תורת כיול – Hebrew" lang="he" hreflang="he" data-title="תורת כיול" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/M%C3%A9rt%C3%A9kt%C3%A9relm%C3%A9let" title="Mértéktérelmélet – Hungarian" lang="hu" hreflang="hu" data-title="Mértéktérelmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/IJktheorie" title="IJktheorie – Dutch" lang="nl" hreflang="nl" data-title="IJktheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B2%E3%83%BC%E3%82%B8%E7%90%86%E8%AB%96" title="ゲージ理論 – Japanese" lang="ja" hreflang="ja" data-title="ゲージ理論" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Gaugeteori" title="Gaugeteori – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Gaugeteori" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Justerteori" title="Justerteori – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Justerteori" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%87%E0%A8%9C_%E0%A8%A5%E0%A8%BF%E0%A8%8A%E0%A8%B0%E0%A9%80" title="ਗੇਜ ਥਿਊਰੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਗੇਜ ਥਿਊਰੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Cechowanie_(fizyka)" title="Cechowanie (fizyka) – Polish" lang="pl" hreflang="pl" data-title="Cechowanie (fizyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_de_gauge" title="Teoria de gauge – Portuguese" lang="pt" hreflang="pt" data-title="Teoria de gauge" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_gauge" title="Teoria gauge – Romanian" lang="ro" hreflang="ro" data-title="Teoria gauge" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Umeritvena_teorija" title="Umeritvena teorija – Slovenian" lang="sl" hreflang="sl" data-title="Umeritvena teorija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Mittakentt%C3%A4teoria" title="Mittakenttäteoria – Finnish" lang="fi" hreflang="fi" data-title="Mittakenttäteoria" data-language-autonym="Suomi" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">September 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Physical theory with fields invariant under the action of local "gauge" Lie groups</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">For a more accessible and less technical introduction to this topic, see <a href="/wiki/Introduction_to_gauge_theory" title="Introduction to gauge theory">Introduction to gauge theory</a>.</div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">This article discusses the physics of gauge theories. 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert-image" typeof="mw:File/Frameless"><a href="/wiki/Feynman_diagram" title="Feynman diagram"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/211px-Feynmann_Diagram_Gluon_Radiation.svg.png" decoding="async" width="211" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/317px-Feynmann_Diagram_Gluon_Radiation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/422px-Feynmann_Diagram_Gluon_Radiation.svg.png 2x" data-file-width="400" data-file-height="250" /></a></span><div class="sidebar-caption"><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></div></td></tr><tr><td class="sidebar-above"> <a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Field_(physics)" title="Field (physics)">Field theory</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">Weak force</a></li> <li><a href="/wiki/Strong_force" class="mw-redirect" title="Strong force">Strong force</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a class="mw-selflink selflink">Gauge theory</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">Symmetries</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">C-symmetry</a></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">P-symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a></li> <li><a href="/wiki/Lorentz_symmetry" class="mw-redirect" title="Lorentz symmetry">Lorentz symmetry</a></li> <li><a href="/wiki/Poincar%C3%A9_symmetry" class="mw-redirect" title="Poincaré symmetry">Poincaré symmetry</a></li> <li><a href="/wiki/Gauge_symmetry_(mathematics)" title="Gauge symmetry (mathematics)">Gauge symmetry</a></li> <li><a href="/wiki/Explicit_symmetry_breaking" title="Explicit symmetry breaking">Explicit symmetry breaking</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Noether_charge" class="mw-redirect" title="Noether charge">Noether charge</a></li> <li><a href="/wiki/Topological_charge" class="mw-redirect" title="Topological charge">Topological charge</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Tools</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomaly</a></li> <li><a href="/wiki/Background_field_method" title="Background field method">Background field method</a></li> <li><a href="/wiki/BRST_quantization" title="BRST quantization">BRST quantization</a></li> <li><a href="/wiki/Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">Correlation function</a></li> <li><a href="/wiki/Crossing_(physics)" title="Crossing (physics)">Crossing</a></li> <li><a href="/wiki/Effective_action" title="Effective action">Effective action</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective field theory</a></li> <li><a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">Expectation value</a></li> <li><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></li> <li><a href="/wiki/Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li> <li><a href="/wiki/LSZ_reduction_formula" title="LSZ reduction formula">LSZ reduction formula</a></li> <li><a href="/wiki/Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">Partition function</a></li> <li><a href="/wiki/Path_Integral_Formulation" class="mw-redirect" title="Path Integral Formulation">Path Integral Formulation</a></li> <li><a href="/wiki/Propagator_(Quantum_Theory)" class="mw-redirect" title="Propagator (Quantum Theory)">Propagator</a></li> <li><a href="/wiki/Quantization_(physics)" title="Quantization (physics)">Quantization</a></li> <li><a href="/wiki/Regularization_(physics)" title="Regularization (physics)">Regularization</a></li> <li><a href="/wiki/Renormalization" title="Renormalization">Renormalization</a></li> <li><a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">Vacuum state</a></li> <li><a href="/wiki/Wick%27s_theorem" title="Wick's theorem">Wick's theorem</a></li> <li><a href="/w/index.php?title=Wightman_Axioms&action=edit&redlink=1" class="new" title="Wightman Axioms (page does not exist)">Wightman Axioms</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a></li> <li><a href="/wiki/Proca_action" title="Proca action">Proca equations</a></li> <li><a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></li> <li><a href="/wiki/Bargmann%E2%80%93Wigner_equations" title="Bargmann–Wigner equations">Bargmann–Wigner equations</a></li> <li><a href="/wiki/Schwinger-Dyson_equation" class="mw-redirect" title="Schwinger-Dyson equation">Schwinger-Dyson equation</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group equation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Incomplete theories</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></li> <li><a href="/wiki/Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Stephen_Louis_Adler" class="mw-redirect" title="Stephen Louis Adler">Adler</a></li> <li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li> <li><a href="/wiki/Alexey_Andreevich_Anselm" class="mw-redirect" title="Alexey Andreevich Anselm">Anselm</a></li> <li><a href="/wiki/Valentine_Bargmann" title="Valentine Bargmann">Bargmann</a></li> <li><a href="/wiki/Carlo_Becchi" title="Carlo Becchi">Becchi</a></li> <li><a href="/wiki/Alexander_Belavin" title="Alexander Belavin">Belavin</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Felix_Berezin" title="Felix Berezin">Berezin</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/James_Bjorken" title="James Bjorken">Bjorken</a></li> <li><a href="/wiki/Konrad_Bleuler" title="Konrad Bleuler">Bleuer</a></li> <li><a href="/wiki/Nikolay_Bogolyubov" title="Nikolay Bogolyubov">Bogoliubov</a></li> <li><a href="/wiki/Stanley_Brodsky" title="Stanley Brodsky">Brodsky</a></li> <li><a href="/wiki/Robert_Brout" title="Robert Brout">Brout</a></li> <li><a href="/wiki/Detlev_Buchholz" title="Detlev Buchholz">Buchholz</a></li> <li><a href="/wiki/Freddy_Cachazo" title="Freddy Cachazo">Cachazo</a></li> <li><a href="/wiki/Curtis_Callan" title="Curtis Callan">Callan</a></li> <li><a href="/wiki/John_Cardy" title="John Cardy">Cardy</a></li> <li><a href="/wiki/Sidney_Coleman" title="Sidney Coleman">Coleman</a></li> <li><a href="/wiki/Alain_Connes" title="Alain Connes">Connes</a></li> <li><a href="/wiki/Roger_Dashen" title="Roger Dashen">Dashen</a></li> <li><a href="/wiki/Bryce_DeWitt" title="Bryce DeWitt">DeWitt</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Sergio_Doplicher" title="Sergio Doplicher">Doplicher</a></li> <li><a href="/wiki/Freeman_Dyson" title="Freeman Dyson">Dyson</a></li> <li><a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">Englert</a></li> <li><a href="/wiki/Ludvig_Faddeev" title="Ludvig Faddeev">Faddeev</a></li> <li><a href="/wiki/Victor_Sergeevich_Fadin" class="mw-redirect" title="Victor Sergeevich Fadin">Fadin</a></li> <li><a href="/wiki/Pierre_Fayet" title="Pierre Fayet">Fayet</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Markus_Fierz" title="Markus Fierz">Fierz</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Paul_Frampton" title="Paul Frampton">Frampton</a></li> <li><a href="/wiki/Harald_Fritzsch" title="Harald Fritzsch">Fritzsch</a></li> <li><a href="/wiki/J%C3%BCrg_Fr%C3%B6hlich" title="Jürg Fröhlich">Fröhlich</a></li> <li><a href="/wiki/Klaus_Fredenhagen" title="Klaus Fredenhagen">Fredenhagen</a></li> <li><a href="/wiki/Wendell_H._Furry" title="Wendell H. Furry">Furry</a></li> <li><a href="/wiki/Sheldon_Glashow" title="Sheldon Glashow">Glashow</a></li> <li><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann</a></li> <li><a href="/wiki/James_Glimm" title="James Glimm">Glimm</a></li> <li><a href="/wiki/Jeffrey_Goldstone" title="Jeffrey Goldstone">Goldstone</a></li> <li><a href="/wiki/Vladimir_Gribov" title="Vladimir Gribov">Gribov</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Suraj_N._Gupta" title="Suraj N. Gupta">Gupta</a></li> <li><a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Guralnik</a></li> <li><a href="/wiki/Rudolf_Haag" title="Rudolf Haag">Haag</a></li> <li><a href="/wiki/C._R._Hagen" title="C. R. Hagen">Hagen</a></li> <li><a href="/wiki/Moo-Young_Han" title="Moo-Young Han">Han</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/Klaus_Hepp" title="Klaus Hepp">Hepp</a></li> <li><a href="/wiki/Peter_Higgs" title="Peter Higgs">Higgs</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li> <li><a href="/wiki/Dmitri_Ivanenko" title="Dmitri Ivanenko">Ivanenko</a></li> <li><a href="/wiki/Roman_Jackiw" title="Roman Jackiw">Jackiw</a></li> <li><a href="/wiki/Arthur_Jaffe" title="Arthur Jaffe">Jaffe</a></li> <li><a href="/wiki/Giovanni_Jona-Lasinio" title="Giovanni Jona-Lasinio">Jona-Lasinio</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Res_Jost" title="Res Jost">Jost</a></li> <li><a href="/wiki/Gunnar_K%C3%A4ll%C3%A9n" title="Gunnar Källén">Källén</a></li> <li><a href="/wiki/Henry_Way_Kendall" title="Henry Way Kendall">Kendall</a></li> <li><a href="/wiki/Toichiro_Kinoshita" title="Toichiro Kinoshita">Kinoshita</a></li> <li><a href="/wiki/Kim_Jihn-eui" title="Kim Jihn-eui">Kim</a></li> <li><a href="/wiki/Igor_R._Klebanov" class="mw-redirect" title="Igor R. Klebanov">Klebanov</a></li> <li><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li> <li><a href="/wiki/Dirk_Kreimer" title="Dirk Kreimer">Kreimer</a></li> <li><a href="/wiki/Eduard_A._Kuraev" title="Eduard A. Kuraev">Kuraev</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Benjamin_W._Lee" title="Benjamin W. Lee">Lee</a></li> <li><a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Lee</a></li> <li><a href="/wiki/Harry_Lehmann" title="Harry Lehmann">Lehmann</a></li> <li><a href="/wiki/Heinrich_Leutwyler" title="Heinrich Leutwyler">Leutwyler</a></li> <li><a href="/wiki/Lev_Lipatov" title="Lev Lipatov">Lipatov</a></li> <li><a href="/wiki/Jan_%C5%81opusza%C5%84ski_(physicist)" title="Jan Łopuszański (physicist)">Łopuszański</a></li> <li><a href="/wiki/Francis_E._Low" title="Francis E. Low">Low</a></li> <li><a href="/wiki/Gerhart_L%C3%BCders" title="Gerhart Lüders">Lüders</a></li> <li><a href="/wiki/Luciano_Maiani" title="Luciano Maiani">Maiani</a></li> <li><a href="/wiki/Ettore_Majorana" title="Ettore Majorana">Majorana</a></li> <li><a href="/wiki/Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li> <li><a href="/wiki/Takeo_Matsubara" title="Takeo Matsubara">Matsubara</a></li> <li><a href="/wiki/Alexander_Arkadyevich_Migdal" title="Alexander Arkadyevich Migdal">Migdal</a></li> <li><a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Mills</a></li> <li><a href="/wiki/Christian_M%C3%B8ller" title="Christian Møller">Møller</a></li> <li><a href="/wiki/Mark_Naimark" title="Mark Naimark">Naimark</a></li> <li><a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Nambu</a></li> <li><a href="/wiki/Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li> <li><a href="/wiki/Kazuhiko_Nishijima" title="Kazuhiko Nishijima">Nishijima</a></li> <li><a href="/wiki/Reinhard_Oehme" title="Reinhard Oehme">Oehme</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Hugh_Osborn" title="Hugh Osborn">Osborn</a></li> <li><a href="/wiki/Konrad_Osterwalder" title="Konrad Osterwalder">Osterwalder</a></li> <li><a href="/wiki/Giorgio_Parisi" title="Giorgio Parisi">Parisi</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Roberto_Peccei" title="Roberto Peccei">Peccei</a></li> <li><a href="/wiki/Michael_Peskin" title="Michael Peskin">Peskin</a></li> <li><a href="/wiki/Jan_Christoph_Plefka" title="Jan Christoph Plefka">Plefka</a></li> <li><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li> <li><a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li> <li><a href="/wiki/Isaak_Pomeranchuk" title="Isaak Pomeranchuk">Pomeranchuk</a></li> <li><a href="/wiki/Victor_Popov" title="Victor Popov">Popov</a></li> <li><a href="/wiki/Alexandru_Proca" title="Alexandru Proca">Proca</a></li> <li><a href="/wiki/Helen_Quinn" title="Helen Quinn">Quinn</a></li> <li><a href="/wiki/Alain_Rouet" title="Alain Rouet">Rouet</a></li> <li><a href="/wiki/Valery_Rubakov" title="Valery Rubakov">Rubakov</a></li> <li><a href="/wiki/David_Ruelle" title="David Ruelle">Ruelle</a></li> <li><a href="/wiki/Jun_John_Sakurai" class="mw-redirect" title="Jun John Sakurai">Sakurai</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Robert_Schrader" title="Robert Schrader">Schrader</a></li> <li><a href="/wiki/Albert_Schwarz" title="Albert Schwarz">Schwarz</a></li> <li><a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Schwinger</a></li> <li><a href="/wiki/Irving_Segal" title="Irving Segal">Segal</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Gordon_Walter_Semenoff" title="Gordon Walter Semenoff">Semenoff</a></li> <li><a href="/wiki/Mikhail_Shifman" title="Mikhail Shifman">Shifman</a></li> <li><a href="/wiki/Dmitry_Shirkov" title="Dmitry Shirkov">Shirkov</a></li> <li><a href="/wiki/Tony_Skyrme" title="Tony Skyrme">Skyrme</a></li> <li><a href="/wiki/Charles_M._Sommerfield" title="Charles M. Sommerfield">Sommerfield</a></li> <li><a href="/wiki/Raymond_Stora" title="Raymond Stora">Stora</a></li> <li><a href="/wiki/Ernst_Stueckelberg" title="Ernst Stueckelberg">Stueckelberg</a></li> <li><a href="/wiki/George_Sudarshan" class="mw-redirect" title="George Sudarshan">Sudarshan</a></li> <li><a href="/wiki/Kurt_Symanzik" title="Kurt Symanzik">Symanzik</a></li> <li><a href="/wiki/Yasushi_Takahashi" title="Yasushi Takahashi">Takahashi</a></li> <li><a href="/wiki/Walter_Thirring" title="Walter Thirring">Thirring</a></li> <li><a href="/wiki/Shin%27ichir%C5%8D_Tomonaga" title="Shin'ichirō Tomonaga">Tomonaga</a></li> <li><a href="/wiki/Igor_Tyutin" title="Igor Tyutin">Tyutin</a></li> <li><a href="/wiki/Arkady_Vainshtein" title="Arkady Vainshtein">Vainshtein</a></li> <li><a href="/wiki/Martinus_Veltman" class="mw-redirect" title="Martinus Veltman">Veltman</a></li> <li><a href="/wiki/Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano</a></li> <li><a href="/wiki/Miguel_%C3%81ngel_Virasoro_(physicist)" title="Miguel Ángel Virasoro (physicist)">Virasoro</a></li> <li><a href="/wiki/John_Clive_Ward" title="John Clive Ward">Ward</a></li> <li><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg</a></li> <li><a href="/wiki/Victor_Weisskopf" title="Victor Weisskopf">Weisskopf</a></li> <li><a href="/wiki/Gregor_Wentzel" title="Gregor Wentzel">Wentzel</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Christof_Wetterich" title="Christof Wetterich">Wetterich</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Gian_Carlo_Wick" title="Gian Carlo Wick">Wick</a></li> <li><a href="/wiki/Arthur_Wightman" title="Arthur Wightman">Wightman</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek</a></li> <li><a href="/wiki/Kenneth_G._Wilson" title="Kenneth G. Wilson">Wilson</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang</a></li> <li><a href="/wiki/Hideki_Yukawa" title="Hideki Yukawa">Yukawa</a></li> <li><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Anthony_Zee" title="Anthony Zee">Zee</a></li> <li><a href="/wiki/Wolfhart_Zimmermann" title="Wolfhart Zimmermann">Zimmermann</a></li> <li><a href="/wiki/Jean_Zinn-Justin" title="Jean Zinn-Justin">Zinn-Justin</a></li> <li><a href="/wiki/Jean-Bernard_Zuber" title="Jean-Bernard Zuber">Zuber</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li></ul> <p><br /> </p> </div></div></div></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:Quantum_field_theory" title="Template:Quantum field theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_field_theory" title="Template talk:Quantum field theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_field_theory" title="Special:EditPage/Template:Quantum field theory"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, a <b>gauge theory</b> is a type of <a href="/wiki/Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">field theory</a> in which the <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a>, and hence the dynamics of the system itself, do not change under <a href="/wiki/Local_symmetry" class="mw-redirect" title="Local symmetry">local transformations</a> according to certain smooth families of operations (<a href="/wiki/Lie_group" title="Lie group">Lie groups</a>). Formally, the Lagrangian is <a href="/wiki/Invariant_(physics)" title="Invariant (physics)">invariant</a> under these transformations. </p><p>The term <i>gauge</i> refers to any specific mathematical formalism to regulate redundant <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a> in the Lagrangian of a physical system. The transformations between possible gauges, called <i>gauge transformations</i>, form a Lie group—referred to as the <i><a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a></i> or the <i>gauge group</i> of the theory. Associated with any Lie group is the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of <a href="/wiki/Group_generator" class="mw-redirect" title="Group generator">group generators</a>. For each group generator there necessarily arises a corresponding field (usually a <a href="/wiki/Vector_field" title="Vector field">vector field</a>) called the <i>gauge field</i>. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called <i>gauge invariance</i>). When such a theory is <a href="/wiki/Canonical_quantization" title="Canonical quantization">quantized</a>, the <a href="/wiki/Quantum" title="Quantum">quanta</a> of the gauge fields are called <i><a href="/wiki/Gauge_boson" title="Gauge boson">gauge bosons</a></i>. If the symmetry group is non-commutative, then the gauge theory is referred to as <i><a href="/wiki/Non-Abelian_group" class="mw-redirect" title="Non-Abelian group">non-abelian</a> gauge theory</i>, the usual example being the <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a>. </p><p>Many powerful theories in physics are described by <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangians</a> that are <a href="/wiki/Invariant_(physics)" title="Invariant (physics)">invariant</a> under some symmetry transformation groups. When they are invariant under a transformation identically performed at <i>every</i> <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> in the <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> in which the physical processes occur, they are said to have a <a href="/wiki/Global_symmetry" class="mw-redirect" title="Global symmetry">global symmetry</a>. <a href="/wiki/Local_symmetry" class="mw-redirect" title="Local symmetry">Local symmetry</a>, the cornerstone of gauge theories, is a stronger constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in spacetime (the same way a constant value can be understood as a function of a certain parameter, the output of which is always the same). </p><p>Gauge theories are important as the successful field theories explaining the dynamics of <a href="/wiki/Elementary_particles" class="mw-redirect" title="Elementary particles">elementary particles</a>. <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a> is an <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> gauge theory with the symmetry group <a href="/wiki/Unitary_group" title="Unitary group">U(1)</a> and has one gauge field, the <a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">electromagnetic four-potential</a>, with the <a href="/wiki/Photon" title="Photon">photon</a> being the gauge boson. The <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> is a non-abelian gauge theory with the symmetry group U(1) × <a href="/wiki/Special_unitary_group#The_group_SU(2)" title="Special unitary group">SU(2)</a> × <a href="/wiki/Special_unitary_group#SU(3)" title="Special unitary group">SU(3)</a> and has a total of twelve gauge bosons: the <a href="/wiki/Photon" title="Photon">photon</a>, three <a href="/wiki/Weak_boson" class="mw-redirect" title="Weak boson">weak bosons</a> and eight <a href="/wiki/Gluons" class="mw-redirect" title="Gluons">gluons</a>. </p><p>Gauge theories are also important in explaining <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitation</a> in the theory of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. Its case is somewhat unusual in that the gauge field is a tensor, the <a href="/wiki/Lanczos_tensor" title="Lanczos tensor">Lanczos tensor</a>. Theories of <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a>, beginning with <a href="/wiki/Gauge_gravitation_theory" title="Gauge gravitation theory">gauge gravitation theory</a>, also postulate the existence of a gauge boson known as the <a href="/wiki/Graviton" title="Graviton">graviton</a>. Gauge symmetries can be viewed as analogues of the <a href="/wiki/Principle_of_general_covariance" class="mw-redirect" title="Principle of general covariance">principle of general covariance</a> of general relativity in which the coordinate system can be chosen freely under arbitrary <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a> of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, <a href="/wiki/Gauge_theory_gravity" title="Gauge theory gravity">gauge theory gravity</a>, replaces the principle of general covariance with a true gauge principle with new gauge fields. </p><p>Historically, these ideas were first stated in the context of <a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">classical electromagnetism</a> and later in <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. However, the modern importance of gauge symmetries appeared first in the <a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">relativistic quantum mechanics</a> of <a href="/wiki/Electron" title="Electron">electrons</a> – <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>, elaborated on below. Today, gauge theories are useful in <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter</a>, <a href="/wiki/Nuclear_physics" title="Nuclear physics">nuclear</a> and <a href="/wiki/High_energy_physics" class="mw-redirect" title="High energy physics">high energy physics</a> among other subfields. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept and the name of gauge theory derives from the work of <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> in 1918.<sup id="cite_ref-Brading_1-0" class="reference"><a href="#cite_note-Brading-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Weyl, in an attempt to generalize the geometrical ideas of <a href="/wiki/General_relativity" title="General relativity">general relativity</a> to include <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, conjectured that <i>Eichinvarianz</i> or invariance under the change of <a href="/wiki/Scale_(measurement)" class="mw-redirect" title="Scale (measurement)">scale</a> (or "gauge") might also be a local symmetry of general relativity. After the development of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, Weyl, <a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Vladimir Fock</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> and <a href="/wiki/Fritz_London" title="Fritz London">Fritz London</a> replaced the simple scale factor with a <a href="/wiki/Complex_number" title="Complex number">complex</a> quantity and turned the scale transformation into a change of <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a>, which is a <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> gauge symmetry. This explained the <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> effect on the <a href="/wiki/Wave_function" title="Wave function">wave function</a> of a <a href="/wiki/Electric_charge" title="Electric charge">charged</a> quantum mechanical <a href="/wiki/Elementary_particle" title="Elementary particle">particle</a>. Weyl's 1929 paper introduced the modern concept of gauge invariance<sup id="cite_ref-O’Raifeartaigh_3-0" class="reference"><a href="#cite_note-O’Raifeartaigh-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> subsequently popularized by <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a> in his 1941 review.<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> In retrospect, <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a>'s formulation, in 1864–65, of <a href="/wiki/Classical_electrodynamics" class="mw-redirect" title="Classical electrodynamics">electrodynamics</a> in "<a href="/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field" title="A Dynamical Theory of the Electromagnetic Field">A Dynamical Theory of the Electromagnetic Field</a>" suggested the possibility of invariance, when he stated that any vector field whose curl vanishes—and can therefore normally be written as a <a href="/wiki/Gradient" title="Gradient">gradient</a> of a function—could be added to the vector potential without affecting the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>. Similarly unnoticed, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> had derived the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a> by postulating the invariance of the <a href="/wiki/Action_(physics)" title="Action (physics)">action</a> under a general coordinate transformation. The importance of these symmetry invariances remained unnoticed until Weyl's work. </p><p>Inspired by Pauli's descriptions of connection between charge conservation and field theory driven by invariance, <a href="/wiki/Chen_Ning_Yang" class="mw-redirect" title="Chen Ning Yang">Chen Ning Yang</a> sought a field theory for <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">atomic nuclei</a> binding based on conservation of nuclear <a href="/wiki/Isospin" title="Isospin">isospin</a>.<sup id="cite_ref-Baggott40_5-0" class="reference"><a href="#cite_note-Baggott40-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 202">: 202 </span></sup> In 1954, Yang and <a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Robert Mills</a> generalized the gauge invariance of electromagnetism, constructing a theory based on the action of the (non-abelian) SU(2) symmetry <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> on the <a href="/wiki/Isospin" title="Isospin">isospin</a> doublet of <a href="/wiki/Proton" title="Proton">protons</a> and <a href="/wiki/Neutron" title="Neutron">neutrons</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This is similar to the action of the <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> group on the <a href="/wiki/Spinor" title="Spinor">spinor</a> <a href="/wiki/Field_(physics)" title="Field (physics)">fields</a> of <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>. </p><p>The <a href="/wiki/Yang-Mills_theory" class="mw-redirect" title="Yang-Mills theory">Yang-Mills theory</a> became the prototype theory to resolve some of the great confusion in <a href="/wiki/Elementary_particle_physics" class="mw-redirect" title="Elementary particle physics">elementary particle physics</a>. This idea later found application in the <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> of the <a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">weak force</a>, and its unification with electromagnetism in the <a href="/wiki/Electroweak" class="mw-redirect" title="Electroweak">electroweak</a> theory. Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called <a href="/wiki/Asymptotic_freedom" title="Asymptotic freedom">asymptotic freedom</a>. Asymptotic freedom was believed to be an important characteristic of strong interactions. This motivated searching for a strong force gauge theory. This theory, now known as <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>, is a gauge theory with the action of the SU(3) group on the <a href="/wiki/Color_charge" title="Color charge">color</a> triplet of <a href="/wiki/Quarks" class="mw-redirect" title="Quarks">quarks</a>. The <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory. </p><p>In the 1970s, <a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Michael Atiyah</a> began studying the mathematics of solutions to the classical <a href="/wiki/Yang%E2%80%93Mills" class="mw-redirect" title="Yang–Mills">Yang–Mills</a> equations. In 1983, Atiyah's student <a href="/wiki/Simon_Donaldson" title="Simon Donaldson">Simon Donaldson</a> built on this work to show that the <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable</a> classification of <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> 4-<a href="/wiki/Manifold" title="Manifold">manifolds</a> is very different from their classification <a href="/wiki/Up_to" title="Up to">up to</a> <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Michael_Freedman" title="Michael Freedman">Michael Freedman</a> used Donaldson's work to exhibit <a href="/wiki/Exotic_R4" title="Exotic R4">exotic <b>R</b><sup>4</sup>s</a>, that is, exotic <a href="/wiki/Differentiable_structure" class="mw-redirect" title="Differentiable structure">differentiable structures</a> on <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean</a> 4-dimensional space. This led to an increasing interest in gauge theory for its own sake, independent of its successes in fundamental physics. In 1994, <a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a> and <a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Nathan Seiberg</a> invented gauge-theoretic techniques based on <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a> that enabled the calculation of certain <a href="/wiki/Topology" title="Topology">topological</a> invariants<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> (the <a href="/wiki/Seiberg%E2%80%93Witten_invariant" class="mw-redirect" title="Seiberg–Witten invariant">Seiberg–Witten invariants</a>). These contributions to mathematics from gauge theory have led to a renewed interest in this area. </p><p>The importance of gauge theories in physics is exemplified in the tremendous success of the mathematical formalism in providing a unified framework to describe the <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theories</a> of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, the <a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">weak force</a> and the <a href="/wiki/Strong_force" class="mw-redirect" title="Strong force">strong force</a>. This theory, known as the <a href="/wiki/Standard_model" class="mw-redirect" title="Standard model">Standard Model</a>, accurately describes experimental predictions regarding three of the four <a href="/wiki/Fundamental_force" class="mw-redirect" title="Fundamental force">fundamental forces</a> of nature, and is a gauge theory with the gauge group <a href="/wiki/SU(3)_%C3%97_SU(2)_%C3%97_U(1)" class="mw-redirect" title="SU(3) × SU(2) × U(1)">SU(3) × SU(2) × U(1)</a>. Modern theories like <a href="/wiki/String_theory" title="String theory">string theory</a>, as well as <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, are, in one way or another, gauge theories. </p> <dl><dd><i>See Jackson and Okun</i><sup id="cite_ref-JacksonOkun_10-0" class="reference"><a href="#cite_note-JacksonOkun-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><i> for early history of gauge and Pickering</i><sup id="cite_ref-Pickering_11-0" class="reference"><a href="#cite_note-Pickering-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><i> for more about the history of gauge and quantum field theories.</i></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=2" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Global_and_local_symmetries">Global and local symmetries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=3" title="Edit section: Global and local symmetries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Global_symmetry">Global symmetry</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=4" title="Edit section: Global symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, the mathematical description of any physical situation usually contains excess <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a>; the same physical situation is equally well described by many equivalent mathematical configurations. For instance, in <a href="/wiki/Newtonian_dynamics" title="Newtonian dynamics">Newtonian dynamics</a>, if two configurations are related by a <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a> (an <a href="/wiki/Inertial" class="mw-redirect" title="Inertial">inertial</a> change of reference frame) they represent the same physical situation. These transformations form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of "<a href="/wiki/Symmetry_in_physics" class="mw-redirect" title="Symmetry in physics">symmetries</a>" of the theory, and a physical situation corresponds not to an individual mathematical configuration but to a class of configurations related to one another by this symmetry group. </p><p>This idea can be generalized to include local as well as global symmetries, analogous to much more abstract "changes of coordinates" in a situation where there is no preferred "<a href="/wiki/Inertial" class="mw-redirect" title="Inertial">inertial</a>" coordinate system that covers the entire physical system. A gauge theory is a mathematical model that has symmetries of this kind, together with a set of techniques for making physical predictions consistent with the symmetries of the model. </p> <div class="mw-heading mw-heading4"><h4 id="Example_of_global_symmetry">Example of global symmetry</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=5" title="Edit section: Example of global symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When a quantity occurring in the mathematical configuration is not just a number but has some geometrical significance, such as a velocity or an axis of rotation, its representation as numbers arranged in a vector or matrix is also changed by a coordinate transformation. For instance, if one description of a pattern of fluid flow states that the fluid velocity in the neighborhood of (<i>x</i>=1, <i>y</i>=0) is 1 m/s in the positive <i>x</i> direction, then a description of the same situation in which the coordinate system has been rotated clockwise by 90 degrees states that the fluid velocity in the neighborhood of (<span class="nowrap"><i>x</i> = 0</span>, <span class="nowrap"><i>y</i>= −1</span>) is 1 m/s in the negative <i>y</i> direction. The coordinate transformation has affected both the coordinate system used to identify the <i>location</i> of the measurement and the basis in which its <i>value</i> is expressed. As long as this transformation is performed globally (affecting the coordinate basis in the same way at every point), the effect on values that represent the <i>rate of change</i> of some quantity along some path in space and time as it passes through point <i>P</i> is the same as the effect on values that are truly local to <i>P</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Local_symmetry">Local symmetry</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=6" title="Edit section: Local symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading5"><h5 id="Use_of_fiber_bundles_to_describe_local_symmetries">Use of fiber bundles to describe local symmetries</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=7" title="Edit section: Use of fiber bundles to describe local symmetries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In order to adequately describe physical situations in more complex theories, it is often necessary to introduce a "coordinate basis" for some of the objects of the theory that do not have this simple relationship to the coordinates used to label points in space and time. (In mathematical terms, the theory involves a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a> in which the fiber at each point of the base space consists of possible coordinate bases for use when describing the values of objects at that point.) In order to spell out a mathematical configuration, one must choose a particular coordinate basis at each point (a <i>local section</i> of the fiber bundle) and express the values of the objects of the theory (usually "<a href="/wiki/Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">fields</a>" in the physicist's sense) using this basis. Two such mathematical configurations are equivalent (describe the same physical situation) if they are related by a transformation of this abstract coordinate basis (a change of local section, or <i>gauge transformation</i>). </p><p>In most gauge theories, the set of possible transformations of the abstract gauge basis at an individual point in space and time is a finite-dimensional Lie group. The simplest such group is <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a>, which appears in the modern formulation of <a href="/wiki/Quantum_electrodynamics#Mathematics" title="Quantum electrodynamics">quantum electrodynamics (QED)</a> via its use of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. QED is generally regarded as the first, and simplest, physical gauge theory. The set of possible gauge transformations of the entire configuration of a given gauge theory also forms a group, the <i>gauge group</i> of the theory. An element of the gauge group can be parameterized by a smoothly varying function from the points of spacetime to the (finite-dimensional) Lie group, such that the value of the function and its derivatives at each point represents the action of the gauge transformation on the fiber over that point. </p><p>A gauge transformation with constant parameter at every point in space and time is analogous to a rigid rotation of the geometric coordinate system; it represents a <a href="/wiki/Global_symmetry" class="mw-redirect" title="Global symmetry">global symmetry</a> of the gauge representation. As in the case of a rigid rotation, this gauge transformation affects expressions that represent the rate of change along a path of some gauge-dependent quantity in the same way as those that represent a truly local quantity. A gauge transformation whose parameter is <i>not</i> a constant function is referred to as a <a href="/wiki/Local_symmetry" class="mw-redirect" title="Local symmetry">local symmetry</a>; its effect on expressions that involve a <a href="/wiki/Derivative" title="Derivative">derivative</a> is qualitatively different from that on expressions that do not. (This is analogous to a non-inertial change of reference frame, which can produce a <a href="/wiki/Coriolis_effect" class="mw-redirect" title="Coriolis effect">Coriolis effect</a>.) </p> <div class="mw-heading mw-heading3"><h3 id="Gauge_fields">Gauge fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=8" title="Edit section: Gauge fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The "gauge covariant" version of a gauge theory accounts for this effect by introducing a <i>gauge field</i> (in mathematical language, an <a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann connection</a>) and formulating all rates of change in terms of the <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a> with respect to this connection. The gauge field becomes an essential part of the description of a mathematical configuration. A configuration in which the gauge field can be eliminated by a gauge transformation has the property that its <a href="/wiki/Field_strength" title="Field strength">field strength</a> (in mathematical language, its <a href="/wiki/Curvature" title="Curvature">curvature</a>) is zero everywhere; a gauge theory is <i>not</i> limited to these configurations. In other words, the distinguishing characteristic of a gauge theory is that the gauge field does not merely compensate for a poor choice of coordinate system; there is generally no gauge transformation that makes the gauge field vanish. </p><p>When analyzing the <a href="/wiki/Dynamics_(physics)" class="mw-redirect" title="Dynamics (physics)">dynamics</a> of a gauge theory, the gauge field must be treated as a dynamical variable, similar to other objects in the description of a physical situation. In addition to its <a href="/wiki/Fundamental_interaction" title="Fundamental interaction">interaction</a> with other objects via the covariant derivative, the gauge field typically contributes <a href="/wiki/Energy" title="Energy">energy</a> in the form of a "self-energy" term. One can obtain the equations for the gauge theory by: </p> <ul><li>starting from a naïve <a href="/wiki/Ansatz" title="Ansatz">ansatz</a> without the gauge field (in which the derivatives appear in a "bare" form);</li> <li>listing those global symmetries of the theory that can be characterized by a continuous parameter (generally an abstract equivalent of a rotation angle);</li> <li>computing the correction terms that result from allowing the symmetry parameter to vary from place to place; and</li> <li>reinterpreting these correction terms as couplings to one or more gauge fields, and giving these fields appropriate self-energy terms and dynamical behavior.</li></ul> <p>This is the sense in which a gauge theory "extends" a global symmetry to a local symmetry, and closely resembles the historical development of the gauge theory of gravity known as <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Physical_experiments">Physical experiments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=9" title="Edit section: Physical experiments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gauge theories used to model the results of physical experiments engage in: </p> <ul><li>limiting the universe of possible configurations to those consistent with the information used to set up the experiment, and then</li> <li>computing the probability distribution of the possible outcomes that the experiment is designed to measure.</li></ul> <p>We cannot express the mathematical descriptions of the "setup information" and the "possible measurement outcomes", or the "boundary conditions" of the experiment, without reference to a particular coordinate system, including a choice of gauge. One assumes an adequate experiment isolated from "external" influence that is itself a gauge-dependent statement. Mishandling gauge dependence calculations in boundary conditions is a frequent source of <a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">anomalies</a>, and approaches to anomaly avoidance classifies gauge theories<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (February 2014)">clarification needed</span></a></i>]</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Continuum_theories">Continuum theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=10" title="Edit section: Continuum theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The two gauge theories mentioned above, continuum electrodynamics and general relativity, are continuum field theories. The techniques of calculation in a <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum theory</a> implicitly assume that: </p> <ul><li>given a completely fixed choice of gauge, the boundary conditions of an individual configuration are completely described</li> <li>given a completely fixed gauge and a complete set of boundary conditions, the least action determines a unique mathematical configuration and therefore a unique physical situation consistent with these bounds</li> <li>fixing the gauge introduces no anomalies in the calculation, due either to gauge dependence in describing partial information about boundary conditions or to incompleteness of the theory.</li></ul> <p>Determination of the likelihood of possible measurement outcomes proceed by: </p> <ul><li>establishing a probability distribution over all physical situations determined by boundary conditions consistent with the setup information</li> <li>establishing a probability distribution of measurement outcomes for each possible physical situation</li> <li><a href="/wiki/Convolution" title="Convolution">convolving</a> these two probability distributions to get a distribution of possible measurement outcomes consistent with the setup information</li></ul> <p>These assumptions have enough validity across a wide range of energy scales and experimental conditions to allow these theories to make accurate predictions about almost all of the phenomena encountered in daily life: light, heat, and electricity, eclipses, spaceflight, etc. They fail only at the smallest and largest scales due to omissions in the theories themselves, and when the mathematical techniques themselves break down, most notably in the case of <a href="/wiki/Turbulence" title="Turbulence">turbulence</a> and other <a href="/wiki/Chaos_(physics)" class="mw-redirect" title="Chaos (physics)">chaotic</a> phenomena. </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_field_theories">Quantum field theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=11" title="Edit section: Quantum field theories"><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/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></div> <p>Other than these classical continuum field theories, the most widely known gauge theories are <a href="/wiki/Quantum_field_theories" class="mw-redirect" title="Quantum field theories">quantum field theories</a>, including <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> and the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of elementary particle physics. The starting point of a quantum field theory is much like that of its continuum analog: a gauge-covariant <a href="/wiki/Action_integral" class="mw-redirect" title="Action integral">action integral</a> that characterizes "allowable" physical situations according to the <a href="/wiki/Principle_of_least_action" class="mw-redirect" title="Principle of least action">principle of least action</a>. However, continuum and quantum theories differ significantly in how they handle the excess degrees of freedom represented by gauge transformations. Continuum theories, and most pedagogical treatments of the simplest quantum field theories, use a <a href="/wiki/Gauge_fixing" title="Gauge fixing">gauge fixing</a> prescription to reduce the orbit of mathematical configurations that represent a given physical situation to a smaller orbit related by a smaller gauge group (the global symmetry group, or perhaps even the trivial group). </p><p>More sophisticated quantum field theories, in particular those that involve a non-abelian gauge group, break the gauge symmetry within the techniques of <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a> by introducing additional fields (the <a href="/wiki/Faddeev%E2%80%93Popov_ghost" title="Faddeev–Popov ghost">Faddeev–Popov ghosts</a>) and counterterms motivated by <a href="/wiki/Anomaly_cancellation" class="mw-redirect" title="Anomaly cancellation">anomaly cancellation</a>, in an approach known as <a href="/wiki/BRST_quantization" title="BRST quantization">BRST quantization</a>. While these concerns are in one sense highly technical, they are also closely related to the nature of measurement, the limits on knowledge of a physical situation, and the interactions between incompletely specified experimental conditions and incompletely understood physical theory.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2014)">citation needed</span></a></i>]</sup> The mathematical techniques that have been developed in order to make gauge theories tractable have found many other applications, from <a href="/wiki/Solid-state_physics" title="Solid-state physics">solid-state physics</a> and <a href="/wiki/Crystallography" title="Crystallography">crystallography</a> to <a href="/wiki/Low-dimensional_topology" title="Low-dimensional topology">low-dimensional topology</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Classical_gauge_theory">Classical gauge theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=12" title="Edit section: Classical gauge theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Classical_electromagnetism">Classical electromagnetism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=13" title="Edit section: Classical electromagnetism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Electrostatics" title="Electrostatics">electrostatics</a>, one can either discuss the electric field, <b>E</b>, or its corresponding <a href="/wiki/Electric_potential" title="Electric potential">electric potential</a>, <i>V</i>. Knowledge of one makes it possible to find the other, except that potentials differing by a constant, <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 V\mapsto V+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">↦</mo> <mi>V</mi> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\mapsto V+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5abcc8f3067a4e7644038e6c93f4f20e4e31b84e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.795ex; height:2.343ex;" alt="{\displaystyle V\mapsto V+C}"></span>, correspond to the same electric field. This is because the electric field relates to <i>changes</i> in the potential from one point in space to another, and the constant <i>C</i> would cancel out when subtracting to find the change in potential. In terms of <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a>, the electric field is the <a href="/wiki/Gradient" title="Gradient">gradient</a> of the potential, <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 \mathbf {E} =-\nabla V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =-\nabla V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13832fe67e844ebe162fa25374a8fe16ea994a5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.387ex; height:2.343ex;" alt="{\displaystyle \mathbf {E} =-\nabla V}"></span>. Generalizing from static electricity to electromagnetism, we have a second potential, the <a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">vector potential</a> <b>A</b>, with </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 {\begin{aligned}\mathbf {E} &=-\nabla V-{\frac {\partial \mathbf {A} }{\partial t}}\\\mathbf {B} &=\nabla \times \mathbf {A} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>V</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {E} &=-\nabla V-{\frac {\partial \mathbf {A} }{\partial t}}\\\mathbf {B} &=\nabla \times \mathbf {A} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab11c7413efe0415f10552efb067a95de8b986ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:18.296ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {E} &=-\nabla V-{\frac {\partial \mathbf {A} }{\partial t}}\\\mathbf {B} &=\nabla \times \mathbf {A} \end{aligned}}}"></span></dd></dl> <p>The general gauge transformations now become not just <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 V\mapsto V+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">↦</mo> <mi>V</mi> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\mapsto V+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5abcc8f3067a4e7644038e6c93f4f20e4e31b84e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.795ex; height:2.343ex;" alt="{\displaystyle V\mapsto V+C}"></span> but </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 {\begin{aligned}\mathbf {A} &\mapsto \mathbf {A} +\nabla f\\V&\mapsto V-{\frac {\partial f}{\partial t}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mi>V</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {A} &\mapsto \mathbf {A} +\nabla f\\V&\mapsto V-{\frac {\partial f}{\partial t}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5a179ab387402c410422e33f50bbd6958f100f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:14.459ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {A} &\mapsto \mathbf {A} +\nabla f\\V&\mapsto V-{\frac {\partial f}{\partial t}}\end{aligned}}}"></span></dd></dl> <p>where <i>f</i> is any twice continuously differentiable function that depends on position and time. The electromagnetic fields remain the same under the gauge transformation. </p> <div class="mw-heading mw-heading3"><h3 id="An_example:_Scalar_O(n)_gauge_theory"><span id="An_example:_Scalar_O.28n.29_gauge_theory"></span>An example: Scalar O(<i>n</i>) gauge theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=14" title="Edit section: An example: Scalar O(n) gauge theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><i>The remainder of this section requires some familiarity with <a href="/wiki/Classical_field_theory" title="Classical field theory">classical</a> or <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, and the use of <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangians</a>.</i></dd></dl> <dl><dd><i>Definitions in this section: <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a>, <a href="/wiki/Gauge_field" class="mw-redirect" title="Gauge field">gauge field</a>, <a href="/wiki/Interaction_Lagrangian" class="mw-redirect" title="Interaction Lagrangian">interaction Lagrangian</a>, <a href="/wiki/Gauge_boson" title="Gauge boson">gauge boson</a>.</i></dd></dl> <p>The following illustrates how local gauge invariance can be "motivated" heuristically starting from global symmetry properties, and how it leads to an interaction between originally non-interacting fields. </p><p>Consider a set of <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> non-interacting real <a href="/wiki/Field_(physics)" title="Field (physics)">scalar fields</a>, with equal masses <i>m</i>. This system is described by an <a href="/wiki/Action_(physics)" title="Action (physics)">action</a> that is the sum of the (usual) action for each scalar 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 \varphi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70503774fb21be77396899900d3aa1e47d8f9e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.32ex; height:2.176ex;" alt="{\displaystyle \varphi _{i}}"></span> </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 {\mathcal {S}}=\int \,\mathrm {d} ^{4}x\sum _{i=1}^{n}\left[{\frac {1}{2}}\partial _{\mu }\varphi _{i}\partial ^{\mu }\varphi _{i}-{\frac {1}{2}}m^{2}\varphi _{i}^{2}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}=\int \,\mathrm {d} ^{4}x\sum _{i=1}^{n}\left[{\frac {1}{2}}\partial _{\mu }\varphi _{i}\partial ^{\mu }\varphi _{i}-{\frac {1}{2}}m^{2}\varphi _{i}^{2}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/033a0351b00a8557fb8e58e0accd37da31a35c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.99ex; height:6.843ex;" alt="{\displaystyle {\mathcal {S}}=\int \,\mathrm {d} ^{4}x\sum _{i=1}^{n}\left[{\frac {1}{2}}\partial _{\mu }\varphi _{i}\partial ^{\mu }\varphi _{i}-{\frac {1}{2}}m^{2}\varphi _{i}^{2}\right]}"></span></dd></dl> <p>The Lagrangian (density) can be compactly written as </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 \ {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\Phi )^{\mathsf {T}}\partial ^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{\mathsf {T}}\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\Phi )^{\mathsf {T}}\partial ^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{\mathsf {T}}\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dca76b497b6504fc12de7829297ad17adc37fc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.464ex; height:5.176ex;" alt="{\displaystyle \ {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\Phi )^{\mathsf {T}}\partial ^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{\mathsf {T}}\Phi }"></span></dd></dl> <p>by introducing a <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</a> of fields </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 \ \Phi ^{\mathsf {T}}=(\varphi _{1},\varphi _{2},\ldots ,\varphi _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Phi ^{\mathsf {T}}=(\varphi _{1},\varphi _{2},\ldots ,\varphi _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/629961d764c164e372d60b83af4783b07e236d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.617ex; height:3.176ex;" alt="{\displaystyle \ \Phi ^{\mathsf {T}}=(\varphi _{1},\varphi _{2},\ldots ,\varphi _{n})}"></span></dd></dl> <p>The term <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 \partial _{\mu }\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80360c30e7b534c1fa6b933fcd8220d130429ae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.136ex; height:2.843ex;" alt="{\displaystyle \partial _{\mu }\Phi }"></span> is the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a> of <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span> along dimension <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span>. </p><p>It is now transparent that the Lagrangian is invariant under the transformation </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 \ \Phi \mapsto \Phi '=G\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">↦</mo> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>G</mi> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Phi \mapsto \Phi '=G\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed44fa0511c6bb3017990abb29beeaa5b6477e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.839ex; height:2.509ex;" alt="{\displaystyle \ \Phi \mapsto \Phi '=G\Phi }"></span></dd></dl> <p>whenever <i>G</i> is a <i>constant</i> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> belonging to the <i>n</i>-by-<i>n</i> <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> O(<i>n</i>). This is seen to preserve the Lagrangian, since the derivative of <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 '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb68da4ee8f61f705569b558e842156ea7548bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.363ex; height:2.509ex;" alt="{\displaystyle \Phi '}"></span> transforms identically to <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span> and both quantities appear inside <a href="/wiki/Dot_product" title="Dot product">dot products</a> in the Lagrangian (orthogonal transformations preserve the dot product). </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 \ (\partial _{\mu }\Phi )\mapsto (\partial _{\mu }\Phi )'=G\partial _{\mu }\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mi>G</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (\partial _{\mu }\Phi )\mapsto (\partial _{\mu }\Phi )'=G\partial _{\mu }\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0418851c4a7ad41c07a37dd29c55870aaedc18d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.831ex; height:3.176ex;" alt="{\displaystyle \ (\partial _{\mu }\Phi )\mapsto (\partial _{\mu }\Phi )'=G\partial _{\mu }\Phi }"></span></dd></dl> <p>This characterizes the <i>global</i> symmetry of this particular Lagrangian, and the symmetry group is often called the <b>gauge group</b>; the mathematical term is <b><a href="/wiki/Structure_group" class="mw-redirect" title="Structure group">structure group</a></b>, especially in the theory of <a href="/wiki/G-structure" class="mw-redirect" title="G-structure">G-structures</a>. Incidentally, <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> implies that invariance under this group of transformations leads to the conservation of the <i>currents</i> </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 \ J_{\mu }^{a}=i\partial _{\mu }\Phi ^{\mathsf {T}}T^{a}\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <mi>i</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ J_{\mu }^{a}=i\partial _{\mu }\Phi ^{\mathsf {T}}T^{a}\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06c6762d317f6eb261e369bb949eb729767a8557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.097ex; height:3.343ex;" alt="{\displaystyle \ J_{\mu }^{a}=i\partial _{\mu }\Phi ^{\mathsf {T}}T^{a}\Phi }"></span></dd></dl> <p>where the <i>T<sup>a</sup></i> matrices are <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generators</a> of the SO(<i>n</i>) group. There is one conserved current for every generator. </p><p>Now, demanding that this Lagrangian should have <i>local</i> O(<i>n</i>)-invariance requires that the <i>G</i> matrices (which were earlier constant) should be allowed to become functions of the <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> <a href="/wiki/Coordinate" class="mw-redirect" title="Coordinate">coordinates</a> <i>x</i>. </p><p>In this case, the <i>G</i> matrices do not "pass through" the derivatives, when <i>G</i> = <i>G</i>(<i>x</i>), </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 \ \partial _{\mu }(G\Phi )\neq G(\partial _{\mu }\Phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>G</mi> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \partial _{\mu }(G\Phi )\neq G(\partial _{\mu }\Phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc41f8e56e14947c9bfa57a79e6bd74951f1ff66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.223ex; height:3.009ex;" alt="{\displaystyle \ \partial _{\mu }(G\Phi )\neq G(\partial _{\mu }\Phi )}"></span></dd></dl> <p>The failure of the derivative to commute with "G" introduces an additional term (in keeping with the product rule), which spoils the invariance of the Lagrangian. In order to rectify this we define a new derivative operator such that the derivative of <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 '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb68da4ee8f61f705569b558e842156ea7548bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.363ex; height:2.509ex;" alt="{\displaystyle \Phi '}"></span> again transforms identically 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 \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span> </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 \ (D_{\mu }\Phi )'=GD_{\mu }\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mi>G</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (D_{\mu }\Phi )'=GD_{\mu }\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e61550e12399c0446761f746849fc6d5de8e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.651ex; height:3.176ex;" alt="{\displaystyle \ (D_{\mu }\Phi )'=GD_{\mu }\Phi }"></span></dd></dl> <p>This new "derivative" is called a <a href="/wiki/Gauge_covariant_derivative" title="Gauge covariant derivative">(gauge) covariant derivative</a> and takes the form </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 \ D_{\mu }=\partial _{\mu }-igA_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <mi>g</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ D_{\mu }=\partial _{\mu }-igA_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a064b3fe5ad1ae10b76deb516ca8f6d58133b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.01ex; height:2.843ex;" alt="{\displaystyle \ D_{\mu }=\partial _{\mu }-igA_{\mu }}"></span></dd></dl> <p>where <i>g</i> is called the coupling constant; a quantity defining the strength of an interaction. After a simple calculation we can see that the <b>gauge field</b> <i>A</i>(<i>x</i>) must transform as follows </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 \ A'_{\mu }=GA_{\mu }G^{-1}-{\frac {i}{g}}(\partial _{\mu }G)G^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi>G</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>g</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>G</mi> <mo stretchy="false">)</mo> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ A'_{\mu }=GA_{\mu }G^{-1}-{\frac {i}{g}}(\partial _{\mu }G)G^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/510661d720ac44d455777334bad5f9575f3fea0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.644ex; height:5.676ex;" alt="{\displaystyle \ A'_{\mu }=GA_{\mu }G^{-1}-{\frac {i}{g}}(\partial _{\mu }G)G^{-1}}"></span></dd></dl> <p>The gauge field is an element of the Lie algebra, and can therefore be expanded as </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 \ A_{\mu }=\sum _{a}A_{\mu }^{a}T^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ A_{\mu }=\sum _{a}A_{\mu }^{a}T^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8ab862768fb418f127c25f8450b9e3b14e86da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.176ex; height:5.509ex;" alt="{\displaystyle \ A_{\mu }=\sum _{a}A_{\mu }^{a}T^{a}}"></span></dd></dl> <p>There are therefore as many gauge fields as there are generators of the Lie algebra. </p><p>Finally, we now have a <i>locally gauge invariant</i> Lagrangian </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 \ {\mathcal {L}}_{\mathrm {loc} }={\frac {1}{2}}(D_{\mu }\Phi )^{\mathsf {T}}D^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{\mathsf {T}}\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {L}}_{\mathrm {loc} }={\frac {1}{2}}(D_{\mu }\Phi )^{\mathsf {T}}D^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{\mathsf {T}}\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456c0cecb94bf9b7c1cfb4590bf7da517a45baba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.977ex; height:5.176ex;" alt="{\displaystyle \ {\mathcal {L}}_{\mathrm {loc} }={\frac {1}{2}}(D_{\mu }\Phi )^{\mathsf {T}}D^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{\mathsf {T}}\Phi }"></span></dd></dl> <p>Pauli uses the term <i>gauge transformation of the first type</i> to mean the transformation of <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span>, while the compensating transformation in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is called a <i>gauge transformation of the second type</i>. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Feynman-Diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Feynman-Diagram.svg/200px-Feynman-Diagram.svg.png" decoding="async" width="200" height="108" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Feynman-Diagram.svg/300px-Feynman-Diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Feynman-Diagram.svg/400px-Feynman-Diagram.svg.png 2x" data-file-width="990" data-file-height="533" /></a><figcaption><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a> of scalar bosons interacting via a gauge boson</figcaption></figure> <p>The difference between this Lagrangian and the original <i>globally gauge-invariant</i> Lagrangian is seen to be the <b>interaction Lagrangian</b> </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 \ {\mathcal {L}}_{\mathrm {int} }=i{\frac {g}{2}}\Phi ^{\mathsf {T}}A_{\mu }^{\mathsf {T}}\partial ^{\mu }\Phi +i{\frac {g}{2}}(\partial _{\mu }\Phi )^{\mathsf {T}}A^{\mu }\Phi -{\frac {g^{2}}{2}}(A_{\mu }\Phi )^{\mathsf {T}}A^{\mu }\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {L}}_{\mathrm {int} }=i{\frac {g}{2}}\Phi ^{\mathsf {T}}A_{\mu }^{\mathsf {T}}\partial ^{\mu }\Phi +i{\frac {g}{2}}(\partial _{\mu }\Phi )^{\mathsf {T}}A^{\mu }\Phi -{\frac {g^{2}}{2}}(A_{\mu }\Phi )^{\mathsf {T}}A^{\mu }\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd6d1b35f14c1d498c7c870c21ffdbcbbd91ddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.578ex; height:5.676ex;" alt="{\displaystyle \ {\mathcal {L}}_{\mathrm {int} }=i{\frac {g}{2}}\Phi ^{\mathsf {T}}A_{\mu }^{\mathsf {T}}\partial ^{\mu }\Phi +i{\frac {g}{2}}(\partial _{\mu }\Phi )^{\mathsf {T}}A^{\mu }\Phi -{\frac {g^{2}}{2}}(A_{\mu }\Phi )^{\mathsf {T}}A^{\mu }\Phi }"></span></dd></dl> <p>This term introduces <a href="/wiki/Fundamental_interaction" title="Fundamental interaction">interactions</a> between the <i>n</i> scalar fields just as a consequence of the demand for local gauge invariance. However, to make this interaction physical and not completely arbitrary, the mediator <i>A</i>(<i>x</i>) needs to propagate in space. That is dealt with in the next section by adding yet another term, <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 {\mathcal {L}}_{\mathrm {gf} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\mathrm {gf} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de2238354dc2e1d8ebd4adf186510607f5d80a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.27ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}_{\mathrm {gf} }}"></span>, to the Lagrangian. In the <a href="/wiki/Quantization_(physics)" title="Quantization (physics)">quantized</a> version of the obtained <a href="/wiki/Classical_field_theory" title="Classical field theory">classical field theory</a>, the <a href="/wiki/Quantum" title="Quantum">quanta</a> of the gauge field <i>A</i>(<i>x</i>) are called <a href="/wiki/Gauge_boson" title="Gauge boson">gauge bosons</a>. The interpretation of the interaction Lagrangian in quantum field theory is of <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a> <a href="/wiki/Boson" title="Boson">bosons</a> interacting by the exchange of these gauge bosons. </p> <div class="mw-heading mw-heading3"><h3 id="The_Yang–Mills_Lagrangian_for_the_gauge_field"><span id="The_Yang.E2.80.93Mills_Lagrangian_for_the_gauge_field"></span>The Yang–Mills Lagrangian for the gauge field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=15" title="Edit section: The Yang–Mills Lagrangian for the gauge field"><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/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a></div> <p>The picture of a classical gauge theory developed in the previous section is almost complete, except for the fact that to define the covariant derivatives <i>D</i>, one needs to know the value of the gauge 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 A(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b078651e6d1a522e8955b73059fbd63e13aec616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle A(x)}"></span> at all spacetime points. Instead of manually specifying the values of this field, it can be given as the solution to a field equation. Further requiring that the Lagrangian that generates this field equation is locally gauge invariant as well, one possible form for the gauge field Lagrangian is </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 {\mathcal {L}}_{\text{gf}}=-{\frac {1}{2}}\operatorname {tr} \left(F^{\mu \nu }F_{\mu \nu }\right)=-{\frac {1}{4}}F^{a\mu \nu }F_{\mu \nu }^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>gf</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\text{gf}}=-{\frac {1}{2}}\operatorname {tr} \left(F^{\mu \nu }F_{\mu \nu }\right)=-{\frac {1}{4}}F^{a\mu \nu }F_{\mu \nu }^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/915a261d55a280167cc902c25229aebf0d07f538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.851ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}_{\text{gf}}=-{\frac {1}{2}}\operatorname {tr} \left(F^{\mu \nu }F_{\mu \nu }\right)=-{\frac {1}{4}}F^{a\mu \nu }F_{\mu \nu }^{a}}"></span></dd></dl> <p>where the <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 F_{\mu \nu }^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9dc92af1933ddc1974715c5c8b26afacfa8c39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.589ex; height:2.843ex;" alt="{\displaystyle F_{\mu \nu }^{a}}"></span> are obtained from potentials <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_{\mu }^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b893e4dc54200319472f0fa9835d0cffc91fea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.967ex; height:2.843ex;" alt="{\displaystyle A_{\mu }^{a}}"></span>, being the components of <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b078651e6d1a522e8955b73059fbd63e13aec616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle A(x)}"></span>, by </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 F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+g\sum _{b,c}f^{abc}A_{\mu }^{b}A_{\nu }^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>+</mo> <mi>g</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mrow> </munder> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+g\sum _{b,c}f^{abc}A_{\mu }^{b}A_{\nu }^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2b14e089d687fc74eeaf11241cd0dcf2c390ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:37.875ex; height:5.843ex;" alt="{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+g\sum _{b,c}f^{abc}A_{\mu }^{b}A_{\nu }^{c}}"></span></dd></dl> <p>and the <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 f^{abc}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{abc}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a85bb95b3c37f4338f39fdf6f8c8d9be34d0e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.84ex; height:3.009ex;" alt="{\displaystyle f^{abc}}"></span> are the <a href="/wiki/Structure_constants" title="Structure constants">structure constants</a> of the Lie algebra of the generators of the gauge group. This formulation of the Lagrangian is called a <b>Yang–Mills action</b>. Other gauge invariant actions also exist (e.g., <a href="/wiki/Nonlinear_electrodynamics" class="mw-redirect" title="Nonlinear electrodynamics">nonlinear electrodynamics</a>, <a href="/wiki/Born%E2%80%93Infeld_action" class="mw-redirect" title="Born–Infeld action">Born–Infeld action</a>, <a href="/wiki/Chern%E2%80%93Simons_model" class="mw-redirect" title="Chern–Simons model">Chern–Simons model</a>, <a href="/wiki/Strong_CP_problem" title="Strong CP problem">theta term</a>, etc.). </p><p>In this Lagrangian term there is no field whose transformation counterweighs the one of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. Invariance of this term under gauge transformations is a particular case of <i>a priori</i> classical (geometrical) symmetry. This symmetry must be restricted in order to perform quantization, the procedure being denominated <a href="/wiki/Gauge_fixing" title="Gauge fixing">gauge fixing</a>, but even after restriction, gauge transformations may be possible.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The complete Lagrangian for the gauge theory is now </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 {\mathcal {L}}={\mathcal {L}}_{\text{loc}}+{\mathcal {L}}_{\text{gf}}={\mathcal {L}}_{\text{global}}+{\mathcal {L}}_{\text{int}}+{\mathcal {L}}_{\text{gf}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>loc</mtext> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>gf</mtext> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>global</mtext> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>int</mtext> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>gf</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\mathcal {L}}_{\text{loc}}+{\mathcal {L}}_{\text{gf}}={\mathcal {L}}_{\text{global}}+{\mathcal {L}}_{\text{int}}+{\mathcal {L}}_{\text{gf}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc36f48a2cd0ed879d5e6561b7c6b253c44055d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.468ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}={\mathcal {L}}_{\text{loc}}+{\mathcal {L}}_{\text{gf}}={\mathcal {L}}_{\text{global}}+{\mathcal {L}}_{\text{int}}+{\mathcal {L}}_{\text{gf}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="An_example:_Electrodynamics">An example: Electrodynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=16" title="Edit section: An example: Electrodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a simple application of the formalism developed in the previous sections, consider the case of <a href="/wiki/Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>, with only the <a href="/wiki/Electron" title="Electron">electron</a> field. The bare-bones action that generates the electron field's <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> is </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 {\mathcal {S}}=\int {\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }\partial _{\mu }-mc^{2}\right)\psi \,\mathrm {d} ^{4}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}=\int {\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }\partial _{\mu }-mc^{2}\right)\psi \,\mathrm {d} ^{4}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b3dcacf3f0c17f582b62230dca1bc356f570ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.654ex; height:5.676ex;" alt="{\displaystyle {\mathcal {S}}=\int {\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }\partial _{\mu }-mc^{2}\right)\psi \,\mathrm {d} ^{4}x}"></span></dd></dl> <p>The global symmetry for this system is </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 \psi \mapsto e^{i\theta }\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \mapsto e^{i\theta }\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc7e0d48bfc718f6751134bc2662df23389e50f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.295ex; height:3.009ex;" alt="{\displaystyle \psi \mapsto e^{i\theta }\psi }"></span></dd></dl> <p>The gauge group here is <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a>, just rotations of the <a href="/wiki/Complex_number" title="Complex number">phase angle</a> of the field, with the particular rotation determined by the constant <span class="texhtml"><i>θ</i></span>. </p><p>"Localising" this symmetry implies the replacement of <span class="texhtml"><i>θ</i></span> by <span class="texhtml"><i>θ</i>(<i>x</i>)</span>. An appropriate covariant derivative is then </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 D_{\mu }=\partial _{\mu }-i{\frac {e}{\hbar }}A_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }=\partial _{\mu }-i{\frac {e}{\hbar }}A_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d13f8d088e7662c7dd8af40db353cec37fc181b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.456ex; height:4.843ex;" alt="{\displaystyle D_{\mu }=\partial _{\mu }-i{\frac {e}{\hbar }}A_{\mu }}"></span></dd></dl> <p>Identifying the "charge" <span class="texhtml"><i>e</i></span> (not to be confused with the mathematical constant <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a> in the symmetry description) with the usual <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> (this is the origin of the usage of the term in gauge theories), and the gauge field <span class="texhtml"><i>A</i>(<i>x</i>)</span> with the four-<a href="/wiki/Vector_potential" title="Vector potential">vector potential</a> of the <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> results in an interaction Lagrangian </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 {\mathcal {L}}_{\text{int}}={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)A_{\mu }(x)=J^{\mu }(x)A_{\mu }(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>int</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\text{int}}={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)A_{\mu }(x)=J^{\mu }(x)A_{\mu }(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d91d5442d6651c9140343750d591294879ac98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:42.176ex; height:4.843ex;" alt="{\displaystyle {\mathcal {L}}_{\text{int}}={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)A_{\mu }(x)=J^{\mu }(x)A_{\mu }(x)}"></span></dd></dl> <p>where <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 J^{\mu }(x)={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{\mu }(x)={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27fc0c3a84ffe66bc11f7b886ea1893cb4cf2c9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.019ex; height:4.843ex;" alt="{\displaystyle J^{\mu }(x)={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)}"></span> is the electric current <a href="/wiki/Four_vector" class="mw-redirect" title="Four vector">four vector</a> in the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac field</a>. The <a href="/wiki/Gauge_principle" title="Gauge principle">gauge principle</a> is therefore seen to naturally introduce the so-called <a href="/wiki/Minimal_coupling" title="Minimal coupling">minimal coupling</a> of the electromagnetic field to the electron field. </p><p>Adding a Lagrangian for the gauge 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 A_{\mu }(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f0dd025b0d02e673f795c8356b5205af4eeac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.105ex; height:3.009ex;" alt="{\displaystyle A_{\mu }(x)}"></span> in terms of the <a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">field strength tensor</a> exactly as in electrodynamics, one obtains the Lagrangian used as the starting point in <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>. </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 {\mathcal {L}}_{\text{QED}}={\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }D_{\mu }-mc^{2}\right)\psi -{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>QED</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\text{QED}}={\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }D_{\mu }-mc^{2}\right)\psi -{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/218bdf908ab080d8edf0f82c2090aae3b53e8b00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.49ex; height:5.676ex;" alt="{\displaystyle {\mathcal {L}}_{\text{QED}}={\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }D_{\mu }-mc^{2}\right)\psi -{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }}"></span></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>, <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, and <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></div> <div class="mw-heading mw-heading2"><h2 id="Mathematical_formalism">Mathematical formalism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=17" title="Edit section: Mathematical formalism"><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">See also: <a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory (mathematics)</a></div> <p>Gauge theories are usually discussed in the language of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>. Mathematically, a <i>gauge</i> is just a choice of a (local) <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> of some <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a>. A <b>gauge transformation</b> is just a transformation between two such sections. </p><p>Although gauge theory is dominated by the study of <a href="/wiki/Connection_form" title="Connection form">connections</a> (primarily because it's mainly studied by <a href="/wiki/High-energy_physics" class="mw-redirect" title="High-energy physics">high-energy physicists</a>), the idea of a connection is not central to gauge theory in general. In fact, a result in general gauge theory shows that <a href="/wiki/Affine_representation" title="Affine representation">affine representations</a> (i.e., affine <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a>) of the gauge transformations can be classified as sections of a <a href="/wiki/Jet_bundle" title="Jet bundle">jet bundle</a> satisfying certain properties. There are representations that transform covariantly pointwise (called by physicists gauge transformations of the first kind), representations that transform as a <a href="/wiki/Connection_form" title="Connection form">connection form</a> (called by physicists gauge transformations of the second kind, an affine representation)—and other more general representations, such as the B field in <a href="/wiki/BF_theory" class="mw-redirect" title="BF theory">BF theory</a>. There are more general <a href="/wiki/Nonlinear_realization" title="Nonlinear realization">nonlinear representations</a> (realizations), but these are extremely complicated. Still, <a href="/wiki/Nonlinear_sigma_model" class="mw-redirect" title="Nonlinear sigma model">nonlinear sigma models</a> transform nonlinearly, so there are applications. </p><p>If there is a <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> <i>P</i> whose <a href="/wiki/Fibre_bundle" class="mw-redirect" title="Fibre bundle">base space</a> is <a href="/wiki/Space" title="Space">space</a> or <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> and <a href="/wiki/Fibre_bundle" class="mw-redirect" title="Fibre bundle">structure group</a> is a Lie group, then the sections of <i>P</i> form a <a href="/wiki/Principal_homogeneous_space" title="Principal homogeneous space">principal homogeneous space</a> of the group of gauge transformations. </p><p><a href="/wiki/Connection_form" title="Connection form">Connections</a> (gauge connection) define this principal bundle, yielding a <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a> ∇ in each <a href="/wiki/Associated_vector_bundle" class="mw-redirect" title="Associated vector bundle">associated vector bundle</a>. If a local frame is chosen (a local basis of sections), then this covariant derivative is represented by the <a href="/wiki/Connection_form" title="Connection form">connection form</a> <i>A</i>, a Lie algebra-valued <a href="/wiki/Differential_form" title="Differential form">1-form</a>, which is called the <b>gauge potential</b> in <a href="/wiki/Physics" title="Physics">physics</a>. This is evidently not an intrinsic but a frame-dependent quantity. The <a href="/wiki/Curvature_form" title="Curvature form">curvature form</a> <i>F</i>, a Lie algebra-valued <a href="/wiki/Differential_form" title="Differential form">2-form</a> that is an intrinsic quantity, is constructed from a connection form by </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 \mathbf {F} =\mathrm {d} \mathbf {A} +\mathbf {A} \wedge \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\mathrm {d} \mathbf {A} +\mathbf {A} \wedge \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf447df86b854d991ceebeb22af476bd5474d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.555ex; height:2.343ex;" alt="{\displaystyle \mathbf {F} =\mathrm {d} \mathbf {A} +\mathbf {A} \wedge \mathbf {A} }"></span></dd></dl> <p>where d stands for the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> 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 \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> stands for the <a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">wedge product</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 \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> is an element of the vector space spanned by the generators <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 T^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8b02d8588d09a61be646fe251831701bbd4986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.822ex; height:2.343ex;" alt="{\displaystyle T^{a}}"></span>, and so the components of <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 \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> do not commute with one another. Hence the wedge product <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 \mathbf {A} \wedge \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \wedge \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412f6fa5b2d80de6c90c807cf4341458f464b724" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.622ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} \wedge \mathbf {A} }"></span> does not vanish.) </p><p>Infinitesimal gauge transformations form a Lie algebra, which is characterized by a smooth Lie-algebra-valued <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>, ε. Under such an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> gauge transformation, </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 \delta _{\varepsilon }\mathbf {A} =[\varepsilon ,\mathbf {A} ]-\mathrm {d} \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>ε</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\varepsilon }\mathbf {A} =[\varepsilon ,\mathbf {A} ]-\mathrm {d} \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d45a28a2639568e520c6af6cb2be657169e8c53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.796ex; height:2.843ex;" alt="{\displaystyle \delta _{\varepsilon }\mathbf {A} =[\varepsilon ,\mathbf {A} ]-\mathrm {d} \varepsilon }"></span></dd></dl> <p>where <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 [\cdot ,\cdot ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\cdot ,\cdot ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28dd4c22d60192519c1c12cf645b040f368db9e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.621ex; height:2.843ex;" alt="{\displaystyle [\cdot ,\cdot ]}"></span> is the Lie bracket. </p><p>One nice thing is that if <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 _{\varepsilon }X=\varepsilon X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mi>X</mi> <mo>=</mo> <mi>ε</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\varepsilon }X=\varepsilon X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/328c14255618fd56b560646d0c868f11cc49f8f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.173ex; height:2.676ex;" alt="{\displaystyle \delta _{\varepsilon }X=\varepsilon X}"></span>, then <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 _{\varepsilon }DX=\varepsilon DX}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mi>D</mi> <mi>X</mi> <mo>=</mo> <mi>ε</mi> <mi>D</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\varepsilon }DX=\varepsilon DX}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b069e18d38e2babe4dfb0c63c76d7e00376523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.021ex; height:2.676ex;" alt="{\displaystyle \delta _{\varepsilon }DX=\varepsilon DX}"></span> where D is the covariant derivative </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 DX\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} X+\mathbf {A} X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>X</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>X</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DX\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} X+\mathbf {A} X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/665fadc0d17b01f00120ca633d8a791bf637a0ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.434ex; height:3.509ex;" alt="{\displaystyle DX\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} X+\mathbf {A} X}"></span></dd></dl> <p>Also, <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 _{\varepsilon }\mathbf {F} =[\varepsilon ,\mathbf {F} ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>ε</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\varepsilon }\mathbf {F} =[\varepsilon ,\mathbf {F} ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f67f59a7bd61d25c5b93b659105798c02956f734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.906ex; height:2.843ex;" alt="{\displaystyle \delta _{\varepsilon }\mathbf {F} =[\varepsilon ,\mathbf {F} ]}"></span>, which means <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 \mathbf {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da18bef8c979f3548bb0d8976f5844012d7b8256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }"></span> transforms covariantly. </p><p>Not all gauge transformations can be generated by <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> gauge transformations in general. An example is when the <a href="/wiki/Fibre_bundle" class="mw-redirect" title="Fibre bundle">base manifold</a> is a <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Manifold" title="Manifold">manifold</a> without <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> such that the <a href="/wiki/Homotopy" title="Homotopy">homotopy</a> class of mappings from that <a href="/wiki/Manifold" title="Manifold">manifold</a> to the Lie group is nontrivial. See <a href="/wiki/Instanton" title="Instanton">instanton</a> for an example. </p><p>The <i>Yang–Mills action</i> is now given by </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 {\frac {1}{4g^{2}}}\int \operatorname {Tr} [*F\wedge F]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>∫</mo> <mi>Tr</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo>∗</mo> <mi>F</mi> <mo>∧</mo> <mi>F</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4g^{2}}}\int \operatorname {Tr} [*F\wedge F]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c423a8c500feae11216ef817412e48aedb5c24e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.249ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{4g^{2}}}\int \operatorname {Tr} [*F\wedge F]}"></span></dd></dl> <p>where * stands for the <a href="/wiki/Hodge_dual" class="mw-redirect" title="Hodge dual">Hodge dual</a> and the integral is defined as in <a href="/wiki/Differential_form" title="Differential form">differential geometry</a>. </p><p>A quantity which is <b>gauge-invariant</b> (i.e., <a href="/wiki/Invariant_(physics)" title="Invariant (physics)">invariant</a> under gauge transformations) is the <a href="/wiki/Wilson_loop" title="Wilson loop">Wilson loop</a>, which is defined over any closed path, γ, as follows: </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 \chi ^{(\rho )}\left({\mathcal {P}}\left\{e^{\int _{\gamma }A}\right\}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>{</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mi>A</mi> </mrow> </msup> <mo>}</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ^{(\rho )}\left({\mathcal {P}}\left\{e^{\int _{\gamma }A}\right\}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4894954286deca5bf08b9dc3ed2759d04f0f32c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.771ex; height:4.843ex;" alt="{\displaystyle \chi ^{(\rho )}\left({\mathcal {P}}\left\{e^{\int _{\gamma }A}\right\}\right)}"></span></dd></dl> <p>where χ is the <a href="/wiki/Group_representation" title="Group representation">character</a> of a complex <a href="/wiki/Group_representation" title="Group representation">representation</a> ρ 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 {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span> represents the path-ordered operator. </p><p>The formalism of gauge theory carries over to a general setting. For example, it is sufficient to ask that a <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> have a <a href="/wiki/Metric_connection" title="Metric connection">metric connection</a>; when one does so, one finds that the metric connection satisfies the Yang–Mills equations of motion. </p> <div class="mw-heading mw-heading2"><h2 id="Quantization_of_gauge_theories">Quantization of gauge theories</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=18" title="Edit section: Quantization of gauge theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gauge theories may be quantized by specialization of methods which are applicable to any <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. However, because of the subtleties imposed by the gauge constraints (see section on Mathematical formalism, above) there are many technical problems to be solved which do not arise in other field theories. At the same time, the richer structure of gauge theories allows simplification of some computations: for example <a href="/wiki/Ward_identities" class="mw-redirect" title="Ward identities">Ward identities</a> connect different <a href="/wiki/Renormalization" title="Renormalization">renormalization</a> constants. </p> <div class="mw-heading mw-heading3"><h3 id="Methods_and_aims">Methods and aims</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=19" title="Edit section: Methods and aims"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first gauge theory quantized was <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> (QED). The first methods developed for this involved gauge fixing and then applying <a href="/wiki/Canonical_quantization" title="Canonical quantization">canonical quantization</a>. The <a href="/wiki/Gupta%E2%80%93Bleuler" class="mw-redirect" title="Gupta–Bleuler">Gupta–Bleuler</a> method was also developed to handle this problem. Non-abelian gauge theories are now handled by a variety of means. Methods for quantization are covered in the article on <a href="/wiki/Quantization_(physics)" title="Quantization (physics)">quantization</a>. </p><p>The main point to quantization is to be able to compute <a href="/wiki/Probability_amplitude" title="Probability amplitude">quantum amplitudes</a> for various processes allowed by the theory. Technically, they reduce to the computations of certain <a href="/wiki/Correlation_functions" class="mw-redirect" title="Correlation functions">correlation functions</a> in the <a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">vacuum state</a>. This involves a <a href="/wiki/Renormalization" title="Renormalization">renormalization</a> of the theory. </p><p>When the <a href="/wiki/Running_coupling" class="mw-redirect" title="Running coupling">running coupling</a> of the theory is small enough, then all required quantities may be computed in <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theory</a>. Quantization schemes intended to simplify such computations (such as <a href="/wiki/Quantization_(physics)#Canonical_quantization" title="Quantization (physics)">canonical quantization</a>) may be called <b>perturbative quantization schemes</b>. At present some of these methods lead to the most precise experimental tests of gauge theories. </p><p>However, in most gauge theories, there are many interesting questions which are non-perturbative. Quantization schemes suited to these problems (such as <a href="/wiki/Lattice_gauge_theory" title="Lattice gauge theory">lattice gauge theory</a>) may be called <b>non-perturbative quantization schemes</b>. Precise computations in such schemes often require <a href="/wiki/Supercomputing" class="mw-redirect" title="Supercomputing">supercomputing</a>, and are therefore less well-developed currently than other schemes. </p> <div class="mw-heading mw-heading3"><h3 id="Anomalies">Anomalies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=20" title="Edit section: Anomalies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some of the symmetries of the classical theory are then seen not to hold in the quantum theory; a phenomenon called an <b><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">anomaly</a></b>. Among the most well known are: </p> <ul><li>The <a href="/wiki/Scale_anomaly" class="mw-redirect" title="Scale anomaly">scale anomaly</a>, which gives rise to a <i>running coupling constant</i>. In QED this gives rise to the phenomenon of the <a href="/wiki/Landau_pole" title="Landau pole">Landau pole</a>. In <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a> (QCD) this leads to <a href="/wiki/Asymptotic_freedom" title="Asymptotic freedom">asymptotic freedom</a>.</li> <li>The <a href="/wiki/Chiral_anomaly" title="Chiral anomaly">chiral anomaly</a> in either chiral or vector field theories with fermions. This has close connection with <a href="/wiki/Topology" title="Topology">topology</a> through the notion of <a href="/wiki/Instanton" title="Instanton">instantons</a>. In QCD this anomaly causes the decay of a <a href="/wiki/Pion" title="Pion">pion</a> to two <a href="/wiki/Photon" title="Photon">photons</a>.</li> <li>The <a href="/wiki/Gauge_anomaly" title="Gauge anomaly">gauge anomaly</a>, which must cancel in any consistent physical theory. In the <a href="/wiki/Electroweak_theory" class="mw-redirect" title="Electroweak theory">electroweak theory</a> this cancellation requires an equal number of <a href="/wiki/Quark" title="Quark">quarks</a> and <a href="/wiki/Lepton" title="Lepton">leptons</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Pure_gauge">Pure gauge</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=21" title="Edit section: Pure gauge"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A pure gauge is the set of field configurations obtained by a <a href="/wiki/Gauge_transformation" class="mw-redirect" title="Gauge transformation">gauge transformation</a> on the null-field configuration, i.e., a gauge transform of zero. So it is a particular "gauge orbit" in the field configuration's space. </p><p>Thus, in the abelian case, where <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_{\mu }(x)\rightarrow A'_{\mu }(x)=A_{\mu }(x)+\partial _{\mu }f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }(x)\rightarrow A'_{\mu }(x)=A_{\mu }(x)+\partial _{\mu }f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fd581ce5e601062ff0d7b8ca83c1b45cacdd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.745ex; height:3.009ex;" alt="{\displaystyle A_{\mu }(x)\rightarrow A'_{\mu }(x)=A_{\mu }(x)+\partial _{\mu }f(x)}"></span>, the pure gauge is just the set of field configurations <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'_{\mu }(x)=\partial _{\mu }f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'_{\mu }(x)=\partial _{\mu }f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65d8f1755b9c61f39a62553b93457f3ec2cf91d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.079ex; height:3.009ex;" alt="{\displaystyle A'_{\mu }(x)=\partial _{\mu }f(x)}"></span> for all <span class="texhtml"><i>f</i>(<i>x</i>)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=22" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Gauge_principle" title="Gauge principle">Gauge principle</a></li> <li><a href="/wiki/Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a></li> <li><a href="/wiki/Coulomb_gauge" class="mw-redirect" title="Coulomb gauge">Coulomb gauge</a></li> <li><a href="/wiki/Electroweak_theory" class="mw-redirect" title="Electroweak theory">Electroweak theory</a></li> <li><a href="/wiki/Gauge_covariant_derivative" title="Gauge covariant derivative">Gauge covariant derivative</a></li> <li><a href="/wiki/Gauge_fixing" title="Gauge fixing">Gauge fixing</a></li> <li><a href="/wiki/Gauge_gravitation_theory" title="Gauge gravitation theory">Gauge gravitation theory</a></li> <li><a href="/wiki/Gauge_group_(mathematics)" title="Gauge group (mathematics)">Gauge group (mathematics)</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Lorenz_gauge" class="mw-redirect" title="Lorenz gauge">Lorenz gauge</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Gluon_field" title="Gluon field">Gluon field</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">Electromagnetic four-potential</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></li> <li><a href="/wiki/Standard_Model_(mathematical_formulation)" class="mw-redirect" title="Standard Model (mathematical formulation)">Standard Model (mathematical formulation)</a></li> <li><a href="/wiki/Symmetry_breaking" title="Symmetry breaking">Symmetry breaking</a></li> <li><a href="/wiki/Symmetry_in_physics" class="mw-redirect" title="Symmetry in physics">Symmetry in physics</a></li> <li><a href="/wiki/Charge_(physics)" title="Charge (physics)">Charge (physics)</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Fock_symmetry_in_theory_of_hydrogen" class="mw-redirect" title="Fock symmetry in theory of hydrogen">Fock symmetry</a></li> <li><a href="/wiki/Ward_identities" class="mw-redirect" title="Ward identities">Ward identities</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_existence_and_mass_gap" title="Yang–Mills existence and mass gap">Yang–Mills existence and mass gap</a></li> <li><a href="/wiki/1964_PRL_symmetry_breaking_papers" title="1964 PRL symmetry breaking papers">1964 PRL symmetry breaking papers</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory (mathematics)</a></li></ul> </div> <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=Gauge_theory&action=edit&section=23" 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 mw-references-columns"><ol class="references"> <li id="cite_note-Brading-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brading_1-0">^</a></b></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="CITEREFBrading2002" class="citation journal cs1"><a href="/wiki/Katherine_Brading" title="Katherine Brading">Brading, Katherine</a> (March 2002). <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=1c6e78e69b177018a8b3013b69d8fe5fb109c1b7">"Which Symmetry? Noether, Weyl, and Conservation of Electric Charge"</a>. <i>Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics</i>. <b>33</b> (1): 3–22. <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/2002SHPMP..33....3B">2002SHPMP..33....3B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS1355-2198%2801%2900033-8">10.1016/S1355-2198(01)00033-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studies+in+History+and+Philosophy+of+Science+Part+B%3A+Studies+in+History+and+Philosophy+of+Modern+Physics&rft.atitle=Which+Symmetry%3F+Noether%2C+Weyl%2C+and+Conservation+of+Electric+Charge&rft.volume=33&rft.issue=1&rft.pages=3-22&rft.date=2002-03&rft_id=info%3Adoi%2F10.1016%2FS1355-2198%2801%2900033-8&rft_id=info%3Abibcode%2F2002SHPMP..33....3B&rft.aulast=Brading&rft.aufirst=Katherine&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fdocument%3Frepid%3Drep1%26type%3Dpdf%26doi%3D1c6e78e69b177018a8b3013b69d8fe5fb109c1b7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacksonOkun2001" class="citation journal cs1">Jackson, J. D.; Okun, L. B. (2001-09-14). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.73.663">"Historical roots of gauge invariance"</a>. <i>Reviews of Modern Physics</i>. <b>73</b> (3): 663–680. <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-ph/0012061">hep-ph/0012061</a></span>. <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%2FRevModPhys.73.663">10.1103/RevModPhys.73.663</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/0034-6861">0034-6861</a>. <q>The discovery of the symmetry under gauge transformations (1 a,b,c) of the quantum mechanical system of a charged particle interacting with electromagnetic fields is due to Fock (1926b)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Historical+roots+of+gauge+invariance&rft.volume=73&rft.issue=3&rft.pages=663-680&rft.date=2001-09-14&rft_id=info%3Aarxiv%2Fhep-ph%2F0012061&rft.issn=0034-6861&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.73.663&rft.aulast=Jackson&rft.aufirst=J.+D.&rft.au=Okun%2C+L.+B.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FRevModPhys.73.663&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-O’Raifeartaigh-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-O’Raifeartaigh_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO’RaifeartaighStraumann2000" class="citation journal cs1">O’Raifeartaigh, Lochlainn; Straumann, Norbert (2000-01-01). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.72.1">"Gauge theory: Historical origins and some modern developments"</a>. <i>Reviews of Modern Physics</i>. <b>72</b> (1): 1–23. <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%2FRevModPhys.72.1">10.1103/RevModPhys.72.1</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/0034-6861">0034-6861</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Gauge+theory%3A+Historical+origins+and+some+modern+developments&rft.volume=72&rft.issue=1&rft.pages=1-23&rft.date=2000-01-01&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.72.1&rft.issn=0034-6861&rft.aulast=O%E2%80%99Raifeartaigh&rft.aufirst=Lochlainn&rft.au=Straumann%2C+Norbert&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FRevModPhys.72.1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPauli1941" class="citation journal cs1"><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli, Wolfgang</a> (1941). <a rel="nofollow" class="external text" href="http://prola.aps.org/abstract/RMP/v13/i3/p203_1">"Relativistic Field Theories of Elementary Particles"</a>. <i>Rev. Mod. Phys</i>. <b>13</b> (3): 203–32. <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/1941RvMP...13..203P">1941RvMP...13..203P</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%2Frevmodphys.13.203">10.1103/revmodphys.13.203</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rev.+Mod.+Phys.&rft.atitle=Relativistic+Field+Theories+of+Elementary+Particles&rft.volume=13&rft.issue=3&rft.pages=203-32&rft.date=1941&rft_id=info%3Adoi%2F10.1103%2Frevmodphys.13.203&rft_id=info%3Abibcode%2F1941RvMP...13..203P&rft.aulast=Pauli&rft.aufirst=Wolfgang&rft_id=http%3A%2F%2Fprola.aps.org%2Fabstract%2FRMP%2Fv13%2Fi3%2Fp203_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-Baggott40-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Baggott40_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaggott2013" class="citation book cs1">Baggott, J. E. (2013). <i>The quantum story: a history in 40 moments</i> (Impression: 3 ed.). Oxford: Oxford Univ. Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-956684-6" title="Special:BookSources/978-0-19-956684-6"><bdi>978-0-19-956684-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+quantum+story%3A+a+history+in+40+moments&rft.place=Oxford&rft.edition=Impression%3A+3&rft.pub=Oxford+Univ.+Press&rft.date=2013&rft.isbn=978-0-19-956684-6&rft.aulast=Baggott&rft.aufirst=J.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYang_C._N.,_Mills_R._L.1954" class="citation journal cs1">Yang C. N., Mills R. L. (1954). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.96.191">"Conservation of Isotopic Spin and Isotopic Gauge Invariance"</a>. <i><a href="/wiki/Phys._Rev." class="mw-redirect" title="Phys. Rev.">Phys. Rev.</a></i> <b>96</b> (1): 191–195. <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/1954PhRv...96..191Y">1954PhRv...96..191Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.96.191">10.1103/PhysRev.96.191</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.&rft.atitle=Conservation+of+Isotopic+Spin+and+Isotopic+Gauge+Invariance&rft.volume=96&rft.issue=1&rft.pages=191-195&rft.date=1954&rft_id=info%3Adoi%2F10.1103%2FPhysRev.96.191&rft_id=info%3Abibcode%2F1954PhRv...96..191Y&rft.au=Yang+C.+N.%2C+Mills+R.+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRev.96.191&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDonaldson1983" class="citation journal cs1">Donaldson, Simon K. (1983). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1983-15090-5">"Self-dual connections and the topology of smooth 4-manifolds"</a>. <i><a href="/wiki/Bull._Amer._Math._Soc." class="mw-redirect" title="Bull. Amer. Math. Soc.">Bull. Amer. Math. Soc.</a></i> <b>8</b> (1): 81–83. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1983-15090-5">10.1090/S0273-0979-1983-15090-5</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0682827">0682827</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Self-dual+connections+and+the+topology+of+smooth+4-manifolds&rft.volume=8&rft.issue=1&rft.pages=81-83&rft.date=1983&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-1983-15090-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0682827%23id-name%3DMR&rft.aulast=Donaldson&rft.aufirst=Simon+K.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0273-0979-1983-15090-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeibergWitten1994a" class="citation cs2"><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg, N.</a>; <a href="/wiki/Edward_Witten" title="Edward Witten">Witten, E.</a> (1994a), "Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory", <i>Nuclear Physics B</i>, <b>426</b> (1): 19–52, <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-th/9407087">hep-th/9407087</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/1994NuPhB.426...19S">1994NuPhB.426...19S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0550-3213%2894%2990124-4">10.1016/0550-3213(94)90124-4</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1293681">1293681</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:14361074">14361074</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuclear+Physics+B&rft.atitle=Electric-magnetic+duality%2C+monopole+condensation%2C+and+confinement+in+N%3D2+supersymmetric+Yang-Mills+theory&rft.volume=426&rft.issue=1&rft.pages=19-52&rft.date=1994&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14361074%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1994NuPhB.426...19S&rft_id=info%3Aarxiv%2Fhep-th%2F9407087&rft_id=info%3Adoi%2F10.1016%2F0550-3213%2894%2990124-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1293681%23id-name%3DMR&rft.aulast=Seiberg&rft.aufirst=N.&rft.au=Witten%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">"Erratum", <i>Nuclear Physics B</i>, <b>430</b> (2): 485–486, 1994, <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/1994NuPhB.430..485.">1994NuPhB.430..485.</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0550-3213%2894%2900449-8">10.1016/0550-3213(94)00449-8</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1303306">1303306</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuclear+Physics+B&rft.atitle=Erratum&rft.volume=430&rft.issue=2&rft.pages=485-486&rft.date=1994&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1303306%23id-name%3DMR&rft_id=info%3Adoi%2F10.1016%2F0550-3213%2894%2900449-8&rft_id=info%3Abibcode%2F1994NuPhB.430..485.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeibergWitten1994b" class="citation cs2"><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg, N.</a>; <a href="/wiki/Edward_Witten" title="Edward Witten">Witten, E.</a> (1994b), "Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD", <i>Nuclear Physics B</i>, <b>431</b> (3): 484–550, <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-th/9408099">hep-th/9408099</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/1994NuPhB.431..484S">1994NuPhB.431..484S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0550-3213%2894%2990214-3">10.1016/0550-3213(94)90214-3</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1306869">1306869</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:17584951">17584951</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuclear+Physics+B&rft.atitle=Monopoles%2C+duality+and+chiral+symmetry+breaking+in+N%3D2+supersymmetric+QCD&rft.volume=431&rft.issue=3&rft.pages=484-550&rft.date=1994&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17584951%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1994NuPhB.431..484S&rft_id=info%3Aarxiv%2Fhep-th%2F9408099&rft_id=info%3Adoi%2F10.1016%2F0550-3213%2894%2990214-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1306869%23id-name%3DMR&rft.aulast=Seiberg&rft.aufirst=N.&rft.au=Witten%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-JacksonOkun-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-JacksonOkun_10-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacksonOkun2001" class="citation journal cs1">Jackson, JD; Okun, LB (2001). "Historical roots of gauge invariance". <i>Reviews of Modern Physics</i>. <b>73</b> (3): 663. <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-ph/0012061">hep-ph/0012061</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/2001RvMP...73..663J">2001RvMP...73..663J</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%2FRevModPhys.73.663">10.1103/RevModPhys.73.663</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:8285663">8285663</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Historical+roots+of+gauge+invariance&rft.volume=73&rft.issue=3&rft.pages=663&rft.date=2001&rft_id=info%3Aarxiv%2Fhep-ph%2F0012061&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8285663%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.73.663&rft_id=info%3Abibcode%2F2001RvMP...73..663J&rft.aulast=Jackson&rft.aufirst=JD&rft.au=Okun%2C+LB&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-Pickering-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pickering_11-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPickering1984" class="citation book cs1">Pickering, A. (1984). <i>Constructing Quarks</i>. <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-66799-5" title="Special:BookSources/0-226-66799-5"><bdi>0-226-66799-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Constructing+Quarks&rft.pub=University+of+Chicago+Press&rft.date=1984&rft.isbn=0-226-66799-5&rft.aulast=Pickering&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">J. J. Sakurai, <i>Advanced Quantum Mechanics</i>, Addison-Wesley, 1967, sect. 1–4.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauge_theory&action=edit&section=24" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>General readers</dt></dl> <ul><li>Schumm, Bruce (2004) <i><a rel="nofollow" class="external text" href="https://archive.org/details/deepdownthingsbr00schu">Deep Down Things</a></i>. Johns Hopkins University Press. Esp. chpt. 8. A serious attempt by a physicist to explain gauge theory and the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> with little formal mathematics.</li></ul> <dl><dt>Texts</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreinerMüller2000" class="citation book cs1">Greiner, Walter; Müller, Berndt (2000). <i>Gauge Theory of Weak Interactions</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-67672-4" title="Special:BookSources/3-540-67672-4"><bdi>3-540-67672-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Theory+of+Weak+Interactions&rft.pub=Springer&rft.date=2000&rft.isbn=3-540-67672-4&rft.aulast=Greiner&rft.aufirst=Walter&rft.au=M%C3%BCller%2C+Berndt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChengLi1983" class="citation book cs1">Cheng, T.-P.; <a href="/w/index.php?title=Ling-Fong_Li&action=edit&redlink=1" class="new" title="Ling-Fong Li (page does not exist)">Li, L.-F.</a> (1983). <i>Gauge Theory of Elementary Particle Physics</i>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-851961-3" title="Special:BookSources/0-19-851961-3"><bdi>0-19-851961-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Theory+of+Elementary+Particle+Physics&rft.pub=Oxford+University+Press&rft.date=1983&rft.isbn=0-19-851961-3&rft.aulast=Cheng&rft.aufirst=T.-P.&rft.au=Li%2C+L.-F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrampton2008" class="citation book cs1"><a href="/wiki/Paul_Frampton" title="Paul Frampton">Frampton, P.</a> (2008). <i>Gauge Field Theories</i> (3rd ed.). <a href="/wiki/Wiley-VCH" title="Wiley-VCH">Wiley-VCH</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Field+Theories&rft.edition=3rd&rft.pub=Wiley-VCH&rft.date=2008&rft.aulast=Frampton&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKane1987" class="citation book cs1">Kane, G.L. (1987). <i>Modern Elementary Particle Physics</i>. Perseus Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-11749-5" title="Special:BookSources/0-201-11749-5"><bdi>0-201-11749-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Elementary+Particle+Physics&rft.pub=Perseus+Books&rft.date=1987&rft.isbn=0-201-11749-5&rft.aulast=Kane&rft.aufirst=G.L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></li></ul> <dl><dt>Articles</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBecchi1997" class="citation arxiv cs1">Becchi, C. (1997). "Introduction to Gauge Theories". <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-ph/9705211">hep-ph/9705211</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Introduction+to+Gauge+Theories&rft.date=1997&rft_id=info%3Aarxiv%2Fhep-ph%2F9705211&rft.aulast=Becchi&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGross1992" class="citation web cs1"><a href="/wiki/David_Gross" title="David Gross">Gross, D.</a> (1992). <a rel="nofollow" class="external text" href="http://psroc.phys.ntu.edu.tw/cjp/download.php?type=paper&vol=30&num=7&page=955">"Gauge theory – Past, Present and Future"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2009-04-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Gauge+theory+%E2%80%93+Past%2C+Present+and+Future&rft.date=1992&rft.aulast=Gross&rft.aufirst=D.&rft_id=http%3A%2F%2Fpsroc.phys.ntu.edu.tw%2Fcjp%2Fdownload.php%3Ftype%3Dpaper%26vol%3D30%26num%3D7%26page%3D955&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson2002" class="citation journal cs1"><a href="/wiki/John_David_Jackson_(physicist)" title="John David Jackson (physicist)">Jackson, J.D.</a> (2002). "From Lorenz to Coulomb and other explicit gauge transformations". <i>Am. J. Phys</i>. <b>70</b> (9): 917–928. <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/physics/0204034">physics/0204034</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/2002AmJPh..70..917J">2002AmJPh..70..917J</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1491265">10.1119/1.1491265</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:119652556">119652556</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Am.+J.+Phys.&rft.atitle=From+Lorenz+to+Coulomb+and+other+explicit+gauge+transformations&rft.volume=70&rft.issue=9&rft.pages=917-928&rft.date=2002&rft_id=info%3Aarxiv%2Fphysics%2F0204034&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119652556%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.1491265&rft_id=info%3Abibcode%2F2002AmJPh..70..917J&rft.aulast=Jackson&rft.aufirst=J.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSvetlichny1999" class="citation arxiv cs1">Svetlichny, George (1999). "Preparation for Gauge Theory". <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/math-ph/9902027">math-ph/9902027</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Preparation+for+Gauge+Theory&rft.date=1999&rft_id=info%3Aarxiv%2Fmath-ph%2F9902027&rft.aulast=Svetlichny&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauge+theory" 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=Gauge_theory&action=edit&section=25" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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interaction</a> <ul><li><a href="/wiki/Color_charge" title="Color charge">Color charge</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Quark_model" title="Quark model">Quark model</a></li></ul></li> <li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a> <ul><li><a href="/wiki/Weak_interaction" title="Weak interaction">Weak interaction</a></li> <li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi's interaction</a></li> <li><a href="/wiki/Weak_hypercharge" title="Weak hypercharge">Weak hypercharge</a></li> <li><a href="/wiki/Weak_isospin" title="Weak isospin">Weak isospin</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><span><img 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title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li> <li><a href="/wiki/Mathematical_formulation_of_the_Standard_Model" title="Mathematical formulation of the Standard Model">Mathematical formulation of the Standard Model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Beyond the<br />Standard Model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Evidence</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hierarchy_problem" title="Hierarchy problem">Hierarchy problem</a></li> <li><a href="/wiki/Dark_matter" title="Dark matter">Dark matter</a></li> <li><a href="/wiki/Cosmological_constant" title="Cosmological constant">Cosmological constant</a> <ul><li><a href="/wiki/Cosmological_constant_problem" title="Cosmological constant problem">problem</a></li></ul></li> <li><a href="/wiki/CP_violation" title="CP violation">Strong CP problem</a></li> <li><a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">Neutrino oscillation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Grand_Unified_Theory" title="Grand Unified Theory">Grand Unified Theory</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">MSSM</a></li> <li><a href="/wiki/Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li> <li><a href="/wiki/Split_supersymmetry" title="Split supersymmetry">Split supersymmetry</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Superstring_theory" title="Superstring theory">Superstring theory</a></li> <li><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li> <li><a href="/wiki/Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</a></li> <li><a href="/wiki/Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a></li> <li><a href="/wiki/Twistor_theory" title="Twistor theory">Twistor theory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Laboratori_Nazionali_del_Gran_Sasso" title="Laboratori Nazionali del Gran Sasso">Gran Sasso</a></li> <li><a href="/wiki/India-based_Neutrino_Observatory" title="India-based Neutrino Observatory">INO</a></li> <li><a href="/wiki/Large_Hadron_Collider" title="Large Hadron Collider">LHC</a></li> <li><a href="/wiki/Sudbury_Neutrino_Observatory" title="Sudbury Neutrino Observatory">SNO</a></li> <li><a href="/wiki/Super-Kamiokande" title="Super-Kamiokande">Super-K</a></li> <li><a href="/wiki/Tevatron" title="Tevatron">Tevatron</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:Standard_Model" title="Category:Standard Model">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Standard_Model_(physics)" class="extiw" title="commons:Category:Standard Model (physics)">Commons</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Industrial_and_applied_mathematics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Industrial_and_applied_mathematics" title="Template:Industrial and applied mathematics"><abbr title="View this 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href="/wiki/Algorithm_design" class="mw-redirect" title="Algorithm design">design</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">analysis</a></li></ul></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Coding_theory" title="Coding theory">Coding theory</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a></li> <li><a href="/wiki/Constraint_satisfaction_problem" title="Constraint satisfaction problem">Constraint satisfaction</a> <ul><li><a href="/wiki/Constraint_programming" title="Constraint programming">Constraint programming</a></li></ul></li> <li><a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Computational logic</a></li> <li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Computational_statistics" title="Computational statistics">Statistics</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Mathematical_software" scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical software</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_arbitrary-precision_arithmetic_software" title="List of arbitrary-precision arithmetic software">Arbitrary-precision arithmetic</a></li> <li><a href="/wiki/List_of_finite_element_software_packages" title="List of finite element software packages">Finite element analysis</a></li> <li><a href="/wiki/Tensor_software" title="Tensor software">Tensor software</a></li> <li><a href="/wiki/List_of_interactive_geometry_software" title="List of interactive geometry software">Interactive geometry software</a></li> <li><a href="/wiki/List_of_optimization_software" title="List of optimization software">Optimization software</a></li> <li><a href="/wiki/List_of_statistical_software" title="List of statistical software">Statistical software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical-analysis software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical libraries</a></li> <li><a href="/wiki/Solver" title="Solver">Solvers</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li> <li><a href="/wiki/Computational_number_theory" title="Computational number theory">Computational number theory</a></li> <li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_theory" title="Approximation theory">Approximation theory</a></li> <li><a href="/wiki/Clifford_analysis" title="Clifford analysis">Clifford analysis</a> <ul><li><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equations</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equations</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/Geometric_analysis" title="Geometric analysis">Geometric analysis</a></li></ul></li> <li><a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical systems</a> <ul><li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li></ul></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> <ul><li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></li></ul></li> <li><a href="/wiki/Harmonic_analysis_(mathematics)" class="mw-redirect" title="Harmonic analysis (mathematics)">Harmonic analysis</a> <ul><li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li></ul></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a> <ul><li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Vector_calculus#Vector_algebra" title="Vector calculus">Vector</a></li></ul></li> <li><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a> <ul><li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></li></ul></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> <ul><li><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical linear algebra</a></li> <li><a href="/wiki/Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li> <li><a href="/wiki/Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variational calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Probability_distribution" title="Probability distribution">Distributions</a> (<a href="/wiki/Random_variable" title="Random variable">random variables</a>)</li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a> / <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">analysis</a></li> <li><a href="/wiki/Functional_integration" title="Functional integration">Path integral</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Stochastic variational calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical<br />physics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a> <ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a></li> <li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a></li></ul></li> <li><a href="/wiki/Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">Field theory</a> <ul><li><a href="/wiki/Classical_field_theory" title="Classical field theory">Classical</a></li> <li><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective</a></li> <li><a class="mw-selflink selflink">Gauge</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum</a></li> <li><a href="/wiki/Statistical_field_theory" title="Statistical field theory">Statistical</a></li> <li><a href="/wiki/Topological_field_theory" class="mw-redirect" title="Topological field theory">Topological</a></li></ul></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a> <ul><li><a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">in quantum mechanics</a></li></ul></li> <li><a href="/wiki/Potential_theory" title="Potential theory">Potential theory</a></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a> <ul><li><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic</a></li> <li><a href="/wiki/Topological_string_theory" title="Topological string theory">Topological</a></li></ul></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a> <ul><li><a href="/wiki/Supersymmetric_quantum_mechanics" title="Supersymmetric quantum mechanics">Supersymmetric quantum mechanics</a></li> <li><a href="/wiki/Supersymmetric_theory_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">Supersymmetric theory of stochastic dynamics</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Algebraic_structures" scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Algebraic_structures" class="mw-redirect" title="Algebraic structures">Algebraic structures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Feynman integral</a></li> <li><a href="/wiki/Poisson_algebra" title="Poisson algebra">Poisson algebra</a></li> <li><a href="/wiki/Quantum_group" title="Quantum group">Quantum group</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group</a></li> <li><a href="/wiki/Particle_physics_and_representation_theory" title="Particle physics and representation theory">Representation theory</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Superalgebra" title="Superalgebra">Superalgebra</a></li> <li><a href="/wiki/Supersymmetry_algebra" title="Supersymmetry algebra">Supersymmetry algebra</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Decision_theory" title="Decision theory">Decision sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice theory</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Sociology</a></li> <li>"<a href="/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a>"</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a> <ul><li><a href="/wiki/Japan_Society_for_Industrial_and_Applied_Mathematics" title="Japan Society for Industrial and Applied Mathematics">Japan Society for Industrial and Applied Mathematics</a></li></ul></li> <li><a href="/wiki/Soci%C3%A9t%C3%A9_de_Math%C3%A9matiques_Appliqu%C3%A9es_et_Industrielles" title="Société de Mathématiques Appliquées et Industrielles">Société de Mathématiques Appliquées et Industrielles</a></li> <li><a href="/wiki/International_Council_for_Industrial_and_Applied_Mathematics" title="International Council for Industrial and Applied Mathematics">International Council for Industrial and Applied Mathematics</a></li> <li><a href="/w/index.php?title=European_Community_on_Computational_Methods_in_Applied_Sciences&action=edit&redlink=1" class="new" title="European Community on Computational Methods in Applied Sciences (page does not exist)">European Community on Computational Methods in Applied Sciences</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><a href="/wiki/Category:Mathematics" title="Category:Mathematics">Category</a></b></li> <li><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a> / <a href="/wiki/Topic_outline_of_mathematics" class="mw-redirect" title="Topic outline of mathematics">outline</a> / <a 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