CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Goldstone boson - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"ce19b698-0f4b-469c-a8a8-e79abb11503a","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Goldstone_boson","wgTitle":"Goldstone boson","wgCurRevisionId":1228630845,"wgRevisionId":1228630845,"wgArticleId":628183,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Webarchive template wayback links","Articles with short description","Short description matches Wikidata","Bosons","Quantum field theory","Mathematical physics","Physics theorems","Subatomic particles with spin 0"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Goldstone_boson","wgRelevantArticleId":628183,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[], "wgRedirectedFrom":"Goldstone_bosons","wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgInternalRedirectTargetUrl":"/wiki/Goldstone_boson","wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1153441","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"], "GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher", "ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Goldstone boson - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Goldstone_boson"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Goldstone_boson&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Goldstone_boson"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Goldstone_boson rootpage-Goldstone_boson skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Goldstone+boson" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Goldstone+boson" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Goldstone+boson" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Goldstone+boson" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Goldstone's_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Goldstone's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Goldstone's theorem</span> </div> </a> <ul id="toc-Goldstone's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-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 Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Natural" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Natural"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Natural</span> </div> </a> <ul id="toc-Natural-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Theory</span> </div> </a> <ul id="toc-Theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Goldstone's_argument" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Goldstone's_argument"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Goldstone's argument</span> </div> </a> <ul id="toc-Goldstone's_argument-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infraparticles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Infraparticles"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Infraparticles</span> </div> </a> <ul id="toc-Infraparticles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Extensions</span> </div> </a> <button aria-controls="toc-Extensions-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 Extensions subsection</span> </button> <ul id="toc-Extensions-sublist" class="vector-toc-list"> <li id="toc-Nonrelativistic_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonrelativistic_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Nonrelativistic theories</span> </div> </a> <ul id="toc-Nonrelativistic_theories-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Breaking_of_spacetime_symmetries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Breaking_of_spacetime_symmetries"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Breaking of spacetime symmetries</span> </div> </a> <ul id="toc-Breaking_of_spacetime_symmetries-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Nambu–Goldstone_fermions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nambu–Goldstone_fermions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Nambu–Goldstone fermions</span> </div> </a> <ul id="toc-Nambu–Goldstone_fermions-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-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">9</span> <span>References</span> </div> </a> <ul id="toc-References-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">Goldstone boson</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 13 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-13" 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">13 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/%D8%A8%D9%88%D8%B2%D9%88%D9%86_%D8%BA%D9%88%D9%84%D8%AF%D8%B3%D8%AA%D9%88%D9%86" title="بوزون غولدستون – Arabic" lang="ar" hreflang="ar" data-title="بوزون غولدستون" 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/Bos%C3%B3_de_Goldstone" title="Bosó de Goldstone – Catalan" lang="ca" hreflang="ca" data-title="Bosó de Goldstone" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Goldstone-Boson" title="Goldstone-Boson – German" lang="de" hreflang="de" data-title="Goldstone-Boson" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Goldstone%27i_bosonid" title="Goldstone'i bosonid – Estonian" lang="et" hreflang="et" data-title="Goldstone'i bosonid" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Bos%C3%B3n_de_Goldstone" title="Bosón de Goldstone – Spanish" lang="es" hreflang="es" data-title="Bosón de Goldstone" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Boson_de_Goldstone" title="Boson de Goldstone – French" lang="fr" hreflang="fr" data-title="Boson de Goldstone" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%A8%EB%93%9C%EC%8A%A4%ED%86%A4_%EB%B3%B4%EC%86%90" title="골드스톤 보손 – Korean" lang="ko" hreflang="ko" data-title="골드스톤 보손" 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/Boson_Goldstone" title="Boson Goldstone – Indonesian" lang="id" hreflang="id" data-title="Boson Goldstone" 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/Bosone_di_Goldstone" title="Bosone di Goldstone – Italian" lang="it" hreflang="it" data-title="Bosone di Goldstone" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8D%97%E9%83%A8%E3%83%BB%E3%82%B4%E3%83%BC%E3%83%AB%E3%83%89%E3%82%B9%E3%83%88%E3%83%BC%E3%83%B3%E7%B2%92%E5%AD%90" title="南部・ゴールドストーン粒子 – Japanese" lang="ja" hreflang="ja" data-title="南部・ゴールドストーン粒子" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%BE%D0%BB%D0%B4%D1%81%D1%82%D0%BE%D1%83%D0%BD%D0%BE%D0%B2%D1%81%D0%BA%D0%B8%D0%B9_%D0%B1%D0%BE%D0%B7%D0%BE%D0%BD" title="Голдстоуновский бозон – Russian" lang="ru" hreflang="ru" data-title="Голдстоуновский бозон" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%BE%D0%B7%D0%BE%D0%BD_%D2%90%D0%BE%D0%BB%D0%B4%D1%81%D1%82%D0%BE%D1%83%D0%BD%D0%B0" title="Бозон Ґолдстоуна – Ukrainian" lang="uk" hreflang="uk" data-title="Бозон Ґолдстоуна" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh badge-Q70894304 mw-list-item" title=""><a href="https://zh.wikipedia.org/wiki/%E6%88%88%E5%BE%B7%E6%96%AF%E9%80%9A%E7%8E%BB%E8%89%B2%E5%AD%90" title="戈德斯通玻色子 – Chinese" lang="zh" hreflang="zh" data-title="戈德斯通玻色子" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1153441#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Goldstone_boson" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Goldstone_boson" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Goldstone_boson"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Goldstone_boson&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Goldstone_boson&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Goldstone_boson"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Goldstone_boson&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Goldstone_boson&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Goldstone_boson" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Goldstone_boson" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Goldstone_boson&oldid=1228630845" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Goldstone_boson&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Goldstone_boson&id=1228630845&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGoldstone_boson"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGoldstone_boson"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Goldstone_boson&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Goldstone_boson&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1153441" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Goldstone_bosons&redirect=no" class="mw-redirect" title="Goldstone bosons">Goldstone bosons</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Massless boson that must be present in a quantum system with spontaneously broken symmetry</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">"Goldstone theorem" redirects here. For the theorem on Feynman diagrams, see <a href="/wiki/Linked-cluster_theorem" class="mw-redirect" title="Linked-cluster theorem">linked-cluster theorem</a>.</div> <p>In <a href="/wiki/Particle_physics" title="Particle physics">particle</a> and <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>, <b>Goldstone bosons</b> or <b>Nambu–Goldstone bosons</b> (<b>NGBs</b>) are <a href="/wiki/Bosons" class="mw-redirect" title="Bosons">bosons</a> that appear necessarily in models exhibiting <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneous breakdown</a> of <a href="/wiki/Continuous_symmetry" title="Continuous symmetry">continuous symmetries</a>. They were discovered by <a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Yoichiro Nambu</a> in <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a> within the context of the <a href="/wiki/BCS_theory" title="BCS theory">BCS superconductivity</a> mechanism,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and subsequently elucidated by <a href="/wiki/Jeffrey_Goldstone" title="Jeffrey Goldstone">Jeffrey Goldstone</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 systematically generalized in the context of <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a> such bosons are <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticles</a> and are known as Anderson–Bogoliubov modes.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><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> </p><p>These <a href="/wiki/Spin_(physics)" title="Spin (physics)">spinless</a> bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the <a href="/wiki/Quantum_number" title="Quantum number">quantum numbers</a> of these. They transform nonlinearly (shift) under the action of these generators, and can thus be excited out of the asymmetric vacuum by these generators. Thus, they can be thought of as the excitations of the field in the broken symmetry directions in group space—and are <a href="/wiki/Massless_particle" title="Massless particle">massless</a> if the spontaneously broken symmetry is not also <a href="/wiki/Explicit_symmetry_breaking" title="Explicit symmetry breaking">broken explicitly</a>. </p><p>If, instead, the symmetry is not exact, i.e. if it is explicitly broken as well as spontaneously broken, then the Nambu–Goldstone <a href="/wiki/Boson" title="Boson">bosons</a> are not massless, though they typically remain relatively light; they are then called <b><a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">pseudo-Goldstone bosons</a></b> or <b>pseudo–Nambu–Goldstone bosons</b> (abbreviated <b>PNGBs</b>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Goldstone's_theorem"><span id="Goldstone.27s_theorem"></span>Goldstone's theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=1" title="Edit section: Goldstone's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Goldstone's theorem</b> examines a generic <a href="/wiki/Continuous_symmetry" title="Continuous symmetry">continuous symmetry</a> which is <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneously broken</a>; i.e., its currents are conserved, but the <a href="/wiki/Ground_state" title="Ground state">ground state</a> is not invariant under the action of the corresponding charges. Then, necessarily, new massless (or light, if the symmetry is not exact) <a href="/wiki/Scalar_field_theory" title="Scalar field theory">scalar</a> particles appear in the spectrum of possible excitations. There is one scalar particle—called a Nambu–Goldstone boson—for each generator of the symmetry that is broken, i.e., that does not preserve the <a href="/wiki/Ground_state" title="Ground state">ground state</a>. The Nambu–Goldstone mode is a long-wavelength fluctuation of the corresponding <a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">order parameter</a>. </p><p>By virtue of their special properties in coupling to the vacuum of the respective symmetry-broken theory, vanishing momentum ("soft") Goldstone bosons involved in field-theoretic amplitudes make such amplitudes vanish ("Adler zeros"). </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Natural">Natural</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=3" title="Edit section: Natural"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>In <a href="/wiki/Fluid" title="Fluid">fluids</a>, the <a href="/wiki/Phonon" title="Phonon">phonon</a> is longitudinal and it is the Goldstone boson of the spontaneously broken <a href="/wiki/Galilean_symmetry" class="mw-redirect" title="Galilean symmetry">Galilean symmetry</a>. In <a href="/wiki/Solid" title="Solid">solids</a>, the situation is more complicated; the Goldstone bosons are the longitudinal and transverse phonons and they happen to be the Goldstone bosons of spontaneously broken Galilean, translational, and rotational symmetry with no simple one-to-one correspondence between the Goldstone modes and the broken symmetries.</li> <li>In <a href="/wiki/Magnet" title="Magnet">magnets</a>, the original rotational symmetry (present in the absence of an external magnetic field) is spontaneously broken such that the magnetization points in a specific direction. The Goldstone bosons then are the <i><a href="/wiki/Magnon" title="Magnon">magnons</a></i>, i.e., spin waves in which the local magnetization direction oscillates.</li> <li>The <i><a href="/wiki/Pion" title="Pion">pions</a></i> are the <a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">pseudo-Goldstone bosons</a> that result from the spontaneous breakdown of the chiral-flavor symmetries of QCD effected by quark condensation due to the strong interaction. These symmetries are further explicitly broken by the masses of the quarks so that the pions are not massless, but their mass is <i>significantly smaller</i> than typical hadron masses.</li> <li>The longitudinal polarization components of the <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W and Z bosons</a> correspond to the Goldstone bosons of the spontaneously broken part of the electroweak symmetry SU(2)⊗U(1), which, however, are not observable.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup> Because this symmetry is gauged, the three would-be Goldstone bosons are absorbed by the three gauge bosons corresponding to the three broken generators; this gives these three gauge bosons a mass and the associated necessary third polarization degree of freedom. This is described in the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> through the <a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a>. An analogous phenomenon occurs in <a href="/wiki/Superconductivity" title="Superconductivity">superconductivity</a>, which served as the original source of inspiration for Nambu, namely, the photon develops a dynamical mass (expressed as magnetic flux exclusion from a superconductor), cf. the <a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau theory</a>.</li> <li><a href="/wiki/Primordial_fluctuations" title="Primordial fluctuations">Primordial fluctuations</a> during <a href="/wiki/Inflation_(cosmology)" class="mw-redirect" title="Inflation (cosmology)">inflation</a> can be viewed as Goldstone bosons arising due to the spontaneous symmetry breaking of <a href="/wiki/Time_translation_symmetry" class="mw-redirect" title="Time translation symmetry">time translation symmetry</a> of a <a href="/wiki/De_Sitter_universe" title="De Sitter universe">de Sitter universe</a>. These fluctuations in the <a href="/wiki/Inflaton" title="Inflaton">inflaton</a> <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> subsequently seed <a href="/wiki/Structure_formation" title="Structure formation">cosmic structure formation</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>Ricciardi and Umezawa proposed in 1967 a general theory (quantum brain) about the possible brain mechanism of memory storage and retrieval in terms of Nambu–Goldstone bosons.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This theory was subsequently extended in 1995 by Giuseppe Vitiello taking into account that the brain is an "open" system (the dissipative quantum model of the brain).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Applications of spontaneous symmetry breaking and of Goldstone's theorem to biological systems, in general, have been published by E. Del Giudice, S. Doglia, M. Milani, and G. Vitiello,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and by E. Del Giudice, G. Preparata and G. Vitiello.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Mari Jibu and Kunio Yasue<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and Giuseppe Vitiello,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> based on these findings, discussed the implications for consciousness.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Theory">Theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=4" title="Edit section: Theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a <a href="/wiki/Complex_number" title="Complex number">complex</a> <a href="/wiki/Scalar_field_theory" title="Scalar field theory">scalar field</a> <span class="texhtml mvar" style="font-style:italic;"><i>ϕ</i></span>, with the constraint that <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 ^{*}\phi =v^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ϕ</mi> <mo>=</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ^{*}\phi =v^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1669f391ff3bc69c4cc04e1b605fc375f3f5a45d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.105ex; height:3.009ex;" alt="{\displaystyle \phi ^{*}\phi =v^{2}}"></span>, a constant. One way to impose a constraint of this sort is by including a <a href="/wiki/Potential" title="Potential">potential</a> interaction term in its <a href="/wiki/Lagrangian_density" class="mw-redirect" title="Lagrangian density">Lagrangian density</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 \lambda (\phi ^{*}\phi -v^{2})^{2}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ϕ</mi> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (\phi ^{*}\phi -v^{2})^{2}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/707c6702327745424ab8fe79c9f964c4423669d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.294ex; height:3.176ex;" alt="{\displaystyle \lambda (\phi ^{*}\phi -v^{2})^{2}~,}"></span></dd></dl> <p>and taking the limit as <span class="texhtml"><i>λ</i> → ∞</span>. This is called the "Abelian nonlinear σ-model".<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> </p><p>The constraint, and the action, below, are invariant under a <i>U</i>(1) phase transformation, <span class="texhtml"><i>δϕ</i>=i<i>εϕ</i></span>. The field can be redefined to give a real <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> (i.e., a spin-zero particle) <span class="texhtml mvar" style="font-style:italic;"><i>θ</i></span> without any constraint 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 \phi =ve^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>v</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =ve^{i\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aecd2fb5bfcec69db0d69ac1f2bd286c4e024f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.266ex; height:3.009ex;" alt="{\displaystyle \phi =ve^{i\theta }}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;"><i>θ</i></span> is the Nambu–Goldstone boson (actually <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\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92ddfd797f65961fda2fcf5230dc00044236149f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.218ex; height:2.176ex;" alt="{\displaystyle v\theta }"></span> is) and the <i>U</i>(1) symmetry transformation effects a shift on <span class="texhtml mvar" style="font-style:italic;"><i>θ</i></span>, namely </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 \theta =\epsilon ~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>θ</mi> <mo>=</mo> <mi>ϵ</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \theta =\epsilon ~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8355c781888c1bc04e7c197b45f9819533d44c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.409ex; height:2.676ex;" alt="{\displaystyle \delta \theta =\epsilon ~,}"></span></dd></dl> <p>but does not preserve the ground state <span class="texhtml">|0〉</span> (i.e. the above infinitesimal transformation <i>does not annihilate it</i>—the hallmark of invariance), as evident in the charge of the current below. </p><p>Thus, the vacuum is degenerate and noninvariant under the action of the spontaneously broken symmetry. </p><p>The corresponding <a href="/wiki/Lagrangian_density" class="mw-redirect" title="Lagrangian density">Lagrangian density</a> is 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 {\mathcal {L}}={\frac {1}{2}}(\partial ^{\mu }\phi ^{*})\partial _{\mu }\phi -m^{2}\phi ^{*}\phi ={\frac {1}{2}}(-ive^{-i\theta }\partial ^{\mu }\theta )(ive^{i\theta }\partial _{\mu }\theta )-m^{2}v^{2},}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi>v</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>v</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial ^{\mu }\phi ^{*})\partial _{\mu }\phi -m^{2}\phi ^{*}\phi ={\frac {1}{2}}(-ive^{-i\theta }\partial ^{\mu }\theta )(ive^{i\theta }\partial _{\mu }\theta )-m^{2}v^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/334ad2a5bf0173a727968d220f29ab5b49ecfd03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:64.06ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial ^{\mu }\phi ^{*})\partial _{\mu }\phi -m^{2}\phi ^{*}\phi ={\frac {1}{2}}(-ive^{-i\theta }\partial ^{\mu }\theta )(ive^{i\theta }\partial _{\mu }\theta )-m^{2}v^{2},}"></span></dd></dl> <p>and thus </p> <dl><dd><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 {v^{2}}{2}}(\partial ^{\mu }\theta )(\partial _{\mu }\theta )-m^{2}v^{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {v^{2}}{2}}(\partial ^{\mu }\theta )(\partial _{\mu }\theta )-m^{2}v^{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b00271dee64bc4042dbefac37b11d38da6d38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.64ex; height:5.676ex;" alt="{\displaystyle ={\frac {v^{2}}{2}}(\partial ^{\mu }\theta )(\partial _{\mu }\theta )-m^{2}v^{2}~.}"></span></dd></dl></dd></dl> <p>Note that the constant 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 m^{2}v^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{2}v^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39e6cd208de27254968802b00afe56e158b9af65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.276ex; height:2.676ex;" alt="{\displaystyle m^{2}v^{2}}"></span> in the Lagrangian density has no physical significance, and the other term in it is simply the kinetic term for a massless scalar. </p><p>The symmetry-induced conserved <i>U</i>(1) current 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 J_{\mu }=v^{2}\partial _{\mu }\theta ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>θ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{\mu }=v^{2}\partial _{\mu }\theta ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d53aa807e09daed0dc2672a837c3ceffdaac81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.57ex; height:3.343ex;" alt="{\displaystyle J_{\mu }=v^{2}\partial _{\mu }\theta ~.}"></span></dd></dl> <p>The charge, <i>Q</i>, resulting from this current shifts <span class="texhtml mvar" style="font-style:italic;"><i>θ</i></span> and the ground state to a new, degenerate, ground state. Thus, a vacuum with <span class="texhtml">〈<i>θ</i>〉 = 0</span> will shift to a <i>different vacuum</i> with <span class="texhtml">〈<i>θ</i>〉 = <i>ε</i></span>. The current connects the original vacuum with the Nambu–Goldstone boson state, <span class="texhtml">〈0|<i>J</i><sub>0</sub>(0)|<i>θ</i>〉≠ 0</span>. </p><p>In general, in a theory with several scalar fields, <span class="texhtml"><i>ϕ</i><sub>j</sub></span>, the Nambu–Goldstone mode <span class="texhtml"><i>ϕ</i><sub>g</sub></span> is <a href="/wiki/Massless_particle" title="Massless particle">massless</a>, and parameterises the curve of possible (degenerate) vacuum states. Its hallmark under the broken symmetry transformation is <i>nonvanishing vacuum expectation</i> <span class="texhtml">〈<i>δϕ<sub>g</sub></i>〉</span>, an <a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">order parameter</a>, for vanishing <span class="texhtml">〈<i>ϕ<sub>g</sub></i>〉 = 0</span>, at some ground state |0〉 chosen at the minimum of the potential, <span class="texhtml">〈∂<i>V</i>/∂<i>ϕ</i><sub>i</sub>〉 = 0</span>. In principle the vacuum should be the minimum of the <a href="/wiki/Effective_action" title="Effective action">effective potential</a> which takes into account quantum effects, however it is equal to the classical potential to first approximation. Symmetry dictates that all variations of the potential with respect to the fields in all symmetry directions vanish. The vacuum value of the first order variation in any direction vanishes as just seen; while the vacuum value of the second order variation must also vanish, as follows. Vanishing vacuum values of field symmetry transformation increments add no new information. </p><p>By contrast, however, <i>nonvanishing vacuum expectations of transformation increments</i>, <span class="texhtml">〈<i>δϕ</i><sub>g</sub>〉</span>, specify the relevant (Goldstone) <i>null eigenvectors of the mass matrix</i>, </p> <div class="equation-box" style="margin: 0 0 0 3.2em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <p><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 \left\langle {\partial ^{2}V \over \partial \phi _{i}\partial \phi _{j}}\right\rangle \langle \delta \phi _{j}\rangle =0~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle {\partial ^{2}V \over \partial \phi _{i}\partial \phi _{j}}\right\rangle \langle \delta \phi _{j}\rangle =0~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d0bb645bea3719b7938fc2e2fbaf5016811f09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.467ex; height:6.509ex;" alt="{\displaystyle \left\langle {\partial ^{2}V \over \partial \phi _{i}\partial \phi _{j}}\right\rangle \langle \delta \phi _{j}\rangle =0~,}"></span> </p> </div> <p>and hence the corresponding zero-mass eigenvalues. </p> <div class="mw-heading mw-heading2"><h2 id="Goldstone's_argument"><span id="Goldstone.27s_argument"></span>Goldstone's argument</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=5" title="Edit section: Goldstone's argument"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The principle behind Goldstone's argument is that the ground state is not unique. Normally, by current conservation, the charge operator for any symmetry current is time-independent, </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 \over dt}Q={d \over dt}\int _{x}J^{0}(x)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dt}Q={d \over dt}\int _{x}J^{0}(x)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11b8ed19b9c488de838295045dee79737a528eec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.586ex; height:5.843ex;" alt="{\displaystyle {d \over dt}Q={d \over dt}\int _{x}J^{0}(x)=0.}"></span></dd></dl> <p>Acting with the charge operator on the vacuum either <i>annihilates the vacuum</i>, if that is symmetric; else, if <i>not</i>, as is the case in spontaneous symmetry breaking, it produces a zero-frequency state out of it, through its shift transformation feature illustrated above. Actually, here, the charge itself is ill-defined, cf. the Fabri–Picasso argument below. </p><p>But its better behaved commutators with fields, that is, the nonvanishing transformation shifts <span class="texhtml">〈<i>δϕ</i><sub>g</sub>〉</span>, are, nevertheless, <i>time-invariant</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 {\frac {d\langle \delta \phi _{g}\rangle }{dt}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\langle \delta \phi _{g}\rangle }{dt}}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b745145548da560b9e6cf1f987e54def0a8735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.225ex; height:5.843ex;" alt="{\displaystyle {\frac {d\langle \delta \phi _{g}\rangle }{dt}}=0,}"></span></dd></dl> <p>thus generating a <span class="texhtml">δ(<i>k</i><sup>0</sup>)</span> in its Fourier transform.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> (This ensures that, inserting a complete set of intermediate states in a nonvanishing current commutator can lead to vanishing time-evolution only when one or more of these states is massless.) </p><p>Thus, if the vacuum is not invariant under the symmetry, action of the charge operator produces a state which is different from the vacuum chosen, but which has zero frequency. This is a long-wavelength oscillation of a field which is nearly stationary: there are physical states with zero frequency, <span class="texhtml"><i>k</i><sup>0</sup></span>, so that the theory cannot have a <a href="/wiki/Mass_gap" title="Mass gap">mass gap</a>. </p><p>This argument is further clarified by taking the limit carefully. If an approximate charge operator acting in a huge but finite region <span class="texhtml mvar" style="font-style:italic;">A</span> is applied to the vacuum, </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 \over dt}Q_{A}={d \over dt}\int _{x}e^{-{\frac {x^{2}}{2A^{2}}}}J^{0}(x)=-\int _{x}e^{-{\frac {x^{2}}{2A^{2}}}}\nabla \cdot J=\int _{x}\nabla \left(e^{-{\frac {x^{2}}{2A^{2}}}}\right)\cdot J,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </msup> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </msup> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi>J</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>J</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dt}Q_{A}={d \over dt}\int _{x}e^{-{\frac {x^{2}}{2A^{2}}}}J^{0}(x)=-\int _{x}e^{-{\frac {x^{2}}{2A^{2}}}}\nabla \cdot J=\int _{x}\nabla \left(e^{-{\frac {x^{2}}{2A^{2}}}}\right)\cdot J,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c6ea174e01afbb10f74be4928b39c08a2e850c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.825ex; height:7.509ex;" alt="{\displaystyle {d \over dt}Q_{A}={d \over dt}\int _{x}e^{-{\frac {x^{2}}{2A^{2}}}}J^{0}(x)=-\int _{x}e^{-{\frac {x^{2}}{2A^{2}}}}\nabla \cdot J=\int _{x}\nabla \left(e^{-{\frac {x^{2}}{2A^{2}}}}\right)\cdot J,}"></span></dd></dl> <p>a state with approximately vanishing time derivative is produced, </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 \left\|{d \over dt}Q_{A}|0\rangle \right\|\approx {\frac {1}{A}}\left\|Q_{A}|0\rangle \right\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>A</mi> </mfrac> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|{d \over dt}Q_{A}|0\rangle \right\|\approx {\frac {1}{A}}\left\|Q_{A}|0\rangle \right\|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3d21cace988573dee6760e4511701e1979d240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.674ex; height:5.843ex;" alt="{\displaystyle \left\|{d \over dt}Q_{A}|0\rangle \right\|\approx {\frac {1}{A}}\left\|Q_{A}|0\rangle \right\|.}"></span></dd></dl> <p>Assuming a nonvanishing mass gap <span class="texhtml"><i>m</i><sub>0</sub></span>, the frequency of any state like the above, which is orthogonal to the vacuum, is at least <span class="texhtml"><i>m</i><sub>0</sub></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 \left\|{\frac {d}{dt}}|\theta \rangle \right\|=\|H|\theta \rangle \|\geq m_{0}\||\theta \rangle \|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">‖</mo> <mo>≥</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">‖</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|{\frac {d}{dt}}|\theta \rangle \right\|=\|H|\theta \rangle \|\geq m_{0}\||\theta \rangle \|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0d5fcfb401d35aff7f9e33d7c69a10be527908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.794ex; height:5.843ex;" alt="{\displaystyle \left\|{\frac {d}{dt}}|\theta \rangle \right\|=\|H|\theta \rangle \|\geq m_{0}\||\theta \rangle \|.}"></span></dd></dl> <p>Letting <span class="texhtml mvar" style="font-style:italic;">A</span> become large leads to a contradiction. Consequently <span class="texhtml mvar" style="font-style:italic;">m</span><sub>0</sub> = 0. However this argument fails when the symmetry is gauged, because then the symmetry generator is only performing a gauge transformation. A gauge transformed state is the same exact state, so that acting with a symmetry generator does not get one out of the vacuum (see <a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a>). </p> <dl><dd><b>Fabri–Picasso Theorem.</b> <span class="texhtml mvar" style="font-style:italic;">Q</span> does not properly exist in the Hilbert space, unless <span class="texhtml"><i>Q</i>|0〉 = 0</span>.</dd></dl> <p>The argument<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> requires both the vacuum and the charge <span class="texhtml mvar" style="font-style:italic;">Q</span> to be translationally invariant, <span class="texhtml"><i>P</i>|0〉 = 0</span>, <span class="texhtml">[<i>P,Q</i>]= 0</span>. </p><p>Consider the correlation function of the charge with itself, </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}\langle 0|QQ|0\rangle &=\int d^{3}x\langle 0|j_{0}(x)Q|0\rangle \\&=\int d^{3}x\left\langle 0\left|e^{iPx}j_{0}(0)e^{-iPx}Q\right|0\right\rangle \\&=\int d^{3}x\left\langle 0\left|e^{iPx}j_{0}(0)e^{-iPx}Qe^{iPx}e^{-iPx}\right|0\right\rangle \\&=\int d^{3}x\left\langle 0\left|j_{0}(0)Q\right|0\right\rangle \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> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Q</mi> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>x</mi> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>x</mi> <mrow> <mo>⟨</mo> <mrow> <mn>0</mn> <mrow> <mo>|</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>P</mi> <mi>x</mi> </mrow> </msup> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>P</mi> <mi>x</mi> </mrow> </msup> <mi>Q</mi> </mrow> <mo>|</mo> </mrow> <mn>0</mn> </mrow> <mo>⟩</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>x</mi> <mrow> <mo>⟨</mo> <mrow> <mn>0</mn> <mrow> <mo>|</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>P</mi> <mi>x</mi> </mrow> </msup> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>P</mi> <mi>x</mi> </mrow> </msup> <mi>Q</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>P</mi> <mi>x</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>P</mi> <mi>x</mi> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mn>0</mn> </mrow> <mo>⟩</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>x</mi> <mrow> <mo>⟨</mo> <mrow> <mn>0</mn> <mrow> <mo>|</mo> <mrow> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>Q</mi> </mrow> <mo>|</mo> </mrow> <mn>0</mn> </mrow> <mo>⟩</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle 0|QQ|0\rangle &=\int d^{3}x\langle 0|j_{0}(x)Q|0\rangle \\&=\int d^{3}x\left\langle 0\left|e^{iPx}j_{0}(0)e^{-iPx}Q\right|0\right\rangle \\&=\int d^{3}x\left\langle 0\left|e^{iPx}j_{0}(0)e^{-iPx}Qe^{iPx}e^{-iPx}\right|0\right\rangle \\&=\int d^{3}x\left\langle 0\left|j_{0}(0)Q\right|0\right\rangle \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d998f86a52f2cfd916ac0a934947b691dfac4ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; width:51.722ex; height:23.176ex;" alt="{\displaystyle {\begin{aligned}\langle 0|QQ|0\rangle &=\int d^{3}x\langle 0|j_{0}(x)Q|0\rangle \\&=\int d^{3}x\left\langle 0\left|e^{iPx}j_{0}(0)e^{-iPx}Q\right|0\right\rangle \\&=\int d^{3}x\left\langle 0\left|e^{iPx}j_{0}(0)e^{-iPx}Qe^{iPx}e^{-iPx}\right|0\right\rangle \\&=\int d^{3}x\left\langle 0\left|j_{0}(0)Q\right|0\right\rangle \end{aligned}}}"></span></dd></dl> <p>so the integrand in the right hand side does not depend on the position. </p><p>Thus, its value is proportional to the total space volume, <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 \|Q|0\rangle \|^{2}=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|Q|0\rangle \|^{2}=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a67a312a2cdc3d5bc2b1263e2d10cf4cc77faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.354ex; height:3.176ex;" alt="{\displaystyle \|Q|0\rangle \|^{2}=\infty }"></span> — unless the symmetry is unbroken, <span class="texhtml"><i>Q</i>|0〉 = 0</span>. Consequently, <span class="texhtml mvar" style="font-style:italic;">Q</span> does not properly exist in the Hilbert space. </p> <div class="mw-heading mw-heading2"><h2 id="Infraparticles">Infraparticles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=6" title="Edit section: Infraparticles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is an arguable loophole in the theorem. If one reads the theorem carefully, it only states that there exist non-<a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">vacuum states</a> with arbitrarily small energies. Take for example a chiral <var>N</var> = 1 <a href="/wiki/Super_QCD" title="Super QCD">super QCD</a> model with a nonzero <a href="/wiki/Sfermion" title="Sfermion">squark</a> <a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">VEV</a> which is <a href="/wiki/Conformal_field_theory" title="Conformal field theory">conformal</a> in the <a href="/wiki/Infrared" title="Infrared">IR</a>. The chiral symmetry is a <a href="/wiki/Global_symmetry" class="mw-redirect" title="Global symmetry">global symmetry</a> which is (partially) spontaneously broken. Some of the "Goldstone bosons" associated with this spontaneous symmetry breaking are charged under the unbroken gauge group and hence, these <a href="/wiki/Composite_particle" class="mw-redirect" title="Composite particle">composite</a> bosons have a continuous <a href="/wiki/Mass_spectrum" title="Mass spectrum">mass spectrum</a> with arbitrarily small masses but yet there is no Goldstone boson with exactly <a href="/wiki/Massless_particle" title="Massless particle">zero mass</a>. In other words, the Goldstone bosons are <a href="/wiki/Infraparticle" title="Infraparticle">infraparticles</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=7" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Nonrelativistic_theories">Nonrelativistic theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=8" title="Edit section: Nonrelativistic theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A version of Goldstone's theorem also applies to <a href="/wiki/Nonrelativistic" class="mw-redirect" title="Nonrelativistic">nonrelativistic</a> theories.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> It essentially states that, for each spontaneously broken symmetry, there corresponds some <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticle</a> which is typically a boson and has no <a href="/wiki/Energy_gap" title="Energy gap">energy gap</a>. In condensed matter these goldstone bosons are also called gapless modes (i.e. states where the energy dispersion relation is like <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 E\propto p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>∝</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\propto p^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75773d615081f86482835781e283e0d2431088de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.262ex; height:2.676ex;" alt="{\displaystyle E\propto p^{n}}"></span> and is zero for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e6ac10fa45fb984d886065f959a6bdd467b5e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=0}"></span>), the nonrelativistic version of the massless particles (i.e. photons where the dispersion relation is 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 E=pc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>p</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=pc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5498982f39427c7573448da10de26a7b3209e47a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.05ex; height:2.509ex;" alt="{\displaystyle E=pc}"></span> and zero for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e6ac10fa45fb984d886065f959a6bdd467b5e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=0}"></span>). Note that the energy in the non relativistic condensed matter case is <span class="texhtml"><i>H</i>−<i>μN</i>−<i>α</i><span style="position:relative; margin-right:-0.75em; right:0.75em; bottom:0.75em;;"><small>→</small></span>⋅<i>P</i><span style="position:relative; margin-right:-0.75em; right:0.75em; bottom:0.75em;;"><small>→</small></span></span> and not <span class="texhtml"><i>H</i></span> as it would be in a relativistic case. However, two <i>different</i> spontaneously broken generators may now give rise to the <i>same</i> Nambu–Goldstone boson. </p><p>As a first example an antiferromagnet has 2 goldstone bosons, a ferromagnet has 1 goldstone bosons, where in both cases we are breaking symmetry from SO(3) to SO(2), for the antiferromagnet the dispersion is <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 E\propto p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>∝</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\propto p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1521d05a726e176c2037ce4d90f4fbb64ec80a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.044ex; height:2.509ex;" alt="{\displaystyle E\propto p}"></span> and the expectation value of the ground state is zero, for the ferromagnet instead the dispersion is <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 E\propto p^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>∝</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\propto p^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9a84373cf9f58783f091bf5850a6777c1b6b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.098ex; height:3.009ex;" alt="{\displaystyle E\propto p^{2}}"></span> and the expectation value of the ground state is not zero, i.e. there is a spontaneously broken symmetry for the ground state <sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>As a second example, in a <a href="/wiki/Superfluid" class="mw-redirect" title="Superfluid">superfluid</a>, both the <i><a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a></i> particle number symmetry and <a href="/wiki/Galilean_symmetry" class="mw-redirect" title="Galilean symmetry">Galilean symmetry</a> are spontaneously broken. However, the <a href="/wiki/Phonon" title="Phonon">phonon</a> is the Goldstone boson for both.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Still in regards to symmetry breaking there is also a close analogy between gapless modes in condensed matter and the Higgs boson, e.g. in the paramagnet to ferromagnet phase transition<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Breaking_of_spacetime_symmetries">Breaking of spacetime symmetries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=9" title="Edit section: Breaking of spacetime symmetries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In contrast to the case of the breaking of internal symmetries, when spacetime symmetries such as <a href="/wiki/Lorentz_symmetry" class="mw-redirect" title="Lorentz symmetry">Lorentz</a>, conformal, rotational, or translational symmetries are broken, the order parameter need not be a scalar field, but may be a tensor field, and the number of independent massless modes may be fewer than the number of spontaneously broken generators. For a theory with an order parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi ({\boldsymbol {r}})\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi ({\boldsymbol {r}})\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6affcfc76eab501da69300db9299a7ee144d8d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.234ex; height:2.843ex;" alt="{\displaystyle \langle \phi ({\boldsymbol {r}})\rangle }"></span> that spontaneously breaks a spacetime symmetry, the number of broken 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> minus the number non-trivial independent solutions <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 c_{a}({\boldsymbol {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{a}({\boldsymbol {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81bf312dbf58226cf39df868c52c6ed1f3bc87e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.148ex; height:2.843ex;" alt="{\displaystyle c_{a}({\boldsymbol {r}})}"></span> to </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 c_{a}({\boldsymbol {r}})T^{a}\langle \phi ({\boldsymbol {r}})\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{a}({\boldsymbol {r}})T^{a}\langle \phi ({\boldsymbol {r}})\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c689eb0c5c5eb93c86c3f3adad94e1fa27559fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.464ex; height:2.843ex;" alt="{\displaystyle c_{a}({\boldsymbol {r}})T^{a}\langle \phi ({\boldsymbol {r}})\rangle =0}"></span></dd></dl> <p>is the number of Goldstone modes that arise.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> For internal symmetries, the above equation has no non-trivial solutions, so the usual Goldstone theorem holds. When solutions do exist, this is because the Goldstone modes are linearly dependent among themselves, in that the resulting mode can be expressed as a gradients of another mode. Since the spacetime dependence of the solutions <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 c_{a}({\boldsymbol {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{a}({\boldsymbol {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81bf312dbf58226cf39df868c52c6ed1f3bc87e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.148ex; height:2.843ex;" alt="{\displaystyle c_{a}({\boldsymbol {r}})}"></span> is in the direction of the unbroken generators, when all translation generators are broken, no non-trivial solutions exist and the number of Goldstone modes is once again exactly the number of broken generators. </p><p>In general, the phonon is effectively the Nambu–Goldstone boson for spontaneously broken translation<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> symmetry. </p> <div class="mw-heading mw-heading2"><h2 id="Nambu–Goldstone_fermions"><span id="Nambu.E2.80.93Goldstone_fermions"></span>Nambu–Goldstone fermions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=10" title="Edit section: Nambu–Goldstone fermions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spontaneously broken global fermionic symmetries, which occur in some <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetric</a> models, lead to Nambu–Goldstone <a href="/wiki/Fermions" class="mw-redirect" title="Fermions">fermions</a>, or <i><a href="/wiki/Goldstinos" class="mw-redirect" title="Goldstinos">goldstinos</a></i>.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> These have spin <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 2 </span></span>⁠</span>, instead of 0, and carry all quantum numbers of the respective supersymmetry generators broken spontaneously. </p><p>Spontaneous supersymmetry breaking smashes up ("reduces") supermultiplet structures into the characteristic <a href="/wiki/Nonlinear_realization" title="Nonlinear realization">nonlinear realizations</a> of broken supersymmetry, so that goldstinos are superpartners of <i>all</i> particles in the theory, of <i>any spin</i>, and the only superpartners, at that. That is, to say, two non-goldstino particles are connected to only goldstinos through supersymmetry transformations, and not to each other, even if they were so connected before the breaking of supersymmetry. As a result, the masses and spin multiplicities of such particles are then arbitrary. </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=Goldstone_boson&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">Pseudo-Goldstone boson</a></li> <li><a href="/wiki/Majoron" title="Majoron">Majoron</a></li> <li><a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li> <li><a href="/wiki/Mermin%E2%80%93Wagner_theorem" title="Mermin–Wagner theorem">Mermin–Wagner theorem</a></li> <li><a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">Vacuum expectation value</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Goldstone_boson&action=edit&section=12" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">In theories with <a href="/wiki/Gauge_symmetry" class="mw-redirect" title="Gauge symmetry">gauge symmetry</a>, the Goldstone bosons are absent. Their degrees of freedom are absorbed ("eaten", gauged out) by <a href="/wiki/Gauge_boson" title="Gauge boson">gauge bosons</a>, through the <a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a>. The latter become massive and their new, longitudinal polarization is provided by the would-be Goldstone boson, in an elaborate rearrangement of degrees of freedom .</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">It corresponds to the <a href="/wiki/Mexican_hat_potential#A_pedagogical_example:_the_Mexican_hat_potential" class="mw-redirect" title="Mexican hat potential">Goldstone sombrero potential</a> where the tip and the sides shoot to infinity, preserving the location of the minimum at its base.</span> </li> </ol></div></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=Goldstone_boson&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFNambu1960" class="citation journal cs1">Nambu, Y (1960). "Quasiparticles and Gauge Invariance in the Theory of Superconductivity". <i>Physical Review</i>. <b>117</b> (3): 648–663. <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/1960PhRv..117..648N">1960PhRv..117..648N</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%2FPhysRev.117.648">10.1103/PhysRev.117.648</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Quasiparticles+and+Gauge+Invariance+in+the+Theory+of+Superconductivity&rft.volume=117&rft.issue=3&rft.pages=648-663&rft.date=1960&rft_id=info%3Adoi%2F10.1103%2FPhysRev.117.648&rft_id=info%3Abibcode%2F1960PhRv..117..648N&rft.aulast=Nambu&rft.aufirst=Y&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" 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="CITEREFGoldstone1961" class="citation journal cs1">Goldstone, J (1961). <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/343400">"Field Theories with Superconductor Solutions"</a>. <i>Nuovo Cimento</i>. <b>19</b> (1): 154–164. <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/1961NCim...19..154G">1961NCim...19..154G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02812722">10.1007/BF02812722</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:120409034">120409034</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuovo+Cimento&rft.atitle=Field+Theories+with+Superconductor+Solutions&rft.volume=19&rft.issue=1&rft.pages=154-164&rft.date=1961&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120409034%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF02812722&rft_id=info%3Abibcode%2F1961NCim...19..154G&rft.aulast=Goldstone&rft.aufirst=J&rft_id=https%3A%2F%2Fcds.cern.ch%2Frecord%2F343400&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldstoneSalamWeinberg1962" class="citation journal cs1">Goldstone, J; Salam, Abdus; Weinberg, Steven (1962). "Broken Symmetries". <i>Physical Review</i>. <b>127</b> (3): 965–970. <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/1962PhRv..127..965G">1962PhRv..127..965G</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%2FPhysRev.127.965">10.1103/PhysRev.127.965</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Broken+Symmetries&rft.volume=127&rft.issue=3&rft.pages=965-970&rft.date=1962&rft_id=info%3Adoi%2F10.1103%2FPhysRev.127.965&rft_id=info%3Abibcode%2F1962PhRv..127..965G&rft.aulast=Goldstone&rft.aufirst=J&rft.au=Salam%2C+Abdus&rft.au=Weinberg%2C+Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" 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="CITEREFAnderson1958" class="citation journal cs1">Anderson, P. W. (1958-05-15). "Coherent Excited States in the Theory of Superconductivity: Gauge Invariance and the Meissner Effect". <i>Physical Review</i>. <b>110</b> (4). American Physical Society (APS): 827–835. <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/1958PhRv..110..827A">1958PhRv..110..827A</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%2Fphysrev.110.827">10.1103/physrev.110.827</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/0031-899X">0031-899X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Coherent+Excited+States+in+the+Theory+of+Superconductivity%3A+Gauge+Invariance+and+the+Meissner+Effect&rft.volume=110&rft.issue=4&rft.pages=827-835&rft.date=1958-05-15&rft.issn=0031-899X&rft_id=info%3Adoi%2F10.1103%2Fphysrev.110.827&rft_id=info%3Abibcode%2F1958PhRv..110..827A&rft.aulast=Anderson&rft.aufirst=P.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson1958" class="citation journal cs1">Anderson, P. W. (1958-12-15). "Random-Phase Approximation in the Theory of Superconductivity". <i>Physical Review</i>. <b>112</b> (6). American Physical Society (APS): 1900–1916. <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/1958PhRv..112.1900A">1958PhRv..112.1900A</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%2Fphysrev.112.1900">10.1103/physrev.112.1900</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/0031-899X">0031-899X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Random-Phase+Approximation+in+the+Theory+of+Superconductivity&rft.volume=112&rft.issue=6&rft.pages=1900-1916&rft.date=1958-12-15&rft.issn=0031-899X&rft_id=info%3Adoi%2F10.1103%2Fphysrev.112.1900&rft_id=info%3Abibcode%2F1958PhRv..112.1900A&rft.aulast=Anderson&rft.aufirst=P.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" 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="CITEREFBogoljubovTolmachovŠirkov1958" class="citation journal cs1">Bogoljubov, N. N.; Tolmachov, V. V.; Širkov, D. V. (1958). "A New Method in the Theory of Superconductivity". <i>Fortschritte der Physik</i>. <b>6</b> (11–12). Wiley: 605–682. <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/1958ForPh...6..605B">1958ForPh...6..605B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fprop.19580061102">10.1002/prop.19580061102</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/0015-8208">0015-8208</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fortschritte+der+Physik&rft.atitle=A+New+Method+in+the+Theory+of+Superconductivity&rft.volume=6&rft.issue=11%E2%80%9312&rft.pages=605-682&rft.date=1958&rft.issn=0015-8208&rft_id=info%3Adoi%2F10.1002%2Fprop.19580061102&rft_id=info%3Abibcode%2F1958ForPh...6..605B&rft.aulast=Bogoljubov&rft.aufirst=N.+N.&rft.au=Tolmachov%2C+V.+V.&rft.au=%C5%A0irkov%2C+D.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" 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="CITEREFBaumannMcAllister2015" class="citation book cs1">Baumann, D.; McAllister, L. (2015). "1". <i>Inflation and String Theory</i>. Cambridge University Press. pp. 5–8. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1107089693" title="Special:BookSources/978-1107089693"><bdi>978-1107089693</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1&rft.btitle=Inflation+and+String+Theory&rft.pages=5-8&rft.pub=Cambridge+University+Press&rft.date=2015&rft.isbn=978-1107089693&rft.aulast=Baumann&rft.aufirst=D.&rft.au=McAllister%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" 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">L.M. Ricciardi, H. Umezawa (1967). Brain and physics of many-body problems. Kybernetik, 4, 44–8. Reprinted in: Globus GG, Pribram K.H., Vitiello G., publishers. Brain and being. Amsterdam: John Benjamins. P. 255–66 (2004).</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">G. Vitiello, (1995). Memory dissipation and capacity in the quantum brain model. Int. J. Mod. Phys. B9, 973-989.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">E. Del Giudice, S. Doglia, M. Milani, G. Vitiello (1985). A quantum field theoretical approach to the collective behavior of biological systems. Nucl. Phys., B251 (FS 13), 375 - 400.</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">E. Del Giudice, S. Doglia, M. Milani, G. Vitiello (1986). Electromagnetic field and spontaneous symmetry breaking in biological matter. Nucl. Phys., B275 (FS 17), 185 - 199.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">E. Del Giudice, G. Preparata, G. Vitiello (1988). Water as a free electron laser. Phys. Rev. Lett., 61, 1085 – 1088.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">M. Jibu, K. Yasue (1995). Quantum brain dynamics and consciousness. Amsterdam: John Benjamins.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Giuseppe Vitiello, My Double Unveiled - The dissipative quantum model of brain. John Benjamins Publ. Co., Amsterdam 2001.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism#Proof_of_the_theorem">Scholarpedia proof of goldstone theorem - kibble</a></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFabriPicasso1966" class="citation journal cs1">Fabri, E.; Picasso, L. E. (1966-03-07). "Quantum Field Theory and Approximate Symmetries". <i>Physical Review Letters</i>. <b>16</b> (10). American Physical Society (APS): 408–410. <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/1966PhRvL..16..408F">1966PhRvL..16..408F</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%2Fphysrevlett.16.408.2">10.1103/physrevlett.16.408.2</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/0031-9007">0031-9007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Quantum+Field+Theory+and+Approximate+Symmetries&rft.volume=16&rft.issue=10&rft.pages=408-410&rft.date=1966-03-07&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2Fphysrevlett.16.408.2&rft_id=info%3Abibcode%2F1966PhRvL..16..408F&rft.aulast=Fabri&rft.aufirst=E.&rft.au=Picasso%2C+L.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.sagredo.eu/lezioni/invar/invar14.pdf">Fabri dispense 1965 </a></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://www.theorie.physik.uni-muenchen.de/activities/lectures/twentyfourth_series/murayama_2/video_murayama_colloquium/index.html">https://www.theorie.physik.uni-muenchen.de/activities/lectures/twentyfourth_series/murayama_2/video_murayama_colloquium/index.html</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220114175338/https://www.theorie.physik.uni-muenchen.de/activities/lectures/twentyfourth_series/murayama_2/video_murayama_colloquium/index.html">Archived</a> 2022-01-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> - min. 30-60</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Haruki Watanabe, Hitoshi Murayama, Unified Description of Nambu Goldstone Bosons without Lorentz invariance Phys. Rev. Lett. 108,251602,2012, <a rel="nofollow" class="external free" href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.251602">https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.251602</a></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220114175338/https://www.theorie.physik.uni-muenchen.de/activities/lectures/twentyfourth_series/murayama_2/video_murayama_colloquium/index.html">"min 42"</a>. Archived from <a rel="nofollow" class="external text" href="https://www.theorie.physik.uni-muenchen.de/activities/lectures/twentyfourth_series/murayama_2/video_murayama_colloquium/index.html">the original</a> on 2022-01-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-01-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=min+42&rft_id=https%3A%2F%2Fwww.theorie.physik.uni-muenchen.de%2Factivities%2Flectures%2Ftwentyfourth_series%2Fmurayama_2%2Fvideo_murayama_colloquium%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.sagredo.eu/lezioni/invar/invar14.pdf">Fabri dispense 1965 </a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoinkaDykeLinghamKinnunen2017" class="citation journal cs1">Hoinka, Sascha; Dyke, Paul; Lingham, Marcus G.; Kinnunen, Jami J.; Bruun, Georg M.; Vale, Chris J. (2017). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/nphys4187">"Goldstone mode and pair-breaking excitations in atomic Fermi superfluids"</a>. <i>Nature Physics</i>. <b>13</b> (10): 943–946. <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/1707.00406">1707.00406</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/2017NatPh..13..943H">2017NatPh..13..943H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphys4187">10.1038/nphys4187</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:59392755">59392755</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Physics&rft.atitle=Goldstone+mode+and+pair-breaking+excitations+in+atomic+Fermi+superfluids&rft.volume=13&rft.issue=10&rft.pages=943-946&rft.date=2017&rft_id=info%3Aarxiv%2F1707.00406&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A59392755%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2Fnphys4187&rft_id=info%3Abibcode%2F2017NatPh..13..943H&rft.aulast=Hoinka&rft.aufirst=Sascha&rft.au=Dyke%2C+Paul&rft.au=Lingham%2C+Marcus+G.&rft.au=Kinnunen%2C+Jami+J.&rft.au=Bruun%2C+Georg+M.&rft.au=Vale%2C+Chris+J.&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2Fnphys4187&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeutwyler" class="citation web cs1">Leutwyler, H. <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/311331/files/9609466.pdf">"Phonons as Goldstone bosons"</a> <span class="cs1-format">(PDF)</span>. <i>cds.cern.ch</i><span class="reference-accessdate">. Retrieved <span class="nowrap">4 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=cds.cern.ch&rft.atitle=Phonons+as+Goldstone+bosons&rft.aulast=Leutwyler&rft.aufirst=H.&rft_id=https%3A%2F%2Fcds.cern.ch%2Frecord%2F311331%2Ffiles%2F9609466.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLykkenSpiropulu2013" class="citation journal cs1">Lykken, Joseph; Spiropulu, Maria (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.1063%2FPT.3.2212">"The future of the Higgs boson"</a>. <i>Physics Today</i>. <b>66</b> (12): 28–33. <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/2013PhT....66l..28L">2013PhT....66l..28L</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/10.1063%2FPT.3.2212">10.1063/PT.3.2212</a></span>. <a href="/wiki/OSTI_(identifier)" class="mw-redirect" title="OSTI (identifier)">OSTI</a> <a rel="nofollow" class="external text" href="https://www.osti.gov/biblio/1131296">1131296</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=The+future+of+the+Higgs+boson&rft.volume=66&rft.issue=12&rft.pages=28-33&rft.date=2013&rft_id=info%3Adoi%2F10.1063%2FPT.3.2212&rft_id=https%3A%2F%2Fwww.osti.gov%2Fbiblio%2F1131296%23id-name%3DOSTI&rft_id=info%3Abibcode%2F2013PhT....66l..28L&rft.aulast=Lykken&rft.aufirst=Joseph&rft.au=Spiropulu%2C+Maria&rft_id=https%3A%2F%2Fdoi.org%2F10.1063%252FPT.3.2212&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLykkenSpiropulu2013" class="citation journal cs1">Lykken, Joseph; Spiropulu, Maria (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.1063%2FPT.3.2212">"The future of the Higgs boson"</a>. <i>Physics Today</i>. <b>66</b> (12): 28–33. <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/2013PhT....66l..28L">2013PhT....66l..28L</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/10.1063%2FPT.3.2212">10.1063/PT.3.2212</a></span>. <a href="/wiki/OSTI_(identifier)" class="mw-redirect" title="OSTI (identifier)">OSTI</a> <a rel="nofollow" class="external text" href="https://www.osti.gov/biblio/1131296">1131296</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=The+future+of+the+Higgs+boson&rft.volume=66&rft.issue=12&rft.pages=28-33&rft.date=2013&rft_id=info%3Adoi%2F10.1063%2FPT.3.2212&rft_id=https%3A%2F%2Fwww.osti.gov%2Fbiblio%2F1131296%23id-name%3DOSTI&rft_id=info%3Abibcode%2F2013PhT....66l..28L&rft.aulast=Lykken&rft.aufirst=Joseph&rft.au=Spiropulu%2C+Maria&rft_id=https%3A%2F%2Fdoi.org%2F10.1063%252FPT.3.2212&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLowManohar2002" class="citation journal cs1">Low, I.; Manohar, A.V. (February 2002). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevLett.88.101602">"Spontaneously Broken Spacetime Symmetries and Goldstone's Theorem"</a>. <i>Phys. Rev. Lett</i>. <b>88</b> (10): 101602–101605. <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/0110285">hep-th/0110285</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/2002PhRvL..88j1602L">2002PhRvL..88j1602L</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%2FPhysRevLett.88.101602">10.1103/PhysRevLett.88.101602</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11909340">11909340</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:15997403">15997403</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+Lett.&rft.atitle=Spontaneously+Broken+Spacetime+Symmetries+and+Goldstone%27s+Theorem&rft.volume=88&rft.issue=10&rft.pages=101602-101605&rft.date=2002-02&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15997403%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2002PhRvL..88j1602L&rft_id=info%3Aarxiv%2Fhep-th%2F0110285&rft_id=info%3Apmid%2F11909340&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.88.101602&rft.aulast=Low&rft.aufirst=I.&rft.au=Manohar%2C+A.V.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevLett.88.101602&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGan2019" class="citation book cs1">Gan, Woon Siong (2019). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-981-13-8751-7_11">"Spontaneous Symmetry Breaking and Phonon as the Goldstone Mode"</a>. <i>Gauge Invariance Approach to Acoustic Fields</i>. pp. 59–62. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-981-13-8751-7_11">10.1007/978-981-13-8751-7_11</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-13-8750-0" title="Special:BookSources/978-981-13-8750-0"><bdi>978-981-13-8750-0</bdi></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:201256113">201256113</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Spontaneous+Symmetry+Breaking+and+Phonon+as+the+Goldstone+Mode&rft.btitle=Gauge+Invariance+Approach+to+Acoustic+Fields&rft.pages=59-62&rft.date=2019&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A201256113%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-981-13-8751-7_11&rft.isbn=978-981-13-8750-0&rft.aulast=Gan&rft.aufirst=Woon+Siong&rft_id=https%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2F978-981-13-8751-7_11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVolkovAkulov1973" class="citation journal cs1">Volkov, D.V.; Akulov, V. (1973). "Is the neutrino a Goldstone particle?". <i>Physics Letters</i>. <b>B46</b> (1): 109–110. <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/1973PhLB...46..109V">1973PhLB...46..109V</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%2F0370-2693%2873%2990490-5">10.1016/0370-2693(73)90490-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters&rft.atitle=Is+the+neutrino+a+Goldstone+particle%3F&rft.volume=B46&rft.issue=1&rft.pages=109-110&rft.date=1973&rft_id=info%3Adoi%2F10.1016%2F0370-2693%2873%2990490-5&rft_id=info%3Abibcode%2F1973PhLB...46..109V&rft.aulast=Volkov&rft.aufirst=D.V.&rft.au=Akulov%2C+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSalam1974" class="citation journal cs1">Salam, A.; et al. (1974). "On Goldstone fermion". <i>Physics Letters</i>. <b>B49</b> (5): 465–467. <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/1974PhLB...49..465S">1974PhLB...49..465S</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%2F0370-2693%2874%2990637-6">10.1016/0370-2693(74)90637-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters&rft.atitle=On+Goldstone+fermion&rft.volume=B49&rft.issue=5&rft.pages=465-467&rft.date=1974&rft_id=info%3Adoi%2F10.1016%2F0370-2693%2874%2990637-6&rft_id=info%3Abibcode%2F1974PhLB...49..465S&rft.aulast=Salam&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGoldstone+boson" class="Z3988"></span></span> </li> </ol></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Particles_in_physics" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align: center;"><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:Particles" title="Template:Particles"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Particles" title="Template talk:Particles"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Particles" title="Special:EditPage/Template:Particles"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Particles_in_physics" style="font-size:114%;margin:0 4em"><a href="/wiki/Particle_physics" title="Particle physics">Particles in physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Elementary_particle" title="Elementary particle">Elementary</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" 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%;text-align: center;"><a href="/wiki/Fermion" title="Fermion">Fermions</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%;font-weight:normal;"><a href="/wiki/Quark" title="Quark">Quarks</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/Up_quark" title="Up quark">Up (quark</a></li> <li><a href="/wiki/Up_antiquark" class="mw-redirect" title="Up antiquark">antiquark)</a></li> <li><a href="/wiki/Down_quark" title="Down quark">Down (quark</a></li> <li><a href="/wiki/Down_antiquark" class="mw-redirect" title="Down antiquark">antiquark)</a></li> <li><a href="/wiki/Charm_quark" title="Charm quark">Charm (quark</a></li> <li><a href="/wiki/Charm_antiquark" class="mw-redirect" title="Charm antiquark">antiquark)</a></li> <li><a href="/wiki/Strange_quark" title="Strange quark">Strange (quark</a></li> <li><a href="/wiki/Strange_antiquark" class="mw-redirect" title="Strange antiquark">antiquark)</a></li> <li><a href="/wiki/Top_quark" title="Top quark">Top (quark</a></li> <li><a href="/wiki/Top_antiquark" class="mw-redirect" title="Top antiquark">antiquark)</a></li> <li><a href="/wiki/Bottom_quark" title="Bottom quark">Bottom (quark</a></li> <li><a href="/wiki/Bottom_antiquark" class="mw-redirect" title="Bottom antiquark">antiquark)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Lepton" title="Lepton">Leptons</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/Electron" title="Electron">Electron</a></li> <li><a href="/wiki/Positron" title="Positron">Positron</a></li> <li><a href="/wiki/Muon" title="Muon">Muon</a></li> <li><a href="/wiki/Muon" title="Muon">Antimuon</a></li> <li><a href="/wiki/Tau_(particle)" title="Tau (particle)">Tau</a></li> <li><a href="/wiki/Tau_(particle)" title="Tau (particle)">Antitau</a></li> <li><a href="/wiki/Neutrino" title="Neutrino">Neutrino</a> <ul><li><a href="/wiki/Electron_neutrino" title="Electron neutrino">Electron neutrino</a></li> <li><a href="/wiki/Neutrino#Antineutrinos" title="Neutrino">Electron antineutrino</a></li> <li><a href="/wiki/Muon_neutrino" title="Muon neutrino">Muon neutrino</a></li> <li><a href="/wiki/Neutrino#Antineutrinos" title="Neutrino">Muon antineutrino</a></li> <li><a href="/wiki/Tau_neutrino" title="Tau neutrino">Tau neutrino</a></li> <li><a href="/wiki/Neutrino#Antineutrinos" title="Neutrino">Tau antineutrino</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Boson" title="Boson">Bosons</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:4em;font-weight:normal; text-align: center;"><a href="/wiki/Gauge_boson" title="Gauge boson">Gauge</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Photon" title="Photon">Photon</a></li> <li><a href="/wiki/Gluon" title="Gluon">Gluon</a></li> <li><a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W and Z bosons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:4em;font-weight:normal; text-align: center;"><a href="/wiki/Scalar_boson" title="Scalar boson">Scalar</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson </a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;;font-weight:normal; text-align: center;"><a href="/wiki/Ghost_(physics)" title="Ghost (physics)">Ghost fields</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/Faddeev%E2%80%93Popov_ghost" title="Faddeev–Popov ghost">Faddeev–Popov ghosts</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Hypothetical_particles" class="mw-redirect" title="Hypothetical particles">Hypothetical</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%;font-weight:normal; text-align: center;"><a href="/wiki/Superpartner" title="Superpartner">Superpartners</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%;font-weight:normal; text-align: center;"><a href="/wiki/Gaugino" title="Gaugino">Gauginos</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/Gluino" title="Gluino">Gluino</a></li> <li><a href="/wiki/Gravitino" title="Gravitino">Gravitino</a></li> <li><a href="/wiki/Photino" title="Photino">Photino</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;">Others</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/Axino" title="Axino">Axino</a></li> <li><a href="/wiki/Chargino" title="Chargino">Chargino</a></li> <li><a href="/wiki/Higgsino" title="Higgsino">Higgsino</a></li> <li><a href="/wiki/Neutralino" title="Neutralino">Neutralino</a></li> <li><a href="/wiki/Sfermion" title="Sfermion">Sfermion</a> (<a href="/wiki/Stop_squark" title="Stop squark">Stop squark</a>)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;">Others</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/Axion" title="Axion">Axion</a></li> <li><a href="/wiki/Curvaton" title="Curvaton">Curvaton</a></li> <li><a href="/wiki/Dilaton" title="Dilaton">Dilaton</a></li> <li><a href="/wiki/Dual_graviton" title="Dual graviton">Dual graviton</a></li> <li><a href="/wiki/Graviphoton" title="Graviphoton">Graviphoton</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Inflaton" title="Inflaton">Inflaton</a></li> <li><a href="/wiki/Leptoquark" title="Leptoquark">Leptoquark</a></li> <li><a href="/wiki/Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></li> <li><a href="/wiki/Majoron" title="Majoron">Majoron</a></li> <li><a href="/wiki/Majorana_fermion" title="Majorana fermion">Majorana fermion</a></li> <li><a href="/wiki/Dark_photon" title="Dark photon">Dark photon</a></li> <li><a href="/wiki/Preon" title="Preon">Preon</a></li> <li><a href="/wiki/Sterile_neutrino" title="Sterile neutrino">Sterile neutrino</a></li> <li><a href="/wiki/Tachyon" title="Tachyon">Tachyon</a></li> <li><a href="/wiki/W%E2%80%B2_and_Z%E2%80%B2_bosons" title="W′ and Z′ bosons">W′ and Z′ bosons</a></li> <li><a href="/wiki/X_and_Y_bosons" title="X and Y bosons">X and Y bosons</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Bound_state" title="Bound state">Composite</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Hadrons" scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Hadron" title="Hadron">Hadrons</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%;font-weight:normal; text-align: center;"><a href="/wiki/Baryon" title="Baryon">Baryons</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/Nucleon" title="Nucleon">Nucleon</a> <ul><li><a href="/wiki/Proton" title="Proton">Proton</a></li> <li><a href="/wiki/Antiproton" title="Antiproton">Antiproton</a></li> <li><a href="/wiki/Neutron" title="Neutron">Neutron</a></li> <li><a href="/wiki/Antineutron" title="Antineutron">Antineutron</a></li></ul></li> <li><a href="/wiki/Delta_baryon" title="Delta baryon">Delta baryon</a></li> <li><a href="/wiki/Lambda_baryon" title="Lambda baryon">Lambda baryon</a></li> <li><a href="/wiki/Sigma_baryon" title="Sigma baryon">Sigma baryon</a></li> <li><a href="/wiki/Xi_baryon" title="Xi baryon">Xi baryon</a></li> <li><a href="/wiki/Omega_baryon" title="Omega baryon">Omega baryon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Meson" title="Meson">Mesons</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/Pion" title="Pion">Pion</a></li> <li><a href="/wiki/Rho_meson" title="Rho meson">Rho meson</a></li> <li><a href="/wiki/Eta_meson" class="mw-redirect" title="Eta meson">Eta and eta prime mesons</a></li> <li><a href="/wiki/Bottom_eta_meson" title="Bottom eta meson">Bottom eta meson</a></li> <li><a href="/wiki/Phi_meson" title="Phi meson">Phi meson</a></li> <li><a href="/wiki/J/psi_meson" title="J/psi meson">J/psi meson</a></li> <li><a href="/wiki/Omega_meson" title="Omega meson">Omega meson</a></li> <li><a href="/wiki/Upsilon_meson" title="Upsilon meson">Upsilon meson</a></li> <li><a href="/wiki/Kaon" title="Kaon">Kaon</a></li> <li><a href="/wiki/B_meson" title="B meson">B meson</a></li> <li><a href="/wiki/D_meson" title="D meson">D meson</a></li> <li><a href="/wiki/Quarkonium" title="Quarkonium">Quarkonium</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Exotic_hadron" title="Exotic hadron">Exotic hadrons</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/Tetraquark" title="Tetraquark">Tetraquark</a> (<a href="/wiki/Double-charm_tetraquark" title="Double-charm tetraquark">Double-charm tetraquark</a>)</li> <li><a href="/wiki/Pentaquark" title="Pentaquark">Pentaquark</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;;font-weight:normal; text-align: center;">Others</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/Atomic_nucleus" title="Atomic nucleus">Atomic nuclei</a></li> <li><a href="/wiki/Atom" title="Atom">Atoms</a></li> <li><a href="/wiki/Exotic_atom" title="Exotic atom">Exotic atoms</a> <ul><li><a href="/wiki/Positronium" title="Positronium">Positronium</a></li> <li><a href="/wiki/Muonium" title="Muonium">Muonium</a></li> <li><a href="/wiki/Tauonium" class="mw-redirect" title="Tauonium">Tauonium</a></li> <li><a href="/wiki/Onium" title="Onium">Onia</a></li> <li><a href="/wiki/Pionium" title="Pionium">Pionium</a></li> <li><a href="/wiki/Protonium" title="Protonium">Protonium</a></li></ul></li> <li><a href="/wiki/Superatom" title="Superatom">Superatoms</a></li> <li><a href="/wiki/Molecule" title="Molecule">Molecules</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Category:Hypothetical_composite_particles" title="Category:Hypothetical composite particles">Hypothetical</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><td colspan="2" class="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%;font-weight:normal;">Baryons</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/Hexaquark" title="Hexaquark">Hexaquark</a></li> <li><a href="/wiki/Heptaquark" title="Heptaquark">Heptaquark</a></li> <li><a href="/wiki/Skyrmion" title="Skyrmion">Skyrmion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mesons</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/Glueball" title="Glueball">Glueball</a></li> <li><a href="/wiki/Theta_meson" title="Theta meson">Theta meson</a></li> <li><a href="/wiki/T_meson" title="T meson">T meson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;;font-weight:normal; text-align: center;">Others</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/Mesonic_molecule" title="Mesonic molecule">Mesonic molecule</a></li> <li><a href="/wiki/Pomeron" title="Pomeron">Pomeron</a></li> <li><a href="/wiki/Diquark" title="Diquark">Diquark</a></li> <li><a href="/wiki/R-hadron" title="R-hadron">R-hadron</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anyon" title="Anyon">Anyon</a></li> <li><a href="/wiki/Davydov_soliton" title="Davydov soliton">Davydov soliton</a></li> <li><a href="/wiki/Dropleton" title="Dropleton">Dropleton</a></li> <li><a href="/wiki/Exciton" title="Exciton">Exciton</a></li> <li><a href="/wiki/Fracton_(subdimensional_particle)" title="Fracton (subdimensional particle)">Fracton</a></li> <li><a href="/wiki/Electron_hole" title="Electron hole">Hole</a></li> <li><a href="/wiki/Magnon" title="Magnon">Magnon</a></li> <li><a href="/wiki/Phonon" title="Phonon">Phonon</a></li> <li><a href="/wiki/Plasmaron" title="Plasmaron">Plasmaron</a></li> <li><a href="/wiki/Plasmon" title="Plasmon">Plasmon</a></li> <li><a href="/wiki/Polariton" title="Polariton">Polariton</a></li> <li><a href="/wiki/Polaron" title="Polaron">Polaron</a></li> <li><a href="/wiki/Roton" title="Roton">Roton</a></li> <li><a href="/wiki/Trion_(physics)" title="Trion (physics)">Trion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_baryons" title="List of baryons">Baryons</a></li> <li><a href="/wiki/List_of_mesons" title="List of mesons">Mesons</a></li> <li><a href="/wiki/List_of_particles" title="List of particles">Particles</a></li> <li><a href="/wiki/List_of_quasiparticles" title="List of quasiparticles">Quasiparticles</a></li> <li><a href="/wiki/Timeline_of_particle_discoveries" title="Timeline of particle discoveries">Timeline of particle discoveries</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_subatomic_physics" title="History of subatomic physics">History of subatomic physics</a> <ul><li><a href="/wiki/Timeline_of_atomic_and_subatomic_physics" title="Timeline of atomic and subatomic physics">timeline</a></li></ul></li> <li><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> <ul><li><a href="/wiki/Mathematical_formulation_of_the_Standard_Model" title="Mathematical formulation of the Standard Model">mathematical formulation</a></li></ul></li> <li><a href="/wiki/Subatomic_particle" title="Subatomic particle">Subatomic particles</a></li> <li><a href="/wiki/Particle" title="Particle">Particles</a></li> <li><a href="/wiki/Antiparticle" title="Antiparticle">Antiparticles</a></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li> <li><a href="/wiki/Eightfold_way_(physics)" title="Eightfold way (physics)">Eightfold way</a> <ul><li><a href="/wiki/Quark_model" title="Quark model">Quark model</a></li></ul></li> <li><a href="/wiki/Exotic_matter" title="Exotic matter">Exotic matter</a></li> <li><a href="/wiki/Massless_particle" title="Massless particle">Massless particle</a></li> <li><a href="/wiki/Relativistic_particle" title="Relativistic particle">Relativistic particle</a></li> <li><a href="/wiki/Virtual_particle" title="Virtual particle">Virtual particle</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li> <li><a href="/wiki/Particle_chauvinism" title="Particle chauvinism">Particle chauvinism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align: center;"><div><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span> <b><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></b></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Goldstone_boson&oldid=1228630845">https://en.wikipedia.org/w/index.php?title=Goldstone_boson&oldid=1228630845</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Bosons" title="Category:Bosons">Bosons</a></li><li><a href="/wiki/Category:Quantum_field_theory" title="Category:Quantum field theory">Quantum field theory</a></li><li><a href="/wiki/Category:Mathematical_physics" title="Category:Mathematical physics">Mathematical physics</a></li><li><a href="/wiki/Category:Physics_theorems" title="Category:Physics theorems">Physics theorems</a></li><li><a href="/wiki/Category:Subatomic_particles_with_spin_0" title="Category:Subatomic particles with spin 0">Subatomic particles with spin 0</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 12 June 2024, at 08:49<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Goldstone_boson&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-djcsx","wgBackendResponseTime":208,"wgPageParseReport":{"limitreport":{"cputime":"0.499","walltime":"0.666","ppvisitednodes":{"value":3080,"limit":1000000},"postexpandincludesize":{"value":112378,"limit":2097152},"templateargumentsize":{"value":3126,"limit":2097152},"expansiondepth":{"value":10,"limit":100},"expensivefunctioncount":{"value":3,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":82149,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 481.408 1 -total"," 36.53% 175.874 13 Template:Cite_journal"," 23.71% 114.127 10 Template:Navbox"," 19.51% 93.916 1 Template:Particles"," 14.30% 68.858 1 Template:Short_description"," 9.59% 46.171 2 Template:Pagetype"," 5.77% 27.763 23 Template:Math"," 5.03% 24.222 1 Template:Redirects_here"," 3.29% 15.817 26 Template:Main_other"," 3.07% 14.778 2 Template:Cite_book"]},"scribunto":{"limitreport-timeusage":{"value":"0.293","limit":"10.000"},"limitreport-memusage":{"value":5688647,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-w7plk","timestamp":"20241122150019","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Goldstone boson","url":"https:\/\/en.wikipedia.org\/wiki\/Goldstone_boson","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1153441","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1153441","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-05-01T06:00:45Z","dateModified":"2024-06-12T08:49:34Z","headline":"massless boson that must be present in a quantum system with spontaneously broken symmetry"}</script> </body> </html>