C-symmetry - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-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-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>C-symmetry - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-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-sticky-header-enabled 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":"e03fd319-6edc-4127-9d2c-4f97d7197e58","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"C-symmetry","wgTitle":"C-symmetry","wgCurRevisionId":1271302481,"wgRevisionId":1271302481,"wgArticleId":151001,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description matches Wikidata","Articles needing additional references from December 2008","All articles needing additional references","All articles with specifically marked weasel-worded phrases","Articles with specifically marked weasel-worded phrases from November 2020","Quantum field theory","Symmetry","Antimatter"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"C-symmetry","wgRelevantArticleId":151001,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"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":40000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q513656","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.math.styles":"ready","ext.cite.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"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.scribunto.logs","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"];</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.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.19"> <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="C-symmetry - 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/C-symmetry"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=C-symmetry&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/C-symmetry"> <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-C-symmetry rootpage-C-symmetry 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" title="Main menu" > <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><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages"><span>Special pages</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/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=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=C-symmetry" 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=C-symmetry" 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/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=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=C-symmetry" 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=C-symmetry" 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-Informal_overview" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Informal_overview"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Informal overview</span> </div> </a> <button aria-controls="toc-Informal_overview-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 Informal overview subsection</span> </button> <ul id="toc-Informal_overview-sublist" class="vector-toc-list"> <li id="toc-In_classical_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_classical_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>In classical fields</span> </div> </a> <ul id="toc-In_classical_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_quantum_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_quantum_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>In quantum theory</span> </div> </a> <ul id="toc-In_quantum_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>In geometry</span> </div> </a> <ul id="toc-In_geometry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Charge_conjugation_for_Dirac_fields" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Charge_conjugation_for_Dirac_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Charge conjugation for Dirac fields</span> </div> </a> <button aria-controls="toc-Charge_conjugation_for_Dirac_fields-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 Charge conjugation for Dirac fields subsection</span> </button> <ul id="toc-Charge_conjugation_for_Dirac_fields-sublist" class="vector-toc-list"> <li id="toc-Charge_conjugation,_chirality,_helicity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Charge_conjugation,_chirality,_helicity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Charge conjugation, chirality, helicity</span> </div> </a> <ul id="toc-Charge_conjugation,_chirality,_helicity-sublist" class="vector-toc-list"> <li id="toc-Weyl_spinors" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Weyl_spinors"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Weyl spinors</span> </div> </a> <ul id="toc-Weyl_spinors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Charge_conjugation_in_the_chiral_basis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Charge_conjugation_in_the_chiral_basis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Charge conjugation in the chiral basis</span> </div> </a> <ul id="toc-Charge_conjugation_in_the_chiral_basis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Majorana_condition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Majorana_condition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Majorana condition</span> </div> </a> <ul id="toc-Majorana_condition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_interpretation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Geometric_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.4</span> <span>Geometric interpretation</span> </div> </a> <ul id="toc-Geometric_interpretation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Charge_conjugation_for_quantized_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Charge_conjugation_for_quantized_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Charge conjugation for quantized fields</span> </div> </a> <ul id="toc-Charge_conjugation_for_quantized_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Charge_reversal_in_electroweak_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Charge_reversal_in_electroweak_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Charge reversal in electroweak theory</span> </div> </a> <ul id="toc-Charge_reversal_in_electroweak_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Scalar_fields" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Scalar_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Scalar fields</span> </div> </a> <ul id="toc-Scalar_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combination_of_charge_and_parity_reversal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Combination_of_charge_and_parity_reversal"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Combination of charge and parity reversal</span> </div> </a> <ul id="toc-Combination_of_charge_and_parity_reversal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_general_settings" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_general_settings"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In general settings</span> </div> </a> <ul id="toc-In_general_settings-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">6</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">7</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">8</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" title="Table of Contents" > <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">C-symmetry</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 20 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-20" 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">20 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Simetria_C" title="Simetria C – Catalan" lang="ca" hreflang="ca" data-title="Simetria C" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/C-symetrie" title="C-symetrie – Czech" lang="cs" hreflang="cs" data-title="C-symetrie" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ladungskonjugation" title="Ladungskonjugation – German" lang="de" hreflang="de" data-title="Ladungskonjugation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Conjugaci%C3%B3n_de_carga" title="Conjugación de carga – Spanish" lang="es" hreflang="es" data-title="Conjugación de carga" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%82%D8%A7%D8%B1%D9%86_%D8%B3%DB%8C" title="تقارن سی – Persian" lang="fa" hreflang="fa" data-title="تقارن سی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Sym%C3%A9trie_C" title="Symétrie C – French" lang="fr" hreflang="fr" data-title="Symétrie C" 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/%EC%A0%84%ED%95%98_%EC%BC%A4%EB%A0%88_%EB%8C%80%EC%B9%AD" title="전하 켤레 대칭 – Korean" lang="ko" hreflang="ko" data-title="전하 켤레 대칭" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Simmetria_C" title="Simmetria C – Italian" lang="it" hreflang="it" data-title="Simmetria C" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%B6lt%C3%A9st%C3%BCkr%C3%B6z%C3%A9s" title="Töltéstükrözés – Hungarian" lang="hu" hreflang="hu" data-title="Töltéstükrözés" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ladingconjugatie" title="Ladingconjugatie – Dutch" lang="nl" hreflang="nl" data-title="Ladingconjugatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%8D%B7%E9%9B%BB%E5%85%B1%E5%BD%B9%E5%A4%89%E6%8F%9B" title="荷電共役変換 – Japanese" lang="ja" hreflang="ja" data-title="荷電共役変換" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ladningssymmetri" title="Ladningssymmetri – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ladningssymmetri" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/C-%E0%A8%B8%E0%A8%AE%E0%A8%BF%E0%A9%B1%E0%A8%9F%E0%A8%B0%E0%A9%80" title="C-ਸਮਿੱਟਰੀ – Punjabi" lang="pa" hreflang="pa" data-title="C-ਸਮਿੱਟਰੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Odbicie_%C5%82adunkowe" title="Odbicie ładunkowe – Polish" lang="pl" hreflang="pl" data-title="Odbicie ładunkowe" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Simetria_C" title="Simetria C – Portuguese" lang="pt" hreflang="pt" data-title="Simetria C" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D1%80%D1%8F%D0%B4%D0%BE%D0%B2%D0%BE%D0%B5_%D1%81%D0%BE%D0%BF%D1%80%D1%8F%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Зарядовое сопряжение – Russian" lang="ru" hreflang="ru" data-title="Зарядовое сопряжение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Charge_conjugation" title="Charge conjugation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Charge conjugation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Simetrija_C" title="Simetrija C – Slovenian" lang="sl" hreflang="sl" data-title="Simetrija C" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B0%D1%80%D1%8F%D0%B4%D0%BE%D0%B2%D0%B5_%D1%81%D0%BF%D1%80%D1%8F%D0%B6%D0%B5%D0%BD%D0%BD%D1%8F" title="Зарядове спряження – Ukrainian" lang="uk" hreflang="uk" data-title="Зарядове спряження" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9B%BB%E8%8D%B7%E5%85%B1%E8%BB%9B%E5%B0%8D%E7%A8%B1" title="電荷共軛對稱 – Chinese" lang="zh" hreflang="zh" data-title="電荷共軛對稱" 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/Q513656#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/C-symmetry" 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:C-symmetry" 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/C-symmetry"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=C-symmetry&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=C-symmetry&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/C-symmetry"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=C-symmetry&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=C-symmetry&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/C-symmetry" 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/C-symmetry" 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="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=C-symmetry&oldid=1271302481" 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=C-symmetry&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=C-symmetry&id=1271302481&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%2FC-symmetry"><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%2FC-symmetry"><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=C-symmetry&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=C-symmetry&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/Q513656" 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"></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">Symmetry of physical laws under a charge-conjugation transformation</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/C-symmetry" title="Special:EditPage/C-symmetry">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22C-symmetry%22">"C-symmetry"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22C-symmetry%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22C-symmetry%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22C-symmetry%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22C-symmetry%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22C-symmetry%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2008</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, <b>charge conjugation</b> is a <a href="/wiki/Transformation_(mathematics)" class="mw-redirect" title="Transformation (mathematics)">transformation</a> that switches all <a href="/wiki/Particle" title="Particle">particles</a> with their corresponding <a href="/wiki/Antiparticle" title="Antiparticle">antiparticles</a>, thus changing the sign of all <a href="/wiki/Charge_(physics)" title="Charge (physics)">charges</a>: not only <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> but also the charges relevant to other forces. The term <b>C-symmetry</b> is an abbreviation of the phrase "charge conjugation symmetry", and is used in discussions of the symmetry of physical laws under charge-conjugation. Other important discrete symmetries are <a href="/wiki/P-symmetry" class="mw-redirect" title="P-symmetry">P-symmetry</a> (parity) and <a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a> (time reversal). </p><p>These discrete symmetries, C, P and T, are symmetries of the equations that describe the known <a href="/wiki/Fundamental_force" class="mw-redirect" title="Fundamental force">fundamental forces</a> of nature: <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, <a href="/wiki/Gravity" title="Gravity">gravity</a>, the <a href="/wiki/Strong_interaction" title="Strong interaction">strong</a> and the <a href="/wiki/Weak_interaction" title="Weak interaction">weak interactions</a>. Verifying whether some given mathematical equation correctly models <a href="/wiki/Nature" title="Nature">nature</a> requires giving physical interpretation not only to <a href="/wiki/Continuous_symmetry" title="Continuous symmetry">continuous symmetries</a>, such as <a href="/wiki/Motion" title="Motion">motion</a> in time, but also to its <a href="/wiki/Discrete_symmetries" class="mw-redirect" title="Discrete symmetries">discrete symmetries</a>, and then determining whether nature adheres to these symmetries. Unlike the continuous symmetries, the interpretation of the discrete symmetries is a bit more intellectually demanding and confusing. An early surprise appeared in the 1950s, when <a href="/wiki/Chien_Shiung_Wu" class="mw-redirect" title="Chien Shiung Wu">Chien Shiung Wu</a> demonstrated that the weak interaction violated P-symmetry. For several decades, it appeared that the combined symmetry CP was preserved, until <a href="/wiki/CP_violation" title="CP violation">CP-violating</a> interactions were discovered. Both discoveries lead to <a href="/wiki/Nobel_Prize" title="Nobel Prize">Nobel Prizes</a>. </p><p>The C-symmetry is particularly troublesome, physically, as the universe is primarily filled with <a href="/wiki/Matter" title="Matter">matter</a>, not <a href="/wiki/Anti-matter" class="mw-redirect" title="Anti-matter">anti-matter</a>, whereas the naive C-symmetry of the physical laws suggests that there should be equal amounts of both. It is currently believed that CP-violation during the early universe can account for the "excess" matter, although the debate is not settled. Earlier textbooks on <a href="/wiki/Cosmology" title="Cosmology">cosmology</a>, predating the 1970s,<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Avoid_weasel_words" class="mw-redirect" title="Wikipedia:Avoid weasel words"><span title="The material near this tag possibly uses too vague attribution or weasel words. (November 2020)">which?</span></a></i>]</sup> routinely suggested that perhaps distant galaxies were made entirely of anti-matter, thus maintaining a net balance of zero in the universe. </p><p>This article focuses on exposing and articulating the C-symmetry of various important equations and theoretical systems, including the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> and the structure of <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. The various <a href="/wiki/Fundamental_particles" class="mw-redirect" title="Fundamental particles">fundamental particles</a> can be classified according to behavior under charge conjugation; this is described in the article on <a href="/wiki/C-parity" class="mw-redirect" title="C-parity">C-parity</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Informal_overview">Informal overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=1" title="Edit section: Informal overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Charge conjugation occurs as a symmetry in three different but closely related settings: a symmetry of the (classical, non-quantized) solutions of several notable differential equations, including the <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a> and the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>, a symmetry of the corresponding quantum fields, and in a general setting, a symmetry in (pseudo-)<a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>. In all three cases, the symmetry is ultimately revealed to be a symmetry under <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>, although exactly what is being conjugated where can be at times obfuscated, depending on notation, coordinate choices and other factors. </p> <div class="mw-heading mw-heading3"><h3 id="In_classical_fields">In classical fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=2" title="Edit section: In classical fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The charge conjugation symmetry is interpreted as that of <a href="/wiki/Electrical_charge" class="mw-redirect" title="Electrical charge">electrical charge</a>, because in all three cases (classical, quantum and geometry), one can construct <a href="/wiki/Noether_current" class="mw-redirect" title="Noether current">Noether currents</a> that resemble those of <a href="/wiki/Classical_electrodynamics" class="mw-redirect" title="Classical electrodynamics">classical electrodynamics</a>. This arises because electrodynamics itself, via <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, can be interpreted as a structure on a <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a>, the so-called <a href="/wiki/Circle_bundle" title="Circle bundle">circle bundle</a>. This provides a geometric interpretation of electromagnetism: the <a href="/wiki/Electromagnetic_potential" class="mw-redirect" title="Electromagnetic potential">electromagnetic potential</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9277f5286335ab99c040c9c9151ab752d3bedc49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.967ex; height:2.843ex;" alt="{\displaystyle A_{\mu }}" /></span> is interpreted as the <a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">gauge connection</a> (the <a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann connection</a>) on the circle bundle. This geometric interpretation then allows (literally almost) anything possessing a complex-number-valued structure to be coupled to the electromagnetic field, provided that this coupling is done in a <a href="/wiki/Gauge-invariant" class="mw-redirect" title="Gauge-invariant">gauge-invariant</a> way. Gauge symmetry, in this geometric setting, is a statement that, as one moves around on the circle, the coupled object must also transform in a "circular way", tracking in a corresponding fashion. More formally, one says that the equations must be gauge invariant under a change of local <a href="/wiki/Coordinate_frame" class="mw-redirect" title="Coordinate frame">coordinate frames</a> on the circle. For U(1), this is just the statement that the system is invariant under multiplication by a phase factor <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^{i\phi (x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\phi (x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353fdaac0375b4d40470f97e8356893ac99602e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle e^{i\phi (x)}}" /></span> that depends on the (space-time) coordinate <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 x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}" /></span> In this geometric setting, charge conjugation can be understood as the discrete symmetry <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 z=(x+iy)\mapsto {\overline {z}}=(x-iy)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=(x+iy)\mapsto {\overline {z}}=(x-iy)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768492cd996193cad4ed319aa5a12af85a442c04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.981ex; height:2.843ex;" alt="{\displaystyle z=(x+iy)\mapsto {\overline {z}}=(x-iy)}" /></span> that performs complex conjugation, that reverses the sense of direction around the circle. </p> <div class="mw-heading mw-heading3"><h3 id="In_quantum_theory">In quantum theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=3" title="Edit section: In quantum theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, charge conjugation can be understood as the exchange of <a href="/wiki/Particle" title="Particle">particles</a> with <a href="/wiki/Anti-particle" class="mw-redirect" title="Anti-particle">anti-particles</a>. To understand this statement, one must have a minimal understanding of what quantum field theory is. In (vastly) simplified terms, it is a technique for performing calculations to obtain solutions for a system of coupled differential equations via <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theory</a>. A key ingredient to this process is the <a href="/wiki/Quantum_field" class="mw-redirect" title="Quantum field">quantum field</a>, one for each of the (free, uncoupled) differential equations in the system. A quantum field is conventionally written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=\int d^{3}p\sum _{\sigma ,n}e^{-ip\cdot x}a\left({\vec {p}},\sigma ,n\right)u\left({\vec {p}},\sigma ,n\right)+e^{ip\cdot x}a^{\dagger }\left({\vec {p}},\sigma ,n\right)v\left({\vec {p}},\sigma ,n\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>p</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo>,</mo> <mi>n</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>p</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>p</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=\int d^{3}p\sum _{\sigma ,n}e^{-ip\cdot x}a\left({\vec {p}},\sigma ,n\right)u\left({\vec {p}},\sigma ,n\right)+e^{ip\cdot x}a^{\dagger }\left({\vec {p}},\sigma ,n\right)v\left({\vec {p}},\sigma ,n\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353a6d8ddc25374284ca6ad2b008f02d30bad0e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:70.145ex; height:6.509ex;" alt="{\displaystyle \psi (x)=\int d^{3}p\sum _{\sigma ,n}e^{-ip\cdot x}a\left({\vec {p}},\sigma ,n\right)u\left({\vec {p}},\sigma ,n\right)+e^{ip\cdot x}a^{\dagger }\left({\vec {p}},\sigma ,n\right)v\left({\vec {p}},\sigma ,n\right)}" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" /></span> is the momentum, <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 \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span> is a spin label, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> is an auxiliary label for other states in the system. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\dagger }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab292dae7b377aefbd977dc19438b2b82da3303c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.192ex; height:2.676ex;" alt="{\displaystyle a^{\dagger }}" /></span> are <a href="/wiki/Creation_and_annihilation_operators" title="Creation and annihilation operators">creation and annihilation operators</a> (<a href="/wiki/Ladder_operator" title="Ladder operator">ladder operators</a>) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66f4b32a0181923cc1337a5634f38241e5c697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.491ex; height:2.009ex;" alt="{\displaystyle u,v}" /></span> are solutions to the (free, non-interacting, uncoupled) differential equation in question. The quantum field plays a central role because, in general, it is not known how to obtain exact solutions to the system of coupled differential questions. However, via perturbation theory, approximate solutions can be constructed as combinations of the free-field solutions. To perform this construction, one has to be able to extract and work with any one given free-field solution, on-demand, when required. The quantum field provides exactly this: it enumerates all possible free-field solutions in a vector space such that any one of them can be singled out at any given time, via the creation and annihilation operators. </p><p>The creation and annihilation operators obey the <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relations</a>, in that the one operator "undoes" what the other "creates". This implies that any given solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\left({\vec {p}},\sigma ,n\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\left({\vec {p}},\sigma ,n\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961da31bfb10a96d83f2a107f1d40a689cb190bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.644ex; height:2.843ex;" alt="{\displaystyle u\left({\vec {p}},\sigma ,n\right)}" /></span> must be paired with its "anti-solution" <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\left({\vec {p}},\sigma ,n\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\left({\vec {p}},\sigma ,n\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1468fd5db5ff4be6b85e1fe1d78537571e3367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.441ex; height:2.843ex;" alt="{\displaystyle v\left({\vec {p}},\sigma ,n\right)}" /></span> so that one undoes or cancels out the other. The pairing is to be performed so that all symmetries are preserved. As one is generally interested in <a href="/wiki/Lorentz_invariance" class="mw-redirect" title="Lorentz invariance">Lorentz invariance</a>, the quantum field contains an integral over all possible Lorentz coordinate frames, written above as an integral over all possible momenta (it is an integral over the fiber of the <a href="/wiki/Frame_bundle" title="Frame bundle">frame bundle</a>). The pairing requires that a given <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\left({\vec {p}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\left({\vec {p}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b8105ed577fb134b9822a6fde834becf3490af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.851ex; height:2.843ex;" alt="{\displaystyle u\left({\vec {p}}\right)}" /></span> is associated with a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\left({\vec {p}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\left({\vec {p}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0a25d49616aa5d4d9a2a90072bfdbafa698d430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.649ex; height:2.843ex;" alt="{\displaystyle v\left({\vec {p}}\right)}" /></span> of the opposite momentum and energy. The quantum field is also a sum over all possible spin states; the dual pairing again matching opposite spins. Likewise for any other quantum numbers, these are also paired as opposites. There is a technical difficulty in carrying out this dual pairing: one must describe what it means for some given solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" /></span> to be "dual to" some other solution <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ba401aee0189f8031d21020a0c640a03339c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.774ex; height:2.009ex;" alt="{\displaystyle v,}" /></span> and to describe it in such a way that it remains consistently dual when integrating over the fiber of the frame bundle, when integrating (summing) over the fiber that describes the spin, and when integrating (summing) over any other fibers that occur in the theory. </p><p>When the fiber to be integrated over is the U(1) fiber of electromagnetism, the dual pairing is such that the direction (orientation) on the fiber is reversed. When the fiber to be integrated over is the SU(3) fiber of the <a href="/wiki/Color_charge" title="Color charge">color charge</a>, the dual pairing again reverses orientation. This "just works" for SU(3) because it has two dual <a href="/wiki/Fundamental_representation" title="Fundamental representation">fundamental representations</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {3} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {3} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2210f71f47c2aef89cebab1b74ee6b14a05fa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {3} }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbf {3} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbf {3} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc137a065a85d90b74412389d43920940c95e0fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.452ex; height:2.843ex;" alt="{\displaystyle {\overline {\mathbf {3} }}}" /></span> which can be naturally paired. This prescription for a quantum field naturally generalizes to any situation where one can enumerate the continuous symmetries of the system, and define duals in a coherent, consistent fashion. The pairing ties together opposite <a href="/wiki/Charge_(physics)" title="Charge (physics)">charges</a> in the fully abstract sense. In physics, a charge is associated with a generator of a continuous symmetry. Different charges are associated with different eigenspaces of the <a href="/wiki/Casimir_invariant" class="mw-redirect" title="Casimir invariant">Casimir invariants</a> of the <a href="/wiki/Universal_enveloping_algebra" title="Universal enveloping algebra">universal enveloping algebra</a> for those symmetries. This is the case for <i>both</i> the Lorentz symmetry of the underlying <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> <a href="/wiki/Manifold_(mathematics)" class="mw-redirect" title="Manifold (mathematics)">manifold</a>, <i>as well as</i> the symmetries of any fibers in the fiber bundle posed above the spacetime manifold. Duality replaces the generator of the symmetry with minus the generator. Charge conjugation is thus associated with reflection along the <a href="/wiki/Line_bundle" title="Line bundle">line bundle</a> or <a href="/wiki/Determinant_bundle" class="mw-redirect" title="Determinant bundle">determinant bundle</a> of the space of symmetries. </p><p>The above then is a sketch of the general idea of a quantum field in quantum field theory. The physical interpretation is that 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 u\left({\vec {p}},\sigma ,n\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\left({\vec {p}},\sigma ,n\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961da31bfb10a96d83f2a107f1d40a689cb190bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.644ex; height:2.843ex;" alt="{\displaystyle u\left({\vec {p}},\sigma ,n\right)}" /></span> correspond to particles, and 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 v\left({\vec {p}},\sigma ,n\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\left({\vec {p}},\sigma ,n\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1468fd5db5ff4be6b85e1fe1d78537571e3367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.441ex; height:2.843ex;" alt="{\displaystyle v\left({\vec {p}},\sigma ,n\right)}" /></span> correspond to antiparticles, and so charge conjugation is a pairing of the two. This sketch also provides enough hints to indicate what charge conjugation might look like in a general geometric setting. There is no particular forced requirement to use perturbation theory, to construct quantum fields that will act as middle-men in a perturbative expansion. Charge conjugation can be given a general setting. </p> <div class="mw-heading mw-heading3"><h3 id="In_geometry">In geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=4" title="Edit section: In geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For general <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a> and <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifolds</a>, one has a <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a>, a <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> and a <a href="/wiki/Metric_tensor" title="Metric tensor">metric</a> that ties the two together. There are several interesting things one can do, when presented with this situation. One is that the smooth structure allows <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> to be posed on the manifold; the <a href="/wiki/Tangent_space" title="Tangent space">tangent</a> and <a href="/wiki/Cotangent_space" title="Cotangent space">cotangent spaces</a> provide enough structure to perform <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">calculus on manifolds</a>. Of key interest is the <a href="/wiki/Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a>, and, with a constant term, what amounts to the Klein–Gordon operator. Cotangent bundles, by their basic construction, are always <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifolds</a>. Symplectic manifolds have <a href="/wiki/Canonical_coordinate" class="mw-redirect" title="Canonical coordinate">canonical coordinates</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b95a7ed3d2b1822eb6a28970329fa7b428219be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.533ex; height:2.009ex;" alt="{\displaystyle x,p}" /></span> interpreted as position and momentum, obeying <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relations</a>. This provides the core infrastructure to extend duality, and thus charge conjugation, to this general setting. </p><p>A second interesting thing one can do is to construct a <a href="/wiki/Spin_structure" title="Spin structure">spin structure</a>. Perhaps the most remarkable thing about this is that it is a very recognizable generalization to a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9769c58523b9b639866a2d48e657d9c26911143a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle (p,q)}" /></span>-dimensional pseudo-Riemannian manifold of the conventional physics concept of <a href="/wiki/Spinor" title="Spinor">spinors</a> living on a (1,3)-dimensional <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a>. The construction passes through a complexified <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> to build a <a href="/wiki/Clifford_bundle" title="Clifford bundle">Clifford bundle</a> and a <a href="/wiki/Spin_manifold" class="mw-redirect" title="Spin manifold">spin manifold</a>. At the end of this construction, one obtains a system that is remarkably familiar, if one is already acquainted with Dirac spinors and the Dirac equation. Several analogies pass through to this general case. First, the <a href="/wiki/Spinor" title="Spinor">spinors</a> are the <a href="/wiki/Weyl_spinor" class="mw-redirect" title="Weyl spinor">Weyl spinors</a>, and they come in complex-conjugate pairs. They are naturally anti-commuting (this follows from the Clifford algebra), which is exactly what one wants to make contact with the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>. Another is the existence of a <a href="/wiki/Chiral_symmetry" class="mw-redirect" title="Chiral symmetry">chiral element</a>, analogous to the <a href="/wiki/Gamma_matrix" class="mw-redirect" title="Gamma matrix">gamma matrix</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94fc10f503ccc16a12b94646d8fff849fab673d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{5}}" /></span> which sorts these spinors into left and right-handed subspaces. The complexification is a key ingredient, and it provides "electromagnetism" in this generalized setting. The spinor bundle doesn't "just" transform under the <a href="/wiki/Pseudo-orthogonal_group" class="mw-redirect" title="Pseudo-orthogonal group">pseudo-orthogonal group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SO(p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SO(p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6609eaa912527e34461522f02d0143988deb1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.355ex; height:2.843ex;" alt="{\displaystyle SO(p,q)}" /></span>, the generalization of the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SO(1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SO(1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75d864d32e8649fb29d72395093e8bdbb3f2a11b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.441ex; height:2.843ex;" alt="{\displaystyle SO(1,3)}" /></span>, but under a bigger group, the complexified <a href="/wiki/Spin_group" title="Spin group">spin group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/813a17eedeee15c77f105ae5b9eb7de487f2ca70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.672ex; height:3.176ex;" alt="{\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q).}" /></span> It is bigger in that it is a <a href="/wiki/Cover_(mathematics)" class="mw-redirect" title="Cover (mathematics)">double covering</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SO(p,q)\times U(1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SO(p,q)\times U(1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2de70860c48d06563e1a4ee946ebb40500d2b41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.596ex; height:2.843ex;" alt="{\displaystyle SO(p,q)\times U(1).}" /></span> </p><p>The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62b00d74ee0cefb86cc052365625abff56d43e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}" /></span> piece can be identified with electromagnetism in several different ways. One way is that the <a href="/wiki/Dirac_operator" title="Dirac operator">Dirac operators</a> on the spin manifold, when squared, contain a piece <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=dA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>d</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=dA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2177791ee25a365cdd5c5c255fc86bb37f48ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.798ex; height:2.176ex;" alt="{\displaystyle F=dA}" /></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> arising from that part of the connection associated with the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62b00d74ee0cefb86cc052365625abff56d43e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}" /></span> piece. This is entirely analogous to what happens when one squares the ordinary Dirac equation in ordinary Minkowski spacetime. A second hint is that this <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62b00d74ee0cefb86cc052365625abff56d43e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}" /></span> piece is associated with the <a href="/wiki/Determinant_bundle" class="mw-redirect" title="Determinant bundle">determinant bundle</a> of the spin structure, effectively tying together the left and right-handed spinors through complex conjugation. </p><p>What remains is to work through the discrete symmetries of the above construction. There are several that appear to generalize <a href="/wiki/P-symmetry" class="mw-redirect" title="P-symmetry">P-symmetry</a> and <a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a>. Identifying the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></span> dimensions with time, and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" /></span> dimensions with space, one can reverse the tangent vectors in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></span> dimensional subspace to get time reversal, and flipping the direction of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" /></span> dimensions corresponds to parity. The C-symmetry can be identified with the reflection on the line bundle. To tie all of these together into a knot, one finally has the concept of <a href="/wiki/Transposition_(mathematics)" class="mw-redirect" title="Transposition (mathematics)">transposition</a>, in that elements of the Clifford algebra can be written in reversed (transposed) order. The net result is that not only do the conventional physics ideas of fields pass over to the general Riemannian setting, but also the ideas of the discrete symmetries. </p><p>There are two ways to react to this. One is to treat it as an interesting curiosity. The other is to realize that, in low dimensions (in low-dimensional spacetime) there are many "accidental" isomorphisms between various <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> and other assorted structures. Being able to examine them in a general setting disentangles these relationships, exposing more clearly "where things come from". </p> <div class="mw-heading mw-heading2"><h2 id="Charge_conjugation_for_Dirac_fields">Charge conjugation for Dirac fields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=5" title="Edit section: Charge conjugation for Dirac fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The laws of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a> (both <a href="/wiki/Classical_physics" title="Classical physics">classical</a> and <a href="/wiki/Quantum" title="Quantum">quantum</a>) are <a href="/wiki/Invariant_(physics)" title="Invariant (physics)">invariant</a> under the exchange of electrical charges with their negatives. For the case of <a href="/wiki/Electron" title="Electron">electrons</a> and <a href="/wiki/Quark" title="Quark">quarks</a>, both of which are <a href="/wiki/Fundamental_particle" class="mw-redirect" title="Fundamental particle">fundamental particle</a> <a href="/wiki/Fermion" title="Fermion">fermion</a> fields, the single-particle field excitations are described by the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</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 (i{\partial \!\!\!{\big /}}-q{A\!\!\!{\big /}}-m)\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> </mrow> <mo>−</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> </mrow> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i{\partial \!\!\!{\big /}}-q{A\!\!\!{\big /}}-m)\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082981492a1c0a261e25b79a4c9793e5e3dbf4af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.602ex; height:3.176ex;" alt="{\displaystyle (i{\partial \!\!\!{\big /}}-q{A\!\!\!{\big /}}-m)\psi =0}" /></span></dd></dl> <p>One wishes to find a charge-conjugate solution </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 (i{\partial \!\!\!{\big /}}+q{A\!\!\!{\big /}}-m)\psi ^{c}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> </mrow> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i{\partial \!\!\!{\big /}}+q{A\!\!\!{\big /}}-m)\psi ^{c}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/361443fe46107cab898701f6d18c7b622686b43d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.547ex; height:3.176ex;" alt="{\displaystyle (i{\partial \!\!\!{\big /}}+q{A\!\!\!{\big /}}-m)\psi ^{c}=0}" /></span></dd></dl> <p>A handful of algebraic manipulations are sufficient to obtain the second from the first.<sup id="cite_ref-Bjorken-Drell-1964_1-0" class="reference"><a href="#cite_note-Bjorken-Drell-1964-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Itzykson-Zuber-1980_2-0" class="reference"><a href="#cite_note-Itzykson-Zuber-1980-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-PS_3-0" class="reference"><a href="#cite_note-PS-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Standard expositions of the Dirac equation demonstrate a conjugate field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\psi }}=\psi ^{\dagger }\gamma ^{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\psi }}=\psi ^{\dagger }\gamma ^{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b589b5b50169bcf917904d2ef597f9b3a89a9de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.183ex; height:3.509ex;" alt="{\displaystyle {\overline {\psi }}=\psi ^{\dagger }\gamma ^{0},}" /></span> interpreted as an anti-particle field, satisfying the complex-transposed Dirac equation </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 {\overline {\psi }}(-i{\partial \!\!\!{\big /}}-q{A\!\!\!{\big /}}-m)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> </mrow> <mo>−</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> </mrow> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\psi }}(-i{\partial \!\!\!{\big /}}-q{A\!\!\!{\big /}}-m)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efd9e570020d42be4b38ff781601ffb3b1f169b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.525ex; height:3.676ex;" alt="{\displaystyle {\overline {\psi }}(-i{\partial \!\!\!{\big /}}-q{A\!\!\!{\big /}}-m)=0}" /></span></dd></dl> <p>Note that some but not all of the signs have flipped. Transposing this back again gives almost the desired form, provided that one can find a 4×4 matrix <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> that transposes the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a> to insert the required sign-change: </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^{-1}\gamma _{\mu }C=-\gamma _{\mu }^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>C</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{-1}\gamma _{\mu }C=-\gamma _{\mu }^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/447c522d8de05c7e0e2812e32f4b800059d6d809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.863ex; height:3.343ex;" alt="{\displaystyle C^{-1}\gamma _{\mu }C=-\gamma _{\mu }^{\textsf {T}}}" /></span></dd></dl> <p>The charge conjugate solution is then given by the <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</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 \psi \mapsto \psi ^{c}=\eta _{c}\,C{\overline {\psi }}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">↦</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \mapsto \psi ^{c}=\eta _{c}\,C{\overline {\psi }}^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d7483313ce07cfd5e16c55a41e5d28187a3fa5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.916ex; height:4.009ex;" alt="{\displaystyle \psi \mapsto \psi ^{c}=\eta _{c}\,C{\overline {\psi }}^{\textsf {T}}}" /></span></dd></dl> <p>The 4×4 matrix <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64528f031cdbe1f52bdaf4ba7a8401108c0d2dc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.413ex; height:2.509ex;" alt="{\displaystyle C,}" /></span> called the charge conjugation matrix, has an explicit form given in the article on <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>. Curiously, this form is not representation-independent, but depends on the specific matrix representation chosen for the <a href="/wiki/Gamma_group" class="mw-redirect" title="Gamma group">gamma group</a> (the subgroup of the <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> capturing the algebraic properties of the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>). This matrix is representation dependent due to a subtle interplay involving the complexification of the <a href="/wiki/Spin_group" title="Spin group">spin group</a> describing the Lorentz covariance of charged particles. The <a href="/wiki/Complex_number" title="Complex number">complex number</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf6da182347dff3657dba35cc4add274134e352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.1ex; height:2.176ex;" alt="{\displaystyle \eta _{c}}" /></span> is an arbitrary phase factor <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 |\eta _{c}|=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\eta _{c}|=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4347a305c15614caa8bf4e183bbea8c74c57955a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.301ex; height:2.843ex;" alt="{\displaystyle |\eta _{c}|=1,}" /></span> generally taken to be <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 \eta _{c}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{c}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399187443104e8cd2ba67d1f9f379467a99487af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.007ex; height:2.676ex;" alt="{\displaystyle \eta _{c}=1.}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Charge_conjugation,_chirality,_helicity"><span id="Charge_conjugation.2C_chirality.2C_helicity"></span>Charge conjugation, chirality, helicity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=6" title="Edit section: Charge conjugation, chirality, helicity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The interplay between chirality and charge conjugation is a bit subtle, and requires articulation. It is often said that charge conjugation does not alter the <a href="/wiki/Chirality_(physics)" title="Chirality (physics)">chirality</a> of particles. This is not the case for <i>fields</i>, the difference arising in the "hole theory" interpretation of particles, where an anti-particle is interpreted as the absence of a particle. This is articulated below. </p><p>Conventionally, <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 \gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94fc10f503ccc16a12b94646d8fff849fab673d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{5}}" /></span> is used as the chirality operator. Under charge conjugation, it transforms as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\gamma _{5}C^{-1}=\gamma _{5}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\gamma _{5}C^{-1}=\gamma _{5}^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2ceafd09b3d0331b819182092441680890654b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.885ex; height:3.343ex;" alt="{\displaystyle C\gamma _{5}C^{-1}=\gamma _{5}^{\textsf {T}}}" /></span></dd></dl> <p>and whether or not <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 \gamma _{5}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{5}^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdf11e9e891067da167220c2a21ef39163c036e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.631ex; height:3.176ex;" alt="{\displaystyle \gamma _{5}^{\textsf {T}}}" /></span> equals <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 \gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94fc10f503ccc16a12b94646d8fff849fab673d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{5}}" /></span> depends on the chosen representation for the gamma matrices. In the Dirac and chiral basis, one does have 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 \gamma _{5}^{\textsf {T}}=\gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{5}^{\textsf {T}}=\gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4befceb69b0190fefd71fca5ae283b1c19945b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.988ex; height:3.176ex;" alt="{\displaystyle \gamma _{5}^{\textsf {T}}=\gamma _{5}}" /></span>, while <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 \gamma _{5}^{\textsf {T}}=-\gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{5}^{\textsf {T}}=-\gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d010f709f1e6d66a27dae99ec8b193698da61cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.796ex; height:3.176ex;" alt="{\displaystyle \gamma _{5}^{\textsf {T}}=-\gamma _{5}}" /></span> is obtained in the Majorana basis. A worked example follows. </p> <div class="mw-heading mw-heading4"><h4 id="Weyl_spinors">Weyl spinors</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=7" title="Edit section: Weyl spinors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the case of massless Dirac spinor fields, chirality is equal to helicity for the positive energy solutions (and minus the helicity for negative energy solutions).<sup id="cite_ref-Itzykson-Zuber-1980_2-1" class="reference"><a href="#cite_note-Itzykson-Zuber-1980-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: § 2-4-3, page 87 ff">: § 2-4-3, page 87 ff </span></sup> One obtains this by writing the massless Dirac equation as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\partial \!\!\!{\big /}\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\partial \!\!\!{\big /}\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecefdc53501ad26b4766c22e55adca8de1196852" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.077ex; height:3.176ex;" alt="{\displaystyle i\partial \!\!\!{\big /}\psi =0}" /></span></dd></dl> <p>Multiplying by <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 \gamma ^{5}\gamma ^{0}=-i\gamma ^{1}\gamma ^{2}\gamma ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}\gamma ^{0}=-i\gamma ^{1}\gamma ^{2}\gamma ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec18359fb8277597efd6653dce6b99b4e4bd6ac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.38ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}\gamma ^{0}=-i\gamma ^{1}\gamma ^{2}\gamma ^{3}}" /></span> one obtains </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 {\epsilon _{ij}}^{m}\sigma ^{ij}\partial _{m}\psi =\gamma _{5}\partial _{t}\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>ψ</mi> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\epsilon _{ij}}^{m}\sigma ^{ij}\partial _{m}\psi =\gamma _{5}\partial _{t}\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6db79e63bb9dcda18b0d7eafca80face4781b227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.257ex; height:3.343ex;" alt="{\displaystyle {\epsilon _{ij}}^{m}\sigma ^{ij}\partial _{m}\psi =\gamma _{5}\partial _{t}\psi }" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{\mu \nu }=i\left[\gamma ^{\mu },\gamma ^{\nu }\right]/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>i</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\mu \nu }=i\left[\gamma ^{\mu },\gamma ^{\nu }\right]/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b313fd46f9dca789d6003848ca4cf297146aaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.639ex; height:2.843ex;" alt="{\displaystyle \sigma ^{\mu \nu }=i\left[\gamma ^{\mu },\gamma ^{\nu }\right]/2}" /></span> is the <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">angular momentum operator</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{ijk}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{ijk}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80cacf43cd468e0f17c0945c0aecaa8d7c57ac99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.278ex; height:2.343ex;" alt="{\displaystyle \epsilon _{ijk}}" /></span> is the <a href="/wiki/Totally_antisymmetric_tensor" class="mw-redirect" title="Totally antisymmetric tensor">totally antisymmetric tensor</a>. This can be brought to a slightly more recognizable form by defining the 3D spin operator <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 \Sigma ^{m}\equiv {\epsilon _{ij}}^{m}\sigma ^{ij},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>≡</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma ^{m}\equiv {\epsilon _{ij}}^{m}\sigma ^{ij},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a9c789d784c013b8c2d34395c79bde375427c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.002ex; height:3.343ex;" alt="{\displaystyle \Sigma ^{m}\equiv {\epsilon _{ij}}^{m}\sigma ^{ij},}" /></span> taking a plane-wave state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=e^{-ik\cdot x}\psi (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=e^{-ik\cdot x}\psi (k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30a63ba6b84f2258b7896c3e7b4296b7923656d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.7ex; height:3.176ex;" alt="{\displaystyle \psi (x)=e^{-ik\cdot x}\psi (k)}" /></span>, applying the on-shell 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 k\cdot k=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot k=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb25b182269f74bbd083fef5cab03ef19eb8cea7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.362ex; height:2.176ex;" alt="{\displaystyle k\cdot k=0}" /></span> and normalizing the momentum to be a 3D unit vector: <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 {\hat {k}}_{i}=k_{i}/k_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {k}}_{i}=k_{i}/k_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48494ae48150d08d736ed36e9c4832f6e2f040f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.548ex; height:3.343ex;" alt="{\displaystyle {\hat {k}}_{i}=k_{i}/k_{0}}" /></span> to write </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(\Sigma \cdot {\hat {k}}\right)\psi =\gamma _{5}\psi ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Σ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mi>ψ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\Sigma \cdot {\hat {k}}\right)\psi =\gamma _{5}\psi ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c0e1b28a7d646ff9232cc2a33def84dee37fb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.342ex; height:4.843ex;" alt="{\displaystyle \left(\Sigma \cdot {\hat {k}}\right)\psi =\gamma _{5}\psi ~.}" /></span></dd></dl> <p>Examining the above, one concludes that angular momentum eigenstates (<a href="/wiki/Helicity_(particle_physics)" title="Helicity (particle physics)">helicity</a> eigenstates) correspond to eigenstates of the <a href="/wiki/Chirality_(physics)" title="Chirality (physics)">chiral operator</a>. This allows the massless Dirac field to be cleanly split into a pair of <a href="/wiki/Weyl_spinor" class="mw-redirect" title="Weyl spinor">Weyl spinors</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\text{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\text{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7518d95340e4d01c3bca68d39f30b4b4b99e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.773ex; height:2.509ex;" alt="{\displaystyle \psi _{\text{L}}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\text{R}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\text{R}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e413d04ebb19353a2bd243b79ccc6d5a75a901b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.602ex; height:2.509ex;" alt="{\displaystyle \psi _{\text{R}},}" /></span> each individually satisfying the <a href="/wiki/Weyl_equation" title="Weyl equation">Weyl equation</a>, but with opposite energy: </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(-p_{0}+\sigma \cdot {\vec {p}}\right)\psi _{\text{R}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>σ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(-p_{0}+\sigma \cdot {\vec {p}}\right)\psi _{\text{R}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ea5fef0b721326eea2ef89c30e9440144f6fd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.618ex; height:2.843ex;" alt="{\displaystyle \left(-p_{0}+\sigma \cdot {\vec {p}}\right)\psi _{\text{R}}=0}" /></span></dd></dl> <p>and </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(p_{0}+\sigma \cdot {\vec {p}}\right)\psi _{\text{L}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>σ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(p_{0}+\sigma \cdot {\vec {p}}\right)\psi _{\text{L}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d810fd6c5938f8e80fcfb15fa9fd8e9652ca2e04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.628ex; height:2.843ex;" alt="{\displaystyle \left(p_{0}+\sigma \cdot {\vec {p}}\right)\psi _{\text{L}}=0}" /></span></dd></dl> <p>Note the freedom one has to equate negative helicity with negative energy, and thus the anti-particle with the particle of opposite helicity. To be clear, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span> here are the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mu }=i\partial _{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\mu }=i\partial _{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0846761e0712cda72fa489ac393b9674d94b9f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:8.841ex; height:2.843ex;" alt="{\displaystyle p_{\mu }=i\partial _{\mu }}" /></span> is the momentum operator. </p> <div class="mw-heading mw-heading4"><h4 id="Charge_conjugation_in_the_chiral_basis">Charge conjugation in the chiral basis</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=8" title="Edit section: Charge conjugation in the chiral basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Taking the <a href="/wiki/Gamma_matrices#Weyl_representation" title="Gamma matrices">Weyl representation</a> of the gamma matrices, one may write a (now taken to be massive) Dirac spinor as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ={\begin{pmatrix}\psi _{\text{L}}\\\psi _{\text{R}}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ={\begin{pmatrix}\psi _{\text{L}}\\\psi _{\text{R}}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f72bf48d19502dff054dc7496786bb514a3ed6bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.739ex; height:6.176ex;" alt="{\displaystyle \psi ={\begin{pmatrix}\psi _{\text{L}}\\\psi _{\text{R}}\end{pmatrix}}}" /></span></dd></dl> <p>The corresponding dual (anti-particle) field 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 {\overline {\psi }}^{\textsf {T}}=\left(\psi ^{\dagger }\gamma ^{0}\right)^{\textsf {T}}={\begin{pmatrix}0&I\\I&0\end{pmatrix}}{\begin{pmatrix}\psi _{\text{L}}^{*}\\\psi _{\text{R}}^{*}\end{pmatrix}}={\begin{pmatrix}\psi _{\text{R}}^{*}\\\psi _{\text{L}}^{*}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>I</mi> </mtd> </mtr> <mtr> <mtd> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\psi }}^{\textsf {T}}=\left(\psi ^{\dagger }\gamma ^{0}\right)^{\textsf {T}}={\begin{pmatrix}0&I\\I&0\end{pmatrix}}{\begin{pmatrix}\psi _{\text{L}}^{*}\\\psi _{\text{R}}^{*}\end{pmatrix}}={\begin{pmatrix}\psi _{\text{R}}^{*}\\\psi _{\text{L}}^{*}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4bf804f0caddc99ad90a25b82869af77d5ef220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:43.66ex; height:6.676ex;" alt="{\displaystyle {\overline {\psi }}^{\textsf {T}}=\left(\psi ^{\dagger }\gamma ^{0}\right)^{\textsf {T}}={\begin{pmatrix}0&I\\I&0\end{pmatrix}}{\begin{pmatrix}\psi _{\text{L}}^{*}\\\psi _{\text{R}}^{*}\end{pmatrix}}={\begin{pmatrix}\psi _{\text{R}}^{*}\\\psi _{\text{L}}^{*}\end{pmatrix}}}" /></span></dd></dl> <p>The charge-conjugate spinors are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{c}={\begin{pmatrix}\psi _{\text{L}}^{c}\\\psi _{\text{R}}^{c}\end{pmatrix}}=\eta _{c}C{\overline {\psi }}^{\textsf {T}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}&0\\0&i\sigma ^{2}\end{pmatrix}}{\begin{pmatrix}\psi _{\text{R}}^{*}\\\psi _{\text{L}}^{*}\end{pmatrix}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}\psi _{\text{R}}^{*}\\i\sigma ^{2}\psi _{\text{L}}^{*}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{c}={\begin{pmatrix}\psi _{\text{L}}^{c}\\\psi _{\text{R}}^{c}\end{pmatrix}}=\eta _{c}C{\overline {\psi }}^{\textsf {T}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}&0\\0&i\sigma ^{2}\end{pmatrix}}{\begin{pmatrix}\psi _{\text{R}}^{*}\\\psi _{\text{L}}^{*}\end{pmatrix}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}\psi _{\text{R}}^{*}\\i\sigma ^{2}\psi _{\text{L}}^{*}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38634f3956b89c119203df9e2d3f6239fd109a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.212ex; height:7.509ex;" alt="{\displaystyle \psi ^{c}={\begin{pmatrix}\psi _{\text{L}}^{c}\\\psi _{\text{R}}^{c}\end{pmatrix}}=\eta _{c}C{\overline {\psi }}^{\textsf {T}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}&0\\0&i\sigma ^{2}\end{pmatrix}}{\begin{pmatrix}\psi _{\text{R}}^{*}\\\psi _{\text{L}}^{*}\end{pmatrix}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}\psi _{\text{R}}^{*}\\i\sigma ^{2}\psi _{\text{L}}^{*}\end{pmatrix}}}" /></span></dd></dl> <p>where, as before, <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 \eta _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf6da182347dff3657dba35cc4add274134e352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.1ex; height:2.176ex;" alt="{\displaystyle \eta _{c}}" /></span> is a phase factor that can be taken to be <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 \eta _{c}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{c}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399187443104e8cd2ba67d1f9f379467a99487af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.007ex; height:2.676ex;" alt="{\displaystyle \eta _{c}=1.}" /></span> Note that the left and right states are inter-changed. This can be restored with a parity transformation. Under <a href="/wiki/P-symmetry" class="mw-redirect" title="P-symmetry">parity</a>, the Dirac spinor transforms as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \left(t,{\vec {x}}\right)\mapsto \psi ^{p}\left(t,{\vec {x}}\right)=\gamma ^{0}\psi \left(t,-{\vec {x}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">↦</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mi>ψ</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \left(t,{\vec {x}}\right)\mapsto \psi ^{p}\left(t,{\vec {x}}\right)=\gamma ^{0}\psi \left(t,-{\vec {x}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155bb8c714da762b10163426fdd89f048ed93c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.652ex; height:3.176ex;" alt="{\displaystyle \psi \left(t,{\vec {x}}\right)\mapsto \psi ^{p}\left(t,{\vec {x}}\right)=\gamma ^{0}\psi \left(t,-{\vec {x}}\right)}" /></span></dd></dl> <p>Under combined charge and parity, one then has </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \left(t,{\vec {x}}\right)\mapsto \psi ^{cp}\left(t,{\vec {x}}\right)={\begin{pmatrix}\psi _{\text{L}}^{cp}\left(t,{\vec {x}}\right)\\\psi _{\text{R}}^{cp}\left(t,{\vec {x}}\right)\end{pmatrix}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}\psi _{\text{L}}^{*}\left(t,-{\vec {x}}\right)\\i\sigma ^{2}\psi _{\text{R}}^{*}\left(t,-{\vec {x}}\right)\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">↦</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>p</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>p</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>p</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \left(t,{\vec {x}}\right)\mapsto \psi ^{cp}\left(t,{\vec {x}}\right)={\begin{pmatrix}\psi _{\text{L}}^{cp}\left(t,{\vec {x}}\right)\\\psi _{\text{R}}^{cp}\left(t,{\vec {x}}\right)\end{pmatrix}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}\psi _{\text{L}}^{*}\left(t,-{\vec {x}}\right)\\i\sigma ^{2}\psi _{\text{R}}^{*}\left(t,-{\vec {x}}\right)\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a58471c0cd8ed0f22a07b52351b9816e695791f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.773ex; height:7.509ex;" alt="{\displaystyle \psi \left(t,{\vec {x}}\right)\mapsto \psi ^{cp}\left(t,{\vec {x}}\right)={\begin{pmatrix}\psi _{\text{L}}^{cp}\left(t,{\vec {x}}\right)\\\psi _{\text{R}}^{cp}\left(t,{\vec {x}}\right)\end{pmatrix}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}\psi _{\text{L}}^{*}\left(t,-{\vec {x}}\right)\\i\sigma ^{2}\psi _{\text{R}}^{*}\left(t,-{\vec {x}}\right)\end{pmatrix}}}" /></span></dd></dl> <p>Conventionally, one takes <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 \eta _{c}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{c}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffc7826aade9d26de389c21801d3062f2598e69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.676ex;" alt="{\displaystyle \eta _{c}=1}" /></span> globally. See however, the note below. </p> <div class="mw-heading mw-heading4"><h4 id="Majorana_condition">Majorana condition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=9" title="Edit section: Majorana condition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Majorana_equation" title="Majorana equation">Majorana condition</a> imposes a constraint between the field and its charge conjugate, namely that they must be equal: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =\psi ^{c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\psi ^{c}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11d91c73192494e25afcad079c782cd89eddb3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.716ex; height:2.676ex;" alt="{\displaystyle \psi =\psi ^{c}.}" /></span> This is perhaps best stated as the requirement that the Majorana spinor must be an eigenstate of the charge conjugation involution. </p><p>Doing so requires some notational care. In many texts discussing charge conjugation, the involution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \mapsto \psi ^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">↦</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \mapsto \psi ^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e0d9c04d82ec4be07b2f4383badddbe17ddfb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.585ex; height:2.676ex;" alt="{\displaystyle \psi \mapsto \psi ^{c}}" /></span> is not given an explicit symbolic name, when applied to <i>single-particle solutions</i> of the Dirac equation. This is in contrast to the case when the <i>quantized field</i> is discussed, where a unitary operator <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 {C}}}"> <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">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" /></span> is defined (as done in a later section, below). For the present section, let the involution be named as <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 {\mathsf {C}}:\psi \mapsto \psi ^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>:</mo> <mi>ψ</mi> <mo stretchy="false">↦</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}:\psi \mapsto \psi ^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db81539627d1b0cf6fe1cfe8437b4b4fb8054d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.007ex; height:2.676ex;" alt="{\displaystyle {\mathsf {C}}:\psi \mapsto \psi ^{c}}" /></span> so 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 {\mathsf {C}}\psi =\psi ^{c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mi>ψ</mi> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\psi =\psi ^{c}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d74fe7d84c4d755c8dcedfed242b6d7c62f5b5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.201ex; height:2.676ex;" alt="{\displaystyle {\mathsf {C}}\psi =\psi ^{c}.}" /></span> Taking this to be a linear operator, one may consider its eigenstates. The Majorana condition singles out one such: <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 {\mathsf {C}}\psi =\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mi>ψ</mi> <mo>=</mo> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\psi =\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee88694eaa93eb4b7acc235026950603f2251b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.257ex; height:2.509ex;" alt="{\displaystyle {\mathsf {C}}\psi =\psi .}" /></span> There are, however, two such eigenstates: <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 {\mathsf {C}}\psi ^{(\pm )}=\pm \psi ^{(\pm )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mo>±</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\psi ^{(\pm )}=\pm \psi ^{(\pm )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f2330d15ee1100c3626418489aa2aa66c98d410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.645ex; height:3.176ex;" alt="{\displaystyle {\mathsf {C}}\psi ^{(\pm )}=\pm \psi ^{(\pm )}.}" /></span> Continuing in the Weyl basis, as above, these eigenstates are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{(+)}={\begin{pmatrix}\psi _{\text{L}}\\i\sigma ^{2}\psi _{\text{L}}^{*}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{(+)}={\begin{pmatrix}\psi _{\text{L}}\\i\sigma ^{2}\psi _{\text{L}}^{*}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56caed1e5fa3c01a893fe375bc873e817c5b8330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.534ex; height:6.509ex;" alt="{\displaystyle \psi ^{(+)}={\begin{pmatrix}\psi _{\text{L}}\\i\sigma ^{2}\psi _{\text{L}}^{*}\end{pmatrix}}}" /></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{(-)}={\begin{pmatrix}i\sigma ^{2}\psi _{\text{R}}^{*}\\\psi _{\text{R}}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{(-)}={\begin{pmatrix}i\sigma ^{2}\psi _{\text{R}}^{*}\\\psi _{\text{R}}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f775962481398431289a5d351845512e2712c461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.717ex; height:6.509ex;" alt="{\displaystyle \psi ^{(-)}={\begin{pmatrix}i\sigma ^{2}\psi _{\text{R}}^{*}\\\psi _{\text{R}}\end{pmatrix}}}" /></span></dd></dl> <p>The Majorana spinor is conventionally taken as just the positive eigenstate, namely <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{(+)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{(+)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a33a320a5b444e090295748d71c5302d02073d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.95ex; height:3.176ex;" alt="{\displaystyle \psi ^{(+)}.}" /></span> The chiral operator <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 \gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94fc10f503ccc16a12b94646d8fff849fab673d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{5}}" /></span> exchanges these two, in that </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 \gamma _{5}{\mathsf {C}}=-{\mathsf {C}}\gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{5}{\mathsf {C}}=-{\mathsf {C}}\gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab99b111ea4264c0d30b69d91b95f2d624505ebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.394ex; height:2.676ex;" alt="{\displaystyle \gamma _{5}{\mathsf {C}}=-{\mathsf {C}}\gamma _{5}}" /></span></dd></dl> <p>This is readily verified by direct substitution. Bear in mind 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 {\mathsf {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699cb4847ecedddeeae2b69e2892ba9987f9b1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle {\mathsf {C}}}" /></span> <i>does <b>not</b> have</i> a 4×4 matrix representation! More precisely, there is no complex 4×4 matrix that can take a complex number to its complex conjugate; this inversion would require an 8×8 real matrix. The physical interpretation of complex conjugation as charge conjugation becomes clear when considering the complex conjugation of scalar fields, described in a subsequent section below. </p><p>The projectors onto the chiral eigenstates can be written as <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_{\text{L}}=\left(1-\gamma _{5}\right)/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{L}}=\left(1-\gamma _{5}\right)/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5fdd1d384c5df2216c6085242d8b861e537eb6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.633ex; height:2.843ex;" alt="{\displaystyle P_{\text{L}}=\left(1-\gamma _{5}\right)/2}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\text{R}}=\left(1+\gamma _{5}\right)/2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{R}}=\left(1+\gamma _{5}\right)/2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c2f1847f94c70d78b86e34655846b1ebabe688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.462ex; height:2.843ex;" alt="{\displaystyle P_{\text{R}}=\left(1+\gamma _{5}\right)/2,}" /></span> and so the above translates 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 P_{\text{L}}{\mathsf {C}}={\mathsf {C}}P_{\text{R}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{L}}{\mathsf {C}}={\mathsf {C}}P_{\text{R}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a20f8d92d248ce08e0d12d1208362b9d3b0f202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.982ex; height:2.509ex;" alt="{\displaystyle P_{\text{L}}{\mathsf {C}}={\mathsf {C}}P_{\text{R}}~.}" /></span></dd></dl> <p>This directly demonstrates that charge conjugation, applied to single-particle complex-number-valued solutions of the Dirac equation flips the chirality of the solution. The projectors onto the charge conjugation eigenspaces are <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^{(+)}=(1+{\mathsf {C}})P_{\text{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{(+)}=(1+{\mathsf {C}})P_{\text{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/743463d0420d2025e8274764a7e4126805885a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.759ex; height:3.343ex;" alt="{\displaystyle P^{(+)}=(1+{\mathsf {C}})P_{\text{L}}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{(-)}=(1-{\mathsf {C}})P_{\text{R}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{(-)}=(1-{\mathsf {C}})P_{\text{R}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d396ca14b5a18786db76a73f799f11ac77c6606" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.588ex; height:3.343ex;" alt="{\displaystyle P^{(-)}=(1-{\mathsf {C}})P_{\text{R}}.}" /></span> </p> <div class="mw-heading mw-heading4"><h4 id="Geometric_interpretation">Geometric interpretation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=10" title="Edit section: Geometric interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The phase factor <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 \ \eta _{c}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \eta _{c}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/873c5076bff2ac19203b7a927a249cdd244a7ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.261ex; height:2.176ex;" alt="{\displaystyle \ \eta _{c}\ }" /></span> can be given a geometric interpretation. It has been noted that, for massive Dirac spinors, the "arbitrary" phase factor <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 \ \eta _{c}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \eta _{c}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/873c5076bff2ac19203b7a927a249cdd244a7ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.261ex; height:2.176ex;" alt="{\displaystyle \ \eta _{c}\ }" /></span> may depend on both the momentum, and the helicity (but not the chirality).<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> This can be interpreted as saying that this phase may vary along the fiber of the <a href="/wiki/Spinor_bundle" title="Spinor bundle">spinor bundle</a>, depending on the local choice of a coordinate frame. Put another way, a spinor field is a local <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> of the spinor bundle, and Lorentz boosts and rotations correspond to movements along the fibers of the corresponding <a href="/wiki/Frame_bundle" title="Frame bundle">frame bundle</a> (again, just a choice of local coordinate frame). Examined in this way, this extra phase freedom can be interpreted as the phase arising from the electromagnetic field. For the <a href="/wiki/Majorana_spinor" class="mw-redirect" title="Majorana spinor">Majorana spinors</a>, the phase would be constrained to not vary under boosts and rotations. </p> <div class="mw-heading mw-heading3"><h3 id="Charge_conjugation_for_quantized_fields">Charge conjugation for quantized fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=11" title="Edit section: Charge conjugation for quantized fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above describes charge conjugation for the single-particle solutions only. When the Dirac field is <a href="/wiki/Second-quantized" class="mw-redirect" title="Second-quantized">second-quantized</a>, as in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, the spinor and electromagnetic fields are described by operators. The charge conjugation involution then manifests as a <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operator</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}"> <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">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" /></span> (in calligraphic font) acting on the particle fields, expressed as<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> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \mapsto \psi ^{c}={\mathcal {C}}\ \psi \ {\mathcal {C}}^{\dagger }=\eta _{c}\ C\ {\overline {\psi }}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">↦</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mtext> </mtext> <mi>ψ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mtext> </mtext> <mi>C</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \mapsto \psi ^{c}={\mathcal {C}}\ \psi \ {\mathcal {C}}^{\dagger }=\eta _{c}\ C\ {\overline {\psi }}^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9fe52859cd572271f96c484fbb4627e0c75767e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.908ex; height:4.009ex;" alt="{\displaystyle \psi \mapsto \psi ^{c}={\mathcal {C}}\ \psi \ {\mathcal {C}}^{\dagger }=\eta _{c}\ C\ {\overline {\psi }}^{\textsf {T}}}" /></span></li> <li><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 {\overline {\psi }}\mapsto {\overline {\psi }}^{c}={\mathcal {C}}\ {\overline {\psi }}\ {\mathcal {C}}^{\dagger }=\eta _{c}^{*}\ \psi ^{\textsf {T}}\ C^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">↦</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <msubsup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mtext> </mtext> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mtext> </mtext> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\psi }}\mapsto {\overline {\psi }}^{c}={\mathcal {C}}\ {\overline {\psi }}\ {\mathcal {C}}^{\dagger }=\eta _{c}^{*}\ \psi ^{\textsf {T}}\ C^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45641929d9079c3ba41db8eac2c3141b009d1e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.63ex; height:3.509ex;" alt="{\displaystyle {\overline {\psi }}\mapsto {\overline {\psi }}^{c}={\mathcal {C}}\ {\overline {\psi }}\ {\mathcal {C}}^{\dagger }=\eta _{c}^{*}\ \psi ^{\textsf {T}}\ C^{-1}}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }\mapsto A_{\mu }^{c}={\mathcal {C}}\ A_{\mu }\ {\mathcal {C}}^{\dagger }=-A_{\mu }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">↦</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mtext> </mtext> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }\mapsto A_{\mu }^{c}={\mathcal {C}}\ A_{\mu }\ {\mathcal {C}}^{\dagger }=-A_{\mu }\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a4663b469d5c2b0673f3ce44cba05736e07950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.672ex; height:3.343ex;" alt="{\displaystyle A_{\mu }\mapsto A_{\mu }^{c}={\mathcal {C}}\ A_{\mu }\ {\mathcal {C}}^{\dagger }=-A_{\mu }\ }" /></span></li></ol> <p>where the non-calligraphic <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\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>C</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ C\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b010f9498beac548bca41502f10621736d581b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.928ex; height:2.176ex;" alt="{\displaystyle \ C\ }" /></span> is the same 4×4 matrix given before. </p> <div class="mw-heading mw-heading3"><h3 id="Charge_reversal_in_electroweak_theory">Charge reversal in electroweak theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=12" title="Edit section: Charge reversal in electroweak theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Charge conjugation does not alter the <a href="/wiki/Chirality_(physics)" title="Chirality (physics)">chirality</a> of particles. A left-handed <a href="/wiki/Neutrino" title="Neutrino">neutrino</a> would be taken by charge conjugation into a left-handed <a href="/wiki/Antineutrino" class="mw-redirect" title="Antineutrino">antineutrino</a>, which does not interact in the Standard Model. This property is what is meant by the "maximal violation" of C-symmetry in the weak interaction. </p><p>Some postulated extensions of the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>, like <a href="/wiki/Left-right_model" class="mw-redirect" title="Left-right model">left-right models</a>, restore this C-symmetry. </p> <div class="mw-heading mw-heading2"><h2 id="Scalar_fields">Scalar fields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=13" title="Edit section: Scalar fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Dirac field has a "hidden" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62b00d74ee0cefb86cc052365625abff56d43e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}" /></span> gauge freedom, allowing it to couple directly to the electromagnetic field without any further modifications to the Dirac equation or the field itself.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> This is not the case for <a href="/wiki/Scalar_field" title="Scalar field">scalar fields</a>, which must be explicitly "complexified" to couple to electromagnetism. This is done by "tensoring in" an additional factor of the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /></span> into the field, or constructing a <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62b00d74ee0cefb86cc052365625abff56d43e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}" /></span>. </p><p>One very conventional technique is simply to start with two real scalar fields, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" /></span> and create a linear combination </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \mathrel {\stackrel {\mathrm {def} }{=}} {\phi +i\chi \over {\sqrt {2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>+</mo> <mi>i</mi> <mi>χ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \mathrel {\stackrel {\mathrm {def} }{=}} {\phi +i\chi \over {\sqrt {2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/443d6173a54803248f957619f0cb2a17b1c331f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.379ex; height:6.343ex;" alt="{\displaystyle \psi \mathrel {\stackrel {\mathrm {def} }{=}} {\phi +i\chi \over {\sqrt {2}}}}" /></span></dd></dl> <p>The charge conjugation <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> is then the mapping <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 {\mathsf {C}}:i\mapsto -i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>:</mo> <mi>i</mi> <mo stretchy="false">↦</mo> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}:i\mapsto -i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f72860eb47ba9116126bad1a2f8a72908951411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.45ex; height:2.343ex;" alt="{\displaystyle {\mathsf {C}}:i\mapsto -i}" /></span> since this is sufficient to reverse the sign on the electromagnetic potential (since this complex number is being used to couple to it). For real scalar fields, charge conjugation is just the identity map: <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 {\mathsf {C}}:\phi \mapsto \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>:</mo> <mi>ϕ</mi> <mo stretchy="false">↦</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}:\phi \mapsto \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c4aae9f75078f7e7193333ed0ac1a555d4d8caf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.807ex; height:2.509ex;" alt="{\displaystyle {\mathsf {C}}:\phi \mapsto \phi }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}:\chi \mapsto \chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>:</mo> <mi>χ</mi> <mo stretchy="false">↦</mo> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}:\chi \mapsto \chi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ec301f1d53bdcf10b707d783d872e06813c5f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.947ex; height:2.509ex;" alt="{\displaystyle {\mathsf {C}}:\chi \mapsto \chi }" /></span> and so, for the complexified field, charge conjugation is just <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}:\psi \mapsto \psi ^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>:</mo> <mi>ψ</mi> <mo stretchy="false">↦</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}:\psi \mapsto \psi ^{*}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb231d355211b56824f39e4114425d986dff667a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.764ex; height:2.676ex;" alt="{\displaystyle {\mathsf {C}}:\psi \mapsto \psi ^{*}.}" /></span> The "mapsto" arrow <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 \mapsto }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↦</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mapsto }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc09de045e7d82eef9fe078e7e7606576640c11b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \mapsto }" /></span> is convenient for tracking "what goes where"; the equivalent older notation is simply to write <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 {\mathsf {C}}\phi =\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mi>ϕ</mi> <mo>=</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\phi =\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9281518a514dc46f866cff232ff2c8b61aa3df68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.355ex; height:2.509ex;" alt="{\displaystyle {\mathsf {C}}\phi =\phi }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}\chi =\chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mi>χ</mi> <mo>=</mo> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\chi =\chi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc5195bd016ab0ff4449cc36ef1dd1ca691ca5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.494ex; height:2.509ex;" alt="{\displaystyle {\mathsf {C}}\chi =\chi }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}\psi =\psi ^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mi>ψ</mi> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\psi =\psi ^{*}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2da04ccb8604746e21ec0e90440ac41cf76783d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.311ex; height:2.676ex;" alt="{\displaystyle {\mathsf {C}}\psi =\psi ^{*}.}" /></span> </p><p>The above describes the conventional construction of a charged scalar field. It is also possible to introduce additional algebraic structure into the fields in other ways. In particular, one may define a "real" field behaving as <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 {\mathsf {C}}:\phi \mapsto -\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>:</mo> <mi>ϕ</mi> <mo stretchy="false">↦</mo> <mo>−</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}:\phi \mapsto -\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ae03ffbaf7da6c26bf1843d0e5ab9985d653cd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.616ex; height:2.509ex;" alt="{\displaystyle {\mathsf {C}}:\phi \mapsto -\phi }" /></span>. As it is real, it cannot couple to electromagnetism by itself, but, when complexified, would result in a charged field that transforms as <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 {\mathsf {C}}:\psi \mapsto -\psi ^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>:</mo> <mi>ψ</mi> <mo stretchy="false">↦</mo> <mo>−</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}:\psi \mapsto -\psi ^{*}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/060d0bac5d7a94d67cb1eb9a6a20338f838b9837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.572ex; height:2.676ex;" alt="{\displaystyle {\mathsf {C}}:\psi \mapsto -\psi ^{*}.}" /></span> Because C-symmetry is a <a href="/wiki/Discrete_symmetry" title="Discrete symmetry">discrete symmetry</a>, one has some freedom to play these kinds of algebraic games in the search for a theory that correctly models some given physical reality. </p><p>In physics literature, a transformation such as <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 {\mathsf {C}}:\phi \mapsto \phi ^{c}=-\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>:</mo> <mi>ϕ</mi> <mo stretchy="false">↦</mo> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}:\phi \mapsto \phi ^{c}=-\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a281b49743d62d6dadf7eeec7a931d2716cae493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.044ex; height:2.676ex;" alt="{\displaystyle {\mathsf {C}}:\phi \mapsto \phi ^{c}=-\phi }" /></span> might be written without any further explanation. The formal mathematical interpretation of this is that the field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" /></span> is an element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \times \mathbb {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \times \mathbb {Z} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beec2cf2a902fb46eb8d00326b574dd6c4040768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.123ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} \times \mathbb {Z} _{2}}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{2}=\{+1,-1\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{2}=\{+1,-1\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7304ed034fd9b27515a71c7067742a8001451981" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.65ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{2}=\{+1,-1\}.}" /></span> Thus, properly speaking, the field should be written as <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 =(r,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =(r,c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3a28541e739b87444405c07ed1e2bc574da551" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.383ex; height:2.843ex;" alt="{\displaystyle \phi =(r,c)}" /></span> which behaves under charge conjugation as <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 {\mathsf {C}}:(r,c)\mapsto (r,-c).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>:</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}:(r,c)\mapsto (r,-c).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d3a8e42e7b8a1b9f523152d770c92e6063cb1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.289ex; height:2.843ex;" alt="{\displaystyle {\mathsf {C}}:(r,c)\mapsto (r,-c).}" /></span> It is very tempting, but not quite formally correct to just multiply these out, to move around the location of this minus sign; this mostly "just works", but a failure to track it properly will lead to confusion. </p> <div class="mw-heading mw-heading2"><h2 id="Combination_of_charge_and_parity_reversal">Combination of charge and parity reversal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=14" title="Edit section: Combination of charge and parity reversal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It was believed for some time that C-symmetry could be combined with the <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a>-inversion transformation (see <a href="/wiki/P-symmetry" class="mw-redirect" title="P-symmetry">P-symmetry</a>) to preserve a combined <a href="/wiki/CP-symmetry" class="mw-redirect" title="CP-symmetry">CP-symmetry</a>. However, violations of this symmetry have been identified in the weak interactions (particularly in the <a href="/wiki/Kaon" title="Kaon">kaons</a> and B <a href="/wiki/Meson" title="Meson">mesons</a>). In the Standard Model, this <a href="/wiki/CP_violation" title="CP violation">CP violation</a> is due to a single phase in the <a href="/wiki/CKM_matrix" class="mw-redirect" title="CKM matrix">CKM matrix</a>. If CP is combined with time reversal (<a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a>), the resulting <a href="/wiki/CPT-symmetry" class="mw-redirect" title="CPT-symmetry">CPT-symmetry</a> can be shown using only the <a href="/wiki/Wightman_axioms" title="Wightman axioms">Wightman axioms</a> to be universally obeyed. </p> <div class="mw-heading mw-heading2"><h2 id="In_general_settings">In general settings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C-symmetry&action=edit&section=15" title="Edit section: In general settings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The analog of charge conjugation can be defined for <a href="/wiki/Higher-dimensional_gamma_matrices" title="Higher-dimensional gamma matrices">higher-dimensional gamma matrices</a>, with an explicit construction for Weyl spinors given in the article on <a href="/wiki/Weyl%E2%80%93Brauer_matrices" title="Weyl–Brauer matrices">Weyl–Brauer matrices</a>. Note, however, spinors as defined abstractly in the representation theory of <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a> are not fields; rather, they should be thought of as existing on a zero-dimensional spacetime. </p><p>The analog of <a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a> follows from <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 \gamma ^{1}\gamma ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{1}\gamma ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e032560547485324ec4f585e6b601c958994423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.668ex; height:3.176ex;" alt="{\displaystyle \gamma ^{1}\gamma ^{3}}" /></span> as the T-conjugation operator for Dirac spinors. Spinors also have an inherent <a href="/wiki/P-symmetry" class="mw-redirect" title="P-symmetry">P-symmetry</a>, obtained by reversing the direction of all of the basis vectors of the <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> from which the spinors are constructed. The relationship to the P and T symmetries for a fermion field on a spacetime manifold are a bit subtle, but can be roughly characterized as follows. When a spinor is constructed via a Clifford algebra, the construction requires a vector space on which to build. By convention, this vector space is the <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> of the spacetime manifold at a given, fixed spacetime point (a single fiber in the <a href="/wiki/Tangent_manifold" class="mw-redirect" title="Tangent manifold">tangent manifold</a>). P and T operations applied to the spacetime manifold can then be understood as also flipping the coordinates of the tangent space as well; thus, the two are glued together. Flipping the parity or the direction of time in one also flips it in the other. This is a convention. One can become unglued by failing to propagate this connection. </p><p>This is done by taking the tangent space as a <a href="/wiki/Vector_space" title="Vector space">vector space</a>, extending it to a <a href="/wiki/Tensor_algebra" title="Tensor algebra">tensor algebra</a>, and then using an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> on the vector space to define a <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a>. Treating each such algebra as a fiber, one obtains a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a> called the <a href="/wiki/Clifford_bundle" title="Clifford bundle">Clifford bundle</a>. Under a change of basis of the tangent space, elements of the Clifford algebra transform according to the <a href="/wiki/Spin_group" title="Spin group">spin group</a>. Building a <a href="/wiki/Principle_fiber_bundle" class="mw-redirect" title="Principle fiber bundle">principle fiber bundle</a> with the spin group as the fiber results in a <a href="/wiki/Spin_structure" title="Spin structure">spin structure</a>. </p><p>All that is missing in the above paragraphs are the <a href="/wiki/Spinor" title="Spinor">spinors</a> themselves. These require the "complexification" of the tangent manifold: tensoring it with the complex plane. Once this is done, the <a href="/wiki/Weyl_spinor" class="mw-redirect" title="Weyl spinor">Weyl spinors</a> can be constructed. These have the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{j}={\frac {1}{\sqrt {2}}}\left(e_{2j}-ie_{2j+1}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{j}={\frac {1}{\sqrt {2}}}\left(e_{2j}-ie_{2j+1}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcbada1b04e1b4911b44b44636edca7f1975e1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.177ex; height:6.176ex;" alt="{\displaystyle w_{j}={\frac {1}{\sqrt {2}}}\left(e_{2j}-ie_{2j+1}\right)}" /></span></dd></dl> <p>where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64afd228cf00e5024b9cdd277462d24ab97b6d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.993ex; height:2.343ex;" alt="{\displaystyle e_{j}}" /></span> are the basis vectors for the vector space <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=T_{p}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=T_{p}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a009a75712f6c7278f5b0d936ea83d11619bb7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.745ex; height:2.843ex;" alt="{\displaystyle V=T_{p}M}" /></span>, the tangent space at point <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\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ad2c18a15749505c928763cd4fdb56f4982816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.542ex; height:2.509ex;" alt="{\displaystyle p\in M}" /></span> in the spacetime manifold <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b865c33e30eb83000cd6387517c66dbbf3c3df9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.089ex; height:2.176ex;" alt="{\displaystyle M.}" /></span> The Weyl spinors, together with their complex conjugates span the tangent space, in the sense that </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 V\otimes \mathbb {C} =W\oplus {\overline {W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>=</mo> <mi>W</mi> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>W</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\otimes \mathbb {C} =W\oplus {\overline {W}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d619b3ec005c86390b8c787a0924c2747f97b578" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.387ex; height:3.176ex;" alt="{\displaystyle V\otimes \mathbb {C} =W\oplus {\overline {W}}}" /></span></dd></dl> <p>The alternating algebra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb22f944a8f2a7565d28d6074f276b6cbafd3fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.986ex; height:2.176ex;" alt="{\displaystyle \wedge W}" /></span> is called the <a href="/w/index.php?title=Spinor_space&action=edit&redlink=1" class="new" title="Spinor space (page does not exist)">spinor space</a>, it is where the spinors live, as well as products of spinors (thus, objects with higher spin values, including vectors and tensors). </p><p><br /> </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=C-symmetry&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/C_parity" title="C parity">C parity</a></li> <li><a href="/wiki/G-parity" title="G-parity">G-parity</a></li> <li><a href="/wiki/Anti-particle" class="mw-redirect" title="Anti-particle">Anti-particle</a></li> <li><a href="/wiki/Antimatter" title="Antimatter">Antimatter</a></li> <li><a href="/wiki/Truly_neutral_particle" title="Truly neutral particle">Truly neutral particle</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=C-symmetry&action=edit&section=17" 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 reflist-lower-alpha"> <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">This freedom is explicitly removed, constrained away in <a href="/wiki/Majorana_spinor" class="mw-redirect" title="Majorana spinor">Majorana spinors</a>.</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=C-symmetry&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-Bjorken-Drell-1964-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bjorken-Drell-1964_1-0">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBjorkenDrell1964" class="citation book cs1">Bjorken, James D. & Drell, Sidney D. (1964). <i>Relativistic Quantum Mechanics</i>. New York, NY: McGraw-Hill. chapter 5.2, pages 66-70.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativistic+Quantum+Mechanics&rft.place=New+York%2C+NY&rft.pages=chapter-5.2%2C+pages-66-70&rft.pub=McGraw-Hill&rft.date=1964&rft.aulast=Bjorken&rft.aufirst=James+D.&rft.au=Drell%2C+Sidney+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AC-symmetry" class="Z3988"></span></span> </li> <li id="cite_note-Itzykson-Zuber-1980-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Itzykson-Zuber-1980_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Itzykson-Zuber-1980_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFItzyksonZuber1980" class="citation book cs1">Itzykson, Claude & Zuber, Jean-Bernard (1980). <i>Quantum Field Theory</i>. New York, NY: McGraw-Hill. chapter 2-4, pages 85 ff.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory&rft.place=New+York%2C+NY&rft.pages=chapter-2-4%2C+pages-85-ff&rft.pub=McGraw-Hill&rft.date=1980&rft.aulast=Itzykson&rft.aufirst=Claude&rft.au=Zuber%2C+Jean-Bernard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AC-symmetry" class="Z3988"></span></span> </li> <li id="cite_note-PS-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-PS_3-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeskin,_M.E.Schroeder,_D.V.1997" class="citation book cs1">Peskin, M.E. & Schroeder, D.V. (1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoqu0000pesk"><i>An Introduction to Quantum Field Theory</i></a></span>. Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-50397-2" title="Special:BookSources/0-201-50397-2"><bdi>0-201-50397-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Quantum+Field+Theory&rft.pub=Addison+Wesley&rft.date=1997&rft.isbn=0-201-50397-2&rft.au=Peskin%2C+M.E.&rft.au=Schroeder%2C+D.V.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoqu0000pesk&rfr_id=info%3Asid%2Fen.wikipedia.org%3AC-symmetry" 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"><a href="#CITEREFItzyksonZuber1980">Itzykson & Zuber (1980)</a>, § 2-4-2 <i>Charge Conjugation</i>, page 86, equation 2-100</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"><a href="#CITEREFBjorkenDrell1964">Bjorken & Drell (1964)</a>, chapter 15</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"><a href="#CITEREFItzyksonZuber1980">Itzykson & Zuber (1980)</a>, § 3-4</span> </li> </ol></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSozzi,_M.S.2008" class="citation book cs1">Sozzi, M.S. (2008). <i>Discrete Symmetries and CP Violation</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-929666-8" title="Special:BookSources/978-0-19-929666-8"><bdi>978-0-19-929666-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Symmetries+and+CP+Violation&rft.pub=Oxford+University+Press&rft.date=2008&rft.isbn=978-0-19-929666-8&rft.au=Sozzi%2C+M.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AC-symmetry" class="Z3988"></span></li></ul> </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="C,_P,_and_T_symmetries22" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><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:C,_P_and_T" title="Template:C, P and T"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:C,_P_and_T" title="Template talk:C, P and T"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:C,_P_and_T" title="Special:EditPage/Template:C, P and T"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="C,_P,_and_T_symmetries22" style="font-size:114%;margin:0 4em">C, P, and T symmetries</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">C-symmetry</a></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">P-symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C_parity" title="C parity">CP</a></li> <li><a href="/wiki/CP_violation#CP-symmetry" title="CP violation">CP symmetry</a></li> <li><a href="/wiki/CPT_symmetry" title="CPT symmetry">CPT symmetry</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chirality_(physics)" title="Chirality (physics)">Chirality</a></li> <li><a href="/wiki/Pin_group" title="Pin group">Pin group</a></li> <li><a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">Symmetry (physics)</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" 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=C-symmetry&oldid=1271302481">https://en.wikipedia.org/w/index.php?title=C-symmetry&oldid=1271302481</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:Quantum_field_theory" title="Category:Quantum field theory">Quantum field theory</a></li><li><a href="/wiki/Category:Symmetry" title="Category:Symmetry">Symmetry</a></li><li><a href="/wiki/Category:Antimatter" title="Category:Antimatter">Antimatter</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: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><li><a href="/wiki/Category:Articles_needing_additional_references_from_December_2008" title="Category:Articles needing additional references from December 2008">Articles needing additional references from December 2008</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:All_articles_with_specifically_marked_weasel-worded_phrases" title="Category:All articles with specifically marked weasel-worded phrases">All articles with specifically marked weasel-worded phrases</a></li><li><a href="/wiki/Category:Articles_with_specifically_marked_weasel-worded_phrases_from_November_2020" title="Category:Articles with specifically marked weasel-worded phrases from November 2020">Articles with specifically marked weasel-worded phrases from November 2020</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 23 January 2025, at 12:27<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=C-symmetry&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"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></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"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div 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"> <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> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-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-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">C-symmetry</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>20 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.next-5f874b7bcc-xvxb4","wgBackendResponseTime":215,"wgPageParseReport":{"limitreport":{"cputime":"0.496","walltime":"0.817","ppvisitednodes":{"value":2070,"limit":1000000},"postexpandincludesize":{"value":25418,"limit":2097152},"templateargumentsize":{"value":2161,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":3,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":31474,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 461.251 1 -total"," 31.65% 145.993 2 Template:Reflist"," 21.71% 100.122 4 Template:Cite_book"," 19.79% 91.281 1 Template:Short_description"," 19.00% 87.653 1 Template:C,_P_and_T"," 18.09% 83.454 1 Template:Navbox"," 14.81% 68.303 1 Template:Refimprove"," 13.30% 61.355 1 Template:Ambox"," 12.53% 57.781 2 Template:Pagetype"," 6.99% 32.248 3 Template:Harvp"]},"scribunto":{"limitreport-timeusage":{"value":"0.248","limit":"10.000"},"limitreport-memusage":{"value":5822725,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFBjorkenDrell1964\"] = 1,\n [\"CITEREFItzyksonZuber1980\"] = 1,\n [\"CITEREFPeskin,_M.E.Schroeder,_D.V.1997\"] = 1,\n [\"CITEREFSozzi,_M.S.2008\"] = 1,\n}\ntemplate_list = table#1 {\n [\"C, P and T\"] = 1,\n [\"Cite book\"] = 4,\n [\"Efn\"] = 1,\n [\"Harvp\"] = 3,\n [\"Notelist\"] = 1,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Refimprove\"] = 1,\n [\"Reflist\"] = 1,\n [\"Rp\"] = 1,\n [\"Short description\"] = 1,\n [\"Which\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.eqiad.next-666f47d5b9-chfqt","timestamp":"20250305223603","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"C-symmetry","url":"https:\/\/en.wikipedia.org\/wiki\/C-symmetry","sameAs":"http:\/\/www.wikidata.org\/entity\/Q513656","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q513656","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":"2002-11-24T11:42:47Z","dateModified":"2025-01-23T12:27:59Z","headline":"symmetry of physical laws under a charge-conjugation transformation"}</script> </body> </html>

CINXE.COM

C-symmetry - Wikipedia