Dirac equation - Wikipedia

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<span>Covariant form and relativistic invariance</span> </div> </a> <ul id="toc-Covariant_form_and_relativistic_invariance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Comparison_with_related_theories" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Comparison_with_related_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Comparison with related theories</span> </div> </a> <button aria-controls="toc-Comparison_with_related_theories-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Comparison with related theories subsection</span> </button> <ul id="toc-Comparison_with_related_theories-sublist" class="vector-toc-list"> <li id="toc-Pauli_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pauli_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Pauli theory</span> </div> </a> <ul id="toc-Pauli_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weyl_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weyl_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Weyl theory</span> </div> </a> <ul id="toc-Weyl_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Physical_interpretation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Physical_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Physical interpretation</span> </div> </a> <button aria-controls="toc-Physical_interpretation-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 Physical interpretation subsection</span> </button> <ul id="toc-Physical_interpretation-sublist" class="vector-toc-list"> <li id="toc-Identification_of_observables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identification_of_observables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Identification of observables</span> </div> </a> <ul id="toc-Identification_of_observables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hole_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hole_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Hole theory</span> </div> </a> <ul id="toc-Hole_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_quantum_field_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_quantum_field_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>In quantum field theory</span> </div> </a> <ul id="toc-In_quantum_field_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_formulation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematical_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Mathematical formulation</span> </div> </a> <button aria-controls="toc-Mathematical_formulation-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 Mathematical formulation subsection</span> </button> <ul id="toc-Mathematical_formulation-sublist" class="vector-toc-list"> <li id="toc-Dirac_adjoint_and_the_adjoint_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dirac_adjoint_and_the_adjoint_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Dirac adjoint and the adjoint equation</span> </div> </a> <ul id="toc-Dirac_adjoint_and_the_adjoint_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Klein–Gordon_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Klein–Gordon_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Klein–Gordon equation</span> </div> </a> <ul id="toc-Klein–Gordon_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conserved_current" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conserved_current"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Conserved current</span> </div> </a> <ul id="toc-Conserved_current-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solutions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Solutions</span> </div> </a> <ul id="toc-Solutions-sublist" class="vector-toc-list"> <li id="toc-Plane-wave_solutions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Plane-wave_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1</span> <span>Plane-wave solutions</span> </div> </a> <ul id="toc-Plane-wave_solutions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lagrangian_formulation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lagrangian_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Lagrangian formulation</span> </div> </a> <ul id="toc-Lagrangian_formulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lorentz_invariance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lorentz_invariance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Lorentz invariance</span> </div> </a> <ul id="toc-Lorentz_invariance-sublist" class="vector-toc-list"> <li id="toc-Further_discussion_of_Lorentz_covariance_of_the_Dirac_equation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Further_discussion_of_Lorentz_covariance_of_the_Dirac_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6.1</span> <span>Further discussion of Lorentz covariance of the Dirac equation</span> </div> </a> <ul id="toc-Further_discussion_of_Lorentz_covariance_of_the_Dirac_equation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Other_formulations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_formulations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Other formulations</span> </div> </a> <button aria-controls="toc-Other_formulations-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 Other formulations subsection</span> </button> <ul id="toc-Other_formulations-sublist" class="vector-toc-list"> <li id="toc-Curved_spacetime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Curved_spacetime"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Curved spacetime</span> </div> </a> <ul id="toc-Curved_spacetime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_algebra_of_physical_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_algebra_of_physical_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>The algebra of physical space</span> </div> </a> <ul id="toc-The_algebra_of_physical_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coupled_Weyl_Spinors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coupled_Weyl_Spinors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Coupled Weyl Spinors</span> </div> </a> <ul id="toc-Coupled_Weyl_Spinors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-U(1)_symmetry" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#U(1)_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>U(1) symmetry</span> </div> </a> <button aria-controls="toc-U(1)_symmetry-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 U(1) symmetry subsection</span> </button> <ul id="toc-U(1)_symmetry-sublist" class="vector-toc-list"> <li id="toc-Vector_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Vector symmetry</span> </div> </a> <ul id="toc-Vector_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gauging_the_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gauging_the_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Gauging the symmetry</span> </div> </a> <ul id="toc-Gauging_the_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Axial_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Axial_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Axial symmetry</span> </div> </a> <ul id="toc-Axial_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extension_to_color_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extension_to_color_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Extension to color symmetry</span> </div> </a> <ul id="toc-Extension_to_color_symmetry-sublist" class="vector-toc-list"> <li id="toc-Physical_applications" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Physical_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.1</span> <span>Physical applications</span> </div> </a> <ul id="toc-Physical_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalisations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Generalisations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.2</span> <span>Generalisations</span> </div> </a> <ul id="toc-Generalisations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <button aria-controls="toc-See_also-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 See also subsection</span> </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Articles_on_the_Dirac_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_on_the_Dirac_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Articles on the Dirac equation</span> </div> </a> <ul id="toc-Articles_on_the_Dirac_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Other equations</span> </div> </a> <ul id="toc-Other_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_topics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_topics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Other topics</span> </div> </a> <ul id="toc-Other_topics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Selected_papers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Selected_papers"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Selected papers</span> </div> </a> <ul id="toc-Selected_papers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Textbooks"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Textbooks</span> </div> </a> <ul id="toc-Textbooks-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" 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">Dirac equation</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 46 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-46" 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">46 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Dirac-vergelyking" title="Dirac-vergelyking – Afrikaans" lang="af" hreflang="af" data-title="Dirac-vergelyking" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Dirac-Gleichung" title="Dirac-Gleichung – Alemannic" lang="gsw" hreflang="gsw" data-title="Dirac-Gleichung" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D8%AF%D9%8A%D8%B1%D8%A7%D9%83" title="معادلة ديراك – Arabic" lang="ar" hreflang="ar" data-title="معادلة ديراك" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A1%E0%A6%BF%E0%A6%B0%E0%A6%BE%E0%A6%95_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="ডিরাক সমীকরণ – Bangla" lang="bn" hreflang="bn" data-title="ডিরাক সমীকরণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D1%9E%D0%BD%D0%B5%D0%BD%D0%BD%D0%B5_%D0%94%D0%B7%D1%96%D1%80%D0%B0%D0%BA%D0%B0" title="Ураўненне Дзірака – Belarusian" lang="be" hreflang="be" data-title="Ураўненне Дзірака" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%94%D0%B8%D1%80%D0%B0%D0%BA" title="Уравнение на Дирак – Bulgarian" lang="bg" hreflang="bg" data-title="Уравнение на Дирак" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equaci%C3%B3_de_Dirac" title="Equació de Dirac – Catalan" lang="ca" hreflang="ca" data-title="Equació de Dirac" 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/Diracova_rovnice" title="Diracova rovnice – Czech" lang="cs" hreflang="cs" data-title="Diracova rovnice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Dirac-ligningen" title="Dirac-ligningen – Danish" lang="da" hreflang="da" data-title="Dirac-ligningen" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dirac-Gleichung" title="Dirac-Gleichung – German" lang="de" hreflang="de" data-title="Dirac-Gleichung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Diraci_v%C3%B5rrand" title="Diraci võrrand – Estonian" lang="et" hreflang="et" data-title="Diraci võrrand" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BE%CE%AF%CF%83%CF%89%CF%83%CE%B7_%CE%9D%CF%84%CE%B9%CF%81%CE%AC%CE%BA" title="Εξίσωση Ντιράκ – Greek" lang="el" hreflang="el" data-title="Εξίσωση Ντιράκ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_de_Dirac" title="Ecuación de Dirac – Spanish" lang="es" hreflang="es" data-title="Ecuación de Dirac" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Diraka_ekvacio" title="Diraka ekvacio – Esperanto" lang="eo" hreflang="eo" data-title="Diraka ekvacio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D9%87_%D8%AF%DB%8C%D8%B1%D8%A7%DA%A9" title="معادله دیراک – Persian" lang="fa" hreflang="fa" data-title="معادله دیراک" 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/%C3%89quation_de_Dirac" title="Équation de Dirac – French" lang="fr" hreflang="fr" data-title="Équation de Dirac" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Cothrom%C3%B3id_Dirac" title="Cothromóid Dirac – Irish" lang="ga" hreflang="ga" data-title="Cothromóid Dirac" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%94%94%EB%9E%99_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="디랙 방정식 – Korean" lang="ko" hreflang="ko" data-title="디랙 방정식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%AB%D6%80%D5%A1%D5%AF%D5%AB_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4" title="Դիրակի հավասարում – Armenian" lang="hy" hreflang="hy" data-title="Դիրակի հավասարում" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_Dirac" title="Persamaan Dirac – Indonesian" lang="id" hreflang="id" data-title="Persamaan Dirac" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_di_Dirac" title="Equazione di Dirac – Italian" lang="it" hreflang="it" data-title="Equazione di Dirac" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%AA_%D7%93%D7%99%D7%A8%D7%90%D7%A7" title="משוואת דיראק – Hebrew" lang="he" hreflang="he" data-title="משוואת דיראק" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%98%E1%83%A0%E1%83%90%E1%83%99%E1%83%98%E1%83%A1_%E1%83%92%E1%83%90%E1%83%9C%E1%83%A2%E1%83%9D%E1%83%9A%E1%83%94%E1%83%91%E1%83%90" title="დირაკის განტოლება – Georgian" lang="ka" hreflang="ka" data-title="დირაკის განტოლება" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Dirac-egyenlet" title="Dirac-egyenlet – Hungarian" lang="hu" hreflang="hu" data-title="Dirac-egyenlet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%A1%E0%A4%BF%E0%A4%B0%E0%A5%85%E0%A4%95_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="डिरॅक समीकरण – Marathi" lang="mr" hreflang="mr" data-title="डिरॅक समीकरण" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Diracvergelijking" title="Diracvergelijking – Dutch" lang="nl" hreflang="nl" data-title="Diracvergelijking" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%87%E3%82%A3%E3%83%A9%E3%83%83%E3%82%AF%E6%96%B9%E7%A8%8B%E5%BC%8F" title="ディラック方程式 – Japanese" lang="ja" hreflang="ja" data-title="ディラック方程式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Dirac-ligning" title="Dirac-ligning – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Dirac-ligning" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Equacion_de_Dirac" title="Equacion de Dirac – Occitan" lang="oc" hreflang="oc" data-title="Equacion de Dirac" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Dirak_tenglamasi" title="Dirak tenglamasi – Uzbek" lang="uz" hreflang="uz" data-title="Dirak tenglamasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A1%E0%A9%80%E0%A8%B0%E0%A8%BE%E0%A8%95_%E0%A8%87%E0%A8%95%E0%A9%81%E0%A8%8F%E0%A8%B8%E0%A8%BC%E0%A8%A8" title="ਡੀਰਾਕ ਇਕੁਏਸ਼ਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਡੀਰਾਕ ਇਕੁਏਸ਼ਨ" 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/R%C3%B3wnanie_Diraca" title="Równanie Diraca – Polish" lang="pl" hreflang="pl" data-title="Równanie Diraca" 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/Equa%C3%A7%C3%A3o_de_Dirac" title="Equação de Dirac – Portuguese" lang="pt" hreflang="pt" data-title="Equação de Dirac" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ecua%C8%9Bia_lui_Dirac" title="Ecuația lui Dirac – Romanian" lang="ro" hreflang="ro" data-title="Ecuația lui Dirac" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%94%D0%B8%D1%80%D0%B0%D0%BA%D0%B0" title="Уравнение Дирака – Russian" lang="ru" hreflang="ru" data-title="Уравнение Дирака" 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/Dirac_equation" title="Dirac equation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Dirac equation" 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/Diracova_ena%C4%8Dba" title="Diracova enačba – Slovenian" lang="sl" hreflang="sl" data-title="Diracova enačba" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diracin_yht%C3%A4l%C3%B6" title="Diracin yhtälö – Finnish" lang="fi" hreflang="fi" data-title="Diracin yhtälö" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Diracekvationen" title="Diracekvationen – Swedish" lang="sv" hreflang="sv" data-title="Diracekvationen" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Ekwasyong_Dirac" title="Ekwasyong Dirac – Tagalog" lang="tl" hreflang="tl" data-title="Ekwasyong Dirac" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%94%D0%B8%D1%80%D0%B0%D0%BA_%D1%82%D0%B8%D0%B3%D0%B5%D0%B7%D0%BB%D3%99%D0%BC%D3%99%D1%81%D0%B5" title="Дирак тигезләмәсе – Tatar" lang="tt" hreflang="tt" data-title="Дирак тигезләмәсе" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Dirac_denklemi" title="Dirac denklemi – Turkish" lang="tr" hreflang="tr" data-title="Dirac denklemi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F_%D0%94%D1%96%D1%80%D0%B0%D0%BA%D0%B0" title="Рівняння Дірака – Ukrainian" lang="uk" hreflang="uk" data-title="Рівняння Дірака" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_Dirac" title="Phương trình Dirac – Vietnamese" lang="vi" hreflang="vi" data-title="Phương trình Dirac" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/Dirac_%E6%96%B9%E7%A8%8B" title="Dirac 方程 – Cantonese" lang="yue" hreflang="yue" data-title="Dirac 方程" data-language-autonym="粵語" data-language-local-name="Cantonese" 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:19.0em;"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><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\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_number" title="Quantum number">Quantum number</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">State</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell's inequality</a></li> <li><a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li></ul> </div> <ul><li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a> <ul><li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Overview</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a class="mw-selflink selflink">Dirac</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced topics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Blackett</a></li> <li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">Bohm</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Davisson</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Hans_Kramers" title="Hans Kramers">Kramers</a></li> <li><a href="/wiki/Willis_Lamb" title="Willis Lamb">Lamb</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Isidor_Isaac_Rabi" title="Isidor Isaac Rabi">Rabi</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Michelle_Simmons" title="Michelle Simmons">Simmons</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics" title="Template:Quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics" title="Template talk:Quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics" title="Special:EditPage/Template:Quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table><p>In <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>, the <b>Dirac equation</b> is a <a href="/wiki/Relativistic_wave_equation" class="mw-redirect" title="Relativistic wave equation">relativistic wave equation</a> derived by British physicist <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a> in 1928. In its <a class="mw-selflink-fragment" href="#Covariant_form_and_relativistic_invariance">free form</a>, or including electromagnetic interactions, it describes all <a href="/wiki/Spin-1/2" title="Spin-1/2">spin-1/2</a> <a href="/wiki/Massive_particle" title="Massive particle">massive particles</a>, called "Dirac particles", such as <a href="/wiki/Electron" title="Electron">electrons</a> and <a href="/wiki/Quark" title="Quark">quarks</a> for which <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a> is a <a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">symmetry</a>. It is consistent with both the principles of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and the theory of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the <a href="/wiki/Fine_structure" title="Fine structure">fine structure</a> of the <a href="/wiki/Hydrogen_spectral_series" title="Hydrogen spectral series">hydrogen spectrum</a> in a completely rigorous way. It has become vital in the building of the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The equation also implied the existence of a new form of matter, <i><a href="/wiki/Antimatter" title="Antimatter">antimatter</a></i>, previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a <i>theoretical</i> justification for the introduction of several component wave functions in <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a>'s <a href="/wiki/Phenomenology_(particle_physics)" class="mw-redirect" title="Phenomenology (particle physics)">phenomenological</a> theory of <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>. The wave functions in the Dirac theory are vectors of four <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> (known as <a href="/wiki/Bispinor" title="Bispinor">bispinors</a>), two of which resemble <a href="/wiki/Pauli_equation" title="Pauli equation">the Pauli wavefunction</a> in the non-relativistic limit, in contrast to the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the <a href="/wiki/Weyl_equation" title="Weyl equation">Weyl equation</a>. </p><p>In the context of <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-<style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> particles. </p><p>Dirac did not fully appreciate the importance of his results; however, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the <a href="/wiki/Positron" title="Positron">positron</a>—represents one of the great triumphs of <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>. This accomplishment has been described as fully on par with the works of <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>, <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a>, and <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a> before him.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The equation has been deemed by some physicists to be the "real seed of modern physics".<sup id="cite_ref-:2_4-0" class="reference"><a href="#cite_note-:2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The equation has also been described as the "centerpiece of relativistic quantum mechanics", with it also stated that "the equation is perhaps the most important one in all of quantum mechanics".<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The Dirac equation is inscribed upon a plaque on the floor of <a href="/wiki/Westminster_Abbey" title="Westminster Abbey">Westminster Abbey</a>. Unveiled on 13 November 1995, the plaque commemorates Dirac's life.<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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Dirac equation in the form originally proposed by <a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a> is:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 291">: 291 </span></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\beta mc^{2}+c\sum _{n=1}^{3}\alpha _{n}p_{n}\right)\psi (x,t)=i\hbar {\frac {\partial \psi (x,t)}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\beta mc^{2}+c\sum _{n=1}^{3}\alpha _{n}p_{n}\right)\psi (x,t)=i\hbar {\frac {\partial \psi (x,t)}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/462b6baf980241a20627f784d6b1c360e908ced3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.985ex; height:7.509ex;" alt="{\displaystyle \left(\beta mc^{2}+c\sum _{n=1}^{3}\alpha _{n}p_{n}\right)\psi (x,t)=i\hbar {\frac {\partial \psi (x,t)}{\partial t}}}"></span> where <span class="texhtml"><i>ψ</i>(<i>x</i>, <i>t</i>)</span> is the <a href="/wiki/Wave_function" title="Wave function">wave function</a> for an <a href="/wiki/Electron" title="Electron">electron</a> of <a href="/wiki/Rest_mass" class="mw-redirect" title="Rest mass">rest mass</a> <span class="texhtml"><i>m</i></span> with <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> coordinates <span class="texhtml"><i>x</i>, <i>t</i></span>. <span class="texhtml"><i>p</i><sub>1</sub>, <i>p</i><sub>2</sub>, <i>p</i><sub>3</sub></span> are the components of the <a href="/wiki/Momentum" title="Momentum">momentum</a>, understood to be the <a href="/wiki/Momentum_operator" title="Momentum operator">momentum operator</a> in the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>. <span class="texhtml"><i>c</i></span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>, and <span class="texhtml"><i>ħ</i></span> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>; these fundamental <a href="/wiki/Physical_constant" title="Physical constant">physical constants</a> reflect special relativity and quantum mechanics, respectively. α and β are 4x4 <a href="/wiki/Gamma_matrices" title="Gamma matrices">Gamma matrices</a> encoding the spin properties of relativistic particles and ensuring the equation's correct transformation under Lorentz transformations.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2025)">citation needed</span></a></i>]</sup> </p><p>Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, thus allowing the atom to be treated in a manner consistent with relativity. He hoped that the corrections introduced this way might have a bearing on the problem of <a href="/wiki/Atomic_spectra" class="mw-redirect" title="Atomic spectra">atomic spectra</a>. </p><p>Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity—which were based on discretizing the <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> stored in the electron's possibly non-circular orbit of the <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">atomic nucleus</a>—had failed, and the new quantum mechanics of <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a>, <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a>, <a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a>, <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a>, and Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter and introduced new mathematical classes of objects that are now essential elements of fundamental physics. </p><p>The new elements in this equation are the four <span class="nowrap">4 × 4</span> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> <span class="texhtml"><i>α</i><sub>1</sub></span>, <span class="texhtml"><i>α</i><sub>2</sub></span>, <span class="texhtml"><i>α</i><sub>3</sub></span> and <span class="texhtml"><i>β</i></span>, and the four-component <a href="/wiki/Wave_function" title="Wave function">wave function</a> <span class="texhtml"><i>ψ</i></span>. There are four components in <span class="texhtml"><i>ψ</i></span> because the evaluation of it at any given point in configuration space is a <a href="/wiki/Bispinor" title="Bispinor">bispinor</a>. It is interpreted as a superposition of a <a href="/wiki/Spin-1/2" title="Spin-1/2">spin-up</a> electron, a spin-down electron, a spin-up positron, and a spin-down positron. </p><p>The <span class="nowrap">4 × 4</span> matrices <span class="texhtml"><i>α</i><sub><i>k</i></sub></span> and <span class="texhtml"><i>β</i></span> are all <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a> and are <a href="/wiki/Involutory_matrix" title="Involutory matrix">involutory</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{i}^{2}=\beta ^{2}=I_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}^{2}=\beta ^{2}=I_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d57f7d7b73cb365bd40f655214fca1e16388275" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.207ex; height:3.343ex;" alt="{\displaystyle \alpha _{i}^{2}=\beta ^{2}=I_{4}}"></span> and they all mutually <a href="/wiki/Anticommute" class="mw-redirect" title="Anticommute">anti-commute</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\alpha _{i}\alpha _{j}+\alpha _{j}\alpha _{i}&=0\quad (i\neq j)\\\alpha _{i}\beta +\beta \alpha _{i}&=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>β</mi> <mo>+</mo> <mi>β</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha _{i}\alpha _{j}+\alpha _{j}\alpha _{i}&=0\quad (i\neq j)\\\alpha _{i}\beta +\beta \alpha _{i}&=0\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28a458b64a8d23784df25a07ae50d07cc14e054e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.213ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\alpha _{i}\alpha _{j}+\alpha _{j}\alpha _{i}&=0\quad (i\neq j)\\\alpha _{i}\beta +\beta \alpha _{i}&=0\end{aligned}}}"></span> </p><p>These matrices and the form of the wave function have a deep mathematical significance. The algebraic structure represented by the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a> had been created some 50 years earlier by the English mathematician <a href="/wiki/William_Kingdon_Clifford" title="William Kingdon Clifford">W. K. Clifford</a>. In turn, Clifford's ideas had emerged from the mid-19th-century work of German mathematician <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a> in his <i>Lineare Ausdehnungslehre</i> (<i>Theory of Linear Expansion</i>). <sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Historical claims need refs (July 2024)">citation needed</span></a></i>]</sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Making_the_Schrödinger_equation_relativistic"><span id="Making_the_Schr.C3.B6dinger_equation_relativistic"></span>Making the Schrödinger equation relativistic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=2" title="Edit section: Making the Schrödinger equation relativistic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Dirac equation is superficially similar to the Schrödinger equation for a massive <a href="/wiki/Free_particle" title="Free particle">free particle</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\phi =i\hbar {\frac {\partial }{\partial t}}\phi ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\phi =i\hbar {\frac {\partial }{\partial t}}\phi ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f06e53e9275e76e36565d145b8ae0bf0a5af425b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.037ex; height:5.843ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\phi =i\hbar {\frac {\partial }{\partial t}}\phi ~.}"></span> </p><p>The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically as they do in the <a href="/wiki/Maxwell_equation" class="mw-redirect" title="Maxwell equation">Maxwell equations</a> that govern the behavior of light — the equations must be differentially of the <i>same order</i> in space and time. In relativity, the momentum and the energies are the space and time parts of a spacetime vector, the <a href="/wiki/Four-momentum" title="Four-momentum">four-momentum</a>, and they are related by the relativistically invariant relation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{2}=m^{2}c^{4}+p^{2}c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{2}=m^{2}c^{4}+p^{2}c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ef6bd7664976b50a3ba51f56615d7f331e72da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.227ex; height:3.009ex;" alt="{\displaystyle E^{2}=m^{2}c^{4}+p^{2}c^{2}}"></span> </p><p>which says that the <a href="/wiki/Four-momentum#Minkowski_norm" title="Four-momentum">length of this four-vector</a> is proportional to the rest mass <span class="texhtml"><i>m</i></span>. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory produces the <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a> describing the propagation of waves, constructed from relativistically invariant objects, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}+\nabla ^{2}\right)\phi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}+\nabla ^{2}\right)\phi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9742ab48dcab62f89bdd2ad27ee2d66bfdd807" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.253ex; height:6.343ex;" alt="{\displaystyle \left(-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}+\nabla ^{2}\right)\phi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\phi }"></span> with the wave function <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> being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. Space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time-derivative in order to solve definite problems. Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the <a href="/wiki/Probability_density_function" title="Probability density function">probability density</a> of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\phi ^{*}\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\phi ^{*}\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/472282fbc5e443b715ed2ca2361544f5f5e5eef4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.125ex; height:2.843ex;" alt="{\displaystyle \rho =\phi ^{*}\phi }"></span> and this density is convected according to the probability current vector <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-{\frac {i\hbar }{2m}}(\phi ^{*}\nabla \phi -\phi \nabla \phi ^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> <mo>−</mo> <mi>ϕ</mi> <mi mathvariant="normal">∇</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-{\frac {i\hbar }{2m}}(\phi ^{*}\nabla \phi -\phi \nabla \phi ^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03ceb1b39c1afeb3c61ba6beb01fa6f4c2d9b5c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.589ex; height:5.343ex;" alt="{\displaystyle J=-{\frac {i\hbar }{2m}}(\phi ^{*}\nabla \phi -\phi \nabla \phi ^{*})}"></span> with the conservation of probability current and density following from the continuity equation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot J+{\frac {\partial \rho }{\partial t}}=0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi>J</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot J+{\frac {\partial \rho }{\partial t}}=0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bac713ddbbeb8d432b914c0c87ebcd2049cee0be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.771ex; height:5.676ex;" alt="{\displaystyle \nabla \cdot J+{\frac {\partial \rho }{\partial t}}=0~.}"></span> </p><p>The fact that the density is <a href="/wiki/Positive-definite_function" title="Positive-definite function">positive definite</a> and convected according to this continuity equation implies that one may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the <a href="/wiki/Conservation_law" title="Conservation law">conservation law</a>. A proper relativistic theory with a probability density current must also share this feature. To maintain the notion of a convected density, one must generalize the Schrödinger expression of the density and current so that space and time derivatives again enter symmetrically in relation to the scalar wave function. The Schrödinger expression can be kept for the current, but the probability density must be replaced by the symmetrically formed expression<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Why? (November 2021)">further explanation needed</span></a></i>]</sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}\partial _{t}\psi -\psi \partial _{t}\psi ^{*}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>ψ</mi> <mo>−</mo> <mi>ψ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}\partial _{t}\psi -\psi \partial _{t}\psi ^{*}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fce48a787a6442bb682c37783704b43535337975" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.753ex; height:5.676ex;" alt="{\displaystyle \rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}\partial _{t}\psi -\psi \partial _{t}\psi ^{*}\right).}"></span> which now becomes the 4th component of a spacetime vector, and the entire <a href="/wiki/Probability_current" title="Probability current">probability 4-current density</a> has the relativistically covariant expression <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{\mu }={\frac {i\hbar }{2m}}\left(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ψ</mi> <mo>−</mo> <mi>ψ</mi> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{\mu }={\frac {i\hbar }{2m}}\left(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d49fea709dc99dad5702ef8fbda8f08dcbe58740" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.252ex; height:5.343ex;" alt="{\displaystyle J^{\mu }={\frac {i\hbar }{2m}}\left(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*}\right).}"></span> </p><p>The continuity equation is as before. Everything is compatible with relativity now, but the expression for the density is no longer positive definite; the initial values of both <span class="texhtml"><i>ψ</i></span> and <span class="texhtml">∂<sub><i>t</i></sub><i>ψ</i></span> may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus, one cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time. </p><p>Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, where it is known as the <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a>, and describes a spinless particle field (e.g. <a href="/wiki/Pi_meson" class="mw-redirect" title="Pi meson">pi meson</a> or <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a>). Historically, Schrödinger himself arrived at this equation before the one that bears his name but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the <i>charge</i> density, which can be positive or negative, and not the probability density. </p> <div class="mw-heading mw-heading3"><h3 id="Dirac's_coup"><span id="Dirac.27s_coup"></span>Dirac's coup</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=3" title="Edit section: Dirac's coup"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dirac thus thought to try an equation that was <i>first order</i> in both space and time. He postulated an equation of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\psi =({\vec {\alpha }}\cdot {\vec {p}}+\beta m)\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>ψ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">→</mo> </mover> </mrow> </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> <mi>β</mi> <mi>m</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\psi =({\vec {\alpha }}\cdot {\vec {p}}+\beta m)\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acdc3c339cb4ba45a865931981be3a4887ea4364" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.414ex; height:2.843ex;" alt="{\displaystyle E\psi =({\vec {\alpha }}\cdot {\vec {p}}+\beta m)\psi }"></span> where the operators <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {\alpha }},\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {\alpha }},\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33cde84bc34e55a5613379a8c6db05e49c308991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.843ex;" alt="{\displaystyle ({\vec {\alpha }},\beta )}"></span> must be independent 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 ({\vec {p}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {p}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab754a7526dc207c6e1810fe527e54cae06f8bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.008ex; height:2.843ex;" alt="{\displaystyle ({\vec {p}},t)}"></span> for linearity and independent 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 ({\vec {x}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {x}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b5a668b21420131ed56f44130efcf94b87a6db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.013ex; height:2.843ex;" alt="{\displaystyle ({\vec {x}},t)}"></span> for space-time homogeneity. These constraints implied additional dynamical variables that 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 ({\vec {\alpha }},\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {\alpha }},\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33cde84bc34e55a5613379a8c6db05e49c308991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.843ex;" alt="{\displaystyle ({\vec {\alpha }},\beta )}"></span> operators will depend upon; from this requirement Dirac concluded that the operators would depend upon 4x4 matrices, related to the Pauli matrices.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 205">: 205 </span></sup> </p><p>One could, for example, formally (i.e. by <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a>, since it is not straightforward to take a <a href="/wiki/Functional_square_root" title="Functional square root">functional square root</a> of the sum of two differential operators) take the <a href="/wiki/Energy%E2%80%93momentum_relation" title="Energy–momentum relation">relativistic expression for the energy</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=c{\sqrt {p^{2}+m^{2}c^{2}}}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=c{\sqrt {p^{2}+m^{2}c^{2}}}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/102c4ccff20e577a7d956237743c7bf048c6ee75" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.652ex; height:4.843ex;" alt="{\displaystyle E=c{\sqrt {p^{2}+m^{2}c^{2}}}~,}"></span> replace <span class="texhtml"><i>p</i></span> by its operator equivalent, expand the square root in an <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible. </p><p>As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator (see also <a href="/wiki/Half_derivative" class="mw-redirect" title="Half derivative">half derivative</a>) thus: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}=\left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)\left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>B</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>C</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>c</mi> </mfrac> </mrow> <mi>D</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>B</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>C</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>c</mi> </mfrac> </mrow> <mi>D</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}=\left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)\left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3a346c5da0315d2feab4e3d651d39aad8f9d4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:77.817ex; height:6.343ex;" alt="{\displaystyle \nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}=\left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)\left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)~.}"></span> </p><p>On multiplying out the right side it is apparent that, in order to get all the cross-terms such as <span class="texhtml">∂<sub><i>x</i></sub>∂<sub><i>y</i></sub></span> to vanish, one must assume <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB+BA=0,~\ldots ~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mo>+</mo> <mi>B</mi> <mi>A</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB+BA=0,~\ldots ~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9196807d4de65df5fcd380fb1cc55f5847ed1c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.808ex; height:2.509ex;" alt="{\displaystyle AB+BA=0,~\ldots ~}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{2}=B^{2}=\dots =1~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}=B^{2}=\dots =1~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61dd42452b962e89203ce3e322726ee32c4ae1c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.024ex; height:2.676ex;" alt="{\displaystyle A^{2}=B^{2}=\dots =1~.}"></span> </p><p>Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a>, immediately understood that these conditions could be met if <span class="texhtml"><i>A</i></span>, <span class="texhtml"><i>B</i></span>, <span class="texhtml"><i>C</i></span> and <span class="texhtml"><i>D</i></span> are <i>matrices</i>, with the implication that the wave function has <i>multiple components</i>. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least <span class="nowrap">4 × 4</span> matrices to set up a system with the properties required — so the wave function had <i>four</i> components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here. </p><p>Given the factorization in terms of these matrices, one can now write down immediately an equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)\psi =\kappa \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>B</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>C</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>c</mi> </mfrac> </mrow> <mi>D</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>=</mo> <mi>κ</mi> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)\psi =\kappa \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ca67277ef4b9c11449c2eac63d9e1ea2b5b4d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.821ex; height:6.176ex;" alt="{\displaystyle \left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)\psi =\kappa \psi }"></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 \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span> to be determined. Applying again the matrix operator on both sides yields <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\nabla ^{2}-{\frac {1}{c^{2}}}\partial _{t}^{2}\right)\psi =\kappa ^{2}\psi ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>=</mo> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\nabla ^{2}-{\frac {1}{c^{2}}}\partial _{t}^{2}\right)\psi =\kappa ^{2}\psi ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c902aea5d09469ef8f8b2edf5a80a3ac6d7a918" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.679ex; height:6.176ex;" alt="{\displaystyle \left(\nabla ^{2}-{\frac {1}{c^{2}}}\partial _{t}^{2}\right)\psi =\kappa ^{2}\psi ~.}"></span> </p><p>Taking <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 \kappa ={\tfrac {mc}{\hbar }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>m</mi> <mi>c</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\tfrac {mc}{\hbar }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a0aa9a303f155614439c17d7fed99435b054fa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.428ex; height:3.343ex;" alt="{\displaystyle \kappa ={\tfrac {mc}{\hbar }}}"></span> shows that all the components of the wave function <i>individually</i> satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}-{\frac {mc}{\hbar }}\right)\psi =0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>B</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>C</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>c</mi> </mfrac> </mrow> <mi>D</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>c</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}-{\frac {mc}{\hbar }}\right)\psi =0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b10520d0846873a01c1298f0afa4a861fd876a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.083ex; height:6.176ex;" alt="{\displaystyle \left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}-{\frac {mc}{\hbar }}\right)\psi =0~.}"></span> </p><p>Setting <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=i\beta \alpha _{1}\,,\,B=i\beta \alpha _{2}\,,\,C=i\beta \alpha _{3}\,,\,D=\beta ~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>i</mi> <mi>β</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mi>B</mi> <mo>=</mo> <mi>i</mi> <mi>β</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mi>C</mi> <mo>=</mo> <mi>i</mi> <mi>β</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mi>D</mi> <mo>=</mo> <mi>β</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=i\beta \alpha _{1}\,,\,B=i\beta \alpha _{2}\,,\,C=i\beta \alpha _{3}\,,\,D=\beta ~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412f1d0602b47b3afa66f4cac9a96a1bea2095e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:41.604ex; height:2.509ex;" alt="{\displaystyle A=i\beta \alpha _{1}\,,\,B=i\beta \alpha _{2}\,,\,C=i\beta \alpha _{3}\,,\,D=\beta ~,}"></span> and because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{2}=\beta ^{2}=I_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{2}=\beta ^{2}=I_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf0d5a37333e867c6c704c8f48140c80ef00a62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.644ex; height:3.009ex;" alt="{\displaystyle D^{2}=\beta ^{2}=I_{4}}"></span>, the Dirac equation is produced as written above. </p> <div class="mw-heading mw-heading3"><h3 id="Covariant_form_and_relativistic_invariance">Covariant form and relativistic invariance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=4" title="Edit section: Covariant form and relativistic invariance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To demonstrate the <a href="/wiki/Lorentz_covariance" title="Lorentz covariance">relativistic invariance</a> of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}D&=\gamma ^{0},\\A&=i\gamma ^{1},\quad B=i\gamma ^{2},\quad C=i\gamma ^{3},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>D</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>B</mi> <mo>=</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>C</mi> <mo>=</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}D&=\gamma ^{0},\\A&=i\gamma ^{1},\quad B=i\gamma ^{2},\quad C=i\gamma ^{3},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/867bd8e48e0564fa853dacadae6a49851c3dd723" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.271ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}D&=\gamma ^{0},\\A&=i\gamma ^{1},\quad B=i\gamma ^{2},\quad C=i\gamma ^{3},\end{aligned}}}"></span> and the equation takes the form (remembering the definition of the covariant components of the <a href="/wiki/4-gradient" class="mw-redirect" title="4-gradient">4-gradient</a> and especially that <span class="texhtml">∂<sub><i>0</i></sub> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>1</i></span><span class="sr-only">/</span><span class="den"><i>c</i></span></span>⁠</span>∂<sub><i>t</i></sub></span>) </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>Dirac equation</b> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ac38859ceaaaaffca56a77087d640db903af72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.541ex; height:3.009ex;" alt="{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0}"></span> </p> </div> <p>where there is an <a href="/wiki/Einstein_notation" title="Einstein notation">implied summation</a> over the values of the twice-repeated index <span class="texhtml"><i>μ</i> = 0, 1, 2, 3</span>, and <span class="texhtml">∂<sub><i>μ</i></sub></span> is the 4-gradient. In practice one often writes the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a> in terms of 2 × 2 sub-matrices taken from the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a> and the 2 × 2 <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>. Explicitly the <a href="/wiki/Gamma_matrices#Dirac_basis" title="Gamma matrices">standard representation</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\quad \gamma ^{1}={\begin{pmatrix}0&\sigma _{x}\\-\sigma _{x}&0\end{pmatrix}},\quad \gamma ^{2}={\begin{pmatrix}0&\sigma _{y}\\-\sigma _{y}&0\end{pmatrix}},\quad \gamma ^{3}={\begin{pmatrix}0&\sigma _{z}\\-\sigma _{z}&0\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\quad \gamma ^{1}={\begin{pmatrix}0&\sigma _{x}\\-\sigma _{x}&0\end{pmatrix}},\quad \gamma ^{2}={\begin{pmatrix}0&\sigma _{y}\\-\sigma _{y}&0\end{pmatrix}},\quad \gamma ^{3}={\begin{pmatrix}0&\sigma _{z}\\-\sigma _{z}&0\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b9cf9918eb959f7aa8f2382ccd8457c597f1220" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:84.226ex; height:6.509ex;" alt="{\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\quad \gamma ^{1}={\begin{pmatrix}0&\sigma _{x}\\-\sigma _{x}&0\end{pmatrix}},\quad \gamma ^{2}={\begin{pmatrix}0&\sigma _{y}\\-\sigma _{y}&0\end{pmatrix}},\quad \gamma ^{3}={\begin{pmatrix}0&\sigma _{z}\\-\sigma _{z}&0\end{pmatrix}}.}"></span> </p><p>The complete system is summarized using the <a href="/wiki/Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a> on spacetime in the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\gamma ^{\mu },\gamma ^{\nu }\right\}=2\eta ^{\mu \nu }I_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo>=</mo> <mn>2</mn> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\gamma ^{\mu },\gamma ^{\nu }\right\}=2\eta ^{\mu \nu }I_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f70ea26c9f892780e0bc946685e111f7063af4d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.852ex; height:2.843ex;" alt="{\displaystyle \left\{\gamma ^{\mu },\gamma ^{\nu }\right\}=2\eta ^{\mu \nu }I_{4}}"></span> where the bracket expression <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a,b\}=ab+ba}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>b</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a,b\}=ab+ba}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26b1381a9421792076cb3dff8671fde43af78bd4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.98ex; height:2.843ex;" alt="{\displaystyle \{a,b\}=ab+ba}"></span> denotes the <a href="/wiki/Anticommutator" class="mw-redirect" title="Anticommutator">anticommutator</a>. These are the defining relations of a <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> over a pseudo-orthogonal 4-dimensional space with <a href="/wiki/Metric_signature" title="Metric signature">metric signature</a> <span class="texhtml">(+ − − −)</span>. The specific Clifford algebra employed in the Dirac equation is known today as the <a href="/wiki/Dirac_algebra" title="Dirac algebra">Dirac algebra</a>. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this <i><a href="/wiki/Geometric_algebra" title="Geometric algebra">geometric algebra</a></i> represents an enormous stride forward in the development of quantum theory. </p><p>The Dirac equation may now be interpreted as an <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> equation, where the rest mass is proportional to an eigenvalue of the <a href="/wiki/4-momentum_operator" class="mw-redirect" title="4-momentum operator">4-momentum operator</a>, the <a href="/wiki/Proportionality_constant" class="mw-redirect" title="Proportionality constant">proportionality constant</a> being the speed of light: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} _{\mathsf {op}}\psi =m\ c\ \psi ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> </msub> <mo>⁡</mo> <mi>ψ</mi> <mo>=</mo> <mi>m</mi> <mtext> </mtext> <mi>c</mi> <mtext> </mtext> <mi>ψ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} _{\mathsf {op}}\psi =m\ c\ \psi ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5230ee65d0b8971dc92c8f2eea7cf62cd35acdc9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.435ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} _{\mathsf {op}}\psi =m\ c\ \psi ~.}"></span> </p><p>Using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial \!\!\!/}\mathrel {\stackrel {\mathrm {def} }{=}} \gamma ^{\mu }\partial _{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> </mrow> <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> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial \!\!\!/}\mathrel {\stackrel {\mathrm {def} }{=}} \gamma ^{\mu }\partial _{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c410c9a4eee038b52a8b0022277f72e32516eb6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.826ex; height:4.009ex;" alt="{\displaystyle {\partial \!\!\!/}\mathrel {\stackrel {\mathrm {def} }{=}} \gamma ^{\mu }\partial _{\mu }}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial \!\!\!{\big /}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial \!\!\!{\big /}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fd2df37b2e7392c51f956efd692ace7b36cd34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.5ex; height:3.176ex;" alt="{\displaystyle {\partial \!\!\!{\big /}}}"></span> is pronounced "d-slash"),<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> according to Feynman slash notation, the Dirac equation becomes: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\partial \!\!\!{\big /}}\psi -mc\psi =0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <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> <mi>ψ</mi> <mo>−</mo> <mi>m</mi> <mi>c</mi> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\partial \!\!\!{\big /}}\psi -mc\psi =0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b863485649ed7a208d575ac4fa8f08b38baf8dfa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.012ex; height:3.176ex;" alt="{\displaystyle i\hbar {\partial \!\!\!{\big /}}\psi -mc\psi =0~.}"></span> </p><p>In practice, physicists often use units of measure such that <span class="texhtml"><i>ħ</i> = <i>c</i> = 1</span>, known as <a href="/wiki/Natural_units" title="Natural units">natural units</a>. The equation then takes the simple form </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>Dirac equation</b> <i> (natural units)</i> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i{\partial \!\!\!{\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 width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <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 /}}-m)\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0638363e10a430afd04e74fd9e5b70048d844c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.767ex; height:3.176ex;" alt="{\displaystyle (i{\partial \!\!\!{\big /}}-m)\psi =0}"></span> </p> </div> <p>A foundational theorem<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. (September 2024)">which?</span></a></i>]</sup> states that if two distinct sets of matrices are given that both satisfy the <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford relations</a>, then they are connected to each other by a <a href="/wiki/Matrix_similarity" title="Matrix similarity">similarity transform</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{\mu \prime }=S^{-1}\gamma ^{\mu }S~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>S</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{\mu \prime }=S^{-1}\gamma ^{\mu }S~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d72ad578822ef84102decb7400985520281248" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.139ex; height:3.176ex;" alt="{\displaystyle \gamma ^{\mu \prime }=S^{-1}\gamma ^{\mu }S~.}"></span> </p><p>If in addition the matrices are all <a href="/wiki/Unitary_transformation" title="Unitary transformation">unitary</a>, as are the Dirac set, then <span class="texhtml"><i>S</i></span> itself is <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary</a>; <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{\mu \prime }=U^{\dagger }\gamma ^{\mu }U~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>U</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{\mu \prime }=U^{\dagger }\gamma ^{\mu }U~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2842ebaddde1bfbaeb06c7850ad61620a342b4ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.371ex; height:3.176ex;" alt="{\displaystyle \gamma ^{\mu \prime }=U^{\dagger }\gamma ^{\mu }U~.}"></span> </p><p>The transformation <span class="texhtml"><i>U</i></span> is unique up to a multiplicative factor of absolute value 1. Let us now imagine a <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator <span class="texhtml"><i>γ</i><sup><i>μ</i></sup>∂<sub><i>μ</i></sub></span> to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the previously mentioned foundational theorem,<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. (September 2024)">which?</span></a></i>]</sup> one may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(iU^{\dagger }\gamma ^{\mu }U\partial _{\mu }^{\prime }-m\right)\psi \left(x^{\prime },t^{\prime }\right)&=0\\U^{\dagger }(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m)U\psi \left(x^{\prime },t^{\prime }\right)&=0~.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>U</mi> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>−</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>U</mi> <mi>ψ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(iU^{\dagger }\gamma ^{\mu }U\partial _{\mu }^{\prime }-m\right)\psi \left(x^{\prime },t^{\prime }\right)&=0\\U^{\dagger }(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m)U\psi \left(x^{\prime },t^{\prime }\right)&=0~.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d904f15e628f7f70a1d7d11e1ede231a4fe1cdc2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.269ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\left(iU^{\dagger }\gamma ^{\mu }U\partial _{\mu }^{\prime }-m\right)\psi \left(x^{\prime },t^{\prime }\right)&=0\\U^{\dagger }(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m)U\psi \left(x^{\prime },t^{\prime }\right)&=0~.\end{aligned}}}"></span> </p><p>If the transformed spinor is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\prime }=U\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>U</mi> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\prime }=U\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae25f7003897abb9601642ddc01e42c1fe9c0e71" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.592ex; height:2.843ex;" alt="{\displaystyle \psi ^{\prime }=U\psi }"></span> then the transformed Dirac equation is produced in a way that demonstrates <a href="/wiki/Manifest_covariance" title="Manifest covariance">manifest relativistic invariance</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m\right)\psi ^{\prime }\left(x^{\prime },t^{\prime }\right)=0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>−</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m\right)\psi ^{\prime }\left(x^{\prime },t^{\prime }\right)=0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5870385beea1c731a4b56c27bc60c51053955a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.296ex; height:3.176ex;" alt="{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m\right)\psi ^{\prime }\left(x^{\prime },t^{\prime }\right)=0~.}"></span> </p><p>Thus, settling on any unitary representation of the gammas is final, provided the spinor is transformed according to the unitary transformation that corresponds to the given Lorentz transformation. </p><p>The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function. The representation shown here is known as the <i>standard</i> representation – in it, the wave function's upper two components go over into Pauli's 2 spinor wave function in the limit of low energies and small velocities in comparison to light. </p><p>The considerations above reveal the origin of the gammas in <i>geometry</i>, hearkening back to Grassmann's original motivation; they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as <span class="texhtml"><i>γ</i><sub><i>μ</i></sub><i>γ</i><sub><i>ν</i></sub></span> represent <i><a href="/wiki/Oriented_surface" class="mw-redirect" title="Oriented surface">oriented surface</a> elements</i>, and so on. With this in mind, one can find the form of the unit volume element on spacetime in terms of the gammas as follows. By definition, it is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {1}{4!}}\epsilon _{\mu \nu \alpha \beta }\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\alpha }\gamma ^{\beta }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> <mi>α</mi> <mi>β</mi> </mrow> </msub> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{4!}}\epsilon _{\mu \nu \alpha \beta }\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\alpha }\gamma ^{\beta }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/294dfd80fad532201bc1e2629e65c1777f57ef81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.115ex; height:5.343ex;" alt="{\displaystyle V={\frac {1}{4!}}\epsilon _{\mu \nu \alpha \beta }\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\alpha }\gamma ^{\beta }.}"></span> </p><p>For this to be an invariant, the <a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">epsilon symbol</a> must be a <a href="/wiki/Tensor" title="Tensor">tensor</a>, and so must contain a factor of <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>g</i></span></span></span>, where <span class="texhtml"><i>g</i></span> is the <a href="/wiki/Determinant" title="Determinant">determinant</a> of the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a>. Since this is negative, that factor is <i>imaginary</i>. Thus <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9e2d73f8d2f07d11ccab24c010ad41f833582e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.671ex; height:3.176ex;" alt="{\displaystyle V=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}.}"></span> </p><p>This matrix is given the special symbol <span class="texhtml"><i>γ</i><sup>5</sup></span>, owing to its importance when one is considering improper transformations of space-time, that is, those that change the orientation of the basis vectors. In the standard representation, it is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{5}={\begin{pmatrix}0&I_{2}\\I_{2}&0\end{pmatrix}}.}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{5}={\begin{pmatrix}0&I_{2}\\I_{2}&0\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faa9123929e25f20509fd5da9321df64f498f091" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.654ex; height:6.176ex;" alt="{\displaystyle \gamma _{5}={\begin{pmatrix}0&I_{2}\\I_{2}&0\end{pmatrix}}.}"></span> </p><p>This matrix will also be found to anticommute with the other four Dirac matrices: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}\gamma ^{\mu }+\gamma ^{\mu }\gamma ^{5}=0}"> <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"> <mi>μ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}\gamma ^{\mu }+\gamma ^{\mu }\gamma ^{5}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a96750a8a6270832e01c3147c16e0307502883" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.776ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}\gamma ^{\mu }+\gamma ^{\mu }\gamma ^{5}=0}"></span> </p><p>It takes a leading role when questions of <i><a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a></i> arise because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime. </p> <div class="mw-heading mw-heading2"><h2 id="Comparison_with_related_theories">Comparison with related theories</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=5" title="Edit section: Comparison with related theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Pauli_theory">Pauli theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=6" title="Edit section: Pauli theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Pauli_equation" title="Pauli equation">Pauli equation</a> and <a href="/wiki/L%C3%A9vy-Leblond_equation" title="Lévy-Leblond equation">Lévy-Leblond equation</a></div> <p>The necessity of introducing half-integer <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> goes back experimentally to the results of the <a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach experiment</a>. A beam of atoms is run through a strong <a href="/wiki/Homogeneity_and_heterogeneity" title="Homogeneity and heterogeneity">inhomogeneous</a> <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>, which then splits into <span class="texhtml"><i>N</i></span> parts depending on the <a href="/wiki/Spin_(physics)" title="Spin (physics)">intrinsic angular momentum</a> of the atoms. It was found that for <a href="/wiki/Silver" title="Silver">silver</a> atoms, the beam was split in two; the <a href="/wiki/Ground_state" title="Ground state">ground state</a> therefore could not be <a href="/wiki/Integer" title="Integer">integer</a>, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into three parts, corresponding to atoms with <span class="texhtml"><i>L<sub>z</sub></i> = −1, 0, +1</span>. The conclusion is that silver atoms have net intrinsic angular momentum of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a> set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the <a href="/wiki/Hamilton%27s_Principle" class="mw-redirect" title="Hamilton's Principle">Hamiltonian</a>, representing a semi-classical coupling of this wave function to an applied magnetic field, as so in <a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI units</a>: (Note that bold faced characters imply <a href="/wiki/Euclidean_vectors" class="mw-redirect" title="Euclidean vectors">Euclidean vectors</a> in 3 <a href="/wiki/Dimensions" class="mw-redirect" title="Dimensions">dimensions</a>, whereas the <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski</a> <a href="/wiki/Four-vector" title="Four-vector">four-vector</a> <span class="texhtml"><i>A</i><sub><i>μ</i></sub></span> can be defined 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 A_{\mu }=\left(\phi /c,-\mathbf {A} \right)~.}"> <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>=</mo> <mrow> <mo>(</mo> <mrow> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }=\left(\phi /c,-\mathbf {A} \right)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add65779dcc87739f596193cb2f212d37015de6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.905ex; height:3.009ex;" alt="{\displaystyle A_{\mu }=\left(\phi /c,-\mathbf {A} \right)~.}"></span>) <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H={\frac {1}{\ 2\ m\ }}\ {\Bigl (}{\boldsymbol {\sigma }}\cdot {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}{\Bigr )}^{2}+e\ \phi ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>e</mi> <mtext> </mtext> <mi>ϕ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H={\frac {1}{\ 2\ m\ }}\ {\Bigl (}{\boldsymbol {\sigma }}\cdot {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}{\Bigr )}^{2}+e\ \phi ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01111c377fb378df71e42e630c2eee2f64443c5f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.884ex; height:5.176ex;" alt="{\displaystyle H={\frac {1}{\ 2\ m\ }}\ {\Bigl (}{\boldsymbol {\sigma }}\cdot {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}{\Bigr )}^{2}+e\ \phi ~.}"></span> </p><p>Here <span class="texhtml"><b>A</b></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 \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> represent the components of the <a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">electromagnetic four-potential</a> in their standard SI units, and the three sigmas are the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual <a href="/wiki/Hamiltonian_mechanics#Hamiltonian_of_a_charged_particle_in_an_electromagnetic_field" title="Hamiltonian mechanics">classical Hamiltonian of a charged particle</a> interacting with an applied field in <a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI units</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H={\frac {1}{\ 2\ m\ }}\ {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}^{2}+e\ \phi -{\frac {e\ \hbar }{\ 2\ m\ }}\ {\boldsymbol {\sigma }}\cdot \mathbf {B} ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>e</mi> <mtext> </mtext> <mi>ϕ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mtext> </mtext> <mi class="MJX-variant">ℏ</mi> </mrow> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H={\frac {1}{\ 2\ m\ }}\ {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}^{2}+e\ \phi -{\frac {e\ \hbar }{\ 2\ m\ }}\ {\boldsymbol {\sigma }}\cdot \mathbf {B} ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/316ba53164b0f2f465958dc07ff5624624903977" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.211ex; height:5.343ex;" alt="{\displaystyle H={\frac {1}{\ 2\ m\ }}\ {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}^{2}+e\ \phi -{\frac {e\ \hbar }{\ 2\ m\ }}\ {\boldsymbol {\sigma }}\cdot \mathbf {B} ~.}"></span> </p><p>This Hamiltonian is now a <span class="nowrap">2 × 2</span> matrix, so the Schrödinger equation based on it must use a two-component wave function. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as <a href="/wiki/Minimal_coupling" title="Minimal coupling">minimal coupling</a>, it takes the form: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Bigl (}\gamma ^{\mu }\ {\bigl (}i\ \hbar \ \partial _{\mu }-e\ A_{\mu }{\bigr )}-m\ c{\Bigr )}\ \psi =0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>i</mi> <mtext> </mtext> <mi class="MJX-variant">ℏ</mi> <mtext> </mtext> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>m</mi> <mtext> </mtext> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mtext> </mtext> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl (}\gamma ^{\mu }\ {\bigl (}i\ \hbar \ \partial _{\mu }-e\ A_{\mu }{\bigr )}-m\ c{\Bigr )}\ \psi =0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bab4cdfbe0c4d02ab0da2391f24efa7122eef851" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.239ex; height:4.843ex;" alt="{\displaystyle {\Bigl (}\gamma ^{\mu }\ {\bigl (}i\ \hbar \ \partial _{\mu }-e\ A_{\mu }{\bigr )}-m\ c{\Bigr )}\ \psi =0~.}"></span> </p><p>A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by <span class="texhtml"><i>i</i></span>, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the <a href="/wiki/Gyromagnetic_ratio" title="Gyromagnetic ratio">gyromagnetic ratio</a> of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. There is more however. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the SI units restored: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}m\ c^{2}-E+e\ \phi \quad &+c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)\\-c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)&m\ c^{2}+E-e\ \phi \end{pmatrix}}{\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>m</mi> <mtext> </mtext> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>E</mi> <mo>+</mo> <mi>e</mi> <mtext> </mtext> <mi>ϕ</mi> <mspace width="1em" /> </mtd> <mtd> <mo>+</mo> <mi>c</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>c</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>m</mi> <mtext> </mtext> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>E</mi> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mi>ϕ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}m\ c^{2}-E+e\ \phi \quad &+c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)\\-c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)&m\ c^{2}+E-e\ \phi \end{pmatrix}}{\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a7df2c0f1a9c2a437b4bdbf48b833fd1ea20353" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.351ex; height:6.509ex;" alt="{\displaystyle {\begin{pmatrix}m\ c^{2}-E+e\ \phi \quad &+c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)\\-c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)&m\ c^{2}+E-e\ \phi \end{pmatrix}}{\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}~.}"></span> so <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(E-e\ \phi )\ \psi _{+}-c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)\ \psi _{-}&=m\ c^{2}\ \psi _{+}\\c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)\ \psi _{+}-\left(E-e\ \phi \right)\ \psi _{-}&=m\ c^{2}\ \psi _{-}\end{aligned}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>−</mo> <mi>c</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mtext> </mtext> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mtext> </mtext> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(E-e\ \phi )\ \psi _{+}-c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)\ \psi _{-}&=m\ c^{2}\ \psi _{+}\\c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)\ \psi _{+}-\left(E-e\ \phi \right)\ \psi _{-}&=m\ c^{2}\ \psi _{-}\end{aligned}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db49956a4ccd1c1fb708957c2151c874864c099" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.343ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}(E-e\ \phi )\ \psi _{+}-c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)\ \psi _{-}&=m\ c^{2}\ \psi _{+}\\c\ {\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\ \mathbf {A} \right)\ \psi _{+}-\left(E-e\ \phi \right)\ \psi _{-}&=m\ c^{2}\ \psi _{-}\end{aligned}}~.}"></span> </p><p>Assuming the field is weak and the motion of the electron non-relativistic, the total energy of the electron is approximately equal to its <a href="/wiki/Rest_energy" class="mw-redirect" title="Rest energy">rest energy</a>, and the momentum going over to the classical value, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E-e\ \phi &\approx m\ c^{2}\\\mathbf {p} &\approx m\ \mathbf {v} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>E</mi> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mi>ϕ</mi> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mi>m</mi> <mtext> </mtext> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mi>m</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E-e\ \phi &\approx m\ c^{2}\\\mathbf {p} &\approx m\ \mathbf {v} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf07de956d7cbcdf8232397a931b233c3beb67e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.198ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}E-e\ \phi &\approx m\ c^{2}\\\mathbf {p} &\approx m\ \mathbf {v} \end{aligned}}}"></span> and so the second equation may be written <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{-}\approx {\frac {1}{\ 2\ m\ c\ }}\ {\boldsymbol {\sigma }}\cdot {\Bigl (}\mathbf {p} -e\ \mathbf {A} {\Bigr )}\ \psi _{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> <mi>c</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mtext> </mtext> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{-}\approx {\frac {1}{\ 2\ m\ c\ }}\ {\boldsymbol {\sigma }}\cdot {\Bigl (}\mathbf {p} -e\ \mathbf {A} {\Bigr )}\ \psi _{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b04123552906137bf8cc1ba74de1a4e1a89c6ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.734ex; height:5.176ex;" alt="{\displaystyle \psi _{-}\approx {\frac {1}{\ 2\ m\ c\ }}\ {\boldsymbol {\sigma }}\cdot {\Bigl (}\mathbf {p} -e\ \mathbf {A} {\Bigr )}\ \psi _{+}}"></span> </p><p>which is of order <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 \ {\tfrac {\ v\ }{c}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mi>v</mi> <mtext> </mtext> </mrow> <mi>c</mi> </mfrac> </mstyle> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\tfrac {\ v\ }{c}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55d726dfbb0dc5a7639bffc7a74d5b3daaba5507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.603ex; height:3.009ex;" alt="{\displaystyle \ {\tfrac {\ v\ }{c}}~.}"></span> Thus, at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}E-m\ c^{2}{\bigr )}\ \psi _{+}={\frac {1}{\ 2\ m\ }}\ {\Bigl [}{\boldsymbol {\sigma }}\cdot {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}{\Bigr ]}^{2}\ \psi _{+}+e\ \phi \ \psi _{+}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>E</mi> <mo>−</mo> <mi>m</mi> <mtext> </mtext> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mi>e</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>+</mo> <mi>e</mi> <mtext> </mtext> <mi>ϕ</mi> <mtext> </mtext> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}E-m\ c^{2}{\bigr )}\ \psi _{+}={\frac {1}{\ 2\ m\ }}\ {\Bigl [}{\boldsymbol {\sigma }}\cdot {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}{\Bigr ]}^{2}\ \psi _{+}+e\ \phi \ \psi _{+}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52505253895c56142b4b0d93ab51910e9e0314c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.834ex; height:5.176ex;" alt="{\displaystyle {\bigl (}E-m\ c^{2}{\bigr )}\ \psi _{+}={\frac {1}{\ 2\ m\ }}\ {\Bigl [}{\boldsymbol {\sigma }}\cdot {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}{\Bigr ]}^{2}\ \psi _{+}+e\ \phi \ \psi _{+}\ }"></span> </p><p>The operator on the left represents the particle's total energy reduced by its rest energy, which is just its classical <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>, so one can recover Pauli's theory upon identifying his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus, the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious <span class="texhtml"><i>i</i></span> that appears in it, and the necessity of a complex wave function, back to the geometry of spacetime through the Dirac algebra. It also highlights why the Schrödinger equation, although ostensibly in the form of a <a href="/wiki/Diffusion_equation" title="Diffusion equation">diffusion equation</a>, actually represents wave propagation. </p><p>It should be strongly emphasized that the entire Dirac spinor represents an <i>irreducible</i> whole. The separation, done here, of the Dirac spinor into large and small components depends on the low-energy approximation being valid. The components that were neglected above, to show that the Pauli theory can be recovered by a low-velocity approximation of Dirac's equation, are necessary to produce new phenomena observed in the relativistic regime – among them <a href="/wiki/Antimatter" title="Antimatter">antimatter</a>, and the <a href="/wiki/Matter_creation" title="Matter creation">creation</a> and <a href="/wiki/Annihilation" title="Annihilation">annihilation</a> of particles. </p> <div class="mw-heading mw-heading3"><h3 id="Weyl_theory">Weyl theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=7" title="Edit section: Weyl theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the massless case <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57f21007575fd03e3be0da20af34d25829cc9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}"></span>, the Dirac equation reduces to the <a href="/wiki/Weyl_equation" title="Weyl equation">Weyl equation</a>, which describes relativistic massless spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> particles.<sup id="cite_ref-Ohlsson2011_11-0" class="reference"><a href="#cite_note-Ohlsson2011-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The theory acquires a second <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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> symmetry: see below. </p> <div class="mw-heading mw-heading2"><h2 id="Physical_interpretation">Physical interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=8" title="Edit section: Physical interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Identification_of_observables">Identification of observables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=9" title="Edit section: Identification of observables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The critical physical question in a quantum theory is this: what are the physically <a href="/wiki/Observable" title="Observable">observable</a> quantities defined by the theory? According to the postulates of quantum mechanics, such quantities are defined by <a href="/wiki/Self-adjoint_operators" class="mw-redirect" title="Self-adjoint operators">self-adjoint operators</a> that act on the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> of possible states of a system. The eigenvalues of these operators are then the possible results of <a href="/wiki/Measurement_problem" title="Measurement problem">measuring</a> the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. To maintain this interpretation on passing to the Dirac theory, the Hamiltonian must be taken to be <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=\gamma ^{0}\left[mc^{2}+c\gamma ^{k}\left(p_{k}-qA_{k}\right)\right]+cqA^{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>q</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>c</mi> <mi>q</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=\gamma ^{0}\left[mc^{2}+c\gamma ^{k}\left(p_{k}-qA_{k}\right)\right]+cqA^{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188deadb17fc5e3a33fa216263e10137a29faf80" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.697ex; height:3.343ex;" alt="{\displaystyle H=\gamma ^{0}\left[mc^{2}+c\gamma ^{k}\left(p_{k}-qA_{k}\right)\right]+cqA^{0}.}"></span> where, as always, there is an <a href="/wiki/Einstein_notation" title="Einstein notation">implied summation</a> over the twice-repeated index <span class="texhtml"><i>k</i> = 1, 2, 3</span>. This looks promising, because one can see by inspection the rest energy of the particle and, in the case of <span class="texhtml"><b>A</b> = 0</span>, the energy of a charge placed in an electric potential <span class="texhtml"><i>cqA</i><sup>0</sup></span>. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=c{\sqrt {\left(\mathbf {p} -q\mathbf {A} \right)^{2}+m^{2}c^{2}}}+qA^{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>q</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=c{\sqrt {\left(\mathbf {p} -q\mathbf {A} \right)^{2}+m^{2}c^{2}}}+qA^{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14fe6b8c15bd318b62cad9db611fb3c7e1eedf28" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:34.121ex; height:4.843ex;" alt="{\displaystyle H=c{\sqrt {\left(\mathbf {p} -q\mathbf {A} \right)^{2}+m^{2}c^{2}}}+qA^{0}.}"></span> </p><p>Thus, the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and one must take great care to correctly identify what is observable in this theory. Much of the apparently paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2020)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Hole_theory">Hole theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=10" title="Edit section: Hole theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The negative <span class="texhtml"><i>E</i></span> solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, they cannot simply be ignored, for once the interaction between the electron and the electromagnetic field is included, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy. Real electrons obviously do not behave in this way, or they would disappear by emitting energy in the form of <a href="/wiki/Photon" title="Photon">photons</a>. </p><p>To cope with this problem, Dirac introduced the hypothesis, known as <b>hole theory</b>, that the <a href="/wiki/Vacuum" title="Vacuum">vacuum</a> is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the <a href="/wiki/Dirac_sea" title="Dirac sea">Dirac sea</a>. Since the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates. </p><p>Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a <b>hole</b> – would behave like a positively charged particle. The hole possesses a <i>positive</i> energy because energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the <a href="/wiki/Positron" title="Positron">positron</a>, experimentally discovered by <a href="/wiki/Carl_David_Anderson" title="Carl David Anderson">Carl Anderson</a> in 1932.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons have to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "<a href="/wiki/Jellium" title="Jellium">jellium</a>" background so that the net electric charge density of the vacuum is zero. In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, a <a href="/wiki/Bogoliubov_transformation" title="Bogoliubov transformation">Bogoliubov transformation</a> on the <a href="/wiki/Creation_and_annihilation_operators" title="Creation and annihilation operators">creation and annihilation operators</a> (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it. </p><p>In certain applications of <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>, however, the underlying concepts of "hole theory" are valid. The sea of <a href="/wiki/Conduction_electron" class="mw-redirect" title="Conduction electron">conduction electrons</a> in an <a href="/wiki/Electrical_conductor" title="Electrical conductor">electrical conductor</a>, called a <a href="/wiki/Composite_fermion#Fermi_sea" title="Composite fermion">Fermi sea</a>, contains electrons with energies up to the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a> of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, and although it too is referred to as an "electron hole", it is distinct from a positron. The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material. </p> <div class="mw-heading mw-heading3"><h3 id="In_quantum_field_theory">In quantum field theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=11" title="Edit section: In quantum field theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Fermionic_field" title="Fermionic field">Fermionic field</a></div> <p>In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theories</a> such as <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>, the Dirac field is subject to a process of <a href="/wiki/Second_quantization" title="Second quantization">second quantization</a>, which resolves some of the paradoxical features of the equation. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation">Mathematical formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=12" title="Edit section: Mathematical formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In its modern formulation for field theory, the Dirac equation is written in terms of a <a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinor</a> 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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </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/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> taking values in a complex vector space described concretely 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 \mathbb {C} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/990797533f68706456ef046fb4e390c6d064f8a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{4}}"></span>, defined on flat spacetime (<a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{1,3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1,3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bbc96ea97da9adfb87ac79a78c8e604c97cd7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.012ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1,3}}"></span>. Its expression also contains <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a> and a parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/501173910e6da8425b4e9d44a4e8643620bc2464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m>0}"></span> interpreted as the mass, as well as other physical constants. Dirac first obtained his equation through a factorization of Einstein's energy-momentum-mass equivalence relation assuming a scalar product of momentum vectors determined by the metric tensor and quantized the resulting relation by associating momenta to their respective operators. </p><p>In terms of a 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 \psi :\mathbb {R} ^{1,3}\rightarrow \mathbb {C} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi :\mathbb {R} ^{1,3}\rightarrow \mathbb {C} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad8f395a20f33e3ce8c723ee28b41d0a08d89aef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.808ex; height:3.009ex;" alt="{\displaystyle \psi :\mathbb {R} ^{1,3}\rightarrow \mathbb {C} ^{4}}"></span>, the Dirac equation is then </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>Dirac equation</b> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi (x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi (x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bb6f75019e72bfa972786424c22af07d6145ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.68ex; height:3.009ex;" alt="{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi (x)=0}"></span> </p> </div> <p>and in <a href="/wiki/Natural_units" title="Natural units">natural units</a>, with <a href="/wiki/Feynman_slash_notation" title="Feynman slash notation">Feynman slash notation</a>, </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>Dirac equation (natural units)</b> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i\partial \!\!\!/-m)\psi (x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i\partial \!\!\!/-m)\psi (x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b6942692584f37d53ac71686a5f630f22d6772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.725ex; height:2.843ex;" alt="{\displaystyle (i\partial \!\!\!/-m)\psi (x)=0}"></span> </p> </div> <p>The gamma matrices are a set of four <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 4\times 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89eb2e0f4ddfe5f30c8016a0f2aa1fb5ecedfe20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 4\times 4}"></span> complex matrices (elements 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 {\text{Mat}}_{4\times 4}(\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>Mat</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Mat}}_{4\times 4}(\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e1dc68dea6f50632bf33e79661283eeef36bde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.84ex; height:2.843ex;" alt="{\displaystyle {\text{Mat}}_{4\times 4}(\mathbb {C} )}"></span>) which satisfy the defining <i>anti</i>-commutation relations: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=2\eta ^{\mu \nu }I_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <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> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mn>2</mn> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=2\eta ^{\mu \nu }I_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ad14fc012327beace837daa6cf8541bf0336d5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.852ex; height:2.843ex;" alt="{\displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=2\eta ^{\mu \nu }I_{4}}"></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 \eta ^{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ^{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027ec2b822550a7f099c1c739a05738ed6be7831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.268ex; height:2.843ex;" alt="{\displaystyle \eta ^{\mu \nu }}"></span> is the Minkowski metric element, and the indices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ,\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489a4ac1b5d76663fa5328e6f3c1d382655cb496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.668ex; height:2.176ex;" alt="{\displaystyle \mu ,\nu }"></span> run over 0,1,2 and 3. These matrices can be realized explicitly under a choice of representation. Two common choices are the Dirac representation and the chiral representation. The Dirac representation is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\quad \gamma ^{i}={\begin{pmatrix}0&\sigma ^{i}\\-\sigma ^{i}&0\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\quad \gamma ^{i}={\begin{pmatrix}0&\sigma ^{i}\\-\sigma ^{i}&0\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aad32fe98cb37bf38bd468b97e03635de094d0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.636ex; height:6.176ex;" alt="{\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\quad \gamma ^{i}={\begin{pmatrix}0&\sigma ^{i}\\-\sigma ^{i}&0\end{pmatrix}},}"></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 \sigma ^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1d0a098aad3c723834e6044b830ce3ddc83407" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.676ex;" alt="{\displaystyle \sigma ^{i}}"></span> are the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>. </p><p>For the chiral representation 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 \gamma ^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/607365b9067701be0de14bc586f165284479fdb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.08ex; height:3.176ex;" alt="{\displaystyle \gamma ^{i}}"></span> are the same, but <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 ^{0}={\begin{pmatrix}0&I_{2}\\I_{2}&0\end{pmatrix}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{0}={\begin{pmatrix}0&I_{2}\\I_{2}&0\end{pmatrix}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315812b661fb7abfcfe0d9e66b700146f1750af3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.31ex; height:6.176ex;" alt="{\displaystyle \gamma ^{0}={\begin{pmatrix}0&I_{2}\\I_{2}&0\end{pmatrix}}~.}"></span> </p><p>The slash notation is a compact notation for <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\!\!\!/:=\gamma ^{\mu }A_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>:=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\!\!\!/:=\gamma ^{\mu }A_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/731586a289bf5d0e78edf1df6c825ae091d5688f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.959ex; height:3.009ex;" alt="{\displaystyle A\!\!\!/:=\gamma ^{\mu }A_{\mu }}"></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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is a four-vector (often it is the four-vector differential 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 \partial _{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be3e5cf12940b60f8569a9927ad28823277177a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.458ex; height:2.843ex;" alt="{\displaystyle \partial _{\mu }}"></span>). The summation over the index <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is implied. </p><p>Alternatively the four coupled linear first-order <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> for the four quantities that make up the wave function can be written as a vector. In <a href="/wiki/Planck_units" title="Planck units">Planck units</a> this becomes:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 6">: 6 </span></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\partial _{x}{\begin{bmatrix}+\psi _{4}\\+\psi _{3}\\-\psi _{2}\\-\psi _{1}\end{bmatrix}}+\partial _{y}{\begin{bmatrix}+\psi _{4}\\-\psi _{3}\\-\psi _{2}\\+\psi _{1}\end{bmatrix}}+i\partial _{z}{\begin{bmatrix}+\psi _{3}\\-\psi _{4}\\-\psi _{1}\\+\psi _{2}\end{bmatrix}}-m{\begin{bmatrix}+\psi _{1}\\+\psi _{2}\\+\psi _{3}\\+\psi _{4}\end{bmatrix}}=i\partial _{t}{\begin{bmatrix}-\psi _{1}\\-\psi _{2}\\+\psi _{3}\\+\psi _{4}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mi>i</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>i</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\partial _{x}{\begin{bmatrix}+\psi _{4}\\+\psi _{3}\\-\psi _{2}\\-\psi _{1}\end{bmatrix}}+\partial _{y}{\begin{bmatrix}+\psi _{4}\\-\psi _{3}\\-\psi _{2}\\+\psi _{1}\end{bmatrix}}+i\partial _{z}{\begin{bmatrix}+\psi _{3}\\-\psi _{4}\\-\psi _{1}\\+\psi _{2}\end{bmatrix}}-m{\begin{bmatrix}+\psi _{1}\\+\psi _{2}\\+\psi _{3}\\+\psi _{4}\end{bmatrix}}=i\partial _{t}{\begin{bmatrix}-\psi _{1}\\-\psi _{2}\\+\psi _{3}\\+\psi _{4}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d55951e4861a058f8d27569eeba031b597fa80" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:66.193ex; height:12.509ex;" alt="{\displaystyle i\partial _{x}{\begin{bmatrix}+\psi _{4}\\+\psi _{3}\\-\psi _{2}\\-\psi _{1}\end{bmatrix}}+\partial _{y}{\begin{bmatrix}+\psi _{4}\\-\psi _{3}\\-\psi _{2}\\+\psi _{1}\end{bmatrix}}+i\partial _{z}{\begin{bmatrix}+\psi _{3}\\-\psi _{4}\\-\psi _{1}\\+\psi _{2}\end{bmatrix}}-m{\begin{bmatrix}+\psi _{1}\\+\psi _{2}\\+\psi _{3}\\+\psi _{4}\end{bmatrix}}=i\partial _{t}{\begin{bmatrix}-\psi _{1}\\-\psi _{2}\\+\psi _{3}\\+\psi _{4}\end{bmatrix}}}"></span> which makes it clearer that it is a set of four partial differential equations with four unknown functions. (Note that the <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a280dfbf6f38f1af288cc21b0ced05dafebf7f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.284ex; height:2.843ex;" alt="{\displaystyle \partial _{y}}"></span>⁠</span> term is not preceded by <span class="texhtml mvar" style="font-style:italic;">i</span> because <span class="texhtml"><i>σ</i><sub><i>y</i></sub></span> is imaginary.) </p> <div class="mw-heading mw-heading3"><h3 id="Dirac_adjoint_and_the_adjoint_equation">Dirac adjoint and the adjoint equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=13" title="Edit section: Dirac adjoint and the adjoint equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Dirac adjoint</b> of the spinor 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 \psi (x)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"></span> is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\psi }}(x)=\psi (x)^{\dagger }\gamma ^{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <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 {\bar {\psi }}(x)=\psi (x)^{\dagger }\gamma ^{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9bbd281b090adb3e753d2ef06febbc33a37002" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.428ex; height:3.176ex;" alt="{\displaystyle {\bar {\psi }}(x)=\psi (x)^{\dagger }\gamma ^{0}.}"></span> Using the property of gamma matrices (which follows straightforwardly from Hermicity properties 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 \gamma ^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9ad28fd0224d333e10919fe473cf40e52ac6c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.503ex; height:2.843ex;" alt="{\displaystyle \gamma ^{\mu }}"></span>) that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\gamma ^{\mu })^{\dagger }=\gamma ^{0}\gamma ^{\mu }\gamma ^{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </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 (\gamma ^{\mu })^{\dagger }=\gamma ^{0}\gamma ^{\mu }\gamma ^{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7996ee9fba5571cef10030a55221e81eafeb079f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.192ex; height:3.176ex;" alt="{\displaystyle (\gamma ^{\mu })^{\dagger }=\gamma ^{0}\gamma ^{\mu }\gamma ^{0},}"></span> one can derive the adjoint Dirac equation by taking the Hermitian conjugate of the Dirac equation and multiplying on the right 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 ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e188650006a8d530e7086eeb8b4c1b9c6ba33ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:3.176ex;" alt="{\displaystyle \gamma ^{0}}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\psi }}(x)(-i\gamma ^{\mu }{\overleftarrow {\partial }}_{\mu }-m)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∂</mi> <mo>←</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}(x)(-i\gamma ^{\mu }{\overleftarrow {\partial }}_{\mu }-m)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1057aea898b8a6b14df249a67dac475eacb5de04" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.594ex; height:4.343ex;" alt="{\displaystyle {\bar {\psi }}(x)(-i\gamma ^{\mu }{\overleftarrow {\partial }}_{\mu }-m)=0}"></span> where the partial derivative <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 {\overleftarrow {\partial }}_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∂</mi> <mo>←</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overleftarrow {\partial }}_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc283c58d4607fd966570167ac8fbeb6b31e20c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.795ex; height:4.343ex;" alt="{\displaystyle {\overleftarrow {\partial }}_{\mu }}"></span> acts from the right on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\psi }}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd7a9de2742ff615699c9b81ae837b1a9cce540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.735ex; height:3.009ex;" alt="{\displaystyle {\bar {\psi }}(x)}"></span>: written in the usual way in terms of a left action of the derivative, we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\partial _{\mu }{\bar {\psi }}(x)\gamma ^{\mu }-m{\bar {\psi }}(x)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\partial _{\mu }{\bar {\psi }}(x)\gamma ^{\mu }-m{\bar {\psi }}(x)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef0f33d67b0f5783ec6f4bab6389663322eec3b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.829ex; height:3.176ex;" alt="{\displaystyle -i\partial _{\mu }{\bar {\psi }}(x)\gamma ^{\mu }-m{\bar {\psi }}(x)=0.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Klein–Gordon_equation"><span id="Klein.E2.80.93Gordon_equation"></span>Klein–Gordon equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=14" title="Edit section: Klein–Gordon equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Applying <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 \!\!\!/+m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>+</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\partial \!\!\!/+m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a545d3c9a1d698023333b9f2415768c79d52efc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.002ex; height:2.843ex;" alt="{\displaystyle i\partial \!\!\!/+m}"></span> to the Dirac equation gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\partial _{\mu }\partial ^{\mu }+m^{2})\psi (x)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\partial _{\mu }\partial ^{\mu }+m^{2})\psi (x)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478fba8c1020fbf3b20417d1bf8b83cebbd6a315" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.329ex; height:3.343ex;" alt="{\displaystyle (\partial _{\mu }\partial ^{\mu }+m^{2})\psi (x)=0.}"></span> That is, each component of the Dirac spinor field satisfies the <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Conserved_current">Conserved current</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=15" title="Edit section: Conserved current"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Conserved_current" title="Conserved current">conserved current</a> of the theory is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a561380bac5b6337ce09eba3ddc53a82b536ed6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.107ex; height:3.009ex;" alt="{\displaystyle J^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .}"></span> </p> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Proof of conservation from Dirac equation</strong> <p>Adding the Dirac and adjoint Dirac equations gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i((\partial _{\mu }{\bar {\psi }})\gamma ^{\mu }\psi +{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ψ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i((\partial _{\mu }{\bar {\psi }})\gamma ^{\mu }\psi +{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/881f845c41f32280aa81fd47f99906553429d52d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.662ex; height:3.176ex;" alt="{\displaystyle i((\partial _{\mu }{\bar {\psi }})\gamma ^{\mu }\psi +{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi )=0}"></span> so by Leibniz rule, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\partial _{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\partial _{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1126a7ee196733f04f5d9ccb65f7ee25a4d9b8ab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.943ex; height:3.176ex;" alt="{\displaystyle i\partial _{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )=0}"></span> </p> </div> <p>Another approach to derive this expression is by variational methods, applying <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> for the global <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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> symmetry to derive the conserved current <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{\mu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{\mu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feaf36c5731316bf031ab47a4d597a6405410d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.396ex; height:2.343ex;" alt="{\displaystyle J^{\mu }.}"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof of conservation from Noether's theorem</strong> <p>Recall the Lagrangian is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bca71b7c4732324eb522b0f6b0a3395a3d4e08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.911ex; height:3.176ex;" alt="{\displaystyle {\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi .}"></span> Under 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 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> symmetry which sends <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi &\mapsto e^{i\alpha }\psi ,\\{\bar {\psi }}&\mapsto e^{-i\alpha }{\bar {\psi }},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ψ</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mi>ψ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi &\mapsto e^{i\alpha }\psi ,\\{\bar {\psi }}&\mapsto e^{-i\alpha }{\bar {\psi }},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb9d5cb140da658c797a5fddd4b24ffda12dc12" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.417ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\psi &\mapsto e^{i\alpha }\psi ,\\{\bar {\psi }}&\mapsto e^{-i\alpha }{\bar {\psi }},\end{aligned}}}"></span> we find the Lagrangian is invariant. </p><p>Now considering the variation parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> to be infinitesimal, we work at first order in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and ignore <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 {O}}{\alpha ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}{\alpha ^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042c7c36b56cddae4e89e2e97f9e643f5fd44b71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.392ex; height:2.676ex;" alt="{\displaystyle {\mathcal {O}}{\alpha ^{2}}}"></span> terms. From the previous discussion we immediately see the explicit variation in the Lagrangian due to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is vanishing, that is under the variation, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\mapsto {\mathcal {L}}+\delta {\mathcal {L}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>+</mo> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\mapsto {\mathcal {L}}+\delta {\mathcal {L}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a9465fae3508ee68fedb4b7131878a20fa60017" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.961ex; height:2.676ex;" alt="{\displaystyle {\mathcal {L}}\mapsto {\mathcal {L}}+\delta {\mathcal {L}},}"></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 \delta {\mathcal {L}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta {\mathcal {L}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/630fe89b387c45ced85724ee86186efa9ec01439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.913ex; height:2.343ex;" alt="{\displaystyle \delta {\mathcal {L}}=0}"></span>. </p><p>As part of Noether's theorem, we find the implicit variation in the Lagrangian due to variation of fields. If the equation of motion for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ,{\bar {\psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ,{\bar {\psi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35de2a319364dc3f6a5802d134230effb394f8b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.143ex; height:2.843ex;" alt="{\displaystyle \psi ,{\bar {\psi }}}"></span> are satisfied, then </p> <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\mathcal {L}}=\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\delta \psi +{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }})}}\delta {\bar {\psi }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>δ</mi> <mi>ψ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta {\mathcal {L}}=\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\delta \psi +{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }})}}\delta {\bar {\psi }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94bdb3ac2f71eaa99d3b79798a1818b5cfe8895e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.275ex; height:7.509ex;" alt="{\displaystyle \delta {\mathcal {L}}=\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\delta \psi +{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }})}}\delta {\bar {\psi }}\right)}"></span></td> <td></td> <td class="nowrap"><span id="math_*" class="reference nourlexpansion" style="font-weight:bold;">*</span></td></tr></tbody></table> <p>This immediately simplifies as there are no partial derivatives 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 {\bar {\psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890adebe2730294079a81b7bf08b2fe0f2c59909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.596ex; height:2.843ex;" alt="{\displaystyle {\bar {\psi }}}"></span> in the Lagrangian. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2003c0fcb8fbf980507d6c7064a31a0d3182681a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.562ex; height:2.676ex;" alt="{\displaystyle \delta \psi }"></span> is the infinitesimal variation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \psi (x)=i\alpha \psi (x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mi>α</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \psi (x)=i\alpha \psi (x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd58a1cdd48c2c81ec572b145e9a44590e10367" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.388ex; height:2.843ex;" alt="{\displaystyle \delta \psi (x)=i\alpha \psi (x).}"></span> We evaluate <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}=i{\bar {\psi }}\gamma ^{\mu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}=i{\bar {\psi }}\gamma ^{\mu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcb89991b05da3f3419cb295786e9080d21bb99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.581ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}=i{\bar {\psi }}\gamma ^{\mu }.}"></span> The equation (<b><a href="#math_*">*</a></b>) finally is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=-\alpha \partial _{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mo>−</mo> <mi>α</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=-\alpha \partial _{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75171b8c0b9400cfac5c37d6523453bf5a1c1e72" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.436ex; height:3.176ex;" alt="{\displaystyle 0=-\alpha \partial _{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )}"></span> </p> </div> <div class="mw-heading mw-heading3"><h3 id="Solutions">Solutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=16" title="Edit section: Solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinor</a> and <a href="#Hole_theory">§ Hole theory</a></div> <p>Since the Dirac operator acts on 4-tuples of <a href="/wiki/Square-integrable_functions" class="mw-redirect" title="Square-integrable functions">square-integrable functions</a>, its solutions should be members of the same <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>. The fact that the energies of the solutions do not have a lower bound is unexpected. </p> <div class="mw-heading mw-heading4"><h4 id="Plane-wave_solutions">Plane-wave solutions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=17" title="Edit section: Plane-wave solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Plane-wave solutions are those arising from an ansatz <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=u(\mathbf {p} )e^{-ip\cdot x}}"> <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> <mi>u</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>p</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=u(\mathbf {p} )e^{-ip\cdot x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32e78db25b7b1459f19aeaf1b60742c451cfcbfa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.761ex; height:3.176ex;" alt="{\displaystyle \psi (x)=u(\mathbf {p} )e^{-ip\cdot x}}"></span> which models a particle with definite 4-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 p=(E_{\mathbf {p} },\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=(E_{\mathbf {p} },\mathbf {p} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8665a66bf73fb744cb5c8e548fdacef6ef07d96a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:11.684ex; height:3.009ex;" alt="{\displaystyle p=(E_{\mathbf {p} },\mathbf {p} )}"></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="{\textstyle E_{\mathbf {p} }={\sqrt {m^{2}+|\mathbf {p} |^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle E_{\mathbf {p} }={\sqrt {m^{2}+|\mathbf {p} |^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686ff49beaac5e46e2692cd6757a36ab1852b4d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:18.835ex; height:4.843ex;" alt="{\textstyle E_{\mathbf {p} }={\sqrt {m^{2}+|\mathbf {p} |^{2}}}.}"></span> </p><p>For this ansatz, the Dirac equation becomes an equation for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(\mathbf {p} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb05a27f493c89513b75f2ff44b5427e55cfef3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.624ex; height:2.843ex;" alt="{\displaystyle u(\mathbf {p} )}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\gamma ^{\mu }p_{\mu }-m\right)u(\mathbf {p} )=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\gamma ^{\mu }p_{\mu }-m\right)u(\mathbf {p} )=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea8f7827274b073057b2c2e48568bdd88784259" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.505ex; height:3.009ex;" alt="{\displaystyle \left(\gamma ^{\mu }p_{\mu }-m\right)u(\mathbf {p} )=0.}"></span> After picking a representation for the gamma matrices <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 ^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9ad28fd0224d333e10919fe473cf40e52ac6c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.503ex; height:2.843ex;" alt="{\displaystyle \gamma ^{\mu }}"></span>, solving this is a matter of solving a system of linear equations. It is a representation-free property of gamma matrices that the solution space is two-dimensional (see <a href="/wiki/Gamma_matrices#Other_representation-free_properties" title="Gamma matrices">here</a>). </p><p>For example, in the chiral representation for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9ad28fd0224d333e10919fe473cf40e52ac6c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.503ex; height:2.843ex;" alt="{\displaystyle \gamma ^{\mu }}"></span>, the solution space is parametrised by 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} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{2}}"></span> 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 \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"></span>, with <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\mathbf {p} )={\begin{pmatrix}{\sqrt {\sigma ^{\mu }p_{\mu }}}\xi \\{\sqrt {{\bar {\sigma }}^{\mu }p_{\mu }}}\xi \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </msqrt> </mrow> <mi>ξ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </msqrt> </mrow> <mi>ξ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(\mathbf {p} )={\begin{pmatrix}{\sqrt {\sigma ^{\mu }p_{\mu }}}\xi \\{\sqrt {{\bar {\sigma }}^{\mu }p_{\mu }}}\xi \end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb5f8207683548dbadef5e820908ad99740d912" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.456ex; height:7.509ex;" alt="{\displaystyle u(\mathbf {p} )={\begin{pmatrix}{\sqrt {\sigma ^{\mu }p_{\mu }}}\xi \\{\sqrt {{\bar {\sigma }}^{\mu }p_{\mu }}}\xi \end{pmatrix}}}"></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 \sigma ^{\mu }=(I_{2},\sigma ^{i}),{\bar {\sigma }}^{\mu }=(I_{2},-\sigma ^{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\mu }=(I_{2},\sigma ^{i}),{\bar {\sigma }}^{\mu }=(I_{2},-\sigma ^{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ac26217f42dcdf3966b0e6d59900cd9ebaed88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.247ex; height:3.176ex;" alt="{\displaystyle \sigma ^{\mu }=(I_{2},\sigma ^{i}),{\bar {\sigma }}^{\mu }=(I_{2},-\sigma ^{i})}"></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 {\sqrt {\cdot }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>⋅</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\cdot }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/023052aea0c7332f3b5a8a581e03a6b4b1119caa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.583ex; height:2.843ex;" alt="{\displaystyle {\sqrt {\cdot }}}"></span> is the Hermitian matrix square-root. </p><p>These plane-wave solutions provide a starting point for canonical quantization. </p> <div class="mw-heading mw-heading3"><h3 id="Lagrangian_formulation">Lagrangian formulation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=18" title="Edit section: Lagrangian formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}=i\hbar c{\overline {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -mc^{2}{\overline {\psi }}\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=i\hbar c{\overline {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -mc^{2}{\overline {\psi }}\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/060bf077a7803b33e65610824acaeb12aeee4844" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.004ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}=i\hbar c{\overline {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -mc^{2}{\overline {\psi }}\psi }"></span> </p><p>If one varies this with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </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/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> one gets the adjoint Dirac equation. Meanwhile, if one varies this with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890adebe2730294079a81b7bf08b2fe0f2c59909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.596ex; height:2.843ex;" alt="{\displaystyle {\bar {\psi }}}"></span> one gets the Dirac equation. </p><p>In natural units and with the slash notation, the action is then </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>Dirac Action</b> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\int d^{4}x\,{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>i</mi> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <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> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int d^{4}x\,{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a50de1deae3d811bf18962611436ca05990b5d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.043ex; height:5.676ex;" alt="{\displaystyle S=\int d^{4}x\,{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi }"></span> </p> </div> <p>For this action, the conserved current <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74fcaa978430bd2a8c9fb6957192b62fa191d9bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.749ex; height:2.343ex;" alt="{\displaystyle J^{\mu }}"></span> above arises as the conserved current corresponding to the global <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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> symmetry through <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> for field theory. Gauging this field theory by changing the symmetry to a local, spacetime point dependent one gives gauge symmetry (really, gauge redundancy). The resultant theory is <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> or QED. See below for a more detailed discussion. </p> <div class="mw-heading mw-heading3"><h3 id="Lorentz_invariance">Lorentz invariance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=19" title="Edit section: Lorentz invariance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Dirac equation is invariant under Lorentz transformations, that is, under the action of the Lorentz group <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 {\text{SO}}(1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SO</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SO}}(1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dac29ed3b32e940a2746f4b549fe0742b1d20628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.269ex; height:2.843ex;" alt="{\displaystyle {\text{SO}}(1,3)}"></span> or strictly <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 {\text{SO}}(1,3)^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SO</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SO}}(1,3)^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78496dc40739f5959f7b0745e927af29bf6c5e6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.78ex; height:3.009ex;" alt="{\displaystyle {\text{SO}}(1,3)^{+}}"></span>, the component connected to the identity. </p><p>For a Dirac spinor viewed concretely as taking values in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/990797533f68706456ef046fb4e390c6d064f8a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{4}}"></span>, the transformation under a Lorentz transformation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> is given by 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 4\times 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89eb2e0f4ddfe5f30c8016a0f2aa1fb5ecedfe20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 4\times 4}"></span> complex 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 S[\Lambda ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[\Lambda ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/077a74b520c1b15b7b8fb3ab6bfd5e42186a8798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.406ex; height:2.843ex;" alt="{\displaystyle S[\Lambda ]}"></span>. There are some subtleties in defining the corresponding <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 S[\Lambda ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[\Lambda ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/077a74b520c1b15b7b8fb3ab6bfd5e42186a8798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.406ex; height:2.843ex;" alt="{\displaystyle S[\Lambda ]}"></span>, as well as a standard abuse of notation. </p><p>Most treatments occur at the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> level. For a more detailed treatment see <a href="/wiki/Lorentz_group#Lie_algebra" title="Lorentz group">here</a>. The Lorentz group 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 4\times 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89eb2e0f4ddfe5f30c8016a0f2aa1fb5ecedfe20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 4\times 4}"></span> <i>real</i> matrices acting on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{1,3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1,3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bbc96ea97da9adfb87ac79a78c8e604c97cd7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.012ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1,3}}"></span> is generated by a set of six matrices <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^{\mu \nu }\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{M^{\mu \nu }\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1eb3dba5f5aafb09220b1ec9d9560a2485d590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.918ex; height:2.843ex;" alt="{\displaystyle \{M^{\mu \nu }\}}"></span> with components <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M^{\mu \nu })^{\rho }{}_{\sigma }=\eta ^{\mu \rho }\delta ^{\nu }{}_{\sigma }-\eta ^{\nu \rho }\delta ^{\mu }{}_{\sigma }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ρ</mi> </mrow> </msup> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>ρ</mi> </mrow> </msup> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M^{\mu \nu })^{\rho }{}_{\sigma }=\eta ^{\mu \rho }\delta ^{\nu }{}_{\sigma }-\eta ^{\nu \rho }\delta ^{\mu }{}_{\sigma }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a94422f6b8296881ef47ac24c1a8bbec802fa3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.396ex; height:2.843ex;" alt="{\displaystyle (M^{\mu \nu })^{\rho }{}_{\sigma }=\eta ^{\mu \rho }\delta ^{\nu }{}_{\sigma }-\eta ^{\nu \rho }\delta ^{\mu }{}_{\sigma }.}"></span> When both 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 \rho ,\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>,</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ,\sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bdc929543888431897809cdd2df86180ff5ed09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.566ex; height:2.176ex;" alt="{\displaystyle \rho ,\sigma }"></span> indices are raised or lowered, these are simply the 'standard basis' of antisymmetric matrices. </p><p>These satisfy the Lorentz algebra commutation relations <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=M^{\mu \sigma }\eta ^{\nu \rho }-M^{\nu \sigma }\eta ^{\mu \rho }+M^{\nu \rho }\eta ^{\mu \sigma }-M^{\mu \rho }\eta ^{\nu \sigma }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>σ</mi> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>ρ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>σ</mi> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ρ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>ρ</mi> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>σ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ρ</mi> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>σ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=M^{\mu \sigma }\eta ^{\nu \rho }-M^{\nu \sigma }\eta ^{\mu \rho }+M^{\nu \rho }\eta ^{\mu \sigma }-M^{\mu \rho }\eta ^{\nu \sigma }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94a9022837b52dcbbcf58bf59339dd5754657dfe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.867ex; height:2.843ex;" alt="{\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=M^{\mu \sigma }\eta ^{\nu \rho }-M^{\nu \sigma }\eta ^{\mu \rho }+M^{\nu \rho }\eta ^{\mu \sigma }-M^{\mu \rho }\eta ^{\nu \sigma }.}"></span> In the article on the <a href="/wiki/Dirac_algebra" title="Dirac algebra">Dirac algebra</a>, it is also found that the spin generators <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\mu \nu }={\frac {1}{4}}[\gamma ^{\mu },\gamma ^{\nu }]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <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> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{\mu \nu }={\frac {1}{4}}[\gamma ^{\mu },\gamma ^{\nu }]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa1d46ce1c9924b2baf13f7d47275d6fba96795" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.928ex; height:5.176ex;" alt="{\displaystyle S^{\mu \nu }={\frac {1}{4}}[\gamma ^{\mu },\gamma ^{\nu }]}"></span> satisfy the Lorentz algebra commutation relations. </p><p>A Lorentz transformation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda =\exp \left({\frac {1}{2}}\omega _{\mu \nu }M^{\mu \nu }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda =\exp \left({\frac {1}{2}}\omega _{\mu \nu }M^{\mu \nu }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2efd2386e6af720ba38a7ab0d22f685d9036eae0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.818ex; height:6.176ex;" alt="{\displaystyle \Lambda =\exp \left({\frac {1}{2}}\omega _{\mu \nu }M^{\mu \nu }\right)}"></span> where the components <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 \omega _{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/938bfe327ff95bcb3b6a76bd7cc181159baf86f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.54ex; height:2.343ex;" alt="{\displaystyle \omega _{\mu \nu }}"></span> are antisymmetric in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ,\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489a4ac1b5d76663fa5328e6f3c1d382655cb496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.668ex; height:2.176ex;" alt="{\displaystyle \mu ,\nu }"></span>. </p><p>The corresponding transformation on spin space is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S[\Lambda ]=\exp \left({\frac {1}{2}}\omega _{\mu \nu }S^{\mu \nu }\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[\Lambda ]=\exp \left({\frac {1}{2}}\omega _{\mu \nu }S^{\mu \nu }\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10722092977f8485627304963fca0083fc6ef228" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.28ex; height:6.176ex;" alt="{\displaystyle S[\Lambda ]=\exp \left({\frac {1}{2}}\omega _{\mu \nu }S^{\mu \nu }\right).}"></span> This is an abuse of notation, but a standard one. The reason is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S[\Lambda ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[\Lambda ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/077a74b520c1b15b7b8fb3ab6bfd5e42186a8798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.406ex; height:2.843ex;" alt="{\displaystyle S[\Lambda ]}"></span> is not a well-defined function 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 \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span>, since there are two different sets of components <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 \omega _{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/938bfe327ff95bcb3b6a76bd7cc181159baf86f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.54ex; height:2.343ex;" alt="{\displaystyle \omega _{\mu \nu }}"></span> (up to equivalence) which give the same <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> but different <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 S[\Lambda ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[\Lambda ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/077a74b520c1b15b7b8fb3ab6bfd5e42186a8798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.406ex; height:2.843ex;" alt="{\displaystyle S[\Lambda ]}"></span>. In practice we implicitly pick one of these <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 \omega _{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/938bfe327ff95bcb3b6a76bd7cc181159baf86f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.54ex; height:2.343ex;" alt="{\displaystyle \omega _{\mu \nu }}"></span> and then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S[\Lambda ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[\Lambda ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/077a74b520c1b15b7b8fb3ab6bfd5e42186a8798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.406ex; height:2.843ex;" alt="{\displaystyle S[\Lambda ]}"></span> is well defined in terms 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 \omega _{\mu \nu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mu \nu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8224ceefae69c6e46f08d195b1b50eee7374491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.187ex; height:2.343ex;" alt="{\displaystyle \omega _{\mu \nu }.}"></span> </p><p>Under a Lorentz transformation, the Dirac equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\gamma ^{\mu }\partial _{\mu }\psi (x)-m\psi (x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\gamma ^{\mu }\partial _{\mu }\psi (x)-m\psi (x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e923264bc0f1f03e2a84fd8ac528e5f923deca1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.21ex; height:3.009ex;" alt="{\displaystyle i\gamma ^{\mu }\partial _{\mu }\psi (x)-m\psi (x)=0}"></span> becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\gamma ^{\mu }((\Lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })S[\Lambda ]\psi (\Lambda ^{-1}x)-mS[\Lambda ]\psi (\Lambda ^{-1}x)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\gamma ^{\mu }((\Lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })S[\Lambda ]\psi (\Lambda ^{-1}x)-mS[\Lambda ]\psi (\Lambda ^{-1}x)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/406788da24426d18a98234247f994455acded045" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.331ex; height:3.343ex;" alt="{\displaystyle i\gamma ^{\mu }((\Lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })S[\Lambda ]\psi (\Lambda ^{-1}x)-mS[\Lambda ]\psi (\Lambda ^{-1}x)=0.}"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Remainder of proof of Lorentz invariance</strong> <p>Multiplying both sides from the left 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 S^{-1}[\Lambda ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{-1}[\Lambda ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/599451b0fac5eb9ffd9fcff587ba85f8189ab397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.761ex; height:3.176ex;" alt="{\displaystyle S^{-1}[\Lambda ]}"></span> and returning the dummy variable to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </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/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle iS[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ]((\Lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })\psi (x)-m\psi (x)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle iS[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ]((\Lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })\psi (x)-m\psi (x)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7020f3da1d164c0669eb5b53d5155f08d42e2bbc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.773ex; height:3.343ex;" alt="{\displaystyle iS[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ]((\Lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })\psi (x)-m\psi (x)=0.}"></span> We'll have shown invariance if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ](\Lambda ^{-1})^{\nu }{}_{\mu }=\gamma ^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ](\Lambda ^{-1})^{\nu }{}_{\mu }=\gamma ^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41bc1b06fd934caccc9008ed8ca291a33432eb05" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.212ex; height:3.343ex;" alt="{\displaystyle S[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ](\Lambda ^{-1})^{\nu }{}_{\mu }=\gamma ^{\nu }}"></span> or equivalently <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ]=\Lambda ^{\mu }{}_{\nu }\gamma ^{\nu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>S</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ]=\Lambda ^{\mu }{}_{\nu }\gamma ^{\nu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d493d98586cc5011c2eeb929b4e26ea6c8a985b1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.717ex; height:3.176ex;" alt="{\displaystyle S[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ]=\Lambda ^{\mu }{}_{\nu }\gamma ^{\nu }.}"></span> This is most easily shown at the algebra level. Supposing the transformations are parametrised by infinitesimal components <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 \omega _{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/938bfe327ff95bcb3b6a76bd7cc181159baf86f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.54ex; height:2.343ex;" alt="{\displaystyle \omega _{\mu \nu }}"></span>, then at first order in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>, on the left-hand side we get <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\omega _{\rho \sigma }(M^{\rho \sigma })^{\mu }{}_{\nu }\gamma ^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\omega _{\rho \sigma }(M^{\rho \sigma })^{\mu }{}_{\nu }\gamma ^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea731e151719d4d00a5273e65ae3ac6e04e6c0f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.508ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\omega _{\rho \sigma }(M^{\rho \sigma })^{\mu }{}_{\nu }\gamma ^{\nu }}"></span> while on the right-hand side we get <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {1}{2}}\omega _{\rho \sigma }S^{\rho \sigma },\gamma ^{\mu }\right]={\frac {1}{2}}\omega _{\rho \sigma }\left[S^{\rho \sigma },\gamma ^{\mu }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msub> <mrow> <mo>[</mo> <mrow> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {1}{2}}\omega _{\rho \sigma }S^{\rho \sigma },\gamma ^{\mu }\right]={\frac {1}{2}}\omega _{\rho \sigma }\left[S^{\rho \sigma },\gamma ^{\mu }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/927a272a97d477df7b517379cf939abfa81fd2ab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.33ex; height:6.176ex;" alt="{\displaystyle \left[{\frac {1}{2}}\omega _{\rho \sigma }S^{\rho \sigma },\gamma ^{\mu }\right]={\frac {1}{2}}\omega _{\rho \sigma }\left[S^{\rho \sigma },\gamma ^{\mu }\right]}"></span> It's a standard exercise to evaluate the commutator on the left-hand side. Writing <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^{\rho \sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{\rho \sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09f53e3ecb985af28069f8f19f6e13ffc857ed28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.521ex; height:2.343ex;" alt="{\displaystyle M^{\rho \sigma }}"></span> in terms of components completes the proof. </p> </div> <p>Associated to Lorentz invariance is a conserved Noether current, or rather a tensor of conserved Noether currents <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 {J}}^{\rho \sigma })^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">J</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {J}}^{\rho \sigma })^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7793f3e718f73b103cc55a8720ddb9a1e778c7fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.117ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {J}}^{\rho \sigma })^{\mu }}"></span>. Similarly, since the equation is invariant under translations, there is a tensor of conserved Noether currents <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d36857e2c75bdfbd5403a04c617a73b65171a9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.815ex; height:2.343ex;" alt="{\displaystyle T^{\mu \nu }}"></span>, which can be identified as the stress-energy tensor of the theory. The Lorentz current <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 {J}}^{\rho \sigma })^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">J</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {J}}^{\rho \sigma })^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7793f3e718f73b103cc55a8720ddb9a1e778c7fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.117ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {J}}^{\rho \sigma })^{\mu }}"></span> can be written in terms of the stress-energy tensor in addition to a tensor representing internal angular momentum. </p> <div class="mw-heading mw-heading4"><h4 id="Further_discussion_of_Lorentz_covariance_of_the_Dirac_equation">Further discussion of Lorentz covariance of the Dirac equation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=20" title="Edit section: Further discussion of Lorentz covariance of the Dirac equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Dirac equation is <a href="/wiki/Lorentz_covariant" class="mw-redirect" title="Lorentz covariant">Lorentz covariant</a>. Articulating this helps illuminate not only the Dirac equation, but also the <a href="/wiki/Majorana_spinor" class="mw-redirect" title="Majorana spinor">Majorana spinor</a> and <a href="/w/index.php?title=Elko_spinor&action=edit&redlink=1" class="new" title="Elko spinor (page does not exist)">Elko spinor</a>, which although closely related, have subtle and important differences. </p><p>Understanding Lorentz covariance is simplified by keeping in mind the geometric character of the process.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Let <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> be a single, fixed point in the <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> <a href="/wiki/Manifold" title="Manifold">manifold</a>. Its location can be expressed in multiple <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">coordinate systems</a>. In the physics literature, these are 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </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/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></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 x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </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/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span>, with the understanding that both <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> </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/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></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 x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </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/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span> describe <i>the same</i> 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 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>, but in different <a href="/wiki/Local_reference_frame" title="Local reference frame">local frames of reference</a> (a <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a> over a small extended patch of spacetime). One can imagine <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> as having a <a href="/wiki/Fiber_(mathematics)" title="Fiber (mathematics)">fiber</a> of different coordinate frames above it. In geometric terms, one says that spacetime can be characterized as a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a>, and specifically, the <a href="/wiki/Frame_bundle" title="Frame bundle">frame bundle</a>. The difference between two points <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> </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/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></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 x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </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/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span> in the same fiber is a combination of <a href="/wiki/Rotation" title="Rotation">rotations</a> and <a href="/wiki/Lorentz_boost" class="mw-redirect" title="Lorentz boost">Lorentz boosts</a>. A choice of coordinate frame is a (local) <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> through that bundle. </p><p>Coupled to the frame bundle is a second bundle, the <a href="/wiki/Spinor_bundle" title="Spinor bundle">spinor bundle</a>. A section through the spinor bundle is just the particle field (the Dirac spinor, in the present case). Different points in the spinor fiber correspond to the same physical object (the fermion) but expressed in different Lorentz frames. Clearly, the frame bundle and the spinor bundle must be tied together in a consistent fashion to get consistent results; formally, one says that the spinor bundle is the <a href="/wiki/Associated_bundle" title="Associated bundle">associated bundle</a>; it is associated to a <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a>, which in the present case is the frame bundle. Differences between points on the fiber correspond to the symmetries of the system. The spinor bundle has two distinct <a href="/wiki/Generator_(mathematics)" title="Generator (mathematics)">generators</a> of its symmetries: the <a href="/wiki/Total_angular_momentum" class="mw-redirect" title="Total angular momentum">total angular momentum</a> and the <a href="/wiki/Intrinsic_angular_momentum" class="mw-redirect" title="Intrinsic angular momentum">intrinsic angular momentum</a>. Both correspond to Lorentz transformations, but in different ways. </p><p>The presentation here follows that of Itzykson and Zuber.<sup id="cite_ref-iz_15-0" class="reference"><a href="#cite_note-iz-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> It is very nearly identical to that of Bjorken and Drell.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> A similar derivation in a <a href="/wiki/General_relativistic" class="mw-redirect" title="General relativistic">general relativistic</a> setting can be found in Weinberg.<sup id="cite_ref-weinberg_17-0" class="reference"><a href="#cite_note-weinberg-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Here we fix our spacetime to be flat, that is, our spacetime is Minkowski space. </p><p>Under a <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</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\mapsto x',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24315cfafa681aec77b353aab9d2e85a63b6e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.605ex; height:2.843ex;" alt="{\displaystyle x\mapsto x',}"></span> the Dirac spinor to transform as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi '(x')=S\psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi '(x')=S\psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3f04e27e1870051b7fabbd51b9c2364072209e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.271ex; height:3.009ex;" alt="{\displaystyle \psi '(x')=S\psi (x)}"></span> It can be shown that an explicit expression for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\exp \left({\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\exp \left({\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b97313a2e0df03f5965f515ea104bc5edb7b6b4b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.98ex; height:6.176ex;" alt="{\displaystyle S=\exp \left({\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)}"></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 \omega ^{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dcfeeb5699089c110d8704c8fed3ed4f1de38d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.54ex; height:2.343ex;" alt="{\displaystyle \omega ^{\mu \nu }}"></span> parameterizes the Lorentz transformation, 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 \sigma _{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2a5225e32ad6a094f47f8bb48dbb742e303797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.422ex; height:2.343ex;" alt="{\displaystyle \sigma _{\mu \nu }}"></span> are the six 4×4 matrices satisfying: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{\mu \nu }={\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]~.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <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> <mo stretchy="false">]</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\mu \nu }={\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e2271999698158d61e356297a43e9821fab684" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.964ex; height:5.176ex;" alt="{\displaystyle \sigma ^{\mu \nu }={\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]~.}"></span> </p><p>This matrix can be interpreted as the <a href="/wiki/Intrinsic_angular_momentum" class="mw-redirect" title="Intrinsic angular momentum">intrinsic angular momentum</a> of the Dirac field. That it deserves this interpretation arises by contrasting it to the generator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db78d877d8430a7d8189010063c2db973726547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.385ex; height:2.843ex;" alt="{\displaystyle J_{\mu \nu }}"></span> of <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a>, having the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\mu \nu }={\frac {1}{2}}\sigma _{\mu \nu }+i(x_{\mu }\partial _{\nu }-x_{\nu }\partial _{\mu })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{\mu \nu }={\frac {1}{2}}\sigma _{\mu \nu }+i(x_{\mu }\partial _{\nu }-x_{\nu }\partial _{\mu })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fdb9ea5f9813eaff6845d395fc42fc0e9f6edd7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.979ex; height:5.176ex;" alt="{\displaystyle J_{\mu \nu }={\frac {1}{2}}\sigma _{\mu \nu }+i(x_{\mu }\partial _{\nu }-x_{\nu }\partial _{\mu })}"></span> This can be interpreted as the <a href="/wiki/Total_angular_momentum" class="mw-redirect" title="Total angular momentum">total angular momentum</a>. It acts on the spinor field as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\prime }(x)=\exp \left({\frac {-i}{2}}\omega ^{\mu \nu }J_{\mu \nu }\right)\psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\prime }(x)=\exp \left({\frac {-i}{2}}\omega ^{\mu \nu }J_{\mu \nu }\right)\psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2692887e350c15a4847193887e479ab812190b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.433ex; height:6.176ex;" alt="{\displaystyle \psi ^{\prime }(x)=\exp \left({\frac {-i}{2}}\omega ^{\mu \nu }J_{\mu \nu }\right)\psi (x)}"></span> Note 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </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/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> above does <i>not</i> have a prime on it: the above is obtained by transforming <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\mapsto x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e2f7f304f1bfe40c8e148a344684b6daac2593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle x\mapsto x'}"></span> obtaining the change to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)\mapsto \psi '(x')}"> <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 stretchy="false">↦</mo> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)\mapsto \psi '(x')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9f8d2f80b11e6492301be671f38b4a5b044355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.288ex; height:3.009ex;" alt="{\displaystyle \psi (x)\mapsto \psi '(x')}"></span> and then returning to the original coordinate system <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'\mapsto x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">↦</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'\mapsto x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8eb773682695f914a4a65ca86e2f34631381e25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle x'\mapsto x}"></span>. </p><p>The geometrical interpretation of the above is that the <a href="/wiki/Frame_field" class="mw-redirect" title="Frame field">frame field</a> is <a href="/wiki/Affine_space" title="Affine space">affine</a>, having no preferred origin. The generator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db78d877d8430a7d8189010063c2db973726547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.385ex; height:2.843ex;" alt="{\displaystyle J_{\mu \nu }}"></span> generates the symmetries of this space: it provides a relabelling of a fixed 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 x~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mtext> </mtext> <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/b4b78e1e0799faa8b859224c9a33e01d4abde274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:1.676ex;" alt="{\displaystyle x~.}"></span> The generator <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 }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2a5225e32ad6a094f47f8bb48dbb742e303797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.422ex; height:2.343ex;" alt="{\displaystyle \sigma _{\mu \nu }}"></span> generates a movement from one point in the fiber to another: a movement 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 x\mapsto x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e2f7f304f1bfe40c8e148a344684b6daac2593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle x\mapsto x'}"></span> with both <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> </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/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></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 x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </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/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span> still corresponding to the same spacetime 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 a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>.</mo> </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/6b803da9c45c1186883bde55107e9ccb102c92c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:1.676ex;" alt="{\displaystyle a.}"></span> These perhaps obtuse remarks can be elucidated with explicit algebra. </p><p>Let <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'=\Lambda x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi mathvariant="normal">Λ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'=\Lambda x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270ddf1e50b881963d5960e52d173b61802fc822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.056ex; height:2.509ex;" alt="{\displaystyle x'=\Lambda x}"></span> be a Lorentz transformation. The Dirac equation is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi (x)-m\psi (x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi (x)-m\psi (x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8074af204563c8e1d08a38ac6739986d9a677220" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.459ex; height:5.509ex;" alt="{\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi (x)-m\psi (x)=0}"></span> If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi ^{\prime }(x^{\prime })-m\psi ^{\prime }(x^{\prime })=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>μ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi ^{\prime }(x^{\prime })-m\psi ^{\prime }(x^{\prime })=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f90b9184495662c4fb060ae80da59262dfa385" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.65ex; height:5.509ex;" alt="{\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi ^{\prime }(x^{\prime })-m\psi ^{\prime }(x^{\prime })=0}"></span> The two spinors <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"> <mi>ψ</mi> </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/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></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 ^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb5ad10b735f22aa3306f320b4617db77474f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.198ex; height:2.843ex;" alt="{\displaystyle \psi ^{\prime }}"></span> should both describe the same physical field, and so should be related by a transformation that does not change any physical observables (charge, current, mass, <i>etc.</i>) The transformation should encode only the change of coordinate frame. It can be shown that such a transformation is a 4×4 <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a>. Thus, one may presume that the relation between the two frames can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\prime }(x^{\prime })=S(\Lambda )\psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\prime }(x^{\prime })=S(\Lambda )\psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6ea51dd7e4f3e87885e097502a0b1f63826617" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.694ex; height:3.009ex;" alt="{\displaystyle \psi ^{\prime }(x^{\prime })=S(\Lambda )\psi (x)}"></span> Inserting this into the transformed equation, the result is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\gamma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}S(\Lambda )\psi (x)-mS(\Lambda )\psi (x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>μ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\gamma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}S(\Lambda )\psi (x)-mS(\Lambda )\psi (x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3646c0b5192ce60f1f927f71de4a1325b9ace24d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.342ex; height:5.509ex;" alt="{\displaystyle i\gamma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}S(\Lambda )\psi (x)-mS(\Lambda )\psi (x)=0}"></span> The coordinates related by Lorentz transformation satisfy: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>μ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/453d3de7ade40b89154be71e0198d8b2ae516b25" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.661ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }}"></span> The original Dirac equation is then regained if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\Lambda )\gamma ^{\mu }S^{-1}(\Lambda )={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu }\gamma ^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\Lambda )\gamma ^{\mu }S^{-1}(\Lambda )={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu }\gamma ^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bba7be07b54f6b2d7511ab29a9bea2dc92e359f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.586ex; height:3.509ex;" alt="{\displaystyle S(\Lambda )\gamma ^{\mu }S^{-1}(\Lambda )={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu }\gamma ^{\nu }}"></span> An explicit expression for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\Lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\Lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd25e13e1ac3334ba077d85e80f25fa5e2b9988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.922ex; height:2.843ex;" alt="{\displaystyle S(\Lambda )}"></span> (equal to the expression given above) can be obtained by considering a Lorentz transformation of infinitesimal rotation near the identity transformation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Lambda ^{\mu }}_{\nu }={g^{\mu }}_{\nu }+{\omega ^{\mu }}_{\nu }\ ,\ {(\Lambda ^{-1})^{\mu }}_{\nu }={g^{\mu }}_{\nu }-{\omega ^{\mu }}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Lambda ^{\mu }}_{\nu }={g^{\mu }}_{\nu }+{\omega ^{\mu }}_{\nu }\ ,\ {(\Lambda ^{-1})^{\mu }}_{\nu }={g^{\mu }}_{\nu }-{\omega ^{\mu }}_{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ceaebf7fbf6ad8b43b08ca39f0c55172febed3c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.531ex; height:3.343ex;" alt="{\displaystyle {\Lambda ^{\mu }}_{\nu }={g^{\mu }}_{\nu }+{\omega ^{\mu }}_{\nu }\ ,\ {(\Lambda ^{-1})^{\mu }}_{\nu }={g^{\mu }}_{\nu }-{\omega ^{\mu }}_{\nu }}"></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 {g^{\mu }}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {g^{\mu }}_{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411bc4b28ba6ae8d9f8d0af96b9946d7c114e102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.445ex; height:2.843ex;" alt="{\displaystyle {g^{\mu }}_{\nu }}"></span> is the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</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 {g^{\mu }}_{\nu }=g^{\mu \nu '}g_{\nu '\nu }={\delta ^{\mu }}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <msup> <mi>ν</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ν</mi> <mo>′</mo> </msup> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {g^{\mu }}_{\nu }=g^{\mu \nu '}g_{\nu '\nu }={\delta ^{\mu }}_{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc5efff8a35053c42c56f88f67df9fda8bb1dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.418ex; height:3.343ex;" alt="{\displaystyle {g^{\mu }}_{\nu }=g^{\mu \nu '}g_{\nu '\nu }={\delta ^{\mu }}_{\nu }}"></span> and is symmetric 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 \omega _{\mu \nu }={\omega ^{\alpha }}_{\nu }g_{\alpha \mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mu \nu }={\omega ^{\alpha }}_{\nu }g_{\alpha \mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9410a0b681d1061c95b89d7fb2a935d2f20d2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.857ex; height:3.009ex;" alt="{\displaystyle \omega _{\mu \nu }={\omega ^{\alpha }}_{\nu }g_{\alpha \mu }}"></span> is antisymmetric. After plugging and chugging, one obtains <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\Lambda )=I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }+{\mathcal {O}}\left(\Lambda ^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\Lambda )=I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }+{\mathcal {O}}\left(\Lambda ^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4b2b110fccc62c9fac41b45a31a91b4d74b544" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.316ex; height:5.176ex;" alt="{\displaystyle S(\Lambda )=I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }+{\mathcal {O}}\left(\Lambda ^{2}\right)}"></span> which is the (infinitesimal) form for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> above and yields the relation <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 }={\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <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> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\mu \nu }={\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0ba5e0514cff9eae8b231e09932002b04cb89d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.736ex; height:5.176ex;" alt="{\displaystyle \sigma ^{\mu \nu }={\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]}"></span> . To obtain the affine relabelling, write <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi '(x')&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x)\\&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x'+{\omega ^{\mu }}_{\nu }\,x^{\prime \,\nu })\\&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }-x_{\mu }^{\prime }\omega ^{\mu \nu }\partial _{\nu }\right)\psi (x')\\&=\left(I+{\frac {-i}{2}}\omega ^{\mu \nu }J_{\mu \nu }\right)\psi (x')\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mspace width="thinmathspace" /> <mi>ν</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi '(x')&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x)\\&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x'+{\omega ^{\mu }}_{\nu }\,x^{\prime \,\nu })\\&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }-x_{\mu }^{\prime }\omega ^{\mu \nu }\partial _{\nu }\right)\psi (x')\\&=\left(I+{\frac {-i}{2}}\omega ^{\mu \nu }J_{\mu \nu }\right)\psi (x')\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8412b657c05849fbcacfbf6f3c69d2cdcfced7dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.838ex; width:44.71ex; height:24.843ex;" alt="{\displaystyle {\begin{aligned}\psi '(x')&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x)\\&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x'+{\omega ^{\mu }}_{\nu }\,x^{\prime \,\nu })\\&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }-x_{\mu }^{\prime }\omega ^{\mu \nu }\partial _{\nu }\right)\psi (x')\\&=\left(I+{\frac {-i}{2}}\omega ^{\mu \nu }J_{\mu \nu }\right)\psi (x')\\\end{aligned}}}"></span> </p><p>After properly antisymmetrizing, one obtains the generator of symmetries <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db78d877d8430a7d8189010063c2db973726547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.385ex; height:2.843ex;" alt="{\displaystyle J_{\mu \nu }}"></span> given earlier. Thus, both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db78d877d8430a7d8189010063c2db973726547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.385ex; height:2.843ex;" alt="{\displaystyle J_{\mu \nu }}"></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 \sigma _{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2a5225e32ad6a094f47f8bb48dbb742e303797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.422ex; height:2.343ex;" alt="{\displaystyle \sigma _{\mu \nu }}"></span> can be said to be the "generators of Lorentz transformations", but with a subtle distinction: the first corresponds to a relabelling of points on the affine <a href="/wiki/Frame_bundle" title="Frame bundle">frame bundle</a>, which forces a translation along the fiber of the spinor on the <a href="/wiki/Spin_bundle" class="mw-redirect" title="Spin bundle">spin bundle</a>, while the second corresponds to translations along the fiber of the spin bundle (taken as a movement <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\mapsto x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e2f7f304f1bfe40c8e148a344684b6daac2593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.958ex; height:2.509ex;" alt="{\displaystyle x\mapsto x'}"></span> along the frame bundle, as well as a movement <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 '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">↦</mo> <msup> <mi>ψ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \mapsto \psi '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5cf42cf86b6815a3d06f3b1a64bb09c2c76dec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.325ex; height:2.843ex;" alt="{\displaystyle \psi \mapsto \psi '}"></span> along the fiber of the spin bundle.) Weinberg provides additional arguments for the physical interpretation of these as total and intrinsic angular momentum.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Other_formulations">Other formulations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=21" title="Edit section: Other formulations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Dirac equation can be formulated in a number of other ways. </p> <div class="mw-heading mw-heading3"><h3 id="Curved_spacetime">Curved spacetime</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=22" title="Edit section: Curved spacetime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This article has developed the Dirac equation in flat spacetime according to special relativity. It is possible to formulate the <a href="/wiki/Dirac_equation_in_curved_spacetime" title="Dirac equation in curved spacetime">Dirac equation in curved spacetime</a>. </p> <div class="mw-heading mw-heading3"><h3 id="The_algebra_of_physical_space">The algebra of physical space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=23" title="Edit section: The algebra of physical space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This article developed the Dirac equation using four-vectors and Schrödinger operators. The <a href="/wiki/Dirac_equation_in_the_algebra_of_physical_space" class="mw-redirect" title="Dirac equation in the algebra of physical space">Dirac equation in the algebra of physical space</a> uses a <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> over the real numbers, a type of geometric algebra. </p> <div class="mw-heading mw-heading3"><h3 id="Coupled_Weyl_Spinors">Coupled Weyl Spinors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=24" title="Edit section: Coupled Weyl Spinors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As mentioned <a class="mw-selflink-fragment" href="#Axial_symmetry">above</a>, the <i>massless</i> Dirac equation immediately reduces to the homogeneous <a href="/wiki/Weyl_equation" title="Weyl equation">Weyl equation</a>. By using the <a href="/wiki/Gamma_matrices#Weyl_(chiral)_basis" title="Gamma matrices">chiral representation of the gamma matrices</a>, the nonzero-mass equation can also be decomposed into a pair of <i>coupled</i> inhomogeneous Weyl equations acting on the first and last pairs of indices of the original four-component spinor, i.e. <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 _{L}\\\psi _{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"> <mi>L</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ={\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b291e6cb29a961cc71c3619e1604eaf0d848ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.777ex; height:6.176ex;" alt="{\displaystyle \psi ={\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}}"></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 \psi _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577ea88983dfe6dd88e20ff7402dce2c7fd981a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.865ex; height:2.509ex;" alt="{\displaystyle \psi _{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 _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacf8c5f2587de7c59bbfd798e11bdbeaa01707b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle \psi _{R}}"></span> are each two-component <a href="/wiki/Weyl_spinor" class="mw-redirect" title="Weyl spinor">Weyl spinors</a>. This is because the skew block form of the chiral gamma matrices means that they swap 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 \psi _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577ea88983dfe6dd88e20ff7402dce2c7fd981a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.865ex; height:2.509ex;" alt="{\displaystyle \psi _{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 _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacf8c5f2587de7c59bbfd798e11bdbeaa01707b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle \psi _{R}}"></span> and apply the two-by-two Pauli matrices to each: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{\mu }{\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}={\begin{pmatrix}\sigma ^{\mu }\psi _{R}\\{\overline {\sigma }}^{\mu }\psi _{L}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{\mu }{\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}={\begin{pmatrix}\sigma ^{\mu }\psi _{R}\\{\overline {\sigma }}^{\mu }\psi _{L}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9fa0456cacd6574be141fca221982cb450c860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.486ex; height:6.176ex;" alt="{\displaystyle \gamma ^{\mu }{\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}={\begin{pmatrix}\sigma ^{\mu }\psi _{R}\\{\overline {\sigma }}^{\mu }\psi _{L}\end{pmatrix}}}"></span>. </p><p>So the Dirac equation </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m){\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</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"> <mi>L</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m){\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c871ccb3f7fbaa7461220d9a7223d423314e27da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.88ex; height:6.176ex;" alt="{\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m){\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}=0}"></span> </p><p>becomes </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\begin{pmatrix}\sigma ^{\mu }\partial _{\mu }\psi _{R}\\{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{L}\end{pmatrix}}=m{\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i{\begin{pmatrix}\sigma ^{\mu }\partial _{\mu }\psi _{R}\\{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{L}\end{pmatrix}}=m{\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d31dfcbe2165e8491972b90aae48315168f4a900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.284ex; height:6.509ex;" alt="{\displaystyle i{\begin{pmatrix}\sigma ^{\mu }\partial _{\mu }\psi _{R}\\{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{L}\end{pmatrix}}=m{\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}}"></span> </p><p>which in turn is equivalent to a pair of inhomogeneous Weyl equations for massless left- and right-<a href="/wiki/Helicity_(particle_physics)" title="Helicity (particle physics)">helicity</a> spinors, where the coupling strength is proportional to the mass: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{R}=m\psi _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{R}=m\psi _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fbf0fec605dfa9a62c63a5b585a17ac6f524f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.81ex; height:3.009ex;" alt="{\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{R}=m\psi _{L}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=m\psi _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=m\psi _{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddbe4690b7c358f1b354244d6d2f07fcadb79586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.926ex; height:3.176ex;" alt="{\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=m\psi _{R}}"></span>.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="In the Penrose source the RHS is divided by \sqrt{2} and there is no imaginary unit on the LHS, but he does not go into the derivation. Other sources -- and the Axial symmetry section above -- seem to agree with the form given here. (June 2023)">clarification needed</span></a></i>]</sup> </p><p>This has been proposed as an intuitive explanation of <a href="/wiki/Zitterbewegung" title="Zitterbewegung">Zitterbewegung</a>, as these massless components would propagate at the speed of light and move in opposite directions, since the helicity is the projection of the spin onto the direction of motion.<sup id="cite_ref-PenroseZigzag_19-0" class="reference"><a href="#cite_note-PenroseZigzag-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Here the role of the "mass" <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> </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/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is not to make the velocity less than the speed of light, but instead controls the average rate at which these reversals occur; specifically, the reversals can be modeled as a <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a>.<sup id="cite_ref-PRL_1984_07_30_20-0" class="reference"><a href="#cite_note-PRL_1984_07_30-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="U(1)_symmetry"><span id="U.281.29_symmetry"></span>U(1) symmetry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=25" title="Edit section: U(1) symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Natural units are used in this section. The coupling constant is labelled by convention 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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>: this parameter can also be viewed as modelling the electron charge. </p> <div class="mw-heading mw-heading3"><h3 id="Vector_symmetry">Vector symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=26" title="Edit section: Vector symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Dirac equation and action admits 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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> symmetry where the 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 \psi ,{\bar {\psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ,{\bar {\psi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35de2a319364dc3f6a5802d134230effb394f8b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.143ex; height:2.843ex;" alt="{\displaystyle \psi ,{\bar {\psi }}}"></span> transform as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi (x)&\mapsto e^{i\alpha }\psi (x),\\{\bar {\psi }}(x)&\mapsto e^{-i\alpha }{\bar {\psi }}(x).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi (x)&\mapsto e^{i\alpha }\psi (x),\\{\bar {\psi }}(x)&\mapsto e^{-i\alpha }{\bar {\psi }}(x).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c67b39358369d132b9d3e219719a265abfe2d177" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.695ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\psi (x)&\mapsto e^{i\alpha }\psi (x),\\{\bar {\psi }}(x)&\mapsto e^{-i\alpha }{\bar {\psi }}(x).\end{aligned}}}"></span> This is a global symmetry, known as 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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> <b>vector</b> symmetry (as opposed to 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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> <b>axial</b> symmetry: see below). By <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> there is a corresponding conserved current: this has been mentioned previously as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{\mu }(x)={\bar {\psi }}(x)\gamma ^{\mu }\psi (x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{\mu }(x)={\bar {\psi }}(x)\gamma ^{\mu }\psi (x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76b1b2a4ddc63a82a0cce9221723ac6aa1762910" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.524ex; height:3.009ex;" alt="{\displaystyle J^{\mu }(x)={\bar {\psi }}(x)\gamma ^{\mu }\psi (x).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Gauging_the_symmetry">Gauging the symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=27" title="Edit section: Gauging the symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></div> <p>If we 'promote' the global symmetry, parametrised by the constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, to a local symmetry, parametrised by a function <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 \alpha :\mathbb {R} ^{1,3}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :\mathbb {R} ^{1,3}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b728b20518ede0c488d78b1605202c63c36869" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.729ex; height:2.676ex;" alt="{\displaystyle \alpha :\mathbb {R} ^{1,3}\to \mathbb {R} }"></span>, or equivalently <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\alpha }:\mathbb {R} ^{1,3}\to {\text{U}}(1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\alpha }:\mathbb {R} ^{1,3}\to {\text{U}}(1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d01c1713b9327c11e9067e5076fdffef20375f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.86ex; height:3.176ex;" alt="{\displaystyle e^{i\alpha }:\mathbb {R} ^{1,3}\to {\text{U}}(1),}"></span> the Dirac equation is no longer invariant: there is a residual derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43941ef40b4eb751a8daea118d0012ea998f5610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.627ex; height:2.843ex;" alt="{\displaystyle \alpha (x)}"></span>. </p><p>The fix proceeds as in <a href="/wiki/Scalar_electrodynamics" title="Scalar electrodynamics">scalar electrodynamics</a>: the partial derivative is promoted to a covariant derivative <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/267983ed233945e70f42935df28e623cfce15c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.148ex; height:2.843ex;" alt="{\displaystyle D_{\mu }}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +ieA_{\mu }\psi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>+</mo> <mi>i</mi> <mi>e</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +ieA_{\mu }\psi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9d83306aba55cd6fce0194e89352c46210ce91" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.583ex; height:2.843ex;" alt="{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +ieA_{\mu }\psi ,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ieA_{\mu }{\bar {\psi }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>i</mi> <mi>e</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ieA_{\mu }{\bar {\psi }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cc41cae7722be7263f30b4491f696aed878db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.831ex; height:3.176ex;" alt="{\displaystyle D_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ieA_{\mu }{\bar {\psi }}.}"></span> The covariant derivative depends on the field being acted on. The newly introduced <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 the 4-vector potential from electrodynamics, but also can be viewed as 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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> <a href="/wiki/Gauge_field" class="mw-redirect" title="Gauge field">gauge field</a> (which, mathematically, is defined as 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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> <a href="/wiki/Principal_connection" class="mw-redirect" title="Principal connection">connection</a>). </p><p>The transformation law under gauge transformations for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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 then the usual <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }(x)\mapsto A_{\mu }(x)+{\frac {1}{e}}\partial _{\mu }\alpha (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>e</mi> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }(x)\mapsto A_{\mu }(x)+{\frac {1}{e}}\partial _{\mu }\alpha (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/431d0e43965aaea6c69303cd88c6bbbc5051dd53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.748ex; height:5.176ex;" alt="{\displaystyle A_{\mu }(x)\mapsto A_{\mu }(x)+{\frac {1}{e}}\partial _{\mu }\alpha (x)}"></span> but can also be derived by asking that covariant derivatives transform under a gauge transformation as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }\psi (x)\mapsto e^{i\alpha (x)}D_{\mu }\psi (x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }\psi (x)\mapsto e^{i\alpha (x)}D_{\mu }\psi (x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5ad9b0987256e70699f11b9b2f1cf45ec71c2d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.015ex; height:3.509ex;" alt="{\displaystyle D_{\mu }\psi (x)\mapsto e^{i\alpha (x)}D_{\mu }\psi (x),}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }{\bar {\psi }}(x)\mapsto e^{-i\alpha (x)}D_{\mu }{\bar {\psi }}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }{\bar {\psi }}(x)\mapsto e^{-i\alpha (x)}D_{\mu }{\bar {\psi }}(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1175112f75abcfcc926e12850b1752bdd507c57f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.459ex; height:3.509ex;" alt="{\displaystyle D_{\mu }{\bar {\psi }}(x)\mapsto e^{-i\alpha (x)}D_{\mu }{\bar {\psi }}(x).}"></span> We then obtain a gauge-invariant Dirac action by promoting the partial derivative to a covariant one: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\int d^{4}x\,{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi =\int d^{4}x\,{\bar {\psi }}\,(i\gamma ^{\mu }D_{\mu }-m)\,\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>i</mi> <mi>D</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <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> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int d^{4}x\,{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi =\int d^{4}x\,{\bar {\psi }}\,(i\gamma ^{\mu }D_{\mu }-m)\,\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b47191d8910a3d1ec1ab3019ce918600bb10a908" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.604ex; height:5.676ex;" alt="{\displaystyle S=\int d^{4}x\,{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi =\int d^{4}x\,{\bar {\psi }}\,(i\gamma ^{\mu }D_{\mu }-m)\,\psi .}"></span> The final step needed to write down a gauge-invariant Lagrangian is to add a Maxwell Lagrangian term, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\text{Maxwell}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Maxwell</mtext> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\text{Maxwell}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97665309b41d549b1aba23d483d599f8031af5b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.535ex; height:6.176ex;" alt="{\displaystyle S_{\text{Maxwell}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }\right].}"></span> Putting these together gives </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>QED Action</b> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi \right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>QED</mtext> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>i</mi> <mi>D</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <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> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>ψ</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi \right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a025c51e08f1173b33252a5ccc3e5357151d4c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.062ex; height:6.176ex;" alt="{\displaystyle S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi \right]}"></span> </p> </div> <p>Expanding out the covariant derivative allows the action to be written in a second useful form: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi -eJ^{\mu }A_{\mu }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>QED</mtext> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>i</mi> <mi mathvariant="normal">∂</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <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> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>−</mo> <mi>e</mi> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi -eJ^{\mu }A_{\mu }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/384edf181c86f4fafcb6a2e4115d8be55f4c404e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.482ex; height:6.176ex;" alt="{\displaystyle S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi -eJ^{\mu }A_{\mu }\right]}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Axial_symmetry">Axial symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=28" title="Edit section: Axial symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Massless</b> Dirac fermions, that is, 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 \psi (x)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"></span> satisfying the Dirac equation 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 m=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57f21007575fd03e3be0da20af34d25829cc9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}"></span>, admit a second, inequivalent <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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> symmetry. </p><p>This is seen most easily by writing the four-component Dirac fermion <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"></span> as a pair of two-component vector fields, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)={\begin{pmatrix}\psi _{1}(x)\\\psi _{2}(x)\end{pmatrix}},}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)={\begin{pmatrix}\psi _{1}(x)\\\psi _{2}(x)\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f40b8a17530679b2f9478d78c0c6472de65efbe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.276ex; height:6.176ex;" alt="{\displaystyle \psi (x)={\begin{pmatrix}\psi _{1}(x)\\\psi _{2}(x)\end{pmatrix}},}"></span> and adopting the <a href="/wiki/Gamma_matrices" title="Gamma matrices">chiral representation</a> for the gamma matrices, 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 i\gamma ^{\mu }\partial _{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\gamma ^{\mu }\partial _{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f332f8c7bddf80d44c5f5220f879ae3497aa65c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.764ex; height:3.009ex;" alt="{\displaystyle i\gamma ^{\mu }\partial _{\mu }}"></span> may be written <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\gamma ^{\mu }\partial _{\mu }={\begin{pmatrix}0&i\sigma ^{\mu }\partial _{\mu }\\i{\bar {\sigma }}^{\mu }\partial _{\mu }\ &0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\gamma ^{\mu }\partial _{\mu }={\begin{pmatrix}0&i\sigma ^{\mu }\partial _{\mu }\\i{\bar {\sigma }}^{\mu }\partial _{\mu }\ &0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8637b048fbb584161c831a8d0b2d45e24a4330fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.566ex; height:6.509ex;" alt="{\displaystyle i\gamma ^{\mu }\partial _{\mu }={\begin{pmatrix}0&i\sigma ^{\mu }\partial _{\mu }\\i{\bar {\sigma }}^{\mu }\partial _{\mu }\ &0\end{pmatrix}}}"></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 \sigma ^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e3f88bdd05d08345945334cf491d972c2132493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.554ex; height:2.343ex;" alt="{\displaystyle \sigma ^{\mu }}"></span> has components <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_{2},\sigma ^{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (I_{2},\sigma ^{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47b285a0b97e645ae64e69402acfb428e044cf24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.051ex; height:3.176ex;" alt="{\displaystyle (I_{2},\sigma ^{i})}"></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 {\bar {\sigma }}^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\sigma }}^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b25596cca3a356fada0453caf72e45a51c1b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.553ex; height:2.343ex;" alt="{\displaystyle {\bar {\sigma }}^{\mu }}"></span> has components <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_{2},-\sigma ^{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (I_{2},-\sigma ^{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1a998f721ea4f9c8b85a4c2fffa1ee749b08b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.859ex; height:3.176ex;" alt="{\displaystyle (I_{2},-\sigma ^{i})}"></span>. </p><p>The Dirac action then takes the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\int d^{4}x\,\psi _{1}^{\dagger }(i\sigma ^{\mu }\partial _{\mu })\psi _{1}+\psi _{2}^{\dagger }(i{\bar {\sigma }}^{\mu }\partial _{\mu })\psi _{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>i</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int d^{4}x\,\psi _{1}^{\dagger }(i\sigma ^{\mu }\partial _{\mu })\psi _{1}+\psi _{2}^{\dagger }(i{\bar {\sigma }}^{\mu }\partial _{\mu })\psi _{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427fdb0111a2a5ad80c80014660aab414c833803" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.171ex; height:5.676ex;" alt="{\displaystyle S=\int d^{4}x\,\psi _{1}^{\dagger }(i\sigma ^{\mu }\partial _{\mu })\psi _{1}+\psi _{2}^{\dagger }(i{\bar {\sigma }}^{\mu }\partial _{\mu })\psi _{2}.}"></span> That is, it decouples into a theory of two <a href="/wiki/Weyl_equation" title="Weyl equation">Weyl spinors</a> or Weyl fermions. </p><p>The earlier vector symmetry is still present, 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 \psi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfdde1da54e02a016fe2a230c58b25dfcc014d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{1}}"></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 _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5083a526766f85c4f39ab695791b0b739f06897" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{2}}"></span> rotate identically. This form of the action makes the second inequivalent <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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> symmetry manifest: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi _{1}(x)&\mapsto e^{i\beta }\psi _{1}(x),\\\psi _{2}(x)&\mapsto e^{-i\beta }\psi _{2}(x).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi _{1}(x)&\mapsto e^{i\beta }\psi _{1}(x),\\\psi _{2}(x)&\mapsto e^{-i\beta }\psi _{2}(x).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac69e529d4a5118d297c03ec8249e6746a268d5b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.529ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\psi _{1}(x)&\mapsto e^{i\beta }\psi _{1}(x),\\\psi _{2}(x)&\mapsto e^{-i\beta }\psi _{2}(x).\end{aligned}}}"></span> This can also be expressed at the level of the Dirac fermion as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)\mapsto \exp(i\beta \gamma ^{5})\psi (x)}"> <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 stretchy="false">↦</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>β</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)\mapsto \exp(i\beta \gamma ^{5})\psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9881423825e2d5c9980c5cf72a9a51f1a1bd25" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.749ex; height:3.176ex;" alt="{\displaystyle \psi (x)\mapsto \exp(i\beta \gamma ^{5})\psi (x)}"></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 \exp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1185b570f67b4221307626254f64f9e619e769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.009ex;" alt="{\displaystyle \exp }"></span> is the exponential map for matrices. </p><p>This isn't the only <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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> symmetry possible, but it is conventional. Any 'linear combination' of the vector and axial symmetries is also 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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> symmetry. </p><p>Classically, the axial symmetry admits a well-formulated gauge theory. But at the quantum level, there is an <a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">anomaly</a>, that is, an obstruction to gauging. </p> <div class="mw-heading mw-heading3"><h3 id="Extension_to_color_symmetry">Extension to color symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=29" title="Edit section: Extension to color symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a></div> <p>We can extend this discussion from an abelian <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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> symmetry to a general non-abelian symmetry under a <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge 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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, the group of <a href="/wiki/Color_charge" title="Color charge">color symmetries</a> for a theory. </p><p>For concreteness, we fix <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 G={\text{SU}}(N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>SU</mtext> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G={\text{SU}}(N)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e0013caf8d63a1716475799928774c2b8f2e8cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.834ex; height:2.843ex;" alt="{\displaystyle G={\text{SU}}(N)}"></span>, the <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a> of matrices acting on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a093c818bd6ea93f8a94d0aed556d907c4ba46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.37ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{N}}"></span>. </p><p>Before this section, <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"></span> could be viewed as a spinor field on Minkowski space, in other words a function <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 :\mathbb {R} ^{1,3}\mapsto \mathbb {C} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">↦</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi :\mathbb {R} ^{1,3}\mapsto \mathbb {C} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d8a53594ee60b9679484c9b88fbdc8e21e02c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.808ex; height:3.009ex;" alt="{\displaystyle \psi :\mathbb {R} ^{1,3}\mapsto \mathbb {C} ^{4}}"></span>, and its components in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/990797533f68706456ef046fb4e390c6d064f8a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{4}}"></span> are labelled by spin indices, conventionally Greek indices taken from the start of the alphabet <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 \alpha ,\beta ,\gamma ,\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma ,\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47c39d15ec4d15fddac17715e6d1b118a539b399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.907ex; height:2.676ex;" alt="{\displaystyle \alpha ,\beta ,\gamma ,\cdots }"></span>. </p><p>Promoting the theory to a gauge theory, informally <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"> <mi>ψ</mi> </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/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> acquires a part transforming like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a093c818bd6ea93f8a94d0aed556d907c4ba46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.37ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{N}}"></span>, and these are labelled by color indices, conventionally Latin indices <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,j,k,\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j,k,\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1befdb28ed15d91de6ddb9b5303fe6e5683d61f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.797ex; height:2.509ex;" alt="{\displaystyle i,j,k,\cdots }"></span>. In total, <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"></span> has <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 4N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc080b972918855f349631dd1d838c538162366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\displaystyle 4N}"></span> components, given in indices 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 \psi ^{i,\alpha }(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>α</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{i,\alpha }(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47aa8c6ec88565764006096bacf8910c3cd586da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.961ex; height:3.176ex;" alt="{\displaystyle \psi ^{i,\alpha }(x)}"></span>. The 'spinor' labels only how the field transforms under spacetime transformations. </p><p>Formally, <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"></span> is valued in a tensor product, that is, it is a function <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 :\mathbb {R} ^{1,3}\to \mathbb {C} ^{4}\otimes \mathbb {C} ^{N}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⊗</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi :\mathbb {R} ^{1,3}\to \mathbb {C} ^{4}\otimes \mathbb {C} ^{N}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b30b2abf34553558590b1d89d06d90a57fc4925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.665ex; height:3.009ex;" alt="{\displaystyle \psi :\mathbb {R} ^{1,3}\to \mathbb {C} ^{4}\otimes \mathbb {C} ^{N}.}"></span> </p><p>Gauging proceeds similarly to the abelian <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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> case, with a few differences. Under a gauge transformation <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:\mathbb {R} ^{1,3}\rightarrow {\text{SU}}(N),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>SU</mtext> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U:\mathbb {R} ^{1,3}\rightarrow {\text{SU}}(N),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9ffe9941b895ff3f70af568a0614106d45ac8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.901ex; height:3.176ex;" alt="{\displaystyle U:\mathbb {R} ^{1,3}\rightarrow {\text{SU}}(N),}"></span> the spinor fields transform as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)\mapsto U(x)\psi (x)}"> <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 stretchy="false">↦</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)\mapsto U(x)\psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55c68a1dbbabe7cc4855bd5f42a1719ab19e257b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.84ex; height:2.843ex;" alt="{\displaystyle \psi (x)\mapsto U(x)\psi (x)}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\psi }}(x)\mapsto {\bar {\psi }}(x)U^{\dagger }(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}(x)\mapsto {\bar {\psi }}(x)U^{\dagger }(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a60098a25029643e08e5417007535ae0c549a432" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.673ex; height:3.176ex;" alt="{\displaystyle {\bar {\psi }}(x)\mapsto {\bar {\psi }}(x)U^{\dagger }(x).}"></span> The matrix-valued gauge field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }}"> <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> or <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 {\text{SU}}(N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SU</mtext> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(N)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40b4f913e99b50055f5d7e7c4a8c9b9e05623cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.909ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(N)}"></span> connection transforms as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }(x)\mapsto U(x)A_{\mu }(x)U(x)^{-1}+{\frac {1}{g}}(\partial _{\mu }U(x))U(x)^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>g</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }(x)\mapsto U(x)A_{\mu }(x)U(x)^{-1}+{\frac {1}{g}}(\partial _{\mu }U(x))U(x)^{-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3180f124063429bad52144d0c4d2f9789bf8803e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.93ex; height:5.676ex;" alt="{\displaystyle A_{\mu }(x)\mapsto U(x)A_{\mu }(x)U(x)^{-1}+{\frac {1}{g}}(\partial _{\mu }U(x))U(x)^{-1},}"></span> and the covariant derivatives defined <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +igA_{\mu }\psi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>+</mo> <mi>i</mi> <mi>g</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +igA_{\mu }\psi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb18e9db6699bbe7fb3249eac5c9930beff5ea4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.616ex; height:2.843ex;" alt="{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +igA_{\mu }\psi ,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ig{\bar {\psi }}A_{\mu }^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>i</mi> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ig{\bar {\psi }}A_{\mu }^{\dagger }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ccaff66b84ceca5a9a5aff9142f60e7eebc53ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.216ex; height:3.509ex;" alt="{\displaystyle D_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ig{\bar {\psi }}A_{\mu }^{\dagger }}"></span> transform as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }\psi (x)\mapsto U(x)D_{\mu }\psi (x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }\psi (x)\mapsto U(x)D_{\mu }\psi (x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4440edfa9d5807820447d171d8eaa6fa5dccd8eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.782ex; height:3.009ex;" alt="{\displaystyle D_{\mu }\psi (x)\mapsto U(x)D_{\mu }\psi (x),}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }{\bar {\psi }}(x)\mapsto (D_{\mu }{\bar {\psi }}(x))U(x)^{\dagger }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }{\bar {\psi }}(x)\mapsto (D_{\mu }{\bar {\psi }}(x))U(x)^{\dagger }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37fe0f81726501dc34203832d20711624ce0e6c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.719ex; height:3.343ex;" alt="{\displaystyle D_{\mu }{\bar {\psi }}(x)\mapsto (D_{\mu }{\bar {\psi }}(x))U(x)^{\dagger }.}"></span> </p><p>Writing down a gauge-invariant action proceeds exactly as 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 {\text{U}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}"></span> case, replacing the Maxwell Lagrangian with the <a href="/wiki/Yang%E2%80%93Mills" class="mw-redirect" title="Yang–Mills">Yang–Mills</a> Lagrangian <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\text{Y-M}}=\int d^{4}x\,-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Y-M</mtext> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tr</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\text{Y-M}}=\int d^{4}x\,-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62c1638a4f48db7494393fba54a524598360ce3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.349ex; height:5.676ex;" alt="{\displaystyle S_{\text{Y-M}}=\int d^{4}x\,-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })}"></span> where the Yang–Mills field strength or curvature is defined here as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }-ig\left[A_{\mu },A_{\nu }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <mi>g</mi> <mrow> <mo>[</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }-ig\left[A_{\mu },A_{\nu }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3adb7fe181bfb570ee03cade284fb340313060b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.424ex; height:3.009ex;" alt="{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }-ig\left[A_{\mu },A_{\nu }\right]}"></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 [\cdot ,\cdot ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\cdot ,\cdot ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28dd4c22d60192519c1c12cf645b040f368db9e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.621ex; height:2.843ex;" alt="{\displaystyle [\cdot ,\cdot ]}"></span> is the matrix commutator. </p><p>The action is then </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>QCD Action</b> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\text{QCD}}=\int d^{4}x\,\left[-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })+{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi \right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>QCD</mtext> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tr</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>i</mi> <mi>D</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <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> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>ψ</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\text{QCD}}=\int d^{4}x\,\left[-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })+{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi \right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee9e110bbf1417a98a76fb27ae401ec25769529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.528ex; height:6.176ex;" alt="{\displaystyle S_{\text{QCD}}=\int d^{4}x\,\left[-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })+{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi \right]}"></span> </p> </div> <div class="mw-heading mw-heading4"><h4 id="Physical_applications">Physical applications</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=30" title="Edit section: Physical applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For physical applications, the case <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=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ab951d779fcdeea3ec188bb5c73518c46b19de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=3}"></span> describes the <a href="/wiki/Quark" title="Quark">quark</a> sector of the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> which models <a href="/wiki/Strong_interactions" class="mw-redirect" title="Strong interactions">strong interactions</a>. Quarks are modelled as Dirac spinors; the gauge field is the <a href="/wiki/Gluon" title="Gluon">gluon</a> field. The case <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=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/405d64b14536deffc3465f1e81b1b7fe9358ad2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=2}"></span> describes part of the <a href="/wiki/Electroweak" class="mw-redirect" title="Electroweak">electroweak</a> sector of the Standard Model. Leptons such as electrons and neutrinos are the Dirac spinors; the gauge field is 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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> gauge boson. </p> <div class="mw-heading mw-heading4"><h4 id="Generalisations">Generalisations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=31" title="Edit section: Generalisations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This expression can be generalised to arbitrary Lie group <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> with connection <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> and a <a href="/wiki/Group_representation" title="Group representation">representation</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 (\rho ,G,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <mi>G</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\rho ,G,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77254725424c2b3b6c4aa7dcf10d2e8e6806a7d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.693ex; height:2.843ex;" alt="{\displaystyle (\rho ,G,V)}"></span>, where the colour part 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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </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/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> is valued in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </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/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>. Formally, the Dirac field is a function <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 :\mathbb {R} ^{1,3}\to \mathbb {C} ^{4}\otimes V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⊗</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi :\mathbb {R} ^{1,3}\to \mathbb {C} ^{4}\otimes V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/322826e026fff506994bd9aaa209cd774fbe41d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.083ex; height:3.009ex;" alt="{\displaystyle \psi :\mathbb {R} ^{1,3}\to \mathbb {C} ^{4}\otimes V.}"></span> </p><p>Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </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/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> transforms under a gauge transformation <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 g:\mathbb {R} ^{1,3}\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} ^{1,3}\to G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b081eedfce3d3fa6bb154cb965d36c96cae96b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.506ex; height:3.009ex;" alt="{\displaystyle g:\mathbb {R} ^{1,3}\to G}"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)\mapsto \rho (g(x))\psi (x)}"> <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 stretchy="false">↦</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)\mapsto \rho (g(x))\psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3940e596dfb0a769ee6001efd8e5842096ac8133" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.185ex; height:2.843ex;" alt="{\displaystyle \psi (x)\mapsto \rho (g(x))\psi (x)}"></span> and the covariant derivative is defined <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +\rho (A_{\mu })\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>+</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +\rho (A_{\mu })\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79e66d4b5c6c2ae4f4ebd96b2c5664b3d380fdf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.062ex; height:3.009ex;" alt="{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +\rho (A_{\mu })\psi }"></span> where here we view <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> as a <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> representation of the Lie 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 {\mathfrak {g}}={\text{L}}(G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}={\text{L}}(G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64bc89999b2bd542faac4d103e2b839dcfc1fe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.359ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}={\text{L}}(G)}"></span> associated to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. </p><p>This theory can be generalised to curved spacetime, but there are subtleties which arise in gauge theory on a general spacetime (or more generally still, a manifold) which, on flat spacetime, can be ignored. This is ultimately due to the <a href="/wiki/Contractible" class="mw-redirect" title="Contractible">contractibility</a> of flat spacetime which allows us to view a gauge field and gauge transformations as defined globally on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{1,3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1,3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bbc96ea97da9adfb87ac79a78c8e604c97cd7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.012ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1,3}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=32" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1216972533">.mw-parser-output .col-begin{border-collapse:collapse;padding:0;color:inherit;width:100%;border:0;margin:0}.mw-parser-output .col-begin-small{font-size:90%}.mw-parser-output .col-break{vertical-align:top;text-align:left}.mw-parser-output .col-break-2{width:50%}.mw-parser-output .col-break-3{width:33.3%}.mw-parser-output .col-break-4{width:25%}.mw-parser-output .col-break-5{width:20%}@media(max-width:720px){.mw-parser-output .col-begin,.mw-parser-output .col-begin>tbody,.mw-parser-output .col-begin>tbody>tr,.mw-parser-output .col-begin>tbody>tr>td{display:block!important;width:100%!important}.mw-parser-output .col-break{padding-left:0!important}}</style><div> <table class="col-begin" role="presentation"> <tbody><tr> <td class="col-break"> <div class="mw-heading mw-heading3"><h3 id="Articles_on_the_Dirac_equation">Articles on the Dirac equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=33" title="Edit section: Articles on the Dirac equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dirac_field" class="mw-redirect" title="Dirac field">Dirac field</a></li> <li><a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinor</a></li> <li><a href="/wiki/Gordon_decomposition" title="Gordon decomposition">Gordon decomposition</a></li> <li><a href="/wiki/Klein_paradox" title="Klein paradox">Klein paradox</a></li> <li><a href="/wiki/Nonlinear_Dirac_equation" title="Nonlinear Dirac equation">Nonlinear Dirac equation</a></li></ul> </td> <td class="col-break"> <div class="mw-heading mw-heading3"><h3 id="Other_equations">Other equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=34" title="Edit section: Other equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Breit_equation" title="Breit equation">Breit equation</a></li> <li><a href="/wiki/Dirac%E2%80%93K%C3%A4hler_equation" title="Dirac–Kähler equation">Dirac–Kähler equation</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger equation</a></li> <li><a href="/wiki/Two-body_Dirac_equations" title="Two-body Dirac equations">Two-body Dirac equations</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl equation</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana equation</a></li></ul> </td> <td class="col-break"> <div class="mw-heading mw-heading3"><h3 id="Other_topics">Other topics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=35" title="Edit section: Other topics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fermionic_field" title="Fermionic field">Fermionic field</a></li> <li><a href="/wiki/Feynman_checkerboard" title="Feynman checkerboard">Feynman checkerboard</a></li> <li><a href="/wiki/Foldy%E2%80%93Wouthuysen_transformation" title="Foldy–Wouthuysen transformation">Foldy–Wouthuysen transformation</a></li> <li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li>ELKO theory</li></ul> <p>  </p> </td></tr></tbody></table></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=Dirac_equation&action=edit&section=36" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=37" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFP.W._Atkins1974" class="citation book cs1">P.W. Atkins (1974). <i>Quanta: A handbook of concepts</i>. Oxford University Press. p. 52. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-855493-6" title="Special:BookSources/978-0-19-855493-6"><bdi>978-0-19-855493-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quanta%3A+A+handbook+of+concepts&rft.pages=52&rft.pub=Oxford+University+Press&rft.date=1974&rft.isbn=978-0-19-855493-6&rft.au=P.W.+Atkins&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGorbarMiranskijShovkovySukhachov2021" class="citation book cs1">Gorbar, Eduard V.; Miranskij, Vladimir A.; Shovkovy, Igor A.; Sukhachov, Pavlo O. (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=heIZEAAAQBAJ&dq=dirac+equation+most+fundamental+equation&pg=PA1"><i>Electronic Properties of Dirac and Weyl Semimetals</i></a>. <a href="/wiki/World_Scientific_Publishing" class="mw-redirect" title="World Scientific Publishing">World Scientific Publishing</a>. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-12-0736-5" title="Special:BookSources/978-981-12-0736-5"><bdi>978-981-12-0736-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electronic+Properties+of+Dirac+and+Weyl+Semimetals&rft.pages=1&rft.pub=World+Scientific+Publishing&rft.date=2021&rft.isbn=978-981-12-0736-5&rft.aulast=Gorbar&rft.aufirst=Eduard+V.&rft.au=Miranskij%2C+Vladimir+A.&rft.au=Shovkovy%2C+Igor+A.&rft.au=Sukhachov%2C+Pavlo+O.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DheIZEAAAQBAJ%26dq%3Ddirac%2Bequation%2Bmost%2Bfundamental%2Bequation%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFT.Hey,_P.Walters2009" class="citation book cs1">T.Hey, P.Walters (2009). <i>The New Quantum Universe</i>. Cambridge University Press. p. 228. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-56457-1" title="Special:BookSources/978-0-521-56457-1"><bdi>978-0-521-56457-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+New+Quantum+Universe&rft.pages=228&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0-521-56457-1&rft.au=T.Hey%2C+P.Walters&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-:2-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-:2_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZichichi2000" class="citation web cs1"><a href="/wiki/Antonino_Zichichi" title="Antonino Zichichi">Zichichi, Antonino</a> (2 March 2000). <a rel="nofollow" class="external text" href="https://physicsworld.com/a/dirac-einstein-and-physics/">"Dirac, Einstein and physics"</a>. <i>Physics World</i><span class="reference-accessdate">. Retrieved <span class="nowrap">22 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+World&rft.atitle=Dirac%2C+Einstein+and+physics&rft.date=2000-03-02&rft.aulast=Zichichi&rft.aufirst=Antonino&rft_id=https%3A%2F%2Fphysicsworld.com%2Fa%2Fdirac-einstein-and-physics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHan2014" class="citation book cs1">Han, Moo-Young (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_AC3CgAAQBAJ&dq=dirac+equation+most+important+equation&pg=PA32"><i>From Photons to Higgs: A Story of Light</i></a> (2nd ed.). <a href="/wiki/World_Scientific_Publishing" class="mw-redirect" title="World Scientific Publishing">World Scientific Publishing</a>. p. 32. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F9071">10.1142/9071</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4579-95-7" title="Special:BookSources/978-981-4579-95-7"><bdi>978-981-4579-95-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Photons+to+Higgs%3A+A+Story+of+Light&rft.pages=32&rft.edition=2nd&rft.pub=World+Scientific+Publishing&rft.date=2014&rft_id=info%3Adoi%2F10.1142%2F9071&rft.isbn=978-981-4579-95-7&rft.aulast=Han&rft.aufirst=Moo-Young&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_AC3CgAAQBAJ%26dq%3Ddirac%2Bequation%2Bmost%2Bimportant%2Bequation%26pg%3DPA32&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGisela_Dirac-Wahrenburg" class="citation web cs1">Gisela Dirac-Wahrenburg. <a rel="nofollow" class="external text" href="http://www.dirac.ch/PaulDirac.html">"Paul Dirac"</a>. Dirac.ch<span class="reference-accessdate">. Retrieved <span class="nowrap">12 July</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Paul+Dirac&rft.pub=Dirac.ch&rft.au=Gisela+Dirac-Wahrenburg&rft_id=http%3A%2F%2Fwww.dirac.ch%2FPaulDirac.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPais2002" class="citation book cs1">Pais, Abraham (2002). <i>Inward bound: of matter and forces in the physical world</i> (Reprint ed.). 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Oxford University Press. p. 255. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-852011-5" title="Special:BookSources/978-0-19-852011-5"><bdi>978-0-19-852011-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Quantum+Mechanics&rft.series=International+Series+of+Monographs+on+Physics&rft.pages=255&rft.edition=4th&rft.pub=Oxford+University+Press&rft.date=1982&rft.isbn=978-0-19-852011-5&rft.aulast=Dirac&rft.aufirst=Paul+A.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuckSudarshan1998" class="citation book cs1">Duck, Ian; Sudarshan, E C G (1998). <a rel="nofollow" class="external text" href="https://www.worldscientific.com/worldscibooks/10.1142/3457"><i>Pauli and the Spin-Statistics Theorem</i></a>. WORLD SCIENTIFIC. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F3457">10.1142/3457</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-3114-9" title="Special:BookSources/978-981-02-3114-9"><bdi>978-981-02-3114-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pauli+and+the+Spin-Statistics+Theorem&rft.pub=WORLD+SCIENTIFIC&rft.date=1998&rft_id=info%3Adoi%2F10.1142%2F3457&rft.isbn=978-981-02-3114-9&rft.aulast=Duck&rft.aufirst=Ian&rft.au=Sudarshan%2C+E+C+G&rft_id=https%3A%2F%2Fwww.worldscientific.com%2Fworldscibooks%2F10.1142%2F3457&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPendleton2012–2013" class="citation book cs1">Pendleton, Brian (2012–2013). <a rel="nofollow" class="external text" href="http://www2.ph.ed.ac.uk/~bjp/qt/rqt.pdf"><i>Quantum Theory</i></a> <span class="cs1-format">(PDF)</span>. section 4.3 "The Dirac Equation". <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www2.ph.ed.ac.uk/~bjp/qt/rqt.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 9 October 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory&rft.pages=section-4.3+%22The+Dirac+Equation%22&rft.date=2012%2F2013&rft.aulast=Pendleton&rft.aufirst=Brian&rft_id=http%3A%2F%2Fwww2.ph.ed.ac.uk%2F~bjp%2Fqt%2Frqt.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-Ohlsson2011-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ohlsson2011_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOhlsson2011" class="citation book cs1"><a href="/wiki/Tommy_Ohlsson" title="Tommy Ohlsson">Ohlsson, Tommy</a> (22 September 2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hRavtAW5EFcC&pg=PA86"><i>Relativistic Quantum Physics: From advanced quantum mechanics to introductory quantum field theory</i></a>. Cambridge University Press. p. 86. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-50432-4" title="Special:BookSources/978-1-139-50432-4"><bdi>978-1-139-50432-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativistic+Quantum+Physics%3A+From+advanced+quantum+mechanics+to+introductory+quantum+field+theory&rft.pages=86&rft.pub=Cambridge+University+Press&rft.date=2011-09-22&rft.isbn=978-1-139-50432-4&rft.aulast=Ohlsson&rft.aufirst=Tommy&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhRavtAW5EFcC%26pg%3DPA86&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPenrose2004" class="citation book cs1">Penrose, Roger (2004). <i>The Road to Reality</i>. Jonathan Cape. p. 625. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-224-04447-8" title="Special:BookSources/0-224-04447-8"><bdi>0-224-04447-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Road+to+Reality&rft.pages=625&rft.pub=Jonathan+Cape&rft.date=2004&rft.isbn=0-224-04447-8&rft.aulast=Penrose&rft.aufirst=Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCollasKlein2019" class="citation book cs1">Collas, Peter; Klein, David (2019). <i>The Dirac Equation in Curved Spacetime: A Guide for Calculations</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-14825-6" title="Special:BookSources/978-3-030-14825-6"><bdi>978-3-030-14825-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Dirac+Equation+in+Curved+Spacetime%3A+A+Guide+for+Calculations&rft.pub=Springer&rft.date=2019&rft.isbn=978-3-030-14825-6&rft.aulast=Collas&rft.aufirst=Peter&rft.au=Klein%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Jurgen Jost, (2002) "Riemannian Geometry and Geometric Analysis (3rd Edition)" Springer Universitext. <i>(See chapter 1 for spin structures and chapter 3 for connections on spin structures)</i></span> </li> <li id="cite_note-iz-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-iz_15-0">^</a></b></span> <span class="reference-text">Claude Itzykson and Jean-Bernard Zuber, (1980) "Quantum Field Theory", McGraw-Hill <i>(See Chapter 2)</i></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">James D. Bjorken, Sidney D. Drell (1964) "Relativistic Quantum Mechanics", McGraw-Hill. <i>(See Chapter 2)</i></span> </li> <li id="cite_note-weinberg-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-weinberg_17-0">^</a></b></span> <span class="reference-text">Steven Weinberg, (1972) "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity", Wiley & Sons <i>(See chapter 12.5, "Tetrad formalism" pages 367ff.)</i>.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Weinberg, "Gravitation", <i>op cit.</i> <i>(See chapter 2.9 "Spin", pages 46-47.)</i></span> </li> <li id="cite_note-PenroseZigzag-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-PenroseZigzag_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPenrose2004" class="citation book cs1">Penrose, Roger (2004). <i>The Road to Reality</i> (Sixth Printing ed.). Alfred A. Knopf. pp. <span class="nowrap">628–</span>632. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-224-04447-8" title="Special:BookSources/0-224-04447-8"><bdi>0-224-04447-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Road+to+Reality&rft.pages=%3Cspan+class%3D%22nowrap%22%3E628-%3C%2Fspan%3E632&rft.edition=Sixth+Printing&rft.pub=Alfred+A.+Knopf&rft.date=2004&rft.isbn=0-224-04447-8&rft.aulast=Penrose&rft.aufirst=Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> <li id="cite_note-PRL_1984_07_30-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-PRL_1984_07_30_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGaveauJacobsonKacSchulman1984" class="citation journal cs1">Gaveau, B.; Jacobson, T.; Kac, M.; Schulman, L. S. (30 July 1984). "Relativistic Extension of the Analogy between Quantum Mechanics and Brownian Motion". <i>Physical Review Letters</i>. <b>53</b> (5): <span class="nowrap">419–</span>422. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1984PhRvL..53..419G">1984PhRvL..53..419G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.53.419">10.1103/PhysRevLett.53.419</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Relativistic+Extension+of+the+Analogy+between+Quantum+Mechanics+and+Brownian+Motion&rft.volume=53&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E419-%3C%2Fspan%3E422&rft.date=1984-07-30&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.53.419&rft_id=info%3Abibcode%2F1984PhRvL..53..419G&rft.aulast=Gaveau&rft.aufirst=B.&rft.au=Jacobson%2C+T.&rft.au=Kac%2C+M.&rft.au=Schulman%2C+L.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Selected_papers">Selected papers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=38" title="Edit section: Selected papers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson1933" class="citation journal cs1">Anderson, Carl (1933). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.43.491">"The Positive Electron"</a>. <i>Physical Review</i>. <b>43</b> (6): 491. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1933PhRv...43..491A">1933PhRv...43..491A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.43.491">10.1103/PhysRev.43.491</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=The+Positive+Electron&rft.volume=43&rft.issue=6&rft.pages=491&rft.date=1933&rft_id=info%3Adoi%2F10.1103%2FPhysRev.43.491&rft_id=info%3Abibcode%2F1933PhRv...43..491A&rft.aulast=Anderson&rft.aufirst=Carl&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRev.43.491&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArminjonF._Reifler2013" class="citation journal cs1">Arminjon, M.; F. Reifler (2013). "Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations". <i>Brazilian Journal of Physics</i>. <b>43</b> (<span class="nowrap">1–</span>2): <span class="nowrap">64–</span>77. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1103.3201">1103.3201</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013BrJPh..43...64A">2013BrJPh..43...64A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs13538-012-0111-0">10.1007/s13538-012-0111-0</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:38235437">38235437</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Brazilian+Journal+of+Physics&rft.atitle=Equivalent+forms+of+Dirac+equations+in+curved+spacetimes+and+generalized+de+Broglie+relations&rft.volume=43&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E64-%3C%2Fspan%3E77&rft.date=2013&rft_id=info%3Aarxiv%2F1103.3201&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A38235437%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs13538-012-0111-0&rft_id=info%3Abibcode%2F2013BrJPh..43...64A&rft.aulast=Arminjon&rft.aufirst=M.&rft.au=F.+Reifler&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDirac1928" class="citation journal cs1">Dirac, P. A. M. (1928). <a rel="nofollow" class="external text" href="http://www.math.ucsd.edu/~nwallach/Dirac1928.pdf">"The Quantum Theory of the Electron"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>. <b>117</b> (778): <span class="nowrap">610–</span>624. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1928RSPSA.117..610D">1928RSPSA.117..610D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1928.0023">10.1098/rspa.1928.0023</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/94981">94981</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150102071809/http://www.math.ucsd.edu/~nwallach/Dirac1928.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2 January 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A%3A+Mathematical%2C+Physical+and+Engineering+Sciences&rft.atitle=The+Quantum+Theory+of+the+Electron&rft.volume=117&rft.issue=778&rft.pages=%3Cspan+class%3D%22nowrap%22%3E610-%3C%2Fspan%3E624&rft.date=1928&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F94981%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1098%2Frspa.1928.0023&rft_id=info%3Abibcode%2F1928RSPSA.117..610D&rft.aulast=Dirac&rft.aufirst=P.+A.+M.&rft_id=http%3A%2F%2Fwww.math.ucsd.edu%2F~nwallach%2FDirac1928.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDirac1930" class="citation journal cs1">Dirac, P. A. M. (1930). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k56219d/f388.table">"A Theory of Electrons and Protons"</a>. <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>. <b>126</b> (801): <span class="nowrap">360–</span>365. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1930RSPSA.126..360D">1930RSPSA.126..360D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1930.0013">10.1098/rspa.1930.0013</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/95359">95359</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A%3A+Mathematical%2C+Physical+and+Engineering+Sciences&rft.atitle=A+Theory+of+Electrons+and+Protons&rft.volume=126&rft.issue=801&rft.pages=%3Cspan+class%3D%22nowrap%22%3E360-%3C%2Fspan%3E365&rft.date=1930&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F95359%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1098%2Frspa.1930.0013&rft_id=info%3Abibcode%2F1930RSPSA.126..360D&rft.aulast=Dirac&rft.aufirst=P.+A.+M.&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k56219d%2Ff388.table&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrischStern1933" class="citation journal cs1">Frisch, R.; Stern, O. (1933). "Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. I". <i>Zeitschrift für Physik</i>. <b>85</b> (<span class="nowrap">1–</span>2): 4. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1933ZPhy...85....4F">1933ZPhy...85....4F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01330773">10.1007/BF01330773</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120793548">120793548</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=%C3%9Cber+die+magnetische+Ablenkung+von+Wasserstoffmolek%C3%BClen+und+das+magnetische+Moment+des+Protons.+I&rft.volume=85&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=4&rft.date=1933&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120793548%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01330773&rft_id=info%3Abibcode%2F1933ZPhy...85....4F&rft.aulast=Frisch&rft.aufirst=R.&rft.au=Stern%2C+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Textbooks">Textbooks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=39" title="Edit section: Textbooks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBjorken,_J_DDrell,_S1964" class="citation book cs1">Bjorken, J D; Drell, S (1964). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/relativisticquan0000bjor"><i>Relativistic Quantum mechanics</i></a></span>. New York, McGraw-Hill.</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.pub=New+York%2C+McGraw-Hill&rft.date=1964&rft.au=Bjorken%2C+J+D&rft.au=Drell%2C+S&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Frelativisticquan0000bjor&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalzen,_FrancisMartin,_Alan1984" class="citation book cs1"><a href="/wiki/Francis_Halzen" title="Francis Halzen">Halzen, Francis</a>; <a href="/wiki/Alan_Martin_(physicist)" title="Alan Martin (physicist)">Martin, Alan</a> (1984). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quarksleptonsint0000halz"><i>Quarks & Leptons: An Introductory Course in Modern Particle Physics</i></a></span>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471887416" title="Special:BookSources/9780471887416"><bdi>9780471887416</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quarks+%26+Leptons%3A+An+Introductory+Course+in+Modern+Particle+Physics&rft.pub=John+Wiley+%26+Sons&rft.date=1984&rft.isbn=9780471887416&rft.au=Halzen%2C+Francis&rft.au=Martin%2C+Alan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquarksleptonsint0000halz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths,_D.J.2008" class="citation book cs1">Griffiths, D.J. (2008). <i>Introduction to Elementary Particles</i> (2nd ed.). Wiley-VCH. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-527-40601-2" title="Special:BookSources/978-3-527-40601-2"><bdi>978-3-527-40601-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Elementary+Particles&rft.edition=2nd&rft.pub=Wiley-VCH&rft.date=2008&rft.isbn=978-3-527-40601-2&rft.au=Griffiths%2C+D.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRae,_Alastair_I._M.Jim_Napolitano2015" class="citation book cs1">Rae, Alastair I. M.; Jim Napolitano (2015). <i>Quantum Mechanics</i> (6th ed.). Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1482299182" title="Special:BookSources/978-1482299182"><bdi>978-1482299182</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.edition=6th&rft.pub=Routledge&rft.date=2015&rft.isbn=978-1482299182&rft.au=Rae%2C+Alastair+I.+M.&rft.au=Jim+Napolitano&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchiff,_L.I.1968" class="citation book cs1">Schiff, L.I. (1968). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quantummechanics0000schi"><i>Quantum Mechanics</i></a></span> (3rd ed.). McGraw-Hill.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1968&rft.au=Schiff%2C+L.I.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantummechanics0000schi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShankar,_R.1994" class="citation book cs1">Shankar, R. (1994). <i>Principles of Quantum Mechanics</i> (2nd ed.). Plenum.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Quantum+Mechanics&rft.edition=2nd&rft.pub=Plenum&rft.date=1994&rft.au=Shankar%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThaller,_B.1992" class="citation book cs1">Thaller, B. (1992). <i>The Dirac Equation</i>. Texts and Monographs in Physics. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Dirac+Equation&rft.series=Texts+and+Monographs+in+Physics&rft.pub=Springer&rft.date=1992&rft.au=Thaller%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+equation" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirac_equation&action=edit&section=40" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output 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aria-labelledby="Quantum_mechanics332" style="padding:3px"><table class="nowraplinks hlist 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics332" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a class="mw-selflink selflink">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> 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Dirac equation - Wikipedia