Majorana equation - Wikipedia

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class="vector-toc-list"> <li id="toc-Purely_real_four-component_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Purely_real_four-component_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Purely real four-component form</span> </div> </a> <ul id="toc-Purely_real_four-component_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Charge-conjugate_four-component_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Charge-conjugate_four-component_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Charge-conjugate four-component form</span> </div> </a> <ul id="toc-Charge-conjugate_four-component_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_two-component_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_two-component_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Complex two-component form</span> </div> </a> <ul id="toc-Complex_two-component_form-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Key_ideas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Key_ideas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Key ideas</span> </div> </a> <ul id="toc-Key_ideas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Two-component_Majorana_equation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Two-component_Majorana_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Two-component Majorana equation</span> </div> </a> <button aria-controls="toc-Two-component_Majorana_equation-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 Two-component Majorana equation subsection</span> </button> <ul id="toc-Two-component_Majorana_equation-sublist" class="vector-toc-list"> <li id="toc-Weyl_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weyl_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Weyl equation</span> </div> </a> <ul id="toc-Weyl_equation-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">3.2</span> <span>Lorentz invariance</span> </div> </a> <ul id="toc-Lorentz_invariance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differentials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differentials"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Differentials</span> </div> </a> <ul id="toc-Differentials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mass_term" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mass_term"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Mass term</span> </div> </a> <ul id="toc-Mass_term-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Left_and_right_Majorana_operators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Left_and_right_Majorana_operators"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Left and right Majorana operators</span> </div> </a> <ul id="toc-Left_and_right_Majorana_operators-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Four-component_Majorana_equation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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<a class="vector-toc-link" href="#Parity"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Parity</span> </div> </a> <ul id="toc-Parity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Solutions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Solutions</span> </div> </a> <button aria-controls="toc-Solutions-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 Solutions subsection</span> </button> <ul id="toc-Solutions-sublist" class="vector-toc-list"> <li id="toc-Spin_eigenstates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spin_eigenstates"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Spin eigenstates</span> </div> </a> <ul id="toc-Spin_eigenstates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Momentum_eigenstates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Momentum_eigenstates"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Momentum eigenstates</span> </div> </a> <ul id="toc-Momentum_eigenstates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Electric_charge" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Electric_charge"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Electric charge</span> </div> </a> <ul id="toc-Electric_charge-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Field_quanta" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Field_quanta"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Field quanta</span> </div> </a> <button aria-controls="toc-Field_quanta-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 Field quanta subsection</span> </button> <ul id="toc-Field_quanta-sublist" class="vector-toc-list"> <li id="toc-Majorana_particle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Majorana_particle"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Majorana particle</span> </div> </a> <ul id="toc-Majorana_particle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Additional_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Additional reading</span> </div> </a> <ul id="toc-Additional_reading-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 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id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 13 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-13" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">13 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D9%85%D8%A7%D9%8A%D9%88%D8%B1%D8%A7%D9%86%D8%A7" title="معادلة مايورانا – Arabic" lang="ar" hreflang="ar" data-title="معادلة مايورانا" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equaci%C3%B3_de_Majorana" title="Equació de Majorana – Catalan" lang="ca" hreflang="ca" data-title="Equació de Majorana" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_de_Majorana" title="Ecuación de Majorana – Spanish" lang="es" hreflang="es" data-title="Ecuación de Majorana" 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/Majorana_ekvacio" title="Majorana ekvacio – Esperanto" lang="eo" hreflang="eo" data-title="Majorana ekvacio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quation_de_Majorana" title="Équation de Majorana – French" lang="fr" hreflang="fr" data-title="Équation de Majorana" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ecuaci%C3%B3n_de_Majorana" title="Ecuación de Majorana – Galician" lang="gl" hreflang="gl" data-title="Ecuación de Majorana" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A7%88%EC%9A%94%EB%9D%BC%EB%82%98_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="마요라나 방정식 – Korean" lang="ko" hreflang="ko" data-title="마요라나 방정식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_di_Majorana" title="Equazione di Majorana – Italian" lang="it" hreflang="it" data-title="Equazione di Majorana" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9E%E3%83%A8%E3%83%A9%E3%83%8A%E6%96%B9%E7%A8%8B%E5%BC%8F" title="マヨラナ方程式 – Japanese" lang="ja" hreflang="ja" data-title="マヨラナ方程式" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnanie_Majorany" title="Równanie Majorany – Polish" lang="pl" hreflang="pl" data-title="Równanie Majorany" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</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%9C%D0%B0%D0%B9%D0%BE%D1%80%D0%B0%D0%BD%D1%8B" title="Уравнение Майораны – Russian" lang="ru" hreflang="ru" data-title="Уравнение Майораны" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F_%D0%9C%D0%B0%D0%B9%D0%BE%D1%80%D0%B0%D0%BD%D0%B8" title="Рівняння Майорани – Ukrainian" lang="uk" hreflang="uk" data-title="Рівняння Майорани" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Relativistic wave description of fermions</div> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, the <b>Majorana equation</b> is a <a href="/wiki/Relativistic_wave_equation" class="mw-redirect" title="Relativistic wave equation">relativistic wave equation</a>. It is named after the Italian physicist <a href="/wiki/Ettore_Majorana" title="Ettore Majorana">Ettore Majorana</a>, who proposed it in 1937 as a means of describing <a href="/wiki/Fermion" title="Fermion">fermions</a> that are their own <a href="/wiki/Antiparticle" title="Antiparticle">antiparticle</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> Particles corresponding to this equation are termed <a href="/wiki/Majorana_particle" class="mw-redirect" title="Majorana particle">Majorana particles</a>, although that term now has a more expansive meaning, referring to any (possibly non-relativistic) fermionic particle that is its own anti-particle (and is therefore electrically neutral). </p><p>There have been proposals that massive <a href="/wiki/Neutrino" title="Neutrino">neutrinos</a> are described by Majorana particles; there are various extensions to the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> that enable this. The article on <a href="/wiki/Majorana_particle" class="mw-redirect" title="Majorana particle">Majorana particles</a> presents status for the experimental searches, including details about neutrinos. This article focuses primarily on the mathematical development of the theory, with attention to its <a href="/wiki/Discrete_symmetry" title="Discrete symmetry">discrete</a> and <a href="/wiki/Continuous_symmetry" title="Continuous symmetry">continuous symmetries</a>. The discrete symmetries are <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a>, <a href="/wiki/Parity_transformation" class="mw-redirect" title="Parity transformation">parity transformation</a> and <a href="/wiki/T-symmetry" title="T-symmetry">time reversal</a>; the continuous symmetry is <a href="/wiki/Lorentz_invariance" class="mw-redirect" title="Lorentz invariance">Lorentz invariance</a>. </p><p>Charge conjugation plays an outsize role, as it is the key symmetry that allows the Majorana particles to be described as electrically neutral. A particularly remarkable aspect is that electrical neutrality allows several global phases to be freely chosen, one each for the left and right <a href="/wiki/Chiral" class="mw-redirect" title="Chiral">chiral</a> fields. This implies that, without explicit constraints on these phases, the Majorana fields are naturally <a href="/wiki/CP_violation" title="CP violation">CP violating</a>. Another aspect of electric neutrality is that the left and right chiral fields can be given distinct masses. That is, electric charge is a <a href="/wiki/Lorentz_invariant" class="mw-redirect" title="Lorentz invariant">Lorentz invariant</a>, and also a <a href="/wiki/Constant_of_motion" title="Constant of motion">constant of motion</a>; whereas chirality is a Lorentz invariant, but is <i>not</i> a constant of motion for massive fields. Electrically neutral fields are thus less constrained than charged fields. Under charge conjugation, the two free global phases appear in the mass terms (as they are Lorentz invariant), and so the Majorana mass is described by a complex matrix, rather than a single number. In short, the discrete symmetries of the Majorana equation are considerably more complicated than those for the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>, where the electrical charge <span class="mwe-math-element"><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 constrains and removes these freedoms. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Majorana equation can be written in several distinct forms: </p> <ul><li>As the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> written so that the <a href="/wiki/Dirac_operator" title="Dirac operator">Dirac operator</a> is purely Hermitian, thus giving purely real solutions.</li> <li>As an operator that relates a four-component <a href="/wiki/Spinor" title="Spinor">spinor</a> to its <a href="/wiki/C-symmetry" title="C-symmetry">charge conjugate</a>.</li> <li>As a 2×2 differential equation acting on a complex two-component spinor, resembling the <a href="/wiki/Weyl_equation" title="Weyl equation">Weyl equation</a> with a properly <a href="/wiki/Lorentz_covariant" class="mw-redirect" title="Lorentz covariant">Lorentz covariant</a> mass term.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <p>These three forms are equivalent, and can be derived from one-another. Each offers slightly different insight into the nature of the equation. The first form emphasises that purely real solutions can be found. The second form clarifies the role of <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a>. The third form provides the most direct contact with the <a href="/wiki/Representation_theory_of_the_Lorentz_group" title="Representation theory of the Lorentz group">representation theory of the Lorentz group</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Purely_real_four-component_form">Purely real four-component form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=2" title="Edit section: Purely real four-component form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The conventional starting point is to state that "the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> can be written in <a href="/wiki/Hermitian_form" class="mw-redirect" title="Hermitian form">Hermitian form</a>", when the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a> are taken in the <i>Majorana representation</i>. The Dirac equation is then written as<sup id="cite_ref-iz_6-0" class="reference"><a href="#cite_note-iz-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\,-i\,{\frac {\partial }{\partial t}}-i\,{\hat {\alpha }}\cdot \nabla +\beta \,m\,\right)\,\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo>+</mo> <mi>β</mi> <mspace width="thinmathspace" /> <mi>m</mi> <mspace width="thinmathspace" /> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\,-i\,{\frac {\partial }{\partial t}}-i\,{\hat {\alpha }}\cdot \nabla +\beta \,m\,\right)\,\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4a805c9cb5d9b5ce3435e4266c7d95d3db91b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.468ex; height:6.176ex;" alt="{\displaystyle \left(\,-i\,{\frac {\partial }{\partial t}}-i\,{\hat {\alpha }}\cdot \nabla +\beta \,m\,\right)\,\psi =0}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682d943d1947245b587f282aba6c88f0870fb302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.176ex;" alt="{\displaystyle {\hat {\alpha }}}"></span> being purely real 4×4 symmetric matrices, 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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> being purely imaginary skew-symmetric; as required to ensure that the operator (that part inside the parentheses) is Hermitian. In this case, purely real 4‑spinor solutions to the equation can be found; these are the <i>Majorana spinors</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Charge-conjugate_four-component_form">Charge-conjugate four-component form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=3" title="Edit section: Charge-conjugate four-component form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Majorana equation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\,{\partial \!\!\!{\big /}}\psi -m\,\psi _{c}=0~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mspace width="thinmathspace" /> <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> <mspace width="thinmathspace" /> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\,{\partial \!\!\!{\big /}}\psi -m\,\psi _{c}=0~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76bb5096ec92d8303a8f1936cff1e1fada789a09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.77ex; height:3.176ex;" alt="{\displaystyle i\,{\partial \!\!\!{\big /}}\psi -m\,\psi _{c}=0~}"></span></dd></dl> <p>with the derivative 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 \!\!\!{\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> written in <a href="/wiki/Feynman_slash_notation" title="Feynman slash notation">Feynman slash notation</a> to include the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a> as well as a summation over the spinor components. The spinor <span class="mwe-math-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 \,\psi _{c}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \,\psi _{c}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e092066e49a3df06ac11fbcb4d04b923f2b3d3fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.232ex; height:2.509ex;" alt="{\textstyle \,\psi _{c}\,}"></span> is the <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugate</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \,\psi \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>ψ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \,\psi \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399b1912c9a0a6474c3f2e80bc8c4837174e28ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\textstyle \,\psi \,.}"></span> By construction, charge conjugates are necessarily given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{c}=\eta _{c}\,C\,{\overline {\psi }}^{\mathsf {T}}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{c}=\eta _{c}\,C\,{\overline {\psi }}^{\mathsf {T}}~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c56bf6ca916c6781e72444b8d035f49383027c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.756ex; height:4.009ex;" alt="{\displaystyle \psi _{c}=\eta _{c}\,C\,{\overline {\psi }}^{\mathsf {T}}~}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,(\cdot )^{\mathsf {T}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mo>⋅</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,(\cdot )^{\mathsf {T}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e66e3391bfec5451f86dc0a4d281a6e1ac4144" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.582ex; height:3.176ex;" alt="{\displaystyle \,(\cdot )^{\mathsf {T}}\,}"></span> denotes the <a href="/wiki/Transpose" title="Transpose">transpose</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\eta _{c}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\eta _{c}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd17700be408bb8ad6e96afbede44ab63063e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.874ex; height:2.176ex;" alt="{\displaystyle \,\eta _{c}\,}"></span> is an arbitrary phase factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,|\eta _{c}|=1\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,|\eta _{c}|=1\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a2adf9c72a3755af955da53f2af07156434cde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.075ex; height:2.843ex;" alt="{\displaystyle \,|\eta _{c}|=1\,,}"></span> conventionally taken 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 \,\eta _{c}=1\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\eta _{c}=1\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6990bb6251b6f9376d073a226d73bb9f52da75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.782ex; height:2.676ex;" alt="{\displaystyle \,\eta _{c}=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 \,C\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,C\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163a8873285c2bf583c476bc5d2a1f7c3b3ecd17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.541ex; height:2.176ex;" alt="{\displaystyle \,C\,}"></span> is a 4×4 matrix, the <i>charge conjugation matrix</i>. The matrix representation 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 \,C\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,C\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163a8873285c2bf583c476bc5d2a1f7c3b3ecd17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.541ex; height:2.176ex;" alt="{\displaystyle \,C\,}"></span> depends on the choice of the representation of the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>. By convention, the conjugate <a href="/wiki/Spinor" title="Spinor">spinor</a> is written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\psi }}=\psi ^{\dagger }\,\gamma ^{0}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\psi }}=\psi ^{\dagger }\,\gamma ^{0}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/981dfdb769616f8809f6fb8d6158d5d3e023efd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.151ex; height:3.509ex;" alt="{\displaystyle {\overline {\psi }}=\psi ^{\dagger }\,\gamma ^{0}~.}"></span></dd></dl> <p>A number of algebraic identities follow from the charge conjugation matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067be67e68f60c53ce83241748d0d6249675c58d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.413ex; height:2.176ex;" alt="{\displaystyle C.}"></span><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> One states that in any representation of the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>, including the Dirac, Weyl, and Majorana representations, 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 \,C\,\gamma _{\mu }=-\gamma _{\mu }^{\mathsf {T}}\,C\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,C\,\gamma _{\mu }=-\gamma _{\mu }^{\mathsf {T}}\,C\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d2231913627460db48b30652b0dd1c4d299d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.047ex; height:3.176ex;" alt="{\displaystyle \,C\,\gamma _{\mu }=-\gamma _{\mu }^{\mathsf {T}}\,C\,}"></span> and so one may write </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{c}=-\eta _{c}\,\gamma ^{0}\,C\,\psi ^{*}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{c}=-\eta _{c}\,\gamma ^{0}\,C\,\psi ^{*}~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51c9fa29162f59326c549a71bf44bba05d4280a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.873ex; height:3.176ex;" alt="{\displaystyle \psi _{c}=-\eta _{c}\,\gamma ^{0}\,C\,\psi ^{*}~}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\psi ^{*}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> </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/d7aca91f9324c00108f7cab4533f995385b47ffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.342ex; height:2.676ex;" alt="{\displaystyle \,\psi ^{*}\,}"></span> is the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\psi \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>ψ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\psi \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/733adc20fee72e1ac9bfd6c35421d0c61e7bc002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle \,\psi \,.}"></span> The charge conjugation matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,C\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,C\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163a8873285c2bf583c476bc5d2a1f7c3b3ecd17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.541ex; height:2.176ex;" alt="{\displaystyle \,C\,}"></span> also has the property that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{-1}=C^{\dagger }=C^{\mathsf {T}}=-C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{-1}=C^{\dagger }=C^{\mathsf {T}}=-C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511bcdff3392886aa2779a543e7cd0731751fed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.91ex; height:2.843ex;" alt="{\displaystyle C^{-1}=C^{\dagger }=C^{\mathsf {T}}=-C}"></span></dd></dl> <p>in all representations (Dirac, chiral, Majorana). From this, and a fair bit of algebra, one may obtain the equivalent equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\,{\partial \!\!\!{\big /}}\psi _{c}-m\,\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mspace width="thinmathspace" /> <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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\,{\partial \!\!\!{\big /}}\psi _{c}-m\,\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2672960cd2cb860a4c498c79b469130093fe6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.189ex; height:3.176ex;" alt="{\displaystyle i\,{\partial \!\!\!{\big /}}\psi _{c}-m\,\psi =0}"></span></dd></dl> <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</strong> <p>This form is not entirely obvious, and so deserves a proof. Starting with </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\,{\partial \!\!\!{\big /}}\psi +m\,\psi _{c}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <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> <mspace width="thinmathspace" /> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\,{\partial \!\!\!{\big /}}\psi +m\,\psi _{c}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf3a23c01c6aac4d3486f9538779b81054ca97f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.997ex; height:3.176ex;" alt="{\displaystyle -i\,{\partial \!\!\!{\big /}}\psi +m\,\psi _{c}=0}"></span></dd></dl> <p>Expand <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\psi _{c}=C\,{\overline {\psi }}^{\mathsf {T}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>C</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\psi _{c}=C\,{\overline {\psi }}^{\mathsf {T}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b62f964717b9a95c2d58cd0cc8172e0cdf13bc64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.463ex; height:3.843ex;" alt="{\displaystyle \,\psi _{c}=C\,{\overline {\psi }}^{\mathsf {T}}\,}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\,{\partial \!\!\!{\big /}}\psi +m\,C\,{\overline {\psi }}^{\mathsf {T}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <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> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\,{\partial \!\!\!{\big /}}\psi +m\,C\,{\overline {\psi }}^{\mathsf {T}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f44df602771a97228d2df2f6af2b55c24330c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.673ex; height:4.176ex;" alt="{\displaystyle -i\,{\partial \!\!\!{\big /}}\psi +m\,C\,{\overline {\psi }}^{\mathsf {T}}=0}"></span></dd></dl> <p>Multiply 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 \,C\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,C\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163a8873285c2bf583c476bc5d2a1f7c3b3ecd17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.541ex; height:2.176ex;" alt="{\displaystyle \,C\,}"></span> use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,C^{2}=-1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,C^{2}=-1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05f1f41b50e9520c8097879dd55030bd1dda22a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.695ex; height:2.843ex;" alt="{\displaystyle \,C^{2}=-1\,}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\,C\,{\partial \!\!\!{\big /}}C^{-1}\,C\,\psi -m\,{\overline {\psi }}^{\mathsf {T}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> <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> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>−</mo> <mi>m</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\,C\,{\partial \!\!\!{\big /}}C^{-1}\,C\,\psi -m\,{\overline {\psi }}^{\mathsf {T}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a04da9b55889c043b7e0293709a3d4ee7ca4b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.344ex; height:4.176ex;" alt="{\displaystyle -i\,C\,{\partial \!\!\!{\big /}}C^{-1}\,C\,\psi -m\,{\overline {\psi }}^{\mathsf {T}}=0}"></span></dd></dl> <p>Charge conjugation transposes the gamma matrices: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +i\,{\partial \!\!\!{\big /}}^{\mathsf {T}}\,C\,\psi -m\,\left(\gamma ^{0}\right)^{\mathsf {T}}\,\psi ^{*}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi>i</mi> <mspace width="thinmathspace" /> <msup> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>−</mo> <mi>m</mi> <mspace width="thinmathspace" /> <msup> <mrow> <mo>(</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +i\,{\partial \!\!\!{\big /}}^{\mathsf {T}}\,C\,\psi -m\,\left(\gamma ^{0}\right)^{\mathsf {T}}\,\psi ^{*}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e7e6959ad2e712f7412bb484a72c52eb94b371b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.202ex; height:3.843ex;" alt="{\displaystyle +i\,{\partial \!\!\!{\big /}}^{\mathsf {T}}\,C\,\psi -m\,\left(\gamma ^{0}\right)^{\mathsf {T}}\,\psi ^{*}=0}"></span></dd></dl> <p>Take the complex conjugate: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\,{\partial \!\!\!{\big /}}^{\dagger }C^{*}\,\psi ^{*}-m\,\left(\gamma ^{0}\right)^{\dagger }\,\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <msup> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <mi>m</mi> <mspace width="thinmathspace" /> <msup> <mrow> <mo>(</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\,{\partial \!\!\!{\big /}}^{\dagger }C^{*}\,\psi ^{*}-m\,\left(\gamma ^{0}\right)^{\dagger }\,\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d84d84f98c980dc8f112459d6e377737995df6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.122ex; height:3.843ex;" alt="{\displaystyle -i\,{\partial \!\!\!{\big /}}^{\dagger }C^{*}\,\psi ^{*}-m\,\left(\gamma ^{0}\right)^{\dagger }\,\psi =0}"></span></dd></dl> <p>The 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 \,\gamma ^{0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> </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/041013df3a58972ff1c5880c9f950e622ecfdaa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.108ex; height:3.176ex;" alt="{\displaystyle \,\gamma ^{0}\,}"></span> is Hermitian, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\left(\gamma ^{0}\right)^{\dagger }=\gamma ^{0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mrow> <mo>(</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>)</mo> </mrow> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\left(\gamma ^{0}\right)^{\dagger }=\gamma ^{0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c749f5ee1650539ab541f9e47e08d6bf67370a88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.633ex; height:3.843ex;" alt="{\displaystyle \,\left(\gamma ^{0}\right)^{\dagger }=\gamma ^{0}\,}"></span> in all three representations (Dirac, chiral, Majorana): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\,{\partial \!\!\!{\big /}}^{\dagger }C^{*}\,\psi ^{*}-m\,\gamma ^{0}\,\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <msup> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <mi>m</mi> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\,{\partial \!\!\!{\big /}}^{\dagger }C^{*}\,\psi ^{*}-m\,\gamma ^{0}\,\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94e270b393bdcdbdee2db362caff456b0aacfcd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.03ex; height:3.843ex;" alt="{\displaystyle -i\,{\partial \!\!\!{\big /}}^{\dagger }C^{*}\,\psi ^{*}-m\,\gamma ^{0}\,\psi =0}"></span></dd></dl> <p>It is also an <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a>, taking the <a href="/wiki/Hermitian_conjugate" class="mw-redirect" title="Hermitian conjugate">Hermitian conjugate</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\gamma ^{0}\,\gamma ^{\mu }\,\gamma ^{0}=\left(\gamma ^{\mu }\right)^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\gamma ^{0}\,\gamma ^{\mu }\,\gamma ^{0}=\left(\gamma ^{\mu }\right)^{\dagger }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482a5e423526a68bcb7e981e003610a0859c7a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.706ex; height:3.343ex;" alt="{\displaystyle \,\gamma ^{0}\,\gamma ^{\mu }\,\gamma ^{0}=\left(\gamma ^{\mu }\right)^{\dagger }}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\,\gamma ^{0}\,{\partial \!\!\!{\big /}}\gamma ^{0}\,C^{*}\,\psi ^{*}-m\,\gamma ^{0}\,\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> <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> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <mi>m</mi> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\,\gamma ^{0}\,{\partial \!\!\!{\big /}}\gamma ^{0}\,C^{*}\,\psi ^{*}-m\,\gamma ^{0}\,\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b7dd7d5e26d1d9c3d4fd063b8603de8334611e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.51ex; height:3.343ex;" alt="{\displaystyle -i\,\gamma ^{0}\,{\partial \!\!\!{\big /}}\gamma ^{0}\,C^{*}\,\psi ^{*}-m\,\gamma ^{0}\,\psi =0}"></span></dd></dl> <p>Multiply 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"> <mspace width="thinmathspace" /> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> </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/041013df3a58972ff1c5880c9f950e622ecfdaa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.108ex; height:3.176ex;" alt="{\displaystyle \,\gamma ^{0}\,}"></span>, note 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 \,\left(\gamma ^{0}\right)^{2}=I\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mrow> <mo>(</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\left(\gamma ^{0}\right)^{2}=I\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674192232560b48fe55ae1789a6a081771d80f83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.563ex; height:3.843ex;" alt="{\displaystyle \,\left(\gamma ^{0}\right)^{2}=I\,}"></span> and make use 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 \,C^{*}=C\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>C</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,C^{*}=C\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e850c5c369ce52318a2a556a37ace836bd1f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.491ex; height:2.343ex;" alt="{\displaystyle \,C^{*}=C\,}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\,{\partial \!\!\!{\big /}}\gamma ^{0}\,C\,\psi ^{*}-m\,\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <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> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\,{\partial \!\!\!{\big /}}\gamma ^{0}\,C\,\psi ^{*}-m\,\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c8a4c547478c55e7972e55ebb62539062e876d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.982ex; height:3.343ex;" alt="{\displaystyle -i\,{\partial \!\!\!{\big /}}\gamma ^{0}\,C\,\psi ^{*}-m\,\psi =0}"></span></dd></dl> <p>The above is just the definition of the conjugate, so conclude that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\,{\partial \!\!\!{\big /}}\psi _{c}-m\,\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mspace width="thinmathspace" /> <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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\,{\partial \!\!\!{\big /}}\psi _{c}-m\,\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2672960cd2cb860a4c498c79b469130093fe6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.189ex; height:3.176ex;" alt="{\displaystyle i\,{\partial \!\!\!{\big /}}\psi _{c}-m\,\psi =0}"></span></dd></dl> </div> <p>A detailed discussion of the physical interpretation of matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> as charge conjugation can be found in the article on <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a>. In short, it is involved in mapping <a href="/wiki/Particle" title="Particle">particles</a> to their <a href="/wiki/Antiparticle" title="Antiparticle">antiparticles</a>, which includes, among other things, the reversal of the <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>. Although <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94a78d3ebc752e9ce9abec7d8ed2dc46c9917e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.457ex; height:2.676ex;" alt="{\displaystyle \psi ^{c}}"></span> is defined as "the charge conjugate" 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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc4043b55bade492740e58cba74198873db1464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.16ex; height:2.509ex;" alt="{\displaystyle \psi ,}"></span> the charge conjugation operator has not one but two eigenvalues. This allows a second spinor, the <a href="/w/index.php?title=ELKO_spinor&action=edit&redlink=1" class="new" title="ELKO spinor (page does not exist)">ELKO spinor</a> to be defined. This is discussed in greater detail below. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_two-component_form">Complex two-component form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=4" title="Edit section: Complex two-component form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">The subscript <i><span class="texhtml">L</span></i> is used throughout this subsection to denote a <i>left</i>-handed chiral spinor.</div> <p>The <i>Majorana operator</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\mathrm {D} _{\text{L}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\mathrm {D} _{\text{L}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/097cffb8a6c01bc0d5f433355d9c27b89e110e6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.456ex; height:2.509ex;" alt="{\displaystyle \,\mathrm {D} _{\text{L}}\,,}"></span> is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{\text{L}}\equiv i\,{\overline {\sigma }}^{\mu }\,\partial _{\mu }+\eta \,m\,\omega \,K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>≡</mo> <mi>i</mi> <mspace width="thinmathspace" /> <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> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>+</mo> <mi>η</mi> <mspace width="thinmathspace" /> <mi>m</mi> <mspace width="thinmathspace" /> <mi>ω</mi> <mspace width="thinmathspace" /> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\text{L}}\equiv i\,{\overline {\sigma }}^{\mu }\,\partial _{\mu }+\eta \,m\,\omega \,K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb40683be79724eec781f09fb06644b057e56617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.561ex; height:3.176ex;" alt="{\displaystyle \mathrm {D} _{\text{L}}\equiv i\,{\overline {\sigma }}^{\mu }\,\partial _{\mu }+\eta \,m\,\omega \,K}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\sigma }}^{\mu }={\begin{bmatrix}\sigma ^{0}&-\sigma ^{1}&-\sigma ^{2}&-\sigma ^{3}\end{bmatrix}}={\begin{bmatrix}I_{2}&-\sigma _{\text{x}}&-\sigma _{\text{y}}&-\sigma _{\text{z}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <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"> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <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> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\sigma }}^{\mu }={\begin{bmatrix}\sigma ^{0}&-\sigma ^{1}&-\sigma ^{2}&-\sigma ^{3}\end{bmatrix}}={\begin{bmatrix}I_{2}&-\sigma _{\text{x}}&-\sigma _{\text{y}}&-\sigma _{\text{z}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39edaddabc13bbe856edde72f5d70c145a764b2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.691ex; margin-bottom: -0.314ex; width:56.502ex; height:3.176ex;" alt="{\displaystyle {\overline {\sigma }}^{\mu }={\begin{bmatrix}\sigma ^{0}&-\sigma ^{1}&-\sigma ^{2}&-\sigma ^{3}\end{bmatrix}}={\begin{bmatrix}I_{2}&-\sigma _{\text{x}}&-\sigma _{\text{y}}&-\sigma _{\text{z}}\end{bmatrix}}}"></span></dd></dl> <p>is a <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> whose components are the 2×2 <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,I_{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,I_{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd744015646a1d26bb737ccfc6dfb355709b8b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.852ex; height:2.509ex;" alt="{\displaystyle \,I_{2}\,}"></span> 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 \,\mu =0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>μ</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\mu =0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a9ed5564bbdcfca69ce265a4f6f87e7ac3f6d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.437ex; height:2.676ex;" alt="{\displaystyle \,\mu =0\,}"></span> and (minus) the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a> 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 \,\mu \in \{1,\,2,\,3\}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>μ</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\mu \in \{1,\,2,\,3\}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fee0ea35a9117fac7cac03aefa8b7edf7774f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.318ex; height:2.843ex;" alt="{\displaystyle \,\mu \in \{1,\,2,\,3\}\,.}"></span> 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 \,\eta \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>η</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\eta \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e10a704e9fe418b374d185c8c3ef4a12c4b7a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.944ex; height:2.176ex;" alt="{\displaystyle \,\eta \,}"></span> is an arbitrary phase factor, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,|\eta |=1\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,|\eta |=1\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0501d95bc7143359b05c01ae34eb65460c107c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.145ex; height:2.843ex;" alt="{\displaystyle \,|\eta |=1\,,}"></span> typically taken to be one: <span class="mwe-math-element"><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 =1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>η</mi> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\eta =1\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf6053106438a0b10ceddfaa6adc03b6c216d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.851ex; height:2.676ex;" alt="{\displaystyle \,\eta =1\,.}"></span> 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 \,\omega \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>ω</mi> <mspace width="thinmathspace" /> </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/f2a7b5716914d99fc1847e0d03500a68ecdec80a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.22ex; height:1.676ex;" alt="{\displaystyle \,\omega \,}"></span> is a 2×2 matrix that can be interpreted as the <a href="/wiki/Symplectic_form" class="mw-redirect" title="Symplectic form">symplectic form</a> for the <a href="/wiki/Symplectic_group" title="Symplectic group">symplectic 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 \,\operatorname {Sp} (2,\mathbb {C} )\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\operatorname {Sp} (2,\mathbb {C} )\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42669ea3721f52d05130564ba9f4003d7f5dd10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.69ex; height:2.843ex;" alt="{\displaystyle \,\operatorname {Sp} (2,\mathbb {C} )\,,}"></span> which is a <a href="/wiki/Cover_(mathematics)" class="mw-redirect" title="Cover (mathematics)">double covering</a> of the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>. It is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =i\,\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>i</mi> <mspace width="thinmathspace" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</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> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =i\,\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60105fc17a838706a11a5dfa2b788f8d20284523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.103ex; height:6.176ex;" alt="{\displaystyle \omega =i\,\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}~,}"></span></dd></dl> <p>which happens to be <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> to the imaginary unit <span class="nowrap">"<span class="texhtml mvar" style="font-style:italic;">i</span>"</span> (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 \omega ^{2}=-I\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>I</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}=-I\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01601073a49975064e122e1c460dcdd5781477c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.965ex; height:2.843ex;" alt="{\displaystyle \omega ^{2}=-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 \,a\,I+b\,\omega \cong a+b\,i\in \mathbb {C} \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>a</mi> <mspace width="thinmathspace" /> <mi>I</mi> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mi>ω</mi> <mo>≅</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,a\,I+b\,\omega \cong a+b\,i\in \mathbb {C} \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8494785e0d114e0ff9d34de1309a915bf3e39cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.108ex; height:2.343ex;" alt="{\displaystyle \,a\,I+b\,\omega \cong a+b\,i\in \mathbb {C} \,}"></span> 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,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2fb1e2e8a8d503638f17fa81297ec46ba7ef59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.167ex; height:2.509ex;" alt="{\displaystyle \,a,b\in \mathbb {R} }"></span>) with the matrix transpose being the analog of <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>. </p><p>Finally, 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 \,K\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>K</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,K\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4155a1de4dfecff464e030bd9bb9b931a4e63ca6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.84ex; height:2.176ex;" alt="{\displaystyle \,K\,}"></span> is a short-hand reminder to take the complex conjugate. The Majorana equation for a left-handed complex-valued two-component spinor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\psi _{\text{L}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\psi _{\text{L}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31bbd4826bbcd5331e7db6116686201927edc009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.547ex; height:2.509ex;" alt="{\displaystyle \,\psi _{\text{L}}\,}"></span> is then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{\text{L}}\psi _{\text{L}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\text{L}}\psi _{\text{L}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696efeb74520ab8dbb71208f1fdb0b8aa3586469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.069ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{\text{L}}\psi _{\text{L}}=0}"></span></dd></dl> <p>or, equivalently, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\,{\overline {\sigma }}^{\mu }\,\partial _{\mu }\psi _{\text{L}}(x)+\eta \,m\,\omega \,\psi _{\text{L}}^{*}(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mspace width="thinmathspace" /> <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> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>η</mi> <mspace width="thinmathspace" /> <mi>m</mi> <mspace width="thinmathspace" /> <mi>ω</mi> <mspace width="thinmathspace" /> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <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\,{\overline {\sigma }}^{\mu }\,\partial _{\mu }\psi _{\text{L}}(x)+\eta \,m\,\omega \,\psi _{\text{L}}^{*}(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb4ece762997bfc4082a52ed40d1687e9c18e80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.446ex; height:3.176ex;" alt="{\displaystyle i\,{\overline {\sigma }}^{\mu }\,\partial _{\mu }\psi _{\text{L}}(x)+\eta \,m\,\omega \,\psi _{\text{L}}^{*}(x)=0}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\psi _{\text{L}}^{*}(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\psi _{\text{L}}^{*}(x)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a08385ab4ff826acd048c606b8ced41b68d4239b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.686ex; height:3.009ex;" alt="{\displaystyle \,\psi _{\text{L}}^{*}(x)\,}"></span> the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\psi _{\text{L}}(x)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\psi _{\text{L}}(x)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3847de09bab11b3456109dd762e3cb42a5a88cd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.333ex; height:2.843ex;" alt="{\displaystyle \,\psi _{\text{L}}(x)\,.}"></span> The subscript <span class="texhtml">L</span> is used throughout this section to denote a <i>left</i>-handed chiral spinor; under a <a href="/wiki/Parity_transformation" class="mw-redirect" title="Parity transformation">parity transformation</a>, this can be taken to a right-handed spinor, and so one also has a right-handed form of the equation. This applies to the four-component equation as well; further details are presented below. </p> <div class="mw-heading mw-heading2"><h2 id="Key_ideas">Key ideas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=5" title="Edit section: Key ideas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some of the properties of the Majorana equation, its solution and its Lagrangian formulation are summarized here. </p> <ul><li>The Majorana equation is similar to the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>, in the sense that it involves four-component spinors, gamma matrices, and mass terms, but includes the <a href="/wiki/C-symmetry" title="C-symmetry">charge conjugate</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="{\textstyle \psi _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a758793d1e507d42dd596f63b5a667d75b041f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.457ex; height:2.509ex;" alt="{\textstyle \psi _{c}}"></span> of a <a href="/wiki/Spinor" title="Spinor">spinor</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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" 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="{\textstyle \psi }"></span>. In contrast, the <a href="/wiki/Weyl_equation" title="Weyl equation">Weyl equation</a> is for two-component spinor without mass.</li> <li>Solutions to the Majorana equation can be interpreted as electrically neutral particles that are their own anti-particle. By convention, the <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a> operator takes particles to their anti-particles, and so the Majorana spinor is conventionally defined as the solution 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 =\psi _{c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\psi _{c}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5caeba5d60bc4d5ff2b7f40bce3c6d04af2034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.716ex; height:2.509ex;" alt="{\displaystyle \psi =\psi _{c}.}"></span> That is, the Majorana spinor is "its own antiparticle". Insofar as charge conjugation takes an electrically charge particle to its anti-particle with opposite charge, one must conclude that the Majorana spinor is electrically neutral.</li> <li>The Majorana equation is <a href="/wiki/Lorentz_invariant" class="mw-redirect" title="Lorentz invariant">Lorentz covariant</a>, and a variety of Lorentz scalars can be constructed from its spinors. This allows several distinct <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangians</a> to be constructed for Majorana fields.</li> <li>When the Lagrangian is expressed in terms of two-component left and right <a href="/wiki/Chiral" class="mw-redirect" title="Chiral">chiral</a> spinors, it may contain three distinct mass terms: left and right Majorana mass terms, and a Dirac mass term. These manifest physically as two distinct masses; this is the key idea of the <a href="/wiki/Seesaw_mechanism" title="Seesaw mechanism">seesaw mechanism</a> for describing low-mass neutrinos with a left-handed coupling to the Standard model, with the right-handed component corresponding to a <a href="/wiki/Sterile_neutrino" title="Sterile neutrino">sterile neutrino</a> at <a href="/wiki/Grand_unification_energy" title="Grand unification energy">GUT-scale</a> masses.</li> <li>The discrete symmetries of <a href="/wiki/C-symmetry" title="C-symmetry">C</a>, <a href="/wiki/P-symmetry" class="mw-redirect" title="P-symmetry">P</a> and <a href="/wiki/T-symmetry" title="T-symmetry">T</a> conjugation are intimately controlled by a freely chosen phase factor on the <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a> operator. This manifests itself as distinct complex phases on the mass terms. This allows both <a href="/wiki/CP_violation" title="CP violation">CP-symmetric and CP-violating</a> Lagrangians to be written.</li> <li>The Majorana fields are <a href="/wiki/CPT_invariant" class="mw-redirect" title="CPT invariant">CPT invariant</a>, but the invariance is, in a sense "freer" than it is for charged particles. This is because charge is necessarily a Lorentz-invariant property, and is thus constrained for charged fields. The neutral Majorana fields are not constrained in this way, and can mix.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Two-component_Majorana_equation">Two-component Majorana equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=6" title="Edit section: Two-component Majorana equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Majorana equation can be written both in terms of a real four-component spinor, and as a complex two-component spinor. Both can be constructed from the <a href="/wiki/Weyl_equation" title="Weyl equation">Weyl equation</a>, with the addition of a properly Lorentz-covariant mass term.<sup id="cite_ref-aste_8-0" class="reference"><a href="#cite_note-aste-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This section provides an explicit construction and articulation. </p> <div class="mw-heading mw-heading3"><h3 id="Weyl_equation">Weyl equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=7" title="Edit section: Weyl equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Weyl equation describes the time evolution of a massless complex-valued two-component <a href="/wiki/Spinor" title="Spinor">spinor</a>. It is conventionally written as<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0}"> <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> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b8f5b9d792eabd9ccbc20a7d63a102fd4208e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.786ex; height:3.009ex;" alt="{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0}"></span></dd></dl> <p>Written out explicitly, it is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{2}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{2}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daaebae05ccc4271610f83dd42a2aaf933fac63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.734ex; height:6.176ex;" alt="{\displaystyle I_{2}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0}"></span></dd></dl> <p>The Pauli four-vector is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{\mu }={\begin{pmatrix}\sigma ^{0}&\sigma ^{1}&\sigma ^{2}&\sigma ^{3}\end{pmatrix}}={\begin{pmatrix}I_{2}&\sigma _{x}&\sigma _{y}&\sigma _{z}\end{pmatrix}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <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> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\mu }={\begin{pmatrix}\sigma ^{0}&\sigma ^{1}&\sigma ^{2}&\sigma ^{3}\end{pmatrix}}={\begin{pmatrix}I_{2}&\sigma _{x}&\sigma _{y}&\sigma _{z}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb43decb1c48458d09f1f26504912f02c533a766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.63ex; height:3.176ex;" alt="{\displaystyle \sigma ^{\mu }={\begin{pmatrix}\sigma ^{0}&\sigma ^{1}&\sigma ^{2}&\sigma ^{3}\end{pmatrix}}={\begin{pmatrix}I_{2}&\sigma _{x}&\sigma _{y}&\sigma _{z}\end{pmatrix}}}"></span></dd></dl> <p>that is, a <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> whose components are the 2 × 2 <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3506ae39df854f347365bae6f326ef4f565be5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{2}}"></span> for <i>μ</i> = 0 and the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a> for <i>μ</i> = 1, 2, 3. Under the <a href="/wiki/Parity_transformation" class="mw-redirect" title="Parity transformation">parity 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 {\vec {x}}\to {\vec {x}}^{\prime }=-{\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}\to {\vec {x}}^{\prime }=-{\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd57358aea5ca84e621392fee0b97702bd3487f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.194ex; height:2.843ex;" alt="{\displaystyle {\vec {x}}\to {\vec {x}}^{\prime }=-{\vec {x}}}"></span> one obtains a dual equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi =0}"> <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> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff51554fc59531739c1cb159798f6b2063cd695d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.785ex; height:3.009ex;" alt="{\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi =0}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\sigma }}^{\mu }={\begin{pmatrix}I_{2}&-\sigma _{x}&-\sigma _{y}&-\sigma _{z}\end{pmatrix}}}"> <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> <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> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\sigma }}^{\mu }={\begin{pmatrix}I_{2}&-\sigma _{x}&-\sigma _{y}&-\sigma _{z}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c687ad86e63ca38062c96dd227df6aab404ad36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.887ex; height:3.176ex;" alt="{\displaystyle {\bar {\sigma }}^{\mu }={\begin{pmatrix}I_{2}&-\sigma _{x}&-\sigma _{y}&-\sigma _{z}\end{pmatrix}}}"></span>. These are two distinct forms of the Weyl equation; their solutions are distinct as well. It can be shown that the solutions have left-handed and right-handed <a href="/wiki/Helicity_(particle_physics)" title="Helicity (particle physics)">helicity</a>, and thus <a href="/wiki/Chirality" title="Chirality">chirality</a>. It is conventional to label these two distinct forms explicitly, thus: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=0\qquad {\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}=0~.}"> <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> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=0\qquad {\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}=0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9b6e21580508493eb0ef074365ab7f177de76d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.145ex; height:3.009ex;" alt="{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=0\qquad {\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}=0~.}"></span></dd></dl> <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=Majorana_equation&action=edit&section=8" title="Edit section: Lorentz invariance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Weyl equation describes a massless particle; the Majorana equation adds a mass term. The mass must be introduced in a <a href="/wiki/Lorentz_invariant" class="mw-redirect" title="Lorentz invariant">Lorentz invariant</a> fashion. This is achieved by observing that the <a href="/wiki/Special_linear_group" title="Special linear group">special linear 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 \operatorname {SL} (2,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>SL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</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 \operatorname {SL} (2,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba7cdf4373aac3a21444fceecdd1101b9dd947a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.429ex; height:2.843ex;" alt="{\displaystyle \operatorname {SL} (2,\mathbb {C} )}"></span> is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the <a href="/wiki/Symplectic_group" title="Symplectic group">symplectic 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 \operatorname {Sp} (2,\mathbb {C} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (2,\mathbb {C} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d25d57937f825fe7c6a17b1024a1d33feb1edd32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.916ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (2,\mathbb {C} ).}"></span> Both of these groups are <a href="/wiki/Cover_(mathematics)" class="mw-redirect" title="Cover (mathematics)">double covers</a> of the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SO} (1,3).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SO} (1,3).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aa5f1e65a99aa3fc87f2f5ee32922377e802b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.916ex; height:2.843ex;" alt="{\displaystyle \operatorname {SO} (1,3).}"></span> The <a href="/wiki/Lorentz_invariance" class="mw-redirect" title="Lorentz invariance">Lorentz invariance</a> of the derivative term (from the Weyl equation) is conventionally worded in terms of the action of the 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 \operatorname {SL} (2,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>SL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</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 \operatorname {SL} (2,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba7cdf4373aac3a21444fceecdd1101b9dd947a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.429ex; height:2.843ex;" alt="{\displaystyle \operatorname {SL} (2,\mathbb {C} )}"></span> on spinors, whereas the Lorentz invariance of the mass term requires invocation of the defining relation for the symplectic group. </p><p>The double-covering of the Lorentz group is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }=S{\overline {\sigma }}_{\nu }S^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <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> <mi>S</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }=S{\overline {\sigma }}_{\nu }S^{\dagger }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056081cd2c785054ee55ab5b26f37c8182050ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.241ex; height:3.343ex;" alt="{\displaystyle {\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }=S{\overline {\sigma }}_{\nu }S^{\dagger }}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda \in \operatorname {SO} (1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo>∈</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda \in \operatorname {SO} (1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977171c23e92171173fa89a79b1cf2383fe3814c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.722ex; height:2.843ex;" alt="{\displaystyle \Lambda \in \operatorname {SO} (1,3)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\in \operatorname {SL} (2,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∈</mo> <mi>SL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</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 S\in \operatorname {SL} (2,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3a0ec78674b056bc7f0e3d313f552641085afa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.769ex; height:2.843ex;" alt="{\displaystyle S\in \operatorname {SL} (2,\mathbb {C} )}"></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 S^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{\dagger }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30011dcdbaf9b267c154008891b2eb6f51dfe50f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.484ex; height:2.676ex;" alt="{\displaystyle S^{\dagger }}"></span> is the <a href="/wiki/Hermitian_transpose" class="mw-redirect" title="Hermitian transpose">Hermitian transpose</a>. This is used to relate the transformation properties of the differentials 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 x\mapsto x^{\prime }=\Lambda x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">Λ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{\prime }=\Lambda x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f664ad11520c9969ce92450b05ab42735b94046" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.999ex; height:2.509ex;" alt="{\displaystyle x\mapsto x^{\prime }=\Lambda x}"></span> to the transformation properties of the spinors. </p><p>The symplectic 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 \operatorname {Sp} (2,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</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 \operatorname {Sp} (2,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72c19d987dd24f1cb69a7f6f58d6e345d762e4e" 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 \operatorname {Sp} (2,\mathbb {C} )}"></span> is defined as the set of all complex 2×2 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 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> that satisfy </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{-1}S^{\textsf {T}}\omega =S^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mi>ω</mi> <mo>=</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{-1}S^{\textsf {T}}\omega =S^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ebf665566883f52ddbdcba4bdd7dae14a72e8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.05ex; height:2.676ex;" alt="{\displaystyle \omega ^{-1}S^{\textsf {T}}\omega =S^{-1}}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</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> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864bb61b276bad0d5f8902aac0cf82c79a53faf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.489ex; height:6.176ex;" alt="{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}"></span></dd></dl> <p>is a <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric matrix</a>. It is used to define a <a href="/wiki/Symplectic_bilinear_form" class="mw-redirect" title="Symplectic bilinear form">symplectic bilinear form</a> 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} ^{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> <mo>.</mo> </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/e42022dd58259f7a15ec663c371aacbf4370aa81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{2}.}"></span> Writing a pair of arbitrary two-vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v\in \mathbb {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>∈</mo> <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 u,v\in \mathbb {C} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f407b9693828787acc49d8d56b5a47a000daa588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.064ex; height:3.009ex;" alt="{\displaystyle u,v\in \mathbb {C} ^{2}}"></span> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\begin{pmatrix}u_{1}\\u_{2}\end{pmatrix}}\qquad v={\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="2em" /> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\begin{pmatrix}u_{1}\\u_{2}\end{pmatrix}}\qquad v={\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70ab0231148f3f31f35b4d1820efa53799ea97fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.21ex; height:6.176ex;" alt="{\displaystyle u={\begin{pmatrix}u_{1}\\u_{2}\end{pmatrix}}\qquad v={\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}}"></span></dd></dl> <p>the symplectic product is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle u,v\rangle =-\langle v,u\rangle =u_{1}v_{2}-u_{2}v_{1}=u^{\textsf {T}}\omega v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mi>ω</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle =-\langle v,u\rangle =u_{1}v_{2}-u_{2}v_{1}=u^{\textsf {T}}\omega v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/524c8f6faf06d775e22036fe746688775727e4f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.931ex; height:3.176ex;" alt="{\displaystyle \langle u,v\rangle =-\langle v,u\rangle =u_{1}v_{2}-u_{2}v_{1}=u^{\textsf {T}}\omega v}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e1ec3184e052a1125fbaf913b6702e9ce7cdf25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.681ex; height:2.676ex;" alt="{\displaystyle u^{\textsf {T}}}"></span> is the transpose 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 u~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2115144da76dd3c111cbf2046e7f49f8d9bce2b" 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 u~.}"></span> This form is invariant under Lorentz transformations, in that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle u,v\rangle =\langle Su,Sv\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mi>u</mi> <mo>,</mo> <mi>S</mi> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle =\langle Su,Sv\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56494ae3b3a7879b5ff4f03500fe385d57c68b29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.698ex; height:2.843ex;" alt="{\displaystyle \langle u,v\rangle =\langle Su,Sv\rangle }"></span></dd></dl> <p>The skew matrix takes the Pauli matrices to minus their transpose: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \sigma _{k}\omega ^{-1}=-\sigma _{k}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \sigma _{k}\omega ^{-1}=-\sigma _{k}^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3639f636c1a44cdc9e01a189cb349d203fc0b96e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.229ex; height:3.343ex;" alt="{\displaystyle \omega \sigma _{k}\omega ^{-1}=-\sigma _{k}^{\textsf {T}}}"></span></dd></dl> <p>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 k=1,2,3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1,2,3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43d8930575f864e1e03f41d9de4edef04e27c368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.512ex; height:2.509ex;" alt="{\displaystyle k=1,2,3.}"></span> The skew matrix can be interpreted as the product of a <a href="/wiki/Parity_transformation" class="mw-redirect" title="Parity transformation">parity transformation</a> and a transposition acting on two-spinors. However, as will be emphasized in a later section, it can also be interpreted as one of the components of the <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a> operator, the other component being <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>. Applying it to the Lorentz transformation yields </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <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> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56547c2c2dead0d18b9a47be83e1d26582717d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.821ex; height:3.843ex;" alt="{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}"></span></dd></dl> <p>These two variants describe the covariance properties of the differentials acting on the left and right spinors, respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Differentials">Differentials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=9" title="Edit section: Differentials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under the <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^{\prime }=\Lambda x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">Λ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{\prime }=\Lambda x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f664ad11520c9969ce92450b05ab42735b94046" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.999ex; height:2.509ex;" alt="{\displaystyle x\mapsto x^{\prime }=\Lambda x}"></span> the differential term transforms as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)\mapsto \sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}(x^{\prime })=\left(S^{-1}\right)^{\dagger }\sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)}"> <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> <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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msup> <mi>σ</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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <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> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>σ</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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)\mapsto \sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}(x^{\prime })=\left(S^{-1}\right)^{\dagger }\sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba88725c765821b250c33a5f5ffdf640c6b3504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:54.862ex; height:5.509ex;" alt="{\displaystyle \sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)\mapsto \sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}(x^{\prime })=\left(S^{-1}\right)^{\dagger }\sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)}"></span></dd></dl> <p>provided that the right-handed field transforms as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }(x^{\prime })=S\psi _{\rm {R}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }(x^{\prime })=S\psi _{\rm {R}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af2537bd595dfa531d81b95d64fbbac9873a33de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.178ex; height:3.176ex;" alt="{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }(x^{\prime })=S\psi _{\rm {R}}(x)}"></span></dd></dl> <p>Similarly, the left-handed differential transforms as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)\mapsto {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {L}}(x^{\prime })=S{\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <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> <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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <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> <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> <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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)\mapsto {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {L}}(x^{\prime })=S{\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4c4a613249aa18d22b3af0ecb89de0b723f904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:49.215ex; height:5.509ex;" alt="{\displaystyle {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)\mapsto {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {L}}(x^{\prime })=S{\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)}"></span></dd></dl> <p>provided that the left-handed spinor transforms as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }(x^{\prime })=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <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> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }(x^{\prime })=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/403c316ef9e4a251333955b41a19fe87c63d7fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.079ex; height:3.843ex;" alt="{\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }(x^{\prime })=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}(x)}"></span></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p>These transformation properties are not particularly "obvious", and so deserve a careful derivation. Begin with the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }(x^{\prime })=R\psi _{\rm {R}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <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>R</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }(x^{\prime })=R\psi _{\rm {R}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8a45b36db8a2e4daee30ba1bb443daa55491b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.443ex; height:3.176ex;" alt="{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }(x^{\prime })=R\psi _{\rm {R}}(x)}"></span></dd></dl> <p>for some unknown <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\in \mathrm {SL} (2,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</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 R\in \mathrm {SL} (2,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f44d870681beb1bc7b199c8fac5a09f0b2c07d02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.034ex; height:2.843ex;" alt="{\displaystyle R\in \mathrm {SL} (2,\mathbb {C} )}"></span> to be determined. The Lorentz transform, in coordinates, is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae706db8c41c750af76d67c46e07b2521e791d3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.477ex; height:2.843ex;" alt="{\displaystyle x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }}"></span></dd></dl> <p>or, equivalently, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\nu }={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }x^{\prime \mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\nu }={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }x^{\prime \mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79cc68d849b4f55fb43d38d5d2a26432f8241188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:16.94ex; height:3.843ex;" alt="{\displaystyle x^{\nu }={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }x^{\prime \mu }}"></span></dd></dl> <p>This leads to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\sigma ^{\mu }\partial _{\mu }^{\prime }\psi _{\rm {R}}(x^{\prime })\\{}={}&\sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}(x^{\prime })\\{}={}&\sigma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}R\psi _{\rm {R}}(x)\\{}={}&\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }{\frac {\partial }{\partial x^{\nu }}}R\psi _{\rm {R}}(x)\\{}={}&\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }\partial _{\nu }R\psi _{\rm {R}}(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 /> <mtd> <msup> <mi>σ</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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <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> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msup> <mi>σ</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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <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> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msup> <mi>σ</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>R</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <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> <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>R</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <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> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mi>R</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sigma ^{\mu }\partial _{\mu }^{\prime }\psi _{\rm {R}}(x^{\prime })\\{}={}&\sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}(x^{\prime })\\{}={}&\sigma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}R\psi _{\rm {R}}(x)\\{}={}&\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }{\frac {\partial }{\partial x^{\nu }}}R\psi _{\rm {R}}(x)\\{}={}&\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }\partial _{\nu }R\psi _{\rm {R}}(x)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c987d7dcdcf9df9b05b61d5d9f135937f6812ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:27.252ex; height:24.176ex;" alt="{\displaystyle {\begin{aligned}&\sigma ^{\mu }\partial _{\mu }^{\prime }\psi _{\rm {R}}(x^{\prime })\\{}={}&\sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}(x^{\prime })\\{}={}&\sigma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}R\psi _{\rm {R}}(x)\\{}={}&\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }{\frac {\partial }{\partial x^{\nu }}}R\psi _{\rm {R}}(x)\\{}={}&\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }\partial _{\nu }R\psi _{\rm {R}}(x)\end{aligned}}}"></span></dd></dl> <p>In order to make use of the Weyl map </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <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> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56547c2c2dead0d18b9a47be83e1d26582717d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.821ex; height:3.843ex;" alt="{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}"></span></dd></dl> <p>a few indexes must be raised and lowered. This is easier said than done, as it invokes the identity </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \Lambda ^{T}\eta =\Lambda ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>η</mi> <mo>=</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta \Lambda ^{T}\eta =\Lambda ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d595b8b912442c2fdc1e172259235b4e9d0ab1c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.385ex; height:3.176ex;" alt="{\displaystyle \eta \Lambda ^{T}\eta =\Lambda ^{-1}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =\operatorname {diag} (+1,-1,-1,-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =\operatorname {diag} (+1,-1,-1,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f21a83c8d523dd76f41b35e2e1ec7979f8ef5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.326ex; height:2.843ex;" alt="{\displaystyle \eta =\operatorname {diag} (+1,-1,-1,-1)}"></span> is the flat-space <a href="/wiki/Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a>. The above identity is often used to define the elements <span class="mwe-math-element"><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 \in \operatorname {SO} (1,3).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo>∈</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda \in \operatorname {SO} (1,3).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd3cdaf1980f51736c3946c88d53ac8929ac0877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.369ex; height:2.843ex;" alt="{\displaystyle \Lambda \in \operatorname {SO} (1,3).}"></span> One takes the transpose: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }={\left(\Lambda ^{-1T}\right)_{\mu }}^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow> <mo>(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }={\left(\Lambda ^{-1T}\right)_{\mu }}^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b39dde0338578807a3db9316218b85bf4e939851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:21.061ex; height:3.843ex;" alt="{\displaystyle {\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }={\left(\Lambda ^{-1T}\right)_{\mu }}^{\nu }}"></span></dd></dl> <p>to write </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }\partial _{\nu }R\psi _{\rm {R}}(x)\\{}={}&\sigma ^{\mu }{\left(\Lambda ^{-1T}\right)_{\mu }}^{\nu }\partial _{\nu }R\psi _{\rm {R}}(x)\\{}={}&\sigma _{\mu }{\Lambda ^{\mu }}_{\nu }\partial ^{\nu }R\psi _{\rm {R}}(x)\\{}={}&\left(S^{-1}\right)^{\dagger }\sigma _{\mu }\partial ^{\mu }S^{-1}R\psi _{\rm {R}}(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 /> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <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> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mi>R</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow> <mo>(</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </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> <mi>R</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <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> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mi>R</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>σ</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> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>R</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }\partial _{\nu }R\psi _{\rm {R}}(x)\\{}={}&\sigma ^{\mu }{\left(\Lambda ^{-1T}\right)_{\mu }}^{\nu }\partial _{\nu }R\psi _{\rm {R}}(x)\\{}={}&\sigma _{\mu }{\Lambda ^{\mu }}_{\nu }\partial ^{\nu }R\psi _{\rm {R}}(x)\\{}={}&\left(S^{-1}\right)^{\dagger }\sigma _{\mu }\partial ^{\mu }S^{-1}R\psi _{\rm {R}}(x)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a47bfef1aa260fb993521e5a70dbd05e0aaac6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.852ex; margin-bottom: -0.319ex; width:27.626ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}&\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }\partial _{\nu }R\psi _{\rm {R}}(x)\\{}={}&\sigma ^{\mu }{\left(\Lambda ^{-1T}\right)_{\mu }}^{\nu }\partial _{\nu }R\psi _{\rm {R}}(x)\\{}={}&\sigma _{\mu }{\Lambda ^{\mu }}_{\nu }\partial ^{\nu }R\psi _{\rm {R}}(x)\\{}={}&\left(S^{-1}\right)^{\dagger }\sigma _{\mu }\partial ^{\mu }S^{-1}R\psi _{\rm {R}}(x)\end{aligned}}}"></span></dd></dl> <p>One thus regains the original form if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R=1,}"> <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> <mi>R</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{-1}R=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8098f2c176b804d48762366fabb5d13e6526f4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.526ex; height:3.009ex;" alt="{\displaystyle S^{-1}R=1,}"></span> that 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 R=S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58ab5b5799605d1e09b4d9f2ac83740efa82545f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.009ex; height:2.176ex;" alt="{\displaystyle R=S.}"></span> Performing the same manipulations for the left-handed equation, one concludes that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }(x^{\prime })=L\psi _{\rm {L}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <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>L</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }(x^{\prime })=L\psi _{\rm {L}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02de82e5449cc1335dc20d4bec0e057fbe645ba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.715ex; height:3.176ex;" alt="{\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }(x^{\prime })=L\psi _{\rm {L}}(x)}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\left(S^{\dagger }\right)^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\left(S^{\dagger }\right)^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7c23f6acb5fe07e08375dad7b7a0cc01a1f16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.275ex; height:3.843ex;" alt="{\displaystyle L=\left(S^{\dagger }\right)^{-1}.}"></span><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> </p> </div> <div class="mw-heading mw-heading3"><h3 id="Mass_term">Mass term</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=10" title="Edit section: Mass term"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of the right handed spinor field transforms as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {R}}^{*}(x)\mapsto \psi _{\rm {R}}^{\prime *}(x^{\prime })=S^{*}\psi _{\rm {R}}^{*}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mo>∗</mo> </mrow> </msubsup> <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> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {R}}^{*}(x)\mapsto \psi _{\rm {R}}^{\prime *}(x^{\prime })=S^{*}\psi _{\rm {R}}^{*}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c00ab91a11f2e6e7b1d02dbe5677eb31d6d41f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.32ex; height:3.176ex;" alt="{\displaystyle \psi _{\rm {R}}^{*}(x)\mapsto \psi _{\rm {R}}^{\prime *}(x^{\prime })=S^{*}\psi _{\rm {R}}^{*}(x)}"></span></dd></dl> <p>The defining relationship 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 \operatorname {Sp} (2,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</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 \operatorname {Sp} (2,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72c19d987dd24f1cb69a7f6f58d6e345d762e4e" 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 \operatorname {Sp} (2,\mathbb {C} )}"></span> can be rewritten 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 \omega S^{*}=\left(S^{\dagger }\right)^{-1}\omega \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>ω</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega S^{*}=\left(S^{\dagger }\right)^{-1}\omega \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1676b8279cecc7261fd33b9599ecddb65e6e5e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.546ex; height:3.843ex;" alt="{\displaystyle \omega S^{*}=\left(S^{\dagger }\right)^{-1}\omega \,.}"></span> From this, one concludes that the skew-complex field transforms as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\omega \psi _{\rm {R}}^{*}(x)\mapsto m\omega \psi _{\rm {R}}^{\prime *}(x^{\prime })=\left(S^{\dagger }\right)^{-1}m\omega \psi _{\rm {R}}^{*}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>m</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mo>∗</mo> </mrow> </msubsup> <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> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>m</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\omega \psi _{\rm {R}}^{*}(x)\mapsto m\omega \psi _{\rm {R}}^{\prime *}(x^{\prime })=\left(S^{\dagger }\right)^{-1}m\omega \psi _{\rm {R}}^{*}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d678c3a0c5c5ffd1e37cd574a944c3f15273599a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.149ex; height:3.843ex;" alt="{\displaystyle m\omega \psi _{\rm {R}}^{*}(x)\mapsto m\omega \psi _{\rm {R}}^{\prime *}(x^{\prime })=\left(S^{\dagger }\right)^{-1}m\omega \psi _{\rm {R}}^{*}(x)}"></span></dd></dl> <p>This is fully compatible with the covariance property of the differential. 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 \eta =e^{i\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =e^{i\phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf2c8478429fd0ba91f036e3f4164b00640fe93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.131ex; height:3.176ex;" alt="{\displaystyle \eta =e^{i\phi }}"></span> to be an arbitrary complex phase factor, the linear combination </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}(x)+\eta m\omega \psi _{\rm {R}}^{*}(x)}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>η</mi> <mi>m</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}(x)+\eta m\omega \psi _{\rm {R}}^{*}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b059352243317c93ed5e8f106c35cdc086c2b35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.498ex; height:3.009ex;" alt="{\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}(x)+\eta m\omega \psi _{\rm {R}}^{*}(x)}"></span></dd></dl> <p>transforms in a covariant fashion. Setting this to zero gives the complex two-component Majorana equation for the right-handed field. Similarly, the left-chiral Majorana equation (including an arbitrary phase factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }"></span>) is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}(x)+\zeta m\omega \psi _{\rm {L}}^{*}(x)=0}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ζ</mi> <mi>m</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <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{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}(x)+\zeta m\omega \psi _{\rm {L}}^{*}(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6385cbbc7eb01e12cc5bd21efae26e8243a00c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.436ex; height:3.176ex;" alt="{\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}(x)+\zeta m\omega \psi _{\rm {L}}^{*}(x)=0}"></span></dd></dl> <p>The left and right chiral versions are related by a <a href="/wiki/Parity_transformation" class="mw-redirect" title="Parity transformation">parity transformation</a>. As shown below, these square to the <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon operator</a> only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =\zeta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mi>ζ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =\zeta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f4687f06b4a468fc1377e34b1b856258dfeccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.01ex; height:2.676ex;" alt="{\displaystyle \eta =\zeta .}"></span> The skew complex conjugate <span class="mwe-math-element"><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 \psi ^{*}=i\sigma ^{2}\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>i</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \psi ^{*}=i\sigma ^{2}\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d49531b11e915c323b673f745d7804346952e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.812ex; height:3.009ex;" alt="{\displaystyle \omega \psi ^{*}=i\sigma ^{2}\psi }"></span> can be recognized as the <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugate</a> form 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> <mtext> </mtext> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ~;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd0ca98e583356e7e53a2d59bae0c8edba589d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.741ex; height:2.509ex;" alt="{\displaystyle \psi ~;}"></span> this is articulated in greater detail below. Thus, the Majorana equation can be read as an equation that connects a spinor to its charge-conjugate form. </p> <div class="mw-heading mw-heading3"><h3 id="Left_and_right_Majorana_operators">Left and right Majorana operators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=11" title="Edit section: Left and right Majorana operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Define a pair of operators, the Majorana operators, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}&=i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K&\mathrm {D} _{\rm {R}}&=i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <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> <mo>+</mo> <mi>ζ</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</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>+</mo> <mi>η</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}&=i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K&\mathrm {D} _{\rm {R}}&=i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97c0aaa9ed37ea02d678cb2fd847cdd4749e61c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.64ex; height:3.176ex;" alt="{\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}&=i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K&\mathrm {D} _{\rm {R}}&=i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K\end{aligned}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is a short-hand reminder to take the complex conjugate. Under Lorentz transformations, these transform as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}\mapsto \mathrm {D} _{\rm {L}}^{\prime }&=S\mathrm {D} _{\rm {L}}S^{\dagger }&\mathrm {D} _{\rm {R}}\mapsto \mathrm {D} _{\rm {R}}^{\prime }&=\left(S^{\dagger }\right)^{-1}\mathrm {D} _{\rm {R}}S^{-1}\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">↦</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">↦</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}\mapsto \mathrm {D} _{\rm {L}}^{\prime }&=S\mathrm {D} _{\rm {L}}S^{\dagger }&\mathrm {D} _{\rm {R}}\mapsto \mathrm {D} _{\rm {R}}^{\prime }&=\left(S^{\dagger }\right)^{-1}\mathrm {D} _{\rm {R}}S^{-1}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df209f5a0c840fd39732671e7562df5b2636daa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:52.364ex; height:3.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}\mapsto \mathrm {D} _{\rm {L}}^{\prime }&=S\mathrm {D} _{\rm {L}}S^{\dagger }&\mathrm {D} _{\rm {R}}\mapsto \mathrm {D} _{\rm {R}}^{\prime }&=\left(S^{\dagger }\right)^{-1}\mathrm {D} _{\rm {R}}S^{-1}\end{aligned}}}"></span></dd></dl> <p>whereas the Weyl spinors transform as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi _{\rm {L}}\mapsto \psi _{\rm {L}}^{\prime }&=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}&\psi _{\rm {R}}\mapsto \psi _{\rm {R}}^{\prime }&=S\psi _{\rm {R}}\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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">↦</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">↦</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi _{\rm {L}}\mapsto \psi _{\rm {L}}^{\prime }&=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}&\psi _{\rm {R}}\mapsto \psi _{\rm {R}}^{\prime }&=S\psi _{\rm {R}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4174f857aaa637842bcefe84fb91253fa50e6511" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:44.451ex; height:3.843ex;" alt="{\displaystyle {\begin{aligned}\psi _{\rm {L}}\mapsto \psi _{\rm {L}}^{\prime }&=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}&\psi _{\rm {R}}\mapsto \psi _{\rm {R}}^{\prime }&=S\psi _{\rm {R}}\end{aligned}}}"></span></dd></dl> <p>just as above. Thus, the matched combinations of these are Lorentz covariant, and one may take </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}\psi _{\rm {L}}&=0&\mathrm {D} _{\rm {R}}\psi _{\rm {R}}&=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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </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}\mathrm {D} _{\rm {L}}\psi _{\rm {L}}&=0&\mathrm {D} _{\rm {R}}\psi _{\rm {R}}&=0\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0270b8b4d68a18cda78e9ad053822ebf345fb57b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.899ex; height:2.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}\psi _{\rm {L}}&=0&\mathrm {D} _{\rm {R}}\psi _{\rm {R}}&=0\end{aligned}}}"></span></dd></dl> <p>as a pair of complex 2-spinor Majorana equations. </p><p>The products <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{\rm {L}}\mathrm {D} _{\rm {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\rm {L}}\mathrm {D} _{\rm {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1790bc80c1feb2dd8d2c74d01646f17ba0473e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.253ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{\rm {L}}\mathrm {D} _{\rm {R}}}"></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 \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45ead9cbc14ee022d53407a8325aeb0b9175cd9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.253ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}}"></span> are both Lorentz covariant. The product is explicitly </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}=\left(i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K\right)\left(i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K\right)=-\left(\partial _{t}^{2}-{\vec {\nabla }}\cdot {\vec {\nabla }}+\eta \zeta ^{*}m^{2}\right)=-\left(\square +\eta \zeta ^{*}m^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <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>+</mo> <mi>η</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <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> <mo>+</mo> <mi>ζ</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>η</mi> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>◻</mi> <mo>+</mo> <mi>η</mi> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}=\left(i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K\right)\left(i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K\right)=-\left(\partial _{t}^{2}-{\vec {\nabla }}\cdot {\vec {\nabla }}+\eta \zeta ^{*}m^{2}\right)=-\left(\square +\eta \zeta ^{*}m^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98e879b609ffd43b8343ab5de994a81ba3322c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:90.794ex; height:4.843ex;" alt="{\displaystyle \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}=\left(i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K\right)\left(i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K\right)=-\left(\partial _{t}^{2}-{\vec {\nabla }}\cdot {\vec {\nabla }}+\eta \zeta ^{*}m^{2}\right)=-\left(\square +\eta \zeta ^{*}m^{2}\right)}"></span></dd></dl> <p>Verifying this requires keeping in mind that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ecdbc4a8b56a82c8b4616e6ffcdc4826610f3d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.569ex; height:2.843ex;" alt="{\displaystyle \omega ^{2}=-1}"></span> and 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 Ki=-iK~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mi>i</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>K</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ki=-iK~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225b1bbf8c9dcc818fea0cd4d89bad81f441f14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.871ex; height:2.343ex;" alt="{\displaystyle Ki=-iK~.}"></span> The RHS reduces to the <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon operator</a> provided 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 \eta \zeta ^{*}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta \zeta ^{*}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/365387964650117c7e0547d298116b3ba1f75431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.603ex; height:2.843ex;" alt="{\displaystyle \eta \zeta ^{*}=1}"></span>, that 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 \eta =\zeta ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mi>ζ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =\zeta ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24a23092b2386e2eb374ad98535649e08a10fe4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.59ex; height:2.676ex;" alt="{\displaystyle \eta =\zeta ~.}"></span> These two Majorana operators are thus "square roots" of the Klein–Gordon operator. </p> <div class="mw-heading mw-heading2"><h2 id="Four-component_Majorana_equation">Four-component Majorana equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=12" title="Edit section: Four-component Majorana equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The real four-component version of the Majorana equation can be constructed from the complex two-component equation as follows. Given the complex 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 _{\rm {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b50f1a5fe1e1eb1e6bbafb30a098d1e022b4c16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.773ex; height:2.509ex;" alt="{\displaystyle \psi _{\rm {L}}}"></span> satisfying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{\rm {L}}\psi _{\rm {L}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\rm {L}}\psi _{\rm {L}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efe10463a16f7ba4a4d5d7084a25b0b1f59749f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.069ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{\rm {L}}\psi _{\rm {L}}=0}"></span> as above, define </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\rm {R}}\equiv -\eta \omega \psi _{\rm {L}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>≡</mo> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\rm {R}}\equiv -\eta \omega \psi _{\rm {L}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374ec25fbff6321444f759e1e3b0053066f12df5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.191ex; height:2.843ex;" alt="{\displaystyle \chi _{\rm {R}}\equiv -\eta \omega \psi _{\rm {L}}^{*}}"></span></dd></dl> <p>Using the algebraic machinery given above, it is not hard to show that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(i\sigma ^{\mu }\partial _{\mu }-\eta m\omega K\right)\chi _{\rm {R}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <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>−</mo> <mi>η</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(i\sigma ^{\mu }\partial _{\mu }-\eta m\omega K\right)\chi _{\rm {R}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/125629c915c03a72535b429d484ee0435d45d803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.73ex; height:3.009ex;" alt="{\displaystyle \left(i\sigma ^{\mu }\partial _{\mu }-\eta m\omega K\right)\chi _{\rm {R}}=0}"></span></dd></dl> <p>Defining a conjugate operator </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }-\eta m\omega K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</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>−</mo> <mi>η</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }-\eta m\omega K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b1897b49cd0b1365392f71c7bbfbe600a1eb99d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.949ex; height:3.009ex;" alt="{\displaystyle \delta _{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }-\eta m\omega K}"></span></dd></dl> <p>The four-component Majorana equation is then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)\left(\psi _{\rm {L}}\oplus \chi _{\rm {R}}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)\left(\psi _{\rm {L}}\oplus \chi _{\rm {R}}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4711d238ebef2284052da0e6c026afbe22bdea3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.126ex; height:2.843ex;" alt="{\displaystyle \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)\left(\psi _{\rm {L}}\oplus \chi _{\rm {R}}\right)=0}"></span></dd></dl> <p>Writing this out in detail, one has </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}={\begin{bmatrix}\mathrm {D} _{\rm {L}}&0\\0&\delta _{\rm {R}}\end{bmatrix}}=i{\begin{bmatrix}I&0\\0&I\end{bmatrix}}\partial _{t}+i{\begin{bmatrix}-\sigma ^{k}&0\\0&\sigma ^{k}\end{bmatrix}}\nabla _{k}+m{\begin{bmatrix}\eta \omega K&0\\0&-\eta \omega K\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>I</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}={\begin{bmatrix}\mathrm {D} _{\rm {L}}&0\\0&\delta _{\rm {R}}\end{bmatrix}}=i{\begin{bmatrix}I&0\\0&I\end{bmatrix}}\partial _{t}+i{\begin{bmatrix}-\sigma ^{k}&0\\0&\sigma ^{k}\end{bmatrix}}\nabla _{k}+m{\begin{bmatrix}\eta \omega K&0\\0&-\eta \omega K\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d597d484da7933fea0472778cfc3fdf28e000ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:76.744ex; height:6.176ex;" alt="{\displaystyle \mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}={\begin{bmatrix}\mathrm {D} _{\rm {L}}&0\\0&\delta _{\rm {R}}\end{bmatrix}}=i{\begin{bmatrix}I&0\\0&I\end{bmatrix}}\partial _{t}+i{\begin{bmatrix}-\sigma ^{k}&0\\0&\sigma ^{k}\end{bmatrix}}\nabla _{k}+m{\begin{bmatrix}\eta \omega K&0\\0&-\eta \omega K\end{bmatrix}}}"></span></dd></dl> <p>Multiplying on the left by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =\gamma ^{0}={\begin{bmatrix}0&I\\I&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <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> <mi>I</mi> </mtd> </mtr> <mtr> <mtd> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =\gamma ^{0}={\begin{bmatrix}0&I\\I&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20c4ffb92d6ca5e8bbd7da703b79b3d02ab6179f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.736ex; height:6.176ex;" alt="{\displaystyle \beta =\gamma ^{0}={\begin{bmatrix}0&I\\I&0\end{bmatrix}}}"></span></dd></dl> <p>brings the above into a matrix form wherein the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a> in the chiral representation can be recognized. This is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)={\begin{bmatrix}0&\delta _{\rm {R}}\\\mathrm {D} _{\rm {L}}&0\end{bmatrix}}=i\beta \partial _{t}+i{\begin{bmatrix}0&\sigma ^{k}\\-\sigma ^{k}&0\end{bmatrix}}\nabla _{k}-m{\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>i</mi> <mi>β</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <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>k</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)={\begin{bmatrix}0&\delta _{\rm {R}}\\\mathrm {D} _{\rm {L}}&0\end{bmatrix}}=i\beta \partial _{t}+i{\begin{bmatrix}0&\sigma ^{k}\\-\sigma ^{k}&0\end{bmatrix}}\nabla _{k}-m{\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c719719c43301582357835fbe727e6fae54e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:73.732ex; height:6.176ex;" alt="{\displaystyle \beta \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)={\begin{bmatrix}0&\delta _{\rm {R}}\\\mathrm {D} _{\rm {L}}&0\end{bmatrix}}=i\beta \partial _{t}+i{\begin{bmatrix}0&\sigma ^{k}\\-\sigma ^{k}&0\end{bmatrix}}\nabla _{k}-m{\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}"></span></dd></dl> <p>That is, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }-m{\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }-m{\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce43828b36262bc1f7cfcabd8600c34c154d70f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.321ex; height:6.176ex;" alt="{\displaystyle \beta \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }-m{\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}"></span></dd></dl> <p>Applying this to the 4-spinor </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {L}}\oplus \chi _{\rm {R}}={\begin{pmatrix}\psi _{\rm {L}}\\\chi _{\rm {R}}\end{pmatrix}}={\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </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> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {L}}\oplus \chi _{\rm {R}}={\begin{pmatrix}\psi _{\rm {L}}\\\chi _{\rm {R}}\end{pmatrix}}={\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e88cab0b385e4a3922bf7ec175b6bbe893fb4266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.145ex; height:6.509ex;" alt="{\displaystyle \psi _{\rm {L}}\oplus \chi _{\rm {R}}={\begin{pmatrix}\psi _{\rm {L}}\\\chi _{\rm {R}}\end{pmatrix}}={\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}}"></span></dd></dl> <p>and recalling 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 \omega ^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ecdbc4a8b56a82c8b4616e6ffcdc4826610f3d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.569ex; height:2.843ex;" alt="{\displaystyle \omega ^{2}=-1}"></span> one finds that the spinor is an eigenstate of the mass term, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}{\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}={\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\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> <mn>0</mn> </mtd> <mtd> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}{\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}={\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab5f96968316fb07feb21d33fe53db2a2286e667" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.535ex; height:6.509ex;" alt="{\displaystyle {\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}{\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}={\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}}"></span></dd></dl> <p>and so, for this particular spinor, the four-component Majorana equation reduces to the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }-m\right){\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}=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> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> </mrow> <mo>)</mo> </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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }-m\right){\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dedd4222f4e4a9162af0a9be6a50669b008501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.47ex; height:6.509ex;" alt="{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }-m\right){\begin{pmatrix}\psi _{\rm {L}}\\-\eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}=0}"></span></dd></dl> <p>The skew matrix can be identified with the <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a> operator (in the <a href="/wiki/Gamma_matrices" title="Gamma matrices">Weyl basis</a>). Explicitly, this is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}={\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}={\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777931cb949a436f2677a7265a6c69351c7548ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.283ex; height:6.176ex;" alt="{\displaystyle {\mathsf {C}}={\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}"></span></dd></dl> <p>Given an arbitrary four-component spinor <span class="mwe-math-element"><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> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674591165409931fc16a182571f72662a79823aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.741ex; height:2.509ex;" alt="{\displaystyle \psi ~,}"></span> its charge conjugate is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}\psi =\psi ^{c}=\eta C{\overline {\psi }}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mi>ψ</mi> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mi>η</mi> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\psi =\psi ^{c}=\eta C{\overline {\psi }}^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ebcc4171e8d582cce8a678509076485a55a45a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.568ex; height:4.009ex;" alt="{\displaystyle {\mathsf {C}}\psi =\psi ^{c}=\eta C{\overline {\psi }}^{\textsf {T}}}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> an ordinary 4×4 matrix, having a form explicitly given in the article on <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>. In conclusion, the 4-component Majorana equation can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}0&=\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi \\&=i\gamma ^{\mu }\partial _{\mu }\psi -m\psi ^{c}\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> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</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> <mi>ψ</mi> <mo>−</mo> <mi>m</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0&=\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi \\&=i\gamma ^{\mu }\partial _{\mu }\psi -m\psi ^{c}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d3e060fcfbf780a5ad4e8e8470ece348821b705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.852ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}0&=\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi \\&=i\gamma ^{\mu }\partial _{\mu }\psi -m\psi ^{c}\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Charge_conjugation_and_parity">Charge conjugation and parity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=13" title="Edit section: Charge conjugation and parity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The charge conjugation operator appears directly in the 4-component version of the Majorana equation. When the spinor field is a charge conjugate of itself, that is, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{c}=\psi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mi>ψ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{c}=\psi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79611ae6d25d1115656afdca686f51b69b5ab9ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.716ex; height:2.676ex;" alt="{\displaystyle \psi ^{c}=\psi ,}"></span> then the Majorana equation reduces to the Dirac equation, and any solution can be interpreted as describing an electrically neutral field. However, the charge conjugation operator has not one, but two distinct eigenstates, one of which is the <a href="/w/index.php?title=ELKO_spinor&action=edit&redlink=1" class="new" title="ELKO spinor (page does not exist)">ELKO spinor</a>; it does <i>not</i> solve the Majorana equation, but rather, a sign-flipped version of it. </p><p>The charge conjugation 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 {\mathsf {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699cb4847ecedddeeae2b69e2892ba9987f9b1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle {\mathsf {C}}}"></span> for a four-component spinor is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}\psi =\psi _{c}=\eta C\left({\overline {\psi }}\right)^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mi>ψ</mi> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>η</mi> <mi>C</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\psi =\psi _{c}=\eta C\left({\overline {\psi }}\right)^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f09558096106fc8f14e9e4eed3aef1a6b892d263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.344ex; height:5.343ex;" alt="{\displaystyle {\mathsf {C}}\psi =\psi _{c}=\eta C\left({\overline {\psi }}\right)^{\textsf {T}}}"></span></dd></dl> <p>A general discussion of the physical interpretation of this operator in terms of <a href="/wiki/Electrical_charge" class="mw-redirect" title="Electrical charge">electrical charge</a> is given in the article on <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a>. Additional discussions are provided by Bjorken & Drell<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> or Itzykson & Zuber.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> In more abstract terms, it is the spinorial equivalent of complex conjugation 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 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> coupling of the electromagnetic field. This can be seen as follows. If one has a single, real <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a>, it cannot couple to electromagnetism; however, a pair of real scalar fields, arranged as a <a href="/wiki/Complex_number" title="Complex number">complex number</a>, can. For scalar fields, charge conjugation is the same as <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>. The <a href="/wiki/Discrete_symmetries" class="mw-redirect" title="Discrete symmetries">discrete symmetries</a> 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 U(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62b00d74ee0cefb86cc052365625abff56d43e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}"></span> gauge theory follows from the "trivial" observation that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *:U(1)\to U(1)\quad e^{i\phi }\mapsto e^{-i\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mo>:</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="1em" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *:U(1)\to U(1)\quad e^{i\phi }\mapsto e^{-i\phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa35edc476988e6f8e1ac9a9e965a101ea964f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.163ex; height:3.176ex;" alt="{\displaystyle *:U(1)\to U(1)\quad e^{i\phi }\mapsto e^{-i\phi }}"></span></dd></dl> <p>is an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> <mo>.</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/ceae79a5061e4276918b82ed149255d364e541b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.401ex; height:2.843ex;" alt="{\displaystyle U(1).}"></span> For spinorial fields, the situation is more confusing. Roughly speaking, however, one can say that the Majorana field is electrically neutral, and that taking an appropriate combination of two Majorana fields can be interpreted as a single electrically charged Dirac field. The charge conjugation operator given above corresponds to the automorphism 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 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> <mo>.</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/ceae79a5061e4276918b82ed149255d364e541b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.401ex; height:2.843ex;" alt="{\displaystyle U(1).}"></span> </p><p>In the above, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> is a 4×4 matrix, given in the article on the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>. Its explicit form is representation-dependent. The 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 {\mathsf {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699cb4847ecedddeeae2b69e2892ba9987f9b1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle {\mathsf {C}}}"></span> cannot be written as a 4×4 matrix, as it is taking the complex conjugate 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>, and complex conjugation cannot be achieved with a complex 4×4 matrix. It can be written as a real 8×8 matrix, presuming one also writes <span class="mwe-math-element"><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> as a purely real 8-component spinor. Letting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> stand for complex conjugation, 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 K(x+iy)=x-iy,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x+iy)=x-iy,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff602d0e142754becb763e6fe86aed58b947cbe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.877ex; height:2.843ex;" alt="{\displaystyle K(x+iy)=x-iy,}"></span> one can then write, for four-component spinors, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}=-\eta \gamma ^{0}CK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>η</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mi>C</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}=-\eta \gamma ^{0}CK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27704ac06b149c68b85a82b0a7c40ff41bde45ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.728ex; height:3.176ex;" alt="{\displaystyle {\mathsf {C}}=-\eta \gamma ^{0}CK}"></span></dd></dl> <p>It is not hard to show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc1001f30c3027a4df4c22089442e1a09b6a2b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.8ex; height:2.676ex;" alt="{\displaystyle {\mathsf {C}}^{2}=1}"></span> and that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}\gamma ^{\mu }{\mathsf {C}}=-\gamma ^{\mu }~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\gamma ^{\mu }{\mathsf {C}}=-\gamma ^{\mu }~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06288b7022cb1bb972b9166cb32d283eaa274f5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.111ex; height:2.843ex;" alt="{\displaystyle {\mathsf {C}}\gamma ^{\mu }{\mathsf {C}}=-\gamma ^{\mu }~.}"></span> It follows from the first identity that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699cb4847ecedddeeae2b69e2892ba9987f9b1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle {\mathsf {C}}}"></span> has two eigenvalues, which may be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}\psi ^{(\pm )}=\pm \psi ^{(\pm )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mo>±</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\psi ^{(\pm )}=\pm \psi ^{(\pm )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbca0a209df8116c48514611cd68045c7b44aa54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.999ex; height:3.176ex;" alt="{\displaystyle {\mathsf {C}}\psi ^{(\pm )}=\pm \psi ^{(\pm )}}"></span></dd></dl> <p>The eigenvectors are readily found in the Weyl basis. From the above, in this basis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699cb4847ecedddeeae2b69e2892ba9987f9b1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle {\mathsf {C}}}"></span> is explicitly </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}={\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <mi>K</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}={\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777931cb949a436f2677a7265a6c69351c7548ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.283ex; height:6.176ex;" alt="{\displaystyle {\mathsf {C}}={\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}}"></span></dd></dl> <p>and thus </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\text{Weyl}}^{(\pm )}={\begin{pmatrix}\psi _{\rm {L}}\\\mp \eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Weyl</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>∓</mo> <mi>η</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\text{Weyl}}^{(\pm )}={\begin{pmatrix}\psi _{\rm {L}}\\\mp \eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d66ae22ab70d979894e590ece5b43fb285d543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.957ex; height:6.509ex;" alt="{\displaystyle \psi _{\text{Weyl}}^{(\pm )}={\begin{pmatrix}\psi _{\rm {L}}\\\mp \eta \omega \psi _{\rm {L}}^{*}\end{pmatrix}}}"></span></dd></dl> <p>Both eigenvectors are clearly solutions to the Majorana equation. However, only the positive eigenvector is a solution to the Dirac equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi ^{(+)}=\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi ^{(+)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <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> </mrow> <mo>)</mo> </mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi ^{(+)}=\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi ^{(+)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5bbe0e9b27b5e4e131fcc3aefc20c67cffd612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.133ex; height:3.509ex;" alt="{\displaystyle 0=\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi ^{(+)}=\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi ^{(+)}}"></span></dd></dl> <p>The negative eigenvector "doesn't work", it has the incorrect sign on the Dirac mass term. It still solves the Klein–Gordon equation, however. The negative eigenvector is termed the <a href="/w/index.php?title=ELKO_spinor&action=edit&redlink=1" class="new" title="ELKO spinor (page does not exist)">ELKO spinor</a>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p>That both eigenstates solve the Klein–Gordon equation follows from the earlier identities for the two-component versions. Defining, as before, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }+\eta m\omega K\qquad \mathrm {D} _{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <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> <mo>+</mo> <mi>η</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> <mspace width="2em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</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>+</mo> <mi>η</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }+\eta m\omega K\qquad \mathrm {D} _{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4685c7866fca38b0be069124b405f69fcae382af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.963ex; height:3.176ex;" alt="{\displaystyle \mathrm {D} _{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }+\eta m\omega K\qquad \mathrm {D} _{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K}"></span></dd></dl> <p>As was previously shown </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}=\mathrm {D} _{\rm {L}}\mathrm {D} _{\rm {R}}=-\left(\square +m^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>◻</mi> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}=\mathrm {D} _{\rm {L}}\mathrm {D} _{\rm {R}}=-\left(\square +m^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81968a4f96794206494f641cd2d5d72beeeb000f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.77ex; height:3.343ex;" alt="{\displaystyle \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}=\mathrm {D} _{\rm {L}}\mathrm {D} _{\rm {R}}=-\left(\square +m^{2}\right)}"></span></dd></dl> <p>The four-component spinor requires the introduction of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }-\eta m\omega K\qquad \delta _{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }-\eta m\omega K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <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> <mo>−</mo> <mi>η</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> <mspace width="2em" /> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</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>−</mo> <mi>η</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }-\eta m\omega K\qquad \delta _{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }-\eta m\omega K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd159b141c7d0513cd14fa83bbc668af21057d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.476ex; height:3.176ex;" alt="{\displaystyle \delta _{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }-\eta m\omega K\qquad \delta _{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }-\eta m\omega K}"></span></dd></dl> <p>which also obey </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {R}}\delta _{\rm {L}}=\delta _{\rm {L}}\delta _{\rm {R}}=-\left(\square +m^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>◻</mi> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {R}}\delta _{\rm {L}}=\delta _{\rm {L}}\delta _{\rm {R}}=-\left(\square +m^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92cde0a5524d38ea67e296da2b4a35a69e608a2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.797ex; height:3.343ex;" alt="{\displaystyle \delta _{\rm {R}}\delta _{\rm {L}}=\delta _{\rm {L}}\delta _{\rm {R}}=-\left(\square +m^{2}\right)}"></span></dd></dl> <p>Therefore </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathrm {D} _{\rm {R}}\oplus \delta _{\rm {L}}\right)\left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=-\left(\square +m^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>◻</mi> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathrm {D} _{\rm {R}}\oplus \delta _{\rm {L}}\right)\left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=-\left(\square +m^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be3a07efbea564cf7bd3955392569383d1c2d1d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.872ex; height:3.343ex;" alt="{\displaystyle \left(\mathrm {D} _{\rm {R}}\oplus \delta _{\rm {L}}\right)\left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=-\left(\square +m^{2}\right)}"></span></dd></dl> <p>The chiral representation requires an extra factor 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 \beta =\gamma ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =\gamma ^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/605c52968b1faf2a6c5cf9f9022caf80f1b68d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.765ex; height:3.176ex;" alt="{\displaystyle \beta =\gamma ^{0}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\qquad \beta \left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mspace width="2em" /> <mi>β</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\qquad \beta \left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e6d7fbd335fda741af881bc24c1313bcf8d891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:58.858ex; height:3.009ex;" alt="{\displaystyle \beta \left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\qquad \beta \left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}}"></span></dd></dl> <p>and so one concludes that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathrm {D} _{\rm {R}}\oplus \delta _{\rm {L}}\right)\left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=\left(i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}\right)\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)=-\left(\square +m^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>◻</mi> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathrm {D} _{\rm {R}}\oplus \delta _{\rm {L}}\right)\left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=\left(i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}\right)\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)=-\left(\square +m^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece18671ba12e73ba65388f01838ffc0c5df1d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:67.235ex; height:3.343ex;" alt="{\displaystyle \left(\mathrm {D} _{\rm {R}}\oplus \delta _{\rm {L}}\right)\left(\mathrm {D} _{\rm {L}}\oplus \delta _{\rm {R}}\right)=\left(i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}\right)\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)=-\left(\square +m^{2}\right)}"></span></dd></dl> <p>That is, both eigenvectors of the charge conjugation operator solve the Klein–Gordon equation. The last identity can also be verified directly, by noting that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}i=-i{\mathsf {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mi>i</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}i=-i{\mathsf {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33bd9db7788a4c90fd5050673f0ac526323e42cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.482ex; height:2.343ex;" alt="{\displaystyle {\mathsf {C}}i=-i{\mathsf {C}}}"></span> and that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}\gamma ^{\mu }=-\gamma ^{\mu }{\mathsf {C}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}\gamma ^{\mu }=-\gamma ^{\mu }{\mathsf {C}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/853f8290f1804c0372f136cfb8bfebda9ca6e4a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.111ex; height:2.843ex;" alt="{\displaystyle {\mathsf {C}}\gamma ^{\mu }=-\gamma ^{\mu }{\mathsf {C}}~.}"></span> </p> </div> <div class="mw-heading mw-heading3"><h3 id="Parity">Parity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=14" title="Edit section: Parity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under parity, the left-handed spinors transform to right-handed spinors. The two eigenvectors of the charge conjugation operator, again in the Weyl basis, are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {{R},{\text{Weyl}}}}^{(\pm )}={\begin{pmatrix}\pm \eta \omega \psi _{\rm {R}}^{*}\\\psi _{\rm {R}}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Weyl</mtext> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>±</mo> <mi>η</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {{R},{\text{Weyl}}}}^{(\pm )}={\begin{pmatrix}\pm \eta \omega \psi _{\rm {R}}^{*}\\\psi _{\rm {R}}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c6a87b770b795cd4ccf714faffca7c072ef810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.806ex; height:6.509ex;" alt="{\displaystyle \psi _{\rm {{R},{\text{Weyl}}}}^{(\pm )}={\begin{pmatrix}\pm \eta \omega \psi _{\rm {R}}^{*}\\\psi _{\rm {R}}\end{pmatrix}}}"></span></dd></dl> <p>As before, both solve the four-component Majorana equation, but only one also solves the Dirac equation. This can be shown by constructing the parity-dual four-component equation. This takes the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81fd74953b0645199f825944ee7fd97cbfa0df5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.106ex; height:3.009ex;" alt="{\displaystyle \beta \left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }-\eta m\omega K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <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> <mo>−</mo> <mi>η</mi> <mi>m</mi> <mi>ω</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }-\eta m\omega K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9ecc51983fcfab8d8519ee338cff0f9de2e596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.882ex; height:3.176ex;" alt="{\displaystyle \delta _{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }-\eta m\omega K}"></span></dd></dl> <p>Given the two-component spinor <span class="mwe-math-element"><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 _{\rm {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2219126634be48c155bbdeea3615c335446df23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.955ex; height:2.509ex;" alt="{\displaystyle \psi _{\rm {R}}}"></span> define its conjugate 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 \chi _{\rm {L}}=-\eta \omega \psi _{\rm {R}}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\rm {L}}=-\eta \omega \psi _{\rm {R}}^{*}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6a0416a0761267b3de6ba43d60ba3e1e74a11c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.838ex; height:2.843ex;" alt="{\displaystyle \chi _{\rm {L}}=-\eta \omega \psi _{\rm {R}}^{*}.}"></span> It is not hard to show 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 \mathrm {D} _{\rm {R}}\psi _{\rm {R}}=-\eta \omega (\delta _{\rm {L}}\chi _{\rm {L}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>η</mi> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\rm {R}}\psi _{\rm {R}}=-\eta \omega (\delta _{\rm {L}}\chi _{\rm {L}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce6838c233c7f53daba573348b7711cab3c6c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.51ex; height:2.843ex;" alt="{\displaystyle \mathrm {D} _{\rm {R}}\psi _{\rm {R}}=-\eta \omega (\delta _{\rm {L}}\chi _{\rm {L}})}"></span> and that therefore, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{\rm {R}}\psi _{\rm {R}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{\rm {R}}\psi _{\rm {R}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a28483a9cac44ea6a40e89f993a89b497819e663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.433ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{\rm {R}}\psi _{\rm {R}}=0}"></span> then also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {L}}\chi _{\rm {L}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {L}}\chi _{\rm {L}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69468fbadd0273783c279409fbee55e39671fea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.267ex; height:2.676ex;" alt="{\displaystyle \delta _{\rm {L}}\chi _{\rm {L}}=0}"></span> and therefore that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)\left(\chi _{\rm {L}}\oplus \psi _{\rm {R}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)\left(\chi _{\rm {L}}\oplus \psi _{\rm {R}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256e32f79294668ff37f1afcbfa63951c1ec10de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.126ex; height:2.843ex;" alt="{\displaystyle 0=\left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)\left(\chi _{\rm {L}}\oplus \psi _{\rm {R}}\right)}"></span></dd></dl> <p>or equivalently </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=(i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}){\begin{pmatrix}\chi _{\rm {L}}\\\psi _{\rm {R}}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=(i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}){\begin{pmatrix}\chi _{\rm {L}}\\\psi _{\rm {R}}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734633a6962ba2cc07123c2df4f53df37893693a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.328ex; height:6.176ex;" alt="{\displaystyle 0=(i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}){\begin{pmatrix}\chi _{\rm {L}}\\\psi _{\rm {R}}\end{pmatrix}}}"></span></dd></dl> <p>This works, 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 {\mathsf {C}}(\chi _{\rm {L}}\oplus \psi _{\rm {R}})=-(\chi _{\rm {L}}\oplus \psi _{\rm {R}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}(\chi _{\rm {L}}\oplus \psi _{\rm {R}})=-(\chi _{\rm {L}}\oplus \psi _{\rm {R}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2ef4b66ee96b439d322c41fdd510e03ca42a54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.031ex; height:2.843ex;" alt="{\displaystyle {\mathsf {C}}(\chi _{\rm {L}}\oplus \psi _{\rm {R}})=-(\chi _{\rm {L}}\oplus \psi _{\rm {R}})}"></span> and so this reduces to the Dirac equation for </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{{\rm {R}},{\text{Weyl}}}^{(-)}=\chi _{\rm {L}}\oplus \psi _{\rm {R}}={\begin{pmatrix}\chi _{\rm {L}}\\\psi _{\rm {R}}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Weyl</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{{\rm {R}},{\text{Weyl}}}^{(-)}=\chi _{\rm {L}}\oplus \psi _{\rm {R}}={\begin{pmatrix}\chi _{\rm {L}}\\\psi _{\rm {R}}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8de983fcf2a701d1ea66c9bc69f431f4c63ac2ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.991ex; height:6.176ex;" alt="{\displaystyle \psi _{{\rm {R}},{\text{Weyl}}}^{(-)}=\chi _{\rm {L}}\oplus \psi _{\rm {R}}={\begin{pmatrix}\chi _{\rm {L}}\\\psi _{\rm {R}}\end{pmatrix}}}"></span></dd></dl> <p>To conclude, and reiterate, the Majorana equation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi =i\gamma ^{\mu }\partial _{\mu }\psi -m\psi _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>=</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> <mi>ψ</mi> <mo>−</mo> <mi>m</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi =i\gamma ^{\mu }\partial _{\mu }\psi -m\psi _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/442ac01b1ff0534d9b0f174c72c848db64bd2063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.813ex; height:3.009ex;" alt="{\displaystyle 0=\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi =i\gamma ^{\mu }\partial _{\mu }\psi -m\psi _{c}}"></span></dd></dl> <p>It has four inequivalent, linearly independent solutions, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {L,R}}^{(\pm )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mo>,</mo> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {L,R}}^{(\pm )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a596e32eed00a4ea92cafd2ba827aff6b0e3815" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.086ex; height:4.009ex;" alt="{\displaystyle \psi _{\rm {L,R}}^{(\pm )}.}"></span> Of these, only two are also solutions to the Dirac equation: namely <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {L}}^{(+)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {L}}^{(+)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb81e9a74bd2f27f6b2e89b5439fb98f9d0823a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.303ex; height:3.676ex;" alt="{\displaystyle \psi _{\rm {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 _{\rm {R}}^{(-)}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {R}}^{(-)}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa7df1db9e0a4047244ed710b1c39fc15bbaf89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.531ex; height:3.676ex;" alt="{\displaystyle \psi _{\rm {R}}^{(-)}~.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Solutions">Solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=15" title="Edit section: Solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Spin_eigenstates">Spin eigenstates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=16" title="Edit section: Spin eigenstates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One convenient starting point for writing the solutions is to work in the rest frame way of the spinors. Writing the quantum Hamiltonian with the conventional sign convention <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=i\partial _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>i</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=i\partial _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc8393929679271d7788a4a7558e1f12f86dad5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.025ex; height:2.509ex;" alt="{\displaystyle H=i\partial _{t}}"></span> leads to the Majorana equation taking the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\partial _{t}\psi =-i{\vec {\alpha }}\cdot \nabla \psi +m\beta \psi _{c}}"> <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>t</mi> </mrow> </msub> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>ψ</mi> <mo>+</mo> <mi>m</mi> <mi>β</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\partial _{t}\psi =-i{\vec {\alpha }}\cdot \nabla \psi +m\beta \psi _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf58b60ee07b63dcdb6139fd295b7a7f3ede205f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.371ex; height:2.676ex;" alt="{\displaystyle i\partial _{t}\psi =-i{\vec {\alpha }}\cdot \nabla \psi +m\beta \psi _{c}}"></span></dd></dl> <p>In the chiral (Weyl) basis, one has that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{0}=\beta ={\begin{pmatrix}0&I\\I&0\end{pmatrix}},\quad {\vec {\alpha }}={\begin{pmatrix}{\vec {\sigma }}&0\\0&-{\vec {\sigma }}\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> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>I</mi> </mtd> </mtr> <mtr> <mtd> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <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> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{0}=\beta ={\begin{pmatrix}0&I\\I&0\end{pmatrix}},\quad {\vec {\alpha }}={\begin{pmatrix}{\vec {\sigma }}&0\\0&-{\vec {\sigma }}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f419ff875c264a0c4f3742abc9d55cc996fdb16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.607ex; height:6.176ex;" alt="{\displaystyle \gamma ^{0}=\beta ={\begin{pmatrix}0&I\\I&0\end{pmatrix}},\quad {\vec {\alpha }}={\begin{pmatrix}{\vec {\sigma }}&0\\0&-{\vec {\sigma }}\end{pmatrix}}}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2fc4b8da9e315ad0648308ed315c4087ec679c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {\sigma }}}"></span> the <a href="/wiki/Pauli_vector" class="mw-redirect" title="Pauli vector">Pauli vector</a>. The sign convention here is consistent with the article <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>. Plugging in the positive charge conjugation eigenstate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\text{Weyl}}^{(+)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Weyl</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\text{Weyl}}^{(+)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/187a4c42dbd59beeaee75f1a8096fd6bf0538186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.49ex; height:4.009ex;" alt="{\displaystyle \psi _{\text{Weyl}}^{(+)}}"></span> given above, one obtains an equation for the two-component spinor </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\partial _{t}\psi _{\rm {L}}=-i{\vec {\sigma }}\cdot \nabla \psi _{\rm {L}}+m(i\sigma _{2}\psi _{\rm {L}}^{*})}"> <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>t</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\partial _{t}\psi _{\rm {L}}=-i{\vec {\sigma }}\cdot \nabla \psi _{\rm {L}}+m(i\sigma _{2}\psi _{\rm {L}}^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bac8497f6556136f3c032f69182b92bd50848a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.709ex; height:3.009ex;" alt="{\displaystyle i\partial _{t}\psi _{\rm {L}}=-i{\vec {\sigma }}\cdot \nabla \psi _{\rm {L}}+m(i\sigma _{2}\psi _{\rm {L}}^{*})}"></span></dd></dl> <p>and likewise </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\partial _{t}(i\sigma _{2}\psi _{\rm {L}}^{*})=+i{\vec {\sigma }}\cdot \nabla (i\sigma _{2}\psi _{\rm {L}}^{*})+m\psi _{\rm {L}}}"> <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>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>m</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\partial _{t}(i\sigma _{2}\psi _{\rm {L}}^{*})=+i{\vec {\sigma }}\cdot \nabla (i\sigma _{2}\psi _{\rm {L}}^{*})+m\psi _{\rm {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312af0c0b9444b5ac588bb36e82cd5ae04986ca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.702ex; height:3.009ex;" alt="{\displaystyle i\partial _{t}(i\sigma _{2}\psi _{\rm {L}}^{*})=+i{\vec {\sigma }}\cdot \nabla (i\sigma _{2}\psi _{\rm {L}}^{*})+m\psi _{\rm {L}}}"></span></dd></dl> <p>These two are in fact the same equation, which can be verified by noting 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 \sigma _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d4b9cd9efc54bcfd04e0a2231913c13f10798d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="{\displaystyle \sigma _{2}}"></span> yields the complex conjugate of the Pauli matrices: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2}\left({\vec {k}}\cdot {\vec {\sigma }}\right)\sigma _{2}=-{\vec {k}}\cdot {\vec {\sigma }}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <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"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{2}\left({\vec {k}}\cdot {\vec {\sigma }}\right)\sigma _{2}=-{\vec {k}}\cdot {\vec {\sigma }}^{*}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa6815e7ce1d14548dd55b18d59a89931aaafdf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.361ex; height:4.843ex;" alt="{\displaystyle \sigma _{2}\left({\vec {k}}\cdot {\vec {\sigma }}\right)\sigma _{2}=-{\vec {k}}\cdot {\vec {\sigma }}^{*}.}"></span></dd></dl> <p>The <a href="/wiki/Plane_wave" title="Plane wave">plane wave</a> solutions can be developed for the energy-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 \left(k_{0},{\vec {k}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(k_{0},{\vec {k}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22201f2df02bc4cf718427a6ecdddf07ecf32b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.286ex; height:4.843ex;" alt="{\displaystyle \left(k_{0},{\vec {k}}\right)}"></span> and are most easily stated in the rest frame. The spin-up rest-frame solution is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {L}}^{(u)}={\begin{pmatrix}e^{-imt}\\e^{imt}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {L}}^{(u)}={\begin{pmatrix}e^{-imt}\\e^{imt}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ab5061569241618edd2220e357b8b701bf6a83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.434ex; height:6.176ex;" alt="{\displaystyle \psi _{\rm {L}}^{(u)}={\begin{pmatrix}e^{-imt}\\e^{imt}\end{pmatrix}}}"></span></dd></dl> <p>while the spin-down solution is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {L}}^{(d)}={\begin{pmatrix}e^{imt}\\-e^{-imt}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {L}}^{(d)}={\begin{pmatrix}e^{imt}\\-e^{-imt}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bab86816d8bd15b9638d81689d4804f89ba9036" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.162ex; height:6.176ex;" alt="{\displaystyle \psi _{\rm {L}}^{(d)}={\begin{pmatrix}e^{imt}\\-e^{-imt}\end{pmatrix}}}"></span></dd></dl> <p>That these are being correctly interpreted can be seen by re-expressing them in the Dirac basis, as <a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinors</a>. In this case, they take the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\text{Dirac}}^{(u)}={\begin{bmatrix}e^{-imt}\\0\\0\\-e^{imt}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Dirac</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\text{Dirac}}^{(u)}={\begin{bmatrix}e^{-imt}\\0\\0\\-e^{imt}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6982e0f5f540b7d55438105755f2504220f49a87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:18.333ex; height:12.843ex;" alt="{\displaystyle \psi _{\text{Dirac}}^{(u)}={\begin{bmatrix}e^{-imt}\\0\\0\\-e^{imt}\end{bmatrix}}}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\text{Dirac}}^{(d)}={\begin{bmatrix}0\\e^{-imt}\\-e^{imt}\\0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Dirac</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\text{Dirac}}^{(d)}={\begin{bmatrix}0\\e^{-imt}\\-e^{imt}\\0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fec500017d69f6626ec2a274309407ed4a5d097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:18.333ex; height:12.843ex;" alt="{\displaystyle \psi _{\text{Dirac}}^{(d)}={\begin{bmatrix}0\\e^{-imt}\\-e^{imt}\\0\end{bmatrix}}}"></span></dd></dl> <p>These are the rest-frame spinors. They can be seen as a linear combination of both the positive and the negative-energy solutions to the Dirac equation. These are the only two solutions; the Majorana equation has only two linearly independent solutions, unlike the Dirac equation, which has four. The doubling of the degrees of freedom of the Dirac equation can be ascribed to the Dirac spinors carrying charge. </p> <div class="mw-heading mw-heading3"><h3 id="Momentum_eigenstates">Momentum eigenstates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=17" title="Edit section: Momentum eigenstates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a general momentum frame, the Majorana spinor can be written as </p> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Majorana_equation&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">December 2020</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Electric_charge">Electric charge</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=18" title="Edit section: Electric charge"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The appearance of 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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" 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="{\textstyle \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="{\textstyle \psi _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a758793d1e507d42dd596f63b5a667d75b041f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.457ex; height:2.509ex;" alt="{\textstyle \psi _{c}}"></span> in the Majorana equation means that the field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" 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="{\textstyle \psi }"></span> cannot be coupled to a charged <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> without violating <a href="/wiki/Charge_conservation" title="Charge conservation">charge conservation</a>, since particles have the opposite charge to their own antiparticles. To satisfy this restriction, <span class="mwe-math-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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" 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="{\textstyle \psi }"></span> must be taken to be electrically neutral. This can be articulated in greater detail. </p><p>The Dirac equation can be written in a purely real form, when the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a> are taken in the Majorana representation. The Dirac equation can then be written as<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-i{\frac {\partial }{\partial t}}-i{\hat {\alpha }}\cdot \nabla +\beta m\right)\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo>+</mo> <mi>β</mi> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(-i{\frac {\partial }{\partial t}}-i{\hat {\alpha }}\cdot \nabla +\beta m\right)\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d5caa37b12ed236a1b942ea04f413cca492d7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.145ex; height:6.176ex;" alt="{\displaystyle \left(-i{\frac {\partial }{\partial t}}-i{\hat {\alpha }}\cdot \nabla +\beta m\right)\psi =0}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682d943d1947245b587f282aba6c88f0870fb302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.176ex;" alt="{\displaystyle {\hat {\alpha }}}"></span> being purely real symmetric matrices, 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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> being purely imaginary skew-symmetric. In this case, purely real solutions to the equation can be found; these are the Majorana spinors. Under the action of <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a>, these transform under the (purely real) <a href="/wiki/Spin_group" title="Spin group">spin group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spin} (1,3).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spin} (1,3).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de86285b6994ebc0af22a287f58735ae289e1c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.339ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spin} (1,3).}"></span> This stands in contrast to the <a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinors</a>, which are only covariant under the action of the complexified spin 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 \operatorname {Spin} ^{\mathbb {C} }(1,3).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Spin</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spin} ^{\mathbb {C} }(1,3).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0ee04317d2d5a9dae4655bbaafda507983238a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.758ex; height:3.176ex;" alt="{\displaystyle \operatorname {Spin} ^{\mathbb {C} }(1,3).}"></span> The interpretation is that complexified spin group encodes the electromagnetic potential, the real spin group does not. </p><p>This can also be stated in a different way: the Dirac equation, and the Dirac spinors contain a sufficient amount of gauge freedom to naturally encode electromagnetic interactions. This can be seen by noting that the electromagnetic potential can very simply be added to the Dirac equation without requiring any additional modifications or extensions to either the equation or the spinor. The location of this extra degree of freedom is pin-pointed by the charge conjugation operator, and the imposition of the Majorana constraint <span class="mwe-math-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 \psi =\psi _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi =\psi _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/499fa44e13e607ff2bca59211b5fdd7d31feb7e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.069ex; height:2.509ex;" alt="{\textstyle \psi =\psi _{c}}"></span> removes this extra degree of freedom. Once removed, there cannot be any coupling to the electromagnetic potential, ergo, the Majorana spinor is necessarily electrically neutral. An electromagnetic coupling can only be obtained by adding back in a complex-number-valued phase factor, and coupling this phase factor to the electromagnetic potential. </p><p>The above can be further sharpened by examining the situation 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 (p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9769c58523b9b639866a2d48e657d9c26911143a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle (p,q)}"></span> spatial dimensions. In this case, the complexified spin 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 \operatorname {Spin} ^{\mathbb {C} }(p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Spin</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spin} ^{\mathbb {C} }(p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85fc68b5db86e3eba1313402012201e22031894c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.025ex; height:3.176ex;" alt="{\displaystyle \operatorname {Spin} ^{\mathbb {C} }(p,q)}"></span> has a <a href="/wiki/Cover_(mathematics)" class="mw-redirect" title="Cover (mathematics)">double covering</a> 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 \operatorname {SO} (p,q)\times S^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SO} (p,q)\times S^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c87f2bfc675a18ad83f600821c4adbf02d98ecb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.599ex; height:3.176ex;" alt="{\displaystyle \operatorname {SO} (p,q)\times S^{1}}"></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 S^{1}\cong U(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>≅</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{1}\cong U(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c26c5197917d290366aa3b3f22208a6eb3daf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.429ex; height:3.176ex;" alt="{\displaystyle S^{1}\cong U(1)}"></span> the circle. The implication is 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 \operatorname {SO} (p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SO} (p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77fdf243cfbc547960ebb0e93766d54c57a3f0e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.183ex; height:2.843ex;" alt="{\displaystyle \operatorname {SO} (p,q)}"></span> encodes the generalized Lorentz transformations (of course), while the circle can be identified 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 \mathrm {U} (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {U} (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e002937bc66667b3508046146eb651b4ffe68b1" 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 \mathrm {U} (1)}"></span> action of the gauge group on electric charges. That is, the gauge-group action of the complexified spin group on a Dirac spinor can be split into a purely-real Lorentzian part, and an electromagnetic part. This can be further elaborated on non-flat (non-Minkowski-flat) <a href="/wiki/Spin_manifold" class="mw-redirect" title="Spin manifold">spin manifolds</a>. In this case, the <a href="/wiki/Dirac_operator" title="Dirac operator">Dirac operator</a> acts on the <a href="/wiki/Spinor_bundle" title="Spinor bundle">spinor bundle</a>. Decomposed into distinct terms, it includes the usual 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+A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>+</mo> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d+A.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf61fa16a27f6cb4af9eeb0c22fe318e12a18f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.446ex; height:2.343ex;" alt="{\displaystyle d+A.}"></span> The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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> field can be seen to arise directly from the curvature of the complexified part of the spin bundle, in that the gauge transformations couple to the complexified part, and not the real-spinor part. 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 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> field corresponds to the electromagnetic potential can be seen by noting that (for example) the square of the Dirac operator is the Laplacian plus the <a href="/wiki/Scalar_curvature" title="Scalar curvature">scalar curvature</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> (of the underlying manifold that the spinor field sits on) plus the (electromagnetic) field strength <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=dA.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>d</mi> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=dA.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6ce7ceb67ec8d20a82e442025835f05182df12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.445ex; height:2.176ex;" alt="{\displaystyle F=dA.}"></span> For the Majorana case, one has only the Lorentz transformations acting on the Majorana spinor; the complexification plays no role. A detailed treatment of these topics can be found in Jost<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> while the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,q)=(1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p,q)=(1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cea63006c0d34f045e07917d6bceb3312085c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.349ex; height:2.843ex;" alt="{\displaystyle (p,q)=(1,3)}"></span> case is articulated in Bleeker.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Unfortunately, neither text explicitly articulates the Majorana spinor in direct form. </p> <div class="mw-heading mw-heading2"><h2 id="Field_quanta">Field quanta</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=19" title="Edit section: Field quanta"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The quanta of the Majorana equation allow for two classes of particles, a neutral particle and its neutral <a href="/wiki/Antiparticle" title="Antiparticle">antiparticle</a>. The frequently applied supplemental condition <span class="mwe-math-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 \Psi =\Psi _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Psi =\Psi _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb37f696537c76c2f23f6f4cae80e9b64b8e3a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.659ex; height:2.509ex;" alt="{\textstyle \Psi =\Psi _{c}}"></span> corresponds to the Majorana spinor. </p> <div class="mw-heading mw-heading3"><h3 id="Majorana_particle">Majorana particle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=20" title="Edit section: Majorana particle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Majorana_particle" class="mw-redirect" title="Majorana particle">Majorana particle</a></div> <p>Particles corresponding to Majorana spinors are known as <a href="/wiki/Majorana_particle" class="mw-redirect" title="Majorana particle">Majorana particles</a>, due to the above self-conjugacy constraint. All the fermions included in the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> have been excluded as <a href="/wiki/Majorana_fermion" title="Majorana fermion">Majorana fermions</a> (since they have non-zero electric charge they cannot be antiparticles of themselves) with the exception of the <a href="/wiki/Neutrino" title="Neutrino">neutrino</a> (which is neutral). </p><p>Theoretically, the neutrino is a possible exception to this pattern. If so, <a href="/wiki/Double_beta_decay#Neutrinoless_double-beta_decay" title="Double beta decay">neutrinoless double-beta decay</a>, as well as a range of lepton-number violating <a href="/wiki/Meson" title="Meson">meson</a> and charged <a href="/wiki/Lepton" title="Lepton">lepton</a> decays, are possible. A number of experiments probing whether the neutrino is a Majorana particle are currently underway.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=21" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Caution: Not all authors use the same conventions for charge conjugation, and so there is plenty of room for subtle sign errors. This article, and the article on <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a>, use the conventions of Itzykson & Zuber, (<i>Quantum Field Theory</i>, see Chapter 2 and Appendix A). These differ very slightly from Bjorken & Drell <i>Relativistic Quantum Mechanics</i> and so allowances must be made if comparing the two.</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">The results presented here are identical to those of Aste, <i>op. cit.</i>, equations 52 and 57, although the derivation performed here is completely different. The double-covering used here is also identical to Aste equations 48, and to the current version (December 2020) of the article on <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>.</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"> Itzykson and Zuber, <i>op. cit.</i> <i>(Chapter 2-4)</i></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"> Itzykson & Zuber, <i>(See Chapter 2-1-2, page 49)</i></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> Ettore Majorana, "Teoria Simmetrica Dell' Elettrone E Del Positrone," <i>Nuovo Cimento</i> <b>14</b> (1937) pp.171–184. <i><a rel="nofollow" class="external text" href="http://fisica.unipv.it/percorsi/pdf/NCMajorana1937_14_171.pdf">PDF Original Italian version</a></i></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"> <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="CITEREFAste2010" class="citation journal cs1">Aste, Andreas (2010). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fsym2041776">"A direct road to Majorana fields"</a>. <i>Symmetry</i>. <b>2010</b> (2): 1776–1809. <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/0806.1690">0806.1690</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/2010Symm....2.1776A">2010Symm....2.1776A</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/10.3390%2Fsym2041776">10.3390/sym2041776</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Symmetry&rft.atitle=A+direct+road+to+Majorana+fields&rft.volume=2010&rft.issue=2&rft.pages=1776-1809&rft.date=2010&rft_id=info%3Aarxiv%2F0806.1690&rft_id=info%3Adoi%2F10.3390%2Fsym2041776&rft_id=info%3Abibcode%2F2010Symm....2.1776A&rft.aulast=Aste&rft.aufirst=Andreas&rft_id=https%3A%2F%2Fdoi.org%2F10.3390%252Fsym2041776&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMajorana+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="CITEREFPal2011" class="citation journal cs1">Pal, Palash B. (2011). "Dirac, Majorana, and Weyl fermions". <i>American Journal of Physics</i>. <b>79</b> (5): 485–498. <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/1006.1718">1006.1718</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/2011AmJPh..79..485P">2011AmJPh..79..485P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.3549729">10.1119/1.3549729</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:118685467">118685467</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Dirac%2C+Majorana%2C+and+Weyl+fermions&rft.volume=79&rft.issue=5&rft.pages=485-498&rft.date=2011&rft_id=info%3Aarxiv%2F1006.1718&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118685467%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.3549729&rft_id=info%3Abibcode%2F2011AmJPh..79..485P&rft.aulast=Pal&rft.aufirst=Palash+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMajorana+equation" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarsch2012" class="citation journal cs1">Marsch, Eckart (2012). <a rel="nofollow" class="external text" href="https://doi.org/10.5402%2F2012%2F760239">"On the Majorana equation: Relations between its complex two-component and real four-component eigenfunctions"</a>. <i>ISRN Mathematical Physics</i>. <b>2012</b>: 1–17. <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/1207.4685">1207.4685</a></span>. <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/10.5402%2F2012%2F760239">10.5402/2012/760239</a></span>. Article 760239.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ISRN+Mathematical+Physics&rft.atitle=On+the+Majorana+equation%3A+Relations+between+its+complex+two-component+and+real+four-component+eigenfunctions&rft.volume=2012&rft.pages=1-17&rft.date=2012&rft_id=info%3Aarxiv%2F1207.4685&rft_id=info%3Adoi%2F10.5402%2F2012%2F760239&rft.aulast=Marsch&rft.aufirst=Eckart&rft_id=https%3A%2F%2Fdoi.org%2F10.5402%252F2012%252F760239&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMajorana+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="CITEREFMarsch2013" class="citation journal cs1">Marsch, Eckart (2013). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/284134592">"A new route to the Majorana equation"</a>. <i>Symmetry</i>. <b>5</b> (4): 271–286. <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/2013Symm....5..271M">2013Symm....5..271M</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/10.3390%2Fsym5040271">10.3390/sym5040271</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Symmetry&rft.atitle=A+new+route+to+the+Majorana+equation&rft.volume=5&rft.issue=4&rft.pages=271-286&rft.date=2013&rft_id=info%3Adoi%2F10.3390%2Fsym5040271&rft_id=info%3Abibcode%2F2013Symm....5..271M&rft.aulast=Marsch&rft.aufirst=Eckart&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F284134592&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMajorana+equation" class="Z3988"></span></span> </li> <li id="cite_note-iz-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-iz_6-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFItzyksonZuber1980" class="citation book cs1">Itzykson, Claude; Zuber, Jean-Bernard (1980). <i>Quantum Field Theory</i>. MacGraw-Hill. §2‑1‑2, page 49.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory&rft.pages=%C2%A72%E2%80%911%E2%80%912%2C+page+49&rft.pub=MacGraw-Hill&rft.date=1980&rft.aulast=Itzykson&rft.aufirst=Claude&rft.au=Zuber%2C+Jean-Bernard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMajorana+equation" class="Z3988"></span></span> </li> <li id="cite_note-aste-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-aste_8-0">^</a></b></span> <span class="reference-text"> Andreas Aste, (2010) "A Direct Road to Majorana Fields", <i>Symmetry</i> <b>2010</b>(2) 1776-1809; doi:10.3390/sym2041776 ISSN 2073-8994.</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">Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-146100-0" title="Special:BookSources/978-0-13-146100-0">978-0-13-146100-0</a></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">The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-57507-2" title="Special:BookSources/978-0-521-57507-2">978-0-521-57507-2</a>.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books/about/An_Introduction_to_Quantum_Field_Theory.html?id=_H-oPv1raioC">An Introduction to Quantum Field Theory</a>, M.E. Peskin, D.V. Schroeder, Addison-Wesley, 1995, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-50397-2" title="Special:BookSources/0-201-50397-2">0-201-50397-2</a></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"> James D. Bjorken, Sidney D. Drell, (1964) "Relativistic Quantum Mechanics", McGraw-Hill <i>(See Chapter 5.2, pages 66-70)</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">Jurgen Jost (2002) "Riemannian geometry and Geometric Analysis (3rd edition) Springer Universitext. <i>(See chapter 1.8 for spin structures, and chapter 3.4 for the Dirac operator.)</i></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">David Bleeker, (1981) "Gauge Theory and Variational Principles" Addison-Wesley <i>(See Chapter 6 for the free Dirac field, and Chapter 7 for the interacting field).</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">A. Franklin, <i>Are There Really Neutrinos?: An Evidential History</i> (Westview Press, 2004), p. 186</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Additional_reading">Additional reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Majorana_equation&action=edit&section=23" title="Edit section: Additional reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>"<a rel="nofollow" class="external text" href="http://www.ejtp.com/majorana.html">Majorana Legacy in Contemporary Physics</a>", <i>Electronic Journal of Theoretical Physics (EJTP)</i> Volume 3, Issue 10 (April 2006) <i>Special issue for the Centenary of Ettore Majorana (1906-1938?)</i>. ISSN 1729-5254</li> <li>Frank Wilczek, (2009) "<a rel="nofollow" class="external text" href="http://www.nature.com/nphys/journal/v5/n9/full/nphys1380.html">Majorana returns</a>", <i>Nature Physics</i> Vol. <b>5</b> pages 614–618.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Majorana_equation&oldid=1184115143">https://en.wikipedia.org/w/index.php?title=Majorana_equation&oldid=1184115143</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Eponymous_equations_of_physics" title="Category:Eponymous equations of physics">Eponymous equations of physics</a></li><li><a href="/wiki/Category:Quantum_field_theory" title="Category:Quantum field theory">Quantum field theory</a></li><li><a href="/wiki/Category:Spinors" title="Category:Spinors">Spinors</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_to_be_expanded_from_December_2020" title="Category:Articles to be expanded from December 2020">Articles to be expanded from December 2020</a></li><li><a href="/wiki/Category:All_articles_to_be_expanded" title="Category:All articles to be expanded">All articles to be expanded</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 8 November 2023, at 12:19<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Majorana equation - Wikipedia