Spinor - Wikipedia

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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Clifford_algebras"> <div class="vector-toc-text"> 2.2 Clifford algebras </div> </a> <ul id="toc-Clifford_algebras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spin_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spin_groups"> <div class="vector-toc-text"> 2.3 Spin groups </div> </a> <ul id="toc-Spin_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spinor_fields_in_physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spinor_fields_in_physics"> <div class="vector-toc-text"> 2.4 Spinor fields in physics </div> </a> <ul id="toc-Spinor_fields_in_physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spinors_in_representation_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spinors_in_representation_theory"> <div class="vector-toc-text"> 2.5 Spinors in representation theory </div> </a> <ul id="toc-Spinors_in_representation_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Attempts_at_intuitive_understanding" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Attempts_at_intuitive_understanding"> <div class="vector-toc-text"> 2.6 Attempts at intuitive understanding </div> </a> <ul id="toc-Attempts_at_intuitive_understanding-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 3 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 4 Examples </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples subsection </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Two_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two_dimensions"> <div class="vector-toc-text"> 4.1 Two dimensions </div> </a> <ul id="toc-Two_dimensions-sublist" class="vector-toc-list"> <li id="toc-Examples_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples_2"> <div class="vector-toc-text"> 4.1.1 Examples </div> </a> <ul id="toc-Examples_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Three_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Three_dimensions"> <div class="vector-toc-text"> 4.2 Three dimensions </div> </a> <ul id="toc-Three_dimensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Explicit_constructions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Explicit_constructions"> <div class="vector-toc-text"> 5 Explicit constructions </div> </a> <button aria-controls="toc-Explicit_constructions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Explicit constructions subsection </button> <ul id="toc-Explicit_constructions-sublist" class="vector-toc-list"> <li id="toc-Component_spinors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Component_spinors"> <div class="vector-toc-text"> 5.1 Component spinors </div> </a> <ul id="toc-Component_spinors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abstract_spinors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Abstract_spinors"> <div class="vector-toc-text"> 5.2 Abstract spinors </div> </a> <ul id="toc-Abstract_spinors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Minimal_ideals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minimal_ideals"> <div class="vector-toc-text"> 5.3 Minimal ideals </div> </a> <ul id="toc-Minimal_ideals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exterior_algebra_construction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exterior_algebra_construction"> <div class="vector-toc-text"> 5.4 Exterior algebra construction </div> </a> <ul id="toc-Exterior_algebra_construction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hermitian_vector_spaces_and_spinors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hermitian_vector_spaces_and_spinors"> <div class="vector-toc-text"> 5.5 Hermitian vector spaces and spinors </div> </a> <ul id="toc-Hermitian_vector_spaces_and_spinors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Clebsch–Gordan_decomposition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Clebsch–Gordan_decomposition"> <div class="vector-toc-text"> 6 Clebsch–Gordan decomposition </div> </a> <button aria-controls="toc-Clebsch–Gordan_decomposition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Clebsch–Gordan decomposition subsection </button> <ul id="toc-Clebsch–Gordan_decomposition-sublist" class="vector-toc-list"> <li id="toc-Even_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Even_dimensions"> <div class="vector-toc-text"> 6.1 Even dimensions </div> </a> <ul id="toc-Even_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Odd_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Odd_dimensions"> <div class="vector-toc-text"> 6.2 Odd dimensions </div> </a> <ul id="toc-Odd_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Consequences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Consequences"> <div class="vector-toc-text"> 6.3 Consequences </div> </a> <ul id="toc-Consequences-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Summary_in_low_dimensions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Summary_in_low_dimensions"> <div class="vector-toc-text"> 7 Summary in low dimensions </div> </a> <ul id="toc-Summary_in_low_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 9 Notes </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"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 10 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Works_cited" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Works_cited"> <div class="vector-toc-text"> 11 Works cited </div> </a> <ul id="toc-Works_cited-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 12 Further reading </div> </a> <ul id="toc-Further_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 vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Spinor</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" 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Available in 23 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-23" aria-hidden="true" > 23 languages </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%AF%D9%88%D9%85_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مدوم (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="مدوم (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D0%BF%D1%96%D0%BD%D0%B0%D1%80" title="Спінар – Belarusian" lang="be" hreflang="be" data-title="Спінар" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espinor" title="Espinor – Catalan" lang="ca" hreflang="ca" data-title="Espinor" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Spinor" title="Spinor – Czech" lang="cs" hreflang="cs" data-title="Spinor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Spinor" title="Spinor – German" lang="de" hreflang="de" data-title="Spinor" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%B1_%CE%BC%CE%B5_%CF%83%CF%84%CE%BF%CE%B9%CF%87%CE%B5%CE%AF%CE%B1_%CE%BC%CE%B9%CE%B3%CE%B1%CE%B4%CE%B9%CE%BA%CE%BF%CF%8D%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%8D%CF%82" title="Διανύσματα με στοιχεία μιγαδικούς αριθμούς – Greek" lang="el" hreflang="el" data-title="Διανύσματα με στοιχεία μιγαδικούς αριθμούς" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espinor" title="Espinor – Spanish" lang="es" hreflang="es" data-title="Espinor" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Spinoro" title="Spinoro – Esperanto" lang="eo" hreflang="eo" data-title="Spinoro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B3%D9%BE%DB%8C%D9%86%D9%88%D8%B1" title="اسپینور – Persian" lang="fa" hreflang="fa" data-title="اسپینور" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Spineur" title="Spineur – French" lang="fr" hreflang="fr" data-title="Spineur" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8A%A4%ED%94%BC%EB%84%88" title="스피너 – Korean" lang="ko" hreflang="ko" data-title="스피너" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spinore" title="Spinore – Italian" lang="it" hreflang="it" data-title="Spinore" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D0%BD%D0%BE%D1%80" title="Спинор – Kazakh" lang="kk" hreflang="kk" data-title="Спинор" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Spinor" title="Spinor – Dutch" lang="nl" hreflang="nl" data-title="Spinor" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B9%E3%83%94%E3%83%8E%E3%83%BC%E3%83%AB" title="スピノール – Japanese" lang="ja" hreflang="ja" data-title="スピノール" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Spinor" title="Spinor – Uzbek" lang="uz" hreflang="uz" data-title="Spinor" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%AA%E0%A8%BF%E0%A9%B1%E0%A8%A8%E0%A9%8C%E0%A8%B0" title="ਸਪਿੱਨੌਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਸਪਿੱਨੌਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Spinor" title="Spinor – Polish" lang="pl" hreflang="pl" data-title="Spinor" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D0%BD%D0%BE%D1%80" title="Спинор – Russian" lang="ru" hreflang="ru" data-title="Спинор" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Spinor" title="Spinor – Slovenian" lang="sl" hreflang="sl" data-title="Spinor" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Spinori" title="Spinori – Finnish" lang="fi" hreflang="fi" data-title="Spinori" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BF%D1%96%D0%BD%D0%BE%D1%80" title="Спінор – Ukrainian" lang="uk" hreflang="uk" data-title="Спінор" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%97%8B%E9%87%8F" title="旋量 – Chinese" lang="zh" hreflang="zh" data-title="旋量" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q604907#sitelinks-wikipedia" 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<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">Non-tensorial representation of the spin group</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spinor_on_the_circle.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Spinor_on_the_circle.png/330px-Spinor_on_the_circle.png" decoding="async" width="330" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Spinor_on_the_circle.png/495px-Spinor_on_the_circle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Spinor_on_the_circle.png/660px-Spinor_on_the_circle.png 2x" data-file-width="844" data-file-height="544" /></a><figcaption>A spinor visualized as a vector pointing along the <a href="/wiki/M%C3%B6bius_band" class="mw-redirect" title="Möbius band">Möbius band</a>, exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.<a href="#cite_note-1">[a]</a></figcaption></figure> In geometry and physics, spinors (pronounced "spinner" IPA <a href="/wiki/Help:IPA/English" title="Help:IPA/English">/spɪnər/</a>) are elements of a <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex</a> <a href="/wiki/Vector_space" title="Vector space">vector space</a> that can be associated with <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>.<a href="#cite_note-2">[b]</a> A spinor transforms linearly when the Euclidean space is subjected to a slight (<a href="/wiki/Infinitesimal_transformation" title="Infinitesimal transformation">infinitesimal</a>) rotation,<a href="#cite_note-3">[c]</a> but unlike <a href="/wiki/Euclidean_vector" title="Euclidean vector">geometric vectors</a> and <a href="/wiki/Tensor" title="Tensor">tensors</a>, a spinor transforms to its negative when the space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">sections</a> of <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundles</a> – in the case of the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> bundle of the <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a>, they thus become "square roots" of <a href="/wiki/Differential_form" title="Differential form">differential forms</a>). It is also possible to associate a substantially similar notion of spinor to <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>, in which case the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a> of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> play the role of rotations. Spinors were introduced in geometry by <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> in 1913.<a href="#cite_note-4">[1]</a><a href="#cite_note-6">[d]</a> In the 1920s physicists discovered that spinors are essential to describe the <a href="/wiki/Intrinsic_angular_momentum" class="mw-redirect" title="Intrinsic angular momentum">intrinsic angular momentum</a>, or "spin", of the <a href="/wiki/Electron" title="Electron">electron</a> and other subatomic particles.<a href="#cite_note-7">[e]</a> Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the <a href="/wiki/Rotation_group" class="mw-redirect" title="Rotation group">rotation group</a>). There are two topologically distinguishable classes (<a href="/wiki/Homotopy_class" class="mw-redirect" title="Homotopy class">homotopy classes</a>) of paths through rotations that result in the same overall rotation, as illustrated by the <a href="/wiki/Belt_trick" class="mw-redirect" title="Belt trick">belt trick</a> puzzle. These two inequivalent classes yield spinor transformations of opposite sign. The <a href="/wiki/Spin_group" title="Spin group">spin group</a> is the group of all rotations keeping track of the class.<a href="#cite_note-8">[f]</a> It doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) <a href="/wiki/Linear_representation" class="mw-redirect" title="Linear representation">linear representation</a> of the spin group, meaning that elements of the spin group <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">act</a> as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class.<a href="#cite_note-9">[g]</a> In mathematical terms, spinors are described by a double-valued <a href="/wiki/Projective_representation" title="Projective representation">projective representation</a> of the rotation group <a href="/wiki/SO(3)" class="mw-redirect" title="SO(3)">SO(3)</a>. Although spinors can be defined purely as elements of a representation space of the spin group (or its <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a>. The Clifford algebra is an <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with.<a href="#cite_note-10">[h]</a> A Clifford space operates on a spinor space, and the elements of a spinor space are spinors.<a href="#cite_note-11">[3]</a> After choosing an <a href="/wiki/Orthonormality" title="Orthonormality">orthonormal</a> basis of Euclidean space, a representation of the Clifford algebra is generated by <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the <a href="/wiki/Column_vectors" class="mw-redirect" title="Column vectors">column vectors</a> on which these matrices act. In three Euclidean dimensions, for instance, the <a href="/wiki/Pauli_spin_matrices" class="mw-redirect" title="Pauli spin matrices">Pauli spin matrices</a> are a set of gamma matrices,<a href="#cite_note-12">[i]</a> and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex<a href="#cite_note-13">[j]</a>) column vectors will either be <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible</a> if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.<a href="#cite_note-14">[k]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=1" title="Edit section: Introduction">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:396px;max-width:396px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage" style="height:146px;overflow:hidden"><a href="/wiki/File:Belt_trick_2.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Belt_trick_2.gif/200px-Belt_trick_2.gif" decoding="async" width="200" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Belt_trick_2.gif/300px-Belt_trick_2.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Belt_trick_2.gif/400px-Belt_trick_2.gif 2x" data-file-width="533" data-file-height="391" /></a></div></div><div class="tsingle" style="width:190px;max-width:190px"><div class="thumbimage" style="height:146px;overflow:hidden"><a href="/wiki/File:Belt_trick_1.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Belt_trick_1.gif/188px-Belt_trick_1.gif" decoding="async" width="188" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Belt_trick_1.gif/282px-Belt_trick_1.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Belt_trick_1.gif/376px-Belt_trick_1.gif 2x" data-file-width="514" data-file-height="402" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">A gradual rotation can be visualized as a ribbon in space.<a href="#cite_note-15">[l]</a> Two gradual rotations with different classes, one through 360° and one through 720° are illustrated here in the <a href="/wiki/Belt_trick" class="mw-redirect" title="Belt trick">belt trick</a> puzzle. A solution of the puzzle is a continuous manipulation of the belt, fixing the endpoints, that untwists it. This is impossible with the 360° rotation, but possible with the 720° rotation. A solution, shown in the second animation, gives an explicit <a href="/wiki/Homotopy" title="Homotopy">homotopy</a> in the rotation group between the 720° rotation and the 0° identity rotation.</div></div></div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Belt_Trick.ogv/220px--Belt_Trick.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="60" data-mwtitle="Belt_Trick.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Belt_Trick.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a9/Belt_Trick.ogv/Belt_Trick.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a9/Belt_Trick.ogv/Belt_Trick.ogv.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a9/Belt_Trick.ogv/Belt_Trick.ogv.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/a/a9/Belt_Trick.ogv" type="video/ogg; codecs="theora"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a9/Belt_Trick.ogv/Belt_Trick.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a9/Belt_Trick.ogv/Belt_Trick.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a9/Belt_Trick.ogv/Belt_Trick.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a9/Belt_Trick.ogv/Belt_Trick.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video><figcaption>An object attached to belts or strings can spin continuously without becoming tangled. Notice that after the cube completes a 360° rotation, the spiral is reversed from its initial configuration. The belts return to their original configuration after spinning a full 720°.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><video id="mwe_player_1" poster="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Antitwister.ogv/220px--Antitwister.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="60" data-mwtitle="Antitwister.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Antitwister.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/c/c4/Antitwister.ogv/Antitwister.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/c/c4/Antitwister.ogv/Antitwister.ogv.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/c/c4/Antitwister.ogv/Antitwister.ogv.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/c/c4/Antitwister.ogv" type="video/ogg; codecs="theora"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/c/c4/Antitwister.ogv/Antitwister.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/c/c4/Antitwister.ogv/Antitwister.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/c/c4/Antitwister.ogv/Antitwister.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/c/c4/Antitwister.ogv/Antitwister.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /></video><figcaption>A more extreme example demonstrating that this works with any number of strings. In the limit, a piece of solid continuous space can rotate in place like this without tearing or intersecting itself</figcaption></figure> What characterizes spinors and distinguishes them from <a href="/wiki/Geometric_vector" class="mw-redirect" title="Geometric vector">geometric vectors</a> and other tensors is subtle. Consider applying a rotation to the coordinates of a system. No object in the system itself has moved, only the coordinates have, so there will always be a compensating change in those coordinate values when applied to any object of the system. Geometrical vectors, for example, have components that will undergo the same rotation as the coordinates. More broadly, any <a href="/wiki/Tensor" title="Tensor">tensor</a> associated with the system (for instance, the <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">stress</a> of some medium) also has coordinate descriptions that adjust to compensate for changes to the coordinate system itself. Spinors do not appear at this level of the description of a physical system, when one is concerned only with the properties of a single isolated rotation of the coordinates. Rather, spinors appear when we imagine that instead of a single rotation, the coordinate system is gradually (<a href="/wiki/Continuous_function" title="Continuous function">continuously</a>) rotated between some initial and final configuration. For any of the familiar and intuitive ("tensorial") quantities associated with the system, the transformation law does not depend on the precise details of how the coordinates arrived at their final configuration. Spinors, on the other hand, are constructed in such a way that makes them sensitive to how the gradual rotation of the coordinates arrived there: They exhibit path-dependence. It turns out that, for any final configuration of the coordinates, there are actually two ("<a href="/wiki/Topology" title="Topology">topologically</a>") inequivalent gradual (continuous) rotations of the coordinate system that result in this same configuration. This ambiguity is called the <a href="/wiki/Homotopy_class" class="mw-redirect" title="Homotopy class">homotopy class</a> of the gradual rotation. The <a href="/wiki/Belt_trick" class="mw-redirect" title="Belt trick">belt trick</a> (shown, in which both ends of the rotated object are physically tethered to an external reference) demonstrates two different rotations, one through an angle of 2π and the other through an angle of 4π, having the same final configurations but different classes. Spinors actually exhibit a sign-reversal that genuinely depends on this homotopy class. This distinguishes them from vectors and other tensors, none of which can feel the class. Spinors can be exhibited as concrete objects using a choice of <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>. In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of <a href="/wiki/Pauli_spin_matrices" class="mw-redirect" title="Pauli spin matrices">Pauli spin matrices</a> corresponding to (<a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">angular momenta</a> about) the three coordinate axes. These are 2×2 matrices with <a href="/wiki/Complex_number" title="Complex number">complex</a> entries, and the two-component complex <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vectors</a> on which these matrices act by <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> are the spinors. In this case, the spin group is isomorphic to the group of 2×2 <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrices</a> with <a href="/wiki/Determinant" title="Determinant">determinant</a> one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves,<a href="#cite_note-16">[m]</a> realizing it as a group of rotations among them,<a href="#cite_note-18">[n]</a> but it also acts on the column vectors (that is, the spinors). More generally, a Clifford algebra can be constructed from any vector space V equipped with a (nondegenerate) <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a>, such as <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> with its standard dot product or <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> with its standard Lorentz metric. The <a href="/wiki/Spin_group" title="Spin group">space of spinors</a> is the space of column vectors with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\lfloor \dim V/2\rfloor }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>dim</mi> <mo>⁡</mo> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\lfloor \dim V/2\rfloor }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d634497e612359dce87a5d6afcfb052592d556d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.89ex; height:2.843ex;" alt="{\displaystyle 2^{\lfloor \dim V/2\rfloor }}" /> components. The orthogonal Lie algebra (i.e., the infinitesimal "rotations") and the spin group associated to the quadratic form are both (canonically) contained in the Clifford algebra, so every Clifford algebra representation also defines a representation of the Lie algebra and the spin group.<a href="#cite_note-19">[o]</a> Depending on the dimension and <a href="/wiki/Metric_signature" title="Metric signature">metric signature</a>, this realization of spinors as column vectors may be <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible</a> or it may decompose into a pair of so-called "half-spin" or Weyl representations.<a href="#cite_note-20">[p]</a> When the vector space V is four-dimensional, the algebra is described by the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>. <div class="mw-heading mw-heading2"><h2 id="Mathematical_definition">Mathematical definition</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=2" title="Edit section: Mathematical definition">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For a more elementary definition, see also: <a href="/wiki/Spinors_in_three_dimensions" title="Spinors in three dimensions">spinors in three dimensions</a></div> The space of spinors is formally defined as the <a href="/wiki/Fundamental_representation" title="Fundamental representation">fundamental representation</a> of the <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a>. (This may or may not decompose into irreducible representations.) The space of spinors may also be defined as a <a href="/wiki/Spin_representation" title="Spin representation">spin representation</a> of the <a href="/wiki/Orthogonal_Lie_algebra" class="mw-redirect" title="Orthogonal Lie algebra">orthogonal Lie algebra</a>. These spin representations are also characterized as the finite-dimensional projective representations of the special orthogonal group that do not factor through linear representations. Equivalently, a spinor is an element of a finite-dimensional <a href="/wiki/Group_representation" title="Group representation">group representation</a> of the <a href="/wiki/Spin_group" title="Spin group">spin group</a> on which the <a href="/wiki/Center_of_a_group" class="mw-redirect" title="Center of a group">center</a> acts non-trivially. <div class="mw-heading mw-heading3"><h3 id="Overview">Overview</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=3" title="Edit section: Overview">edit</a>]</div> There are essentially two frameworks for viewing the notion of a spinor: the representation theoretic point of view and the geometric point of view. <div class="mw-heading mw-heading4"><h4 id="Representation_theoretic_point_of_view">Representation theoretic point of view</h4>[<a href="/w/index.php?title=Spinor&action=edit&section=4" title="Edit section: Representation theoretic point of view">edit</a>]</div> From a <a href="/wiki/Representation_theory" title="Representation theory">representation theoretic</a> point of view, one knows beforehand that there are some representations of the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> that cannot be formed by the usual tensor constructions. These missing representations are then labeled the <a href="/wiki/Spin_representation" title="Spin representation">spin representations</a>, and their constituents spinors. From this view, a spinor must belong to a <a href="/wiki/Group_representation" title="Group representation">representation</a> of the <a href="/wiki/Covering_space" title="Covering space">double cover</a> of the <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">rotation group</a> SO(n,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />), or more generally of a double cover of the <a href="/wiki/Generalized_special_orthogonal_group" class="mw-redirect" title="Generalized special orthogonal group">generalized special orthogonal group</a> SO+(p, q, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />) on spaces with a <a href="/wiki/Metric_signature" title="Metric signature">metric signature</a> of (p, q). These double covers are <a href="/wiki/Lie_groups" class="mw-redirect" title="Lie groups">Lie groups</a>, called the <a href="/wiki/Spin_group" title="Spin group">spin groups</a> Spin(n) or Spin(p, q). All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield double-valued <a href="/wiki/Projective_representation" title="Projective representation">projective representations</a> of the groups themselves. (This means that the action of a particular rotation on vectors in the quantum Hilbert space is only defined up to a sign.) In summary, given a representation specified by the data <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V,{\text{Spin}}(p,q),\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spin</mtext> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V,{\text{Spin}}(p,q),\rho )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fef5252f27eb56775a4fc44f962d43c519a46555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.473ex; height:2.843ex;" alt="{\displaystyle (V,{\text{Spin}}(p,q),\rho )}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" /> is a vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6419d3aa99701ca996737b17a5e1174d53e6c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {R} }" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" /> is a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho :{\text{Spin}}(p,q)\rightarrow {\text{GL}}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spin</mtext> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>GL</mtext> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho :{\text{Spin}}(p,q)\rightarrow {\text{GL}}(V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7fd02e4046a42a4f4c04148716c17968ba4ac25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.234ex; height:2.843ex;" alt="{\displaystyle \rho :{\text{Spin}}(p,q)\rightarrow {\text{GL}}(V)}" />, a spinor is an element of the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" />. <div class="mw-heading mw-heading4"><h4 id="Geometric_point_of_view">Geometric point of view</h4>[<a href="/w/index.php?title=Spinor&action=edit&section=5" title="Edit section: Geometric point of view">edit</a>]</div> From a geometrical point of view, one can explicitly construct the spinors and then examine how they behave under the action of the relevant Lie groups. This latter approach has the advantage of providing a concrete and elementary description of what a spinor is. However, such a description becomes unwieldy when complicated properties of the spinors, such as <a href="/wiki/Fierz_identity" title="Fierz identity">Fierz identities</a>, are needed. <div class="mw-heading mw-heading3"><h3 id="Clifford_algebras">Clifford algebras</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=6" title="Edit section: Clifford algebras">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></div> The language of <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a><a href="#cite_note-21">[5]</a> (sometimes called <a href="/wiki/Geometric_algebra" title="Geometric algebra">geometric algebras</a>) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the <a href="/wiki/Classification_of_Clifford_algebras" title="Classification of Clifford algebras">classification of Clifford algebras</a>. It largely removes the need for ad hoc constructions. In detail, let V be a finite-dimensional complex vector space with nondegenerate symmetric bilinear form g. The Clifford algebra Cℓ(V, g) is the algebra generated by V along with the anticommutation relation xy + yx = 2g(x, y). It is an abstract version of the algebra generated by the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma</a> or <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>. If V = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}" />, with the standard form g(x, y) = xTy = x1y1 + ... + xnyn we denote the Clifford algebra by Cℓn(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />). Since by the choice of an orthonormal basis every complex vector space with non-degenerate form is isomorphic to this standard example, this notation is abused more generally if dim<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />(V) = n. If n = 2k is even, Cℓn(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />) is isomorphic as an algebra (in a non-unique way) to the algebra Mat(2k, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />) of 2k × 2k complex matrices (by the <a href="/wiki/Artin%E2%80%93Wedderburn_theorem" class="mw-redirect" title="Artin–Wedderburn theorem">Artin–Wedderburn theorem</a> and the easy to prove fact that the Clifford algebra is <a href="/wiki/Central_simple_algebra" title="Central simple algebra">central simple</a>). If n = 2k + 1 is odd, Cℓ2k+1(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />) is isomorphic to the algebra Mat(2k, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />) ⊕ Mat(2k, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />) of two copies of the 2k × 2k complex matrices. Therefore, in either case Cℓ(V, g) has a unique (up to isomorphism) irreducible representation (also called simple <a href="/wiki/Clifford_module" title="Clifford module">Clifford module</a>), commonly denoted by Δ, of dimension 2[n/2]. Since the Lie algebra so(V, g) is embedded as a Lie subalgebra in Cℓ(V, g) equipped with the Clifford algebra <a href="/wiki/Commutator" title="Commutator">commutator</a> as Lie bracket, the space Δ is also a Lie algebra representation of so(V, g) called a <a href="/wiki/Spin_representation" title="Spin representation">spin representation</a>. If n is odd, this Lie algebra representation is irreducible. If n is even, it splits further[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>] into two irreducible representations Δ = Δ+ ⊕ Δ− called the Weyl or half-spin representations. Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the <a href="/wiki/Clifford_algebra#Spinors" title="Clifford algebra">Clifford algebra</a> article for more details. <div class="mw-heading mw-heading3"><h3 id="Spin_groups">Spin groups</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=7" title="Edit section: Spin groups">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spin_representations_do_not_lift.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Spin_representations_do_not_lift.svg/220px-Spin_representations_do_not_lift.svg.png" decoding="async" width="220" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Spin_representations_do_not_lift.svg/330px-Spin_representations_do_not_lift.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Spin_representations_do_not_lift.svg/440px-Spin_representations_do_not_lift.svg.png 2x" data-file-width="118" data-file-height="114" /></a><figcaption>The spin representation Δ is a vector space equipped with a representation of the spin group that does not factor through a representation of the (special) orthogonal group. The vertical arrows depict a <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a>.</figcaption></figure> Spinors form a <a href="/wiki/Vector_space" title="Vector space">vector space</a>, usually over the <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>, equipped with a linear <a href="/wiki/Group_representation" title="Group representation">group representation</a> of the <a href="/wiki/Spin_group" title="Spin group">spin group</a> that does not factor through a representation of the group of rotations (see diagram). The spin group is the <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">group of rotations</a> keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>, but the simply connected spin group is its <a href="/wiki/Double_covering_group" class="mw-redirect" title="Double covering group">double cover</a>. So for every rotation there are two elements of the spin group that represent it. <a href="/wiki/Geometric_vector" class="mw-redirect" title="Geometric vector">Geometric vectors</a> and other <a href="/wiki/Tensor" title="Tensor">tensors</a> cannot feel the difference between these two elements, but they produce opposite signs when they affect any spinor under the representation. Thinking of the elements of the spin group as <a href="/wiki/Homotopy_classes" class="mw-redirect" title="Homotopy classes">homotopy classes</a> of one-parameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a one-parameter family of rotations is visualized as a ribbon in space, with the arc length parameter of that ribbon being the parameter (its tangent, normal, binormal frame actually gives the rotation), then these two distinct homotopy classes are visualized in the two states of the <a href="/wiki/Belt_trick" class="mw-redirect" title="Belt trick">belt trick</a> puzzle (above). The space of spinors is an auxiliary vector space that can be constructed explicitly in coordinates, but ultimately only exists up to isomorphism in that there is no "natural" construction of them that does not rely on arbitrary choices such as coordinate systems. A notion of spinors can be associated, as such an auxiliary mathematical object, with any vector space equipped with a <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> such as <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> with its standard <a href="/wiki/Dot_product" title="Dot product">dot product</a>, or <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> with its <a href="/wiki/Lorentz_metric" class="mw-redirect" title="Lorentz metric">Lorentz metric</a>. In the latter case, the "rotations" include the <a href="/wiki/Lorentz_boost" class="mw-redirect" title="Lorentz boost">Lorentz boosts</a>, but otherwise the theory is substantially similar.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Spinor_fields_in_physics">Spinor fields in physics</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=8" title="Edit section: Spinor fields in physics">edit</a>]</div> The constructions given above, in terms of Clifford algebra or representation theory, can be thought of as defining spinors as geometric objects in zero-dimensional <a href="/wiki/Space-time" class="mw-redirect" title="Space-time">space-time</a>. To obtain the spinors of physics, such as the <a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinor</a>, one extends the construction to obtain a <a href="/wiki/Spin_structure" title="Spin structure">spin structure</a> on 4-dimensional space-time (<a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>). Effectively, one starts with the <a href="/wiki/Tangent_manifold" class="mw-redirect" title="Tangent manifold">tangent manifold</a> of space-time, each point of which is a 4-dimensional vector space with SO(3,1) symmetry, and then builds the <a href="/wiki/Spin_group" title="Spin group">spin group</a> at each point. The neighborhoods of points are endowed with concepts of smoothness and differentiability: the standard construction is one of a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a>, the fibers of which are affine spaces transforming under the spin group. After constructing the fiber bundle, one may then consider differential equations, such as the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>, or the <a href="/wiki/Weyl_equation" title="Weyl equation">Weyl equation</a> on the fiber bundle. These equations (Dirac or Weyl) have solutions that are <a href="/wiki/Plane_wave" title="Plane wave">plane waves</a>, having symmetries characteristic of the fibers, i.e. having the symmetries of spinors, as obtained from the (zero-dimensional) Clifford algebra/spin representation theory described above. Such plane-wave solutions (or other solutions) of the differential equations can then properly be called <a href="/wiki/Fermion" title="Fermion">fermions</a>; fermions have the algebraic qualities of spinors. By general convention, the terms "fermion" and "spinor" are often used interchangeably in physics, as synonyms of one-another.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] It appears that all <a href="/wiki/Fundamental_particle" class="mw-redirect" title="Fundamental particle">fundamental particles</a> in nature that are spin-1/2 are described by the Dirac equation, with the possible exception of the <a href="/wiki/Neutrino" title="Neutrino">neutrino</a>. There does not seem to be any a priori reason why this would be the case. A perfectly valid choice for spinors would be the non-complexified version of Cℓ2,2(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />), the <a href="/wiki/Majorana_spinor" class="mw-redirect" title="Majorana spinor">Majorana spinor</a>.<a href="#cite_note-22">[6]</a> There also does not seem to be any particular prohibition to having <a href="/wiki/Weyl_spinor" class="mw-redirect" title="Weyl spinor">Weyl spinors</a> appear in nature as fundamental particles. The Dirac, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra.<a href="#cite_note-23">[7]</a> Dirac and Weyl spinors are complex representations while Majorana spinors are real representations. Weyl spinors are insufficient to describe massive particles, such as <a href="/wiki/Electron" title="Electron">electrons</a>, since the Weyl plane-wave solutions necessarily travel at the speed of light; for massive particles, the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> is needed. The initial construction of the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of particle physics starts with both the electron and the neutrino as massless Weyl spinors; the <a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a> gives electrons a mass; the classical <a href="/wiki/Neutrino" title="Neutrino">neutrino</a> remained massless, and was thus an example of a Weyl spinor.<a href="#cite_note-24">[q]</a> However, because of observed <a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">neutrino oscillation</a>, it is now believed that they are not Weyl spinors, but perhaps instead Majorana spinors.<a href="#cite_note-25">[8]</a> It is not known whether Weyl spinor fundamental particles exist in nature. The situation for <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a> is different: one can construct two and three-dimensional "spacetimes" in a large variety of different physical materials, ranging from <a href="/wiki/Semiconductor" title="Semiconductor">semiconductors</a> to far more exotic materials. In 2015, an international team led by <a href="/wiki/Princeton_University" title="Princeton University">Princeton University</a> scientists announced that they had found a <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticle</a> that behaves as a Weyl fermion.<a href="#cite_note-26">[9]</a> <div class="mw-heading mw-heading3"><h3 id="Spinors_in_representation_theory">Spinors in representation theory</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=9" title="Edit section: Spinors in representation theory">edit</a>]</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/Spin_representation" title="Spin representation">Spin representation</a></div> One major mathematical application of the construction of spinors is to make possible the explicit construction of <a href="/wiki/Linear_representation" class="mw-redirect" title="Linear representation">linear representations</a> of the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a> of the <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal groups</a>, and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the <a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index theorem</a>, and to provide constructions in particular for <a href="/wiki/Discrete_series" class="mw-redirect" title="Discrete series">discrete series</a> representations of <a href="/wiki/Semisimple_group" class="mw-redirect" title="Semisimple group">semisimple groups</a>. The spin representations of the special orthogonal Lie algebras are distinguished from the <a href="/wiki/Tensor" title="Tensor">tensor</a> representations given by <a href="/wiki/Young_symmetrizer" title="Young symmetrizer">Weyl's construction</a> by the <a href="/wiki/Weight_(representation_theory)" title="Weight (representation theory)">weights</a>. Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the <a href="/wiki/Spin_representation" title="Spin representation">spin representation</a> article. <div class="mw-heading mw-heading3"><h3 id="Attempts_at_intuitive_understanding">Attempts at intuitive understanding</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=10" title="Edit section: Attempts at intuitive understanding">edit</a>]</div> The spinor can be described, in simple terms, as "vectors of a space the transformations of which are related in a particular way to rotations in physical space".<a href="#cite_note-27">[10]</a> Stated differently: <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">Spinors ... provide a linear representation of the group of <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a> in a space with any number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> of dimensions, each spinor having <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0fa1eebc5674c68166a18a335b39288ed2eb325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.266ex; height:2.343ex;" alt="{\displaystyle 2^{\nu }}" /> components where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2\nu +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mi>ν</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2\nu +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02fb584fa8bec4ac60d3eeb0b3ae8496199e1c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.891ex; height:2.343ex;" alt="{\displaystyle n=2\nu +1}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2654d0632cb4b4cd30cfddb80a75ba7d743216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.395ex; height:2.176ex;" alt="{\displaystyle 2\nu }" />.<a href="#cite_note-cartan-1966-quote-5">[2]</a></blockquote> Several ways of illustrating everyday analogies have been formulated in terms of the <a href="/wiki/Plate_trick" title="Plate trick">plate trick</a>, <a href="/wiki/Tangloids" title="Tangloids">tangloids</a> and other examples of <a href="/wiki/Orientation_entanglement" title="Orientation entanglement">orientation entanglement</a>. Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by <a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Michael Atiyah</a>'s statement that is recounted by Dirac's biographer Graham Farmelo: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote">No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the "square root" of geometry and, just as understanding the <a href="/wiki/Square_root_of_%E2%88%921" class="mw-redirect" title="Square root of −1">square root of −1</a> took centuries, the same might be true of spinors.<a href="#cite_note-28">[11]</a></blockquote> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=11" title="Edit section: History">edit</a>]</div> The most general mathematical form of spinors was discovered by <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> in 1913.<a href="#cite_note-29">[12]</a> The word "spinor" was coined by <a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Paul Ehrenfest</a> in his work on <a href="/wiki/Quantum_physics" class="mw-redirect" title="Quantum physics">quantum physics</a>.<a href="#cite_note-30">[13]</a> Spinors were first applied to <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a> by <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a> in 1927, when he introduced his <a href="/wiki/Pauli_matrices" title="Pauli matrices">spin matrices</a>.<a href="#cite_note-31">[14]</a> The following year, <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a> discovered the fully <a href="/wiki/Special_relativity" title="Special relativity">relativistic</a> theory of <a href="/wiki/Electron" title="Electron">electron</a> <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> by showing the connection between spinors and the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>.<a href="#cite_note-32">[15]</a> By the 1930s, Dirac, <a href="/wiki/Piet_Hein_(Denmark)" class="mw-redirect" title="Piet Hein (Denmark)">Piet Hein</a> and others at the <a href="/wiki/Niels_Bohr_Institute" title="Niels Bohr Institute">Niels Bohr Institute</a> (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as <a href="/wiki/Tangloids" title="Tangloids">Tangloids</a> to teach and model the calculus of spinors. Spinor spaces were represented as <a href="/wiki/Left_ideal" class="mw-redirect" title="Left ideal">left ideals</a> of a matrix algebra in 1930, by <a href="/w/index.php?title=Gustave_Juvett&action=edit&redlink=1" class="new" title="Gustave Juvett (page does not exist)">Gustave Juvett</a><a href="#cite_note-33">[16]</a> and by <a href="/wiki/Fritz_Sauter" title="Fritz Sauter">Fritz Sauter</a>.<a href="#cite_note-34">[17]</a><a href="#cite_note-lounesto-1995-p151-35">[18]</a> More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a <a href="/wiki/Minimal_ideal" title="Minimal ideal">minimal left ideal</a> in Mat(2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />).<a href="#cite_note-37">[r]</a><a href="#cite_note-lounesto-2001-p148f-p327f-38">[20]</a> In 1947 <a href="/wiki/Marcel_Riesz" title="Marcel Riesz">Marcel Riesz</a> constructed spinor spaces as elements of a minimal left ideal of <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a>. In 1966/1967, <a href="/wiki/David_Hestenes" title="David Hestenes">David Hestenes</a><a href="#cite_note-39">[21]</a><a href="#cite_note-40">[22]</a> replaced spinor spaces by the <a href="/wiki/Even_subalgebra" class="mw-redirect" title="Even subalgebra">even subalgebra</a> Cℓ01,3(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />) of the <a href="/wiki/Spacetime_algebra" title="Spacetime algebra">spacetime algebra</a> Cℓ1,3(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />).<a href="#cite_note-lounesto-1995-p151-35">[18]</a><a href="#cite_note-lounesto-2001-p148f-p327f-38">[20]</a> As of the 1980s, the theoretical physics group at <a href="/wiki/Birkbeck_College" class="mw-redirect" title="Birkbeck College">Birkbeck College</a> around <a href="/wiki/David_Bohm" title="David Bohm">David Bohm</a> and <a href="/wiki/Basil_Hiley" title="Basil Hiley">Basil Hiley</a> has been developing <a href="/wiki/Basil_Hiley#Implicate_orders,_pre-space_and_algebraic_structures" title="Basil Hiley">algebraic approaches to quantum theory</a> that build on Sauter and Riesz' identification of spinors with minimal left ideals. <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=12" title="Edit section: Examples">edit</a>]</div> Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra Cℓp, q(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />). This is an algebra built up from an orthonormal basis of n = p + q mutually orthogonal vectors under addition and multiplication, p of which have norm +1 and q of which have norm −1, with the product rule for the basis vectors <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}e_{j}={\begin{cases}+1&i=j,\,i\in (1,\ldots ,p)\\-1&i=j,\,i\in (p+1,\ldots ,n)\\-e_{j}e_{i}&i\neq j.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mi>i</mi> <mo>=</mo> <mi>j</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>i</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi>i</mi> <mo>=</mo> <mi>j</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>i</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}e_{j}={\begin{cases}+1&i=j,\,i\in (1,\ldots ,p)\\-1&i=j,\,i\in (p+1,\ldots ,n)\\-e_{j}e_{i}&i\neq j.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7bb0526bf1c69aae1ca3133e4e894e4dc46e2fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:41.277ex; height:8.843ex;" alt="{\displaystyle e_{i}e_{j}={\begin{cases}+1&i=j,\,i\in (1,\ldots ,p)\\-1&i=j,\,i\in (p+1,\ldots ,n)\\-e_{j}e_{i}&i\neq j.\end{cases}}}" /> <div class="mw-heading mw-heading3"><h3 id="Two_dimensions">Two dimensions</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=13" title="Edit section: Two dimensions">edit</a>]</div> The Clifford algebra Cℓ2,0(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ1 and σ2, and one unit <a href="/wiki/Pseudoscalar" title="Pseudoscalar">pseudoscalar</a> i = σ1σ2. From the definitions above, it is evident that (σ1)2 = (σ2)2 = 1, and (σ1σ2)(σ1σ2) = −σ1σ1σ2σ2 = −1. The even subalgebra Cℓ02,0(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />), spanned by even-graded basis elements of Cℓ2,0(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and σ1σ2. As a real algebra, Cℓ02,0(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />) is isomorphic to the field of <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />. As a result, it admits a conjugation operation (analogous to <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugation</a>), sometimes called the reverse of a Clifford element, defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b478020c54d9d0ffe746b56175967bf3058a35a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.624ex; height:2.843ex;" alt="{\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}}" /> which, by the Clifford relations, can be written <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}=a-b\sigma _{1}\sigma _{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}=a-b\sigma _{1}\sigma _{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47f5af5b9cd3885d9982b68c049b3f3631911f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.2ex; height:2.843ex;" alt="{\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}=a-b\sigma _{1}\sigma _{2}.}" /> The action of an even Clifford element γ ∈ Cℓ02,0(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />) on vectors, regarded as 1-graded elements of Cℓ2,0(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />), is determined by mapping a general vector u = a1σ1 + a2σ2 to the vector <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (u)=\gamma u\gamma ^{*},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>γ</mi> <mi>u</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (u)=\gamma u\gamma ^{*},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f390f9d95e95eecd15482cea53ae5ee26b84e24" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.073ex; height:2.843ex;" alt="{\displaystyle \gamma (u)=\gamma u\gamma ^{*},}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94d9c54dd2c77e4571ed51bd635158b22c204a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:2.843ex;" alt="{\displaystyle \gamma ^{*}}" /> is the conjugate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" />, and the product is Clifford multiplication. In this situation, a spinor<a href="#cite_note-41">[s]</a> is an ordinary complex number. The action of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" /> on a spinor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" /> is given by ordinary complex multiplication: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (\phi )=\gamma \phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>γ</mi> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (\phi )=\gamma \phi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60002d8d1af4b2d36f494ebd33c0dfc6cfaf61b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.85ex; height:2.843ex;" alt="{\displaystyle \gamma (\phi )=\gamma \phi .}" /> An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (u)=\gamma u\gamma ^{*}=\gamma ^{2}u.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>γ</mi> <mi>u</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (u)=\gamma u\gamma ^{*}=\gamma ^{2}u.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd3498f2f7f9ea565bad34197f49377406328e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.835ex; height:3.176ex;" alt="{\displaystyle \gamma (u)=\gamma u\gamma ^{*}=\gamma ^{2}u.}" /> On the other hand, in comparison with its action on spinors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (\phi )=\gamma \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>γ</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (\phi )=\gamma \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3c366d3970de98bbdca34b665682cd128133ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.203ex; height:2.843ex;" alt="{\displaystyle \gamma (\phi )=\gamma \phi }" />, the action of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" /> on ordinary vectors appears as the square of its action on spinors. Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of θ corresponds to γ2 = exp(θ σ1σ2), so that the corresponding action on spinors is via γ = ± exp(θ σ1σ2/2). In general, because of <a href="/wiki/Branch_cut" class="mw-redirect" title="Branch cut">logarithmic branching</a>, it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued. In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of language</a>, the two are often conflated. One may then talk about "the action of a spinor on a vector". In a general setting, such statements are meaningless. But in dimensions 2 and 3 (as applied, for example, to <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>) they make sense. <div class="mw-heading mw-heading4"><h4 id="Examples_2">Examples</h4>[<a href="/w/index.php?title=Spinor&action=edit&section=14" title="Edit section: Examples">edit</a>]</div> <ul><li>The even-graded element <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\tfrac {1}{\sqrt {2}}}(1-\sigma _{1}\sigma _{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\tfrac {1}{\sqrt {2}}}(1-\sigma _{1}\sigma _{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/663bab211bf447973ae626e48d4dce92f4a285e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.963ex; height:4.176ex;" alt="{\displaystyle \gamma ={\tfrac {1}{\sqrt {2}}}(1-\sigma _{1}\sigma _{2})}" /> corresponds to a vector rotation of 90° from σ1 around towards σ2, which can be checked by confirming that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}(1-\sigma _{1}\sigma _{2})\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}(1-\sigma _{2}\sigma _{1})=a_{1}\sigma _{2}-a_{2}\sigma _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}(1-\sigma _{1}\sigma _{2})\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}(1-\sigma _{2}\sigma _{1})=a_{1}\sigma _{2}-a_{2}\sigma _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f96ce85e42d006a011228f1918fdd0b513bef5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:52.576ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}(1-\sigma _{1}\sigma _{2})\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}(1-\sigma _{2}\sigma _{1})=a_{1}\sigma _{2}-a_{2}\sigma _{1}}" /> It corresponds to a spinor rotation of only 45°, however: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {2}}}(1-\sigma _{1}\sigma _{2})\{a_{1}+a_{2}\sigma _{1}\sigma _{2}\}={\frac {a_{1}+a_{2}}{\sqrt {2}}}+{\frac {-a_{1}+a_{2}}{\sqrt {2}}}\sigma _{1}\sigma _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2}}}(1-\sigma _{1}\sigma _{2})\{a_{1}+a_{2}\sigma _{1}\sigma _{2}\}={\frac {a_{1}+a_{2}}{\sqrt {2}}}+{\frac {-a_{1}+a_{2}}{\sqrt {2}}}\sigma _{1}\sigma _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb132f02019f89badb29febc13ce451ba5f4b67" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:57.098ex; height:6.009ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2}}}(1-\sigma _{1}\sigma _{2})\{a_{1}+a_{2}\sigma _{1}\sigma _{2}\}={\frac {a_{1}+a_{2}}{\sqrt {2}}}+{\frac {-a_{1}+a_{2}}{\sqrt {2}}}\sigma _{1}\sigma _{2}}" /></li> <li>Similarly the even-graded element γ = −σ1σ2 corresponds to a vector rotation of 180°: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\sigma _{1}\sigma _{2})\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}(-\sigma _{2}\sigma _{1})=-a_{1}\sigma _{1}-a_{2}\sigma _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\sigma _{1}\sigma _{2})\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}(-\sigma _{2}\sigma _{1})=-a_{1}\sigma _{1}-a_{2}\sigma _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b463faf28524b66b008b4272f3afba50eeff27" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.336ex; height:2.843ex;" alt="{\displaystyle (-\sigma _{1}\sigma _{2})\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}(-\sigma _{2}\sigma _{1})=-a_{1}\sigma _{1}-a_{2}\sigma _{2}}" /> but a spinor rotation of only 90°:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\sigma _{1}\sigma _{2})\{a_{1}+a_{2}\sigma _{1}\sigma _{2}\}=a_{2}-a_{1}\sigma _{1}\sigma _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\sigma _{1}\sigma _{2})\{a_{1}+a_{2}\sigma _{1}\sigma _{2}\}=a_{2}-a_{1}\sigma _{1}\sigma _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea10716e632f2a721157b86a5e08887cf5d17eee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.147ex; height:2.843ex;" alt="{\displaystyle (-\sigma _{1}\sigma _{2})\{a_{1}+a_{2}\sigma _{1}\sigma _{2}\}=a_{2}-a_{1}\sigma _{1}\sigma _{2}}" /></li> <li>Continuing on further, the even-graded element γ = −1 corresponds to a vector rotation of 360°: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}\,(-1)=a_{1}\sigma _{1}+a_{2}\sigma _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}\,(-1)=a_{1}\sigma _{1}+a_{2}\sigma _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea83023dfc745ea23caf62c64407372c15717c91" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.714ex; height:2.843ex;" alt="{\displaystyle (-1)\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}\,(-1)=a_{1}\sigma _{1}+a_{2}\sigma _{2}}" /> but a spinor rotation of 180°.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Three_dimensions">Three dimensions</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=15" title="Edit section: Three dimensions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Spinors_in_three_dimensions" title="Spinors in three dimensions">Spinors in three dimensions</a> and <a href="/wiki/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">Quaternions and spatial rotation</a></div> The Clifford algebra Cℓ3,0(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, <a href="/wiki/Pauli_matrices" title="Pauli matrices">σ1, σ2 and σ3</a>, the three unit bivectors σ1σ2, σ2σ3, σ3σ1 and the <a href="/wiki/Pseudoscalar" title="Pseudoscalar">pseudoscalar</a> i = σ1σ2σ3. It is straightforward to show that (σ1)2 = (σ2)2 = (σ3)2 = 1, and (σ1σ2)2 = (σ2σ3)2 = (σ3σ1)2 = (σ1σ2σ3)2 = −1. The sub-algebra of even-graded elements is made up of scalar dilations, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u'=\rho ^{\left({\frac {1}{2}}\right)}u\rho ^{\left({\frac {1}{2}}\right)}=\rho u,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mi>u</mi> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>=</mo> <mi>ρ</mi> <mi>u</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u'=\rho ^{\left({\frac {1}{2}}\right)}u\rho ^{\left({\frac {1}{2}}\right)}=\rho u,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04f2a3aa833707ea63fe10176696f9155119018" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.52ex; height:4.343ex;" alt="{\displaystyle u'=\rho ^{\left({\frac {1}{2}}\right)}u\rho ^{\left({\frac {1}{2}}\right)}=\rho u,}" /> and vector rotations <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u'=\gamma u\gamma ^{*},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>γ</mi> <mi>u</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u'=\gamma u\gamma ^{*},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65714c95b31176621ad444b13210137260b50bdc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.686ex; height:3.009ex;" alt="{\displaystyle u'=\gamma u\gamma ^{*},}" /> where <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{aligned}\gamma &=\cos \left({\frac {\theta }{2}}\right)-\{a_{1}\sigma _{2}\sigma _{3}+a_{2}\sigma _{3}\sigma _{1}+a_{3}\sigma _{1}\sigma _{2}\}\sin \left({\frac {\theta }{2}}\right)\\&=\cos \left({\frac {\theta }{2}}\right)-i\{a_{1}\sigma _{1}+a_{2}\sigma _{2}+a_{3}\sigma _{3}\}\sin \left({\frac {\theta }{2}}\right)\\&=\cos \left({\frac {\theta }{2}}\right)-iv\sin \left({\frac {\theta }{2}}\right)\end{aligned}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>i</mi> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>i</mi> <mi>v</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{aligned}\gamma &=\cos \left({\frac {\theta }{2}}\right)-\{a_{1}\sigma _{2}\sigma _{3}+a_{2}\sigma _{3}\sigma _{1}+a_{3}\sigma _{1}\sigma _{2}\}\sin \left({\frac {\theta }{2}}\right)\\&=\cos \left({\frac {\theta }{2}}\right)-i\{a_{1}\sigma _{1}+a_{2}\sigma _{2}+a_{3}\sigma _{3}\}\sin \left({\frac {\theta }{2}}\right)\\&=\cos \left({\frac {\theta }{2}}\right)-iv\sin \left({\frac {\theta }{2}}\right)\end{aligned}}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/267c7af0fda8511814176c1af9ac00c41731bd40" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:56.359ex; height:18.509ex;" alt="{\displaystyle \left.{\begin{aligned}\gamma &=\cos \left({\frac {\theta }{2}}\right)-\{a_{1}\sigma _{2}\sigma _{3}+a_{2}\sigma _{3}\sigma _{1}+a_{3}\sigma _{1}\sigma _{2}\}\sin \left({\frac {\theta }{2}}\right)\\&=\cos \left({\frac {\theta }{2}}\right)-i\{a_{1}\sigma _{1}+a_{2}\sigma _{2}+a_{3}\sigma _{3}\}\sin \left({\frac {\theta }{2}}\right)\\&=\cos \left({\frac {\theta }{2}}\right)-iv\sin \left({\frac {\theta }{2}}\right)\end{aligned}}\right\}}" /></td> <td></td> <td class="nowrap">1</td></tr></tbody></table> corresponds to a vector rotation through an angle θ about an axis defined by a unit vector v = a1σ1 + a2σ2 + a3σ3. As a special case, it is easy to see that, if v = σ3, this reproduces the σ1σ2 rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the σ3 direction invariant, since <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\cos \left({\frac {\theta }{2}}\right)-i\sigma _{3}\sin \left({\frac {\theta }{2}}\right)\right]\sigma _{3}\left[\cos \left({\frac {\theta }{2}}\right)+i\sigma _{3}\sin \left({\frac {\theta }{2}}\right)\right]=\left[\cos ^{2}\left({\frac {\theta }{2}}\right)+\sin ^{2}\left({\frac {\theta }{2}}\right)\right]\sigma _{3}=\sigma _{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[\cos \left({\frac {\theta }{2}}\right)-i\sigma _{3}\sin \left({\frac {\theta }{2}}\right)\right]\sigma _{3}\left[\cos \left({\frac {\theta }{2}}\right)+i\sigma _{3}\sin \left({\frac {\theta }{2}}\right)\right]=\left[\cos ^{2}\left({\frac {\theta }{2}}\right)+\sin ^{2}\left({\frac {\theta }{2}}\right)\right]\sigma _{3}=\sigma _{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf41676a84bad923febc2a5ba2ed73ab21e6b834" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:90.706ex; height:6.176ex;" alt="{\displaystyle \left[\cos \left({\frac {\theta }{2}}\right)-i\sigma _{3}\sin \left({\frac {\theta }{2}}\right)\right]\sigma _{3}\left[\cos \left({\frac {\theta }{2}}\right)+i\sigma _{3}\sin \left({\frac {\theta }{2}}\right)\right]=\left[\cos ^{2}\left({\frac {\theta }{2}}\right)+\sin ^{2}\left({\frac {\theta }{2}}\right)\right]\sigma _{3}=\sigma _{3}.}" /> The bivectors σ2σ3, σ3σ1 and σ1σ2 are in fact <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton's</a> <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> i, j, and k, discovered in 1843: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {i} &=-\sigma _{2}\sigma _{3}=-i\sigma _{1}\\\mathbf {j} &=-\sigma _{3}\sigma _{1}=-i\sigma _{2}\\\mathbf {k} &=-\sigma _{1}\sigma _{2}=-i\sigma _{3}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {i} &=-\sigma _{2}\sigma _{3}=-i\sigma _{1}\\\mathbf {j} &=-\sigma _{3}\sigma _{1}=-i\sigma _{2}\\\mathbf {k} &=-\sigma _{1}\sigma _{2}=-i\sigma _{3}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c87034fb096738a33da19372c00e362379194f4b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:19.923ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {i} &=-\sigma _{2}\sigma _{3}=-i\sigma _{1}\\\mathbf {j} &=-\sigma _{3}\sigma _{1}=-i\sigma _{2}\\\mathbf {k} &=-\sigma _{1}\sigma _{2}=-i\sigma _{3}\end{aligned}}}" /> With the identification of the even-graded elements with the algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }" /> of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself.<a href="#cite_note-42">[t]</a> Thus the (real<a href="#cite_note-43">[u]</a>) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication. Note that the expression (1) for a vector rotation through an angle θ, the angle appearing in γ was halved. Thus the spinor rotation γ(ψ) = γψ (ordinary quaternionic multiplication) will rotate the spinor ψ through an angle one-half the measure of the angle of the corresponding vector rotation. Once again, the problem of lifting a vector rotation to a spinor rotation is two-valued: the expression (1) with (180° + θ/2) in place of θ/2 will produce the same vector rotation, but the negative of the spinor rotation. The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes. <div class="mw-heading mw-heading2"><h2 id="Explicit_constructions">Explicit constructions</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=16" title="Edit section: Explicit constructions">edit</a>]</div> A space of spinors can be constructed explicitly with concrete and abstract constructions. The equivalence of these constructions is a consequence of the uniqueness of the spinor representation of the complex Clifford algebra. For a complete example in dimension 3, see <a href="/wiki/Spinors_in_three_dimensions" title="Spinors in three dimensions">spinors in three dimensions</a>. <div class="mw-heading mw-heading3"><h3 id="Component_spinors">Component spinors</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=17" title="Edit section: Component spinors">edit</a>]</div> Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra Cℓ(V, g) can be defined as follows. Choose an orthonormal basis e1 ... en for V i.e. g(eμeν) = ημν where ημμ = ±1 and ημν = 0 for μ ≠ ν. Let k = ⌊n/2⌋. Fix a set of 2k × 2k matrices γ1 ... γn such that γμγν + γνγμ = 2ημν1 (i.e. fix a convention for the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>). Then the assignment eμ → γμ extends uniquely to an algebra homomorphism Cℓ(V, g) → Mat(2k, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />) by sending the monomial eμ1 ⋅⋅⋅ eμk in the Clifford algebra to the product γμ1 ⋅⋅⋅ γμk of matrices and extending linearly. The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =\mathbb {C} ^{2^{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =\mathbb {C} ^{2^{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa1ff934ed5f5d6135a893d0471046dc797f195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.626ex; height:3.176ex;" alt="{\displaystyle \Delta =\mathbb {C} ^{2^{k}}}" /> on which the gamma matrices act is now a space of spinors. One needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli sigma matrices</a> gives rise to the familiar two component spinors used in non relativistic <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. Likewise using the 4 × 4 Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativistic <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. In general, in order to define gamma matrices of the required kind, one can use the <a href="/wiki/Weyl%E2%80%93Brauer_matrices" title="Weyl–Brauer matrices">Weyl–Brauer matrices</a>. In this construction the representation of the Clifford algebra Cℓ(V, g), the Lie algebra so(V, g), and the Spin group Spin(V, g), all depend on the choice of the orthonormal basis and the choice of the gamma matrices. This can cause confusion over conventions, but invariants like traces are independent of choices. In particular, all physically observable quantities must be independent of such choices. In this construction a spinor can be represented as a vector of 2k complex numbers and is denoted with spinor indices (usually α, β, γ). In the physics literature, such <a href="/wiki/Abstract_indices" class="mw-redirect" title="Abstract indices">indices</a> are often used to denote spinors even when an abstract spinor construction is used. <div class="mw-heading mw-heading3"><h3 id="Abstract_spinors">Abstract spinors</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=18" title="Edit section: Abstract spinors">edit</a>]</div> There are at least two different, but essentially equivalent, ways to define spinors abstractly. One approach seeks to identify the minimal ideals for the left action of Cℓ(V, g) on itself. These are subspaces of the Clifford algebra of the form Cℓ(V, g)ω, admitting the evident action of Cℓ(V, g) by left-multiplication: c : xω → cxω. There are two variations on this theme: one can either find a primitive element ω that is a <a href="/wiki/Nilpotent" title="Nilpotent">nilpotent</a> element of the Clifford algebra, or one that is an <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a>. The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it.<a href="#cite_note-44">[23]</a> In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself. The second approach is to construct a vector space using a distinguished subspace of V, and then specify the action of the Clifford algebra externally to that vector space. In either approach, the fundamental notion is that of an <a href="/wiki/Isotropic_subspace" class="mw-redirect" title="Isotropic subspace">isotropic subspace</a> W. Each construction depends on an initial freedom in choosing this subspace. In physical terms, this corresponds to the fact that there is no measurement protocol that can specify a basis of the spin space, even if a preferred basis of V is given. As above, we let (V, g) be an n-dimensional complex vector space equipped with a nondegenerate bilinear form. If V is a real vector space, then we replace V by its <a href="/wiki/Complexification" title="Complexification">complexification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488efafc27b11de008a8bb600fe0a083fab76068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.725ex; height:2.509ex;" alt="{\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }" /> and let g denote the induced bilinear form on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488efafc27b11de008a8bb600fe0a083fab76068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.725ex; height:2.509ex;" alt="{\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }" />. Let W be a maximal isotropic subspace, i.e. a maximal subspace of V such that g|W = 0. If n = 2k is even, then let W′ be an isotropic subspace complementary to W. If n = 2k + 1 is odd, let W′ be a maximal isotropic subspace with W ∩ W′ = 0, and let U be the orthogonal complement of W ⊕ W′. In both the even- and odd-dimensional cases W and W′ have dimension k. In the odd-dimensional case, U is one-dimensional, spanned by a unit vector u. <div class="mw-heading mw-heading3"><h3 id="Minimal_ideals">Minimal ideals</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=19" title="Edit section: Minimal ideals">edit</a>]</div> Since W′ is isotropic, multiplication of elements of W′ inside Cℓ(V, g) is <a href="/wiki/Alternative_algebra" title="Alternative algebra">skew</a>. Hence vectors in W′ anti-commute, and Cℓ(W′, g|W′) = Cℓ(W′, 0) is just the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> Λ∗W′. Consequently, the k-fold product of W′ with itself, W′k, is one-dimensional. Let ω be a generator of W′k. In terms of a basis w′1, ..., w′k of in W′, one possibility is to set <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =w'_{1}w'_{2}\cdots w'_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <msubsup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>⋯</mo> <msubsup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =w'_{1}w'_{2}\cdots w'_{k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc67063ae8e7e85e1dc063c02a9ad00954f84a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.878ex; height:2.843ex;" alt="{\displaystyle \omega =w'_{1}w'_{2}\cdots w'_{k}.}" /> Note that ω2 = 0 (i.e., ω is nilpotent of order 2), and moreover, w′ω = 0 for all w′ ∈ W′. The following facts can be proven easily: <ol><li>If n = 2k, then the left ideal Δ = Cℓ(V, g)ω is a minimal left ideal. Furthermore, this splits into the two spin spaces Δ+ = Cℓevenω and Δ− = Cℓoddω on restriction to the action of the even Clifford algebra.</li> <li>If n = 2k + 1, then the action of the unit vector u on the left ideal Cℓ(V, g)ω decomposes the space into a pair of isomorphic irreducible eigenspaces (both denoted by Δ), corresponding to the respective eigenvalues +1 and −1.</li></ol> In detail, suppose for instance that n is even. Suppose that I is a non-zero left ideal contained in Cℓ(V, g)ω. We shall show that I must be equal to Cℓ(V, g)ω by proving that it contains a nonzero scalar multiple of ω. Fix a basis wi of W and a complementary basis wi′ of W′ so that <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;">wiwj′ +wj′wi = δij, and</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">(wi)2 = 0, (wi′)2 = 0.</div> Note that any element of I must have the form αω, by virtue of our assumption that I ⊂ Cℓ(V, g) ω. Let αω ∈ I be any such element. Using the chosen basis, we may write <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}+\sum _{j}B_{j}w'_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>⋯</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}+\sum _{j}B_{j}w'_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54269582f7d2a99a2794a8637dda80697bd101be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:43.047ex; height:6.009ex;" alt="{\displaystyle \alpha =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}+\sum _{j}B_{j}w'_{j}}" /> where the ai1...ip are scalars, and the Bj are auxiliary elements of the Clifford algebra. Observe now that the product <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \omega =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}\omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mi>ω</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>⋯</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msub> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \omega =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}\omega .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdce618aa9432ffd91667ef48b12130bdf3bd4f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.756ex; height:6.009ex;" alt="{\displaystyle \alpha \omega =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}\omega .}" /> Pick any nonzero monomial a in the expansion of α with maximal homogeneous degree in the elements wi: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=a_{i_{1}\dots i_{\text{max}}}w_{i_{1}}\dots w_{i_{\text{max}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>…</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a_{i_{1}\dots i_{\text{max}}}w_{i_{1}}\dots w_{i_{\text{max}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9626852d843d9c9c30e5df6c0d4faa5bc3cd85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.234ex; height:2.343ex;" alt="{\displaystyle a=a_{i_{1}\dots i_{\text{max}}}w_{i_{1}}\dots w_{i_{\text{max}}}}" /> (no summation implied), then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w'_{i_{\text{max}}}\cdots w'_{i_{1}}\alpha \omega =a_{i_{1}\dots i_{\text{max}}}\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mrow> <mo>′</mo> </msubsup> <mo>⋯</mo> <msubsup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>′</mo> </msubsup> <mi>α</mi> <mi>ω</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mrow> </msub> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w'_{i_{\text{max}}}\cdots w'_{i_{1}}\alpha \omega =a_{i_{1}\dots i_{\text{max}}}\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf755d3f00b845e960d73c12d6e5881ab7e5697" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:27.384ex; height:3.176ex;" alt="{\displaystyle w'_{i_{\text{max}}}\cdots w'_{i_{1}}\alpha \omega =a_{i_{1}\dots i_{\text{max}}}\omega }" /> is a nonzero scalar multiple of ω, as required. Note that for n even, this computation also shows that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =\mathrm {C} \ell (W)\omega =\left(\Lambda ^{*}W\right)\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> <mi>ω</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>W</mi> </mrow> <mo>)</mo> </mrow> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =\mathrm {C} \ell (W)\omega =\left(\Lambda ^{*}W\right)\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1d93a9855e984eed5f58b32008b94f470f86ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.216ex; height:2.843ex;" alt="{\displaystyle \Delta =\mathrm {C} \ell (W)\omega =\left(\Lambda ^{*}W\right)\omega }" /> as a vector space. In the last equality we again used that W is isotropic. In physics terms, this shows that Δ is built up like a <a href="/wiki/Fock_space" title="Fock space">Fock space</a> by <a href="/wiki/Creation_and_annihilation" class="mw-redirect" title="Creation and annihilation">creating</a> spinors using anti-commuting creation operators in W acting on a vacuum ω. <div class="mw-heading mw-heading3"><h3 id="Exterior_algebra_construction">Exterior algebra construction</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=20" title="Edit section: Exterior algebra construction">edit</a>]</div> The computations with the minimal ideal construction suggest that a spinor representation can also be defined directly using the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> Λ∗ W = ⊕j Λj W of the isotropic subspace W. Let Δ = Λ∗ W denote the exterior algebra of W considered as vector space only. This will be the spin representation, and its elements will be referred to as spinors.<a href="#cite_note-45">[24]</a><a href="#cite_note-46">[25]</a> The action of the Clifford algebra on Δ is defined first by giving the action of an element of V on Δ, and then showing that this action respects the Clifford relation and so extends to a <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> of the full Clifford algebra into the <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> End(Δ) by the <a href="/wiki/Clifford_algebra#Universal_property_and_construction" title="Clifford algebra">universal property of Clifford algebras</a>. The details differ slightly according to whether the dimension of V is even or odd. When dim(V) is even, V = W ⊕ W′ where W′ is the chosen isotropic complement. Hence any v ∈ V decomposes uniquely as v = w + w′ with w ∈ W and w′ ∈ W′. The action of v on a spinor is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(v)w_{1}\wedge \cdots \wedge w_{n}=\left(\epsilon (w)+i\left(w'\right)\right)\left(w_{1}\wedge \cdots \wedge w_{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>ϵ</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mrow> <mo>(</mo> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(v)w_{1}\wedge \cdots \wedge w_{n}=\left(\epsilon (w)+i\left(w'\right)\right)\left(w_{1}\wedge \cdots \wedge w_{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a462e8307ee6270d43cdd969bcf4ea6595673e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.632ex; height:3.009ex;" alt="{\displaystyle c(v)w_{1}\wedge \cdots \wedge w_{n}=\left(\epsilon (w)+i\left(w'\right)\right)\left(w_{1}\wedge \cdots \wedge w_{n}\right)}" /> where i(w′) is <a href="/wiki/Interior_product" title="Interior product">interior product</a> with w′ using the nondegenerate quadratic form to identify V with V∗, and ε(w) denotes the <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a>. This action is sometimes called the Clifford product. It may be verified that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(u)\,c(v)+c(v)\,c(u)=2\,g(u,v)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(u)\,c(v)+c(v)\,c(u)=2\,g(u,v)\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/554e378f254f531a0dbbb56f8103f739cee0b761" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.892ex; height:2.843ex;" alt="{\displaystyle c(u)\,c(v)+c(v)\,c(u)=2\,g(u,v)\,,}" /> and so c respects the Clifford relations and extends to a homomorphism from the Clifford algebra to End(Δ). The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group<a href="#cite_note-47">[26]</a> (the half-spin representations, or Weyl spinors) via <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{+}=\Lambda ^{\text{even}}W,\,\Delta _{-}=\Lambda ^{\text{odd}}W.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>=</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>even</mtext> </mrow> </msup> <mi>W</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>=</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>odd</mtext> </mrow> </msup> <mi>W</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{+}=\Lambda ^{\text{even}}W,\,\Delta _{-}=\Lambda ^{\text{odd}}W.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbfc4860fa4dc00b66a9d4c869c4ceae7981d3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.611ex; height:3.009ex;" alt="{\displaystyle \Delta _{+}=\Lambda ^{\text{even}}W,\,\Delta _{-}=\Lambda ^{\text{odd}}W.}" /> When dim(V) is odd, V = W ⊕ U ⊕ W′, where U is spanned by a unit vector u orthogonal to W. The Clifford action c is defined as before on W ⊕ W′, while the Clifford action of (multiples of) u is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(u)\alpha ={\begin{cases}\alpha &{\hbox{if }}\alpha \in \Lambda ^{\text{even}}W\\-\alpha &{\hbox{if }}\alpha \in \Lambda ^{\text{odd}}W\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>α</mi> <mo>∈</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>even</mtext> </mrow> </msup> <mi>W</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>α</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>α</mi> <mo>∈</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>odd</mtext> </mrow> </msup> <mi>W</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(u)\alpha ={\begin{cases}\alpha &{\hbox{if }}\alpha \in \Lambda ^{\text{even}}W\\-\alpha &{\hbox{if }}\alpha \in \Lambda ^{\text{odd}}W\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f90f45f13b393d5da7aeef0cf6d13305d9662850" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.635ex; height:6.176ex;" alt="{\displaystyle c(u)\alpha ={\begin{cases}\alpha &{\hbox{if }}\alpha \in \Lambda ^{\text{even}}W\\-\alpha &{\hbox{if }}\alpha \in \Lambda ^{\text{odd}}W\end{cases}}}" /> As before, one verifies that c respects the Clifford relations, and so induces a homomorphism. <div class="mw-heading mw-heading3"><h3 id="Hermitian_vector_spaces_and_spinors">Hermitian vector spaces and spinors</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=21" title="Edit section: Hermitian vector spaces and spinors">edit</a>]</div> If the vector space V has extra structure that provides a decomposition of its complexification into two maximal isotropic subspaces, then the definition of spinors (by either method) becomes natural. The main example is the case that the real vector space V is a <a href="/wiki/Hermitian_form" class="mw-redirect" title="Hermitian form">hermitian vector space</a> (V, h), i.e., V is equipped with a <a href="/wiki/Linear_complex_structure" title="Linear complex structure">complex structure</a> J that is an <a href="/wiki/Orthogonal_transformation" title="Orthogonal transformation">orthogonal transformation</a> with respect to the inner product g on V. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488efafc27b11de008a8bb600fe0a083fab76068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.725ex; height:2.509ex;" alt="{\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }" /> splits in the ±i eigenspaces of J. These eigenspaces are isotropic for the complexification of g and can be identified with the complex vector space (V, J) and its complex conjugate (V, −J). Therefore, for a hermitian vector space (V, h) the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda _{\mathbb {C} }^{\cdot }{\bar {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda _{\mathbb {C} }^{\cdot }{\bar {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92788e0461a16cbc4e754d6e61e57b58243ce23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.819ex; height:3.176ex;" alt="{\displaystyle \Lambda _{\mathbb {C} }^{\cdot }{\bar {V}}}" /> (as well as its complex conjugate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda _{\mathbb {C} }^{\cdot }V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> </msubsup> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda _{\mathbb {C} }^{\cdot }V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb23cd9e9724920714f95c9e7c94c2955802072b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.819ex; height:2.843ex;" alt="{\displaystyle \Lambda _{\mathbb {C} }^{\cdot }V}" />) is a spinor space for the underlying real euclidean vector space. With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an <a href="/wiki/Almost_Hermitian_manifold" class="mw-redirect" title="Almost Hermitian manifold">almost Hermitian manifold</a> and is the reason why every <a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">almost complex manifold</a> (in particular every <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a>) has a <a href="/wiki/Spin-c_structure" class="mw-redirect" title="Spin-c structure">Spinc structure</a>. Likewise, every complex vector bundle on a manifold carries a Spinc structure.<a href="#cite_note-48">[27]</a> <div class="mw-heading mw-heading2"><h2 id="Clebsch–Gordan_decomposition">Clebsch–Gordan decomposition</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=22" title="Edit section: Clebsch–Gordan decomposition">edit</a>]</div> A number of <a href="/wiki/Clebsch%E2%80%93Gordan_coefficients" title="Clebsch–Gordan coefficients">Clebsch–Gordan decompositions</a> are possible on the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> of one spin representation with another.<a href="#cite_note-49">[28]</a> These decompositions express the tensor product in terms of the alternating representations of the orthogonal group. For the real or complex case, the alternating representations are <ul><li>Γr = ΛrV, the representation of the orthogonal group on skew tensors of rank r.</li></ul> In addition, for the real orthogonal groups, there are three <a href="/wiki/Character_theory" title="Character theory">characters</a> (one-dimensional representations) <ul><li>σ+ : O(p, q) → {−1, +1} given by σ+(R) = −1, if R reverses the spatial orientation of V, +1, if R preserves the spatial orientation of V. (The spatial character.)</li> <li>σ− : O(p, q) → {−1, +1} given by σ−(R) = −1, if R reverses the temporal orientation of V, +1, if R preserves the temporal orientation of V. (The temporal character.)</li> <li>σ = σ+σ− . (The orientation character.)</li></ul> The Clebsch–Gordan decomposition allows one to define, among other things: <ul><li>An action of spinors on vectors.</li> <li>A <a href="/wiki/Hermitian_metric" class="mw-redirect" title="Hermitian metric">Hermitian metric</a> on the complex representations of the real spin groups.</li> <li>A <a href="/wiki/Dirac_operator" title="Dirac operator">Dirac operator</a> on each spin representation.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Even_dimensions">Even dimensions</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=23" title="Edit section: Even dimensions">edit</a>]</div> If n = 2k is even, then the tensor product of Δ with the <a href="/wiki/Contragredient_representation" class="mw-redirect" title="Contragredient representation">contragredient representation</a> decomposes as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{n}\Gamma _{p}\cong \bigoplus _{p=0}^{k-1}\left(\Gamma _{p}\oplus \sigma \Gamma _{p}\right)\oplus \Gamma _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊗</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>≅</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≅</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⊕</mo> <mi>σ</mi> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⊕</mo> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{n}\Gamma _{p}\cong \bigoplus _{p=0}^{k-1}\left(\Gamma _{p}\oplus \sigma \Gamma _{p}\right)\oplus \Gamma _{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44847ed0e5eefa962cba502cc86fbf3e0b157610" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.656ex; height:7.676ex;" alt="{\displaystyle \Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{n}\Gamma _{p}\cong \bigoplus _{p=0}^{k-1}\left(\Gamma _{p}\oplus \sigma \Gamma _{p}\right)\oplus \Gamma _{k}}" /> which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elements αω ⊗ βω′. The rightmost formulation follows from the transformation properties of the <a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a>. Note that on restriction to the even Clifford algebra, the paired summands Γp ⊕ σΓp are isomorphic, but under the full Clifford algebra they are not. There is a natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\alpha \omega )^{*}=\omega \left(\alpha ^{*}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <mi>ω</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>ω</mi> <mrow> <mo>(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha \omega )^{*}=\omega \left(\alpha ^{*}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6dc2616509329da9df38ed4bb3aee3f6cb9da4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.113ex; height:2.843ex;" alt="{\displaystyle (\alpha \omega )^{*}=\omega \left(\alpha ^{*}\right).}" /> So Δ ⊗ Δ also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta _{+}\otimes \Delta _{+}^{*}\cong \Delta _{-}\otimes \Delta _{-}^{*}&\cong \bigoplus _{p=0}^{k}\Gamma _{2p}\\\Delta _{+}\otimes \Delta _{-}^{*}\cong \Delta _{-}\otimes \Delta _{+}^{*}&\cong \bigoplus _{p=0}^{k-1}\Gamma _{2p+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> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>≅</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>≅</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>≅</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>≅</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta _{+}\otimes \Delta _{+}^{*}\cong \Delta _{-}\otimes \Delta _{-}^{*}&\cong \bigoplus _{p=0}^{k}\Gamma _{2p}\\\Delta _{+}\otimes \Delta _{-}^{*}\cong \Delta _{-}\otimes \Delta _{+}^{*}&\cong \bigoplus _{p=0}^{k-1}\Gamma _{2p+1}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef2d15310558a57ebea9d602bb0fb04f4b57166" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:35.748ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\Delta _{+}\otimes \Delta _{+}^{*}\cong \Delta _{-}\otimes \Delta _{-}^{*}&\cong \bigoplus _{p=0}^{k}\Gamma _{2p}\\\Delta _{+}\otimes \Delta _{-}^{*}\cong \Delta _{-}\otimes \Delta _{+}^{*}&\cong \bigoplus _{p=0}^{k-1}\Gamma _{2p+1}\end{aligned}}}" /> For the complex representations of the real Clifford algebras, the associated <a href="/wiki/Reality_structure" class="mw-redirect" title="Reality structure">reality structure</a> on the complex Clifford algebra descends to the space of spinors (via the explicit construction in terms of minimal ideals, for instance). In this way, we obtain the complex conjugate Δ of the representation Δ, and the following isomorphism is seen to hold: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>≅</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8ae63789550e6eafb2d65043fa85a78f9abf73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.863ex; height:3.009ex;" alt="{\displaystyle {\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}}" /> In particular, note that the representation Δ of the orthochronous spin group is a <a href="/wiki/Unitary_representation" title="Unitary representation">unitary representation</a>. In general, there are Clebsch–Gordan decompositions <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \otimes {\bar {\Delta }}\cong \bigoplus _{p=0}^{k}\left(\sigma _{-}\Gamma _{p}\oplus \sigma _{+}\Gamma _{p}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>≅</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⊕</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \otimes {\bar {\Delta }}\cong \bigoplus _{p=0}^{k}\left(\sigma _{-}\Gamma _{p}\oplus \sigma _{+}\Gamma _{p}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fbeb1d7dcc8e018771536b632a0972440f7dab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.092ex; height:7.676ex;" alt="{\displaystyle \Delta \otimes {\bar {\Delta }}\cong \bigoplus _{p=0}^{k}\left(\sigma _{-}\Gamma _{p}\oplus \sigma _{+}\Gamma _{p}\right).}" /> In metric signature (p, q), the following isomorphisms hold for the conjugate half-spin representations <ul><li>If q is even, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\Delta }}_{+}\cong \sigma _{-}\otimes \Delta _{+}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>≅</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\Delta }}_{+}\cong \sigma _{-}\otimes \Delta _{+}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094d3c5fee50051593d05471cadf2daf21dd124f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.67ex; height:3.343ex;" alt="{\displaystyle {\bar {\Delta }}_{+}\cong \sigma _{-}\otimes \Delta _{+}^{*}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\Delta }}_{-}\cong \sigma _{-}\otimes \Delta _{-}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>≅</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\Delta }}_{-}\cong \sigma _{-}\otimes \Delta _{-}^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15bbf37869a1164ded6f6902bb4c8f8560bf767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.317ex; height:3.343ex;" alt="{\displaystyle {\bar {\Delta }}_{-}\cong \sigma _{-}\otimes \Delta _{-}^{*}.}" /></li> <li>If q is odd, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\Delta }}_{+}\cong \sigma _{-}\otimes \Delta _{-}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>≅</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\Delta }}_{+}\cong \sigma _{-}\otimes \Delta _{-}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fcff16b41eb8d79114f23c89bd87fae75e7f00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.67ex; height:3.343ex;" alt="{\displaystyle {\bar {\Delta }}_{+}\cong \sigma _{-}\otimes \Delta _{-}^{*}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\Delta }}_{-}\cong \sigma _{-}\otimes \Delta _{+}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>≅</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\Delta }}_{-}\cong \sigma _{-}\otimes \Delta _{+}^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05fdda604d5760c5b3a53ddd23aff37020e75f32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.317ex; height:3.343ex;" alt="{\displaystyle {\bar {\Delta }}_{-}\cong \sigma _{-}\otimes \Delta _{+}^{*}.}" /></li></ul> Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the half-spin representations Δ± ⊗ Δ±. <div class="mw-heading mw-heading3"><h3 id="Odd_dimensions">Odd dimensions</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=24" title="Edit section: Odd dimensions">edit</a>]</div> If n = 2k + 1 is odd, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{k}\Gamma _{2p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊗</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>≅</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{k}\Gamma _{2p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce77d980a71d439d45b1bd03008943c75e59b7ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.743ex; height:7.676ex;" alt="{\displaystyle \Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{k}\Gamma _{2p}.}" /> In the real case, once again the isomorphism holds <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>≅</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11042b67cf0e4ebd0687f62154fabead4180ed8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.509ex; height:3.009ex;" alt="{\displaystyle {\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}.}" /> Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \otimes {\bar {\Delta }}\cong \sigma _{-}\Gamma _{0}\oplus \sigma _{+}\Gamma _{1}\oplus \dots \oplus \sigma _{\pm }\Gamma _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>≅</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \otimes {\bar {\Delta }}\cong \sigma _{-}\Gamma _{0}\oplus \sigma _{+}\Gamma _{1}\oplus \dots \oplus \sigma _{\pm }\Gamma _{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea85dc8c1f368185083230c142b22ce024e92740" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.125ex; height:3.009ex;" alt="{\displaystyle \Delta \otimes {\bar {\Delta }}\cong \sigma _{-}\Gamma _{0}\oplus \sigma _{+}\Gamma _{1}\oplus \dots \oplus \sigma _{\pm }\Gamma _{k}}" /> <div class="mw-heading mw-heading3"><h3 id="Consequences">Consequences</h3>[<a href="/w/index.php?title=Spinor&action=edit&section=25" title="Edit section: Consequences">edit</a>]</div> There are many far-reaching consequences of the Clebsch–Gordan decompositions of the spinor spaces. The most fundamental of these pertain to Dirac's theory of the electron, among whose basic requirements are <ul><li>A manner of regarding the product of two spinors ϕψ as a scalar. In physical terms, a spinor should determine a <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitude</a> for the <a href="/wiki/Quantum_state" title="Quantum state">quantum state</a>.</li> <li>A manner of regarding the product ψϕ as a vector. This is an essential feature of Dirac's theory, which ties the spinor formalism to the geometry of physical space.</li> <li>A manner of regarding a spinor as acting upon a vector, by an expression such as ψvψ. In physical terms, this represents an <a href="/wiki/Electric_current" title="Electric current">electric current</a> of Maxwell's <a href="/wiki/Electromagnetic_theory" class="mw-redirect" title="Electromagnetic theory">electromagnetic theory</a>, or more generally a <a href="/wiki/Probability_current" title="Probability current">probability current</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Summary_in_low_dimensions">Summary in low dimensions</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=26" title="Edit section: Summary in low dimensions">edit</a>]</div> <ul><li>In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a <a href="/wiki/Real_representation" title="Real representation">real</a> 1-dimensional representation that does not transform.</li> <li>In 2 Euclidean dimensions, the left-handed and the right-handed Weyl spinor are 1-component <a href="/wiki/Complex_representation" title="Complex representation">complex representations</a>, i.e. complex numbers that get multiplied by e±iφ/2 under a rotation by angle φ.</li> <li>In 3 Euclidean dimensions, the single spinor representation is 2-dimensional and <a href="/wiki/Quaternionic_representation" title="Quaternionic representation">quaternionic</a>. The existence of spinors in 3 dimensions follows from the isomorphism of the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> SU(2) ≅ Spin(3) that allows us to define the action of Spin(3) on a complex 2-component column (a spinor); the generators of SU(2) can be written as <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>.</li> <li>In 4 Euclidean dimensions, the corresponding isomorphism is Spin(4) ≅ SU(2) × SU(2). There are two inequivalent <a href="/wiki/Quaternionic_representation" title="Quaternionic representation">quaternionic</a> 2-component Weyl spinors and each of them transforms under one of the SU(2) factors only.</li> <li>In 5 Euclidean dimensions, the relevant isomorphism is Spin(5) ≅ USp(4) ≅ Sp(2) that implies that the single spinor representation is 4-dimensional and quaternionic.</li> <li>In 6 Euclidean dimensions, the isomorphism Spin(6) ≅ SU(4) guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another.</li> <li>In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no isomorphisms to a Lie algebra from another series (A or C) exist from this dimension on.</li> <li>In 8 Euclidean dimensions, there are two Weyl–Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of <a href="/wiki/Spin(8)" class="mw-redirect" title="Spin(8)">Spin(8)</a> called <a href="/wiki/Triality" title="Triality">triality</a>.</li> <li>In d + 8 dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in d dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. See <a href="/wiki/Bott_periodicity" class="mw-redirect" title="Bott periodicity">Bott periodicity</a>.</li> <li>In spacetimes with p spatial and q time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the (p + q)-dimensional Euclidean space, but the reality projections mimic the structure in |p − q| Euclidean dimensions. For example, in 3 + 1 dimensions there are two non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphism SL(2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />) ≅ Spin(3,1).</li></ul> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th rowspan="2"><a href="/wiki/Metric_signature" title="Metric signature">Metric signature</a> </th> <th colspan="2">Weyl, complex </th> <th rowspan="2">Conjugacy </th> <th rowspan="2">Dirac, complex </th> <th colspan="2">Majorana–Weyl, real </th> <th rowspan="2">Majorana, real </th></tr> <tr> <th>Left-handed </th> <th>Right-handed </th> <th>Left-handed </th> <th>Right-handed </th></tr> <tr> <td>(2,0)</td> <td>1</td> <td>1</td> <td>Mutual</td> <td>2</td> <td>–</td> <td>–</td> <td>2 </td></tr> <tr> <td>(1,1)</td> <td>1</td> <td>1</td> <td>Self</td> <td>2</td> <td>1</td> <td>1</td> <td>2 </td></tr> <tr> <td>(3,0)</td> <td>–</td> <td>–</td> <td>–</td> <td>2</td> <td>–</td> <td>–</td> <td>– </td></tr> <tr> <td>(2,1)</td> <td>–</td> <td>–</td> <td>–</td> <td>2</td> <td>–</td> <td>–</td> <td>2 </td></tr> <tr> <td>(4,0)</td> <td>2</td> <td>2</td> <td>Self</td> <td>4</td> <td>–</td> <td>–</td> <td>– </td></tr> <tr> <td>(3,1)</td> <td>2</td> <td>2</td> <td>Mutual</td> <td>4</td> <td>–</td> <td>–</td> <td>4 </td></tr> <tr> <td>(5,0)</td> <td>–</td> <td>–</td> <td>–</td> <td>4</td> <td>–</td> <td>–</td> <td>– </td></tr> <tr> <td>(4,1)</td> <td>–</td> <td>–</td> <td>–</td> <td>4</td> <td>–</td> <td>–</td> <td>– </td></tr> <tr> <td>(6,0)</td> <td>4</td> <td>4</td> <td>Mutual</td> <td>8</td> <td>–</td> <td>–</td> <td>8 </td></tr> <tr> <td>(5,1)</td> <td>4</td> <td>4</td> <td>Self</td> <td>8</td> <td>–</td> <td>–</td> <td>– </td></tr> <tr> <td>(7,0)</td> <td>–</td> <td>–</td> <td>–</td> <td>8</td> <td>–</td> <td>–</td> <td>8 </td></tr> <tr> <td>(6,1)</td> <td>–</td> <td>–</td> <td>–</td> <td>8</td> <td>–</td> <td>–</td> <td>– </td></tr> <tr> <td>(8,0)</td> <td>8</td> <td>8</td> <td>Self</td> <td>16</td> <td>8</td> <td>8</td> <td>16 </td></tr> <tr> <td>(7,1)</td> <td>8</td> <td>8</td> <td>Mutual</td> <td>16</td> <td>–</td> <td>–</td> <td>16 </td></tr> <tr> <td>(9,0)</td> <td>–</td> <td>–</td> <td>–</td> <td>16</td> <td>–</td> <td>–</td> <td>16 </td></tr> <tr> <td>(8,1)</td> <td>–</td> <td>–</td> <td>–</td> <td>16</td> <td>–</td> <td>–</td> <td>16 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=27" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 27em;"> <ul><li><a href="/wiki/Anyon" title="Anyon">Anyon</a></li> <li><a href="/wiki/Dirac_equation_in_the_algebra_of_physical_space" class="mw-redirect" title="Dirac equation in the algebra of physical space">Dirac equation in the algebra of physical space</a></li> <li><a href="/wiki/Eigenspinor" title="Eigenspinor">Eigenspinor</a></li> <li><a href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan theory</a></li> <li><a href="/wiki/Projective_representation" title="Projective representation">Projective representation</a></li> <li><a href="/wiki/Pure_spinor" title="Pure spinor">Pure spinor</a></li> <li><a href="/wiki/Spin-1/2" title="Spin-1/2">Spin-1/2</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor bundle</a></li> <li><a href="/wiki/Supercharge" title="Supercharge">Supercharge</a></li> <li><a href="/wiki/Twistor_theory" title="Twistor theory">Twistor theory</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=28" title="Edit section: Notes">edit</a>]</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-columns references-column-width reflist-lower-alpha" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> Spinors in three dimensions are points of a <a href="/wiki/Line_bundle" title="Line bundle">line bundle</a> over a <a href="/wiki/Conic" class="mw-redirect" title="Conic">conic</a> in the <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a>. In this picture, which is associated to spinors of a three-dimensional <a href="/wiki/Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean space</a> of signature (1,2), the conic is an ordinary real conic (here the circle), the line bundle is the Möbius bundle, and the spin group is SL2(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />). In Euclidean signature, the projective plane, conic and line bundle are over the complex instead, and this picture is just a real slice. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> Spinors can always be defined over the complex numbers. However, in some signatures there exist real spinors. Details can be found in <a href="/wiki/Spin_representation" title="Spin representation">spin representation</a>. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> A formal definition of spinors at this level is that the space of spinors is a <a href="/wiki/Linear_representation" class="mw-redirect" title="Linear representation">linear representation</a> of the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of <a href="/wiki/Infinitesimal_rotation" class="mw-redirect" title="Infinitesimal rotation">infinitesimal rotations</a> of a <a href="/wiki/Spin_representation" title="Spin representation">certain kind</a>. </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> "Spinors were first used under that name, by physicists, in the field of Quantum Mechanics. In their most general form, spinors were discovered in 1913 by the author of this work, in his investigations on the linear representations of simple groups*; they provide a linear representation of the group of rotations in a space with any number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> of dimensions, each spinor having <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0fa1eebc5674c68166a18a335b39288ed2eb325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.266ex; height:2.343ex;" alt="{\displaystyle 2^{\nu }}" /> components where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2\nu +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mi>ν</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2\nu +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02fb584fa8bec4ac60d3eeb0b3ae8496199e1c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.891ex; height:2.343ex;" alt="{\displaystyle n=2\nu +1}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2654d0632cb4b4cd30cfddb80a75ba7d743216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.395ex; height:2.176ex;" alt="{\displaystyle 2\nu }" />."<a href="#cite_note-cartan-1966-quote-5">[2]</a> The star (*) refers to Cartan (1913). </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> More precisely, it is the <a href="/wiki/Fermion" title="Fermion">fermions</a> of <a href="/wiki/Spin-1/2" title="Spin-1/2">spin-1/2</a> that are described by spinors, which is true both in the relativistic and non-relativistic theory. The wavefunction of the non-relativistic electron has values in 2-component spinors transforming under 3-dimensional infinitesimal rotations. The relativistic <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> for the electron is an equation for 4-component spinors transforming under infinitesimal Lorentz transformations, for which a substantially similar theory of spinors exists. </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Formally, the spin group is the group of <a href="/wiki/Relative_homotopy_class" class="mw-redirect" title="Relative homotopy class">relative homotopy classes</a> with fixed endpoints in the rotation group. </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> More formally, the space of spinors can be defined as an (<a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible</a>) representation of the spin group that does not factor through a representation of the rotation group (in general, the connected component of the identity of the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a>). </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a> is a name for the Clifford algebra in an applied setting. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> The Pauli matrices correspond to <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">angular momenta</a> operators about the three coordinate axes. This makes them slightly atypical gamma matrices because in addition to their anticommutation relation they also satisfy commutation relations. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> The <a href="/wiki/Metric_signature" title="Metric signature">metric signature</a> relevant as well if we are concerned with real spinors. See <a href="/wiki/Spin_representation" title="Spin representation">spin representation</a>. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> Whether the representation decomposes depends on whether they are regarded as representations of the spin group (or its Lie algebra), in which case it decomposes in even but not odd dimensions, or the Clifford algebra when it is the other way around. Other structures than this decomposition can also exist; precise criteria are covered at <a href="/wiki/Spin_representation" title="Spin representation">spin representation</a> and <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a>. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> The <a href="/wiki/TNB_frame" class="mw-redirect" title="TNB frame">TNB frame</a> of the ribbon defines a rotation continuously for each value of the arc length parameter. </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> This is the set of 2×2 complex <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">traceless</a> <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">hermitian matrices</a>. </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> Except for a <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\pm 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>±</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\pm 1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eeb8470fc14dbbc80503092df4c0c36d56e66ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.296ex; height:2.843ex;" alt="{\displaystyle \{\pm 1\}}" /> corresponding to the two different elements of the spin group that go to the same rotation.<a href="#cite_note-17">[4]</a> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> So the ambiguity in identifying the spinors themselves persists from the point of view of the group theory, and still depends on choices. </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> The Clifford algebra can be given an even/odd <a href="/wiki/Graded_algebra" class="mw-redirect" title="Graded algebra">grading</a> from the parity of the degree in the gammas, and the spin group and its Lie algebra both lie in the even part. Whether here by "representation" we mean representations of the spin group or the Clifford algebra will affect the determination of their reducibility. Other structures than this splitting can also exist; precise criteria are covered at <a href="/wiki/Spin_representation" title="Spin representation">spin representation</a> and <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a>. </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> More precisely, the electron starts out as two massless Weyl spinors, left and right-handed. Upon symmetry breaking, both gain a mass, and are coupled to form a Dirac spinor. </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> The matrices of dimension N × N in which only the elements of the left column are non-zero form a left ideal in the N × N matrix algebra Mat(N, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />) – multiplying such a matrix M from the left with any N × N matrix A gives the result AM that is again an N × N matrix in which only the elements of the left column are non-zero. Moreover, it can be shown that it is a minimal left ideal.<a href="#cite_note-36">[19]</a> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> These are the right-handed Weyl spinors in two dimensions. For the left-handed Weyl spinors, the representation is via γ(ϕ) = γϕ. The Majorana spinors are the common underlying real representation for the Weyl representations. </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> Since, for a <a href="/wiki/Skew_field" class="mw-redirect" title="Skew field">skew field</a>, the kernel of the representation must be trivial. So inequivalent representations can only arise via an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a> of the skew-field. In this case, there are a pair of equivalent representations: γ(ϕ) = γϕ, and its quaternionic conjugate γ(ϕ) = ϕγ. </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> The complex spinors are obtained as the representations of the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} \otimes _{\mathbb {R} }\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} \otimes _{\mathbb {R} }\mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87333035eee79dcfb7042925014511dfc5247b4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.745ex; height:2.509ex;" alt="{\displaystyle \mathbb {H} \otimes _{\mathbb {R} }\mathbb {C} }" /> = Mat2(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />). These are considered in more detail in <a href="/wiki/Spinors_in_three_dimensions" title="Spinors in three dimensions">spinors in three dimensions</a>. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=29" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFCartan1913">Cartan 1913</a>. </li> <li id="cite_note-cartan-1966-quote-5">^ <a href="#cite_ref-cartan-1966-quote_5-0">a</a> <a href="#cite_ref-cartan-1966-quote_5-1">b</a> Quote from Elie Cartan: The Theory of Spinors, Hermann, Paris, 1966, first sentence of the Introduction section at the beginning of the book, before page numbers start. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <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="CITEREFRukhsan-Ul-Haq2016" class="citation journal cs1">Rukhsan-Ul-Haq (December 2016). <a rel="nofollow" class="external text" href="https://www.ias.ac.in/article/fulltext/reso/021/12/1105-1117">"Geometry of Spin: Clifford Algebraic Approach"</a>. 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"Opérateurs de Dirac et équations de Maxwell". <a href="/wiki/Commentarii_Mathematici_Helvetici" title="Commentarii Mathematici Helvetici">Commentarii Mathematici Helvetici</a> (in French). 2: 225–235. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01214461">10.1007/BF01214461</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:121226923">121226923</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Commentarii+Mathematici+Helvetici&rft.atitle=Op%C3%A9rateurs+de+Dirac+et+%C3%A9quations+de+Maxwell&rft.volume=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E225-%3C%2Fspan%3E235&rft.date=1930&rft_id=info%3Adoi%2F10.1007%2FBF01214461&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121226923%23id-name%3DS2CID&rft.aulast=Juvet&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSauter1930" class="citation journal cs1">Sauter, F. (1930). "Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren". Zeitschrift für Physik. 63 (11–12): 803–814. <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/1930ZPhy...63..803S">1930ZPhy...63..803S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01339277">10.1007/BF01339277</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:122940202">122940202</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=L%C3%B6sung+der+Diracschen+Gleichungen+ohne+Spezialisierung+der+Diracschen+Operatoren&rft.volume=63&rft.issue=%3Cspan+class%3D%22nowrap%22%3E11%E2%80%93%3C%2Fspan%3E12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E803-%3C%2Fspan%3E814&rft.date=1930&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122940202%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01339277&rft_id=info%3Abibcode%2F1930ZPhy...63..803S&rft.aulast=Sauter&rft.aufirst=F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"> </li> <li id="cite_note-lounesto-1995-p151-35">^ <a href="#cite_ref-lounesto-1995-p151_35-0">a</a> <a href="#cite_ref-lounesto-1995-p151_35-1">b</a> Pertti Lounesto: <a href="/wiki/Albert_Crumeyrolle" title="Albert Crumeyrolle">Crumeyrolle</a>'s bivectors and spinors, pp. 137–166, In: Rafał Abłamowicz, Pertti Lounesto (eds.): Clifford algebras and spinor structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992), <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/0-7923-3366-7" title="Special:BookSources/0-7923-3366-7">0-7923-3366-7</a>, 1995, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DnyUDg483kEC&pg=PA151">p. 151</a> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> See also: Pertti Lounesto: Clifford algebras and spinors, London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, <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/978-0-521-00551-7" title="Special:BookSources/978-0-521-00551-7">978-0-521-00551-7</a>, p. 52 </li> <li id="cite_note-lounesto-2001-p148f-p327f-38">^ <a href="#cite_ref-lounesto-2001-p148f-p327f_38-0">a</a> <a href="#cite_ref-lounesto-2001-p148f-p327f_38-1">b</a> Pertti Lounesto: Clifford algebras and spinors, London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, <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/978-0-521-00551-7" title="Special:BookSources/978-0-521-00551-7">978-0-521-00551-7</a>, p. 148 f. and <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DTecU6UpkSgC&pg=PA327">p. 327 f.</a> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> D. Hestenes: Space–Time Algebra, Gordon and Breach, New York, 1966, 1987, 1992 </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHestenes1967" class="citation journal cs1">Hestenes, D. (1967). <a rel="nofollow" class="external text" href="https://davidhestenes.net/geocalc/pdf/RealSpinorFields.pdf">"Real spinor fields"</a> (PDF). <a href="/wiki/Journal_of_Mathematical_Physics" title="Journal of Mathematical Physics">J. Math. Phys.</a> 8 (4): 798–808. <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/1967JMP.....8..798H">1967JMP.....8..798H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1705279">10.1063/1.1705279</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:13371668">13371668</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Math.+Phys.&rft.atitle=Real+spinor+fields&rft.volume=8&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E798-%3C%2Fspan%3E808&rft.date=1967&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13371668%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1063%2F1.1705279&rft_id=info%3Abibcode%2F1967JMP.....8..798H&rft.aulast=Hestenes&rft.aufirst=D.&rft_id=https%3A%2F%2Fdavidhestenes.net%2Fgeocalc%2Fpdf%2FRealSpinorFields.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> This construction is due to Cartan (1913). The treatment here is based on <a href="#CITEREFChevalley1996">Chevalley (1996)</a>. </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> One source for this subsection is <a href="#CITEREFFultonHarris1991">Fulton & Harris (1991)</a>. </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> Jurgen Jost, "Riemannian Geometry and Geometric Analysis" (2002) Springer-Verlag Universitext <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/3-540-42627-2" title="Special:BookSources/3-540-42627-2">3-540-42627-2</a>. See chapter 1. </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> Via the even-graded Clifford algebra. </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <a href="#CITEREFLawsonMichelsohn1989">Lawson & Michelsohn 1989</a>, Appendix D. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <a href="#CITEREFBrauerWeyl1935">Brauer & Weyl 1935</a>. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Works_cited">Works cited</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=30" title="Edit section: Works cited">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrauerWeyl1935" class="citation journal cs1"><a href="/wiki/Richard_Brauer" title="Richard Brauer">Brauer, Richard</a>; <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl, Hermann</a> (1935). "Spinors in n dimensions". American Journal of Mathematics. 57 (2). The Johns Hopkins University Press: 425–449. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2371218">10.2307/2371218</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2371218">2371218</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+Mathematics&rft.atitle=Spinors+in+n+dimensions&rft.volume=57&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E425-%3C%2Fspan%3E449&rft.date=1935&rft_id=info%3Adoi%2F10.2307%2F2371218&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2371218%23id-name%3DJSTOR&rft.aulast=Brauer&rft.aufirst=Richard&rft.au=Weyl%2C+Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCartan1913" class="citation journal cs1"><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan, Élie</a> (1913). <a rel="nofollow" class="external text" href="http://archive.numdam.org/article/BSMF_1913__41__53_1.pdf">"Les groupes projectifs qui ne laissent invariante aucune multiplicité plane"</a> (PDF). Bull. Soc. Math. Fr. 41: 53–96. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.916">10.24033/bsmf.916</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Soc.+Math.+Fr.&rft.atitle=Les+groupes+projectifs+qui+ne+laissent+invariante+aucune+multiplicit%C3%A9+plane&rft.volume=41&rft.pages=%3Cspan+class%3D%22nowrap%22%3E53-%3C%2Fspan%3E96&rft.date=1913&rft_id=info%3Adoi%2F10.24033%2Fbsmf.916&rft.aulast=Cartan&rft.aufirst=%C3%89lie&rft_id=http%3A%2F%2Farchive.numdam.org%2Farticle%2FBSMF_1913__41__53_1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChevalley1996" class="citation book cs1"><a href="/wiki/Claude_Chevalley" title="Claude Chevalley">Chevalley, Claude</a> (1996) [1954]. The Algebraic Theory of Spinors and Clifford Algebras (reprint ed.). Columbia University Press (1954); Springer (1996). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-57063-9" title="Special:BookSources/978-3-540-57063-9"><bdi>978-3-540-57063-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Algebraic+Theory+of+Spinors+and+Clifford+Algebras&rft.edition=reprint&rft.pub=Columbia+University+Press+%281954%29%3B+Springer+%281996%29&rft.date=1996&rft.isbn=978-3-540-57063-9&rft.aulast=Chevalley&rft.aufirst=Claude&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDirac1928" class="citation journal cs1"><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac, Paul M.</a> (1928). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1928.0023">"The quantum theory of the electron"</a>. Proceedings of the Royal Society of London A. 117 (778): 610–624. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1928RSPSA.117..610D">1928RSPSA.117..610D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1928.0023">10.1098/rspa.1928.0023</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/94981">94981</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London+A&rft.atitle=The+quantum+theory+of+the+electron&rft.volume=117&rft.issue=778&rft.pages=%3Cspan+class%3D%22nowrap%22%3E610-%3C%2Fspan%3E624&rft.date=1928&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F94981%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1098%2Frspa.1928.0023&rft_id=info%3Abibcode%2F1928RSPSA.117..610D&rft.aulast=Dirac&rft.aufirst=Paul+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frspa.1928.0023&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFultonHarris1991" class="citation book cs1"><a href="/wiki/William_Fulton_(mathematician)" title="William Fulton (mathematician)">Fulton, William</a>; <a href="/wiki/Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, Joe</a> (1991). Representation Theory: A first course. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, Readings in Mathematics. Vol. 129. New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-0979-9">10.1007/978-1-4612-0979-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97495-4" title="Special:BookSources/0-387-97495-4"><bdi>0-387-97495-4</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1153249">1153249</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Representation+Theory%3A+A+first+course&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics%2C+Readings+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1991&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1153249%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-0979-9&rft.isbn=0-387-97495-4&rft.aulast=Fulton&rft.aufirst=William&rft.au=Harris%2C+Joe&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLawsonMichelsohn1989" class="citation book cs1"><a href="/wiki/H._Blaine_Lawson" title="H. Blaine Lawson">Lawson, H. Blaine</a>; <a href="/wiki/Marie-Louise_Michelsohn" title="Marie-Louise Michelsohn">Michelsohn, Marie-Louise</a> (1989). Spin Geometry. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08542-0" title="Special:BookSources/0-691-08542-0"><bdi>0-691-08542-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spin+Geometry&rft.pub=Princeton+University+Press&rft.date=1989&rft.isbn=0-691-08542-0&rft.aulast=Lawson&rft.aufirst=H.+Blaine&rft.au=Michelsohn%2C+Marie-Louise&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPauli1927" class="citation journal cs1"><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli, Wolfgang</a> (1927). "Zur Quantenmechanik des magnetischen Elektrons". Zeitschrift für Physik. 43 (9–10): 601–632. <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/1927ZPhy...43..601P">1927ZPhy...43..601P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01397326">10.1007/BF01397326</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:128228729">128228729</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=Zur+Quantenmechanik+des+magnetischen+Elektrons&rft.volume=43&rft.issue=%3Cspan+class%3D%22nowrap%22%3E9%E2%80%93%3C%2Fspan%3E10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E601-%3C%2Fspan%3E632&rft.date=1927&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A128228729%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01397326&rft_id=info%3Abibcode%2F1927ZPhy...43..601P&rft.aulast=Pauli&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTomonaga1998" class="citation book cs1">Tomonaga, Sin-Itiro (1998). "Lecture 7: The quantity which is neither vector nor tensor". The Story of Spin. University of Chicago Press. p. 129. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-80794-0" title="Special:BookSources/0-226-80794-0"><bdi>0-226-80794-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lecture+7%3A+The+quantity+which+is+neither+vector+nor+tensor&rft.btitle=The+Story+of+Spin&rft.pages=129&rft.pub=University+of+Chicago+Press&rft.date=1998&rft.isbn=0-226-80794-0&rft.aulast=Tomonaga&rft.aufirst=Sin-Itiro&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Spinor&action=edit&section=31" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCartan1981" class="citation book cs1"><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan, Élie</a> (1981) [1966]. The Theory of Spinors (reprint ed.). Paris, FR: Hermann (1966); Dover Publications (1981). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-64070-9" title="Special:BookSources/978-0-486-64070-9"><bdi>978-0-486-64070-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Spinors&rft.place=Paris%2C+FR&rft.edition=reprint&rft.pub=Hermann+%281966%29%3B+Dover+Publications+%281981%29&rft.date=1981&rft.isbn=978-0-486-64070-9&rft.aulast=Cartan&rft.aufirst=%C3%89lie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGilkey1984" class="citation book cs1"><a href="/wiki/Peter_B._Gilkey" title="Peter B. Gilkey">Gilkey, Peter B.</a> (1984). <a rel="nofollow" class="external text" href="http://www.emis.de/monographs/gilkey/index.html">Invariance Theory: The heat equation, and the Atiyah–Singer index theorem</a>. Publish or Perish. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-914098-20-9" title="Special:BookSources/0-914098-20-9"><bdi>0-914098-20-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Invariance+Theory%3A+The+heat+equation%2C+and+the+Atiyah%E2%80%93Singer+index+theorem&rft.pub=Publish+or+Perish&rft.date=1984&rft.isbn=0-914098-20-9&rft.aulast=Gilkey&rft.aufirst=Peter+B.&rft_id=http%3A%2F%2Fwww.emis.de%2Fmonographs%2Fgilkey%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHarvey1990" class="citation book cs1">Harvey, F. Reese (1990). Spinors and Calibrations. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-329650-4" title="Special:BookSources/978-0-12-329650-4"><bdi>978-0-12-329650-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spinors+and+Calibrations&rft.pub=Academic+Press&rft.date=1990&rft.isbn=978-0-12-329650-4&rft.aulast=Harvey&rft.aufirst=F.+Reese&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHitchin1974" class="citation journal cs1"><a href="/wiki/Nigel_Hitchin" title="Nigel Hitchin">Hitchin, Nigel J.</a> (1974). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0001-8708%2874%2990021-8">"Harmonic spinors"</a>. <a href="/wiki/Advances_in_Mathematics" title="Advances in Mathematics">Advances in Mathematics</a>. 14: 1–55. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0001-8708%2874%2990021-8">10.1016/0001-8708(74)90021-8</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0358873">0358873</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Mathematics&rft.atitle=Harmonic+spinors&rft.volume=14&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E55&rft.date=1974&rft_id=info%3Adoi%2F10.1016%2F0001-8708%2874%2990021-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D358873%23id-name%3DMR&rft.aulast=Hitchin&rft.aufirst=Nigel+J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0001-8708%252874%252990021-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPenroseRindler1988" class="citation book cs1"><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose, Roger</a>; Rindler, W. (1988). Spinor and twistor methods in space-time geometry. Spinors and Space-Time. Vol. 2. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-34786-6" title="Special:BookSources/0-521-34786-6"><bdi>0-521-34786-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spinor+and+twistor+methods+in+space-time+geometry&rft.series=Spinors+and+Space-Time&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=0-521-34786-6&rft.aulast=Penrose&rft.aufirst=Roger&rft.au=Rindler%2C+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpinor" class="Z3988"></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline 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class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinate_system" title="Coordinate system">Coordinate system</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li> <li><a href="/wiki/Dyadics" title="Dyadics">Dyadic algebra</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="/wiki/Physics" title="Physics">Physics</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li> <li><a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a></li> <li><a href="/wiki/Index_notation" title="Index notation">Index notation</a></li> <li><a href="/wiki/Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li> <li><a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li> <li><a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li> <li><a href="/wiki/Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li> <li><a href="/wiki/Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li> <li><a href="/wiki/Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor definitions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a></li> <li><a href="/wiki/Tensor_density" title="Tensor density">Tensor density</a></li> <li><a href="/wiki/Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li> <li><a href="/wiki/Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li> <li><a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li> <li><a href="/wiki/Tensor_operator" title="Tensor operator">Tensor operator</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li> <li><a href="/wiki/Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li> <li><a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a></li> <li><a href="/wiki/Symmetrization" title="Symmetrization">Symmetrization</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Connection form</a></li> <li><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Dimension" title="Dimension">Dimension</a></li> <li><a href="/wiki/Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Pseudotensor" title="Pseudotensor">Pseudotensor</a></li> <li><a class="mw-selflink selflink">Spinor</a></li> <li><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li> <li><a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li> <li><a href="/wiki/Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li> <li><a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li> <li><a href="/wiki/Tullio_Levi-Civita" 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