Lorentz group - Wikipedia

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class="vector-toc-list"> <li id="toc-Physics_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physics_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Physics definition</span> </div> </a> <ul id="toc-Physics_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Mathematical definition</span> </div> </a> <ul id="toc-Mathematical_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Mathematical properties</span> </div> </a> <ul id="toc-Mathematical_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connected_components" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connected_components"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Connected components</span> </div> </a> <ul id="toc-Connected_components-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Restricted_Lorentz_group" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Restricted_Lorentz_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Restricted Lorentz group</span> </div> </a> <button aria-controls="toc-Restricted_Lorentz_group-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Restricted Lorentz group subsection</span> </button> <ul id="toc-Restricted_Lorentz_group-sublist" class="vector-toc-list"> <li id="toc-Surfaces_of_transitivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surfaces_of_transitivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Surfaces of transitivity</span> </div> </a> <ul id="toc-Surfaces_of_transitivity-sublist" class="vector-toc-list"> <li id="toc-As_symmetric_spaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#As_symmetric_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>As symmetric spaces</span> </div> </a> <ul id="toc-As_symmetric_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Representations_of_the_Lorentz_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representations_of_the_Lorentz_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Representations of the Lorentz group</span> </div> </a> <ul id="toc-Representations_of_the_Lorentz_group-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Homomorphisms_and_isomorphisms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Homomorphisms_and_isomorphisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Homomorphisms and isomorphisms</span> </div> </a> <button aria-controls="toc-Homomorphisms_and_isomorphisms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Homomorphisms and isomorphisms subsection</span> </button> <ul id="toc-Homomorphisms_and_isomorphisms-sublist" class="vector-toc-list"> <li id="toc-Weyl_representation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weyl_representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Weyl representation</span> </div> </a> <ul id="toc-Weyl_representation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notational_conventions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notational_conventions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Notational conventions</span> </div> </a> <ul id="toc-Notational_conventions-sublist" class="vector-toc-list"> <li id="toc-Proof" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Proof</span> </div> </a> <ul id="toc-Proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Symplectic_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symplectic_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Symplectic group</span> </div> </a> <ul id="toc-Symplectic_group-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Covering_groups" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Covering_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Covering groups</span> </div> </a> <ul id="toc-Covering_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Topology</span> </div> </a> <ul id="toc-Topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conjugacy_classes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Conjugacy_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Conjugacy classes</span> </div> </a> <button aria-controls="toc-Conjugacy_classes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Conjugacy classes subsection</span> </button> <ul id="toc-Conjugacy_classes-sublist" class="vector-toc-list"> <li id="toc-Elliptic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elliptic"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Elliptic</span> </div> </a> <ul id="toc-Elliptic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hyperbolic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hyperbolic"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Hyperbolic</span> </div> </a> <ul id="toc-Hyperbolic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Loxodromic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Loxodromic"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Loxodromic</span> </div> </a> <ul id="toc-Loxodromic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parabolic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parabolic"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Parabolic</span> </div> </a> <ul id="toc-Parabolic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Appearance_of_the_night_sky" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Appearance_of_the_night_sky"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Appearance of the night sky</span> </div> </a> <ul id="toc-Appearance_of_the_night_sky-sublist" class="vector-toc-list"> <li id="toc-Projective_geometry_and_different_views_of_the_2-sphere" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Projective_geometry_and_different_views_of_the_2-sphere"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.1</span> <span>Projective geometry and different views of the 2-sphere</span> </div> </a> <ul id="toc-Projective_geometry_and_different_views_of_the_2-sphere-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Lie_algebra" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Lie_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Lie algebra</span> </div> </a> <button aria-controls="toc-Lie_algebra-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Lie algebra subsection</span> </button> <ul id="toc-Lie_algebra-sublist" class="vector-toc-list"> <li id="toc-Commutation_relations_of_the_Lorentz_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Commutation_relations_of_the_Lorentz_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Commutation relations of the Lorentz algebra</span> </div> </a> <ul id="toc-Commutation_relations_of_the_Lorentz_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generators_of_boosts_and_rotations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generators_of_boosts_and_rotations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Generators of boosts and rotations</span> </div> </a> <ul id="toc-Generators_of_boosts_and_rotations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generators_of_the_Möbius_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generators_of_the_Möbius_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Generators of the Möbius group</span> </div> </a> <ul id="toc-Generators_of_the_Möbius_group-sublist" class="vector-toc-list"> <li id="toc-Worked_example:_rotation_about_the_y-axis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Worked_example:_rotation_about_the_y-axis"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3.1</span> <span>Worked example: rotation about the y-axis</span> </div> </a> <ul id="toc-Worked_example:_rotation_about_the_y-axis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Subgroups_of_the_Lorentz_group" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Subgroups_of_the_Lorentz_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Subgroups of the Lorentz group</span> </div> </a> <ul id="toc-Subgroups_of_the_Lorentz_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization_to_higher_dimensions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalization_to_higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Generalization to higher dimensions</span> </div> </a> <ul id="toc-Generalization_to_higher_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"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reading_List" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Reading_List"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Reading List</span> </div> </a> <ul id="toc-Reading_List-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Lorentz group</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" 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Available in 18 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-18" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">18 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_%D9%84%D9%88%D8%B1%D9%86%D8%AA%D8%B2" title="زمرة لورنتز – Arabic" lang="ar" hreflang="ar" data-title="زمرة لورنتز" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_%D0%9B%D0%BE%D1%80%D1%8D%D0%BD%D1%86%D0%B0" title="Група Лорэнца – Belarusian" lang="be" hreflang="be" data-title="Група Лорэнца" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_de_Lorentz" title="Grup de Lorentz – Catalan" lang="ca" hreflang="ca" data-title="Grup de Lorentz" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Lorentzova_grupa" title="Lorentzova grupa – Czech" lang="cs" hreflang="cs" data-title="Lorentzova grupa" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lorentz-Gruppe" title="Lorentz-Gruppe – German" lang="de" hreflang="de" data-title="Lorentz-Gruppe" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_de_Lorentz" title="Grupo de Lorentz – Spanish" lang="es" hreflang="es" data-title="Grupo de Lorentz" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_de_Lorentz" title="Groupe de Lorentz – French" lang="fr" hreflang="fr" data-title="Groupe de Lorentz" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A1%9C%EB%9F%B0%EC%B8%A0_%EA%B5%B0" title="로런츠 군 – Korean" lang="ko" hreflang="ko" data-title="로런츠 군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_di_Lorentz" title="Gruppo di Lorentz – Italian" lang="it" hreflang="it" data-title="Gruppo di Lorentz" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lorentz-groep" title="Lorentz-groep – Dutch" lang="nl" hreflang="nl" data-title="Lorentz-groep" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AD%E3%83%BC%E3%83%AC%E3%83%B3%E3%83%84%E7%BE%A4" title="ローレンツ群 – Japanese" lang="ja" hreflang="ja" data-title="ローレンツ群" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A9%8C%E0%A8%B0%E0%A9%B0%E0%A8%9F%E0%A8%9C%E0%A8%BC_%E0%A8%97%E0%A8%B0%E0%A9%81%E0%A9%B1%E0%A8%AA" title="ਲੌਰੰਟਜ਼ ਗਰੁੱਪ – Punjabi" lang="pa" hreflang="pa" data-title="ਲੌਰੰਟਜ਼ ਗਰੁੱਪ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_Lorentza" title="Grupa Lorentza – Polish" lang="pl" hreflang="pl" data-title="Grupa Lorentza" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_de_Lorentz" title="Grupo de Lorentz – Portuguese" lang="pt" hreflang="pt" data-title="Grupo de Lorentz" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%BF%D0%B0_%D0%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86%D0%B0" title="Группа Лоренца – Russian" lang="ru" hreflang="ru" data-title="Группа Лоренца" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_%D0%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86%D0%B0" title="Група Лоренца – Ukrainian" lang="uk" hreflang="uk" data-title="Група Лоренца" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_bi%E1%BA%BFn_%C4%91%E1%BB%95i_Lorentz" title="Nhóm biến đổi Lorentz – Vietnamese" lang="vi" hreflang="vi" data-title="Nhóm biến đổi Lorentz" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8B%9E%E4%BE%96%E8%8C%B2%E7%BE%A4" title="勞侖茲群 – Chinese" lang="zh" hreflang="zh" data-title="勞侖茲群" data-language-autonym="中文" data-language-local-name="Chinese" 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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">Lie group of Lorentz transformations</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist 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nomobile nowraplinks" style="width:20.0em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → <b>Group theory</b></span><br /><a href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert"><span typeof="mw:File"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></span></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Basic notions</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <i><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></i></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_group" title="Simple group">simple</a></li> <li><a href="/wiki/Finite_group" title="Finite group">finite</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">infinite</a></li> <li><a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li> <li><a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative</a></li> <li><a href="/wiki/Additive_group" title="Additive group">additive</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">abelian</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">solvable</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Z<sub><i>n</i></sub></li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> S<sub><i>n</i></sub></li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub><i>n</i></sub></li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> D<sub><i>n</i></sub></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li> <li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul> <ul><li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul> <ul><li><a href="/wiki/P-group" title="P-group"><i>p</i>-group</a></li> <li><a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul> <ul><li><a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul> <ul><li><a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></th></tr><tr><td class="sidebar-content"> <ul><li>cyclic</li> <li>alternating</li> <li><a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Lie type</a></li> <li><a href="/wiki/Sporadic_group" title="Sporadic group">sporadic</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">Lattices</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Integer" title="Integer">Integers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Free_group" title="Free group">Free group</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li><li>SL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li></ul></div></div> <ul><li><a href="/wiki/Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li> <li><a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice</a></li> <li><a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Hyperbolic group</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Topological_group" title="Topological group">Topological</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li></ul> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></li></ul> <ul><li><a class="mw-selflink selflink">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal</a></li></ul> <ul><li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Infinite_dimensional_Lie_group" class="mw-redirect" title="Infinite dimensional Lie group">Infinite dimensional Lie group</a> <div class="hlist"><ul><li>O(∞)</li><li>SU(∞)</li><li>Sp(∞)</li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Algebraic_group" title="Algebraic group">Algebraic groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul> <ul><li><a href="/wiki/Reductive_group" title="Reductive group">Reductive group</a></li></ul> <ul><li><a href="/wiki/Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_theory_sidebar" title="Template:Group theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_theory_sidebar" title="Template talk:Group theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_theory_sidebar" title="Special:EditPage/Template:Group theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hendrik_Antoon_Lorentz.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Hendrik_Antoon_Lorentz.jpg/300px-Hendrik_Antoon_Lorentz.jpg" decoding="async" width="300" height="429" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/33/Hendrik_Antoon_Lorentz.jpg 1.5x" data-file-width="420" data-file-height="600" /></a><figcaption><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Hendrik Antoon Lorentz</a> (1853–1928), after whom the Lorentz group is named.</figcaption></figure> <p>In <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Lorentz group</b> is the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of all <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a> of <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a>, the <a href="/wiki/Classical_field_theory" title="Classical field theory">classical</a> and <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum</a> setting for all (non-gravitational) <a href="/wiki/Physics" title="Physics">physical phenomena</a>. The Lorentz group is named for the <a href="/wiki/Dutch_people" title="Dutch people">Dutch</a> physicist <a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Hendrik Lorentz</a>. </p><p>For example, the following laws, equations, and theories respect Lorentz symmetry: </p> <ul><li>The <a href="/wiki/Kinematics" title="Kinematics">kinematical laws</a> of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's field equations</a> in the theory of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a></li> <li>The <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> in the theory of the <a href="/wiki/Electron" title="Electron">electron</a></li> <li>The <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of particle physics</li></ul> <p>The Lorentz group expresses the fundamental <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> of space and time of all known fundamental <a href="/wiki/Laws_of_science" class="mw-redirect" title="Laws of science">laws of nature</a>. In small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as special relativity. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_properties">Basic properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=1" title="Edit section: Basic properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Lorentz group is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>—the group of all <a href="/wiki/Isometry#Generalizations" title="Isometry">isometries</a> of <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a>. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is the <a href="/wiki/Group_action_(mathematics)#Fixed_points_and_stabilizer_subgroups" class="mw-redirect" title="Group action (mathematics)">isotropy subgroup</a> with respect to the origin of the <a href="/wiki/Isometry_group" title="Isometry group">isometry group</a> of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the <b>homogeneous Lorentz group</b> while the Poincaré group is sometimes called the <i>inhomogeneous Lorentz group</i>. Lorentz transformations are examples of <a href="/wiki/Linear_map" title="Linear map">linear transformations</a>; general isometries of Minkowski spacetime are <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformations</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Physics_definition">Physics definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=2" title="Edit section: Physics definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Assume two <a href="/wiki/Inertial_reference_frames" class="mw-redirect" title="Inertial reference frames">inertial reference frames</a> <span class="texhtml">(<i>t</i>, <i>x</i>, <i>y</i>, <i>z</i>)</span> and <span class="texhtml">(<i>t</i>′, <i>x</i>′, <i>y</i>′, <i>z</i>′)</span>, and two points <span class="texhtml"><i>P</i><sub>1</sub></span>, <span class="texhtml"><i>P</i><sub>2</sub></span>, the Lorentz group is the set of all the transformations between the two reference frames that preserve the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> propagating between the two points: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}(\Delta t')^{2}-(\Delta x')^{2}-(\Delta y')^{2}-(\Delta z')^{2}=c^{2}(\Delta t)^{2}-(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>z</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}(\Delta t')^{2}-(\Delta x')^{2}-(\Delta y')^{2}-(\Delta z')^{2}=c^{2}(\Delta t)^{2}-(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce47dc77492776b8ee97845e5b25ec39aec7171a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:74.23ex; height:3.176ex;" alt="{\displaystyle c^{2}(\Delta t')^{2}-(\Delta x')^{2}-(\Delta y')^{2}-(\Delta z')^{2}=c^{2}(\Delta t)^{2}-(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2}}"></span></dd></dl> <p>In matrix form these are all the linear transformations <span class="texhtml">Λ</span> such that: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{\textsf {T}}\eta \Lambda =\eta \qquad \eta =\operatorname {diag} (1,-1,-1,-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mi>η</mi> <mi mathvariant="normal">Λ</mi> <mo>=</mo> <mi>η</mi> <mspace width="2em" /> <mi>η</mi> <mo>=</mo> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda ^{\textsf {T}}\eta \Lambda =\eta \qquad \eta =\operatorname {diag} (1,-1,-1,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2e1b7b83c05dee741f69df2836191564a71364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.178ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{\textsf {T}}\eta \Lambda =\eta \qquad \eta =\operatorname {diag} (1,-1,-1,-1)}"></span></dd></dl> <p>These are then called <a href="/wiki/Lorentz_transformations" class="mw-redirect" title="Lorentz transformations">Lorentz transformations</a> </p> <div class="mw-heading mw-heading3"><h3 id="Mathematical_definition">Mathematical definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=3" title="Edit section: Mathematical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematically, the Lorentz group may be described as the <a href="/wiki/Indefinite_orthogonal_group" title="Indefinite orthogonal group">indefinite orthogonal group</a> <span class="texhtml">O(1, 3)</span>, the <a href="/wiki/Matrix_Lie_group" class="mw-redirect" title="Matrix Lie group">matrix Lie group</a> that preserves the <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t,x,y,z)\mapsto t^{2}-x^{2}-y^{2}-z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t,x,y,z)\mapsto t^{2}-x^{2}-y^{2}-z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfc539a353993e4a33ba39726d1de63190e3e92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.096ex; height:3.176ex;" alt="{\displaystyle (t,x,y,z)\mapsto t^{2}-x^{2}-y^{2}-z^{2}}"></span></dd></dl> <p>on <span class="texhtml"><b>R</b><sup>4</sup></span> (the vector space equipped with this quadratic form is sometimes written <span class="texhtml"><b>R</b><sup>1,3</sup></span>). This quadratic form is, when put on matrix form (see <i><a href="/wiki/Classical_group#O(p,_q)_and_O(n)_–_the_orthogonal_groups" title="Classical group">Classical orthogonal group</a></i>), interpreted in physics as the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> of Minkowski spacetime. </p> <div class="mw-heading mw-heading3"><h3 id="Mathematical_properties">Mathematical properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=4" title="Edit section: Mathematical properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Lorentz group is a six-<a href="/wiki/Dimension" title="Dimension">dimensional</a> <a href="/wiki/Compact_space" title="Compact space">noncompact</a> <a href="/wiki/Non-abelian_group" title="Non-abelian group">non-abelian</a> <a href="/wiki/Real_Lie_group" class="mw-redirect" title="Real Lie group">real Lie group</a> that is not <a href="/wiki/Connected_space" title="Connected space">connected</a>. The four <a href="/wiki/Connected_component_(topology)" class="mw-redirect" title="Connected component (topology)">connected components</a> are not <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Identity_component" title="Identity component">identity component</a> (i.e., the component containing the identity element) of the Lorentz group is itself a group, and is often called the <b>restricted Lorentz group</b>, and is denoted <span class="texhtml">SO<sup>+</sup>(1, 3)</span>. The restricted Lorentz group consists of those Lorentz transformations that preserve both the <a href="/wiki/Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a> of space and the direction of time. Its <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> has order 2, and its universal cover, the <a href="/wiki/Spin_group#Indefinite_signature" title="Spin group">indefinite spin group</a> <span class="texhtml">Spin(1, 3)</span>, is isomorphic to both the <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a> <span class="texhtml">SL(2, <b>C</b>)</span> and to the <a href="/wiki/Symplectic_group" title="Symplectic group">symplectic group</a> <span class="texhtml">Sp(2, <b>C</b>)</span>. These isomorphisms allow the Lorentz group to act on a large number of mathematical structures important to physics, most notably <a href="/wiki/Spinor" title="Spinor">spinors</a>. Thus, in <a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">relativistic quantum mechanics</a> and in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, it is very common to call <span class="texhtml">SL(2, <b>C</b>)</span> the Lorentz group, with the understanding that <span class="texhtml">SO<sup>+</sup>(1, 3)</span> is a specific representation (the vector representation) of it. </p><p>A recurrent representation of the action of the Lorentz group on Minkowski space uses <a href="/wiki/Biquaternion" title="Biquaternion">biquaternions</a>, which form a <a href="/wiki/Composition_algebra" title="Composition algebra">composition algebra</a>. The isometry property of Lorentz transformations holds according to the composition property <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |pq|=|p|\times |q|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |pq|=|p|\times |q|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe667e2394725b0860a31f08bd1d6ca120cb694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.298ex; height:2.843ex;" alt="{\displaystyle |pq|=|p|\times |q|}"></span>⁠</span>. </p><p>Another property of the Lorentz group is <i>conformality</i> or preservation of angles. Lorentz boosts act by <a href="/wiki/Hyperbolic_rotation" class="mw-redirect" title="Hyperbolic rotation">hyperbolic rotation</a> of a spacetime plane, and such "rotations" preserve <a href="/wiki/Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angle</a>, the measure of <a href="/wiki/Rapidity" title="Rapidity">rapidity</a> used in relativity. Therefore, the Lorentz group is a subgroup of the <a href="/wiki/Conformal_group#Conformal_group_of_spacetime" title="Conformal group">conformal group of spacetime</a>. </p><p>Note that this article refers to <span class="texhtml">O(1, 3)</span> as the "Lorentz group", <span class="texhtml">SO(1, 3)</span> as the "proper Lorentz group", and <span class="texhtml">SO<sup>+</sup>(1, 3)</span> as the "restricted Lorentz group". Many authors (especially in physics) use the name "Lorentz group" for <span class="texhtml">SO(1, 3)</span> (or sometimes even <span class="texhtml">SO<sup>+</sup>(1, 3)</span>) rather than <span class="texhtml">O(1, 3)</span>. When reading such authors it is important to keep clear exactly which they are referring to. </p> <div class="mw-heading mw-heading3"><h3 id="Connected_components">Connected components</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=5" title="Edit section: Connected components"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:World_line.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/World_line.svg/300px-World_line.svg.png" decoding="async" width="300" height="306" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/World_line.svg/450px-World_line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/World_line.svg/600px-World_line.svg.png 2x" data-file-width="481" data-file-height="491" /></a><figcaption>Light cone in 2D space plus a time dimension.</figcaption></figure> <p>Because it is a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>, the Lorentz group <span class="texhtml">O(1, 3)</span> is a group and also has a topological description as a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>. As a manifold, it has four connected components. Intuitively, this means that it consists of four topologically separated pieces. </p><p>The four connected components can be categorized by two transformation properties its elements have: </p> <ul><li>Some elements are reversed under time-inverting Lorentz transformations, for example, a future-pointing <a href="/wiki/Timelike_vector" class="mw-redirect" title="Timelike vector">timelike vector</a> would be inverted to a past-pointing vector</li> <li>Some elements have orientation reversed by <b>improper Lorentz transformations</b>, for example, certain <a href="/wiki/Vierbein" class="mw-redirect" title="Vierbein">vierbein</a> (tetrads)</li></ul> <p>Lorentz transformations that preserve the direction of time are called <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="orthochronous"></span><span class="vanchor-text">orthochronous</span></span></b>. The subgroup of orthochronous transformations is often denoted <span class="texhtml">O<sup>+</sup>(1, 3)</span>. Those that preserve orientation are called <b>proper</b>, and as linear transformations they have determinant <span class="texhtml">+1</span>. (The improper Lorentz transformations have determinant <span class="texhtml">−1</span>.) The subgroup of proper Lorentz transformations is denoted <span class="texhtml">SO(1, 3)</span>. </p><p>The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the <b>proper, orthochronous Lorentz group</b> or <b>restricted Lorentz group</b>, and is denoted by <span class="texhtml">SO<sup>+</sup>(1, 3)</span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </p><p>The set of the four connected components can be given a group structure as the <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> <span class="texhtml">O(1, 3) / SO<sup>+</sup>(1, 3)</span>, which is isomorphic to the <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a>. Every element in <span class="texhtml">O(1, 3)</span> can be written as the <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of a proper, orthochronous transformation and an element of the <a href="/wiki/Discrete_group" title="Discrete group">discrete group</a> </p> <dl><dd><span class="texhtml">{1, <i>P</i>, <i>T</i>, <i>PT</i>}</span></dd></dl> <p>where <i>P</i> and <i>T</i> are the <a href="/wiki/P-symmetry" class="mw-redirect" title="P-symmetry">parity</a> and <a href="/wiki/T-symmetry" title="T-symmetry">time reversal</a> operators: </p> <dl><dd><span class="texhtml"><i>P</i> = diag(1, −1, −1, −1)</span></dd> <dd><span class="texhtml"><i>T</i> = diag(−1, 1, 1, 1)</span>.</dd></dl> <p>Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two bits of information, which pick out one of the four connected components. This pattern is typical of finite-dimensional Lie groups. </p> <div class="mw-heading mw-heading2"><h2 id="Restricted_Lorentz_group">Restricted Lorentz group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=6" title="Edit section: Restricted Lorentz group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The restricted Lorentz group <span class="texhtml">SO<sup>+</sup>(1, 3)</span> is the <a href="/wiki/Identity_component" title="Identity component">identity component</a> of the Lorentz group, which means that it consists of all Lorentz transformations that can be connected to the identity by a <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> curve lying in the group. The restricted Lorentz group is a connected <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of the full Lorentz group with the same dimension, in this case with dimension six. </p><p>The restricted Lorentz group is generated by ordinary <a href="/wiki/Coordinate_rotation" class="mw-redirect" title="Coordinate rotation">spatial rotations</a> and <a href="/wiki/Lorentz_boost" class="mw-redirect" title="Lorentz boost">Lorentz boosts</a> (which are rotations in a hyperbolic space that includes a time-like direction<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>). Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation (specified by <a href="/wiki/Charts_on_SO(3)" title="Charts on SO(3)">3 real parameters</a>) and a boost (also specified by 3 real parameters), it takes 6 real parameters to specify an arbitrary proper orthochronous Lorentz transformation. This is one way to understand why the restricted Lorentz group is six-dimensional. (See also the <a href="#Lie_algebra">Lie algebra of the Lorentz group</a>.) </p><p>The set of all rotations forms a <a href="/wiki/Lie_subgroup" class="mw-redirect" title="Lie subgroup">Lie subgroup</a> isomorphic to the ordinary <a href="/wiki/Rotation_group_SO(3)" class="mw-redirect" title="Rotation group SO(3)">rotation group <span class="texhtml">SO(3)</span></a>. The set of all boosts, however, does <i>not</i> form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of non-colinear boosts is equivalent to a boost and a rotation, and this relates to <a href="/wiki/Thomas_rotation" class="mw-redirect" title="Thomas rotation">Thomas rotation</a>.) A boost in some direction, or a rotation about some axis, generates a <a href="/wiki/One-parameter_subgroup" class="mw-redirect" title="One-parameter subgroup">one-parameter subgroup</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Surfaces_of_transitivity">Surfaces of transitivity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=7" title="Edit section: Surfaces of transitivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/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 img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:490px;max-width:490px"><div class="trow"><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:HyperboloidOfOneSheet.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/HyperboloidOfOneSheet.png/150px-HyperboloidOfOneSheet.png" decoding="async" width="150" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/HyperboloidOfOneSheet.png/225px-HyperboloidOfOneSheet.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/HyperboloidOfOneSheet.png/300px-HyperboloidOfOneSheet.png 2x" data-file-width="385" data-file-height="426" /></a></span></div><div class="thumbcaption">Hyperboloid of one sheet</div></div><div class="tsingle" style="width:180px;max-width:180px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:DoubleCone.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/DoubleCone.png/178px-DoubleCone.png" decoding="async" width="178" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/DoubleCone.png/267px-DoubleCone.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/DoubleCone.png/356px-DoubleCone.png 2x" data-file-width="1350" data-file-height="1274" /></a></span></div><div class="thumbcaption">Common conical surface</div></div><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:HyperboloidOfTwoSheets.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/HyperboloidOfTwoSheets.png/150px-HyperboloidOfTwoSheets.png" decoding="async" width="150" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/HyperboloidOfTwoSheets.png/225px-HyperboloidOfTwoSheets.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/HyperboloidOfTwoSheets.png/300px-HyperboloidOfTwoSheets.png 2x" data-file-width="406" data-file-height="456" /></a></span></div><div class="thumbcaption">Hyperboloid of two sheets</div></div></div></div></div> <p>If a group <span class="texhtml"><i>G</i></span> acts on a space <span class="texhtml"><i>V</i></span>, then a surface <span class="texhtml"><i>S</i> ⊂ <i>V</i></span> is a <b>surface of transitivity</b> if <span class="texhtml"><i>S</i></span> is invariant under <span class="texhtml"><i>G</i></span> (i.e., <span class="texhtml">∀<i>g</i> ∈ <i>G</i>, ∀<i>s</i> ∈ <i>S</i>: <i>gs</i> ∈ <i>S</i></span>) and for any two points <span class="texhtml"><i>s</i><sub>1</sub>, <i>s</i><sub>2</sub> ∈ <i>S</i></span> there is a <span class="texhtml"><i>g</i> ∈ <i>G</i></span> such that <span class="texhtml"><i>gs</i><sub>1</sub> = <i>s</i><sub>2</sub></span>. By definition of the Lorentz group, it preserves the quadratic form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09c436b64dd3671256337bf2deeb654e273f932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.779ex; height:3.176ex;" alt="{\displaystyle Q(x)=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.}"></span></dd></dl> <p>The surfaces of transitivity of the orthochronous Lorentz group <span class="texhtml">O<sup>+</sup>(1, 3)</span>, <span class="texhtml"><i>Q</i>(<i>x</i>) = const.</span> acting on flat <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> <span class="texhtml"><b>R</b><sup>1,3</sup></span> are the following:<sup id="cite_ref-Gelfand_1_4-0" class="reference"><a href="#cite_note-Gelfand_1-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li><span class="texhtml"><i>Q</i>(<i>x</i>) > 0, x<sub>0</sub> > 0</span> is the upper branch of a <a href="/wiki/Hyperboloid" title="Hyperboloid">hyperboloid</a> of two sheets. Points on this sheet are separated from the origin by a future <a href="/wiki/Time-like" class="mw-redirect" title="Time-like">time-like</a> vector.</li> <li><span class="texhtml"><i>Q</i>(<i>x</i>) > 0, x<sub>0</sub> < 0</span> is the lower branch of this hyperboloid. Points on this sheet are the past <a href="/wiki/Time-like" class="mw-redirect" title="Time-like">time-like</a> vectors.</li> <li><span class="texhtml"><i>Q</i>(<i>x</i>) = 0, x<sub>0</sub> > 0</span> is the upper branch of the <a href="/wiki/Light_cone" title="Light cone">light cone</a>, the future light cone.</li> <li><span class="texhtml"><i>Q</i>(<i>x</i>) = 0, x<sub>0</sub> < 0</span> is the lower branch of the light cone, the past light cone.</li> <li><span class="texhtml"><i>Q</i>(<i>x</i>) < 0</span> is a hyperboloid of one sheet. Points on this sheet are <a href="/wiki/Space-like" class="mw-redirect" title="Space-like">space-like</a> separated from the origin.</li> <li>The origin <span class="texhtml"><i>x</i><sub>0</sub> = <i>x</i><sub>1</sub> = <i>x</i><sub>2</sub> = <i>x</i><sub>3</sub> = 0</span>.</li></ul> <p>These surfaces are <span class="nowrap"><span class="texhtml">3</span>-dimensional</span>, so the images are not faithful, but they are faithful for the corresponding facts about <span class="texhtml">O<sup>+</sup>(1, 2)</span>. For the full Lorentz group, the surfaces of transitivity are only four since the transformation <span class="texhtml"><i>T</i></span> takes an upper branch of a hyperboloid (cone) to a lower one and vice versa. </p> <div class="mw-heading mw-heading4"><h4 id="As_symmetric_spaces">As symmetric spaces</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=8" title="Edit section: As symmetric spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An equivalent way to formulate the above surfaces of transitivity is as a <a href="/wiki/Symmetric_space" title="Symmetric space">symmetric space</a> in the sense of Lie theory. For example, the upper sheet of the hyperboloid can be written as the quotient space <span class="texhtml">SO<sup>+</sup>(1, 3) / SO(3)</span>, due to the <a href="/wiki/Orbit-stabilizer_theorem" class="mw-redirect" title="Orbit-stabilizer theorem">orbit-stabilizer theorem</a>. Furthermore, this upper sheet also provides a model for three-dimensional <a href="/wiki/Hyperbolic_space" title="Hyperbolic space">hyperbolic space</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Representations_of_the_Lorentz_group">Representations of the Lorentz group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=9" title="Edit section: Representations of the Lorentz group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These observations constitute a good starting point for finding all <a href="/wiki/Representation_theory_of_the_Lorentz_group#Infinite-dimensional_unitary_representations" title="Representation theory of the Lorentz group">infinite-dimensional unitary representations</a> of the Lorentz group, in fact, of the Poincaré group, using the method of <a href="/wiki/Induced_representation" title="Induced representation">induced representations</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> One begins with a "standard vector", one for each surface of transitivity, and then ask which subgroup preserves these vectors. These subgroups are called <a href="/wiki/Little_group" class="mw-redirect" title="Little group">little groups</a> by physicists. The problem is then essentially reduced to the easier problem of finding representations of the little groups. For example, a standard vector in one of the hyperbolas of two sheets could be suitably chosen as <span class="texhtml">(<i>m</i>, 0, 0, 0)</span>. For each <span class="texhtml"><i>m</i> ≠ 0</span>, the vector pierces exactly one sheet. In this case the little group is <span class="texhtml">SO(3)</span>, the <a href="/wiki/Rotation_group" class="mw-redirect" title="Rotation group">rotation group</a>, all of whose representations are known. The precise infinite-dimensional unitary representation under which a particle transforms is part of its classification. Not all representations can correspond to physical particles (as far as is known). Standard vectors on the one-sheeted hyperbolas would correspond to <a href="/wiki/Tachyon" title="Tachyon">tachyons</a>. Particles on the light cone are <a href="/wiki/Photon" title="Photon">photons</a>, and more hypothetically, <a href="/wiki/Graviton" title="Graviton">gravitons</a>. The "particle" corresponding to the origin is the vacuum. </p> <div class="mw-heading mw-heading2"><h2 id="Homomorphisms_and_isomorphisms">Homomorphisms and isomorphisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=10" title="Edit section: Homomorphisms and isomorphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks" style="width:20.0em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → <b>Group theory</b></span><br /><a href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert"><span typeof="mw:File"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></span></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Basic notions</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <i><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></i></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_group" title="Simple group">simple</a></li> <li><a href="/wiki/Finite_group" title="Finite group">finite</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">infinite</a></li> <li><a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li> <li><a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative</a></li> <li><a href="/wiki/Additive_group" title="Additive group">additive</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">abelian</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">solvable</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Z<sub><i>n</i></sub></li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> S<sub><i>n</i></sub></li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub><i>n</i></sub></li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> D<sub><i>n</i></sub></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li> <li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul> <ul><li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul> <ul><li><a href="/wiki/P-group" title="P-group"><i>p</i>-group</a></li> <li><a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul> <ul><li><a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul> <ul><li><a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></th></tr><tr><td class="sidebar-content"> <ul><li>cyclic</li> <li>alternating</li> <li><a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Lie type</a></li> <li><a href="/wiki/Sporadic_group" title="Sporadic group">sporadic</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">Lattices</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Integer" title="Integer">Integers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Free_group" title="Free group">Free group</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li><li>SL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li></ul></div></div> <ul><li><a href="/wiki/Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li> <li><a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice</a></li> <li><a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Hyperbolic group</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Topological_group" title="Topological group">Topological</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li></ul> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></li></ul> <ul><li><a class="mw-selflink selflink">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal</a></li></ul> <ul><li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Infinite_dimensional_Lie_group" class="mw-redirect" title="Infinite dimensional Lie group">Infinite dimensional Lie group</a> <div class="hlist"><ul><li>O(∞)</li><li>SU(∞)</li><li>Sp(∞)</li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Algebraic_group" title="Algebraic group">Algebraic groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul> <ul><li><a href="/wiki/Reductive_group" title="Reductive group">Reductive group</a></li></ul> <ul><li><a href="/wiki/Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_theory_sidebar" title="Template:Group theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_theory_sidebar" title="Template talk:Group theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_theory_sidebar" title="Special:EditPage/Template:Group theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>Several other groups are either homomorphic or isomorphic to the restricted Lorentz group <span class="texhtml">SO<sup>+</sup>(1, 3)</span>. These homomorphisms play a key role in explaining various phenomena in physics. </p> <ul><li>The <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a> <span class="texhtml">SL(2, <b>C</b>)</span> is a <a href="/wiki/Cover_(mathematics)" class="mw-redirect" title="Cover (mathematics)">double covering</a> of the restricted Lorentz group. This relationship is widely used to express the <a href="/wiki/Lorentz_invariance" class="mw-redirect" title="Lorentz invariance">Lorentz invariance</a> of the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> and the covariance of spinors. In other words, the (restricted) Lorentz group is isomorphic to <span class="texhtml">SL(2, <b>C</b>) / Z<sub>2</sub></span></li> <li>The <a href="/wiki/Symplectic_group" title="Symplectic group">symplectic group</a> <span class="texhtml">Sp(2, <b>C</b>)</span> is isomorphic to <span class="texhtml">SL(2, <b>C</b>)</span>; it is used to construct <a href="/wiki/Weyl_spinor" class="mw-redirect" title="Weyl spinor">Weyl spinors</a>, as well as to explain how spinors can have a mass.</li> <li>The <a href="/wiki/Spin_group" title="Spin group">spin group</a> <span class="texhtml">Spin(1, 3)</span> is isomorphic to <span class="texhtml">SL(2, <b>C</b>)</span>; it is used to explain spin and spinors in terms of the <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a>, thus making it clear how to generalize the Lorentz group to general settings in <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, including theories of <a href="/wiki/Supergravity" title="Supergravity">supergravity</a> and <a href="/wiki/String_theory" title="String theory">string theory</a>.</li> <li>The restricted Lorentz group is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the <a href="/wiki/Projective_special_linear_group" class="mw-redirect" title="Projective special linear group">projective special linear group</a> <span class="texhtml">PSL(2, <b>C</b>)</span> which is, in turn, isomorphic to the <a href="/wiki/M%C3%B6bius_group" class="mw-redirect" title="Möbius group">Möbius group</a>, the <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> of <a href="/wiki/Conformal_geometry" title="Conformal geometry">conformal geometry</a> on the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a>. This relationship is central to the classification of the subgroups of the Lorentz group according to an earlier classification scheme developed for the Möbius group.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Weyl_representation">Weyl representation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=11" title="Edit section: Weyl representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Weyl representation</b> or <a href="/wiki/Representation_theory_of_the_Lorentz_group#The_covering_group_SL(2,_C)" title="Representation theory of the Lorentz group"><b>spinor map</b></a> is a pair of <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> <a href="/wiki/Homomorphism" title="Homomorphism">homomorphisms</a> from <span class="texhtml">SL(2, <b>C</b>)</span> to <span class="texhtml">SO<sup>+</sup>(1, 3)</span>. They form a matched pair under parity transformations, corresponding to left and right <a href="/wiki/Chiral" class="mw-redirect" title="Chiral">chiral</a> spinors. </p><p>One may define an action of <span class="texhtml">SL(2, <b>C</b>)</span> on Minkowski spacetime by writing a point of spacetime as a two-by-two <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian matrix</a> in the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X}}={\begin{bmatrix}ct+z&x-iy\\x+iy&ct-z\end{bmatrix}}=ct1\!\!1+x\sigma _{x}+y\sigma _{y}+z\sigma _{z}=ct1\!\!1+{\vec {x}}\cdot {\vec {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mi>t</mi> <mo>+</mo> <mi>z</mi> </mtd> <mtd> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mtd> <mtd> <mi>c</mi> <mi>t</mi> <mo>−</mo> <mi>z</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>c</mi> <mi>t</mi> <mn>1</mn> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>y</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>z</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>c</mi> <mi>t</mi> <mn>1</mn> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}={\begin{bmatrix}ct+z&x-iy\\x+iy&ct-z\end{bmatrix}}=ct1\!\!1+x\sigma _{x}+y\sigma _{y}+z\sigma _{z}=ct1\!\!1+{\vec {x}}\cdot {\vec {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaff740f723f91f6788d4eb50eb9c80a28c34dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.485ex; height:6.176ex;" alt="{\displaystyle {\overline {X}}={\begin{bmatrix}ct+z&x-iy\\x+iy&ct-z\end{bmatrix}}=ct1\!\!1+x\sigma _{x}+y\sigma _{y}+z\sigma _{z}=ct1\!\!1+{\vec {x}}\cdot {\vec {\sigma }}}"></span></dd></dl> <p>in terms of <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>. </p><p>This presentation, the Weyl presentation, satisfies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \,{\overline {X}}=(ct)^{2}-x^{2}-y^{2}-z^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \,{\overline {X}}=(ct)^{2}-x^{2}-y^{2}-z^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a2c9e4795d29ccd997b9835a9125330649931e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.855ex; height:3.509ex;" alt="{\displaystyle \det \,{\overline {X}}=(ct)^{2}-x^{2}-y^{2}-z^{2}.}"></span></dd></dl> <p>Therefore, one has identified the space of Hermitian matrices (which is four-dimensional, as a <i>real</i> vector space) with Minkowski spacetime, in such a way that the <a href="/wiki/Determinant" title="Determinant">determinant</a> of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime. An element <span class="texhtml"><i>S</i> ∈ SL(2, <b>C</b>)</span> acts on the space of Hermitian matrices via </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X}}\mapsto S{\overline {X}}S^{\dagger }~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">↦</mo> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}\mapsto S{\overline {X}}S^{\dagger }~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5967ab357db268453da9e0194e54883ef9f17397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.087ex; height:3.343ex;" alt="{\displaystyle {\overline {X}}\mapsto S{\overline {X}}S^{\dagger }~,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{\dagger }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30011dcdbaf9b267c154008891b2eb6f51dfe50f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.484ex; height:2.676ex;" alt="{\displaystyle S^{\dagger }}"></span> is the <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Hermitian transpose</a> of <span class="texhtml"><i>S</i></span>. This action preserves the determinant and so <span class="texhtml">SL(2, <b>C</b>)</span> acts on Minkowski spacetime by (linear) isometries. The parity-inverted form of the above is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=ct1\!\!1-{\vec {x}}\cdot {\vec {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>c</mi> <mi>t</mi> <mn>1</mn> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=ct1\!\!1-{\vec {x}}\cdot {\vec {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42234ce4896e3d9df163a4c505192a6f58dd004e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.655ex; height:2.509ex;" alt="{\displaystyle X=ct1\!\!1-{\vec {x}}\cdot {\vec {\sigma }}}"></span></dd></dl> <p>which transforms as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mapsto \left(S^{-1}\right)^{\dagger }XS^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">↦</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mi>X</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\mapsto \left(S^{-1}\right)^{\dagger }XS^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b67ce0af685e9039d0eb4eba3212bb59293384f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.375ex; height:3.843ex;" alt="{\displaystyle X\mapsto \left(S^{-1}\right)^{\dagger }XS^{-1}}"></span></dd></dl> <p>That this is the correct transformation follows by noting that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X}}X=\left(c^{2}t^{2}-{\vec {x}}\cdot {\vec {x}}\right)1\!\!1=\left(c^{2}t^{2}-x^{2}-y^{2}-z^{2}\right)1\!\!1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mi>X</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mn>1</mn> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mn>1</mn> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mn>1</mn> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}X=\left(c^{2}t^{2}-{\vec {x}}\cdot {\vec {x}}\right)1\!\!1=\left(c^{2}t^{2}-x^{2}-y^{2}-z^{2}\right)1\!\!1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc1afa24d79c57e67d6f2b697c1d9a1b22e5f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.797ex; height:3.676ex;" alt="{\displaystyle {\overline {X}}X=\left(c^{2}t^{2}-{\vec {x}}\cdot {\vec {x}}\right)1\!\!1=\left(c^{2}t^{2}-x^{2}-y^{2}-z^{2}\right)1\!\!1}"></span></dd></dl> <p>remains invariant under the above pair of transformations. </p><p>These maps are <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a>, and <a href="/wiki/Kernel_(group_theory)" class="mw-redirect" title="Kernel (group theory)">kernel</a> of either map is the two element subgroup <span class="texhtml">±<i>I</i></span>. By the <a href="/wiki/Isomorphism_theorem#First_isomorphism_theorem" class="mw-redirect" title="Isomorphism theorem">first isomorphism theorem</a>, the quotient group <span class="texhtml">PSL(2, <b>C</b>) = SL(2, <b>C</b>) / {±<i>I</i>}</span> is isomorphic to <span class="texhtml">SO<sup>+</sup>(1, 3)</span>. </p><p>The parity map swaps these two coverings. It corresponds to Hermitian conjugation being an automorphism of <span class="texhtml">SL(2, <b>C</b>)</span>. These two distinct coverings corresponds to the two distinct <a href="/wiki/Chiral" class="mw-redirect" title="Chiral">chiral</a> actions of the Lorentz group on <a href="/wiki/Spinor" title="Spinor">spinors</a>. The non-overlined form corresponds to right-handed spinors transforming as <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{R}\mapsto S\psi _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">↦</mo> <mi>S</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{R}\mapsto S\psi _{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dcb92afac479c0e9c41f4c92d7ba8fbc053ae61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.099ex; height:2.509ex;" alt="{\displaystyle \psi _{R}\mapsto S\psi _{R}}"></span>⁠</span>, while the overline form corresponds to left-handed spinors transforming as <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{L}\mapsto \left(S^{\dagger }\right)^{-1}\psi _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">↦</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{L}\mapsto \left(S^{\dagger }\right)^{-1}\psi _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0aa35e01e5f350351289cb5683069ef68c4ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.29ex; height:3.843ex;" alt="{\displaystyle \psi _{L}\mapsto \left(S^{\dagger }\right)^{-1}\psi _{L}}"></span>⁠</span>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> </p><p>It is important to observe that this pair of coverings does <i>not</i> survive quantization; when quantized, this leads to the peculiar phenomenon of the <a href="/wiki/Chiral_anomaly" title="Chiral anomaly">chiral anomaly</a>. The classical (i.e., non-quantized) symmetries of the Lorentz group are broken by quantization; this is the content of the <a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Notational_conventions">Notational conventions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=12" title="Edit section: Notational conventions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In physics, it is conventional to denote a Lorentz transformation <span class="texhtml">Λ ∈ SO<sup>+</sup>(1, 3)</span> as <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Lambda ^{\mu }}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Lambda ^{\mu }}_{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4fcc87f4e42e30d8e2ff84220b541801b2cdda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.94ex; height:2.676ex;" alt="{\displaystyle {\Lambda ^{\mu }}_{\nu }}"></span>⁠</span>, thus showing the matrix with spacetime indexes <span class="texhtml"><i>μ</i>, <i>ν</i> = 0, 1, 2, 3</span>. A four-vector can be created from the Pauli matrices in two different ways: as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{\mu }=(I,{\vec {\sigma }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\mu }=(I,{\vec {\sigma }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9384256cc294abe2538979ac371e6eff0bfc2951" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.997ex; height:2.843ex;" alt="{\displaystyle \sigma ^{\mu }=(I,{\vec {\sigma }})}"></span> and as <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\sigma }}^{\mu }=\left(I,-{\vec {\sigma }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\sigma }}^{\mu }=\left(I,-{\vec {\sigma }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5765d90eaf8fe9cc50090ad53ee995c5da8b74ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.921ex; height:3.009ex;" alt="{\displaystyle {\overline {\sigma }}^{\mu }=\left(I,-{\vec {\sigma }}\right)}"></span>⁠</span>. The two forms are related by a <a href="/wiki/P-symmetry" class="mw-redirect" title="P-symmetry">parity transformation</a>. Note that <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\sigma }}_{\mu }=\sigma ^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\sigma }}_{\mu }=\sigma ^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51947d72c45e832463cde5adf9baff3090eb7df2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.322ex; height:3.009ex;" alt="{\displaystyle {\overline {\sigma }}_{\mu }=\sigma ^{\mu }}"></span>⁠</span>. </p><p>Given a Lorentz transformation <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\mu }\mapsto x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\mu }\mapsto x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e87267e1440ed3f15e1b3c2a29b8155d84326f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.644ex; height:2.843ex;" alt="{\displaystyle x^{\mu }\mapsto x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }}"></span>⁠</span>, the double-covering of the orthochronous Lorentz group by <span class="texhtml"><i>S</i> ∈ SL(2, <b>C</b>)</span> given above can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\prime \mu }{\overline {\sigma }}_{\mu }={\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }x^{\nu }=Sx^{\nu }{\overline {\sigma }}_{\nu }S^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>μ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>S</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\prime \mu }{\overline {\sigma }}_{\mu }={\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }x^{\nu }=Sx^{\nu }{\overline {\sigma }}_{\nu }S^{\dagger }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1292531c0063d973c76191e5ce6bf83666df0378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.881ex; height:3.343ex;" alt="{\displaystyle x^{\prime \mu }{\overline {\sigma }}_{\mu }={\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }x^{\nu }=Sx^{\nu }{\overline {\sigma }}_{\nu }S^{\dagger }}"></span></dd></dl> <p>Dropping the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684350815d8cc05d6862ce3edf1fb819c1774b46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.553ex; height:2.343ex;" alt="{\displaystyle x^{\mu }}"></span> this takes the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }=S{\overline {\sigma }}_{\nu }S^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mi>S</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }=S{\overline {\sigma }}_{\nu }S^{\dagger }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056081cd2c785054ee55ab5b26f37c8182050ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.241ex; height:3.343ex;" alt="{\displaystyle {\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }=S{\overline {\sigma }}_{\nu }S^{\dagger }}"></span></dd></dl> <p>The parity conjugate form is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56547c2c2dead0d18b9a47be83e1d26582717d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.821ex; height:3.843ex;" alt="{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Proof">Proof</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=13" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>That the above is the correct form for indexed notation is not immediately obvious, partly because, when working in indexed notation, it is quite easy to accidentally confuse a Lorentz transform with its inverse, or its transpose. This confusion arises due to the identity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \Lambda ^{\textsf {T}}\eta =\Lambda ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mi>η</mi> <mo>=</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta \Lambda ^{\textsf {T}}\eta =\Lambda ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d9144cbea4a0d631266c8073e7b8af2da720fea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.348ex; height:3.176ex;" alt="{\displaystyle \eta \Lambda ^{\textsf {T}}\eta =\Lambda ^{-1}}"></span> being difficult to recognize when written in indexed form. Lorentz transforms are <i>not</i> tensors under Lorentz transformations! Thus a direct proof of this identity is useful, for establishing its correctness. It can be demonstrated by starting with the identity </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \sigma ^{k}\omega ^{-1}=-\left(\sigma ^{k}\right)^{\textsf {T}}=-\left(\sigma ^{k}\right)^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \sigma ^{k}\omega ^{-1}=-\left(\sigma ^{k}\right)^{\textsf {T}}=-\left(\sigma ^{k}\right)^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff911d2048f7bcb2e496c634654ab75a75b130ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.961ex; height:3.843ex;" alt="{\displaystyle \omega \sigma ^{k}\omega ^{-1}=-\left(\sigma ^{k}\right)^{\textsf {T}}=-\left(\sigma ^{k}\right)^{*}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,2,3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1,2,3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f91eb32681441793a4f3edee319505ca8afaa4f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.865ex; height:2.509ex;" alt="{\displaystyle k=1,2,3}"></span> so that the above are just the usual Pauli matrices, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot )^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>⋅</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cdot )^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef96f713ce0a8546ab0b1624e5e6d1e83ea5e445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.808ex; height:3.176ex;" alt="{\displaystyle (\cdot )^{\textsf {T}}}"></span> is the matrix transpose, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot )^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>⋅</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cdot )^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424a546bb662e64642817b1ec0411ef2b1a00471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.51ex; height:2.843ex;" alt="{\displaystyle (\cdot )^{*}}"></span> is complex conjugation. The matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864bb61b276bad0d5f8902aac0cf82c79a53faf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.489ex; height:6.176ex;" alt="{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}"></span></dd></dl> <p>Written as the four-vector, the relationship is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mu }^{\textsf {T}}=\sigma _{\mu }^{*}=\omega {\overline {\sigma }}_{\mu }\omega ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mi>ω</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mu }^{\textsf {T}}=\sigma _{\mu }^{*}=\omega {\overline {\sigma }}_{\mu }\omega ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea7d5593f7a2fa72412726399c2d16a2a7aa8f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.324ex; height:3.343ex;" alt="{\displaystyle \sigma _{\mu }^{\textsf {T}}=\sigma _{\mu }^{*}=\omega {\overline {\sigma }}_{\mu }\omega ^{-1}}"></span></dd></dl> <p>This transforms as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sigma _{\mu }^{\textsf {T}}{\Lambda ^{\mu }}_{\nu }&=\omega {\overline {\sigma }}_{\mu }\omega ^{-1}{\Lambda ^{\mu }}_{\nu }\\&=\omega S\;{\overline {\sigma }}_{\nu }\,S^{\dagger }\omega ^{-1}\\&=\left(\omega S\omega ^{-1}\right)\,\left(\omega {\overline {\sigma }}_{\nu }\omega ^{-1}\right)\,\left(\omega S^{\dagger }\omega ^{-1}\right)\\&=\left(S^{-1}\right)^{\textsf {T}}\,\sigma _{\nu }^{\textsf {T}}\,\left(S^{-1}\right)^{*}\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> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ω</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ω</mi> <mi>S</mi> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>S</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> <mspace width="thinmathspace" /> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{\mu }^{\textsf {T}}{\Lambda ^{\mu }}_{\nu }&=\omega {\overline {\sigma }}_{\mu }\omega ^{-1}{\Lambda ^{\mu }}_{\nu }\\&=\omega S\;{\overline {\sigma }}_{\nu }\,S^{\dagger }\omega ^{-1}\\&=\left(\omega S\omega ^{-1}\right)\,\left(\omega {\overline {\sigma }}_{\nu }\omega ^{-1}\right)\,\left(\omega S^{\dagger }\omega ^{-1}\right)\\&=\left(S^{-1}\right)^{\textsf {T}}\,\sigma _{\nu }^{\textsf {T}}\,\left(S^{-1}\right)^{*}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d5dea3dbacd8248dbaa710fe815dfe4835db8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:40.616ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{\mu }^{\textsf {T}}{\Lambda ^{\mu }}_{\nu }&=\omega {\overline {\sigma }}_{\mu }\omega ^{-1}{\Lambda ^{\mu }}_{\nu }\\&=\omega S\;{\overline {\sigma }}_{\nu }\,S^{\dagger }\omega ^{-1}\\&=\left(\omega S\omega ^{-1}\right)\,\left(\omega {\overline {\sigma }}_{\nu }\omega ^{-1}\right)\,\left(\omega S^{\dagger }\omega ^{-1}\right)\\&=\left(S^{-1}\right)^{\textsf {T}}\,\sigma _{\nu }^{\textsf {T}}\,\left(S^{-1}\right)^{*}\end{aligned}}}"></span></dd></dl> <p>Taking one more transpose, one gets </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56547c2c2dead0d18b9a47be83e1d26582717d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.821ex; height:3.843ex;" alt="{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Symplectic_group">Symplectic group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=14" title="Edit section: Symplectic group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The symplectic group <span class="texhtml">Sp(2, <b>C</b>)</span> is isomorphic to <span class="texhtml">SL(2, <b>C</b>)</span>. This isomorphism is constructed so as to preserve a <a href="/wiki/Symplectic_bilinear_form" class="mw-redirect" title="Symplectic bilinear form">symplectic bilinear form</a> on <span class="texhtml"><b>C</b><sup>2</sup></span>, that is, to leave the form invariant under Lorentz transformations. This may be articulated as follows. The symplectic group is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (2,\mathbf {C} )=\left\{S\in \operatorname {GL} (2,\mathbf {C} ):S^{\textsf {T}}\omega S=\omega \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>S</mi> <mo>∈</mo> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mi>ω</mi> <mi>S</mi> <mo>=</mo> <mi>ω</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (2,\mathbf {C} )=\left\{S\in \operatorname {GL} (2,\mathbf {C} ):S^{\textsf {T}}\omega S=\omega \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b4f06e0812185d8970a06661c9591c36c33d080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.185ex; height:3.343ex;" alt="{\displaystyle \operatorname {Sp} (2,\mathbf {C} )=\left\{S\in \operatorname {GL} (2,\mathbf {C} ):S^{\textsf {T}}\omega S=\omega \right\}}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864bb61b276bad0d5f8902aac0cf82c79a53faf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.489ex; height:6.176ex;" alt="{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}"></span></dd></dl> <p>Other common notations are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b18c96ea10eaa3025f7a665ee95869690e0656a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.488ex; height:1.676ex;" alt="{\displaystyle \omega =\epsilon }"></span> for this element; sometimes <span class="texhtml"><i>J</i></span> is used, but this invites confusion with the idea of <a href="/wiki/Almost_complex_structure" class="mw-redirect" title="Almost complex structure">almost complex structures</a>, which are not the same, as they transform differently. </p><p>Given a pair of Weyl spinors (two-component spinors) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}~,\quad v={\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}~,\quad v={\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02368930903e0d6464067ef02e544d666a93ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.57ex; height:6.176ex;" alt="{\displaystyle u={\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}~,\quad v={\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}}"></span></dd></dl> <p>the invariant bilinear form is conventionally written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle u,v\rangle =-\langle v,u\rangle =u_{1}v_{2}-u_{2}v_{1}=u^{\textsf {T}}\omega v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mi>ω</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle =-\langle v,u\rangle =u_{1}v_{2}-u_{2}v_{1}=u^{\textsf {T}}\omega v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/524c8f6faf06d775e22036fe746688775727e4f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.931ex; height:3.176ex;" alt="{\displaystyle \langle u,v\rangle =-\langle v,u\rangle =u_{1}v_{2}-u_{2}v_{1}=u^{\textsf {T}}\omega v}"></span></dd></dl> <p>This form is invariant under the Lorentz group, so that for <span class="texhtml"><i>S</i> ∈ SL(2, <b>C</b>)</span> one has </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Su,Sv\rangle =\langle u,v\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mi>u</mi> <mo>,</mo> <mi>S</mi> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle Su,Sv\rangle =\langle u,v\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6df789987c4c57849aa75ab5ba51e63123e3a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.698ex; height:2.843ex;" alt="{\displaystyle \langle Su,Sv\rangle =\langle u,v\rangle }"></span></dd></dl> <p>This defines a kind of "scalar product" of spinors, and is commonly used to defined a Lorentz-invariant <a href="/wiki/Mass" title="Mass">mass</a> term in <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangians</a>. There are several notable properties to be called out that are important to physics. One is that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ecdbc4a8b56a82c8b4616e6ffcdc4826610f3d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.569ex; height:2.843ex;" alt="{\displaystyle \omega ^{2}=-1}"></span> and so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{-1}=\omega ^{\textsf {T}}=\omega ^{\dagger }=-\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{-1}=\omega ^{\textsf {T}}=\omega ^{\dagger }=-\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/685284f37d88c1f2fbcdc53c59feb17265344f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.533ex; height:2.843ex;" alt="{\displaystyle \omega ^{-1}=\omega ^{\textsf {T}}=\omega ^{\dagger }=-\omega }"></span> </p><p>The defining relation can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega S^{\textsf {T}}\omega ^{-1}=S^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega S^{\textsf {T}}\omega ^{-1}=S^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9463b29232f8a5bff34a2e16007d08509fc525c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.05ex; height:2.676ex;" alt="{\displaystyle \omega S^{\textsf {T}}\omega ^{-1}=S^{-1}}"></span></dd></dl> <p>which closely resembles the defining relation for the Lorentz group </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \Lambda ^{\textsf {T}}\eta ^{-1}=\Lambda ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta \Lambda ^{\textsf {T}}\eta ^{-1}=\Lambda ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353788cd21f2814d743062fca7fb274830a1d47c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.685ex; height:3.176ex;" alt="{\displaystyle \eta \Lambda ^{\textsf {T}}\eta ^{-1}=\Lambda ^{-1}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =\operatorname {diag} (+1,-1,-1,-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =\operatorname {diag} (+1,-1,-1,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f21a83c8d523dd76f41b35e2e1ec7979f8ef5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.326ex; height:2.843ex;" alt="{\displaystyle \eta =\operatorname {diag} (+1,-1,-1,-1)}"></span> is the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> for <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> and of course, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda \in \operatorname {SO} (1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo>∈</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda \in \operatorname {SO} (1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977171c23e92171173fa89a79b1cf2383fe3814c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.722ex; height:2.843ex;" alt="{\displaystyle \Lambda \in \operatorname {SO} (1,3)}"></span> as before. </p> <div class="mw-heading mw-heading2"><h2 id="Covering_groups">Covering groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=15" title="Edit section: Covering groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since <span class="texhtml">SL(2, <b>C</b>)</span> is simply connected, it is the <a href="/wiki/Universal_cover" class="mw-redirect" title="Universal cover">universal covering group</a> of the restricted Lorentz group <span class="texhtml">SO<sup>+</sup>(1, 3)</span>. By restriction, there is a homomorphism <span class="texhtml">SU(2) → SO(3)</span>. Here, the <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a> SU(2), which is isomorphic to the group of unit <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>, is also simply connected, so it is the covering group of the rotation group <span class="texhtml">SO(3)</span>. Each of these <a href="/wiki/Covering_map" class="mw-redirect" title="Covering map">covering maps</a> are twofold covers in the sense that precisely two elements of the covering group map to each element of the quotient. One often says that the restricted Lorentz group and the rotation group are <b>doubly connected</b>. This means that the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of the each group is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the two-element <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> <span class="texhtml">Z<sub>2</sub></span>. </p><p>Twofold coverings are characteristic of <a href="/wiki/Spin_group" title="Spin group">spin groups</a>. Indeed, in addition to the double coverings </p> <dl><dd><span class="texhtml">Spin<sup>+</sup>(1, 3) = SL(2, <b>C</b>) → SO<sup>+</sup>(1, 3)</span></dd> <dd><span class="texhtml">Spin(3) = SU(2) → SO(3)</span></dd></dl> <p>we have the double coverings </p> <dl><dd><span class="texhtml"><a href="/wiki/Pin_group" title="Pin group">Pin(1, 3)</a> → O(1, 3)</span></dd> <dd><span class="texhtml">Spin(1, 3) → SO(1, 3)</span></dd> <dd><span class="texhtml">Spin<sup>+</sup>(1, 2) = SU(1, 1) → SO(1, 2)</span></dd></dl> <p>These spinorial <a href="/wiki/Double_covering_group" class="mw-redirect" title="Double covering group">double coverings</a> are constructed from <a href="/wiki/Clifford_algebras" class="mw-redirect" title="Clifford algebras">Clifford algebras</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Topology">Topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=16" title="Edit section: Topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The left and right groups in the double covering </p> <dl><dd><span class="texhtml">SU(2) → SO(3)</span></dd></dl> <p>are <a href="/wiki/Deformation_retract" class="mw-redirect" title="Deformation retract">deformation retracts</a> of the left and right groups, respectively, in the double covering </p> <dl><dd><span class="texhtml">SL(2, <b>C</b>) → SO<sup>+</sup>(1, 3)</span>.</dd></dl> <p>But the homogeneous space <span class="texhtml">SO<sup>+</sup>(1, 3) / SO(3)</span> is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to <a href="/wiki/Hyperbolic_3-space" class="mw-redirect" title="Hyperbolic 3-space">hyperbolic 3-space</a> <span class="texhtml"><i>H</i><sup>3</sup></span>, so we have exhibited the restricted Lorentz group as a <a href="/wiki/Fiber_bundle" title="Fiber bundle">principal fiber bundle</a> with fibers <span class="texhtml">SO(3)</span> and base <span class="texhtml"><i>H</i><sup>3</sup></span>. Since the latter is homeomorphic to <span class="texhtml"><b>R</b><sup>3</sup></span>, while <span class="texhtml">SO(3)</span> is homeomorphic to three-dimensional <a href="/wiki/Real_projective_space" title="Real projective space">real projective space</a> <span class="texhtml"><b>R</b>P<sup>3</sup></span>, we see that the restricted Lorentz group is <i>locally</i> homeomorphic to the product of <span class="texhtml"><b>R</b>P<sup>3</sup></span> with <span class="texhtml"><b>R</b><sup>3</sup></span>. Since the base space is contractible, this can be extended to a global homeomorphism.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Can this be extended to a diffeomorphism? If not, why not? Any references for this? (December 2020)">clarification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Conjugacy_classes">Conjugacy classes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=17" title="Edit section: Conjugacy classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because the restricted Lorentz group <span class="texhtml">SO<sup>+</sup>(1, 3)</span> is isomorphic to the Möbius group <span class="texhtml">PSL(2, <b>C</b>)</span>, its <a href="/wiki/Conjugacy_classes" class="mw-redirect" title="Conjugacy classes">conjugacy classes</a> also fall into five classes: </p> <ul><li><b>Elliptic</b> transformations</li> <li><b>Hyperbolic</b> transformations</li> <li><b>Loxodromic</b> transformations</li> <li><b>Parabolic</b> transformations</li> <li>The trivial <b>identity</b> transformation</li></ul> <p>In the article on <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>, it is explained how this classification arises by considering the <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a> of Möbius transformations in their action on the Riemann sphere, which corresponds here to <a href="/wiki/Null_vector" title="Null vector">null</a> <a href="/wiki/Eigenspace" class="mw-redirect" title="Eigenspace">eigenspaces</a> of restricted Lorentz transformations in their action on Minkowski spacetime. </p><p>An example of each type is given in the subsections below, along with the effect of the <a href="/wiki/One-parameter_subgroup" class="mw-redirect" title="One-parameter subgroup">one-parameter subgroup</a> it generates (e.g., on the appearance of the night sky). </p><p>The Möbius transformations are the <a href="/wiki/Conformal_map" title="Conformal map">conformal transformations</a> of the Riemann sphere (or celestial sphere). Then conjugating with an arbitrary element of <span class="texhtml">SL(2, <b>C</b>)</span> obtains the following examples of arbitrary elliptic, hyperbolic, loxodromic, and parabolic (restricted) Lorentz transformations, respectively. The effect on the <b>flow lines</b> of the corresponding one-parameter subgroups is to transform the pattern seen in the examples by some conformal transformation. For example, an elliptic Lorentz transformation can have any two distinct fixed points on the celestial sphere, but points still flow along circular arcs from one fixed point toward the other. The other cases are similar. </p> <div class="mw-heading mw-heading3"><h3 id="Elliptic">Elliptic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=18" title="Edit section: Elliptic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An elliptic element of <span class="texhtml">SL(2, <b>C</b>)</span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}={\begin{bmatrix}\exp \left({\frac {i}{2}}\theta \right)&0\\0&\exp \left(-{\frac {i}{2}}\theta \right)\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}={\begin{bmatrix}\exp \left({\frac {i}{2}}\theta \right)&0\\0&\exp \left(-{\frac {i}{2}}\theta \right)\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4a75102c8525df2a829e28fb5eb8b5b15e0747" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.099ex; height:7.509ex;" alt="{\displaystyle P_{1}={\begin{bmatrix}\exp \left({\frac {i}{2}}\theta \right)&0\\0&\exp \left(-{\frac {i}{2}}\theta \right)\end{bmatrix}}}"></span></dd></dl> <p>and has fixed points <span class="texhtml mvar" style="font-style:italic;">ξ</span> = 0, ∞. Writing the action as <span class="texhtml"><i>X</i> ↦ <i>P</i><sub>1</sub> <i>X P</i><sub>1</sub><sup>†</sup></span> and collecting terms, the spinor map converts this to the (restricted) Lorentz transformation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1}={\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&\sin(\theta )&0\\0&-\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{bmatrix}}=\exp \left(\theta {\begin{bmatrix}0&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&0\end{bmatrix}}\right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{1}={\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&\sin(\theta )&0\\0&-\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{bmatrix}}=\exp \left(\theta {\begin{bmatrix}0&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&0\end{bmatrix}}\right)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba479d18e7ecb2edca9e84556e8c2133b2645d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:63.411ex; height:12.843ex;" alt="{\displaystyle Q_{1}={\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&\sin(\theta )&0\\0&-\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{bmatrix}}=\exp \left(\theta {\begin{bmatrix}0&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&0\end{bmatrix}}\right)~.}"></span></dd></dl> <p>This transformation then represents a rotation about the <span class="texhtml mvar" style="font-style:italic;">z</span> axis, exp(<span class="texhtml"><i>iθJ<sub>z</sub></i></span>). The one-parameter subgroup it generates is obtained by taking <span class="texhtml mvar" style="font-style:italic;">θ</span> to be a real variable, the rotation angle, instead of a constant. </p><p>The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same two fixed points, the North and South poles. The transformations move all other points around latitude circles so that this group yields a continuous counter-clockwise rotation about the <span class="texhtml mvar" style="font-style:italic;">z</span> axis as <span class="texhtml mvar" style="font-style:italic;">θ</span> increases. The <i>angle doubling</i> evident in the spinor map is a characteristic feature of <i>spinorial double coverings</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Hyperbolic">Hyperbolic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=19" title="Edit section: Hyperbolic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A hyperbolic element of <span class="texhtml">SL(2, <b>C</b>)</span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}={\begin{bmatrix}\exp \left({\frac {\eta }{2}}\right)&0\\0&\exp \left(-{\frac {\eta }{2}}\right)\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}={\begin{bmatrix}\exp \left({\frac {\eta }{2}}\right)&0\\0&\exp \left(-{\frac {\eta }{2}}\right)\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0abc6e952256ba838f54cfba106e02645174e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.928ex; height:7.509ex;" alt="{\displaystyle P_{2}={\begin{bmatrix}\exp \left({\frac {\eta }{2}}\right)&0\\0&\exp \left(-{\frac {\eta }{2}}\right)\end{bmatrix}}}"></span></dd></dl> <p>and has fixed points <span class="texhtml mvar" style="font-style:italic;">ξ</span> = 0, ∞. Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a dilation from the origin. </p><p>The spinor map converts this to the Lorentz transformation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{2}={\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&1&0&0\\0&0&1&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}=\exp \left(\eta {\begin{bmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{bmatrix}}\right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{2}={\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&1&0&0\\0&0&1&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}=\exp \left(\eta {\begin{bmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{bmatrix}}\right)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfbe26c7dec0e9d8dd5a650057a90871389623e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:62.485ex; height:12.843ex;" alt="{\displaystyle Q_{2}={\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&1&0&0\\0&0&1&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}=\exp \left(\eta {\begin{bmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{bmatrix}}\right)~.}"></span></dd></dl> <p>This transformation represents a boost along the <span class="texhtml mvar" style="font-style:italic;">z</span> axis with <a href="/wiki/Rapidity" title="Rapidity">rapidity</a> <span class="texhtml mvar" style="font-style:italic;">η</span>. The one-parameter subgroup it generates is obtained by taking <span class="texhtml mvar" style="font-style:italic;">η</span> to be a real variable, instead of a constant. The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same fixed points (the North and South poles), and they move all other points along <a href="/wiki/Longitude" title="Longitude">longitudes</a> away from the South pole and toward the North pole. </p> <div class="mw-heading mw-heading3"><h3 id="Loxodromic">Loxodromic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=20" title="Edit section: Loxodromic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A loxodromic element of <span class="texhtml">SL(2, <b>C</b>)</span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}=P_{2}P_{1}=P_{1}P_{2}={\begin{bmatrix}\exp \left({\frac {1}{2}}(\eta +i\theta )\right)&0\\0&\exp \left(-{\frac {1}{2}}(\eta +i\theta )\right)\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>η</mi> <mo>+</mo> <mi>i</mi> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>η</mi> <mo>+</mo> <mi>i</mi> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}=P_{2}P_{1}=P_{1}P_{2}={\begin{bmatrix}\exp \left({\frac {1}{2}}(\eta +i\theta )\right)&0\\0&\exp \left(-{\frac {1}{2}}(\eta +i\theta )\right)\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026ad231cef18841aba36d9e50866faeb4fa22f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.725ex; height:7.509ex;" alt="{\displaystyle P_{3}=P_{2}P_{1}=P_{1}P_{2}={\begin{bmatrix}\exp \left({\frac {1}{2}}(\eta +i\theta )\right)&0\\0&\exp \left(-{\frac {1}{2}}(\eta +i\theta )\right)\end{bmatrix}}}"></span></dd></dl> <p>and has fixed points <span class="texhtml mvar" style="font-style:italic;">ξ</span> = 0, ∞. The spinor map converts this to the Lorentz transformation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{3}=Q_{2}Q_{1}=Q_{1}Q_{2}={\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&\cos(\theta )&\sin(\theta )&0\\0&-\sin(\theta )&\cos(\theta )&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}=\exp {\begin{bmatrix}0&0&0&\eta \\0&0&\theta &0\\0&-\theta &0&0\\\eta &0&0&0\end{bmatrix}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>η</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>θ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>θ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>η</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{3}=Q_{2}Q_{1}=Q_{1}Q_{2}={\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&\cos(\theta )&\sin(\theta )&0\\0&-\sin(\theta )&\cos(\theta )&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}=\exp {\begin{bmatrix}0&0&0&\eta \\0&0&\theta &0\\0&-\theta &0&0\\\eta &0&0&0\end{bmatrix}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b93161e84c8e5ce1df9e4ac9f06f769471e3384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:88.79ex; height:13.176ex;" alt="{\displaystyle Q_{3}=Q_{2}Q_{1}=Q_{1}Q_{2}={\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&\cos(\theta )&\sin(\theta )&0\\0&-\sin(\theta )&\cos(\theta )&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}=\exp {\begin{bmatrix}0&0&0&\eta \\0&0&\theta &0\\0&-\theta &0&0\\\eta &0&0&0\end{bmatrix}}~.}"></span></dd></dl> <p>The one-parameter subgroup this generates is obtained by replacing <span class="texhtml"><i>η</i> + i<i>θ</i></span> with any real multiple of this complex constant. (If <span class="texhtml"><i>η</i></span>, <span class="texhtml"><i>θ</i></span> vary independently, then a <i>two-dimensional</i> <a href="/wiki/Abelian_group" title="Abelian group">abelian subgroup</a> is obtained, consisting of simultaneous rotations about the <span class="texhtml mvar" style="font-style:italic;">z</span> axis and boosts along the <span class="texhtml mvar" style="font-style:italic;">z</span>-axis; in contrast, the <i>one-dimensional</i> subgroup discussed here consists of those elements of this two-dimensional subgroup such that the <b>rapidity</b> of the boost and <b>angle</b> of the rotation have a <i>fixed ratio</i>.) </p><p>The corresponding continuous transformations of the celestial sphere (excepting the identity) all share the same two fixed points (the North and South poles). They move all other points away from the South pole and toward the North pole (or vice versa), along a family of curves called <a href="/wiki/Rhumb_line" title="Rhumb line"><b>loxodromes</b></a>. Each loxodrome spirals infinitely often around each pole. </p> <div class="mw-heading mw-heading3"><h3 id="Parabolic">Parabolic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=21" title="Edit section: Parabolic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A parabolic element of <span class="texhtml">SL(2, <b>C</b>)</span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}={\begin{bmatrix}1&\alpha \\0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}={\begin{bmatrix}1&\alpha \\0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf058c82d43299b02d44068bb712609e8806931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.824ex; height:6.176ex;" alt="{\displaystyle P_{4}={\begin{bmatrix}1&\alpha \\0&1\end{bmatrix}}}"></span></dd></dl> <p>and has the single fixed point <span class="texhtml mvar" style="font-style:italic;">ξ</span> = ∞ on the Riemann sphere. Under stereographic projection, it appears as an ordinary <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> along the <a href="/wiki/Real_axis" class="mw-redirect" title="Real axis">real axis</a>. </p><p>The spinor map converts this to the matrix (representing a Lorentz transformation) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}Q_{4}&={\begin{bmatrix}1+{\frac {1}{2}}\vert \alpha \vert ^{2}&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&-{\frac {1}{2}}\vert \alpha \vert ^{2}\\\operatorname {Re} (\alpha )&1&0&-\operatorname {Re} (\alpha )\\-\operatorname {Im} (\alpha )&0&1&\operatorname {Im} (\alpha )\\{\frac {1}{2}}\vert \alpha \vert ^{2}&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&1-{\frac {1}{2}}\vert \alpha \vert ^{2}\end{bmatrix}}\\[6pt]&=\exp {\begin{bmatrix}0&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&0\\\operatorname {Re} (\alpha )&0&0&-\operatorname {Re} (\alpha )\\-\operatorname {Im} (\alpha )&0&0&\operatorname {Im} (\alpha )\\0&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&0\end{bmatrix}}~.\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="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo fence="false" stretchy="false">|</mo> <mi>α</mi> <msup> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo fence="false" stretchy="false">|</mo> <mi>α</mi> <msup> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo fence="false" stretchy="false">|</mo> <mi>α</mi> <msup> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo fence="false" stretchy="false">|</mo> <mi>α</mi> <msup> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Q_{4}&={\begin{bmatrix}1+{\frac {1}{2}}\vert \alpha \vert ^{2}&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&-{\frac {1}{2}}\vert \alpha \vert ^{2}\\\operatorname {Re} (\alpha )&1&0&-\operatorname {Re} (\alpha )\\-\operatorname {Im} (\alpha )&0&1&\operatorname {Im} (\alpha )\\{\frac {1}{2}}\vert \alpha \vert ^{2}&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&1-{\frac {1}{2}}\vert \alpha \vert ^{2}\end{bmatrix}}\\[6pt]&=\exp {\begin{bmatrix}0&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&0\\\operatorname {Re} (\alpha )&0&0&-\operatorname {Re} (\alpha )\\-\operatorname {Im} (\alpha )&0&0&\operatorname {Im} (\alpha )\\0&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&0\end{bmatrix}}~.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe74e84f87ed4999b00c40a37587136c1737f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.005ex; width:53.54ex; height:29.176ex;" alt="{\displaystyle {\begin{aligned}Q_{4}&={\begin{bmatrix}1+{\frac {1}{2}}\vert \alpha \vert ^{2}&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&-{\frac {1}{2}}\vert \alpha \vert ^{2}\\\operatorname {Re} (\alpha )&1&0&-\operatorname {Re} (\alpha )\\-\operatorname {Im} (\alpha )&0&1&\operatorname {Im} (\alpha )\\{\frac {1}{2}}\vert \alpha \vert ^{2}&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&1-{\frac {1}{2}}\vert \alpha \vert ^{2}\end{bmatrix}}\\[6pt]&=\exp {\begin{bmatrix}0&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&0\\\operatorname {Re} (\alpha )&0&0&-\operatorname {Re} (\alpha )\\-\operatorname {Im} (\alpha )&0&0&\operatorname {Im} (\alpha )\\0&\operatorname {Re} (\alpha )&-\operatorname {Im} (\alpha )&0\end{bmatrix}}~.\end{aligned}}}"></span></dd></dl> <p>This generates a two-parameter abelian subgroup, which is obtained by considering <span class="texhtml mvar" style="font-style:italic;">α</span> a complex variable rather than a constant. The corresponding continuous transformations of the celestial sphere (except for the identity transformation) move points along a family of circles that are all tangent at the North pole to a certain <a href="/wiki/Great_circle" title="Great circle">great circle</a>. All points other than the North pole itself move along these circles. </p><p>Parabolic Lorentz transformations are often called <b>null rotations</b>. Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations (elliptic, hyperbolic, loxodromic, parabolic), it is illustrated here how to determine the effect of an example of a parabolic Lorentz transformation on Minkowski spacetime. </p><p>The matrix given above yields the transformation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}\rightarrow {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}+\operatorname {Re} (\alpha )\;{\begin{bmatrix}x\\t-z\\0\\x\end{bmatrix}}-\operatorname {Im} (\alpha )\;{\begin{bmatrix}y\\0\\z-t\\y\end{bmatrix}}+{\frac {\vert \alpha \vert ^{2}}{2}}\;{\begin{bmatrix}t-z\\0\\0\\t-z\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> <mo>−</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>−</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo>−</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">|</mo> <mi>α</mi> <msup> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>t</mi> <mo>−</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> <mo>−</mo> <mi>z</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}\rightarrow {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}+\operatorname {Re} (\alpha )\;{\begin{bmatrix}x\\t-z\\0\\x\end{bmatrix}}-\operatorname {Im} (\alpha )\;{\begin{bmatrix}y\\0\\z-t\\y\end{bmatrix}}+{\frac {\vert \alpha \vert ^{2}}{2}}\;{\begin{bmatrix}t-z\\0\\0\\t-z\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6406f21fa39f0c15f2b6306d99ed7c657c5649d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:67.726ex; height:12.509ex;" alt="{\displaystyle {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}\rightarrow {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}+\operatorname {Re} (\alpha )\;{\begin{bmatrix}x\\t-z\\0\\x\end{bmatrix}}-\operatorname {Im} (\alpha )\;{\begin{bmatrix}y\\0\\z-t\\y\end{bmatrix}}+{\frac {\vert \alpha \vert ^{2}}{2}}\;{\begin{bmatrix}t-z\\0\\0\\t-z\end{bmatrix}}.}"></span></dd></dl> <p>Now, without loss of generality, pick <span class="texhtml">Im(<i>α</i>) = 0</span>. Differentiating this transformation with respect to the now real group parameter <span class="texhtml mvar" style="font-style:italic;">α</span> and evaluating at <span class="texhtml"><i>α</i> = 0</span> produces the corresponding vector field (first order linear partial differential operator), </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,\left(\partial _{t}+\partial _{z}\right)+(t-z)\,\partial _{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,\left(\partial _{t}+\partial _{z}\right)+(t-z)\,\partial _{x}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82aca42b3573ac7a4a4ebce16418071043c283c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.909ex; height:2.843ex;" alt="{\displaystyle x\,\left(\partial _{t}+\partial _{z}\right)+(t-z)\,\partial _{x}.}"></span></dd></dl> <p>Apply this to a function <span class="texhtml"><i>f</i>(<i>t</i>, <i>x</i>, <i>y</i>, <i>z</i>)</span>, and demand that it stays invariant; i.e., it is annihilated by this transformation. The solution of the resulting first order linear partial differential equation can be expressed in the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t,x,y,z)=F\left(y,\,t-z,\,t^{2}-x^{2}-z^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>t</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> <mspace width="thinmathspace" /> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t,x,y,z)=F\left(y,\,t-z,\,t^{2}-x^{2}-z^{2}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc4031a5db4785f296fe76929fe9f07b919f2cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.862ex; height:3.343ex;" alt="{\displaystyle f(t,x,y,z)=F\left(y,\,t-z,\,t^{2}-x^{2}-z^{2}\right),}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">F</span> is an <i>arbitrary</i> smooth function. The arguments of <span class="texhtml mvar" style="font-style:italic;">F</span> give three <i>rational invariants</i> describing how points (events) move under this parabolic transformation, as they themselves do not move, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=c_{1},~~~~t-z=c_{2},~~~~t^{2}-x^{2}-z^{2}=c_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>t</mi> <mo>−</mo> <mi>z</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=c_{1},~~~~t-z=c_{2},~~~~t^{2}-x^{2}-z^{2}=c_{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4f61ec92b8202357ae56caeef614a6a402fe72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.865ex; height:3.009ex;" alt="{\displaystyle y=c_{1},~~~~t-z=c_{2},~~~~t^{2}-x^{2}-z^{2}=c_{3}.}"></span></dd></dl> <p>Choosing real values for the constants on the right hand sides yields three conditions, and thus specifies a curve in Minkowski spacetime. This curve is an orbit of the transformation. </p><p>The form of the rational invariants shows that these flowlines (orbits) have a simple description: suppressing the inessential coordinate <span class="texhtml mvar" style="font-style:italic;">y</span>, each orbit is the intersection of a <i>null plane</i>, <span class="texhtml"> <i>t</i> = <i>z + c</i><sub>2</sub></span>, with a <i>hyperboloid</i>, <span class="texhtml"><i>t<sup>2</sup> − x<sup>2</sup> − z<sup>2</sup></i> = <i>c<sub>3</sub></i></span>. The case <span class="texhtml mvar" style="font-style:italic;">c</span><sub>3</sub> = 0 has the hyperboloid degenerate to a light cone with the orbits becoming parabolas lying in corresponding null planes. </p><p>A particular null line lying on the light cone is left <i>invariant</i>; this corresponds to the unique (double) fixed point on the Riemann sphere mentioned above. The other null lines through the origin are "swung around the cone" by the transformation. Following the motion of one such null line as <span class="texhtml mvar" style="font-style:italic;">α</span> increases corresponds to following the motion of a point along one of the circular flow lines on the celestial sphere, as described above. </p><p>A choice <span class="texhtml">Re(<i>α</i>) = 0</span> instead, produces similar orbits, now with the roles of <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> interchanged. </p><p> Parabolic transformations lead to the gauge symmetry of massless particles (such as <a href="/wiki/Photon" title="Photon">photons</a>) with <a href="/wiki/Helicity_(particle_physics)" title="Helicity (particle physics)">helicity</a> |<span class="texhtml mvar" style="font-style:italic;">h</span>| ≥ 1. In the above explicit example, a massless particle moving in the <span class="texhtml mvar" style="font-style:italic;">z</span> direction, so with 4-momentum <span class="texhtml"><b>P</b> = (<i>p</i>, 0, 0, <i>p</i>)</span>, is not affected at all by the <span class="texhtml mvar" style="font-style:italic;">x</span>-boost and <span class="texhtml mvar" style="font-style:italic;">y</span>-rotation combination <span class="texhtml"><i>K<sub>x</sub> − J<sub>y</sub></i></span> defined below, in the "little group" of its motion. This is evident from the explicit transformation law discussed: like any light-like vector, <i><b>P</b></i> itself is now invariant; i.e., all traces or effects of <span class="texhtml mvar" style="font-style:italic;">α</span> have disappeared. <span class="texhtml"><i>c</i><sub>1</sub> = <i>c</i><sub>2</sub> = <i>c</i><sub>3</sub> = 0</span>, in the special case discussed. (The other similar generator, <span class="texhtml"><i>K</i><sub><i>y</i></sub> + <i>J</i><sub><i>x</i></sub></span> as well as it and <span class="texhtml"><i>J</i><sub><i>z</i></sub></span> comprise altogether the little group of the light-like vector, isomorphic to <span class="texhtml"><i>E</i>(2)</span>.)</p><figure typeof="mw:File/Thumb"><a href="/wiki/File:Lorentz_boost_on_light_cone_and_celestial_circle.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Lorentz_boost_on_light_cone_and_celestial_circle.gif/487px-Lorentz_boost_on_light_cone_and_celestial_circle.gif" decoding="async" width="487" height="274" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e9/Lorentz_boost_on_light_cone_and_celestial_circle.gif 1.5x" data-file-width="600" data-file-height="337" /></a><figcaption>The action of a Lorentz boost in the x-direction on the light-cone and 'celestial circle' in 1+2 spacetime. After applying the Lorentz boost matrix to the whole space, the celestial circle must be recovered by rescaling each point to <span class="texhtml"><i>t</i> = 1</span>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Appearance_of_the_night_sky">Appearance of the night sky</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=22" title="Edit section: Appearance of the night sky"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This isomorphism has the consequence that Möbius transformations of the Riemann sphere represent the way that Lorentz transformations change the appearance of the night sky, as seen by an observer who is maneuvering at relativistic velocities relative to the "fixed stars". </p><p>Suppose the "fixed stars" live in Minkowski spacetime and are modeled by points on the celestial sphere. Then a given point on the celestial sphere can be associated with <span class="texhtml"><i>ξ</i> = <i>u</i> + <i>iv</i></span>, a complex number that corresponds to the point on the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a>, and can be identified with a null vector (a <a href="/wiki/Minkowski_space#Causal_structure" title="Minkowski space">light-like vector</a>) in Minkowski space </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}u^{2}+v^{2}+1\\2u\\-2v\\u^{2}+v^{2}-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <mi>v</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}u^{2}+v^{2}+1\\2u\\-2v\\u^{2}+v^{2}-1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcabd28e3fe5366c5ac3ecf38e72ba40261fe44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:15.261ex; height:12.843ex;" alt="{\displaystyle {\begin{bmatrix}u^{2}+v^{2}+1\\2u\\-2v\\u^{2}+v^{2}-1\end{bmatrix}}}"></span></dd></dl> <p>or, in the Weyl representation (the spinor map), the Hermitian matrix </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=2{\begin{bmatrix}u^{2}+v^{2}&u+iv\\u-iv&1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi>u</mi> <mo>+</mo> <mi>i</mi> <mi>v</mi> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> <mo>−</mo> <mi>i</mi> <mi>v</mi> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=2{\begin{bmatrix}u^{2}+v^{2}&u+iv\\u-iv&1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f953fc67b6cd714e0e1aeac70a5c48d959227f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.007ex; height:6.176ex;" alt="{\displaystyle N=2{\begin{bmatrix}u^{2}+v^{2}&u+iv\\u-iv&1\end{bmatrix}}.}"></span></dd></dl> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Lorentz_boost_on_the_celestial_sphere.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Lorentz_boost_on_the_celestial_sphere.gif/491px-Lorentz_boost_on_the_celestial_sphere.gif" decoding="async" width="491" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/fc/Lorentz_boost_on_the_celestial_sphere.gif 1.5x" data-file-width="600" data-file-height="337" /></a><figcaption>The action of a Lorentz boost in the negative z-direction on the spacelike projection of the celestial sphere (in some choice of orthonormal frame). Again, after the Lorentz boost matrix is applied to the whole space, the celestial sphere must be recovered by rescaling back to <span class="texhtml"><i>t</i> = 1</span>, or equivalently <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>| = 1</span>.</figcaption></figure> <p>The set of real scalar multiples of this null vector, called a <i>null line</i> through the origin, represents a <i>line of sight</i> from an observer at a particular place and time (an arbitrary event we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars. Then the points of the <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a> (equivalently, lines of sight) are identified with certain Hermitian matrices. </p> <div class="mw-heading mw-heading4"><h4 id="Projective_geometry_and_different_views_of_the_2-sphere">Projective geometry and different views of the 2-sphere</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=23" title="Edit section: Projective geometry and different views of the 2-sphere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This picture emerges cleanly in the language of projective geometry. The (restricted) Lorentz group acts on the <b>projective celestial sphere</b>. This is the space of non-zero null vectors with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29a2960e88369263fe3cfe00ccbfeb83daee212a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t>0}"></span> under the given quotient for projective spaces: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t,x,y,z)\sim (t',x',y',z')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t,x,y,z)\sim (t',x',y',z')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1b8de6ed8b2b18c5d3536a68093fc90fede2e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.493ex; height:3.009ex;" alt="{\displaystyle (t,x,y,z)\sim (t',x',y',z')}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t',x',y',z')=(\lambda t,\lambda x,\lambda y,\lambda z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>t</mi> <mo>,</mo> <mi>λ</mi> <mi>x</mi> <mo>,</mo> <mi>λ</mi> <mi>y</mi> <mo>,</mo> <mi>λ</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t',x',y',z')=(\lambda t,\lambda x,\lambda y,\lambda z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197a7ee39bf589f1afeca741367daecbb0424aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.914ex; height:3.009ex;" alt="{\displaystyle (t',x',y',z')=(\lambda t,\lambda x,\lambda y,\lambda z)}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea25afc0351140f919cf791c49c1964b8b081de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda >0}"></span>. This is referred to as the celestial sphere as this allows us to rescale the time coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> to 1 after acting using a Lorentz transformation, ensuring the space-like part sits on the unit sphere. </p><p>From the Möbius side, <span class="texhtml">SL(2, <b>C</b>)</span> acts on complex projective space <span class="texhtml"><b>C</b>P<sup>1</sup></span>, which can be shown to be diffeomorphic to the 2-sphere – this is sometimes referred to as the Riemann sphere. The quotient on projective space leads to a quotient on the group <span class="texhtml">SL(2, <b>C</b>)</span>. </p><p>Finally, these two can be linked together by using the complex projective vector to construct a null-vector. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"></span> is a <span class="texhtml"><b>C</b>P<sup>1</sup></span> projective vector, it can be tensored with its Hermitian conjugate to produce a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"></span> Hermitian matrix. From elsewhere in this article we know this space of matrices can be viewed as 4-vectors. The space of matrices coming from turning each projective vector in the Riemann sphere into a matrix is known as the <a href="/wiki/Bloch_sphere" title="Bloch sphere">Bloch sphere</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Lie_algebra">Lie algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=24" title="Edit section: Lie algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" 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nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Lie_group" title="Lie group">Lie groups</a> and <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a></th></tr><tr><td class="sidebar-image" style="padding-bottom:0.9em;"><span typeof="mw:File/Frameless"><a href="/wiki/File:E8Petrie.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/180px-E8Petrie.svg.png" decoding="async" width="180" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/270px-E8Petrie.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/360px-E8Petrie.svg.png 2x" data-file-width="1476" data-file-height="1451" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Classical_group" title="Classical group">Classical groups</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li> <li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li> <li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li> <li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li> <li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li> <li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Simple_Lie_group" title="Simple Lie group">Simple Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks hlist" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="font-weight:normal; font-style:italic;"> Classical</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_Lie_group#A_series" title="Simple Lie group">A<sub><i>n</i></sub></a></li> <li><a href="/wiki/Simple_Lie_group#B_series" title="Simple Lie group">B<sub><i>n</i></sub></a></li> <li><a href="/wiki/Simple_Lie_group#C_series" title="Simple Lie group">C<sub><i>n</i></sub></a></li> <li><a href="/wiki/Simple_Lie_group#D_series" title="Simple Lie group">D<sub><i>n</i></sub></a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-weight:normal; font-style:italic;"> Exceptional</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Table_of_Lie_groups" title="Table of Lie groups">Other Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li> <li><a class="mw-selflink selflink">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal group</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li> <li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie group–Lie algebra correspondence</a></li> <li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">Exponential map</a></li> <li><a href="/wiki/Adjoint_representation" title="Adjoint representation">Adjoint representation</a></li> <li><div class="hlist"><ul><li><a href="/wiki/Killing_form" title="Killing form">Killing form</a></li><li><a href="/wiki/Index_of_a_Lie_algebra" title="Index of a Lie algebra">Index</a></li></ul></div></li> <li><a href="/wiki/Simple_Lie_algebra" title="Simple Lie algebra">Simple Lie algebra</a></li> <li><a href="/wiki/Loop_algebra" title="Loop algebra">Loop algebra</a></li> <li><a href="/wiki/Affine_Lie_algebra" title="Affine Lie algebra">Affine Lie algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Semisimple_Lie_algebra" title="Semisimple Lie algebra">Semisimple Lie algebra</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Dynkin_diagram" title="Dynkin diagram">Dynkin diagrams</a></li> <li><a href="/wiki/Cartan_subalgebra" title="Cartan subalgebra">Cartan subalgebra</a></li> <li><div class="hlist"><ul><li><a href="/wiki/Root_system" title="Root system">Root system</a></li><li><a href="/wiki/Weyl_group" title="Weyl group">Weyl group</a></li></ul></div></li> <li><div class="hlist"><ul><li><a href="/wiki/Real_form_(Lie_theory)" title="Real form (Lie theory)">Real form</a></li><li><a href="/wiki/Complexification_(Lie_group)" title="Complexification (Lie group)">Complexification</a></li></ul></div></li> <li><a href="/wiki/Split_Lie_algebra" title="Split Lie algebra">Split Lie algebra</a></li> <li><a href="/wiki/Compact_Lie_algebra" title="Compact Lie algebra">Compact Lie algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Representation_of_a_Lie_group" title="Representation of a Lie group">Lie group representation</a></li> <li><a href="/wiki/Lie_algebra_representation" title="Lie algebra representation">Lie algebra representation</a></li> <li><a href="/wiki/Representation_theory_of_semisimple_Lie_algebras" title="Representation theory of semisimple Lie algebras">Representation theory of semisimple Lie algebras</a></li> <li><a href="/wiki/Representations_of_classical_Lie_groups" title="Representations of classical Lie groups">Representations of classical Lie groups</a></li> <li><a href="/wiki/Theorem_of_the_highest_weight" title="Theorem of the highest weight">Theorem of the highest weight</a></li> <li><a href="/wiki/Borel%E2%80%93Weil%E2%80%93Bott_theorem" title="Borel–Weil–Bott theorem">Borel–Weil–Bott theorem</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Lie groups in <a href="/wiki/Physics" title="Physics">physics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Particle_physics_and_representation_theory" title="Particle physics and representation theory">Particle physics and representation theory</a></li> <li><a href="/wiki/Representation_theory_of_the_Lorentz_group" title="Representation theory of the Lorentz group">Lorentz group representations</a></li> <li><a href="/wiki/Representation_theory_of_the_Poincar%C3%A9_group" title="Representation theory of the Poincaré group">Poincaré group representations</a></li> <li><a href="/wiki/Representation_theory_of_the_Galilean_group" title="Representation theory of the Galilean group">Galilean group representations</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></li> <li><a href="/wiki/Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li> <li><a href="/wiki/Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a></li> <li><a href="/wiki/Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a></li> <li><a href="/wiki/Armand_Borel" title="Armand Borel">Armand Borel</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below plainlist"> <ul><li><a href="/wiki/Glossary_of_Lie_groups_and_Lie_algebras" title="Glossary of Lie groups and Lie algebras">Glossary</a></li> <li><a href="/wiki/Table_of_Lie_groups" title="Table of Lie groups">Table of Lie groups</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Lie_groups" title="Template:Lie groups"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Lie_groups" title="Template talk:Lie groups"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Lie_groups" title="Special:EditPage/Template:Lie groups"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>As with any Lie group, a useful way to study many aspects of the Lorentz group is via its <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>. Since the Lorentz group <span class="texhtml">SO(1, 3)</span> is a <a href="/wiki/Lie_group#Matrix_Lie_groups" title="Lie group">matrix Lie group</a>, its corresponding Lie algebra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39939e04003dd2b08e12f527207e2ca6827db8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:7.376ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(1,3)}"></span> is a matrix Lie algebra, which may be computed as<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(1,3)=\left\{4\times 4\,\,\,\mathbf {R} {\text{-valued matrices}}\,X\mid e^{tX}\in \mathrm {SO} (1,3)\,\mathrm {for} \,\mathrm {all} \,t\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mn>4</mn> <mo>×</mo> <mn>4</mn> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>-valued matrices</mtext> </mrow> <mspace width="thinmathspace" /> <mi>X</mi> <mo>∣</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>X</mi> </mrow> </msup> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> </mrow> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(1,3)=\left\{4\times 4\,\,\,\mathbf {R} {\text{-valued matrices}}\,X\mid e^{tX}\in \mathrm {SO} (1,3)\,\mathrm {for} \,\mathrm {all} \,t\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3abe6f4c9d0e118ee349fa1c23c8f8ea6071944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.041ex; width:64.186ex; height:3.343ex;" alt="{\displaystyle {\mathfrak {so}}(1,3)=\left\{4\times 4\,\,\,\mathbf {R} {\text{-valued matrices}}\,X\mid e^{tX}\in \mathrm {SO} (1,3)\,\mathrm {for} \,\mathrm {all} \,t\right\}}"></span>.</dd></dl> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is the diagonal matrix with diagonal entries <span class="texhtml">(1, −1, −1, −1)</span>, then the Lie algebra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {o}}(1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {o}}(1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c63975a74a0684ea613b0709833834c2c48441eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {o}}(1,3)}"></span> consists of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\times 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89eb2e0f4ddfe5f30c8016a0f2aa1fb5ecedfe20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 4\times 4}"></span> matrices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> such that<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta X\eta =-X^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mi>X</mi> <mi>η</mi> <mo>=</mo> <mo>−</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta X\eta =-X^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e2fc7a29a58ef5f853da0349014df3a1ab4966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.574ex; height:3.176ex;" alt="{\displaystyle \eta X\eta =-X^{\textsf {T}}}"></span>.</dd></dl> <p>Explicitly, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39939e04003dd2b08e12f527207e2ca6827db8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:7.376ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(1,3)}"></span> consists of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\times 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89eb2e0f4ddfe5f30c8016a0f2aa1fb5ecedfe20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 4\times 4}"></span> matrices of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}0&a&b&c\\a&0&d&e\\b&-d&0&f\\c&-e&-f&0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>d</mi> </mtd> <mtd> <mi>e</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mo>−</mo> <mi>d</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mo>−</mo> <mi>e</mi> </mtd> <mtd> <mo>−</mo> <mi>f</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}0&a&b&c\\a&0&d&e\\b&-d&0&f\\c&-e&-f&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b2ccea309b2381c989d25b007afe2358e85b683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:20.405ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}0&a&b&c\\a&0&d&e\\b&-d&0&f\\c&-e&-f&0\end{pmatrix}}}"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c,d,e,f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c,d,e,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8e309fdcaa616a3df686996309ed454dd18639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.982ex; height:2.509ex;" alt="{\displaystyle a,b,c,d,e,f}"></span> are arbitrary real numbers. This Lie algebra is six dimensional. The subalgebra of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39939e04003dd2b08e12f527207e2ca6827db8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:7.376ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(1,3)}"></span> consisting of elements in which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> equal to zero is isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"></span>. </p><p>The full Lorentz group <span class="texhtml">O(1, 3)</span>, the proper Lorentz group <span class="texhtml">SO(1, 3)</span> and the proper orthochronous Lorentz group <span class="texhtml">SO<sup>+</sup>(1, 3)</span> (the component connected to the identity) all have the same Lie algebra, which is typically denoted <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39939e04003dd2b08e12f527207e2ca6827db8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:7.376ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(1,3)}"></span>⁠</span>. </p><p>Since the identity component of the Lorentz group is isomorphic to a finite quotient of <span class="texhtml">SL(2, <b>C</b>)</span> (see the section above on the connection of the Lorentz group to the Möbius group), the Lie algebra of the Lorentz group is isomorphic to the Lie algebra <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed49583941c2ea409dc43127c8cec405dad74f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:7.659ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )}"></span>⁠</span>. As a complex Lie algebra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed49583941c2ea409dc43127c8cec405dad74f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:7.659ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )}"></span> is three dimensional, but is six dimensional when viewed as a real Lie algebra. </p> <div class="mw-heading mw-heading3"><h3 id="Commutation_relations_of_the_Lorentz_algebra">Commutation relations of the Lorentz algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=25" title="Edit section: Commutation relations of the Lorentz algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard basis matrices can be indexed as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eec1d62d5b95fbd5d27730facf99aa69a38c159f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.593ex; height:2.343ex;" alt="{\displaystyle M^{\mu \nu }}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ,\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489a4ac1b5d76663fa5328e6f3c1d382655cb496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.668ex; height:2.176ex;" alt="{\displaystyle \mu ,\nu }"></span> take values in <span class="texhtml">{0, 1, 2, 3}</span>. These arise from taking only one of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,\cdots ,f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,\cdots ,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389009c929799a5d770afe633240a5ccbdf1aa63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.718ex; height:2.509ex;" alt="{\displaystyle a,b,\cdots ,f}"></span> to be one, and others zero, in turn. The components can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M^{\mu \nu })_{\rho \sigma }=\delta ^{\mu }{}_{\rho }\delta ^{\nu }{}_{\sigma }-\delta ^{\nu }{}_{\rho }\delta ^{\mu }{}_{\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M^{\mu \nu })_{\rho \sigma }=\delta ^{\mu }{}_{\rho }\delta ^{\nu }{}_{\sigma }-\delta ^{\nu }{}_{\rho }\delta ^{\mu }{}_{\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426d8be6a77381ef5faed184c48ab136a5ebb587" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.742ex; height:3.009ex;" alt="{\displaystyle (M^{\mu \nu })_{\rho \sigma }=\delta ^{\mu }{}_{\rho }\delta ^{\nu }{}_{\sigma }-\delta ^{\nu }{}_{\rho }\delta ^{\mu }{}_{\sigma }}"></span>.</dd></dl> <p>The commutation relations are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=M^{\mu \sigma }\eta ^{\nu \rho }-M^{\nu \sigma }\eta ^{\mu \rho }+M^{\nu \rho }\eta ^{\mu \sigma }-M^{\mu \rho }\eta ^{\nu \sigma }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>σ</mi> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>ρ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>σ</mi> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ρ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>ρ</mi> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>σ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ρ</mi> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>σ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=M^{\mu \sigma }\eta ^{\nu \rho }-M^{\nu \sigma }\eta ^{\mu \rho }+M^{\nu \rho }\eta ^{\mu \sigma }-M^{\mu \rho }\eta ^{\nu \sigma }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94a9022837b52dcbbcf58bf59339dd5754657dfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.867ex; height:2.843ex;" alt="{\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=M^{\mu \sigma }\eta ^{\nu \rho }-M^{\nu \sigma }\eta ^{\mu \rho }+M^{\nu \rho }\eta ^{\mu \sigma }-M^{\mu \rho }\eta ^{\nu \sigma }.}"></span></dd></dl> <p>There are different possible choices of convention in use. In physics, it is common to include a factor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> with the basis elements, which gives a factor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> in the commutation relations. </p><p>Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{0i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{0i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffec250c895f0d4930dce7b337c3d7dd9a4cdddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.12ex; height:2.676ex;" alt="{\displaystyle M^{0i}}"></span> generate boosts and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c7d8cc7b7a7e75db0e41e9884a562d16408620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.976ex; height:2.676ex;" alt="{\displaystyle M^{ij}}"></span> generate rotations. </p><p>The structure constants for the Lorentz algebra can be read off from the commutation relations. Any set of basis elements which satisfy these relations form a representation of the Lorentz algebra. </p> <div class="mw-heading mw-heading3"><h3 id="Generators_of_boosts_and_rotations">Generators of boosts and rotations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=26" title="Edit section: Generators of boosts and rotations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Lorentz group can be thought of as a subgroup of the <a href="/wiki/Diffeomorphism_group" class="mw-redirect" title="Diffeomorphism group">diffeomorphism group</a> of <span class="texhtml"><b>R</b><sup>4</sup></span> and therefore its Lie algebra can be identified with vector fields on <span class="texhtml"><b>R</b><sup>4</sup></span>. In particular, the vectors that generate isometries on a space are its <a href="/wiki/Killing_vector" class="mw-redirect" title="Killing vector">Killing vectors</a>, which provides a convenient alternative to the <a href="/wiki/Maurer%E2%80%93Cartan_form#Intrinsic_construction" title="Maurer–Cartan form">left-invariant vector field</a> for calculating the Lie algebra. We can write down a set of six <a href="/wiki/Lie_algebra#Generators_and_dimension" title="Lie algebra">generators</a>: </p> <ul><li>Vector fields on <span class="texhtml"><b>R</b><sup>4</sup></span> generating three rotations <span class="texhtml"><i>i<b>J</b></i></span>, <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -y\partial _{x}+x\partial _{y}\equiv iJ_{z}~,\qquad -z\partial _{y}+y\partial _{z}\equiv iJ_{x}~,\qquad -x\partial _{z}+z\partial _{x}\equiv iJ_{y}~;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>y</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>x</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>≡</mo> <mi>i</mi> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mo>−</mo> <mi>z</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>y</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>≡</mo> <mi>i</mi> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mo>−</mo> <mi>x</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <mi>z</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>≡</mo> <mi>i</mi> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mtext> </mtext> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -y\partial _{x}+x\partial _{y}\equiv iJ_{z}~,\qquad -z\partial _{y}+y\partial _{z}\equiv iJ_{x}~,\qquad -x\partial _{z}+z\partial _{x}\equiv iJ_{y}~;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e06094d17a0f9a45ba9e10ef68bb87c8049a29ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:67.49ex; height:2.843ex;" alt="{\displaystyle -y\partial _{x}+x\partial _{y}\equiv iJ_{z}~,\qquad -z\partial _{y}+y\partial _{z}\equiv iJ_{x}~,\qquad -x\partial _{z}+z\partial _{x}\equiv iJ_{y}~;}"></span></dd></dl></li> <li>Vector fields on <span class="texhtml"><b>R</b><sup>4</sup></span> generating three boosts <span class="texhtml"><i>i<b>K</b></i></span>, <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\partial _{t}+t\partial _{x}\equiv iK_{x}~,\qquad y\partial _{t}+t\partial _{y}\equiv iK_{y}~,\qquad z\partial _{t}+t\partial _{z}\equiv iK_{z}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>t</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>≡</mo> <mi>i</mi> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mi>y</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>t</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>≡</mo> <mi>i</mi> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mi>z</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>t</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>≡</mo> <mi>i</mi> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\partial _{t}+t\partial _{x}\equiv iK_{x}~,\qquad y\partial _{t}+t\partial _{y}\equiv iK_{y}~,\qquad z\partial _{t}+t\partial _{z}\equiv iK_{z}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6363571abbadd98ef49160b712121b87c6f4b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.733ex; height:2.843ex;" alt="{\displaystyle x\partial _{t}+t\partial _{x}\equiv iK_{x}~,\qquad y\partial _{t}+t\partial _{y}\equiv iK_{y}~,\qquad z\partial _{t}+t\partial _{z}\equiv iK_{z}.}"></span></dd></dl></li></ul> <p>The factor of <span class="texhtml"><i>i</i></span> appears to ensure that the generators of rotations are Hermitian. </p><p>It may be helpful to briefly recall here how to obtain a one-parameter group from a <a href="/wiki/Vector_field" title="Vector field">vector field</a>, written in the form of a first order <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a> <a href="/wiki/Partial_differential_operator" class="mw-redirect" title="Partial differential operator">partial differential operator</a> such as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}=-y\partial _{x}+x\partial _{y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>y</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>x</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=-y\partial _{x}+x\partial _{y}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e0fbef8b4bcef45e03f4d0757ccce17b4eb34ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.173ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}=-y\partial _{x}+x\partial _{y}.}"></span></dd></dl> <p>The corresponding initial value problem (consider <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8388798e62b6e00c44025dd269b3d3b520d29095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.476ex; height:2.843ex;" alt="{\displaystyle r=(x,y)}"></span> a function of a scalar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> and solve <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\lambda }r={\mathcal {L}}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\lambda }r={\mathcal {L}}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ed863118948f0e6093f3055a94f46fe8ed4236" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.225ex; height:2.509ex;" alt="{\displaystyle \partial _{\lambda }r={\mathcal {L}}r}"></span> with some initial conditions) is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial x}{\partial \lambda }}=-y,\;{\frac {\partial y}{\partial \lambda }}=x,\;x(0)=x_{0},\;y(0)=y_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial x}{\partial \lambda }}=-y,\;{\frac {\partial y}{\partial \lambda }}=x,\;x(0)=x_{0},\;y(0)=y_{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61585fa0e8a999b1bd82c331f6ad67d28f0304fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:42.396ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial x}{\partial \lambda }}=-y,\;{\frac {\partial y}{\partial \lambda }}=x,\;x(0)=x_{0},\;y(0)=y_{0}.}"></span></dd></dl> <p>The solution can be written </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(\lambda )=x_{0}\cos(\lambda )-y_{0}\sin(\lambda ),\;y(\lambda )=x_{0}\sin(\lambda )+y_{0}\cos(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(\lambda )=x_{0}\cos(\lambda )-y_{0}\sin(\lambda ),\;y(\lambda )=x_{0}\sin(\lambda )+y_{0}\cos(\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce66f0af5fab8787f558e9a544774c340594d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.666ex; height:2.843ex;" alt="{\displaystyle x(\lambda )=x_{0}\cos(\lambda )-y_{0}\sin(\lambda ),\;y(\lambda )=x_{0}\sin(\lambda )+y_{0}\cos(\lambda )}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&\cos(\lambda )&-\sin(\lambda )&0\\0&\sin(\lambda )&\cos(\lambda )&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}t_{0}\\x_{0}\\y_{0}\\z_{0}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&\cos(\lambda )&-\sin(\lambda )&0\\0&\sin(\lambda )&\cos(\lambda )&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}t_{0}\\x_{0}\\y_{0}\\z_{0}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d4fe73c98844117f5c4f2a732c21293b77c61b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:42.152ex; height:12.843ex;" alt="{\displaystyle {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&\cos(\lambda )&-\sin(\lambda )&0\\0&\sin(\lambda )&\cos(\lambda )&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}t_{0}\\x_{0}\\y_{0}\\z_{0}\end{bmatrix}}}"></span></dd></dl> <p>where we easily recognize the one-parameter matrix group of rotations <span class="texhtml">exp(<i>iλJ</i><sub><i>z</i></sub>)</span> about the z-axis. </p><p>Differentiating with respect to the group parameter <span class="texhtml mvar" style="font-style:italic;">λ</span> and setting it <span class="texhtml"><i>λ</i> = 0</span> in that result, we recover the standard matrix, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle iJ_{z}={\begin{bmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{bmatrix}}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle iJ_{z}={\begin{bmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{bmatrix}}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240d607080411cc37f7079a8c24181a58939476d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:24.698ex; height:12.509ex;" alt="{\displaystyle iJ_{z}={\begin{bmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{bmatrix}}~,}"></span></dd></dl> <p>which corresponds to the vector field we started with. This illustrates how to pass between matrix and vector field representations of elements of the Lie algebra. The <a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">exponential map</a> plays this special role not only for the Lorentz group but for Lie groups in general. </p><p>Reversing the procedure in the previous section, we see that the Möbius transformations that correspond to our six generators arise from exponentiating respectively <span class="texhtml"><i>η</i>/2</span> (for the three boosts) or <span class="texhtml"><i>iθ</i>/2</span> (for the three rotations) times the three <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}},\;\;\sigma _{2}={\begin{bmatrix}0&-i\\i&0\end{bmatrix}},\;\;\sigma _{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}},\;\;\sigma _{2}={\begin{bmatrix}0&-i\\i&0\end{bmatrix}},\;\;\sigma _{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edbae100c3560bbac2cf075e3facc5ab0dec6ada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.554ex; height:6.176ex;" alt="{\displaystyle \sigma _{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}},\;\;\sigma _{2}={\begin{bmatrix}0&-i\\i&0\end{bmatrix}},\;\;\sigma _{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Generators_of_the_Möbius_group"><span id="Generators_of_the_M.C3.B6bius_group"></span>Generators of the Möbius group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=27" title="Edit section: Generators of the Möbius group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another generating set arises via the isomorphism to the Möbius group. The following table lists the six generators, in which </p> <ul><li>The first column gives a generator of the flow under the Möbius action (after stereographic projection from the Riemann sphere) as a <i>real</i> vector field on the Euclidean plane.</li> <li>The second column gives the corresponding one-parameter subgroup of Möbius transformations.</li> <li>The third column gives the corresponding one-parameter subgroup of Lorentz transformations (the image under our homomorphism of preceding one-parameter subgroup).</li> <li>The fourth column gives the corresponding generator of the flow under the Lorentz action as a real vector field on Minkowski spacetime.</li></ul> <p>Notice that the generators consist of </p> <ul><li>Two parabolics (null rotations)</li> <li>One hyperbolic (boost in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06bbc68eb647fa5766f644368eba8942c4b72868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.236ex; height:2.509ex;" alt="{\displaystyle \partial _{z}}"></span> direction)</li> <li>Three elliptics (rotations about the <i>x</i>, <i>y</i>, <i>z</i> axes, respectively)</li></ul> <table class="wikitable" style="margin:auto; text-align:center;"> <tbody><tr> <th>Vector field on <span class="texhtml"><b>R</b><sup>2</sup></span> </th> <th>One-parameter subgroup of <span class="texhtml">SL(2, <b>C</b>)</span>, <br />representing Möbius transformations </th> <th>One-parameter subgroup of <span class="texhtml">SO<sup>+</sup>(1, 3)</span>, <br />representing Lorentz transformations </th> <th>Vector field on <span class="texhtml"><b>R</b><sup>1,3</sup></span> </th></tr> <tr> <th colspan="4" style="background-color:#ddddff;">Parabolic </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{u}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{u}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2035be40f90fe7cff559085d786143a118c6fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle \partial _{u}\,\!}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&\alpha \\0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&\alpha \\0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4accb57a3fa386474d656a7b740498e0ca14b45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.179ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&\alpha \\0&1\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1+{\frac {1}{2}}\alpha ^{2}&\alpha &0&-{\frac {1}{2}}\alpha ^{2}\\\alpha &1&0&-\alpha \\0&0&1&0\\{\frac {1}{2}}\alpha ^{2}&\alpha &0&1-{\frac {1}{2}}\alpha ^{2}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi>α</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi>α</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1+{\frac {1}{2}}\alpha ^{2}&\alpha &0&-{\frac {1}{2}}\alpha ^{2}\\\alpha &1&0&-\alpha \\0&0&1&0\\{\frac {1}{2}}\alpha ^{2}&\alpha &0&1-{\frac {1}{2}}\alpha ^{2}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c5963210eb6d306bd5a3a8d9325986e7a02ed20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:29.876ex; height:14.176ex;" alt="{\displaystyle {\begin{bmatrix}1+{\frac {1}{2}}\alpha ^{2}&\alpha &0&-{\frac {1}{2}}\alpha ^{2}\\\alpha &1&0&-\alpha \\0&0&1&0\\{\frac {1}{2}}\alpha ^{2}&\alpha &0&1-{\frac {1}{2}}\alpha ^{2}\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}X_{1}=x&(\partial _{t}+\partial _{z})+{}\\&(t-z)\partial _{x}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}X_{1}=x&(\partial _{t}+\partial _{z})+{}\\&(t-z)\partial _{x}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13fdcb1f1271e094ade3ece2a7f8038beb6201eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.945ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}X_{1}=x&(\partial _{t}+\partial _{z})+{}\\&(t-z)\partial _{x}\end{aligned}}}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{v}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{v}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09de455297876adb174050aef1614063b8db2b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.651ex; height:2.509ex;" alt="{\displaystyle \partial _{v}\,\!}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&i\alpha \\0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>i</mi> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&i\alpha \\0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec27466dbb4555219f172d0b71546b848629eb85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.982ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&i\alpha \\0&1\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1+{\frac {1}{2}}\alpha ^{2}&0&\alpha &-{\frac {1}{2}}\alpha ^{2}\\0&1&0&0\\\alpha &0&1&-\alpha \\{\frac {1}{2}}\alpha ^{2}&0&\alpha &1-{\frac {1}{2}}\alpha ^{2}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>α</mi> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>α</mi> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1+{\frac {1}{2}}\alpha ^{2}&0&\alpha &-{\frac {1}{2}}\alpha ^{2}\\0&1&0&0\\\alpha &0&1&-\alpha \\{\frac {1}{2}}\alpha ^{2}&0&\alpha &1-{\frac {1}{2}}\alpha ^{2}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b565299623895491597ecc36b4745dcbc56de32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:29.876ex; height:14.176ex;" alt="{\displaystyle {\begin{bmatrix}1+{\frac {1}{2}}\alpha ^{2}&0&\alpha &-{\frac {1}{2}}\alpha ^{2}\\0&1&0&0\\\alpha &0&1&-\alpha \\{\frac {1}{2}}\alpha ^{2}&0&\alpha &1-{\frac {1}{2}}\alpha ^{2}\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}X_{2}=y&(\partial _{t}+\partial _{z})+{}\\&(t-z)\partial _{y}\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>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}X_{2}=y&(\partial _{t}+\partial _{z})+{}\\&(t-z)\partial _{y}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adda065aab7e1933023e6fe55bfa8ce4433955f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.771ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}X_{2}=y&(\partial _{t}+\partial _{z})+{}\\&(t-z)\partial _{y}\end{aligned}}}"></span> </td></tr> <tr> <th colspan="4" style="background-color:#ddddff;">Hyperbolic </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\left(u\partial _{u}+v\partial _{v}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>+</mo> <mi>v</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\left(u\partial _{u}+v\partial _{v}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ac58e415e85352ea2422794ac290fb9d0f94c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.164ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\left(u\partial _{u}+v\partial _{v}\right)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\exp \left({\frac {\eta }{2}}\right)&0\\0&\exp \left(-{\frac {\eta }{2}}\right)\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\exp \left({\frac {\eta }{2}}\right)&0\\0&\exp \left(-{\frac {\eta }{2}}\right)\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffcdf243a0dc64016fe72ec83ea6a2ac088de119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.283ex; height:7.509ex;" alt="{\displaystyle {\begin{bmatrix}\exp \left({\frac {\eta }{2}}\right)&0\\0&\exp \left(-{\frac {\eta }{2}}\right)\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&1&0&0\\0&0&1&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&1&0&0\\0&0&1&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a1d38e82700a15a609432bb1eecfe99dc97085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:27.91ex; height:12.843ex;" alt="{\displaystyle {\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&1&0&0\\0&0&1&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}=z\partial _{t}+t\partial _{z}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>z</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>t</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}=z\partial _{t}+t\partial _{z}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e87376f686868f3377a6695370e10374c45cd51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:15.529ex; height:2.509ex;" alt="{\displaystyle X_{3}=z\partial _{t}+t\partial _{z}\,\!}"></span> </td></tr> <tr> <th colspan="4" style="background-color:#ddddff;">Elliptic </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\left(-v\partial _{u}+u\partial _{v}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>v</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\left(-v\partial _{u}+u\partial _{v}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7c6f40eec6d219dab09beb02aa7315fcecb399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.972ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\left(-v\partial _{u}+u\partial _{v}\right)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\exp \left({\frac {i\theta }{2}}\right)&0\\0&\exp \left({\frac {-i\theta }{2}}\right)\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\exp \left({\frac {i\theta }{2}}\right)&0\\0&\exp \left({\frac {-i\theta }{2}}\right)\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee687f3ab037fd9ea8395e6322d75e84d9c6a281" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:24.458ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}\exp \left({\frac {i\theta }{2}}\right)&0\\0&\exp \left({\frac {-i\theta }{2}}\right)\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&-\sin(\theta )&0\\0&\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&-\sin(\theta )&0\\0&\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c3c51f0dc951b79837dc5872a0c608f8856a313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:27.106ex; height:12.843ex;" alt="{\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&-\sin(\theta )&0\\0&\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{4}=-y\partial _{x}+x\partial _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>y</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>x</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{4}=-y\partial _{x}+x\partial _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd0f3855d20223d9e7ad282ca336154d406f324c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.901ex; height:2.843ex;" alt="{\displaystyle X_{4}=-y\partial _{x}+x\partial _{y}}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v^{2}-u^{2}-1}{2}}\partial _{u}-uv\,\partial _{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>−</mo> <mi>u</mi> <mi>v</mi> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v^{2}-u^{2}-1}{2}}\partial _{u}-uv\,\partial _{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad5db6ad34d93ecc6869ed1821f3edf44125c6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.601ex; height:5.676ex;" alt="{\displaystyle {\frac {v^{2}-u^{2}-1}{2}}\partial _{u}-uv\,\partial _{v}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</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> <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> <mtd> <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> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e21fa057e90b1bfe19cb490199c4f7a9cecf83a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:23.204ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/246323695fdbc43c3706e2b0bb5e9ab2b2c31ad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:27.106ex; height:12.843ex;" alt="{\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{5}=-x\partial _{z}+z\partial _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>x</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <mi>z</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{5}=-x\partial _{z}+z\partial _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e8ee9ec28e58735efb8ef22db61fcdf0893a45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.786ex; height:2.509ex;" alt="{\displaystyle X_{5}=-x\partial _{z}+z\partial _{x}}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle uv\,\partial _{u}+{\frac {1-u^{2}+v^{2}}{2}}\partial _{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mi>v</mi> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle uv\,\partial _{u}+{\frac {1-u^{2}+v^{2}}{2}}\partial _{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f7ec12eea6f3ac0774163efdb9e1a4c4afdcdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.601ex; height:5.676ex;" alt="{\displaystyle uv\,\partial _{u}+{\frac {1-u^{2}+v^{2}}{2}}\partial _{v}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&i\sin \left({\frac {\theta }{2}}\right)\\i\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>i</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> <mtr> <mtd> <mi>i</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> <mtd> <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> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&i\sin \left({\frac {\theta }{2}}\right)\\i\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c9883f1b9c541d3253999966b2c865738479e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:23.132ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&i\sin \left({\frac {\theta }{2}}\right)\\i\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos(\theta )&-\sin(\theta )\\0&0&\sin(\theta )&\cos(\theta )\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos(\theta )&-\sin(\theta )\\0&0&\sin(\theta )&\cos(\theta )\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94419ba0d5388255296a67c889e175ef2ee40040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:27.106ex; height:12.843ex;" alt="{\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos(\theta )&-\sin(\theta )\\0&0&\sin(\theta )&\cos(\theta )\end{bmatrix}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{6}=-z\partial _{y}+y\partial _{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>z</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>y</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{6}=-z\partial _{y}+y\partial _{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a74b0b969ecdb95b80140a45e35864ef2060c6d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.489ex; height:2.843ex;" alt="{\displaystyle X_{6}=-z\partial _{y}+y\partial _{z}}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Worked_example:_rotation_about_the_y-axis">Worked example: rotation about the y-axis</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=28" title="Edit section: Worked example: rotation about the y-axis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Start with </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2}={\begin{bmatrix}0&i\\-i&0\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{2}={\begin{bmatrix}0&i\\-i&0\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2ef60cbcacc8588505cc41219924334453af26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.429ex; height:6.176ex;" alt="{\displaystyle \sigma _{2}={\begin{bmatrix}0&i\\-i&0\end{bmatrix}}.}"></span></dd></dl> <p>Exponentiate: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \left({\frac {i\theta }{2}}\,\sigma _{2}\right)={\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <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> </mtd> <mtd> <mo>−</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> <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> <mtd> <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> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left({\frac {i\theta }{2}}\,\sigma _{2}\right)={\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32dbda2e0fa060a42e421af929785b20f5e93e58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:39.421ex; height:9.843ex;" alt="{\displaystyle \exp \left({\frac {i\theta }{2}}\,\sigma _{2}\right)={\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}.}"></span></dd></dl> <p>This element of <span class="texhtml">SL(2, <b>C</b>)</span> represents the one-parameter subgroup of (elliptic) Möbius transformations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \mapsto \xi '={\frac {\cos \left({\frac {\theta }{2}}\right)\,\xi -\sin \left({\frac {\theta }{2}}\right)}{\sin \left({\frac {\theta }{2}}\right)\,\xi +\cos \left({\frac {\theta }{2}}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo stretchy="false">↦</mo> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> <mspace width="thinmathspace" /> <mi>ξ</mi> <mo>−</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> </mrow> <mrow> <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> <mspace width="thinmathspace" /> <mi>ξ</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> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \mapsto \xi '={\frac {\cos \left({\frac {\theta }{2}}\right)\,\xi -\sin \left({\frac {\theta }{2}}\right)}{\sin \left({\frac {\theta }{2}}\right)\,\xi +\cos \left({\frac {\theta }{2}}\right)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c712171aaeb4de5724b3f026605434f884e979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:30.035ex; height:10.176ex;" alt="{\displaystyle \xi \mapsto \xi '={\frac {\cos \left({\frac {\theta }{2}}\right)\,\xi -\sin \left({\frac {\theta }{2}}\right)}{\sin \left({\frac {\theta }{2}}\right)\,\xi +\cos \left({\frac {\theta }{2}}\right)}}.}"></span></dd></dl> <p>Next, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {d\xi '}{d\theta }}\right|_{\theta =0}=-{\frac {1+\xi ^{2}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>ξ</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {d\xi '}{d\theta }}\right|_{\theta =0}=-{\frac {1+\xi ^{2}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8130fee0110c0e167fec0df83e42e53095e58dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.001ex; height:6.509ex;" alt="{\displaystyle \left.{\frac {d\xi '}{d\theta }}\right|_{\theta =0}=-{\frac {1+\xi ^{2}}{2}}.}"></span></dd></dl> <p>The corresponding vector field on <span class="texhtml"><b>C</b></span> (thought of as the image of <span class="texhtml"><i>S</i><sup>2</sup></span> under stereographic projection) is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {1+\xi ^{2}}{2}}\,\partial _{\xi }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {1+\xi ^{2}}{2}}\,\partial _{\xi }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e967a39337dbf6d9a91ee6601f020bd19812a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.964ex; height:5.676ex;" alt="{\displaystyle -{\frac {1+\xi ^{2}}{2}}\,\partial _{\xi }.}"></span></dd></dl> <p>Writing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =u+iv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mi>u</mi> <mo>+</mo> <mi>i</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi =u+iv}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58840f90456cc4f6e0239ad936142fbe69b973c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.229ex; height:2.509ex;" alt="{\displaystyle \xi =u+iv}"></span>, this becomes the vector field on <span class="texhtml"><b>R</b><sup>2</sup></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {1+u^{2}-v^{2}}{2}}\,\partial _{u}-uv\,\partial _{v}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>−</mo> <mi>u</mi> <mi>v</mi> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {1+u^{2}-v^{2}}{2}}\,\partial _{u}-uv\,\partial _{v}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff06ef53399d96148a64b9299cd4fa56d0148cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.443ex; height:5.676ex;" alt="{\displaystyle -{\frac {1+u^{2}-v^{2}}{2}}\,\partial _{u}-uv\,\partial _{v}.}"></span></dd></dl> <p>Returning to our element of <span class="texhtml">SL(2, <b>C</b>)</span>, writing out the action <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mapsto PXP^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">↦</mo> <mi>P</mi> <mi>X</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\mapsto PXP^{\dagger }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30dade75a424089685d82dcc446c850dd58eb13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.103ex; height:2.676ex;" alt="{\displaystyle X\mapsto PXP^{\dagger }}"></span> and collecting terms, we find that the image under the spinor map is the element of <span class="texhtml">SO<sup>+</sup>(1, 3)</span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7edad7eb737325b6992cd6ee0e88c4d29786d11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:27.753ex; height:12.843ex;" alt="{\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{bmatrix}}.}"></span></dd></dl> <p>Differentiating with respect to <span class="texhtml"><i>θ</i></span> at <span class="texhtml"><i>θ</i> = 0</span>, yields the corresponding vector field on <span class="texhtml"><b>R</b><sup>1,3</sup></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\partial _{x}-x\partial _{z}.\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <mi>x</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\partial _{x}-x\partial _{z}.\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39834b9247f37158c89cdf27a5a904a0c271aa70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:10.935ex; height:2.509ex;" alt="{\displaystyle z\partial _{x}-x\partial _{z}.\,\!}"></span></dd></dl> <p>This is evidently the generator of counterclockwise rotation about the <span class="texhtml"><i>y</i></span>-axis. </p> <div class="mw-heading mw-heading2"><h2 id="Subgroups_of_the_Lorentz_group">Subgroups of the Lorentz group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=29" title="Edit section: Subgroups of the Lorentz group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The subalgebras of the Lie algebra of the Lorentz group can be enumerated, up to conjugacy, from which the <a href="/wiki/Closed_subgroup" class="mw-redirect" title="Closed subgroup">closed subgroups</a> of the restricted Lorentz group can be listed, up to conjugacy. (See the book by Hall cited below for the details.) These can be readily expressed in terms of the generators <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> given in the table above. </p><p>The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}"></span> generates a one-parameter subalgebra of parabolics <span class="texhtml">SO(0, 1)</span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f17eb6a51e16ea92736c904a92d8a78e73a598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{3}}"></span> generates a one-parameter subalgebra of boosts <span class="texhtml">SO(1, 1)</span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae165d68e73491207cb065c2b9e7b90f78cf2e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{4}}"></span> generates a one-parameter of rotations <span class="texhtml">SO(2)</span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}+aX_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3}+aX_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc3e5e7dd6b924ea2e1220ba647c42d4382180b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.027ex; height:2.509ex;" alt="{\displaystyle X_{3}+aX_{4}}"></span> (for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}"></span>) generates a one-parameter subalgebra of loxodromic transformations.</li></ul> <p>(Strictly speaking the last corresponds to infinitely many classes, since distinct <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> give different classes.) The two-dimensional subalgebras are: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6099d6fb3c34ad5e22fad9c79c40c4ebfee1ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.991ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2}}"></span> generate an abelian subalgebra consisting entirely of parabolics,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e12cf2b6d1a3f9b8320bd482a7a0c18d1a4a71a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.991ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{3}}"></span> generate a nonabelian subalgebra isomorphic to the Lie algebra of the <a href="/wiki/Affine_group" title="Affine group">affine group</a> <span class="texhtml">Aff(1)</span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3},X_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{3},X_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ddd72b572edd8d6f9465ab3fc4563a1e3ad6db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.991ex; height:2.509ex;" alt="{\displaystyle X_{3},X_{4}}"></span> generate an abelian subalgebra consisting of boosts, rotations, and loxodromics all sharing the same pair of fixed points.</li></ul> <p>The three-dimensional subalgebras use the <a href="/wiki/Bianchi_classification" title="Bianchi classification">Bianchi classification</a> scheme: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4126aa4015fd02d5478fd2d09601afd33286509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.003ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3}}"></span> generate a <b>Bianchi V</b> subalgebra, isomorphic to the Lie algebra of <span class="texhtml">Hom(2)</span>, the group of <i>euclidean homotheties</i>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d69d21a5e941714d1c2abe1aef3afad13b7e9a5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.003ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{4}}"></span> generate a <b>Bianchi VII<sub>0</sub></b> subalgebra, isomorphic to the Lie algebra of <span class="texhtml"><i>E</i>(2)</span>, the <a href="/wiki/Euclidean_group" title="Euclidean group">euclidean group</a>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3}+aX_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3}+aX_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/043218df541abf22893dac560369ff8916d7e12a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.052ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3}+aX_{4}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}"></span>, generate a <b>Bianchi VII<sub>a</sub></b> subalgebra,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{3},X_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{3},X_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/537b513354ba14e92776cee5b3d7dda77ef7446c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.003ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{3},X_{5}}"></span> generate a <b>Bianchi VIII</b> subalgebra, isomorphic to the Lie algebra of <span class="texhtml">SL(2, <b>R</b>)</span>, the group of isometries of the <a href="/wiki/Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">hyperbolic plane</a>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{4},X_{5},X_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{4},X_{5},X_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21c8b18612ad63e3d3f9dbe2dfd9cb8c80ec749c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.003ex; height:2.509ex;" alt="{\displaystyle X_{4},X_{5},X_{6}}"></span> generate a <b>Bianchi IX</b> subalgebra, isomorphic to the Lie algebra of <span class="texhtml">SO(3)</span>, the rotation group.</li></ul> <p>The <a href="/wiki/Bianchi_classification" title="Bianchi classification">Bianchi types</a> refer to the classification of three-dimensional Lie algebras by the Italian mathematician <a href="/wiki/Luigi_Bianchi" title="Luigi Bianchi">Luigi Bianchi</a>. </p><p>The four-dimensional subalgebras are all conjugate to </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3},X_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3},X_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbe282436962742f9850b44f04ae273cb0bfe86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.016ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3},X_{4}}"></span> generate a subalgebra isomorphic to the Lie algebra of <span class="texhtml">Sim(2)</span>, the group of Euclidean <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similitudes</a>.</li></ul> <p>The subalgebras form a lattice (see the figure), and each subalgebra generates by exponentiation a <a href="/wiki/Closed_subgroup" class="mw-redirect" title="Closed subgroup">closed subgroup</a> of the restricted Lie group. From these, all subgroups of the Lorentz group can be constructed, up to conjugation, by multiplying by one of the elements of the Klein four-group. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lorentz_group_subalgebra_lattice.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Lorentz_group_subalgebra_lattice.png/300px-Lorentz_group_subalgebra_lattice.png" decoding="async" width="300" height="291" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/b2/Lorentz_group_subalgebra_lattice.png 1.5x" data-file-width="446" data-file-height="433" /></a><figcaption>The lattice of subalgebras of the Lie algebra <span class="texhtml">SO(1, 3)</span>, up to conjugacy.</figcaption></figure> <p>As with any connected Lie group, the coset spaces of the closed subgroups of the restricted Lorentz group, or <a href="/wiki/Homogeneous_spaces" class="mw-redirect" title="Homogeneous spaces">homogeneous spaces</a>, have considerable mathematical interest. A few, brief descriptions: </p> <ul><li>The group <span class="texhtml">Sim(2)</span> is the stabilizer of a <i>null line</i>; i.e., of a point on the Riemann sphere—so the homogeneous space <span class="texhtml">SO<sup>+</sup>(1, 3) / Sim(2)</span> is the <a href="/wiki/Kleinian_geometry" class="mw-redirect" title="Kleinian geometry">Kleinian geometry</a> that represents <a href="/wiki/Conformal_geometry" title="Conformal geometry">conformal geometry</a> on the sphere <span class="texhtml"><i>S</i><sup>2</sup></span>.</li> <li>The (identity component of the) Euclidean group <span class="texhtml">SE(2)</span> is the stabilizer of a <a href="/wiki/Null_vector" title="Null vector">null vector</a>, so the homogeneous space <span class="texhtml">SO<sup>+</sup>(1, 3) / SE(2)</span> is the <a href="/wiki/Momentum_space" class="mw-redirect" title="Momentum space">momentum space</a> of a massless particle; geometrically, this Kleinian geometry represents the <i>degenerate</i> geometry of the light cone in Minkowski spacetime.</li> <li>The rotation group <span class="texhtml">SO(3)</span> is the stabilizer of a <a href="/wiki/Timelike_vector" class="mw-redirect" title="Timelike vector">timelike vector</a>, so the homogeneous space <span class="texhtml">SO<sup>+</sup>(1, 3) / SO(3)</span> is the <a href="/wiki/Momentum_space" class="mw-redirect" title="Momentum space">momentum space</a> of a massive particle; geometrically, this space is none other than three-dimensional <a href="/wiki/Hyperbolic_space" title="Hyperbolic space">hyperbolic space</a> <span class="texhtml"><i>H</i><sup>3</sup></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Generalization_to_higher_dimensions">Generalization to higher dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=30" title="Edit section: Generalization to higher dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Indefinite_orthogonal_group" title="Indefinite orthogonal group">Indefinite orthogonal group</a></div> <p>The concept of the Lorentz group has a natural generalization to spacetime of any number of dimensions. Mathematically, the Lorentz group of (<i>n</i> + 1)-dimensional Minkowski space is the <a href="/wiki/Indefinite_orthogonal_group" title="Indefinite orthogonal group">indefinite orthogonal group</a> <span class="texhtml">O(<i>n</i>, 1)</span> of linear transformations of <span class="texhtml"><b>R</b><sup><i>n</i>+1</sup></span> that preserves the quadratic form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},\ldots ,x_{n},x_{n+1})\mapsto x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}-x_{n+1}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\ldots ,x_{n},x_{n+1})\mapsto x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}-x_{n+1}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2553d50a252ad8c0d36998b6b512931c7d0642a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:51.33ex; height:3.343ex;" alt="{\displaystyle (x_{1},x_{2},\ldots ,x_{n},x_{n+1})\mapsto x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}-x_{n+1}^{2}.}"></span></dd></dl> <p>The group <span class="texhtml">O(1, <i>n</i>)</span> preserves the quadratic form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},\ldots ,x_{n},x_{n+1})\mapsto x_{1}^{2}-x_{2}^{2}-\cdots -x_{n+1}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mo>⋯</mo> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\ldots ,x_{n},x_{n+1})\mapsto x_{1}^{2}-x_{2}^{2}-\cdots -x_{n+1}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79b35530b4dd8ec6bf8870723b1cc81b911b1e84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:45.295ex; height:3.343ex;" alt="{\displaystyle (x_{1},x_{2},\ldots ,x_{n},x_{n+1})\mapsto x_{1}^{2}-x_{2}^{2}-\cdots -x_{n+1}^{2}}"></span></dd></dl> <p><span class="texhtml">O(1, <i>n</i>)</span> is isomorphic to <span class="texhtml">O(<i>n</i>, 1)</span>, and both presentations of the Lorentz group are in use in the theoretical physics community. The former is more common in literature related to gravity, while the latter is more common in particle physics literature. </p><p>A common notation for the vector space <span class="texhtml"><b>R</b><sup><i>n</i>+1</sup></span>, equipped with this choice of quadratic form, is <span class="texhtml"><b>R</b><sup>1,<i>n</i></sup></span>. </p><p>Many of the properties of the Lorentz group in four dimensions (where <span class="nowrap"><i>n</i> = 3</span>) generalize straightforwardly to arbitrary <span class="texhtml"><i>n</i></span>. For instance, the Lorentz group <span class="texhtml">O(<i>n</i>, 1)</span> has four connected components, and it acts by conformal transformations on the celestial <span class="texhtml">(<i>n</i> − 1)</span>-sphere in <span class="texhtml">(<i>n</i> + 1)</span>-dimensional Minkowski space. The identity component <span class="texhtml">SO<sup>+</sup>(<i>n</i>, 1)</span> is an <span class="texhtml">SO(<i>n</i>)</span>-bundle over hyperbolic <span class="texhtml"><i>n</i></span>-space <span class="texhtml"><i>H</i><sup><i>n</i></sup></span>. </p><p>The low-dimensional cases <span class="texhtml"><i>n</i> = 1</span> and <span class="texhtml"><i>n</i> = 2</span> are often useful as "toy models" for the physical case <span class="texhtml"><i>n</i> = 3</span>, while higher-dimensional Lorentz groups are used in physical theories such as <a href="/wiki/String_theory" title="String theory">string theory</a> that posit the existence of hidden dimensions. The Lorentz group <span class="texhtml">O(<i>n</i>, 1)</span> is also the isometry group of <span class="texhtml"><i>n</i></span>-dimensional <a href="/wiki/De_Sitter_space" title="De Sitter space">de Sitter space</a> <span class="texhtml">dS<sub><i>n</i></sub></span>, which may be realized as the homogeneous space <span class="texhtml">O(<i>n</i>, 1) / O(<i>n</i> − 1, 1)</span>. In particular <span class="texhtml">O(4, 1)</span> is the isometry group of the <a href="/wiki/De_Sitter_universe" title="De Sitter universe">de Sitter universe</a> <span class="texhtml">dS<sub>4</sub></span>, a cosmological model. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=31" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/Representation_theory_of_the_Lorentz_group" title="Representation theory of the Lorentz group">Lorentz group representation theory</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a></li> <li><a href="/wiki/M%C3%B6bius_group" class="mw-redirect" title="Möbius group">Möbius group</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Indefinite_orthogonal_group" title="Indefinite orthogonal group">Indefinite orthogonal group</a></li> <li><a href="/wiki/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">Quaternions and spatial rotation</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=32" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Note that some authors refer to <span class="texhtml">SO(1, 3)</span> or even <span class="texhtml">O(1, 3)</span> when they mean <span class="texhtml">SO<sup>+</sup>(1, 3)</span>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">See the article <a href="/wiki/Weyl_equation" title="Weyl equation">Weyl equation</a> for explicit derivations.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=33" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeinberg2002">Weinberg 2002</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Varićak V 1910 "Theory of Relativity and Lobachevskian geometry", Phys Z 1910 §3 'Lorentz-Einstein transformation as translation'. Engl.tr in Wikipedia</span> </li> <li id="cite_note-Gelfand_1-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gelfand_1_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGelfandMinlosShapiro1963">Gelfand, Minlos & Shapiro 1963</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFWigner1939">Wigner 1939</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> Definition 3.18</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> Proposition 3.25</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Reading_List">Reading List</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz_group&action=edit&section=34" title="Edit section: Reading List"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><a href="/wiki/Emil_Artin" title="Emil Artin">Emil Artin</a> (1957) <a rel="nofollow" class="external text" href="https://archive.org/details/geometricalgebra033556mbp/page/n115/mode/2up?view=theater"><i>Geometric Algebra</i>, chapter III: Symplectic and Orthogonal Geometry</a> via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>, covers orthogonal groups <span class="texhtml">O(<i>p</i>, <i>q</i>)</span></li> <li><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="CITEREFCarmeli,_Moshe1977" class="citation book cs1"><a href="/wiki/Moshe_Carmeli" title="Moshe Carmeli">Carmeli, Moshe</a> (1977). <i>Group Theory and General Relativity, Representations of the Lorentz Group and Their Applications to the Gravitational Field</i>. McGraw-Hill, New York. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-009986-9" title="Special:BookSources/978-0-07-009986-9"><bdi>978-0-07-009986-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Group+Theory+and+General+Relativity%2C+Representations+of+the+Lorentz+Group+and+Their+Applications+to+the+Gravitational+Field&rft.pub=McGraw-Hill%2C+New+York&rft.date=1977&rft.isbn=978-0-07-009986-9&rft.au=Carmeli%2C+Moshe&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span> A canonical reference; <i>see chapters 1–6</i> for representations of the Lorentz group.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrankel,_Theodore2004" class="citation book cs1">Frankel, Theodore (2004). <i>The Geometry of Physics (2nd Ed.)</i>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-53927-2" title="Special:BookSources/978-0-521-53927-2"><bdi>978-0-521-53927-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Physics+%282nd+Ed.%29&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=978-0-521-53927-2&rft.au=Frankel%2C+Theodore&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span> An excellent resource for Lie theory, fiber bundles, spinorial coverings, and many other topics.</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). <i>Representation theory. A first course</i>. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. <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/978-0-387-97495-8" title="Special:BookSources/978-0-387-97495-8"><bdi>978-0-387-97495-8</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>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/246650103">246650103</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.+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=info%3Aoclcnum%2F246650103&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=978-0-387-97495-8&rft.aulast=Fulton&rft.aufirst=William&rft.au=Harris%2C+Joe&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span> <i>See Lecture 11</i> for the irreducible representations of <span class="texhtml">SL(2, <b>C</b>)</span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelfandMinlosShapiro1963" class="citation cs2"><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gelfand, I.M.</a>; <a href="/wiki/Robert_Adol%27fovich_Minlos" class="mw-redirect" title="Robert Adol'fovich Minlos">Minlos, R.A.</a>; Shapiro, Z.Ya. (1963), <i>Representations of the Rotation and Lorentz Groups and their Applications</i>, New York: Pergamon Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Representations+of+the+Rotation+and+Lorentz+Groups+and+their+Applications&rft.place=New+York&rft.pub=Pergamon+Press&rft.date=1963&rft.aulast=Gelfand&rft.aufirst=I.M.&rft.au=Minlos%2C+R.A.&rft.au=Shapiro%2C+Z.Ya.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2015" class="citation cs2">Hall, Brian C. (2015), <i>Lie Groups, Lie Algebras, and Representations: An Elementary Introduction</i>, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3319134666" title="Special:BookSources/978-3319134666"><bdi>978-3319134666</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Groups%2C+Lie+Algebras%2C+and+Representations%3A+An+Elementary+Introduction&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=2015&rft.isbn=978-3319134666&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall,_G._S.2004" class="citation book cs1">Hall, G. S. (2004). <i>Symmetries and Curvature Structure in General Relativity</i>. Singapore: World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-1051-9" title="Special:BookSources/978-981-02-1051-9"><bdi>978-981-02-1051-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetries+and+Curvature+Structure+in+General+Relativity&rft.place=Singapore&rft.pub=World+Scientific&rft.date=2004&rft.isbn=978-981-02-1051-9&rft.au=Hall%2C+G.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span> <i>See Chapter 6</i> for the subalgebras of the Lie algebra of the Lorentz group.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHatcher,_Allen2002" class="citation book cs1">Hatcher, Allen (2002). <i>Algebraic topology</i>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-79540-1" title="Special:BookSources/978-0-521-79540-1"><bdi>978-0-521-79540-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+topology&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-0-521-79540-1&rft.au=Hatcher%2C+Allen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span> <i>See also</i> the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://pi.math.cornell.edu/~hatcher/AT/ATpage.html">"online version"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">July 3,</span> 2005</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=online+version&rft_id=http%3A%2F%2Fpi.math.cornell.edu%2F~hatcher%2FAT%2FATpage.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span> <i>See Section 1.3</i> for a beautifully illustrated discussion of covering spaces. <i>See Section 3D</i> for the topology of rotation groups.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisnerThorneWheeler1973" class="citation book cs1"><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner, Charles</a>; <a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne, Kip S.</a>; <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John</a> (1973). <a href="/wiki/Gravitation_(book)" title="Gravitation (book)"><i>Gravitation</i></a>. <a href="/wiki/W._H._Freeman_and_Company" title="W. H. Freeman and Company">W. H. Freeman and Company</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-0344-0" title="Special:BookSources/978-0-7167-0344-0"><bdi>978-0-7167-0344-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pub=W.+H.+Freeman+and+Company&rft.date=1973&rft.isbn=978-0-7167-0344-0&rft.aulast=Misner&rft.aufirst=Charles&rft.au=Thorne%2C+Kip+S.&rft.au=Wheeler%2C+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span> §41.3</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNaber,_Gregory1992" class="citation book cs1">Naber, Gregory (1992). <i>The Geometry of Minkowski Spacetime</i>. New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0486432359" title="Special:BookSources/978-0486432359"><bdi>978-0486432359</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+Geometry+of+Minkowski+Spacetime&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1992&rft.isbn=978-0486432359&rft.au=Naber%2C+Gregory&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span> (Dover reprint edition.) An excellent reference on Minkowski spacetime and the Lorentz group.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeedham,_Tristan1997" class="citation book cs1"><a href="/wiki/Tristan_Needham" title="Tristan Needham">Needham, Tristan</a> (1997). <i>Visual Complex Analysis</i>. Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853446-4" title="Special:BookSources/978-0-19-853446-4"><bdi>978-0-19-853446-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Visual+Complex+Analysis&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=1997&rft.isbn=978-0-19-853446-4&rft.au=Needham%2C+Tristan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span> <i>See Chapter 3</i> for a superbly illustrated discussion of Möbius transformations.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg2002" class="citation cs2"><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg, S.</a> (2002), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quantumtheoryoff00stev"><i>The Quantum Theory of Fields</i></a></span>, vol. 1, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55001-7" title="Special:BookSources/978-0-521-55001-7"><bdi>978-0-521-55001-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Quantum+Theory+of+Fields&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-0-521-55001-7&rft.aulast=Weinberg&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumtheoryoff00stev&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWigner1939" class="citation cs2"><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner, E. P.</a> (1939), "On unitary representations of the inhomogeneous Lorentz group", <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>, <b>40</b> (1): 149–204, <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/1939AnMat..40..149W">1939AnMat..40..149W</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968551">10.2307/1968551</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/1968551">1968551</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=1503456">1503456</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:121773411">121773411</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=On+unitary+representations+of+the+inhomogeneous+Lorentz+group&rft.volume=40&rft.issue=1&rft.pages=149-204&rft.date=1939&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121773411%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1939AnMat..40..149W&rft_id=info%3Adoi%2F10.2307%2F1968551&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1968551%23id-name%3DJSTOR&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1503456%23id-name%3DMR&rft.aulast=Wigner&rft.aufirst=E.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorentz+group" class="Z3988"></span></li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Lorentz_group&oldid=1258812721">https://en.wikipedia.org/w/index.php?title=Lorentz_group&oldid=1258812721</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Lie_groups" title="Category:Lie groups">Lie groups</a></li><li><a href="/wiki/Category:Special_relativity" title="Category:Special relativity">Special relativity</a></li><li><a href="/wiki/Category:Group_theory" title="Category:Group theory">Group theory</a></li><li><a href="/wiki/Category:Hendrik_Lorentz" title="Category:Hendrik Lorentz">Hendrik Lorentz</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_December_2020" title="Category:Wikipedia articles needing clarification from December 2020">Wikipedia articles needing clarification from December 2020</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 21 November 2024, at 19:34<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Lorentz group - Wikipedia