Chiralità (fisica)

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href="#Le_equazioni_del_moto_per_gli_spinori"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Le equazioni del moto per gli spinori</span> </div> </a> <ul id="toc-Le_equazioni_del_moto_per_gli_spinori-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conferme_sperimentali" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conferme_sperimentali"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Conferme sperimentali</span> </div> </a> <ul id="toc-Conferme_sperimentali-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item 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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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Chiralità (fisica)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vai a una voce in un'altra lingua. Disponibile in 17 lingue" > <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-17" 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">17 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quiralitat_(f%C3%ADsica)" title="Quiralitat (física) - catalano" lang="ca" hreflang="ca" data-title="Quiralitat (física)" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kiralitet_(fysik)" title="Kiralitet (fysik) - danese" lang="da" hreflang="da" data-title="Kiralitet (fysik)" data-language-autonym="Dansk" data-language-local-name="danese" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Chiralit%C3%A4t_(Physik)" title="Chiralität (Physik) - tedesco" lang="de" hreflang="de" data-title="Chiralität (Physik)" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Chirality_(physics)" title="Chirality (physics) - inglese" lang="en" hreflang="en" data-title="Chirality (physics)" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Quiralidad_(f%C3%ADsica)" title="Quiralidad (física) - spagnolo" lang="es" hreflang="es" data-title="Quiralidad (física)" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kiralitate_(fisika)" title="Kiralitate (fisika) - basco" lang="eu" hreflang="eu" data-title="Kiralitate (fisika)" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%E2%80%8C%D8%B3%D8%A7%D9%86%DB%8C_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="دست‌سانی (فیزیک) - persiano" lang="fa" hreflang="fa" data-title="دست‌سانی (فیزیک)" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AB%E3%82%A4%E3%83%A9%E3%83%AA%E3%83%86%E3%82%A3" title="カイラリティ - giapponese" lang="ja" hreflang="ja" data-title="カイラリティ" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A5%D0%B8%D1%80%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Хиралност - macedone" lang="mk" hreflang="mk" data-title="Хиралност" data-language-autonym="Македонски" data-language-local-name="macedone" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9A%E0%A9%80%E0%A8%B0%E0%A9%88%E0%A8%B2%E0%A8%BF%E0%A8%9F%E0%A9%80_(%E0%A8%AD%E0%A9%8C%E0%A8%A4%E0%A8%BF%E0%A8%95_%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਚੀਰੈਲਿਟੀ (ਭੌਤਿਕ ਵਿਗਿਆਨ) - punjabi" lang="pa" hreflang="pa" data-title="ਚੀਰੈਲਿਟੀ (ਭੌਤਿਕ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Quiralidade_(f%C3%ADsica)" title="Quiralidade (física) - portoghese" lang="pt" hreflang="pt" data-title="Quiralidade (física)" data-language-autonym="Português" data-language-local-name="portoghese" 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%A5%D0%B8%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Хиральность (физика) - russo" lang="ru" hreflang="ru" data-title="Хиральность (физика)" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kiralnost_(fizika)" title="Kiralnost (fizika) - sloveno" lang="sl" hreflang="sl" data-title="Kiralnost (fizika)" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%B8%D1%80%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Хиралност (физика) - serbo" lang="sr" hreflang="sr" data-title="Хиралност (физика)" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kiralite_(fizik)" title="Kiralite (fizik) - turco" lang="tr" hreflang="tr" data-title="Kiralite (fizik)" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A5%D1%96%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%96%D1%81%D1%82%D1%8C_(%D1%84%D1%96%D0%B7%D0%B8%D0%BA%D0%B0)" title="Хіральність (фізика) - ucraino" lang="uk" hreflang="uk" data-title="Хіральність (фізика)" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%89%8B%E5%BE%B5%E6%80%A7" title="手徵性 - cinese" lang="zh" hreflang="zh" data-title="手徵性" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2305173#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespace"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Chiralit%C3%A0_(fisica)" title="Vedi la voce [c]" accesskey="c"><span>Voce</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussione:Chiralit%C3%A0_(fisica)" rel="discussion" title="Vedi le discussioni relative a questa pagina [t]" accesskey="t"><span>Discussione</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambia versione linguistica" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">italiano</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet 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class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&action=history" title="Versioni precedenti di questa pagina [h]" accesskey="h"><span>Cronologia</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Strumenti" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Strumenti</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Strumenti</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">nascondi</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Altre opzioni" > <div 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class="vector-menu-heading"> Generale </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciale:PuntanoQui/Chiralit%C3%A0_(fisica)" title="Elenco di tutte le pagine che sono collegate a questa [j]" accesskey="j"><span>Puntano qui</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciale:ModificheCorrelate/Chiralit%C3%A0_(fisica)" rel="nofollow" title="Elenco delle ultime modifiche alle pagine collegate a questa [k]" accesskey="k"><span>Modifiche correlate</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speciale:PagineSpeciali" title="Elenco di tutte le pagine speciali [q]" accesskey="q"><span>Pagine speciali</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&oldid=141828253" title="Collegamento permanente a questa versione di questa pagina"><span>Link permanente</span></a></li><li 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class="vector-menu-heading"> Stampa/esporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speciale:Libro&bookcmd=book_creator&referer=Chiralit%C3%A0+%28fisica%29"><span>Crea un libro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speciale:DownloadAsPdf&page=Chiralit%C3%A0_%28fisica%29&action=show-download-screen"><span>Scarica come PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&printable=yes" title="Versione stampabile di questa pagina [p]" accesskey="p"><span>Versione stampabile</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In altri progetti </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" 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lang="it" dir="ltr"><p>La <b>chiralità</b>, dal <a href="/wiki/Lingua_greca_antica" title="Lingua greca antica">greco</a> <i>χείρ</i> (khéir), "mano",<b><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></b> è una proprietà che distingue i <a href="/wiki/Sistema_fisico" class="mw-redirect" title="Sistema fisico">sistemi fisici</a> in destrorsi e sinistrorsi: un sistema fisico possiede una chiralità se sotto una trasformazione di <a href="/wiki/Parit%C3%A0_(fisica)" title="Parità (fisica)">parità</a> si trasforma nel sistema con la chiralità opposta.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Le <a href="/wiki/Forza" title="Forza">forze</a> che agiscono su di un sistema fisico possono modificare o meno la chiralità; l'<a href="/wiki/Interazione" title="Interazione">interazione</a> che trasforma un sistema con chiralità definita in un altro con la stessa chiralità si dice <a href="/w/index.php?title=Trasformazione_chirale&action=edit&redlink=1" class="new" title="Trasformazione chirale (la pagina non esiste)">trasformazione chirale</a>. </p><p>Si può dimostrare che l'<a href="/wiki/Elicit%C3%A0" title="Elicità">elicità</a> di una particella di <a href="/wiki/Spin" title="Spin">spin</a> 1/2 tende alla sua chiralità nel limite di massa nulla, ovvero nel limite in cui l'energia della particella è molto maggiore della sua massa. Dal momento che gli esperimenti <a href="/wiki/Super-Kamiokande" title="Super-Kamiokande">Super-Kamiokande</a> e <a href="/wiki/OPERA" title="OPERA">OPERA</a> (assieme ad altri) hanno dimostrato che anche i <a href="/wiki/Neutrino" title="Neutrino">neutrini</a> hanno una massa diversa da zero<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-PhysRevD_4-0" class="reference"><a href="#cite_note-PhysRevD-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>, non si conoscono particelle di spin 1/2 a massa nulla, e quindi con chiralità assolutamente definita. D'altra parte nel <a href="/wiki/Modello_standard" title="Modello standard">modello standard</a> i fermioni avrebbero tutti massa nulla grazie all'<a href="/wiki/Invarianza_(fisica)" title="Invarianza (fisica)">invarianza</a> sotto parità, altrimenti detta <b>simmetria chirale</b>, necessaria alla <a href="/wiki/Coerenza_(logica_matematica)" title="Coerenza (logica matematica)">coerenza</a> della teoria. L'acquisizione di massa avverrebbe tramite <a href="/wiki/Rottura_spontanea_di_simmetria" title="Rottura spontanea di simmetria">rottura spontanea di simmetria</a> dovuta al <a href="/wiki/Campo_di_Higgs" class="mw-redirect" title="Campo di Higgs">campo di Higgs</a>, che conserva al contempo la <a href="/wiki/Simmetria_di_gauge" title="Simmetria di gauge">simmetria di gauge</a> del modello. </p><p>La chiralità è strettamente legata alla parità: infatti in una teoria simmetrica sotto parità le componenti a chiralità positiva e negativa dei campi devono essere trattate allo stesso modo. Nel 1957, in un esperimento condotto da <a href="/wiki/Chien-Shiung_Wu" class="mw-redirect" title="Chien-Shiung Wu">Chien-Shiung Wu</a> è stato dimostrato che nei decadimenti deboli la <a href="/wiki/Parit%C3%A0_(fisica)" title="Parità (fisica)">parità</a> viene violata, aprendo la strada alla <a href="/w/index.php?title=Teoria_V-A&action=edit&redlink=1" class="new" title="Teoria V-A (la pagina non esiste)">teoria V-A</a> delle <a href="/wiki/Interazione_debole" title="Interazione debole">interazioni deboli</a>: mentre la forza elettromagnetica, la forza forte e quella gravitazionale si accoppiano ugualmente alle particelle con chiralità negativa e positiva (cioè a particelle sinistrorse e destrorse), l'interazione debole si accoppia esclusivamente alle particelle sinistrorse. </p><p>Le correnti chirali di elettroni sono state per la prima volta osservate in un esperimento del 2023.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definizione">Definizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&veaction=edit&section=1" title="Modifica la sezione Definizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&action=edit&section=1" title="Edit section's source code: Definizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La chiralità ha radici profonde nella <a href="/wiki/Teoria_delle_rappresentazioni" title="Teoria delle rappresentazioni">teoria delle rappresentazioni</a> del <a href="/wiki/Gruppo_di_Lorentz" title="Gruppo di Lorentz">gruppo di Lorentz</a>. Dato che il gruppo di Lorentz è isomorfo al prodotto diretto di due gruppi <a href="/wiki/SU(2)" class="mw-redirect" title="SU(2)">SU(2)</a>, le rappresentazioni più semplici dopo quella scalare (1,1) sono degli oggetti che si trasformano come un vettore a due dimensioni per uno dei due gruppi e come uno scalare per l'altro, cioè (2,1) e (1,2). Queste due possibilità definiscono gli <a href="/wiki/Spinore_di_Weyl" class="mw-redirect" title="Spinore di Weyl">spinori di Weyl</a>, che vengono identificati come spinori <i>left</i> o <i>right</i>. Uno <a href="/wiki/Spinore_di_Dirac" title="Spinore di Dirac">spinore di Dirac</a> è costruito a partire da due spinori di Weyl: l'<a href="/wiki/Equazione_di_Dirac" title="Equazione di Dirac">equazione di Dirac</a> stessa non è altro che un operatore di proiezione che impone che, nel <a href="/wiki/Sistema_di_riferimento" title="Sistema di riferimento">sistema di riferimento</a> a riposo del campo, le componenti left e right siano identiche; le autofunzioni della matrice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84291213468b401be7afb67f4fc09b273af23cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}}"></span> sono invece proprio i due spinori di Weyl, come sarà mostrato in dettaglio. </p> <div class="mw-heading mw-heading2"><h2 id="La_quinta_matrice_gamma_e_autofunzioni">La quinta matrice gamma e autofunzioni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&veaction=edit&section=2" title="Modifica la sezione La quinta matrice gamma e autofunzioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&action=edit&section=2" title="Edit section's source code: La quinta matrice gamma e autofunzioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Matrice" title="Matrice">matrice</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84291213468b401be7afb67f4fc09b273af23cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}}"></span> (<a href="/wiki/Gamma_di_Dirac" title="Gamma di Dirac">Gamma di Dirac</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>=</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45ec1c937f35bcb250cc6809b3f8fb65e9cd56d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.571ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}}"></span>) è detta <a href="/wiki/Operatore_(fisica)" title="Operatore (fisica)">operatore</a> di chiralità. Poiché l'operatore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84291213468b401be7afb67f4fc09b273af23cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}}"></span> è un <a href="/wiki/Operatore_hermitiano" class="mw-redirect" title="Operatore hermitiano">operatore hermitiano</a> esso è <a href="/wiki/Diagonalizzabilit%C3%A0" title="Diagonalizzabilità">diagonalizzabile</a> e dalla proprietà che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {(\gamma ^{5})}^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(\gamma ^{5})}^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3182098603911f7eb1eaf52ea639125e30c084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.459ex; height:3.676ex;" alt="{\displaystyle {(\gamma ^{5})}^{2}=1}"></span> </p><p>segue che gli <a href="/wiki/Autovalori" class="mw-redirect" title="Autovalori">autovalori</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84291213468b401be7afb67f4fc09b273af23cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}}"></span> sono +1 e -1; ne consegue che possiamo indicare con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577ea88983dfe6dd88e20ff7402dce2c7fd981a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.865ex; height:2.509ex;" alt="{\displaystyle \psi _{L}}"></span> e con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacf8c5f2587de7c59bbfd798e11bdbeaa01707b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle \psi _{R}}"></span> le <a href="/wiki/Autofunzione" title="Autofunzione">autofunzioni</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84291213468b401be7afb67f4fc09b273af23cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}}"></span> con <a href="/wiki/Autovalori" class="mw-redirect" title="Autovalori">autovalori</a> -1 e +1 rispettivamente; ossia: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}\psi _{L}=-1\psi _{L};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}\psi _{L}=-1\psi _{L};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6630f909b8e6c9bedc5cf6c659ff630a788a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.779ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}\psi _{L}=-1\psi _{L};}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}\psi _{R}=+1\psi _{R}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}\psi _{R}=+1\psi _{R}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3577d0013af844b6a21916116202c3a856a30d51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.036ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}\psi _{R}=+1\psi _{R}.}"></span></dd></dl> <p>In questo caso si stanno utilizzando le convenzioni </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{c}^{5}={\begin{pmatrix}-I_{2}&0\\0&I_{2}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</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 \gamma _{c}^{5}={\begin{pmatrix}-I_{2}&0\\0&I_{2}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21533300214a119348d9d140774d8e76d2f5d001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.891ex; height:6.176ex;" alt="{\displaystyle \gamma _{c}^{5}={\begin{pmatrix}-I_{2}&0\\0&I_{2}\end{pmatrix}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ={\binom {\psi _{L}}{\psi _{R}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ={\binom {\psi _{L}}{\psi _{R}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fac992fe05b81bde8a523775e911799290e1853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.026ex; height:6.176ex;" alt="{\displaystyle \psi ={\binom {\psi _{L}}{\psi _{R}}}}"></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{c}^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{c}^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4d5e9efecfc5451453b2f0a1c4ab351a8efca31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:3.009ex;" alt="{\displaystyle \gamma _{c}^{5}}"></span> è la matrice Gamma di Dirac in rappresentazione chirale e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3506ae39df854f347365bae6f326ef4f565be5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{2}}"></span> è l'identità di dimensioni 2x2 </p><p>A partire da uno <a href="/wiki/Spinore" title="Spinore">spinore</a> generico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> è possibile scrivere le due autofunzioni di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84291213468b401be7afb67f4fc09b273af23cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}}"></span> nel seguente modo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{L}={\frac {1-\gamma ^{5}}{2}}\psi ;}"> <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>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>ψ</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{L}={\frac {1-\gamma ^{5}}{2}}\psi ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82999a52805f5d638347ce08e636570f85e46661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.296ex; height:5.676ex;" alt="{\displaystyle \psi _{L}={\frac {1-\gamma ^{5}}{2}}\psi ;}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{R}={\frac {1+\gamma ^{5}}{2}}\psi ;}"> <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>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>ψ</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{R}={\frac {1+\gamma ^{5}}{2}}\psi ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df76f25b4d955af44ff526846a10a4cde7cf8f03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.424ex; height:5.676ex;" alt="{\displaystyle \psi _{R}={\frac {1+\gamma ^{5}}{2}}\psi ;}"></span></dd></dl> <p>segue pertanto che uno spinore generico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> può essere scomposto nella somma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =\psi _{L}+\psi _{R}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\psi _{L}+\psi _{R}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6bddafdf43e72bc36a35d80b7da4de06d76336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.956ex; height:2.509ex;" alt="{\displaystyle \psi =\psi _{L}+\psi _{R}.}"></span></dd></dl> <p>Si possono definire due operatori di proiezione di chiralità: </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_{L}={\frac {1-\gamma ^{5}}{2}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{L}={\frac {1-\gamma ^{5}}{2}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d0663a1a19fa04773a4938e27ab41d4b043e82d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.762ex; height:5.676ex;" alt="{\displaystyle P_{L}={\frac {1-\gamma ^{5}}{2}};}"></span></dd></dl> <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_{R}={\frac {1+\gamma ^{5}}{2}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{R}={\frac {1+\gamma ^{5}}{2}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed459b7039c72048ebc729cbc58bfde3d6b35feb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.89ex; height:5.676ex;" alt="{\displaystyle P_{R}={\frac {1+\gamma ^{5}}{2}};}"></span></dd></dl> <p>che soddisfano alle seguenti proprietà: </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_{L}+P_{R}=1;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{L}+P_{R}=1;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87ca151cf62ad52c65ce0e48100d1a5d69bbe77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.564ex; height:2.509ex;" alt="{\displaystyle P_{L}+P_{R}=1;}"></span></dd></dl> <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_{L})^{2}=P_{L};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P_{L})^{2}=P_{L};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1f11077fa91a30c8b8af46467330bf9372e122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.296ex; height:3.176ex;" alt="{\displaystyle (P_{L})^{2}=P_{L};}"></span></dd></dl> <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_{R})^{2}=P_{R};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P_{R})^{2}=P_{R};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30054e8747e598705dddf985309ed80ff5bbf9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.553ex; height:3.176ex;" alt="{\displaystyle (P_{R})^{2}=P_{R};}"></span></dd></dl> <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_{L}P_{R}=P_{R}P_{L}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{L}P_{R}=P_{R}P_{L}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcf9bca07d9d449b4163f863dfcb9cd9bb3db87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.637ex; height:2.509ex;" alt="{\displaystyle P_{L}P_{R}=P_{R}P_{L}=0.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Le_equazioni_del_moto_per_gli_spinori">Le equazioni del moto per gli spinori</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&veaction=edit&section=3" title="Modifica la sezione Le equazioni del moto per gli spinori" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&action=edit&section=3" title="Edit section's source code: Le equazioni del moto per gli spinori"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> è una generica soluzione dell'<a href="/wiki/Equazione_di_Dirac" title="Equazione di Dirac">equazione di Dirac</a> libera, abbiamo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }\right)\psi _{L}(x)=\left(i\gamma ^{\mu }\partial _{\mu }\right){\frac {1-\gamma ^{5}}{2}}\psi (x)={\frac {1+\gamma ^{5}}{2}}\left(i\gamma ^{\mu }\partial _{\mu }\right)\psi (x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }\right)\psi _{L}(x)=\left(i\gamma ^{\mu }\partial _{\mu }\right){\frac {1-\gamma ^{5}}{2}}\psi (x)={\frac {1+\gamma ^{5}}{2}}\left(i\gamma ^{\mu }\partial _{\mu }\right)\psi (x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa813c8332c2a390b92d525b2c085cfe6fefb74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.765ex; height:5.676ex;" alt="{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }\right)\psi _{L}(x)=\left(i\gamma ^{\mu }\partial _{\mu }\right){\frac {1-\gamma ^{5}}{2}}\psi (x)={\frac {1+\gamma ^{5}}{2}}\left(i\gamma ^{\mu }\partial _{\mu }\right)\psi (x),}"></span></dd></dl> <p>ovvero: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }\right)\psi _{L}(x)={\frac {1+\gamma ^{5}}{2}}m\psi (x)=m\psi _{R}(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }\right)\psi _{L}(x)={\frac {1+\gamma ^{5}}{2}}m\psi (x)=m\psi _{R}(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df48065a8a151055b7bb2651d626a5f1b8afc2b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.845ex; height:5.676ex;" alt="{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }\right)\psi _{L}(x)={\frac {1+\gamma ^{5}}{2}}m\psi (x)=m\psi _{R}(x),}"></span></dd></dl> <p>e analogamente: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }\right)\psi _{R}(x)=m\psi _{L}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }\right)\psi _{R}(x)=m\psi _{L}(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aac59b59010911b66b45908074133c1fd66e7aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.881ex; height:3.009ex;" alt="{\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }\right)\psi _{R}(x)=m\psi _{L}(x).}"></span></dd></dl> <p>Queste sono due <a href="/wiki/Equazioni_del_moto" class="mw-redirect" title="Equazioni del moto">equazioni del moto</a> per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577ea88983dfe6dd88e20ff7402dce2c7fd981a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.865ex; height:2.509ex;" alt="{\displaystyle \psi _{L}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacf8c5f2587de7c59bbfd798e11bdbeaa01707b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle \psi _{R}}"></span>. Esse sono accoppiate dal termine di <a href="/wiki/Massa_(fisica)" title="Massa (fisica)">massa</a> e si disaccoppiano solo se m = 0. Tale proprietà è da mettere in relazione con il fatto che la chiralità è un buon <a href="/wiki/Numero_quantico" title="Numero quantico">numero quantico</a> (ossia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84291213468b401be7afb67f4fc09b273af23cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:3.176ex;" alt="{\displaystyle \gamma ^{5}}"></span> commuta con l'<a href="/wiki/Equazione_di_Dirac#Proprietà_dell'hamiltoniana_di_Dirac" title="Equazione di Dirac">Hamiltoniana di Dirac</a>) solo se m = 0. </p><p>Le proprietà peculiari delle particelle con <a href="/wiki/Spin" title="Spin">spin</a> 1/2 con <a href="/wiki/Massa_(fisica)" title="Massa (fisica)">massa</a> nulla derivano dall'invarianza della densità di <a href="/wiki/Lagrangiana" title="Lagrangiana">lagrangiana</a> di Dirac per trasformazioni chirali. Definita come trasformazione chirale la seguente trasformazione </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \longrightarrow \psi '=\gamma ^{5}\psi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">⟶</mo> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mi>ψ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \longrightarrow \psi '=\gamma ^{5}\psi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa966117d8998b78252cb13cf3ff5a904726140a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.4ex; height:3.176ex;" alt="{\displaystyle \psi \longrightarrow \psi '=\gamma ^{5}\psi ,}"></span></dd></dl> <p>la lagrangiana di Dirac si trasforma nel modo seguente: </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 L_{D}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi \longrightarrow L'_{D}={\bar {\psi }}(-\gamma ^{5})\left(i\gamma ^{\mu }\partial _{\mu }-m\right)(\gamma ^{5})\psi =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">⟶</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{D}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi \longrightarrow L'_{D}={\bar {\psi }}(-\gamma ^{5})\left(i\gamma ^{\mu }\partial _{\mu }-m\right)(\gamma ^{5})\psi =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb77c601e1fe139266b1621732a37ecdfba293e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:62.866ex; height:3.343ex;" alt="{\displaystyle L_{D}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi \longrightarrow L'_{D}={\bar {\psi }}(-\gamma ^{5})\left(i\gamma ^{\mu }\partial _{\mu }-m\right)(\gamma ^{5})\psi =}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }+m\right)(\gamma ^{5})^{2}\psi ={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }+m\right)\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }+m\right)(\gamma ^{5})^{2}\psi ={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }+m\right)\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a1a7a6cca3714a4cfb403a655635aa2dbe67b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.423ex; height:3.343ex;" alt="{\displaystyle ={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }+m\right)(\gamma ^{5})^{2}\psi ={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }+m\right)\psi }"></span></dd></dl> <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 L'_{D}=L_{D}+2m{\bar {\psi }}\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L'_{D}=L_{D}+2m{\bar {\psi }}\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c247554544dae215c44bf7887959dddf297b9f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.249ex; height:3.176ex;" alt="{\displaystyle L'_{D}=L_{D}+2m{\bar {\psi }}\psi .}"></span></dd></dl> <p>Si ha quindi l'<a href="/wiki/Invarianza_(fisica)" title="Invarianza (fisica)">invarianza</a> per trasformazioni di chiralità <a href="/wiki/Se_e_solo_se" title="Se e solo se">se e solo se</a> m = 0. Come si può vedere si ha che per una particella di spin 1/2 con massa m = 0 le equazioni per gli spinori <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577ea88983dfe6dd88e20ff7402dce2c7fd981a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.865ex; height:2.509ex;" alt="{\displaystyle \psi _{L}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacf8c5f2587de7c59bbfd798e11bdbeaa01707b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle \psi _{R}}"></span> sono disaccoppiate. </p> <div class="mw-heading mw-heading2"><h2 id="Conferme_sperimentali">Conferme sperimentali</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&veaction=edit&section=4" title="Modifica la sezione Conferme sperimentali" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&action=edit&section=4" title="Edit section's source code: Conferme sperimentali"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nel 2024 è stata riprodotta in laboratorio la prima corrente chirale. Il fenomeno è stato ottenuto con materiali particolari che, quando vengono colpiti dalla luce, causano l'emissione di elettroni tutti con lo stesso <a href="/wiki/Spin" title="Spin">spin</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&veaction=edit&section=5" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&action=edit&section=5" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.grecoantico.com/dizionario-greco-antico.php?parola=%CF%87%CE%B5%CE%AF%CF%81"><span style="font-style:italic;">DIZIONARIO GRECO ANTICO - Greco antico - Italiano</span></a>, su <span style="font-style:italic;">grecoantico.com</span>. <small>URL consultato il 19 marzo 2024</small>.</cite></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> <span style="font-style:italic;">Particles and nuclei: an introduction to the physical concepts</span>, 5ª ed., Springer, 2006, p. 145, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-3-540-36683-6" title="Speciale:RicercaISBN/978-3-540-36683-6">978-3-540-36683-6</a>.</cite></span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"> <cite class="citation pubblicazione" style="font-style:normal"> A. Goobar, S. Hannestad, E. Mörtsell, H. Tu, <a rel="nofollow" class="external text" href="https://oadoi.org/10.1088/1475-7516/2006/06/019"><span style="font-style:italic;">The neutrino mass bound from WMAP 3 year data, the baryon acoustic peak, the SNLS supernovae and the Lyman-α forest</span></a>, in <span style="font-style:italic;"><a href="/w/index.php?title=Journal_of_Cosmology_and_Astroparticle_Physics&action=edit&redlink=1" class="new" title="Journal of Cosmology and Astroparticle Physics (la pagina non esiste)">Journal of Cosmology and Astroparticle Physics</a></span>, vol. 606, 2006, p. 19, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1088%2F1475-7516%2F2006%2F06%2F019">10.1088/1475-7516/2006/06/019</a>.</cite></span> </li> <li id="cite_note-PhysRevD-4"><a href="#cite_ref-PhysRevD_4-0"><b>^</b></a> <span class="reference-text"> <cite class="citation pubblicazione" style="font-style:normal"> J. Schechter, J.W.F. Valle, <a rel="nofollow" class="external text" href="https://oadoi.org/10.1103/PhysRevD.22.2227"><span style="font-style:italic;">Neutrino Masses in SU(2) x U(1) Theories</span></a>, in <span style="font-style:italic;"><a href="/wiki/Physical_Review" title="Physical Review">Physical Review D</a></span>, vol. 22, 1980, p. 2227, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1103%2FPhysRevD.22.2227">10.1103/PhysRevD.22.2227</a>.</cite></span> </li> <li id="cite_note-5"><a href="#cite_ref-5"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.ansa.it/campania/notizie/2024/02/07/un-nuovo-stato-della-materia-apre-a-unelettronica-inedita_a0881e72-87df-434b-93cf-9a0ea11a6c66.html"><span style="font-style:italic;">Un nuovo stato della materia apre a un'elettronica inedita</span></a>, su <span style="font-style:italic;">ansa.it</span>, 7 febbraio 2024.</cite></span> </li> <li id="cite_note-6"><a href="#cite_ref-6"><b>^</b></a> <span class="reference-text"><cite class="citation pubblicazione" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Federico Mazzola, Wojciech Brzezicki e Maria Teresa Mercaldo, <a rel="nofollow" class="external text" href="https://www.nature.com/articles/s41586-024-07033-8"><span style="font-style:italic;">Signatures of a surface spin–orbital chiral metal</span></a>, in <span style="font-style:italic;">Nature</span>, vol. 626, n. 8000, 2024-02, pp. 752–758, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1038%2Fs41586-024-07033-8">10.1038/s41586-024-07033-8</a>. <small>URL consultato il 19 marzo 2024</small>.</cite></span> </li> <li id="cite_note-7"><a href="#cite_ref-7"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.ansa.it/canale_scienza/notizie/fisica_matematica/2024/10/26/prodotta-la-prima-corrente-chirale-con-elettroni-ordinati_db44b9c8-f101-4311-8bfe-14ebb2449e52.html"><span style="font-style:italic;">Prodotta la prima corrente chirale, con elettroni ordinati - Fisica e Matematica - Ansa.it</span></a>, su <span style="font-style:italic;">Agenzia ANSA</span>, 26 ottobre 2024. <small>URL consultato il 27 ottobre 2024</small>.</cite></span> </li> <li id="cite_note-8"><a href="#cite_ref-8"><b>^</b></a> <span class="reference-text"><cite class="citation pubblicazione" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Federico Mazzola, Wojciech Brzezicki e Maria Teresa Mercaldo, <a rel="nofollow" class="external text" href="https://www.nature.com/articles/s41586-024-07033-8"><span style="font-style:italic;">Signatures of a surface spin–orbital chiral metal</span></a>, in <span style="font-style:italic;">Nature</span>, vol. 626, n. 8000, 2024-02, pp. 752–758, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1038%2Fs41586-024-07033-8">10.1038/s41586-024-07033-8</a>. <small>URL consultato il 27 ottobre 2024</small>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&veaction=edit&section=6" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&action=edit&section=6" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Marie_Curie" title="Marie Curie">Marie Curie</a> (1955): <i>Pierre Curie</i>, Parigi, Éditions Dënoel; traduzione italiana <a href="/w/index.php?title=CUEN&action=edit&redlink=1" class="new" title="CUEN (la pagina non esiste)">CUEN</a>, Napoli, 1998. L'edizione originale è del 1925.</li> <li><a href="/wiki/Pierre_Curie" title="Pierre Curie">Pierre Curie</a> (1894): <i>Sur la symétrie dans les phénomenes physiques, symétrie d'un champ eléctrique et d'un champ magnétique</i>, Journal de Physique 3me serie 3, 393- 415.</li> <li>István Hargittai e Magdolna Hargittai (1995): <i>Symmetry Through the Eyes of a Chemist</i>, 2ª edizione, New York, Kluwer.</li> <li>István Hargittai e Magdolna Hargittai (2000): <i>In Our Own Image</i>, New York, Kluwer. Jenann, Ismael (2001): Essays on Symmetry, New York, Garland.</li> <li>Alan Holden (1971): <i>Shapes, Space and Symmetry</i>, New York, Columbia University Press; ristampa New York, Dover, 1991.</li> <li>Joe Rosen (1975): <i>Symmetry Discovered</i>, Londra, Cambridge University Press; ristampa New York, Dover, 2000.</li> <li>Joe Rosen (1983): <i>A Symmetry Primer for Scientists</i>, New York, <a href="/wiki/John_Wiley_%26_Sons" title="John Wiley & Sons">John Wiley & Sons</a>.</li> <li>Alexei Vasil'evich Shubnikov e Vladimir Alexandrovich Koptsik (1974): <i>Symmetry in Science and Art</i>, New York, Plenum Press.</li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> (1952): <i>Symmetry</i>. Princeton University Press, 1952. <a href="/wiki/Speciale:RicercaISBN/0691023743" class="internal mw-magiclink-isbn">ISBN 0-691-02374-3</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&veaction=edit&section=7" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Chiralit%C3%A0_(fisica)&action=edit&section=7" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Equazione_di_Dirac" title="Equazione di Dirac">Equazione di Dirac</a></li> <li><a href="/wiki/Fotone" title="Fotone">Fotone</a></li> <li><a href="/wiki/Gamma_di_Dirac" title="Gamma di Dirac">Gamma di Dirac</a></li> <li><a href="/wiki/Legge_di_conservazione" title="Legge di conservazione">Legge di conservazione</a></li> <li><a href="/wiki/Simmetria_(fisica)" title="Simmetria (fisica)">Simmetria (fisica)</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4385706-1">4385706-1</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:P_physics.svg" class="mw-file-description" title="Quantistica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/P_physics.svg/25px-P_physics.svg.png" decoding="async" width="25" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/P_physics.svg/38px-P_physics.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/P_physics.svg/50px-P_physics.svg.png 2x" data-file-width="400" data-file-height="360" /></a></span> <b><a href="/wiki/Portale:Quantistica" title="Portale:Quantistica">Portale Quantistica</a></b>: accedi alle voci di Wikipedia che trattano di quantistica</div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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Chiralità (fisica) - Wikipedia