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T-symmetry - Wikipedia
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subsection</span> </button> <ul id="toc-Macroscopic_phenomena-sublist" class="vector-toc-list"> <li id="toc-The_second_law_of_thermodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_second_law_of_thermodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>The second law of thermodynamics</span> </div> </a> <ul id="toc-The_second_law_of_thermodynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Big_Bang" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Big_Bang"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Big Bang</span> </div> </a> <ul id="toc-Big_Bang-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Black_holes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Black_holes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Black holes</span> </div> </a> <ul id="toc-Black_holes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kinetic_consequences:_detailed_balance_and_Onsager_reciprocal_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kinetic_consequences:_detailed_balance_and_Onsager_reciprocal_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Kinetic consequences: detailed balance and Onsager reciprocal relations</span> </div> </a> <ul id="toc-Kinetic_consequences:_detailed_balance_and_Onsager_reciprocal_relations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effect_of_time_reversal_on_some_variables_of_classical_physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Effect_of_time_reversal_on_some_variables_of_classical_physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Effect of time reversal on some variables of classical physics</span> </div> </a> <ul id="toc-Effect_of_time_reversal_on_some_variables_of_classical_physics-sublist" class="vector-toc-list"> <li id="toc-Even" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Even"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5.1</span> <span>Even</span> </div> </a> <ul id="toc-Even-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Odd" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Odd"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5.2</span> <span>Odd</span> </div> </a> <ul id="toc-Odd-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example:_Magnetic_Field_and_Onsager_reciprocal_relations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example:_Magnetic_Field_and_Onsager_reciprocal_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5.3</span> <span>Example: Magnetic Field and Onsager reciprocal relations</span> </div> </a> <ul id="toc-Example:_Magnetic_Field_and_Onsager_reciprocal_relations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Microscopic_phenomena:_time_reversal_invariance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Microscopic_phenomena:_time_reversal_invariance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Microscopic phenomena: time reversal invariance</span> </div> </a> <button aria-controls="toc-Microscopic_phenomena:_time_reversal_invariance-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 Microscopic phenomena: time reversal invariance subsection</span> </button> <ul id="toc-Microscopic_phenomena:_time_reversal_invariance-sublist" class="vector-toc-list"> <li id="toc-Time_reversal_in_quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time_reversal_in_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Time reversal in quantum mechanics</span> </div> </a> <ul id="toc-Time_reversal_in_quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formal_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Formal notation</span> </div> </a> <ul id="toc-Formal_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Anti-unitary_representation_of_time_reversal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Anti-unitary_representation_of_time_reversal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Anti-unitary representation of time reversal</span> </div> </a> <ul id="toc-Anti-unitary_representation_of_time_reversal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electric_dipole_moments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electric_dipole_moments"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Electric dipole moments</span> </div> </a> <ul id="toc-Electric_dipole_moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kramers'_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kramers'_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Kramers' theorem</span> </div> </a> <ul id="toc-Kramers'_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time_reversal_of_the_known_dynamical_laws" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time_reversal_of_the_known_dynamical_laws"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Time reversal of the known dynamical laws</span> </div> </a> <ul id="toc-Time_reversal_of_the_known_dynamical_laws-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time_reversal_of_noninvasive_measurements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time_reversal_of_noninvasive_measurements"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Time reversal of noninvasive measurements</span> </div> </a> <ul id="toc-Time_reversal_of_noninvasive_measurements-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Inline_citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inline_citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Inline citations</span> </div> </a> <ul id="toc-Inline_citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>General references</span> </div> </a> <ul id="toc-General_references-sublist" class="vector-toc-list"> </ul> </li> </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 " 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Available in 21 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-21" 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">21 languages</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/Simetria_T" title="Simetria T – Catalan" lang="ca" hreflang="ca" data-title="Simetria T" data-language-autonym="Català" data-language-local-name="Catalan" 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/T-symmetri" title="T-symmetri – Danish" lang="da" hreflang="da" data-title="T-symmetri" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zeitumkehr_(Physik)" title="Zeitumkehr (Physik) – German" lang="de" hreflang="de" data-title="Zeitumkehr (Physik)" 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/Simetr%C3%ADa_temporal" title="Simetría temporal – Spanish" lang="es" hreflang="es" data-title="Simetría temporal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%82%D8%A7%D8%B1%D9%86_%D9%86%D8%B3%D8%A8%D8%AA_%D8%A8%D9%87_%D8%B2%D9%85%D8%A7%D9%86" title="تقارن نسبت به زمان – Persian" lang="fa" hreflang="fa" data-title="تقارن نسبت به زمان" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Sym%C3%A9trie_T" title="Symétrie T – French" lang="fr" hreflang="fr" data-title="Symétrie T" 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/%EC%8B%9C%EA%B0%84_%EC%97%AD%EC%A0%84_%EB%8C%80%EC%B9%AD" title="시간 역전 대칭 – Korean" lang="ko" hreflang="ko" data-title="시간 역전 대칭" 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/Simmetria_temporale" title="Simmetria temporale – Italian" lang="it" hreflang="it" data-title="Simmetria temporale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Id%C5%91t%C3%BCkr%C3%B6z%C3%A9s" title="Időtükrözés – Hungarian" lang="hu" hreflang="hu" data-title="Időtükrözés" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Simetri_T" title="Simetri T – Malay" lang="ms" hreflang="ms" data-title="Simetri T" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/T-symmetrie" title="T-symmetrie – Dutch" lang="nl" hreflang="nl" data-title="T-symmetrie" 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/%E6%99%82%E9%96%93%E5%8F%8D%E8%BB%A2%E5%AF%BE%E7%A7%B0%E6%80%A7" title="時間反転対称性 – Japanese" lang="ja" hreflang="ja" data-title="時間反転対称性" 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/T-%E0%A8%B8%E0%A8%AE%E0%A8%BF%E0%A9%B1%E0%A8%9F%E0%A8%B0%E0%A9%80" title="T-ਸਮਿੱਟਰੀ – Punjabi" lang="pa" hreflang="pa" data-title="T-ਸਮਿੱਟਰੀ" 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/Parzysto%C5%9B%C4%87_T" title="Parzystość T – Polish" lang="pl" hreflang="pl" data-title="Parzystość T" 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/Simetria_temporal" title="Simetria temporal – Portuguese" lang="pt" hreflang="pt" data-title="Simetria temporal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Simetrie_T" title="Simetrie T – Romanian" lang="ro" hreflang="ro" data-title="Simetrie T" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/T-%D1%81%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="T-симметрия – Russian" lang="ru" hreflang="ru" data-title="T-симметрия" data-language-autonym="Русский" data-language-local-name="Russian" 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/Simetrija_T" title="Simetrija T – Slovenian" lang="sl" hreflang="sl" data-title="Simetrija T" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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%92%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D1%81%D0%BA%D0%B0_%D0%B8%D0%BD%D0%B2%D0%B5%D1%80%D0%B7%D0%B8%D1%98%D0%B0" title="Временска инверзија – Serbian" lang="sr" hreflang="sr" data-title="Временска инверзија" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B2%D0%BE%D1%80%D0%BE%D1%82%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Зворотність – Ukrainian" lang="uk" hreflang="uk" data-title="Зворотність" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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%99%82%E9%96%93%E5%8F%8D%E6%BC%94%E5%B0%8D%E7%A8%B1" title="時間反演對稱 – Chinese" lang="zh" hreflang="zh" data-title="時間反演對稱" 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template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Time_sidebar" title="Template talk:Time sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Time_sidebar" title="Special:EditPage/Template:Time sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>T-symmetry</b> or <b>time reversal symmetry</b> is the theoretical <a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">symmetry of physical laws</a> under the <a href="/wiki/Transformation_(mathematics)" class="mw-redirect" title="Transformation (mathematics)">transformation</a> of <a href="/wiki/Time" title="Time">time</a> reversal, </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:t\mapsto -t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>t</mi> <mo stretchy="false">↦<!-- ↦ --></mo> <mo>−<!-- − --></mo> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:t\mapsto -t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0674953b780c5b1372e6942e33c9a796961d5d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.322ex; height:2.343ex;" alt="{\displaystyle T:t\mapsto -t.}"></span></dd></dl> <p>Since the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a> states that <a href="/wiki/Entropy" title="Entropy">entropy</a> increases as time flows toward the future, in general, the macroscopic <a href="/wiki/Universe" title="Universe">universe</a> does not show symmetry under time reversal. In other words, time is said to be non-symmetric, or asymmetric, except for special equilibrium states when the second law of thermodynamics predicts the time symmetry to hold. However, quantum <a href="/wiki/Weak_measurement" title="Weak measurement">noninvasive measurements</a> are predicted to violate time symmetry even in equilibrium,<sup id="cite_ref-non-time_1-0" class="reference"><a href="#cite_note-non-time-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> contrary to their classical counterparts, although this has not yet been experimentally confirmed. </p><p>Time <i>asymmetries</i> (see <a href="/wiki/Arrow_of_time" title="Arrow of time">Arrow of time</a>) generally are caused by one of three categories: </p> <ol><li>intrinsic to the dynamic <a href="/wiki/Physical_law" class="mw-redirect" title="Physical law">physical law</a> (e.g., for the <a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">weak force</a>)</li> <li>due to the <a href="/wiki/Big_Bang" title="Big Bang">initial conditions of the universe</a> (e.g., for the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>)</li> <li>due to <a href="/wiki/Weak_measurement" title="Weak measurement">measurements</a> (e.g., for the noninvasive measurements)</li></ol> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Macroscopic_phenomena">Macroscopic phenomena</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=1" title="Edit section: Macroscopic phenomena"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="The_second_law_of_thermodynamics">The second law of thermodynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=2" title="Edit section: The second law of thermodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Frame"><a href="/wiki/File:Teeter-totter.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/a2/Teeter-totter.png" decoding="async" width="216" height="340" class="mw-file-element" data-file-width="216" data-file-height="340" /></a><figcaption>A toy called the <a href="/wiki/Teeter-totter" class="mw-redirect" title="Teeter-totter">teeter-totter</a> illustrates, in cross-section, the two aspects of time reversal invariance. When set into motion atop a pedestal (rocking side to side, as in the image), the figure oscillates for a very long time.<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="Text claims that the item in the image should rock back an forth. However, the item in the image appears to be unstable unlike the a teeter totters that the link shows. (March 2024)">clarification needed</span></a></i>]</sup> The toy is engineered to minimize friction and illustrate the reversibility of <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a>. However, the mechanically stable state of the toy is when the figure falls down from the pedestal into one of arbitrarily many positions. This is an illustration of the law of increase of <a href="/wiki/Entropy" title="Entropy">entropy</a> through <a href="/wiki/Boltzmann" class="mw-redirect" title="Boltzmann">Boltzmann</a>'s identification of the logarithm of the number of states with the entropy.</figcaption></figure> <p>Daily experience shows that T-symmetry does not hold for the behavior of bulk materials. Of these macroscopic laws, most notable is the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>. Many other phenomena, such as the relative motion of bodies with friction, or viscous motion of fluids, reduce to this, because the underlying mechanism is the dissipation of usable energy (for example, kinetic energy) into heat. </p><p>The question of whether this time-asymmetric dissipation is really inevitable has been considered by many physicists, often in the context of <a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a>. The name comes from a <a href="/wiki/Thought_experiment" title="Thought experiment">thought experiment</a> described by <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> in which a microscopic demon guards a gate between two halves of a room. It only lets slow molecules into one half, only fast ones into the other. By eventually making one side of the room cooler than before and the other hotter, it seems to reduce the <a href="/wiki/Entropy" title="Entropy">entropy</a> of the room, and reverse the arrow of time. Many analyses have been made of this; all show that when the entropy of room and demon are taken together, this total entropy does increase. Modern analyses of this problem have taken into account <a href="/wiki/Claude_E._Shannon" class="mw-redirect" title="Claude E. Shannon">Claude E. Shannon</a>'s relation between <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">entropy and information</a>. Many interesting results in modern computing are closely related to this problem—<a href="/wiki/Reversible_computing" title="Reversible computing">reversible computing</a>, <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computing</a> and <a href="/wiki/Physical_limits_to_computing" class="mw-redirect" title="Physical limits to computing">physical limits to computing</a>, are examples. These seemingly metaphysical questions are today, in these ways, slowly being converted into hypotheses of the physical sciences. </p><p>The current consensus hinges upon the Boltzmann–Shannon identification of the logarithm of <a href="/wiki/Phase_space" title="Phase space">phase space</a> volume with the negative of <a href="/wiki/Information_theory" title="Information theory">Shannon information</a>, and hence to <a href="/wiki/Entropy" title="Entropy">entropy</a>. In this notion, a fixed initial state of a macroscopic system corresponds to relatively low entropy because the coordinates of the molecules of the body are constrained. As the system evolves in the presence of <a href="/wiki/Dissipation" title="Dissipation">dissipation</a>, the molecular coordinates can move into larger volumes of phase space, becoming more uncertain, and thus leading to increase in entropy. </p> <div class="mw-heading mw-heading3"><h3 id="Big_Bang">Big Bang</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=3" title="Edit section: Big Bang"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One resolution to irreversibility is to say that the constant increase of entropy we observe happens <i>only</i> because of the initial state of our universe. Other possible states of the universe (for example, a universe at <a href="/wiki/Heat_death_of_the_Universe" class="mw-redirect" title="Heat death of the Universe">heat death</a> equilibrium) would actually result in no increase of entropy. In this view, the apparent T-asymmetry of our universe is a problem in <a href="/wiki/Physical_cosmology" title="Physical cosmology">cosmology</a>: why did the universe start with a low entropy? This view, supported by cosmological observations (such as the <a href="/wiki/Isotropy" title="Isotropy">isotropy</a> of the <a href="/wiki/Cosmic_microwave_background" title="Cosmic microwave background">cosmic microwave background</a>) connects this problem to the question of <i>initial conditions</i> of the universe. </p> <div class="mw-heading mw-heading3"><h3 id="Black_holes">Black holes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=4" title="Edit section: Black holes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The laws of gravity seem to be time reversal invariant in classical mechanics; however, specific solutions need not be. </p><p>An object can cross through the <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a> of a <a href="/wiki/Black_hole" title="Black hole">black hole</a> from the outside, and then fall rapidly to the central region where our understanding of physics breaks down. Since within a black hole the forward light-cone is directed towards the center and the backward light-cone is directed outward, it is not even possible to define time-reversal in the usual manner. The only way anything can escape from a black hole is as <a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a>. </p><p>The time reversal of a black hole would be a hypothetical object known as a <a href="/wiki/White_hole" title="White hole">white hole</a>. From the outside they appear similar. While a black hole has a beginning and is inescapable, a white hole has an ending and cannot be entered. The forward light-cones of a white hole are directed outward; and its backward light-cones are directed towards the center. </p><p>The event horizon of a black hole may be thought of as a surface moving outward at the local speed of light and is just on the edge between escaping and falling back. The event horizon of a white hole is a surface moving inward at the local speed of light and is just on the edge between being swept outward and succeeding in reaching the center. They are two different kinds of horizons—the horizon of a white hole is like the horizon of a black hole turned inside-out. </p><p>The modern view of black hole irreversibility is to relate it to the second law of thermodynamics, since black holes are viewed as <a href="/wiki/Black_hole_thermodynamics" title="Black hole thermodynamics">thermodynamic objects</a>. For example, according to the <a href="/wiki/Gauge%E2%80%93gravity_duality" class="mw-redirect" title="Gauge–gravity duality">gauge–gravity duality</a> conjecture, all microscopic processes in a black hole are reversible, and only the collective behavior is irreversible, as in any other macroscopic, thermal system.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2010)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Kinetic_consequences:_detailed_balance_and_Onsager_reciprocal_relations">Kinetic consequences: detailed balance and Onsager reciprocal relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=5" title="Edit section: Kinetic consequences: detailed balance and Onsager reciprocal relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In physical and <a href="/wiki/Chemical_kinetics" title="Chemical kinetics">chemical kinetics</a>, T-symmetry of the mechanical microscopic equations implies two important laws: the principle of <a href="/wiki/Detailed_balance" title="Detailed balance">detailed balance</a> and the <a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a>. T-symmetry of the microscopic description together with its kinetic consequences are called <a href="/wiki/Microscopic_reversibility" title="Microscopic reversibility">microscopic reversibility</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Effect_of_time_reversal_on_some_variables_of_classical_physics">Effect of time reversal on some variables of classical physics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=6" title="Edit section: Effect of time reversal on some variables of classical physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Even">Even</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=7" title="Edit section: Even"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Classical variables that do not change upon time reversal include: </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 {\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}"></span>, position of a particle in three-space</dd> <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span>, acceleration of the particle</dd> <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 {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span>, force on the particle</dd> <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>, energy of the particle</dd> <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, electric potential (voltage)</dd> <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 {\vec {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc18ae485a72f148e85ccbeff2b3dcdd4f5f3f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\vec {E}}}"></span>, electric field</dd> <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 {\vec {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4bab6670be71833b1e405dab053af4f29dda49d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.843ex;" alt="{\displaystyle {\vec {D}}}"></span>, electric displacement</dd> <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>, density of electric charge</dd> <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 {\vec {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765f1dd50e122eb3e565c9bfee85de8f74d47f27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.778ex; height:2.843ex;" alt="{\displaystyle {\vec {P}}}"></span>, electric polarization</dd> <dd><a href="/wiki/Energy_density" title="Energy density">Energy density</a> of the electromagnetic field</dd> <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_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9de5760ded748359e36c7fb067c45f5e9642e890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.835ex; height:2.843ex;" alt="{\displaystyle T_{ij}}"></span>, <a href="/wiki/Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell stress tensor</a></dd> <dd>All masses, charges, coupling constants, and other physical constants, except those associated with the weak force.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Odd">Odd</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=8" title="Edit section: Odd"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Classical variables that time reversal negates include: </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}"> <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>, the time when an event occurs</dd> <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 {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span>, velocity of a particle</dd> <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 {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}"></span>, linear momentum of a particle</dd> <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 {\vec {l}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {l}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c457ef3d8e8bdf1f2ec773bb2523e6174dac0893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.287ex; height:2.843ex;" alt="{\displaystyle {\vec {l}}}"></span>, angular momentum of a particle (both orbital and spin)</dd> <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 {\vec {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391292ffadc65b0cde3e96f23afcdb811619dd95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:3.009ex;" alt="{\displaystyle {\vec {A}}}"></span>, electromagnetic vector potential</dd> <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 {\vec {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ae7d80cab55b606de217162280b2279142bbb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {B}}}"></span>, magnetic field</dd> <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 {\vec {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17aa4b2a53bc35011373e1bfe86baf779b521329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\vec {H}}}"></span>, magnetic auxiliary field</dd> <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 {\vec {j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {j}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce1ed1de8493f7cc7d856ca5427cf311b1597f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.094ex; height:3.176ex;" alt="{\displaystyle {\vec {j}}}"></span>, density of electric current</dd> <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 {\vec {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b903a0e453efca49cfda6ba33edb465de21a6b1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.843ex;" alt="{\displaystyle {\vec {M}}}"></span>, magnetization</dd> <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 {\vec {S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {S}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71a6b104c40975c738d5f0e22d445ebd509eb81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.538ex; height:3.009ex;" alt="{\displaystyle {\vec {S}}}"></span>, <a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a></dd> <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 {P}}}"> <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">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>, power (rate of work done).</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Example:_Magnetic_Field_and_Onsager_reciprocal_relations">Example: Magnetic Field and Onsager reciprocal relations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=9" title="Edit section: Example: Magnetic Field and Onsager reciprocal relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let us consider the example of a system of charged particles subject to a constant external magnetic field: in this case the canonical time reversal operation that reverses the velocities and the time <span class="mwe-math-element"><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> and keeps the coordinates untouched is no more a symmetry for the system. Under this consideration, it seems that only Onsager–Casimir reciprocal relations could hold;<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> these equalities relate two different systems, one subject 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 {\vec {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ae7d80cab55b606de217162280b2279142bbb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {B}}}"></span> and another 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 -{\vec {B}}}"> <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>B</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\vec {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/810de81e39aec5f05cde29f3ae2d99a532a9a480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.572ex; height:3.009ex;" alt="{\displaystyle -{\vec {B}}}"></span>, and so their utility is limited. However, it was proved that it is possible to find other time reversal operations which preserve the dynamics and so Onsager reciprocal relations;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> in conclusion, one cannot state that the presence of a magnetic field always breaks T-symmetry. </p> <div class="mw-heading mw-heading2"><h2 id="Microscopic_phenomena:_time_reversal_invariance">Microscopic phenomena: time reversal invariance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=10" title="Edit section: Microscopic phenomena: time reversal invariance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most systems are asymmetric under time reversal, but there may be phenomena with symmetry. In classical mechanics, a velocity <i>v</i> reverses under the operation of <i>T</i>, but an acceleration does not.<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> Therefore, one models dissipative phenomena through terms that are odd in <i>v</i>. However, delicate experiments in which known sources of dissipation are removed reveal that the laws of mechanics are time reversal invariant. Dissipation itself is originated in the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>. </p><p>The motion of a charged body in a magnetic field, <i>B</i> involves the velocity through the <a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a> term <i>v</i>×<i>B</i>, and might seem at first to be asymmetric under <i>T</i>. A closer look assures us that <i>B</i> also changes sign under time reversal. This happens because a magnetic field is produced by an electric current, <i>J</i>, which reverses sign under <i>T</i>. Thus, the motion of classical charged particles in <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic fields</a> is also time reversal invariant. (Despite this, it is still useful to consider the time-reversal non-invariance in a <i>local</i> sense when the external field is held fixed, as when the <a href="/wiki/Magneto-optic_effect" title="Magneto-optic effect">magneto-optic effect</a> is analyzed. This allows one to analyze the conditions under which optical phenomena that locally break time-reversal, such as <a href="/wiki/Faraday_isolator" class="mw-redirect" title="Faraday isolator">Faraday isolators</a> and <a rel="nofollow" class="external text" href="http://magnetooptics.phy.bme.hu/research/topics/optical-properties-of-multiferroic-materials/">directional dichroism</a>, can occur.) </p><p>In <a href="/wiki/Physics" title="Physics">physics</a> one separates the laws of motion, called <a href="/wiki/Kinematics" title="Kinematics">kinematics</a>, from the laws of force, called <a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">dynamics</a>. Following the classical kinematics of <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a>, the kinematics of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> is built in such a way that it presupposes nothing about the time reversal symmetry of the dynamics. In other words, if the dynamics are invariant, then the kinematics will allow it to remain invariant; if the dynamics is not, then the kinematics will also show this. The structure of the quantum laws of motion are richer, and we examine these next. </p> <div class="mw-heading mw-heading3"><h3 id="Time_reversal_in_quantum_mechanics">Time reversal in quantum mechanics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=11" title="Edit section: Time reversal in quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Frame"><a href="/wiki/File:Parity_1drep.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/fc/Parity_1drep.png" decoding="async" width="258" height="426" class="mw-file-element" data-file-width="258" data-file-height="426" /></a><figcaption>Two-dimensional representations of <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a> are given by a pair of quantum states that go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that all <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible representations</a> of parity are one-dimensional. <b>Kramers' theorem</b> states that time reversal need not have this property because it is represented by an anti-unitary operator.</figcaption></figure> <p>This section contains a discussion of the three most important properties of time reversal in quantum mechanics; chiefly, </p> <ol><li>that it must be represented as an anti-unitary operator,</li> <li>that it protects non-degenerate quantum states from having an <a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">electric dipole moment</a>,</li> <li>that it has two-dimensional representations with the property <span class="nowrap"><i>T</i><sup>2</sup> = −1</span> (for <a href="/wiki/Fermion" title="Fermion">fermions</a>).</li></ol> <p>The strangeness of this result is clear if one compares it with parity. If parity transforms a pair of <a href="/wiki/Quantum_states" class="mw-redirect" title="Quantum states">quantum states</a> into each other, then the sum and difference of these two basis states are states of good parity. Time reversal does not behave like this. It seems to violate the theorem that all <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a> be represented by one-dimensional irreducible representations. The reason it does this is that it is represented by an anti-unitary operator. It thus opens the way to <a href="/wiki/Spinor" title="Spinor">spinors</a> in quantum mechanics. </p><p>On the other hand, the notion of quantum-mechanical time reversal turns out to be a useful tool for the development of physically motivated <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computing</a> and <a href="/wiki/Quantum_simulator" title="Quantum simulator">simulation</a> settings, providing, at the same time, relatively simple tools to assess their <a href="/wiki/Computational_complexity" title="Computational complexity">complexity</a>. For instance, quantum-mechanical time reversal was used to develop novel <a href="/wiki/Boson_sampling" title="Boson sampling">boson sampling</a> schemes<sup id="cite_ref-Chakhmakhchyan2017_7-0" class="reference"><a href="#cite_note-Chakhmakhchyan2017-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and to prove the duality between two fundamental optical operations, <a href="/wiki/Beam_splitter#Quantum_mechanical_description" title="Beam splitter">beam splitter</a> and <a href="/wiki/Squeezed_coherent_state#Operator_representation" title="Squeezed coherent state">squeezing</a> transformations.<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-heading3"><h3 id="Formal_notation">Formal notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=12" title="Edit section: Formal notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In formal mathematical presentations of T-symmetry, three different kinds of notation for <b>T</b> need to be carefully distinguished: the <b>T</b> that is an <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a>, capturing the actual reversal of the time coordinate, the <b>T</b> that is an ordinary finite dimensional matrix, acting on <a href="/wiki/Spinor" title="Spinor">spinors</a> and vectors, and the <b>T</b> that is an operator on an infinite-dimensional <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>. </p><p>For a <a href="/wiki/Real_number" title="Real number">real</a> (not <a href="/wiki/Complex_number" title="Complex number">complex</a>) classical (unquantized) <a href="/wiki/Scalar_field" title="Scalar field">scalar field</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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ<!-- ϕ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>, the time reversal <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> can simply be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}\phi (t,{\vec {x}})=\phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>ϕ<!-- ϕ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}\phi (t,{\vec {x}})=\phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dad28cabeb3d0dd59506ccb1e6121f781aa9867a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.557ex; height:3.009ex;" alt="{\displaystyle {\mathsf {T}}\phi (t,{\vec {x}})=\phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})}"></span></dd></dl> <p>as time reversal leaves the scalar value at a fixed spacetime point unchanged, up to an overall sign <span class="mwe-math-element"><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=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo>±<!-- ± --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64de17d8811b8b810f5ef7573efd10016f7938c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.16ex; height:2.176ex;" alt="{\displaystyle s=\pm 1}"></span>. A slightly more formal way to write this is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}:\phi (t,{\vec {x}})\mapsto \phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mo>:</mo> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">↦<!-- ↦ --></mo> <msup> <mi>ϕ<!-- ϕ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}:\phi (t,{\vec {x}})\mapsto \phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb0f2c3184f42af0660a91716c124fc0b7613db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.01ex; height:3.009ex;" alt="{\displaystyle {\mathsf {T}}:\phi (t,{\vec {x}})\mapsto \phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})}"></span></dd></dl> <p>which has the advantage of emphasizing that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7860e4d66c67ce669b88f07a796b143537193daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle {\mathsf {T}}}"></span> is a <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">map</a>, and thus the "mapsto" notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mapsto ~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↦<!-- ↦ --></mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mapsto ~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe600dd05f0fa80c9b2c5061f0c19d8a11004f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.196ex; height:2.176ex;" alt="{\displaystyle \mapsto ~,}"></span> whereas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϕ<!-- ϕ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95d32cfb216128b7d93ae7cd4301cc4a8a76c2f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.478ex; height:3.009ex;" alt="{\displaystyle \phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})}"></span> is a factual statement relating the old and new fields to one-another. </p><p>Unlike scalar fields, <a href="/wiki/Spinor" title="Spinor">spinor</a> and <a href="/wiki/Vector_field" title="Vector field">vector fields</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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- 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> might have a non-trivial behavior under time reversal. In this case, one has to write </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}:\psi (t,{\vec {x}})\mapsto \psi ^{\prime }(-t,{\vec {x}})=T\psi (t,{\vec {x}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mo>:</mo> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">↦<!-- ↦ --></mo> <msup> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}:\psi (t,{\vec {x}})\mapsto \psi ^{\prime }(-t,{\vec {x}})=T\psi (t,{\vec {x}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72266d78296790a50b64ca8fefafaa2052174f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.939ex; height:3.009ex;" alt="{\displaystyle {\mathsf {T}}:\psi (t,{\vec {x}})\mapsto \psi ^{\prime }(-t,{\vec {x}})=T\psi (t,{\vec {x}})}"></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 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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is just an ordinary <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. For <a href="/wiki/Complex_number" title="Complex number">complex</a> fields, <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a> may be required, for which the mapping <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K:(x+iy)\mapsto (x-iy)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦<!-- ↦ --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K:(x+iy)\mapsto (x-iy)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db3f28738a813eb360d5b5028ea7235437025d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.492ex; height:2.843ex;" alt="{\displaystyle K:(x+iy)\mapsto (x-iy)}"></span> can be thought of as a 2x2 matrix. For a <a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinor</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> cannot be written as a 4x4 matrix, because, in fact, complex conjugation is indeed required; however, it can be written as an 8x8 matrix, acting on the 8 real components of a Dirac spinor. </p><p>In the general setting, there is no <i>ab initio</i> value to be given 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 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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>; its actual form depends on the specific equation or equations which are being examined. In general, one simply states that the equations must be time-reversal invariant, and then solves for the explicit value 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 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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> that achieves this goal. In some cases, generic arguments can be made. Thus, for example, for spinors in three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, or four-dimensional <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>, an explicit transformation can be given. It is conventionally given 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 T=e^{i\pi J_{y}}K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>π<!-- π --></mi> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </msup> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=e^{i\pi J_{y}}K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f00fb53a19077d00dd36d0d45666e7a2a82402b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.366ex; height:2.676ex;" alt="{\displaystyle T=e^{i\pi J_{y}}K}"></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 J_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/782c237016b7230689619e010bf692a0adc79b38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.34ex; height:2.843ex;" alt="{\displaystyle J_{y}}"></span> is the y-component of the <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">angular momentum operator</a> 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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is complex conjugation, as before. This form follows whenever the spinor can be described with a linear <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> that is first-order in the time derivative, which is generally the case in order for something to be validly called "a spinor". </p><p>The formal notation now makes it clear how to extend time-reversal to an arbitrary <a href="/wiki/Tensor_field" title="Tensor field">tensor field</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 \psi _{abc\cdots }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>⋯<!-- ⋯ --></mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{abc\cdots }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9c43c653344217f99e15b6e73b244b15a67669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.958ex; height:2.509ex;" alt="{\displaystyle \psi _{abc\cdots }}"></span> In this case, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}:\psi _{abc\cdots }(t,{\vec {x}})\mapsto \psi _{abc\cdots }^{\prime }(-t,{\vec {x}})={T_{a}}^{d}\,{T_{b}}^{e}\,{T_{c}}^{f}\cdots \psi _{def\cdots }(t,{\vec {x}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mo>:</mo> <msub> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>⋯<!-- ⋯ --></mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">↦<!-- ↦ --></mo> <msubsup> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>⋯<!-- ⋯ --></mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msup> <mo>⋯<!-- ⋯ --></mo> <msub> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> <mo>⋯<!-- ⋯ --></mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}:\psi _{abc\cdots }(t,{\vec {x}})\mapsto \psi _{abc\cdots }^{\prime }(-t,{\vec {x}})={T_{a}}^{d}\,{T_{b}}^{e}\,{T_{c}}^{f}\cdots \psi _{def\cdots }(t,{\vec {x}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/075fdc95b743cd6c134d44b91c2555024f3b5bf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.75ex; height:3.343ex;" alt="{\displaystyle {\mathsf {T}}:\psi _{abc\cdots }(t,{\vec {x}})\mapsto \psi _{abc\cdots }^{\prime }(-t,{\vec {x}})={T_{a}}^{d}\,{T_{b}}^{e}\,{T_{c}}^{f}\cdots \psi _{def\cdots }(t,{\vec {x}})}"></span></dd></dl> <p>Covariant tensor indexes will transform 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 {T_{a}}^{b}={(T^{-1})_{b}}^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T_{a}}^{b}={(T^{-1})_{b}}^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1dd48b7415a98e09725e52ab26bad915d69014a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.397ex; height:3.176ex;" alt="{\displaystyle {T_{a}}^{b}={(T^{-1})_{b}}^{a}}"></span> and so on. For quantum fields, there is also a third <b>T</b>, written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}},}"> <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">T</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090b979b782d97475fac5729494456eb986dfe44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.676ex;" alt="{\displaystyle {\mathcal {T}},}"></span> which is actually an infinite dimensional operator acting on a Hilbert space. It acts on quantized fields <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ<!-- 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/f5471531a3fe80741a839bc98d49fae862a6439a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }"></span> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}:\Psi (t,{\vec {x}})\mapsto \Psi ^{\prime }(-t,{\vec {x}})={\mathcal {T}}\Psi (t,{\vec {x}}){\mathcal {T}}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mo>:</mo> <mi mathvariant="normal">Ψ<!-- Ψ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">↦<!-- ↦ --></mo> <msup> <mi mathvariant="normal">Ψ<!-- Ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mi mathvariant="normal">Ψ<!-- Ψ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}:\Psi (t,{\vec {x}})\mapsto \Psi ^{\prime }(-t,{\vec {x}})={\mathcal {T}}\Psi (t,{\vec {x}}){\mathcal {T}}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb9ec40b8f686d4a998239ce60d22ff1fad8a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.593ex; height:3.176ex;" alt="{\displaystyle {\mathsf {T}}:\Psi (t,{\vec {x}})\mapsto \Psi ^{\prime }(-t,{\vec {x}})={\mathcal {T}}\Psi (t,{\vec {x}}){\mathcal {T}}^{-1}}"></span></dd></dl> <p>This can be thought of as a special case of a tensor with one covariant, and one contravariant index, and thus two <span class="mwe-math-element"><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 {T}}}"> <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">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span>'s are required. </p><p>All three of these symbols capture the idea of time-reversal; they differ with respect to the specific <a href="/wiki/Space_(mathematics)" title="Space (mathematics)">space</a> that is being acted on: functions, vectors/spinors, or infinite-dimensional operators. The remainder of this article is not cautious to distinguish these three; the <i>T</i> that appears below is meant to be either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7860e4d66c67ce669b88f07a796b143537193daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle {\mathsf {T}}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}},}"> <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">T</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090b979b782d97475fac5729494456eb986dfe44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.676ex;" alt="{\displaystyle {\mathcal {T}},}"></span> depending on context, left for the reader to infer. </p> <div class="mw-heading mw-heading3"><h3 id="Anti-unitary_representation_of_time_reversal">Anti-unitary representation of time reversal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=13" title="Edit section: Anti-unitary representation of time reversal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Eugene Wigner</a> showed that a symmetry operation <i>S</i> of a Hamiltonian is represented, in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> either by a <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operator</a>, <span class="nowrap"><i>S</i> = <i>U</i></span>, or an <a href="/wiki/Antiunitary" class="mw-redirect" title="Antiunitary">antiunitary</a> one, <span class="nowrap"><i>S</i> = <i>UK</i></span> where <i>U</i> is unitary, and <i>K</i> denotes <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>. These are the only operations that act on Hilbert space so as to preserve the <i>length</i> of the projection of any one state-vector onto another state-vector. </p><p>Consider the <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a> operator. Acting on the position, it reverses the directions of space, so that <span class="nowrap"><i>PxP</i><sup>−1</sup> = −<i>x</i></span>. Similarly, it reverses the direction of <i>momentum</i>, so that <span class="nowrap"><i>PpP</i><sup>−1</sup> = −<i>p</i></span>, where <i>x</i> and <i>p</i> are the position and momentum operators. This preserves the <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutator</a> <span class="nowrap">[<i>x</i>, <i>p</i>] = <i>iħ</i></span>, where <i>ħ</i> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>, only if <i>P</i> is chosen to be unitary, <span class="nowrap"><i>PiP</i><sup>−1</sup> = <i>i</i></span>. </p><p>On the other hand, the <i>time reversal</i> operator <i>T</i>, it does nothing to the x-operator, <span class="nowrap"><i>TxT</i><sup>−1</sup> = <i>x</i></span>, but it reverses the direction of p, so that <span class="nowrap"><i>TpT</i><sup>−1</sup> = −<i>p</i></span>. The canonical commutator is invariant only if <i>T</i> is chosen to be anti-unitary, i.e., <span class="nowrap"><i>TiT</i><sup>−1</sup> = −<i>i</i></span>. </p><p>Another argument involves energy, the time-component of the four-momentum. If time reversal were implemented as a unitary operator, it would reverse the sign of the energy just as space-reversal reverses the sign of the momentum. This is not possible, because, unlike momentum, energy is always positive. Since energy in quantum mechanics is defined as the phase factor exp(–<i>iEt</i>) that one gets when one moves forward in time, the way to reverse time while preserving the sign of the energy is to also reverse the sense of "<i>i</i>", so that the sense of phases is reversed. </p><p>Similarly, any operation that reverses the sense of phase, which changes the sign of <i>i</i>, will turn positive energies into negative energies unless it also changes the direction of time. So every antiunitary symmetry in a theory with positive energy must reverse the direction of time. Every antiunitary operator can be written as the product of the time reversal operator and a unitary operator that does not reverse time. </p><p>For a <a href="/wiki/Elementary_particle" title="Elementary particle">particle</a> with spin <i>J</i>, one can use the representation </p> <dl><dd><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=e^{-i\pi J_{y}/\hbar }K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mi>i</mi> <mi>π<!-- π --></mi> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ<!-- ℏ --></mi> </mrow> </msup> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=e^{-i\pi J_{y}/\hbar }K,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae47e5ced042230cff0672b254c64a5a68ea8db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.037ex; height:3.176ex;" alt="{\displaystyle T=e^{-i\pi J_{y}/\hbar }K,}"></span></dd></dl></dd></dl> <p>where <i>J</i><sub><i>y</i></sub> is the <i>y</i>-component of the spin, and use of <span class="nowrap"><i>TJT</i><sup>−1</sup> = −<i>J</i></span> has been made. </p> <div class="mw-heading mw-heading3"><h3 id="Electric_dipole_moments">Electric dipole moments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=14" title="Edit section: Electric dipole moments"><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/Electron_electric_dipole_moment" title="Electron electric dipole moment">Electron electric dipole moment</a></div> <p>This has an interesting consequence on the <a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">electric dipole moment</a> (EDM) of any particle. The EDM is defined through the shift in the energy of a state when it is put in an external electric field: <span class="nowrap">Δ<i>e</i> = d·<i>E</i> + <i>E</i>·δ·<i>E</i></span>, where <i>d</i> is called the EDM and δ, the induced dipole moment. One important property of an EDM is that the energy shift due to it changes sign under a parity transformation. However, since <b>d</b> is a vector, its expectation value in a state |ψ⟩ must be proportional to ⟨ψ| <i>J</i> |ψ⟩, that is the expected spin. Thus, under time reversal, an invariant state must have vanishing EDM. In other words, a non-vanishing EDM signals both <i>P</i> and <i>T</i> symmetry-breaking.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Some molecules, such as water, must have EDM irrespective of whether <b>T</b> is a symmetry. This is correct; if a quantum system has degenerate ground states that transform into each other under parity, then time reversal need not be broken to give EDM. </p><p>Experimentally observed bounds on the <a href="/wiki/Neutron_electric_dipole_moment" title="Neutron electric dipole moment">electric dipole moment of the nucleon</a> currently set stringent limits on the violation of time reversal symmetry in the <a href="/wiki/Strong_interactions" class="mw-redirect" title="Strong interactions">strong interactions</a>, and their modern theory: <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>. Then, using the <a href="/wiki/CPT_invariance" class="mw-redirect" title="CPT invariance">CPT invariance</a> of a relativistic <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, this puts <a href="/wiki/CryoEDM" title="CryoEDM">strong bounds</a> on <a href="/wiki/Strong_CP_violation" class="mw-redirect" title="Strong CP violation">strong CP violation</a>. </p><p>Experimental bounds on the <a href="/wiki/Electron_electric_dipole_moment" title="Electron electric dipole moment">electron electric dipole moment</a> also place limits on theories of particle physics and their parameters.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Kramers'_theorem"><span id="Kramers.27_theorem"></span>Kramers' theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=15" title="Edit section: Kramers' theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Kramers%27_theorem" title="Kramers' theorem">Kramers' theorem</a></div> <p>For <i>T</i>, which is an anti-unitary <i>Z</i><sub>2</sub> symmetry generator </p> <dl><dd><i>T</i><sup>2</sup> = <i>UKUK</i> = <i>UU</i><sup>*</sup> = <i>U</i> (<i>U</i><sup>T</sup>)<sup>−1</sup> = Φ,</dd></dl> <p>where Φ is a diagonal matrix of phases. As a result, <span class="nowrap"><i>U</i> = Φ<i>U</i><sup>T</sup></span> and <span class="nowrap"><i>U</i><sup>T</sup> = <i>U</i>Φ</span>, showing that </p> <dl><dd><i>U</i> = Φ <i>U</i> Φ.</dd></dl> <p>This means that the entries in Φ are ±1, as a result of which one may have either <span class="nowrap"><i>T</i><sup>2</sup> = ±1</span>. This is specific to the anti-unitarity of <i>T</i>. For a unitary operator, such as the <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a>, any phase is allowed. </p><p>Next, take a Hamiltonian invariant under <i>T</i>. Let |<i>a</i>⟩ and <i>T</i>|<i>a</i>⟩ be two quantum states of the same energy. Now, if <span class="nowrap"><i>T</i><sup>2</sup> = −1</span>, then one finds that the states are orthogonal: a result called <b>Kramers' theorem</b>. This implies that if <span class="nowrap"><i>T</i><sup>2</sup> = −1</span>, then there is a twofold degeneracy in the state. This result in non-relativistic <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> presages the <a href="/wiki/Spin_statistics_theorem" class="mw-redirect" title="Spin statistics theorem">spin statistics theorem</a> of <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. </p><p><a href="/wiki/Quantum_state" title="Quantum state">Quantum states</a> that give unitary representations of time reversal, i.e., have <b> <span class="nowrap"><i>T</i><sup>2</sup> = 1</span></b>, are characterized by a <a href="/wiki/Multiplicative_quantum_number" title="Multiplicative quantum number">multiplicative quantum number</a>, sometimes called the <b>T-parity</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Time_reversal_of_the_known_dynamical_laws">Time reversal of the known dynamical laws</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=16" title="Edit section: Time reversal of the known dynamical laws"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a> codified the basic laws of dynamics into the <a href="/wiki/Standard_model" class="mw-redirect" title="Standard model">standard model</a>. This is formulated as a <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> that has <a href="/wiki/CPT_symmetry" title="CPT symmetry">CPT symmetry</a>, i.e., the laws are invariant under simultaneous operation of time reversal, <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a> and <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a>. However, time reversal itself is seen not to be a symmetry (this is usually called <a href="/wiki/CP_violation" title="CP violation">CP violation</a>). There are two possible origins of this asymmetry, one through the <a href="/wiki/CKM_matrix" class="mw-redirect" title="CKM matrix">mixing</a> of different <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">flavours</a> of quarks in their <a href="/wiki/Weak_interaction" title="Weak interaction">weak decays</a>, the second through a direct CP violation in strong interactions. The first is seen in experiments, the second is strongly constrained by the non-observation of the <a href="/wiki/Neutron_electric_dipole_moment" title="Neutron electric dipole moment">EDM of a neutron</a>. </p><p>Time reversal violation is unrelated to the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>, because due to the conservation of the <a href="/wiki/CPT_symmetry" title="CPT symmetry">CPT symmetry</a>, the effect of time reversal is to rename <a href="/wiki/Elementary_particle" title="Elementary particle">particles</a> as <a href="/wiki/Antiparticle" title="Antiparticle">antiparticles</a> and <i>vice versa</i>. Thus the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a> is thought to originate in the <a href="/wiki/Initial_conditions" class="mw-redirect" title="Initial conditions">initial conditions</a> in the universe. </p> <div class="mw-heading mw-heading3"><h3 id="Time_reversal_of_noninvasive_measurements">Time reversal of noninvasive measurements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=17" title="Edit section: Time reversal of noninvasive measurements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Strong measurements</a> (both classical and quantum) are certainly disturbing, causing asymmetry due to the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>. However, <a href="/wiki/Weak_measurement" title="Weak measurement">noninvasive measurements</a> should not disturb the evolution, so they are expected to be time-symmetric. Surprisingly, it is true only in classical physics but not in quantum physics, even in a thermodynamically invariant equilibrium state.<sup id="cite_ref-non-time_1-1" class="reference"><a href="#cite_note-non-time-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This type of asymmetry is independent of <a href="/wiki/CPT_symmetry" title="CPT symmetry">CPT symmetry</a> but has not yet been confirmed experimentally due to extreme conditions of the checking proposal. </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=T-symmetry&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Arrow_of_time" title="Arrow of time">Arrow of time</a></li> <li><a href="/wiki/Causality_(physics)" title="Causality (physics)">Causality (physics)</a></li> <li>Computing applications <ul><li><a href="/wiki/Limits_of_computation" title="Limits of computation">Limits of computation</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Reversible_computing" title="Reversible computing">Reversible computing</a></li></ul></li> <li><a href="/wiki/Standard_model" class="mw-redirect" title="Standard model">Standard model</a> <ul><li><a href="/wiki/CKM_matrix" class="mw-redirect" title="CKM matrix">CKM matrix</a></li> <li><a href="/wiki/CP_violation" title="CP violation">CP violation</a></li> <li><a href="/wiki/CPT_invariance" class="mw-redirect" title="CPT invariance">CPT invariance</a></li> <li><a href="/wiki/Neutrino_mass" class="mw-redirect" title="Neutrino mass">Neutrino mass</a></li> <li><a href="/wiki/Strong_CP_problem" title="Strong CP problem">Strong CP problem</a></li> <li><a href="/wiki/Wheeler%E2%80%93Feynman_absorber_theory" title="Wheeler–Feynman absorber theory">Wheeler–Feynman absorber theory</a></li></ul></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Microscopic_reversibility" title="Microscopic reversibility">Microscopic reversibility</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second law of thermodynamics</a></li> <li><a href="/wiki/Time_translation_symmetry" class="mw-redirect" title="Time translation symmetry">Time translation symmetry</a></li></ul> <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=T-symmetry&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Inline_citations">Inline citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=20" title="Edit section: Inline citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-non-time-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-non-time_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-non-time_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBednorzFrankeBelzig2013" class="citation journal cs1">Bednorz, Adam; Franke, Kurt; Belzig, Wolfgang (February 2013). "Noninvasiveness and time symmetry of weak measurements". <i>New Journal of Physics</i>. <b>15</b> (2): 023043. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1108.1305">1108.1305</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013NJPh...15b3043B">2013NJPh...15b3043B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F15%2F2%2F023043">10.1088/1367-2630/15/2/023043</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:17583996">17583996</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+Journal+of+Physics&rft.atitle=Noninvasiveness+and+time+symmetry+of+weak+measurements&rft.volume=15&rft.issue=2&rft.pages=023043&rft.date=2013-02&rft_id=info%3Aarxiv%2F1108.1305&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17583996%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1367-2630%2F15%2F2%2F023043&rft_id=info%3Abibcode%2F2013NJPh...15b3043B&rft.aulast=Bednorz&rft.aufirst=Adam&rft.au=Franke%2C+Kurt&rft.au=Belzig%2C+Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-symmetry" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKubo1957" class="citation journal cs1">Kubo, Ryogo (15 June 1957). "Statistical-Mechanical Theory of Irreversible Processes. I. 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"Time reversal symmetry in time-dependent correlation functions for systems in a constant magnetic field". <i>EPL (Europhysics Letters)</i>. <b>108</b> (6): 60004. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1209%2F0295-5075%2F108%2F60004">10.1209/0295-5075/108/60004</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:121427119">121427119</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=EPL+%28Europhysics+Letters%29&rft.atitle=Time+reversal+symmetry+in+time-dependent+correlation+functions+for+systems+in+a+constant+magnetic+field&rft.volume=108&rft.issue=6&rft.pages=60004&rft.date=2015&rft_id=info%3Adoi%2F10.1209%2F0295-5075%2F108%2F60004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121427119%23id-name%3DS2CID&rft.aulast=Bonella&rft.aufirst=Sara&rft.au=Ciccotti%2C+Giovanni&rft.au=Rondoni%2C+Lamberto&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-symmetry" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuoBenentiCasatiWang2020" class="citation journal cs1">Luo, Rongxiang; Benenti, Giuliano; Casati, Giulio; Wang, Jiao (2020). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevResearch.2.022009">"Onsager reciprocal relations with broken time-reversal symmetry"</a>. <i>Physical Review Research</i>. <b>2</b> (2): 022009. <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/2020PhRvR...2b2009L">2020PhRvR...2b2009L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevResearch.2.022009">10.1103/PhysRevResearch.2.022009</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Research&rft.atitle=Onsager+reciprocal+relations+with+broken+time-reversal+symmetry&rft.volume=2&rft.issue=2&rft.pages=022009&rft.date=2020&rft_id=info%3Adoi%2F10.1103%2FPhysRevResearch.2.022009&rft_id=info%3Abibcode%2F2020PhRvR...2b2009L&rft.aulast=Luo&rft.aufirst=Rongxiang&rft.au=Benenti%2C+Giuliano&rft.au=Casati%2C+Giulio&rft.au=Wang%2C+Jiao&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRevResearch.2.022009&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-symmetry" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarboneRondoni2020" class="citation journal cs1">Carbone, Davide; 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Liedl, Klaus R.; Rode, Bernd M. 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"Bidirectional molecular dynamics: Interpretation in terms of a modern formulation of classical mechanics". <i>Journal of Computational Chemistry</i>. <b>17</b> (13): 1564–1570. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F%28SICI%291096-987X%28199610%2917%3A13%3C1564%3A%3AAID-JCC8%3E3.0.CO%3B2-Q">10.1002/(SICI)1096-987X(199610)17:13<1564::AID-JCC8>3.0.CO;2-Q</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Chemistry&rft.atitle=Bidirectional+molecular+dynamics%3A+Interpretation+in+terms+of+a+modern+formulation+of+classical+mechanics&rft.volume=17&rft.issue=13&rft.pages=1564-1570&rft.date=1996&rft_id=info%3Adoi%2F10.1002%2F%28SICI%291096-987X%28199610%2917%3A13%3C1564%3A%3AAID-JCC8%3E3.0.CO%3B2-Q&rft.aulast=Kerdcharoen&rft.aufirst=Teerakiat&rft.au=Liedl%2C+Klaus+R.&rft.au=Rode%2C+Bernd+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-symmetry" class="Z3988"></span></span> </li> <li id="cite_note-Chakhmakhchyan2017-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Chakhmakhchyan2017_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChakhmakhchyanCerf2017" class="citation journal cs1">Chakhmakhchyan, Levon; Cerf, Nicolas (2017). 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"Simulating arbitrary Gaussian circuits with linear optics". <i>Physical Review A</i>. <b>98</b> (6): 062314. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1803.11534">1803.11534</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018PhRvA..98f2314C">2018PhRvA..98f2314C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.98.062314">10.1103/PhysRevA.98.062314</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:119227039">119227039</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+A&rft.atitle=Simulating+arbitrary+Gaussian+circuits+with+linear+optics&rft.volume=98&rft.issue=6&rft.pages=062314&rft.date=2018&rft_id=info%3Aarxiv%2F1803.11534&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119227039%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.98.062314&rft_id=info%3Abibcode%2F2018PhRvA..98f2314C&rft.aulast=Chakhmakhchyan&rft.aufirst=Levon&rft.au=Cerf%2C+Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-symmetry" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKhriplovichLamoreaux2012" class="citation book cs1">Khriplovich, Iosip B.; Lamoreaux, Steve K. (2012). <i>CP violation without strangeness : electric dipole moments of particles, atoms, and molecules</i>. [S.l.]: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-64577-8" title="Special:BookSources/978-3-642-64577-8"><bdi>978-3-642-64577-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=CP+violation+without+strangeness+%3A+electric+dipole+moments+of+particles%2C+atoms%2C+and+molecules.&rft.place=%5BS.l.%5D&rft.pub=Springer&rft.date=2012&rft.isbn=978-3-642-64577-8&rft.aulast=Khriplovich&rft.aufirst=Iosip+B.&rft.au=Lamoreaux%2C+Steve+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-symmetry" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIbrahimItaniNath2014" class="citation journal cs1">Ibrahim, Tarik; Itani, Ahmad; Nath, Pran (12 Aug 2014). "Electron EDM as a Sensitive Probe of PeV Scale Physics". <i>Physical Review D</i>. <b>90</b> (5): 055006. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1406.0083">1406.0083</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014PhRvD..90e5006I">2014PhRvD..90e5006I</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.90.055006">10.1103/PhysRevD.90.055006</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:118880896">118880896</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Electron+EDM+as+a+Sensitive+Probe+of+PeV+Scale+Physics&rft.volume=90&rft.issue=5&rft.pages=055006&rft.date=2014-08-12&rft_id=info%3Aarxiv%2F1406.0083&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118880896%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.90.055006&rft_id=info%3Abibcode%2F2014PhRvD..90e5006I&rft.aulast=Ibrahim&rft.aufirst=Tarik&rft.au=Itani%2C+Ahmad&rft.au=Nath%2C+Pran&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-symmetry" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKimCarosi2010" class="citation journal cs1">Kim, Jihn E.; Carosi, Gianpaolo (4 March 2010). "Axions and the strong CP problem". <i>Reviews of Modern Physics</i>. <b>82</b> (1): 557–602. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0807.3125">0807.3125</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010RvMP...82..557K">2010RvMP...82..557K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.82.557">10.1103/RevModPhys.82.557</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Axions+and+the+strong+CP+problem&rft.volume=82&rft.issue=1&rft.pages=557-602&rft.date=2010-03-04&rft_id=info%3Aarxiv%2F0807.3125&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.82.557&rft_id=info%3Abibcode%2F2010RvMP...82..557K&rft.aulast=Kim&rft.aufirst=Jihn+E.&rft.au=Carosi%2C+Gianpaolo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-symmetry" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-symmetry&action=edit&section=21" title="Edit section: General references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Maxwell's demon: entropy, information, computing, edited by H.S.Leff and A.F. Rex (IOP publishing, 1990) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7503-0057-4" title="Special:BookSources/0-7503-0057-4">0-7503-0057-4</a></li> <li>Maxwell's demon, 2: entropy, classical and quantum information, edited by H.S.Leff and A.F. Rex (IOP publishing, 2003) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7503-0759-5" title="Special:BookSources/0-7503-0759-5">0-7503-0759-5</a></li> <li>The emperor's new mind: concerning computers, minds, and the laws of physics, by Roger Penrose (Oxford university press, 2002) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-286198-0" title="Special:BookSources/0-19-286198-0">0-19-286198-0</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSozzi,_M.S.2008" class="citation book cs1">Sozzi, M.S. (2008). <i>Discrete symmetries and CP violation</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-929666-8" title="Special:BookSources/978-0-19-929666-8"><bdi>978-0-19-929666-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+symmetries+and+CP+violation&rft.pub=Oxford+University+Press&rft.date=2008&rft.isbn=978-0-19-929666-8&rft.au=Sozzi%2C+M.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-symmetry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirss,_R._R.1964" class="citation book cs1">Birss, R. R. (1964). <i>Symmetry and Magnetism</i>. John Wiley & Sons, Inc., New York.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetry+and+Magnetism&rft.pub=John+Wiley+%26+Sons%2C+Inc.%2C+New+York&rft.date=1964&rft.au=Birss%2C+R.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-symmetry" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://magnetooptics.phy.bme.hu/research/topics/optical-properties-of-multiferroic-materials/">Multiferroic</a> materials with time-reversal breaking optical properties</li> <li>CP violation, by I.I. Bigi and A.I. Sanda (Cambridge University Press, 2000) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-44349-0" title="Special:BookSources/0-521-44349-0">0-521-44349-0</a></li> <li><a rel="nofollow" class="external text" href="http://pdg.lbl.gov/2004/reviews/cpviolrpp.pdf">Particle Data Group on CP violation</a> <ul><li>the <a rel="nofollow" class="external text" href="http://www-public.slac.stanford.edu/babar/">Babar</a> experiment in <a href="/wiki/SLAC" class="mw-redirect" title="SLAC">SLAC</a></li> <li>the <a rel="nofollow" class="external text" href="http://belle.kek.jp">BELLE</a> experiment in <a href="/wiki/KEK" title="KEK">KEK</a></li> <li>the <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050404165826/http://kpasa.fnal.gov:8080/public/ktev.html">KTeV</a> experiment in <a href="/wiki/Fermilab" title="Fermilab">Fermilab</a></li> <li>the <a rel="nofollow" class="external text" href="http://cplear.web.cern.ch/cplear/Welcome.html">CPLEAR</a> experiment in <a href="/wiki/CERN" title="CERN">CERN</a></li></ul></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output 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.hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="C,_P,_and_T_symmetries" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:C,_P_and_T" title="Template:C, P and T"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:C,_P_and_T" title="Template talk:C, P and T"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:C,_P_and_T" title="Special:EditPage/Template:C, P and T"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="C,_P,_and_T_symmetries" style="font-size:114%;margin:0 4em">C, P, and T symmetries</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C-symmetry" title="C-symmetry">C-symmetry</a></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">P-symmetry</a></li> <li><a class="mw-selflink selflink">T-symmetry</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C_parity" title="C parity">CP</a></li> <li><a href="/wiki/CP_violation#CP-symmetry" title="CP violation">CP symmetry</a></li> <li><a href="/wiki/CPT_symmetry" title="CPT symmetry">CPT symmetry</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chirality_(physics)" title="Chirality (physics)">Chirality</a></li> <li><a href="/wiki/Pin_group" title="Pin group">Pin group</a></li> <li><a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">Symmetry (physics)</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"></div><div role="navigation" class="navbox" aria-labelledby="Time" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Time_topics" title="Template:Time topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Time_topics" title="Template talk:Time topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Time_topics" title="Special:EditPage/Template:Time topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Time" style="font-size:114%;margin:0 4em"><a href="/wiki/Time" title="Time">Time</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Past" title="Past">Past</a></li> <li><a href="/wiki/Present" title="Present">Present</a></li> <li><a href="/wiki/Future" title="Future">Future</a></li> <li><a href="/wiki/Eternity" title="Eternity">Eternity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Horology" class="mw-redirect" title="Horology">Measurement</a><br />and <a href="/wiki/Time_standard" title="Time standard">standards</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Chronometry" title="Chronometry">Chronometry</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinated_Universal_Time" title="Coordinated Universal Time">UTC</a></li> <li><a href="/wiki/Universal_Time" title="Universal Time">UT</a></li> <li><a href="/wiki/International_Atomic_Time" title="International Atomic Time">TAI</a></li> <li><a href="/wiki/Unit_of_time" title="Unit of time">Unit of time</a></li> <li><a href="/wiki/Orders_of_magnitude_(time)" title="Orders of magnitude (time)">Orders of magnitude (time)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/System_of_measurement" class="mw-redirect" title="System of measurement">Measurement<br />systems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Italian_six-hour_clock" title="Italian six-hour clock">Italian six-hour clock</a></li> <li><a href="/wiki/Thai_six-hour_clock" title="Thai six-hour clock">Thai six-hour clock</a></li> <li><a href="/wiki/12-hour_clock" title="12-hour clock">12-hour clock</a></li> <li><a href="/wiki/24-hour_clock" title="24-hour clock">24-hour clock</a></li> <li><a href="/wiki/Relative_hour" title="Relative hour">Relative hour</a></li> <li><a href="/wiki/Daylight_saving_time" title="Daylight saving time">Daylight saving time</a></li> <li><a href="/wiki/Traditional_Chinese_timekeeping" title="Traditional Chinese timekeeping">Chinese</a></li> <li><a href="/wiki/Decimal_time" title="Decimal time">Decimal</a></li> <li><a href="/wiki/Hexadecimal_time" title="Hexadecimal time">Hexadecimal</a></li> <li><a href="/wiki/Hindu_units_of_time" title="Hindu units of time">Hindu</a></li> <li><a href="/wiki/Metric_time" title="Metric time">Metric</a></li> <li><a href="/wiki/Roman_timekeeping" title="Roman timekeeping">Roman</a></li> <li><a href="/wiki/Sidereal_time" title="Sidereal time">Sidereal</a></li> <li><a href="/wiki/Solar_time" title="Solar time">Solar</a></li> <li><a href="/wiki/Time_zone" title="Time zone">Time zone</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Calendar" title="Calendar">Calendars</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calendar#Systems" title="Calendar">Main types</a> <ul><li><a href="/wiki/Solar_calendar" title="Solar calendar">Solar</a></li> <li><a href="/wiki/Lunar_calendar" title="Lunar calendar">Lunar</a></li> <li><a href="/wiki/Lunisolar_calendar" title="Lunisolar calendar">Lunisolar</a></li></ul></li> <li><a href="/wiki/Gregorian_calendar" title="Gregorian calendar">Gregorian</a></li> <li><a href="/wiki/Julian_calendar" title="Julian calendar">Julian</a></li> <li><a href="/wiki/Hebrew_calendar" title="Hebrew calendar">Hebrew</a></li> <li><a href="/wiki/Islamic_calendar" title="Islamic calendar">Islamic</a></li> <li><a href="/wiki/Solar_Hijri_calendar" title="Solar Hijri calendar">Solar Hijri</a></li> <li><a href="/wiki/Chinese_calendar" title="Chinese calendar">Chinese</a></li> <li><a href="/wiki/Hindu_calendar" title="Hindu calendar">Hindu Panchang</a></li> <li><a href="/wiki/Maya_calendar" title="Maya calendar">Maya</a></li> <li><i><a href="/wiki/List_of_calendars" title="List of calendars">List</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Clock" title="Clock">Clocks</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Clock#Types" title="Clock">Main types</a> <ul><li><a href="/wiki/Astronomical_clock" title="Astronomical clock">astronomical</a> <ul><li><a href="/wiki/Astrarium" title="Astrarium">astrarium</a></li></ul></li> <li><a href="/wiki/Atomic_clock" title="Atomic clock">atomic</a> <ul><li><a href="/wiki/Quantum_clock" class="mw-redirect" title="Quantum clock">quantum</a></li></ul></li> <li><a href="/wiki/Hourglass" title="Hourglass">hourglass</a></li> <li><a href="/wiki/Marine_chronometer" title="Marine chronometer">marine</a></li> <li><a href="/wiki/Sundial" title="Sundial">sundial</a></li> <li><a href="/wiki/Watch" title="Watch">watch</a> <ul><li><a href="/wiki/Mechanical_watch" title="Mechanical watch">mechanical</a></li> <li><a href="/wiki/Stopwatch" title="Stopwatch">stopwatch</a></li></ul></li> <li><a href="/wiki/Water_clock" title="Water clock">water-based</a></li></ul></li> <li><a href="/wiki/Cuckoo_clock" title="Cuckoo clock">Cuckoo clock</a></li> <li><a href="/wiki/Digital_clock" title="Digital clock">Digital clock</a></li> <li><a href="/wiki/Grandfather_clock" title="Grandfather clock">Grandfather clock</a></li> <li><i><a href="/wiki/History_of_timekeeping_devices" title="History of timekeeping devices">History</a></i> <ul><li><i><a href="/wiki/Timeline_of_time_measurement_inventions" title="Timeline of time measurement inventions">Timeline</a></i></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Chronology" title="Chronology">Chronology</a></li><li><a href="/wiki/History" title="History">History</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astronomical_chronology" title="Astronomical chronology">Astronomical chronology</a></li> <li><a href="/wiki/Big_History" title="Big History">Big History</a></li> <li><a href="/wiki/Calendar_era" title="Calendar era">Calendar era</a></li> <li><a href="/wiki/Deep_time" title="Deep time">Deep time</a></li> <li><a href="/wiki/Periodization" title="Periodization">Periodization</a></li> <li><a href="/wiki/Regnal_year" title="Regnal year">Regnal year</a></li> <li><a href="/wiki/Timeline" title="Timeline">Timeline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Philosophy_of_space_and_time" title="Philosophy of space and time">Philosophy of time</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/A_series_and_B_series" title="A series and B series">A series and B series</a></li> <li><a href="/wiki/B-theory_of_time" title="B-theory of time">B-theory of time</a></li> <li><a href="/wiki/Chronocentrism" title="Chronocentrism">Chronocentrism</a></li> <li><a href="/wiki/Duration_(philosophy)" title="Duration (philosophy)">Duration</a></li> <li><a href="/wiki/Endurantism" title="Endurantism">Endurantism</a></li> <li><a href="/wiki/Eternal_return" title="Eternal return">Eternal return</a></li> <li><a href="/wiki/Eternalism_(philosophy_of_time)" title="Eternalism (philosophy of time)">Eternalism</a></li> <li><a href="/wiki/Event_(philosophy)" title="Event (philosophy)">Event</a></li> <li><a href="/wiki/Perdurantism" title="Perdurantism">Perdurantism</a></li> <li><a href="/wiki/Philosophical_presentism" title="Philosophical presentism">Presentism</a></li> <li><a href="/wiki/Temporal_finitism" title="Temporal finitism">Temporal finitism</a></li> <li><a href="/wiki/Temporal_parts" title="Temporal parts">Temporal parts</a></li> <li><i><a href="/wiki/The_Unreality_of_Time" title="The Unreality of Time">The Unreality of Time</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Category:Time_in_religion" title="Category:Time in religion">Religion</a></li><li><a href="/wiki/Template:Time_in_religion_and_mythology" title="Template:Time in religion and mythology">Mythology</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ages_of_Man" title="Ages of Man">Ages of Man</a></li> <li><a href="/wiki/Destiny" title="Destiny">Destiny</a></li> <li><a href="/wiki/Immortality" title="Immortality">Immortality</a></li> <li><a href="/wiki/The_Dreaming" title="The Dreaming">Dreamtime</a></li> <li><a href="/wiki/K%C4%81la" title="Kāla">Kāla</a></li> <li><a href="/wiki/Time_and_fate_deities" title="Time and fate deities">Time and fate deities</a> <ul><li><a href="/wiki/Father_Time" title="Father Time">Father Time</a></li></ul></li> <li><a href="/wiki/Wheel_of_time" title="Wheel of time">Wheel of time</a> <ul><li><a href="/wiki/Kalachakra" title="Kalachakra">Kalachakra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_perception" title="Time perception">Human experience</a><br />and <a href="/wiki/Time-use_research" title="Time-use research">use of time</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronemics" title="Chronemics">Chronemics</a></li> <li><a href="/wiki/Generation_time" title="Generation time">Generation time</a></li> <li><a href="/wiki/Mental_chronometry" title="Mental chronometry">Mental chronometry</a></li> <li><a href="/wiki/Duration_(music)" title="Duration (music)">Music</a> <ul><li><a href="/wiki/Tempo" title="Tempo">tempo</a></li> <li><a href="/wiki/Time_signature" title="Time signature">time signature</a></li></ul></li> <li><a href="/wiki/Rosy_retrospection" title="Rosy retrospection">Rosy retrospection</a></li> <li><a href="/wiki/Tense%E2%80%93aspect%E2%80%93mood" title="Tense–aspect–mood">Tense–aspect–mood</a></li> <li><a href="/wiki/Time_management" title="Time management">Time management</a></li> <li><a href="/wiki/Yesterday_(time)" title="Yesterday (time)">Yesterday</a> – <a href="/wiki/Present" title="Present">Today</a> – <a href="/wiki/Tomorrow_(time)" title="Tomorrow (time)">Tomorrow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Time in <a href="/wiki/Science" title="Science">science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Geology" title="Geology">Geology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Geologic_time_scale" title="Geologic time scale">Geological time</a> <ul><li><a href="/wiki/Age_(geology)" class="mw-redirect" title="Age (geology)">age</a></li> <li><a href="/wiki/Chronozone" title="Chronozone">chron</a></li> <li><a href="/wiki/Eon_(geology)" class="mw-redirect" title="Eon (geology)">eon</a></li> <li><a href="/wiki/Epoch_(geology)" class="mw-redirect" title="Epoch (geology)">epoch</a></li> <li><a href="/wiki/Era_(geology)" class="mw-redirect" title="Era (geology)">era</a></li> <li><a href="/wiki/Geological_period" class="mw-redirect" title="Geological period">period</a></li></ul></li> <li><a href="/wiki/Geochronology" title="Geochronology">Geochronology</a></li> <li><a href="/wiki/Geological_history_of_Earth" title="Geological history of Earth">Geological history of Earth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Time_in_physics" title="Time in physics">Physics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_space_and_time" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Arrow_of_time" title="Arrow of time">Arrow of time</a></li> <li><a href="/wiki/Chronon" title="Chronon">Chronon</a></li> <li><a href="/wiki/Coordinate_time" title="Coordinate time">Coordinate time</a></li> <li><a href="/wiki/Instant" title="Instant">Instant</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></li> <li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a></li> <li><a href="/wiki/Time_domain" title="Time domain">Time domain</a></li> <li><a href="/wiki/Time_translation_symmetry" class="mw-redirect" title="Time translation symmetry">Time translation symmetry</a></li> <li><a class="mw-selflink selflink">Time reversal symmetry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;">Other fields</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronological_dating" title="Chronological dating">Chronological dating</a></li> <li><a href="/wiki/Chronobiology" title="Chronobiology">Chronobiology</a> <ul><li><a href="/wiki/Circadian_rhythm" title="Circadian rhythm">Circadian rhythms</a></li></ul></li> <li><a href="/wiki/Chemical_clock" title="Chemical clock">Clock reaction</a></li> <li><a href="/wiki/Glottochronology" title="Glottochronology">Glottochronology</a></li> <li><a href="/wiki/Time_geography" title="Time geography">Time geography</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Leap_year" title="Leap year">Leap year</a></li> <li><a href="/wiki/Memory" title="Memory">Memory</a></li> <li><a href="/wiki/Moment_(unit)" title="Moment (unit)">Moment</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/System_time" title="System time">System time</a></li> <li><i><a href="/wiki/Tempus_fugit" title="Tempus fugit">Tempus fugit</a></i></li> <li><a href="/wiki/Time_capsule" title="Time capsule">Time capsule</a></li> <li><a href="/wiki/Time_immemorial" title="Time immemorial">Time immemorial</a></li> <li><a href="/wiki/Time_travel" title="Time travel">Time travel</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Time" title="Category:Time">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a href="https://commons.wikimedia.org/wiki/Category:Time" class="extiw" title="commons:Category:Time">Commons</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Time_measurement_and_standards" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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:Time_measurement_and_standards" title="Template:Time measurement and standards"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Time_measurement_and_standards" title="Template talk:Time measurement and standards"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Time_measurement_and_standards" title="Special:EditPage/Template:Time measurement and standards"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Time_measurement_and_standards" style="font-size:114%;margin:0 4em"><a href="/wiki/Time" title="Time">Time measurement</a> and <a href="/wiki/Time_standard" title="Time standard">standards</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Chronometry" title="Chronometry">Chronometry</a></li> <li><a href="/wiki/Orders_of_magnitude_(time)" title="Orders of magnitude (time)">Orders of magnitude</a></li> <li><a href="/wiki/Time_metrology" class="mw-redirect" title="Time metrology">Metrology</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">International standards</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinated_Universal_Time" title="Coordinated Universal Time">Coordinated Universal Time</a> <ul><li><a href="/wiki/UTC_offset" title="UTC offset">offset</a></li></ul></li> <li><a href="/wiki/Universal_Time" title="Universal Time">UT</a></li> <li><a href="/wiki/%CE%94T_(timekeeping)" title="ΔT (timekeeping)">ΔT</a></li> <li><a href="/wiki/DUT1" title="DUT1">DUT1</a></li> <li><a href="/wiki/International_Earth_Rotation_and_Reference_Systems_Service" title="International Earth Rotation and Reference Systems Service">International Earth Rotation and Reference Systems Service</a></li> <li><a href="/wiki/ISO_31-1" title="ISO 31-1">ISO 31-1</a></li> <li><a href="/wiki/ISO_8601" title="ISO 8601">ISO 8601</a></li> <li><a href="/wiki/International_Atomic_Time" title="International Atomic Time">International Atomic Time</a></li> <li><a href="/wiki/12-hour_clock" title="12-hour clock">12-hour clock</a></li> <li><a href="/wiki/24-hour_clock" title="24-hour clock">24-hour clock</a></li> <li><a href="/wiki/Barycentric_Coordinate_Time" title="Barycentric Coordinate Time">Barycentric Coordinate Time</a></li> <li><a href="/wiki/Barycentric_Dynamical_Time" title="Barycentric Dynamical Time">Barycentric Dynamical Time</a></li> <li><a href="/wiki/Civil_time" title="Civil time">Civil time</a></li> <li><a href="/wiki/Daylight_saving_time" title="Daylight saving time">Daylight saving time</a></li> <li><a href="/wiki/Geocentric_Coordinate_Time" title="Geocentric Coordinate Time">Geocentric Coordinate Time</a></li> <li><a href="/wiki/International_Date_Line" title="International Date Line">International Date Line</a></li> <li><a href="/wiki/IERS_Reference_Meridian" title="IERS Reference Meridian">IERS Reference Meridian</a></li> <li><a href="/wiki/Leap_second" title="Leap second">Leap second</a></li> <li><a href="/wiki/Solar_time" title="Solar time">Solar time</a></li> <li><a href="/wiki/Terrestrial_Time" title="Terrestrial Time">Terrestrial Time</a></li> <li><a href="/wiki/Time_zone" title="Time zone">Time zone</a></li> <li><a href="/wiki/180th_meridian" title="180th meridian">180th meridian</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="9" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Hourglass" title="Hourglass"><img alt="template illustration" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Marine_sandglass_MMM.jpg/75px-Marine_sandglass_MMM.jpg" decoding="async" width="75" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Marine_sandglass_MMM.jpg/113px-Marine_sandglass_MMM.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Marine_sandglass_MMM.jpg/150px-Marine_sandglass_MMM.jpg 2x" data-file-width="1637" data-file-height="3819" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Time_zone" title="Time zone"><img alt="template illustration" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png/75px-Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png" decoding="async" width="75" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png/113px-Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png/150px-Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png 2x" data-file-width="496" data-file-height="1007" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Obsolete standards</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ephemeris_time" title="Ephemeris time">Ephemeris time</a></li> <li><a href="/wiki/Greenwich_Mean_Time" title="Greenwich Mean Time">Greenwich Mean Time</a></li> <li><a href="/wiki/Prime_meridian" title="Prime meridian">Prime meridian</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_in_physics" title="Time in physics">Time in physics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_space_and_time" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></li> <li><a href="/wiki/Chronon" title="Chronon">Chronon</a></li> <li><a href="/wiki/Continuous_signal" class="mw-redirect" title="Continuous signal">Continuous signal</a></li> <li><a href="/wiki/Coordinate_time" title="Coordinate time">Coordinate time</a></li> <li><a href="/wiki/Cosmological_decade" title="Cosmological decade">Cosmological decade</a></li> <li><a href="/wiki/Discrete_time_and_continuous_time" title="Discrete time and continuous time">Discrete time and continuous time</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a></li> <li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li> <li><a href="/wiki/Time_domain" title="Time domain">Time domain</a></li> <li><a href="/wiki/Time-translation_symmetry" title="Time-translation symmetry">Time-translation symmetry</a></li> <li><a class="mw-selflink selflink">T-symmetry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Horology" class="mw-redirect" title="Horology">Horology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Clock" title="Clock">Clock</a></li> <li><a href="/wiki/Astrarium" title="Astrarium">Astrarium</a></li> <li><a href="/wiki/Atomic_clock" title="Atomic clock">Atomic clock</a></li> <li><a href="/wiki/Complication_(horology)" title="Complication (horology)">Complication</a></li> <li><a href="/wiki/History_of_timekeeping_devices" title="History of timekeeping devices">History of timekeeping devices</a></li> <li><a href="/wiki/Hourglass" title="Hourglass">Hourglass</a></li> <li><a href="/wiki/Marine_chronometer" title="Marine chronometer">Marine chronometer</a></li> <li><a href="/wiki/Marine_sandglass" title="Marine sandglass">Marine sandglass</a></li> <li><a href="/wiki/Radio_clock" title="Radio clock">Radio clock</a></li> <li><a href="/wiki/Watch" title="Watch">Watch</a> <ul><li><a href="/wiki/Stopwatch" title="Stopwatch">stopwatch</a></li></ul></li> <li><a href="/wiki/Water_clock" title="Water clock">Water clock</a></li> <li><a href="/wiki/Sundial" title="Sundial">Sundial</a></li> <li><a href="/wiki/Dialing_scales" title="Dialing scales">Dialing scales</a></li> <li><a href="/wiki/Equation_of_time" title="Equation of time">Equation of time</a></li> <li><a href="/wiki/History_of_sundials" title="History of sundials">History of sundials</a></li> <li><a href="/wiki/Schema_for_horizontal_dials" title="Schema for horizontal dials">Sundial markup schema</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Calendar" title="Calendar">Calendar</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gregorian_calendar" title="Gregorian calendar">Gregorian</a></li> <li><a href="/wiki/Hebrew_calendar" title="Hebrew calendar">Hebrew</a></li> <li><a href="/wiki/Hindu_calendar" title="Hindu calendar">Hindu</a></li> <li><a href="/wiki/Holocene_calendar" title="Holocene calendar">Holocene</a></li> <li><a href="/wiki/Islamic_calendar" title="Islamic calendar">Islamic</a> (lunar Hijri)</li> <li><a href="/wiki/Julian_calendar" title="Julian calendar">Julian</a></li> <li><a href="/wiki/Solar_Hijri_calendar" title="Solar Hijri calendar">Solar Hijri</a></li> <li><a href="/wiki/Astronomical_year_numbering" title="Astronomical year numbering">Astronomical</a></li> <li><a href="/wiki/Dominical_letter" title="Dominical letter">Dominical letter</a></li> <li><a href="/wiki/Epact" title="Epact">Epact</a></li> <li><a href="/wiki/Equinox" title="Equinox">Equinox</a></li> <li><a href="/wiki/Intercalation_(timekeeping)" title="Intercalation (timekeeping)">Intercalation</a></li> <li><a href="/wiki/Julian_day" title="Julian day">Julian day</a></li> <li><a href="/wiki/Leap_year" title="Leap year">Leap year</a></li> <li><a href="/wiki/Lunar_calendar" title="Lunar calendar">Lunar</a></li> <li><a href="/wiki/Lunisolar_calendar" title="Lunisolar calendar">Lunisolar</a></li> <li><a href="/wiki/Solar_calendar" title="Solar calendar">Solar</a></li> <li><a href="/wiki/Solstice" title="Solstice">Solstice</a></li> <li><a href="/wiki/Tropical_year" title="Tropical year">Tropical year</a></li> <li><a href="/wiki/Determination_of_the_day_of_the_week" title="Determination of the day of the week">Weekday determination</a></li> <li><a href="/wiki/Names_of_the_days_of_the_week" title="Names of the days of the week">Weekday names</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Archaeology and geology</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronological_dating" title="Chronological dating">Chronological dating</a></li> <li><a href="/wiki/Geologic_time_scale" title="Geologic time scale">Geologic time scale</a></li> <li><a href="/wiki/International_Commission_on_Stratigraphy" title="International Commission on Stratigraphy">International Commission on Stratigraphy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Astronomical_chronology" title="Astronomical chronology">Astronomical chronology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Galactic_year" title="Galactic year">Galactic year</a></li> <li><a href="/wiki/Nuclear_timescale" title="Nuclear timescale">Nuclear timescale</a></li> <li><a href="/wiki/Precession" title="Precession">Precession</a></li> <li><a href="/wiki/Sidereal_time" title="Sidereal time">Sidereal time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Unit_of_time" title="Unit of time">units of time</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Instant" title="Instant">Instant</a></li> <li><a href="/wiki/Flick_(time)" title="Flick (time)">Flick</a></li> <li><a href="/wiki/Shake_(unit)" title="Shake (unit)">Shake</a></li> <li><a href="/wiki/Jiffy_(time)" title="Jiffy (time)">Jiffy</a></li> <li><a href="/wiki/Second" title="Second">Second</a></li> <li><a href="/wiki/Minute" title="Minute">Minute</a></li> <li><a href="/wiki/Moment_(unit)" title="Moment (unit)">Moment</a></li> <li><a href="/wiki/Hour" title="Hour">Hour</a></li> <li><a href="/wiki/Day" title="Day">Day</a></li> <li><a href="/wiki/Week" title="Week">Week</a></li> <li><a href="/wiki/Fortnight" title="Fortnight">Fortnight</a></li> <li><a href="/wiki/Month" title="Month">Month</a></li> <li><a href="/wiki/Year" title="Year">Year</a></li> <li><a href="/wiki/Olympiad" title="Olympiad">Olympiad</a></li> <li><a href="/wiki/Lustrum" title="Lustrum">Lustrum</a></li> <li><a href="/wiki/Decade" title="Decade">Decade</a></li> <li><a href="/wiki/Century" title="Century">Century</a></li> <li><a href="/wiki/Saeculum" title="Saeculum">Saeculum</a></li> <li><a href="/wiki/Millennium" title="Millennium">Millennium</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronology" title="Chronology">Chronology</a></li> <li><a href="/wiki/Duration_(philosophy)" title="Duration (philosophy)">Duration</a> <ul><li><a href="/wiki/Duration_(music)" title="Duration (music)">music</a></li></ul></li> <li><a href="/wiki/Mental_chronometry" title="Mental chronometry">Mental chronometry</a></li> <li><a href="/wiki/Decimal_time" title="Decimal time">Decimal time</a></li> <li><a href="/wiki/Metric_time" title="Metric time">Metric time</a></li> <li><a href="/wiki/System_time" title="System time">System time</a></li> <li><a href="/wiki/Time_metrology" class="mw-redirect" title="Time metrology">Time metrology</a></li> <li><a href="/wiki/Time_value_of_money" title="Time value of money">Time value of money</a></li> <li><a href="/wiki/Timekeeper" title="Timekeeper">Timekeeper</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q142270#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" 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