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class="vector-toc-numb">2</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Possible_causes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Possible_causes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Possible causes</span> </div> </a> <button aria-controls="toc-Possible_causes-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 Possible causes subsection</span> </button> <ul id="toc-Possible_causes-sublist" class="vector-toc-list"> <li id="toc-Vacuum_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vacuum_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Vacuum energy</span> </div> </a> <ul id="toc-Vacuum_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_van_der_Waals_force" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_van_der_Waals_force"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Relativistic van der Waals force</span> </div> </a> <ul id="toc-Relativistic_van_der_Waals_force-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Effects" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Effects"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Effects</span> </div> </a> <ul id="toc-Effects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_of_Casimir_effect_assuming_zeta-regularization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Derivation_of_Casimir_effect_assuming_zeta-regularization"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Derivation of Casimir effect assuming zeta-regularization</span> </div> </a> <button aria-controls="toc-Derivation_of_Casimir_effect_assuming_zeta-regularization-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 Derivation of Casimir effect assuming zeta-regularization subsection</span> </button> <ul id="toc-Derivation_of_Casimir_effect_assuming_zeta-regularization-sublist" class="vector-toc-list"> <li id="toc-More_recent_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#More_recent_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>More recent theory</span> </div> </a> <ul id="toc-More_recent_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Measurement" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Measurement</span> </div> </a> <ul id="toc-Measurement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regularization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Regularization"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Regularization</span> </div> </a> <ul id="toc-Regularization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalities" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalities"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Generalities</span> </div> </a> <ul id="toc-Generalities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dynamical_Casimir_effect" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dynamical_Casimir_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Dynamical Casimir effect</span> </div> </a> <ul id="toc-Dynamical_Casimir_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Repulsive_forces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Repulsive_forces"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Repulsive forces</span> </div> </a> <ul id="toc-Repulsive_forces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Speculative_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Speculative_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Speculative applications</span> </div> </a> <ul id="toc-Speculative_applications-sublist" class="vector-toc-list"> </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">12</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">13</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Further reading</span> </div> </a> <button aria-controls="toc-Further_reading-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 Further reading subsection</span> </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Introductory_readings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Introductory_readings"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.1</span> <span>Introductory readings</span> </div> </a> <ul id="toc-Introductory_readings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Papers,_books_and_lectures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Papers,_books_and_lectures"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.2</span> <span>Papers, books and lectures</span> </div> </a> <ul id="toc-Papers,_books_and_lectures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Temperature_dependence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Temperature_dependence"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.3</span> <span>Temperature dependence</span> </div> </a> <ul id="toc-Temperature_dependence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Casimir effect</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 37 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-37" 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">37 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%A3%D8%AB%D9%8A%D8%B1_%D9%83%D8%A7%D8%B2%D9%8A%D9%85%D9%8A%D8%B1" title="تأثير كازيمير – Arabic" lang="ar" hreflang="ar" data-title="تأثير كازيمير" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D1%84%D0%B5%D0%BA%D1%82_%D0%9A%D0%B0%D0%B7%D1%96%D0%BC%D1%96%D1%80%D0%B0" title="Эфект Казіміра – Belarusian" lang="be" hreflang="be" data-title="Эфект Казіміра" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D1%84%D0%B5%D0%BA%D1%82_%D0%BD%D0%B0_%D0%9A%D0%B0%D0%B7%D0%B8%D0%BC%D0%B8%D1%80" title="Ефект на Казимир – Bulgarian" lang="bg" hreflang="bg" data-title="Ефект на Казимир" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Efecte_Casimir" title="Efecte Casimir – Catalan" lang="ca" hreflang="ca" data-title="Efecte Casimir" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Casimir%C5%AFv_jev" title="Casimirův jev – Czech" lang="cs" hreflang="cs" data-title="Casimirův jev" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Casimir-effekten" title="Casimir-effekten – Danish" lang="da" hreflang="da" data-title="Casimir-effekten" 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/Casimir-Effekt" title="Casimir-Effekt – German" lang="de" hreflang="de" data-title="Casimir-Effekt" 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/Efecto_Casimir" title="Efecto Casimir – Spanish" lang="es" hreflang="es" data-title="Efecto Casimir" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Efiko_de_Casimir" title="Efiko de Casimir – Esperanto" lang="eo" hreflang="eo" data-title="Efiko de Casimir" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Casimir_efektua" title="Casimir efektua – Basque" lang="eu" hreflang="eu" data-title="Casimir efektua" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AB%D8%B1_%DA%A9%D8%A7%D8%B3%DB%8C%D9%85%DB%8C%D8%B1" title="اثر کاسیمیر – Persian" lang="fa" hreflang="fa" data-title="اثر کاسیمیر" 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/Effet_Casimir" title="Effet Casimir – French" lang="fr" hreflang="fr" data-title="Effet Casimir" 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%B9%B4%EC%8B%9C%EB%AF%B8%EB%A5%B4_%ED%9A%A8%EA%B3%BC" title="카시미르 효과 – Korean" lang="ko" hreflang="ko" data-title="카시미르 효과" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%A1%D5%A6%D5%AB%D5%B4%D5%AB%D6%80%D5%AB_%D5%A7%D6%86%D5%A5%D5%AF%D5%BF" title="Կազիմիրի էֆեկտ – Armenian" lang="hy" hreflang="hy" data-title="Կազիմիրի էֆեկտ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Kas%C3%ADm%C3%ADr_hrif" title="Kasímír hrif – Icelandic" lang="is" hreflang="is" data-title="Kasímír hrif" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Effetto_Casimir" title="Effetto Casimir – Italian" lang="it" hreflang="it" data-title="Effetto Casimir" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A4%D7%A7%D7%98_%D7%A7%D7%96%D7%99%D7%9E%D7%99%D7%A8" title="אפקט קזימיר – Hebrew" lang="he" hreflang="he" data-title="אפקט קזימיר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B2%BE%E0%B2%B8%E0%B2%BF%E0%B2%AE%E0%B2%BF%E0%B2%B0%E0%B3%8D_%E0%B2%AA%E0%B2%B0%E0%B2%BF%E0%B2%A3%E0%B2%BE%E0%B2%AE" title="ಕಾಸಿಮಿರ್ ಪರಿಣಾಮ – Kannada" lang="kn" hreflang="kn" data-title="ಕಾಸಿಮಿರ್ ಪರಿಣಾಮ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kazimira_efekts" title="Kazimira efekts – Latvian" lang="lv" hreflang="lv" data-title="Kazimira efekts" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Casimir-effektus" title="Casimir-effektus – Hungarian" lang="hu" hreflang="hu" data-title="Casimir-effektus" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%B0%D0%B7%D0%B8%D0%BC%D0%B8%D1%80%D0%BE%D0%B2_%D0%B5%D1%84%D0%B5%D0%BA%D1%82" title="Казимиров ефект – Macedonian" lang="mk" hreflang="mk" data-title="Казимиров ефект" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Casimireffect" title="Casimireffect – Dutch" lang="nl" hreflang="nl" data-title="Casimireffect" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AB%E3%82%B7%E3%83%9F%E3%83%BC%E3%83%AB%E5%8A%B9%E6%9E%9C" title="カシミール効果 – Japanese" lang="ja" hreflang="ja" data-title="カシミール効果" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Efekt_Casimira" title="Efekt Casimira – Polish" lang="pl" hreflang="pl" data-title="Efekt Casimira" 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/Efeito_Casimir" title="Efeito Casimir – Portuguese" lang="pt" hreflang="pt" data-title="Efeito Casimir" 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/Efectul_Casimir" title="Efectul Casimir – Romanian" lang="ro" hreflang="ro" data-title="Efectul Casimir" 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/%D0%AD%D1%84%D1%84%D0%B5%D0%BA%D1%82_%D0%9A%D0%B0%D0%B7%D0%B8%D0%BC%D0%B8%D1%80%D0%B0" title="Эффект Казимира – Russian" lang="ru" hreflang="ru" data-title="Эффект Казимира" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Efekti_i_Kazimirit" title="Efekti i Kazimirit – Albanian" lang="sq" hreflang="sq" data-title="Efekti i Kazimirit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Casimir_effect" title="Casimir effect – Simple English" lang="en-simple" hreflang="en-simple" data-title="Casimir effect" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Casimirov_jav" title="Casimirov jav – Slovak" lang="sk" hreflang="sk" data-title="Casimirov jav" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Casimirjev_pojav" title="Casimirjev pojav – Slovenian" lang="sl" hreflang="sl" data-title="Casimirjev pojav" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Casimirin_ilmi%C3%B6" title="Casimirin ilmiö – Finnish" lang="fi" hreflang="fi" data-title="Casimirin ilmiö" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Casimireffekten" title="Casimireffekten – Swedish" lang="sv" hreflang="sv" data-title="Casimireffekten" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D0%B0%D0%B7%D0%B8%D0%BC%D0%B8%D1%80_%D1%8D%D1%84%D1%84%D0%B5%D0%BA%D1%82%D1%8B" title="Казимир эффекты – Tatar" lang="tt" hreflang="tt" data-title="Казимир эффекты" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Casimir_kuvveti" title="Casimir kuvveti – Turkish" lang="tr" 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:19.0em;"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_number" title="Quantum number">Quantum number</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">State</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell's inequality</a></li> <li><a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li></ul> </div> <ul><li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a> <ul><li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Overview</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced topics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Blackett</a></li> <li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">Bohm</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Davisson</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Hans_Kramers" title="Hans Kramers">Kramers</a></li> <li><a href="/wiki/Willis_Lamb" title="Willis Lamb">Lamb</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Isidor_Isaac_Rabi" title="Isidor Isaac Rabi">Rabi</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Michelle_Simmons" title="Michelle Simmons">Simmons</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics" title="Template:Quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics" title="Template talk:Quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics" title="Special:EditPage/Template:Quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, the <b>Casimir effect</b> (or <b>Casimir force</b>)<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a physical <a href="/wiki/Force_(physics)" class="mw-redirect" title="Force (physics)">force</a> acting on the macroscopic boundaries of a confined space which arises from the <a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">quantum fluctuations</a> of a <a href="/wiki/Field_(physics)" title="Field (physics)">field</a>. The term <b>Casimir pressure</b> is sometimes used when it is described in units of force per unit area.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It is named after the Dutch physicist <a href="/wiki/Hendrik_Casimir" title="Hendrik Casimir">Hendrik Casimir</a>, who predicted the effect for <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetic</a> systems in 1948. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm/220px--Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="165" data-durationhint="31" data-mwtitle="Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2f/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="640" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2f/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="960" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2f/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1440" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/2/2f/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm" type="video/webm; codecs="vp9"" data-width="2048" data-height="1536" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2f/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="192" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2f/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="320" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2f/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="480" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2f/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm/Ag_Micromirrors_in_solution_plasmonics_Casimir_effect_Brownian_motion.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="480" data-height="360" /></video></span><figcaption>Video of silver micromirrors in solution under optical darkfield microscope demonstrating Brownian motion, Casimir effect and colorful scattering of surface plasmons</figcaption></figure> <p>In the same year Casimir, together with <a href="/wiki/Dirk_Polder" title="Dirk Polder">Dirk Polder</a>, described a similar effect experienced by a neutral atom in the vicinity of a macroscopic interface which is called the Casimir–Polder force.<sup id="cite_ref-:3_4-0" class="reference"><a href="#cite_note-:3-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Their result is a generalization of the <a href="/wiki/London_dispersion_force" title="London dispersion force">London</a>–<a href="/wiki/Van_der_Waals_force" title="Van der Waals force">van der Waals force</a> and includes <a href="/wiki/Retarded_potential" title="Retarded potential">retardation</a> due to the finite <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. The fundamental principles leading to the London–van der Waals force, the Casimir force, and the Casimir–Polder force can be formulated on the same footing.<sup id="cite_ref-:2_5-0" class="reference"><a href="#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In 1997 a direct experiment by Steven K. Lamoreaux quantitatively measured the Casimir force to be within 5% of the value predicted by the theory.<sup id="cite_ref-Lamoureaux1997_7-0" class="reference"><a href="#cite_note-Lamoureaux1997-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The Casimir effect can be understood by the idea that the presence of macroscopic material interfaces, such as <a href="/wiki/Electrical_conductor" title="Electrical conductor">electrical conductors</a> and <a href="/wiki/Dielectric" title="Dielectric">dielectrics</a>, alter the <a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">vacuum expectation value</a> of the energy of the <a href="/wiki/Second_quantization" title="Second quantization">second-quantized</a> <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>.<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><sup id="cite_ref-Mohideen_9-0" class="reference"><a href="#cite_note-Mohideen-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Since the value of this energy depends on the shapes and positions of the materials, the Casimir effect manifests itself as a force between such objects. </p><p>Any <a href="/wiki/Transmission_medium" title="Transmission medium">medium</a> supporting <a href="/wiki/Oscillation" title="Oscillation">oscillations</a> has an analogue of the Casimir effect. For example, beads on a string<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> as well as plates submerged in turbulent water<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> or gas<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> illustrate the Casimir force. </p><p>In modern <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>, the Casimir effect plays an important role in the <a href="/wiki/Nucleon#Chiral_bag_model" title="Nucleon">chiral bag model</a> of the <a href="/wiki/Nucleon" title="Nucleon">nucleon</a>; in <a href="/wiki/Applied_physics" title="Applied physics">applied physics</a> it is significant in some aspects of emerging <a href="/wiki/Microtechnologies" class="mw-redirect" title="Microtechnologies">microtechnologies</a> and <a href="/wiki/Nanotechnologies" class="mw-redirect" title="Nanotechnologies">nanotechnologies</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Physical_properties">Physical properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=1" title="Edit section: Physical properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The typical example is of two <a href="/wiki/Electric_charge" title="Electric charge">uncharged</a> conductive plates in a <a href="/wiki/Vacuum" title="Vacuum">vacuum</a>, placed a few nanometers apart. In a <a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">classical</a> description, the lack of an external field means that no field exists between the plates, and no force connects them.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> When this field is instead studied using the <a href="/wiki/QED_vacuum" title="QED vacuum">quantum electrodynamic vacuum</a>, it is seen that the plates do affect the <a href="/wiki/Virtual_particle" title="Virtual particle">virtual photons</a> that constitute the field, and generate a net force<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> – either an attraction or a repulsion depending on the plates' specific arrangement. Although the Casimir effect can be expressed in terms of virtual particles interacting with the objects, it is best described and more easily calculated in terms of the <a href="/wiki/Zero-point_energy" title="Zero-point energy">zero-point energy</a> of a <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantized field</a> in the intervening space between the objects. This force has been measured and is a striking example of an effect captured formally by <a href="/wiki/Second_quantization" title="Second quantization">second quantization</a>.<sup id="cite_ref-Physics_World_17-0" class="reference"><a href="#cite_note-Physics_World-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The treatment of boundary conditions in these calculations is controversial. In fact, "Casimir's original goal was to compute the <a href="/wiki/Van_der_Waals_force" title="Van der Waals force">van der Waals force</a> between <a href="/wiki/Dipolar_polarization" class="mw-redirect" title="Dipolar polarization">polarizable molecules</a>" of the conductive plates. Thus it can be interpreted without any reference to the zero-point energy (vacuum energy) of quantum fields.<sup id="cite_ref-:0_19-0" class="reference"><a href="#cite_note-:0-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Because the strength of the force falls off rapidly with distance, it is measurable only when the distance between the objects is small. This force becomes so strong that it becomes the dominant force between uncharged conductors at submicron scales. In fact, at separations of 10 nm – about 100 times the typical size of an atom – the Casimir effect produces the equivalent of about 1 <a href="/wiki/Atmosphere_(unit)" class="mw-redirect" title="Atmosphere (unit)">atmosphere of pressure</a> (the precise value depends on surface geometry and other factors).<sup id="cite_ref-Physics_World_17-1" class="reference"><a href="#cite_note-Physics_World-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Netherlands" title="Netherlands">Dutch</a> physicists Hendrik Casimir and <a href="/wiki/Dirk_Polder" title="Dirk Polder">Dirk Polder</a> at <a href="/wiki/Philips_Natuurkundig_Laboratorium" title="Philips Natuurkundig Laboratorium">Philips Research Labs</a> proposed the existence of a force between two polarizable atoms and between such an atom and a conducting plate in 1947;<sup id="cite_ref-:3_4-1" class="reference"><a href="#cite_note-:3-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> this special form is called the Casimir–Polder force. After a conversation with <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a>, who suggested it had something to do with zero-point energy, Casimir alone formulated the theory predicting a force between neutral conducting plates in 1948.<sup id="cite_ref-:1_20-0" class="reference"><a href="#cite_note-:1-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> This latter phenomenon is called the Casimir effect. </p><p>Predictions of the force were later extended to finite-conductivity metals and dielectrics, while later calculations considered more general geometries. Experiments before 1997 observed the force qualitatively, and indirect validation of the predicted Casimir energy was made by measuring the thickness of <a href="/wiki/Liquid_helium" title="Liquid helium">liquid helium</a> films. Finally, in 1997 Lamoreaux's direct experiment quantitatively measured the force to within 5% of the value predicted by the theory.<sup id="cite_ref-Lamoureaux1997_7-1" class="reference"><a href="#cite_note-Lamoureaux1997-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Subsequent experiments approached an accuracy of a few percent. </p> <div class="mw-heading mw-heading2"><h2 id="Possible_causes">Possible causes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=3" title="Edit section: Possible causes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Vacuum_energy">Vacuum energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=4" title="Edit section: Vacuum energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert-image" typeof="mw:File/Frameless"><a href="/wiki/Feynman_diagram" title="Feynman diagram"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/211px-Feynmann_Diagram_Gluon_Radiation.svg.png" decoding="async" width="211" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/317px-Feynmann_Diagram_Gluon_Radiation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/422px-Feynmann_Diagram_Gluon_Radiation.svg.png 2x" data-file-width="400" data-file-height="250" /></a></span><div class="sidebar-caption"><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></div></td></tr><tr><td class="sidebar-above"> <a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Field_(physics)" title="Field (physics)">Field theory</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">Weak force</a></li> <li><a href="/wiki/Strong_force" class="mw-redirect" title="Strong force">Strong force</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">Symmetries</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">C-symmetry</a></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">P-symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a></li> <li><a href="/wiki/Lorentz_symmetry" class="mw-redirect" title="Lorentz symmetry">Lorentz symmetry</a></li> <li><a href="/wiki/Poincar%C3%A9_symmetry" class="mw-redirect" title="Poincaré symmetry">Poincaré symmetry</a></li> <li><a href="/wiki/Gauge_symmetry_(mathematics)" title="Gauge symmetry (mathematics)">Gauge symmetry</a></li> <li><a href="/wiki/Explicit_symmetry_breaking" title="Explicit symmetry breaking">Explicit symmetry breaking</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Noether_charge" class="mw-redirect" title="Noether charge">Noether charge</a></li> <li><a href="/wiki/Topological_charge" class="mw-redirect" title="Topological charge">Topological charge</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Tools</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomaly</a></li> <li><a href="/wiki/Background_field_method" title="Background field method">Background field method</a></li> <li><a href="/wiki/BRST_quantization" title="BRST quantization">BRST quantization</a></li> <li><a href="/wiki/Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">Correlation function</a></li> <li><a href="/wiki/Crossing_(physics)" title="Crossing (physics)">Crossing</a></li> <li><a href="/wiki/Effective_action" title="Effective action">Effective action</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective field theory</a></li> <li><a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">Expectation value</a></li> <li><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></li> <li><a href="/wiki/Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li> <li><a href="/wiki/LSZ_reduction_formula" title="LSZ reduction formula">LSZ reduction formula</a></li> <li><a href="/wiki/Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">Partition function</a></li> <li><a href="/wiki/Path_Integral_Formulation" class="mw-redirect" title="Path Integral Formulation">Path Integral Formulation</a></li> <li><a href="/wiki/Propagator_(Quantum_Theory)" class="mw-redirect" title="Propagator (Quantum Theory)">Propagator</a></li> <li><a href="/wiki/Quantization_(physics)" title="Quantization (physics)">Quantization</a></li> <li><a href="/wiki/Regularization_(physics)" title="Regularization (physics)">Regularization</a></li> <li><a href="/wiki/Renormalization" title="Renormalization">Renormalization</a></li> <li><a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">Vacuum state</a></li> <li><a href="/wiki/Wick%27s_theorem" title="Wick's theorem">Wick's theorem</a></li> <li><a href="/w/index.php?title=Wightman_Axioms&action=edit&redlink=1" class="new" title="Wightman Axioms (page does not exist)">Wightman Axioms</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a></li> <li><a href="/wiki/Proca_action" title="Proca action">Proca equations</a></li> <li><a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></li> <li><a href="/wiki/Bargmann%E2%80%93Wigner_equations" title="Bargmann–Wigner equations">Bargmann–Wigner equations</a></li> <li><a href="/wiki/Schwinger-Dyson_equation" class="mw-redirect" title="Schwinger-Dyson equation">Schwinger-Dyson equation</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group equation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Incomplete theories</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></li> <li><a href="/wiki/Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Stephen_Louis_Adler" class="mw-redirect" title="Stephen Louis Adler">Adler</a></li> <li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li> <li><a href="/wiki/Alexey_Andreevich_Anselm" class="mw-redirect" title="Alexey Andreevich Anselm">Anselm</a></li> <li><a href="/wiki/Valentine_Bargmann" title="Valentine Bargmann">Bargmann</a></li> <li><a href="/wiki/Carlo_Becchi" title="Carlo Becchi">Becchi</a></li> <li><a href="/wiki/Alexander_Belavin" title="Alexander Belavin">Belavin</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Felix_Berezin" title="Felix Berezin">Berezin</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/James_Bjorken" title="James Bjorken">Bjorken</a></li> <li><a href="/wiki/Konrad_Bleuler" title="Konrad Bleuler">Bleuer</a></li> <li><a href="/wiki/Nikolay_Bogolyubov" title="Nikolay Bogolyubov">Bogoliubov</a></li> <li><a href="/wiki/Stanley_Brodsky" title="Stanley Brodsky">Brodsky</a></li> <li><a href="/wiki/Robert_Brout" title="Robert Brout">Brout</a></li> <li><a href="/wiki/Detlev_Buchholz" title="Detlev Buchholz">Buchholz</a></li> <li><a href="/wiki/Freddy_Cachazo" title="Freddy Cachazo">Cachazo</a></li> <li><a href="/wiki/Curtis_Callan" title="Curtis Callan">Callan</a></li> <li><a href="/wiki/Sidney_Coleman" title="Sidney Coleman">Coleman</a></li> <li><a href="/wiki/Alain_Connes" title="Alain Connes">Connes</a></li> <li><a href="/wiki/Roger_Dashen" title="Roger Dashen">Dashen</a></li> <li><a href="/wiki/Bryce_DeWitt" title="Bryce DeWitt">DeWitt</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Sergio_Doplicher" title="Sergio Doplicher">Doplicher</a></li> <li><a href="/wiki/Freeman_Dyson" title="Freeman Dyson">Dyson</a></li> <li><a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">Englert</a></li> <li><a href="/wiki/Ludvig_Faddeev" title="Ludvig Faddeev">Faddeev</a></li> <li><a href="/wiki/Victor_Sergeevich_Fadin" class="mw-redirect" title="Victor Sergeevich Fadin">Fadin</a></li> <li><a href="/wiki/Pierre_Fayet" title="Pierre Fayet">Fayet</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Markus_Fierz" title="Markus Fierz">Fierz</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Paul_Frampton" title="Paul Frampton">Frampton</a></li> <li><a href="/wiki/Harald_Fritzsch" title="Harald Fritzsch">Fritzsch</a></li> <li><a href="/wiki/J%C3%BCrg_Fr%C3%B6hlich" title="Jürg Fröhlich">Fröhlich</a></li> <li><a href="/wiki/Klaus_Fredenhagen" title="Klaus Fredenhagen">Fredenhagen</a></li> <li><a href="/wiki/Wendell_H._Furry" title="Wendell H. Furry">Furry</a></li> <li><a href="/wiki/Sheldon_Glashow" title="Sheldon Glashow">Glashow</a></li> <li><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gelfand</a></li> <li><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann</a></li> <li><a href="/wiki/James_Glimm" title="James Glimm">Glimm</a></li> <li><a href="/wiki/Jeffrey_Goldstone" title="Jeffrey Goldstone">Goldstone</a></li> <li><a href="/wiki/Vladimir_Gribov" title="Vladimir Gribov">Gribov</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Suraj_N._Gupta" title="Suraj N. Gupta">Gupta</a></li> <li><a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Guralnik</a></li> <li><a href="/wiki/Rudolf_Haag" title="Rudolf Haag">Haag</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/Klaus_Hepp" title="Klaus Hepp">Hepp</a></li> <li><a href="/wiki/Peter_Higgs" title="Peter Higgs">Higgs</a></li> <li><a href="/wiki/C._R._Hagen" title="C. R. Hagen">Hagen</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li> <li><a href="/wiki/Dmitri_Ivanenko" title="Dmitri Ivanenko">Ivanenko</a></li> <li><a href="/wiki/Roman_Jackiw" title="Roman Jackiw">Jackiw</a></li> <li><a href="/wiki/Arthur_Jaffe" title="Arthur Jaffe">Jaffe</a></li> <li><a href="/wiki/Giovanni_Jona-Lasinio" title="Giovanni Jona-Lasinio">Jona-Lasinio</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Res_Jost" title="Res Jost">Jost</a></li> <li><a href="/wiki/Gunnar_K%C3%A4ll%C3%A9n" title="Gunnar Källén">Källén</a></li> <li><a href="/wiki/Henry_Way_Kendall" title="Henry Way Kendall">Kendall</a></li> <li><a href="/wiki/Toichiro_Kinoshita" title="Toichiro Kinoshita">Kinoshita</a></li> <li><a href="/wiki/Igor_R._Klebanov" class="mw-redirect" title="Igor R. Klebanov">Klebanov</a></li> <li><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li> <li><a href="/wiki/Dirk_Kreimer" title="Dirk Kreimer">Kreimer</a></li> <li><a href="/wiki/Eduard_A._Kuraev" title="Eduard A. Kuraev">Kuraev</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Lee</a></li> <li><a href="/wiki/Harry_Lehmann" title="Harry Lehmann">Lehmann</a></li> <li><a href="/wiki/Heinrich_Leutwyler" title="Heinrich Leutwyler">Leutwyler</a></li> <li><a href="/wiki/Lev_Lipatov" title="Lev Lipatov">Lipatov</a></li> <li><a href="/wiki/Jan_%C5%81opusza%C5%84ski_(physicist)" title="Jan Łopuszański (physicist)">Łopuszański</a></li> <li><a href="/wiki/Francis_E._Low" title="Francis E. Low">Low</a></li> <li><a href="/wiki/Gerhart_L%C3%BCders" title="Gerhart Lüders">Lüders</a></li> <li><a href="/wiki/Luciano_Maiani" title="Luciano Maiani">Maiani</a></li> <li><a href="/wiki/Ettore_Majorana" title="Ettore Majorana">Majorana</a></li> <li><a href="/wiki/Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li> <li><a href="/wiki/Alexander_Arkadyevich_Migdal" title="Alexander Arkadyevich Migdal">Migdal</a></li> <li><a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Mills</a></li> <li><a href="/wiki/Christian_M%C3%B8ller" title="Christian Møller">Møller</a></li> <li><a href="/wiki/Mark_Naimark" title="Mark Naimark">Naimark</a></li> <li><a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Nambu</a></li> <li><a href="/wiki/Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li> <li><a href="/wiki/Kazuhiko_Nishijima" title="Kazuhiko Nishijima">Nishijima</a></li> <li><a href="/wiki/Reinhard_Oehme" title="Reinhard Oehme">Oehme</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Konrad_Osterwalder" title="Konrad Osterwalder">Osterwalder</a></li> <li><a href="/wiki/Giorgio_Parisi" title="Giorgio Parisi">Parisi</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Michael_Peskin" title="Michael Peskin">Peskin</a></li> <li><a href="/wiki/Jan_Christoph_Plefka" title="Jan Christoph Plefka">Plefka</a></li> <li><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li> <li><a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li> <li><a href="/wiki/Isaak_Pomeranchuk" title="Isaak Pomeranchuk">Pomeranchuk</a></li> <li><a href="/wiki/Victor_Popov" title="Victor Popov">Popov</a></li> <li><a href="/wiki/Alexandru_Proca" title="Alexandru Proca">Proca</a></li> <li><a href="/wiki/Valery_Rubakov" title="Valery Rubakov">Rubakov</a></li> <li><a href="/wiki/David_Ruelle" title="David Ruelle">Ruelle</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Robert_Schrader" title="Robert Schrader">Schrader</a></li> <li><a href="/wiki/Albert_Schwarz" title="Albert Schwarz">Schwarz</a></li> <li><a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Schwinger</a></li> <li><a href="/wiki/Irving_Segal" title="Irving Segal">Segal</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Gordon_Walter_Semenoff" title="Gordon Walter Semenoff">Semenoff</a></li> <li><a href="/wiki/Mikhail_Shifman" title="Mikhail Shifman">Shifman</a></li> <li><a href="/wiki/Dmitry_Shirkov" title="Dmitry Shirkov">Shirkov</a></li> <li><a href="/wiki/Tony_Skyrme" title="Tony Skyrme">Skyrme</a></li> <li><a href="/wiki/Charles_M._Sommerfield" title="Charles M. Sommerfield">Sommerfield</a></li> <li><a href="/wiki/Raymond_Stora" title="Raymond Stora">Stora</a></li> <li><a href="/wiki/Ernst_Stueckelberg" title="Ernst Stueckelberg">Stueckelberg</a></li> <li><a href="/wiki/George_Sudarshan" class="mw-redirect" title="George Sudarshan">Sudarshan</a></li> <li><a href="/wiki/Kurt_Symanzik" title="Kurt Symanzik">Symanzik</a></li> <li><a href="/wiki/Walter_Thirring" title="Walter Thirring">Thirring</a></li> <li><a href="/wiki/Shin%27ichir%C5%8D_Tomonaga" title="Shin'ichirō Tomonaga">Tomonaga</a></li> <li><a href="/wiki/Igor_Tyutin" title="Igor Tyutin">Tyutin</a></li> <li><a href="/wiki/Arkady_Vainshtein" title="Arkady Vainshtein">Vainshtein</a></li> <li><a href="/wiki/Martinus_Veltman" class="mw-redirect" title="Martinus Veltman">Veltman</a></li> <li><a href="/wiki/Miguel_%C3%81ngel_Virasoro_(physicist)" title="Miguel Ángel Virasoro (physicist)">Virasoro</a></li> <li><a href="/wiki/John_Clive_Ward" title="John Clive Ward">Ward</a></li> <li><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg</a></li> <li><a href="/wiki/Victor_Weisskopf" title="Victor Weisskopf">Weisskopf</a></li> <li><a href="/wiki/Gregor_Wentzel" title="Gregor Wentzel">Wentzel</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Christof_Wetterich" title="Christof Wetterich">Wetterich</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Gian_Carlo_Wick" title="Gian Carlo Wick">Wick</a></li> <li><a href="/wiki/Arthur_Wightman" title="Arthur Wightman">Wightman</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek</a></li> <li><a href="/wiki/Kenneth_G._Wilson" title="Kenneth G. Wilson">Wilson</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang</a></li> <li><a href="/wiki/Hideki_Yukawa" title="Hideki Yukawa">Yukawa</a></li> <li><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Anthony_Zee" title="Anthony Zee">Zee</a></li> <li><a href="/wiki/Wolfhart_Zimmermann" title="Wolfhart Zimmermann">Zimmermann</a></li> <li><a href="/wiki/Jean_Zinn-Justin" title="Jean Zinn-Justin">Zinn-Justin</a></li> <li><a href="/wiki/Jean-Bernard_Zuber" title="Jean-Bernard Zuber">Zuber</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li></ul> <p><br /> </p> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_field_theory" title="Template:Quantum field theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_field_theory" title="Template talk:Quantum field theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_field_theory" title="Special:EditPage/Template:Quantum field theory"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <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/Vacuum_energy" title="Vacuum energy">Vacuum energy</a></div> <p>The causes of the Casimir effect are described by quantum field theory, which states that all of the various fundamental <a href="/wiki/Field_(physics)" title="Field (physics)">fields</a>, such as the <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>, must be quantized at each and every point in space. In a simplified view, a "field" in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field can be visualized as the displacement of a ball from its rest position. Vibrations in this field propagate and are governed by the appropriate <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a> for the particular field in question. The second quantization of quantum field theory requires that each such ball-spring combination be quantized, that is, that the strength of the field be quantized at each point in space. At the most basic level, the field at each point in space is a <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">simple harmonic oscillator</a>, and its quantization places a <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> at each point. Excitations of the field correspond to the <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a> of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>. However, even the vacuum has a vastly complex structure, so all calculations of quantum field theory must be made in relation to this model of the vacuum. </p><p>The vacuum has, implicitly, all of the properties that a particle may have: <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> or <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> in the case of <a href="/wiki/Light" title="Light">light</a>, <a href="/wiki/Energy" title="Energy">energy</a>, and so on. On average, most of these properties cancel out: the vacuum is, after all, "empty" in this sense. One important exception is the <a href="/wiki/Vacuum_energy" title="Vacuum energy">vacuum energy</a> or the <a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">vacuum expectation value</a> of the energy. The quantization of a simple harmonic oscillator states that the lowest possible energy or zero-point energy that such an oscillator may have is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {E}={\tfrac {1}{2}}\hbar \omega \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {E}={\tfrac {1}{2}}\hbar \omega \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f72c3945693bc37a2f2d408a8fbc4e19975d85a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.318ex; height:3.509ex;" alt="{\displaystyle {E}={\tfrac {1}{2}}\hbar \omega \,.}"></span> </p><p>Summing over all possible oscillators at all points in space gives an infinite quantity. Since only <i>differences</i> in energy are physically measurable (with the notable exception of gravitation, which remains <a href="/wiki/Quantum_gravity#Nonrenormalizability_of_gravity" title="Quantum gravity">beyond the scope of quantum field theory</a>), this infinity may be considered a feature of the mathematics rather than of the physics. This argument is the underpinning of the theory of <a href="/wiki/Renormalization" title="Renormalization">renormalization</a>. Dealing with infinite quantities in this way was a <a href="/wiki/Renormalization#Attitudes_and_interpretation" title="Renormalization">cause of widespread unease among quantum field theorists</a> before the development in the 1970s of the <a href="/wiki/Renormalization_group" title="Renormalization group">renormalization group</a>, a mathematical formalism for scale transformations that provides a natural basis for the process. </p><p>When the scope of the physics is widened to include gravity, the interpretation of this formally infinite quantity remains problematic. There is currently <a href="/wiki/Cosmological_constant_problem" title="Cosmological constant problem">no compelling explanation</a> as to why it should not result in a <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a> that is many orders of magnitude larger than observed.<sup id="cite_ref-Rugh_22-0" class="reference"><a href="#cite_note-Rugh-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> However, since we do not yet have any fully coherent <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum theory of gravity</a>, there is likewise no compelling reason as to why it should instead actually result in the value of the cosmological constant that we observe.<sup id="cite_ref-Bianchi_23-0" class="reference"><a href="#cite_note-Bianchi-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>The Casimir effect for <a href="/wiki/Fermion" title="Fermion">fermions</a> can be understood as the <a href="/wiki/Spectral_asymmetry" title="Spectral asymmetry">spectral asymmetry</a> of the <a href="/wiki/(%E2%88%921)F" title="(−1)F">fermion operator <span class="texhtml">(−1)<sup><i>F</i></sup></span></a>, where it is known as the <a href="/wiki/Witten_index" title="Witten index">Witten index</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Relativistic_van_der_Waals_force">Relativistic van der Waals force</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=5" title="Edit section: Relativistic van der Waals force"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alternatively, a 2005 paper by <a href="/wiki/Robert_Jaffe_(physicist)" title="Robert Jaffe (physicist)">Robert Jaffe</a> of MIT states that "Casimir effects can be formulated and Casimir forces can be computed without reference to zero-point energies. They are relativistic, quantum forces between charges and currents. The Casimir force (per unit area) between parallel plates vanishes as alpha, the fine structure constant, goes to zero, and the standard result, which appears to be independent of alpha, corresponds to the alpha approaching infinity limit", and that "The Casimir force is simply the (relativistic, <a href="/wiki/Retarded_potential" title="Retarded potential">retarded</a>) van der Waals force between the metal plates."<sup id="cite_ref-:0_19-1" class="reference"><a href="#cite_note-:0-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Casimir and Polder's original paper used this method to derive the Casimir–Polder force. In 1978, Schwinger, DeRadd, and Milton published a similar derivation for the Casimir effect between two parallel plates.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> More recently, Nikolic proved from first principles of <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> that the Casimir force does not originate from the vacuum energy of the electromagnetic field,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and explained in simple terms why the fundamental microscopic origin of Casimir force lies in van der Waals forces.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Effects">Effects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=6" title="Edit section: Effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Casimir's observation was that the <a href="/wiki/Canonical_quantization" title="Canonical quantization">second-quantized</a> quantum electromagnetic field, in the presence of bulk bodies such as metals or <a href="/wiki/Dielectric" title="Dielectric">dielectrics</a>, must obey the same <a href="/wiki/Boundary_value_problem" title="Boundary value problem">boundary conditions</a> that the classical electromagnetic field must obey. In particular, this affects the calculation of the vacuum energy in the presence of a <a href="/wiki/Electrical_conductor" title="Electrical conductor">conductor</a> or dielectric. </p><p>Consider, for example, the calculation of the vacuum expectation value of the electromagnetic field inside a metal cavity, such as, for example, a <a href="/wiki/Cavity_magnetron" title="Cavity magnetron">radar cavity</a> or a <a href="/wiki/Microwave" title="Microwave">microwave</a> <a href="/wiki/Waveguide" title="Waveguide">waveguide</a>. In this case, the correct way to find the zero-point energy of the field is to sum the energies of the <a href="/wiki/Standing_wave" title="Standing wave">standing waves</a> of the cavity. To each and every possible standing wave corresponds an energy; say the energy of the <span class="texhtml mvar" style="font-style:italic;">n</span>th standing wave is <span class="texhtml mvar" style="font-style:italic;">E<sub>n</sub></span>. The vacuum expectation value of the energy of the electromagnetic field in the cavity is then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle E\rangle ={\tfrac {1}{2}}\sum _{n}E_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle E\rangle ={\tfrac {1}{2}}\sum _{n}E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d28d391d62c960981e80478c80a0a3b32b2b8fe7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.404ex; height:5.509ex;" alt="{\displaystyle \langle E\rangle ={\tfrac {1}{2}}\sum _{n}E_{n}}"></span> </p><p>with the sum running over all possible values of <span class="texhtml mvar" style="font-style:italic;">n</span> enumerating the standing waves. The factor of <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> is present because the zero-point energy of the <span class="texhtml mvar" style="font-style:italic;">n</span>th mode is <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>E<sub>n</sub></i></span>, where <span class="texhtml mvar" style="font-style:italic;">E<sub>n</sub></span> is the energy increment for the <span class="texhtml mvar" style="font-style:italic;">n</span>th mode. (It is the same <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> as appears in the equation <span class="texhtml"><i>E</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>ħω</i></span>.) Written in this way, this sum is clearly divergent; however, it can be used to create finite expressions. </p><p>In particular, one may ask how the zero-point energy depends on the shape <span class="texhtml mvar" style="font-style:italic;">s</span> of the cavity. Each energy level <span class="texhtml mvar" style="font-style:italic;">E<sub>n</sub></span> depends on the shape, and so one should write <span class="texhtml"><i>E<sub>n</sub></i>(<i>s</i>)</span> for the energy level, and <span class="texhtml"><span class="nowrap">⟨<i>E</i>(<i>s</i>)⟩</span></span> for the vacuum expectation value. At this point comes an important observation: The force at point <span class="texhtml mvar" style="font-style:italic;">p</span> on the wall of the cavity is equal to the change in the vacuum energy if the shape <span class="texhtml mvar" style="font-style:italic;">s</span> of the wall is perturbed a little bit, say by <span class="texhtml"><i>δs</i></span>, at <span class="texhtml mvar" style="font-style:italic;">p</span>. That is, one has </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(p)=-\left.{\frac {\delta \langle E(s)\rangle }{\delta s}}\right\vert _{p}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mi>δ</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(p)=-\left.{\frac {\delta \langle E(s)\rangle }{\delta s}}\right\vert _{p}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38f5e2131d93b8583b18742d9bbe827ca1926ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.735ex; height:6.843ex;" alt="{\displaystyle F(p)=-\left.{\frac {\delta \langle E(s)\rangle }{\delta s}}\right\vert _{p}\,.}"></span> </p><p>This value is finite in many practical calculations.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>Attraction between the plates can be easily understood by focusing on the one-dimensional situation. Suppose that a moveable conductive plate is positioned at a short distance <span class="texhtml mvar" style="font-style:italic;">a</span> from one of two widely separated plates (distance <span class="texhtml mvar" style="font-style:italic;">l</span> apart). With <span class="texhtml"><i>a</i> ≪ <i>l</i></span>, the states within the slot of width <span class="texhtml mvar" style="font-style:italic;">a</span> are highly constrained so that the energy <span class="texhtml mvar" style="font-style:italic;">E</span> of any one mode is widely separated from that of the next. This is not the case in the large region <span class="texhtml mvar" style="font-style:italic;">l</span> where there is a large number of states (about <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>l</i></span><span class="sr-only">/</span><span class="den"><i>a</i></span></span>⁠</span></span>) with energy evenly spaced between <span class="texhtml mvar" style="font-style:italic;">E</span> and the next mode in the narrow slot, or in other words, all slightly larger than <span class="texhtml mvar" style="font-style:italic;">E</span>. Now on shortening <span class="texhtml mvar" style="font-style:italic;">a</span> by an amount <span class="texhtml mvar" style="font-style:italic;">da</span> (which is negative), the mode in the narrow slot shrinks in wavelength and therefore increases in energy proportional to <span class="texhtml">−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>da</i></span><span class="sr-only">/</span><span class="den"><i>a</i></span></span>⁠</span></span>, whereas all the <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>l</i></span><span class="sr-only">/</span><span class="den"><i>a</i></span></span>⁠</span></span> states that lie in the large region lengthen and correspondingly decrease their energy by an amount proportional to <span class="texhtml">−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>da</i></span><span class="sr-only">/</span><span class="den"><i>l</i></span></span>⁠</span></span> (note the different denominator). The two effects nearly cancel, but the net change is slightly negative, because the energy of all the <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>l</i></span><span class="sr-only">/</span><span class="den"><i>a</i></span></span>⁠</span></span> modes in the large region are slightly larger than the single mode in the slot. Thus the force is attractive: it tends to make <span class="texhtml mvar" style="font-style:italic;">a</span> slightly smaller, the plates drawing each other closer, across the thin slot. </p> <div class="mw-heading mw-heading2"><h2 id="Derivation_of_Casimir_effect_assuming_zeta-regularization">Derivation of Casimir effect assuming zeta-regularization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=7" title="Edit section: Derivation of Casimir effect assuming zeta-regularization"><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">See <b><a href="https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension" class="extiw" title="v:Quantum mechanics/Casimir effect in one dimension">Wikiversity</a></b> for an elementary calculation in one dimension.</div> <p>In the original calculation done by Casimir, he considered the space between a pair of conducting metal plates at distance <span class="texhtml mvar" style="font-style:italic;">a</span> apart. In this case, the standing waves are particularly easy to calculate, because the transverse component of the electric field and the normal component of the magnetic field must vanish on the surface of a conductor. Assuming the plates lie parallel to the <span class="texhtml mvar" style="font-style:italic;">xy</span>-plane, the standing waves are </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{n}(x,y,z;t)=e^{-i\omega _{n}t}e^{ik_{x}x+ik_{y}y}\sin(k_{n}z)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>;</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>t</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <mi>i</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n}(x,y,z;t)=e^{-i\omega _{n}t}e^{ik_{x}x+ik_{y}y}\sin(k_{n}z)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d3e18f956a887b83c01c38a742271321010fe2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.455ex; height:3.176ex;" alt="{\displaystyle \psi _{n}(x,y,z;t)=e^{-i\omega _{n}t}e^{ik_{x}x+ik_{y}y}\sin(k_{n}z)\,,}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">ψ</span> stands for the electric component of the electromagnetic field, and, for brevity, the polarization and the magnetic components are ignored here. Here, <span class="texhtml mvar" style="font-style:italic;">k<sub>x</sub></span> and <span class="texhtml mvar" style="font-style:italic;">k<sub>y</sub></span> are the <a href="/wiki/Wavenumber" title="Wavenumber">wavenumbers</a> in directions parallel to the plates, and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{n}={\frac {n\pi }{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> </mrow> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{n}={\frac {n\pi }{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7957824a7e82ade9e48875b8c0e34790a0af52d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.091ex; height:4.676ex;" alt="{\displaystyle k_{n}={\frac {n\pi }{a}}}"></span> </p><p>is the wavenumber perpendicular to the plates. Here, <span class="texhtml mvar" style="font-style:italic;">n</span> is an integer, resulting from the requirement that <span class="texhtml mvar" style="font-style:italic;">ψ</span> vanish on the metal plates. The frequency of this wave is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{n}=c{\sqrt {{k_{x}}^{2}+{k_{y}}^{2}+{\frac {n^{2}\pi ^{2}}{a^{2}}}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{n}=c{\sqrt {{k_{x}}^{2}+{k_{y}}^{2}+{\frac {n^{2}\pi ^{2}}{a^{2}}}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165540d9bfcfbd9995e71a092471dec3d9046997" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.234ex; height:7.509ex;" alt="{\displaystyle \omega _{n}=c{\sqrt {{k_{x}}^{2}+{k_{y}}^{2}+{\frac {n^{2}\pi ^{2}}{a^{2}}}}}\,,}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">c</span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. The vacuum energy is then the sum over all possible excitation modes. Since the area of the plates is large, we may sum by integrating over two of the dimensions in <span class="texhtml mvar" style="font-style:italic;">k</span>-space. The assumption of <a href="/wiki/Van_Hove_singularity#Theory" title="Van Hove singularity">periodic boundary conditions</a> yields, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle E\rangle ={\frac {\hbar }{2}}\cdot 2\int {\frac {A\,dk_{x}\,dk_{y}}{(2\pi )^{2}}}\sum _{n=1}^{\infty }\omega _{n}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mn>2</mn> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle E\rangle ={\frac {\hbar }{2}}\cdot 2\int {\frac {A\,dk_{x}\,dk_{y}}{(2\pi )^{2}}}\sum _{n=1}^{\infty }\omega _{n}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a9ab81a228c3ea08e5b644183e9471012cba6b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.892ex; height:6.843ex;" alt="{\displaystyle \langle E\rangle ={\frac {\hbar }{2}}\cdot 2\int {\frac {A\,dk_{x}\,dk_{y}}{(2\pi )^{2}}}\sum _{n=1}^{\infty }\omega _{n}\,,}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">A</span> is the area of the metal plates, and a factor of 2 is introduced for the two possible polarizations of the wave. This expression is clearly infinite, and to proceed with the calculation, it is convenient to introduce a <a href="/wiki/Regularization_(physics)" title="Regularization (physics)">regulator</a> (discussed in greater detail below). The regulator will serve to make the expression finite, and in the end will be removed. The <a href="/wiki/Zeta_function_regularization" title="Zeta function regularization">zeta-regulated</a> version of the energy per unit-area of the plate is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\langle E(s)\rangle }{A}}=\hbar \int {\frac {dk_{x}\,dk_{y}}{(2\pi )^{2}}}\sum _{n=1}^{\infty }\omega _{n}\left|\omega _{n}\right|^{-s}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow> <mo>|</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\langle E(s)\rangle }{A}}=\hbar \int {\frac {dk_{x}\,dk_{y}}{(2\pi )^{2}}}\sum _{n=1}^{\infty }\omega _{n}\left|\omega _{n}\right|^{-s}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec58b2acc0d96d634784bebbd1e964d116b76a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.06ex; height:6.843ex;" alt="{\displaystyle {\frac {\langle E(s)\rangle }{A}}=\hbar \int {\frac {dk_{x}\,dk_{y}}{(2\pi )^{2}}}\sum _{n=1}^{\infty }\omega _{n}\left|\omega _{n}\right|^{-s}\,.}"></span> </p><p>In the end, the limit <span class="texhtml"><i>s</i> → 0</span> is to be taken. Here <span class="texhtml mvar" style="font-style:italic;">s</span> is just a <a href="/wiki/Complex_number" title="Complex number">complex number</a>, not to be confused with the shape discussed previously. This integral sum is finite for <span class="texhtml mvar" style="font-style:italic;">s</span> <a href="/wiki/Real_number" title="Real number">real</a> and larger than 3. The sum has a <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">pole</a> at <span class="texhtml"><i>s</i> = 3</span>, but may be <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytically continued</a> to <span class="texhtml"><i>s</i> = 0</span>, where the expression is finite. The above expression simplifies to: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\langle E(s)\rangle }{A}}={\frac {\hbar c^{1-s}}{4\pi ^{2}}}\sum _{n}\int _{0}^{\infty }2\pi q\,dq\left|q^{2}+{\frac {\pi ^{2}n^{2}}{a^{2}}}\right|^{\frac {1-s}{2}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mn>2</mn> <mi>π</mi> <mi>q</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <msup> <mrow> <mo>|</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\langle E(s)\rangle }{A}}={\frac {\hbar c^{1-s}}{4\pi ^{2}}}\sum _{n}\int _{0}^{\infty }2\pi q\,dq\left|q^{2}+{\frac {\pi ^{2}n^{2}}{a^{2}}}\right|^{\frac {1-s}{2}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3900c06924d83cba4eebace2af616f082cbce9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.14ex; height:8.009ex;" alt="{\displaystyle {\frac {\langle E(s)\rangle }{A}}={\frac {\hbar c^{1-s}}{4\pi ^{2}}}\sum _{n}\int _{0}^{\infty }2\pi q\,dq\left|q^{2}+{\frac {\pi ^{2}n^{2}}{a^{2}}}\right|^{\frac {1-s}{2}}\,,}"></span> </p><p>where <a href="/wiki/Polar_coordinate_system" title="Polar coordinate system">polar coordinates</a> <span class="texhtml"><i>q</i><sup>2</sup> = <i>k<sub>x</sub></i><sup>2</sup> + <i>k<sub>y</sub></i><sup>2</sup></span> were introduced to turn the <a href="/wiki/Multiple_integral" title="Multiple integral">double integral</a> into a single integral. The <span class="texhtml mvar" style="font-style:italic;">q</span> in front is the Jacobian, and the <span class="texhtml">2<i>π</i></span> comes from the angular integration. The integral converges if <span class="texhtml">Re(<i>s</i>) > 3</span>, resulting in </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\langle E(s)\rangle }{A}}=-{\frac {\hbar c^{1-s}\pi ^{2-s}}{2a^{3-s}}}{\frac {1}{3-s}}\sum _{n}\left|n\right|^{3-s}=-{\frac {\hbar c^{1-s}\pi ^{2-s}}{2a^{3-s}(3-s)}}\sum _{n}{\frac {1}{\left|n\right|^{s-3}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>−</mo> <mi>s</mi> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msup> <mrow> <mo>|</mo> <mi>n</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow> <mo>|</mo> <mi>n</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\langle E(s)\rangle }{A}}=-{\frac {\hbar c^{1-s}\pi ^{2-s}}{2a^{3-s}}}{\frac {1}{3-s}}\sum _{n}\left|n\right|^{3-s}=-{\frac {\hbar c^{1-s}\pi ^{2-s}}{2a^{3-s}(3-s)}}\sum _{n}{\frac {1}{\left|n\right|^{s-3}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8fa1a8e81397513f05de3e9d1eb049905a8a434" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:68.703ex; height:6.843ex;" alt="{\displaystyle {\frac {\langle E(s)\rangle }{A}}=-{\frac {\hbar c^{1-s}\pi ^{2-s}}{2a^{3-s}}}{\frac {1}{3-s}}\sum _{n}\left|n\right|^{3-s}=-{\frac {\hbar c^{1-s}\pi ^{2-s}}{2a^{3-s}(3-s)}}\sum _{n}{\frac {1}{\left|n\right|^{s-3}}}\,.}"></span> </p><p>The sum diverges at <span class="texhtml mvar" style="font-style:italic;">s</span> in the neighborhood of zero, but if the damping of large-frequency excitations corresponding to analytic continuation of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> to <span class="texhtml"><i>s</i> = 0</span> is assumed to make sense physically in some way, then one has </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\langle E\rangle }{A}}=\lim _{s\to 0}{\frac {\langle E(s)\rangle }{A}}=-{\frac {\hbar c\pi ^{2}}{6a^{3}}}\zeta (-3)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>6</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\langle E\rangle }{A}}=\lim _{s\to 0}{\frac {\langle E(s)\rangle }{A}}=-{\frac {\hbar c\pi ^{2}}{6a^{3}}}\zeta (-3)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c365489df2945c0e2b538b1d9367234fc06bc034" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.817ex; height:6.009ex;" alt="{\displaystyle {\frac {\langle E\rangle }{A}}=\lim _{s\to 0}{\frac {\langle E(s)\rangle }{A}}=-{\frac {\hbar c\pi ^{2}}{6a^{3}}}\zeta (-3)\,.}"></span> </p><p>But <span class="texhtml"><i>ζ</i>(−3) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">120</span></span>⁠</span></span> and so one obtains </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\langle E\rangle }{A}}=-{\frac {\hbar c\pi ^{2}}{720a^{3}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>720</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\langle E\rangle }{A}}=-{\frac {\hbar c\pi ^{2}}{720a^{3}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4ae7d4b7b16f505290867adcdcac877cd527ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.969ex; height:6.009ex;" alt="{\displaystyle {\frac {\langle E\rangle }{A}}=-{\frac {\hbar c\pi ^{2}}{720a^{3}}}\,.}"></span> </p><p>The analytic continuation has evidently lost an additive positive infinity, somehow exactly accounting for the zero-point energy (not included above) outside the slot between the plates, but which changes upon plate movement within a closed system. The Casimir force per unit area <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>F</i><sub>c</sub></span><span class="sr-only">/</span><span class="den"><i>A</i></span></span>⁠</span></span> for idealized, perfectly conducting plates with vacuum between them is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{\mathrm {c} }}{A}}=-{\frac {d}{da}}{\frac {\langle E\rangle }{A}}=-{\frac {\hbar c\pi ^{2}}{240a^{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>240</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{\mathrm {c} }}{A}}=-{\frac {d}{da}}{\frac {\langle E\rangle }{A}}=-{\frac {\hbar c\pi ^{2}}{240a^{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d3cc87ca86bebff52dadce17cfeb2cb3d3188d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.417ex; height:6.009ex;" alt="{\displaystyle {\frac {F_{\mathrm {c} }}{A}}=-{\frac {d}{da}}{\frac {\langle E\rangle }{A}}=-{\frac {\hbar c\pi ^{2}}{240a^{4}}}}"></span> </p><p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">ħ</span> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>,</li> <li><span class="texhtml mvar" style="font-style:italic;">c</span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>,</li> <li><span class="texhtml mvar" style="font-style:italic;">a</span> is the <a href="/wiki/Distance" title="Distance">distance</a> between the two plates</li></ul> <p>The force is negative, indicating that the force is attractive: by moving the two plates closer together, the energy is lowered. The presence of <span class="texhtml mvar" style="font-style:italic;">ħ</span> shows that the Casimir force per unit area <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>F</i><sub>c</sub></span><span class="sr-only">/</span><span class="den"><i>A</i></span></span>⁠</span></span> is very small, and that furthermore, the force is inherently of quantum-mechanical origin. </p><p>By <a href="/wiki/Integral" title="Integral">integrating</a> the equation above it is possible to calculate the energy required to separate to infinity the two plates as: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}U_{E}(a)&=\int F(a)\,da=\int -\hbar c\pi ^{2}{\frac {A}{240a^{4}}}\,da\\[4pt]&=\hbar c\pi ^{2}{\frac {A}{720a^{3}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>∫</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>a</mi> <mo>=</mo> <mo>∫</mo> <mo>−</mo> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mrow> <mn>240</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>a</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mrow> <mn>720</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U_{E}(a)&=\int F(a)\,da=\int -\hbar c\pi ^{2}{\frac {A}{240a^{4}}}\,da\\[4pt]&=\hbar c\pi ^{2}{\frac {A}{720a^{3}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ee5b5fe6831ed1cf8fa9069c471b7e45e62b3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:41.787ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}U_{E}(a)&=\int F(a)\,da=\int -\hbar c\pi ^{2}{\frac {A}{240a^{4}}}\,da\\[4pt]&=\hbar c\pi ^{2}{\frac {A}{720a^{3}}}\end{aligned}}}"></span> </p><p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">ħ</span> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>,</li> <li><span class="texhtml mvar" style="font-style:italic;">c</span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>,</li> <li><span class="texhtml mvar" style="font-style:italic;">A</span> is the <a href="/wiki/Area" title="Area">area</a> of one of the plates,</li> <li><span class="texhtml mvar" style="font-style:italic;">a</span> is the <a href="/wiki/Distance" title="Distance">distance</a> between the two plates</li></ul> <p>In Casimir's original derivation,<sup id="cite_ref-:1_20-1" class="reference"><a href="#cite_note-:1-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> a moveable conductive plate is positioned at a short distance <span class="texhtml mvar" style="font-style:italic;">a</span> from one of two widely separated plates (distance <span class="texhtml mvar" style="font-style:italic;">L</span> apart). The zero-point energy on <i>both</i> sides of the plate is considered. Instead of the above <i>ad hoc</i> analytic continuation assumption, non-convergent sums and integrals are computed using <a href="/wiki/Euler%E2%80%93Maclaurin_summation" class="mw-redirect" title="Euler–Maclaurin summation">Euler–Maclaurin summation</a> with a regularizing function (e.g., exponential regularization) not so anomalous as <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>ω<sub>n</sub></i></span>|<sup>−<i>s</i></sup></span> in the above.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="More_recent_theory">More recent theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=8" title="Edit section: More recent theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Casimir's analysis of idealized metal plates was generalized to arbitrary dielectric and realistic metal plates by <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Evgeny Lifshitz</a> and his students.<sup id="cite_ref-:2_5-1" class="reference"><a href="#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Using this approach, complications of the bounding surfaces, such as the modifications to the Casimir force due to finite conductivity, can be calculated numerically using the tabulated complex dielectric functions of the bounding materials. Lifshitz's theory for two metal plates reduces to Casimir's idealized <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>a</i><sup>4</sup></span></span>⁠</span></span> force law for large separations <span class="texhtml mvar" style="font-style:italic;">a</span> much greater than the <a href="/wiki/Skin_depth" class="mw-redirect" title="Skin depth">skin depth</a> of the metal, and conversely reduces to the <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>a</i><sup>3</sup></span></span>⁠</span></span> force law of the <a href="/wiki/London_dispersion_force" title="London dispersion force">London dispersion force</a> (with a coefficient called a <a href="/wiki/Hamaker_constant" title="Hamaker constant">Hamaker constant</a>) for small <span class="texhtml mvar" style="font-style:italic;">a</span>, with a more complicated dependence on <span class="texhtml mvar" style="font-style:italic;">a</span> for intermediate separations determined by the <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">dispersion</a> of the materials.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>Lifshitz's result was subsequently generalized to arbitrary multilayer planar geometries as well as to anisotropic and magnetic materials, but for several decades the calculation of Casimir forces for non-planar geometries remained limited to a few idealized cases admitting analytical solutions.<sup id="cite_ref-Rodriguez11-review_31-0" class="reference"><a href="#cite_note-Rodriguez11-review-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> For example, the force in the experimental sphere–plate geometry was computed with an approximation (due to Derjaguin) that the sphere radius <span class="texhtml mvar" style="font-style:italic;">R</span> is much larger than the separation <span class="texhtml mvar" style="font-style:italic;">a</span>, in which case the nearby surfaces are nearly parallel and the parallel-plate result can be adapted to obtain an approximate <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>R</i></span><span class="sr-only">/</span><span class="den"><i>a</i><sup>3</sup></span></span>⁠</span></span> force (neglecting both skin-depth and <a href="/wiki/Orders_of_approximation" class="mw-redirect" title="Orders of approximation">higher-order</a> curvature effects).<sup id="cite_ref-Rodriguez11-review_31-1" class="reference"><a href="#cite_note-Rodriguez11-review-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> However, in the 2010s a number of authors developed and demonstrated a variety of numerical techniques, in many cases adapted from classical <a href="/wiki/Computational_electromagnetics" title="Computational electromagnetics">computational electromagnetics</a>, that are capable of accurately calculating Casimir forces for arbitrary geometries and materials, from simple finite-size effects of finite plates to more complicated phenomena arising for patterned surfaces or objects of various shapes.<sup id="cite_ref-Rodriguez11-review_31-2" class="reference"><a href="#cite_note-Rodriguez11-review-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Reid2013_33-0" class="reference"><a href="#cite_note-Reid2013-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Measurement">Measurement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=9" title="Edit section: Measurement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of the first experimental tests was conducted by Marcus Sparnaay at Philips in <a href="/wiki/Eindhoven" title="Eindhoven">Eindhoven</a> (Netherlands), in 1958, in a delicate and difficult experiment with parallel plates, obtaining results not in contradiction with the Casimir theory,<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> but with large experimental errors. </p><p>The Casimir effect was measured more accurately in 1997 by Steve K. Lamoreaux of <a href="/wiki/Los_Alamos_National_Laboratory" title="Los Alamos National Laboratory">Los Alamos National Laboratory</a>,<sup id="cite_ref-Lamoureaux1997_7-2" class="reference"><a href="#cite_note-Lamoureaux1997-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and by Umar Mohideen and Anushree Roy of the <a href="/wiki/University_of_California,_Riverside" title="University of California, Riverside">University of California, Riverside</a>.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> In practice, rather than using two parallel plates, which would require phenomenally accurate alignment to ensure they were parallel, the experiments use one plate that is flat and another plate that is a part of a <a href="/wiki/Sphere" title="Sphere">sphere</a> with a very large <a href="/wiki/Radius" title="Radius">radius</a>. </p><p>In 2001, a group (Giacomo Bressi, Gianni Carugno, Roberto Onofrio and Giuseppe Ruoso) at the <a href="/wiki/University_of_Padua" title="University of Padua">University of Padua</a> (Italy) finally succeeded in measuring the Casimir force between parallel plates using <a href="/wiki/Microelectromechanical_system_oscillator#Resonators" title="Microelectromechanical system oscillator">microresonators</a>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> Numerous variations of these experiments are summarized in the 2009 review by Klimchitskaya.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>In 2013, a conglomerate of scientists from <a href="/wiki/Hong_Kong_University_of_Science_and_Technology" title="Hong Kong University of Science and Technology">Hong Kong University of Science and Technology</a>, <a href="/wiki/University_of_Florida" title="University of Florida">University of Florida</a>, <a href="/wiki/Harvard_University" title="Harvard University">Harvard University</a>, <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a>, and <a href="/wiki/Oak_Ridge_National_Laboratory" title="Oak Ridge National Laboratory">Oak Ridge National Laboratory</a> demonstrated a compact integrated silicon chip that can measure the Casimir force.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> The integrated chip defined by electron-beam lithography does not need extra alignment, making it an ideal platform for measuring Casimir force between complex geometries. In 2017 and 2021, the same group from <a href="/wiki/Hong_Kong_University_of_Science_and_Technology" title="Hong Kong University of Science and Technology">Hong Kong University of Science and Technology</a> demonstrated the non-monotonic Casimir force<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> and distance-independent Casimir force,<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> respectively, using this on-chip platform. </p> <div class="mw-heading mw-heading2"><h2 id="Regularization">Regularization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=10" title="Edit section: Regularization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In order to be able to perform calculations in the general case, it is convenient to introduce a <a href="/wiki/Regularization_(physics)" title="Regularization (physics)">regulator</a> in the summations. This is an artificial device, used to make the sums finite so that they can be more easily manipulated, followed by the taking of a limit so as to remove the regulator. </p><p>The <a href="/wiki/Heat_kernel_regularization" class="mw-redirect" title="Heat kernel regularization">heat kernel</a> or <a href="/wiki/Exponential_function" title="Exponential function">exponentially</a> regulated sum is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle E(t)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}|\exp {\bigl (}-t|\omega _{n}|{\bigr )}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo>−</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle E(t)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}|\exp {\bigl (}-t|\omega _{n}|{\bigr )}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e5ee663815b6ba77d278ccd5659536fa761835" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.853ex; height:6.343ex;" alt="{\displaystyle \langle E(t)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}|\exp {\bigl (}-t|\omega _{n}|{\bigr )}\,,}"></span> </p><p>where the limit <span class="texhtml"><i>t</i> → 0<sup>+</sup></span> is taken in the end. The divergence of the sum is typically manifested as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle E(t)\rangle ={\frac {C}{t^{3}}}+{\textrm {finite}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>finite</mtext> </mrow> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle E(t)\rangle ={\frac {C}{t^{3}}}+{\textrm {finite}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd1ff5e746de9a003f3bc4c64267745fddc3267" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.525ex; height:5.676ex;" alt="{\displaystyle \langle E(t)\rangle ={\frac {C}{t^{3}}}+{\textrm {finite}}\,}"></span> </p><p>for three-dimensional cavities. The infinite part of the sum is associated with the bulk constant <span class="texhtml mvar" style="font-style:italic;">C</span> which <i>does not</i> depend on the shape of the cavity. The interesting part of the sum is the finite part, which is shape-dependent. The <a href="/wiki/Gaussian_function" title="Gaussian function">Gaussian</a> regulator </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle E(t)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}|\exp \left(-t^{2}|\omega _{n}|^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle E(t)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}|\exp \left(-t^{2}|\omega _{n}|^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92912c74d548e94e0ece48e71a3f27cdf1faedb1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.154ex; height:6.343ex;" alt="{\displaystyle \langle E(t)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}|\exp \left(-t^{2}|\omega _{n}|^{2}\right)}"></span> </p><p>is better suited to numerical calculations because of its superior convergence properties, but is more difficult to use in theoretical calculations. Other, suitably smooth, regulators may be used as well. The <a href="/wiki/Zeta_function_regulator" class="mw-redirect" title="Zeta function regulator">zeta function regulator</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle E(s)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}||\omega _{n}|^{-s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle E(s)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}||\omega _{n}|^{-s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/534dda440d6b50d37958ff79f3aaabb135ee7b98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.215ex; height:6.343ex;" alt="{\displaystyle \langle E(s)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}||\omega _{n}|^{-s}}"></span> </p><p>is completely unsuited for numerical calculations, but is quite useful in theoretical calculations. In particular, divergences show up as poles in the <a href="/wiki/Complex_plane" title="Complex plane">complex <span class="texhtml mvar" style="font-style:italic;">s</span> plane</a>, with the bulk divergence at <span class="texhtml"><i>s</i> = 4</span>. This sum may be <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytically continued</a> past this pole, to obtain a finite part at <span class="texhtml"><i>s</i> = 0</span>. </p><p>Not every cavity configuration necessarily leads to a finite part (the lack of a pole at <span class="texhtml"><i>s</i> = 0</span>) or shape-independent infinite parts. In this case, it should be understood that additional physics has to be taken into account. In particular, at extremely large frequencies (above the <a href="/wiki/Plasma_frequency" class="mw-redirect" title="Plasma frequency">plasma frequency</a>), metals become transparent to <a href="/wiki/Photon" title="Photon">photons</a> (such as <a href="/wiki/X-ray" title="X-ray">X-rays</a>), and dielectrics show a frequency-dependent cutoff as well. This frequency dependence acts as a natural regulator. There are a variety of bulk effects in <a href="/wiki/Solid_state_physics" class="mw-redirect" title="Solid state physics">solid state physics</a>, mathematically very similar to the Casimir effect, where the <a href="/wiki/Cutoff_frequency" title="Cutoff frequency">cutoff frequency</a> comes into explicit play to keep expressions finite. (These are discussed in greater detail in <i>Landau and Lifshitz</i>, "Theory of Continuous Media".<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. (May 2022)">citation needed</span></a></i>]</sup>) </p> <div class="mw-heading mw-heading2"><h2 id="Generalities">Generalities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=11" title="Edit section: Generalities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Casimir effect can also be computed using the mathematical mechanisms of <a href="/wiki/Functional_integral" class="mw-redirect" title="Functional integral">functional integrals</a> of quantum field theory, although such calculations are considerably more abstract, and thus difficult to comprehend. In addition, they can be carried out only for the simplest of geometries. However, the formalism of quantum field theory makes it clear that the vacuum expectation value summations are in a certain sense summations over so-called "virtual particles". </p><p>More interesting is the understanding that the sums over the energies of standing waves should be formally understood as sums over the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of a <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>. This allows atomic and molecular effects, such as the <a href="/wiki/Van_der_Waals_force" title="Van der Waals force">Van der Waals force</a>, to be understood as a variation on the theme of the Casimir effect. Thus one considers the Hamiltonian of a system as a function of the arrangement of objects, such as atoms, in <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a>. The change in the zero-point energy as a function of changes of the configuration can be understood to result in forces acting between the objects. </p><p>In the <a href="/wiki/Chiral_bag_model" class="mw-redirect" title="Chiral bag model">chiral bag model</a> of the nucleon, the Casimir energy plays an important role in showing the mass of the nucleon is independent of the bag radius. In addition, the spectral asymmetry is interpreted as a non-zero vacuum expectation value of the <a href="/wiki/Baryon_number" title="Baryon number">baryon number</a>, cancelling the <a href="/wiki/Topological_winding_number" class="mw-redirect" title="Topological winding number">topological winding number</a> of the <a href="/wiki/Pion" title="Pion">pion</a> field surrounding the nucleon. </p><p>A "pseudo-Casimir" effect can be found in <a href="/wiki/Liquid_crystal" title="Liquid crystal">liquid crystal</a> systems, where the boundary conditions imposed through anchoring by rigid walls give rise to a long-range force, analogous to the force that arises between conducting plates.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Dynamical_Casimir_effect">Dynamical Casimir effect</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=12" title="Edit section: Dynamical Casimir effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The dynamical Casimir effect is the production of particles and energy from an accelerated <i>moving mirror</i>. This reaction was predicted by certain numerical solutions to <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> equations made in the 1970s.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> In May 2011 an announcement was made by researchers at the <a href="/wiki/Chalmers_University_of_Technology" title="Chalmers University of Technology">Chalmers University of Technology</a>, in Gothenburg, Sweden, of the detection of the dynamical Casimir effect. In their experiment, microwave photons were generated out of the vacuum in a superconducting microwave resonator. These researchers used a modified <a href="/wiki/SQUID" title="SQUID">SQUID</a> to change the effective length of the resonator in time, mimicking a mirror moving at the required relativistic velocity. If confirmed this would be the first experimental verification of the dynamical Casimir effect.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> In March 2013 an article appeared on the <a href="/wiki/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America" title="Proceedings of the National Academy of Sciences of the United States of America">PNAS</a> scientific journal describing an experiment that demonstrated the dynamical Casimir effect in a Josephson metamaterial.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> In July 2019 an article was published describing an experiment providing evidence of optical dynamical Casimir effect in a dispersion-oscillating fibre.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> In 2020, <a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Frank Wilczek</a> et al., proposed a resolution to the <a href="/wiki/Information_loss_paradox" class="mw-redirect" title="Information loss paradox">information loss paradox</a> associated with the moving mirror model of the dynamical Casimir effect.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> Constructed within the framework of <a href="/wiki/Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">quantum field theory in curved spacetime</a>, the dynamical Casimir effect (moving mirror) has been used to help understand the <a href="/wiki/Unruh_effect" title="Unruh effect">Unruh effect</a>.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Repulsive_forces">Repulsive forces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=13" title="Edit section: Repulsive forces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are few instances wherein the Casimir effect can give rise to repulsive forces between uncharged objects. Evgeny Lifshitz showed (theoretically) that in certain circumstances (most commonly involving liquids), repulsive forces can arise.<sup id="cite_ref-DLP_50-0" class="reference"><a href="#cite_note-DLP-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> This has sparked interest in applications of the Casimir effect toward the development of levitating devices. An experimental demonstration of the Casimir-based repulsion predicted by Lifshitz was carried out by Munday et al.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> who described it as "<i>quantum levitation</i>". Other scientists have also suggested the use of <a href="/wiki/Gain_media" class="mw-redirect" title="Gain media">gain media</a> to achieve a similar levitation effect,<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> though this is controversial because these materials seem to violate fundamental causality constraints and the requirement of thermodynamic equilibrium (<a href="/wiki/Kramers%E2%80%93Kronig_relations" title="Kramers–Kronig relations">Kramers–Kronig relations</a>). Casimir and Casimir–Polder repulsion can in fact occur for sufficiently anisotropic electrical bodies; for a review of the issues involved with repulsion see Milton et al.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> A notable recent development on repulsive Casimir forces relies on using chiral materials. Q.-D. Jiang at Stockholm University and Nobel Laureate Frank Wilczek at MIT show that chiral "lubricant" can generate repulsive, enhanced, and tunable Casimir interactions.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p><p>Timothy Boyer showed in his work published in 1968<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> that a conductor with spherical symmetry will also show this repulsive force, and the result is independent of radius. Further work shows that the repulsive force can be generated with materials of carefully chosen dielectrics.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Speculative_applications">Speculative applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=14" title="Edit section: Speculative applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It has been suggested that the Casimir forces have application in nanotechnology,<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> in particular silicon integrated circuit technology based micro- and nanoelectromechanical systems, and so-called Casimir oscillators.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><p>In 1995 and 1998 Maclay et al.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> published the first models of a <a href="/wiki/Microelectromechanical_system" class="mw-redirect" title="Microelectromechanical system">microelectromechanical system</a> (MEMS) with Casimir forces. While not exploiting the Casimir force for useful work, the papers drew attention from the MEMS community due to the revelation that Casimir effect needs to be considered as a vital factor in the future design of MEMS. In particular, Casimir effect might be the critical factor in the <a href="/wiki/Stiction" title="Stiction">stiction</a> failure of MEMS.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (February 2024)">page needed</span></a></i>]</sup> </p><p>In 2001, Capasso et al. showed how the force can be used to control the mechanical motion of a MEMS device, The researchers suspended a polysilicon plate from a torsional rod – a twisting horizontal bar just a few microns in diameter. When they brought a metallized sphere close up to the plate, the attractive Casimir force between the two objects made the plate rotate. They also studied the dynamical behaviour of the MEMS device by making the plate oscillate. The Casimir force reduced the rate of oscillation and led to nonlinear phenomena, such as <a href="/wiki/Hysteresis" title="Hysteresis">hysteresis</a> and <a href="/wiki/Bistability" title="Bistability">bistability</a> in the frequency response of the oscillator. According to the team, the system's behaviour agreed well with theoretical calculations.<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p><p>The Casimir effect shows that quantum field theory allows the energy density in very small regions of space to be negative relative to the ordinary vacuum energy, and the energy densities cannot be arbitrarily negative as the theory breaks down at atomic distances.<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 175">: 175 </span></sup> <sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> Such prominent physicists such as <a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Stephen Hawking</a><sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Kip_Thorne" title="Kip Thorne">Kip Thorne</a>,<sup id="cite_ref-time_travel_68-0" class="reference"><a href="#cite_note-time_travel-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> have speculated that such effects might make it possible to stabilize a <a href="/wiki/Wormhole#Traversable_wormholes" title="Wormhole">traversable wormhole</a>. </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=Casimir_effect&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span 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href="/w/index.php?title=Casimir_effect&action=edit&section=16" title="Edit section: References"><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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLamoreaux2005" class="citation journal cs1">Lamoreaux, Steven K. (2005). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1088/0034-4885/68/1/R04/meta">"The Casimir force: Background, experiments, and applications"</a>. <i>Reports on Progress in Physics</i>. <b>68</b> (1): 201–236. <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/2005RPPh...68..201L">2005RPPh...68..201L</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%2F0034-4885%2F68%2F1%2Fr04">10.1088/0034-4885/68/1/r04</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:21131414">21131414</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reports+on+Progress+in+Physics&rft.atitle=The+Casimir+force%3A+Background%2C+experiments%2C+and+applications&rft.volume=68&rft.issue=1&rft.pages=201-236&rft.date=2005&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A21131414%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0034-4885%2F68%2F1%2Fr04&rft_id=info%3Abibcode%2F2005RPPh...68..201L&rft.aulast=Lamoreaux&rft.aufirst=Steven+K.&rft_id=https%3A%2F%2Fiopscience.iop.org%2Farticle%2F10.1088%2F0034-4885%2F68%2F1%2FR04%2Fmeta&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" 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="CITEREFMostepanenkoTrunov1988" class="citation journal cs1">Mostepanenko, V M; Trunov, N N (30 November 1988). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1070/PU1988v031n11ABEH005641">"The Casimir effect and its applications"</a>. <i>Soviet Physics Uspekhi</i>. <b>31</b> (11): 965–987. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1070%2FPU1988v031n11ABEH005641">10.1070/PU1988v031n11ABEH005641</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0038-5670">0038-5670</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Soviet+Physics+Uspekhi&rft.atitle=The+Casimir+effect+and+its+applications&rft.volume=31&rft.issue=11&rft.pages=965-987&rft.date=1988-11-30&rft_id=info%3Adoi%2F10.1070%2FPU1988v031n11ABEH005641&rft.issn=0038-5670&rft.aulast=Mostepanenko&rft.aufirst=V+M&rft.au=Trunov%2C+N+N&rft_id=https%3A%2F%2Fiopscience.iop.org%2Farticle%2F10.1070%2FPU1988v031n11ABEH005641&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlimchitskayaMohideenMostepanenko2009" class="citation journal cs1">Klimchitskaya, G. L.; Mohideen, U.; Mostepanenko, V. M. (21 December 2009). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.81.1827">"The Casimir force between real materials: Experiment and theory"</a>. <i>Reviews of Modern Physics</i>. <b>81</b> (4): 1827–1885. <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/0902.4022">0902.4022</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/2009RvMP...81.1827K">2009RvMP...81.1827K</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.81.1827">10.1103/RevModPhys.81.1827</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0034-6861">0034-6861</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=The+Casimir+force+between+real+materials%3A+Experiment+and+theory&rft.volume=81&rft.issue=4&rft.pages=1827-1885&rft.date=2009-12-21&rft_id=info%3Aarxiv%2F0902.4022&rft.issn=0034-6861&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.81.1827&rft_id=info%3Abibcode%2F2009RvMP...81.1827K&rft.aulast=Klimchitskaya&rft.aufirst=G.+L.&rft.au=Mohideen%2C+U.&rft.au=Mostepanenko%2C+V.+M.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FRevModPhys.81.1827&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-:3-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-:3_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:3_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCasimirPolder1948" class="citation journal cs1"><a href="/wiki/Hendrik_Casimir" title="Hendrik Casimir">Casimir, H. B. G.</a>; <a href="/wiki/Dirk_Polder" title="Dirk Polder">Polder, D.</a> (15 February 1948). "The Influence of Retardation on the London–van der Waals Forces". <i><a href="/wiki/Physical_Review" title="Physical Review">Physical Review</a></i>. <b>73</b> (4): 360–372. <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/1948PhRv...73..360C">1948PhRv...73..360C</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%2FPhysRev.73.360">10.1103/PhysRev.73.360</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-899X">0031-899X</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&rft.atitle=The+Influence+of+Retardation+on+the+London%E2%80%93van+der+Waals+Forces&rft.volume=73&rft.issue=4&rft.pages=360-372&rft.date=1948-02-15&rft.issn=0031-899X&rft_id=info%3Adoi%2F10.1103%2FPhysRev.73.360&rft_id=info%3Abibcode%2F1948PhRv...73..360C&rft.aulast=Casimir&rft.aufirst=H.+B.+G.&rft.au=Polder%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-:2-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDzyaloshinskiiLifshitzPitaevskii1961" class="citation journal cs1">Dzyaloshinskii, I E; Lifshitz, E M; Pitaevskii, Lev P (1961). "General Theory of van der Waals' Forces". <i>Soviet Physics Uspekhi</i>. <b>4</b> (2): 153. <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/1961SvPhU...4..153D">1961SvPhU...4..153D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1070%2FPU1961v004n02ABEH003330">10.1070/PU1961v004n02ABEH003330</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Soviet+Physics+Uspekhi&rft.atitle=General+Theory+of+van+der+Waals%27+Forces&rft.volume=4&rft.issue=2&rft.pages=153&rft.date=1961&rft_id=info%3Adoi%2F10.1070%2FPU1961v004n02ABEH003330&rft_id=info%3Abibcode%2F1961SvPhU...4..153D&rft.aulast=Dzyaloshinskii&rft.aufirst=I+E&rft.au=Lifshitz%2C+E+M&rft.au=Pitaevskii%2C+Lev+P&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIntravaiaBehunin2012" class="citation journal cs1">Intravaia, Francesco; Behunin, Ryan (28 December 2012). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevA.86.062517">"Casimir effect as a sum over modes in dissipative systems"</a>. <i>Physical Review A</i>. <b>86</b> (6): 062517. <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/1209.6072">1209.6072</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/2012PhRvA..86f2517I">2012PhRvA..86f2517I</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.86.062517">10.1103/PhysRevA.86.062517</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</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:119211980">119211980</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=Casimir+effect+as+a+sum+over+modes+in+dissipative+systems&rft.volume=86&rft.issue=6&rft.pages=062517&rft.date=2012-12-28&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119211980%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2012PhRvA..86f2517I&rft_id=info%3Aarxiv%2F1209.6072&rft.issn=1050-2947&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.86.062517&rft.aulast=Intravaia&rft.aufirst=Francesco&rft.au=Behunin%2C+Ryan&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevA.86.062517&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-Lamoureaux1997-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lamoureaux1997_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lamoureaux1997_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lamoureaux1997_7-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLamoreaux1997" class="citation journal cs1">Lamoreaux, S. K. (1997). "Demonstration of the Casimir Force in the 0.6 to 6 μm Range". <i>Physical Review Letters</i>. <b>78</b> (1): 5–8. <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/1997PhRvL..78....5L">1997PhRvL..78....5L</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%2FPhysRevLett.78.5">10.1103/PhysRevLett.78.5</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:25323874">25323874</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+Letters&rft.atitle=Demonstration+of+the+Casimir+Force+in+the+0.6+to+6+%CE%BCm+Range&rft.volume=78&rft.issue=1&rft.pages=5-8&rft.date=1997&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A25323874%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.78.5&rft_id=info%3Abibcode%2F1997PhRvL..78....5L&rft.aulast=Lamoreaux&rft.aufirst=S.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">E. L. Losada" <a rel="nofollow" class="external text" href="http://particulas.cnea.gov.ar/workshops/silafae/data/226.pdf">Functional Approach to the Fermionic Casimir Effect</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110531174312/http://particulas.cnea.gov.ar/workshops/silafae/data/226.pdf">Archived</a> 31 May 2011 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>"</span> </li> <li id="cite_note-Mohideen-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mohideen_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichael_BordagGalina_Leonidovna_KlimchitskayaUmar_Mohideen2009" class="citation book cs1">Michael Bordag; Galina Leonidovna Klimchitskaya; Umar Mohideen (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CqE1f_s5PgYC&pg=PA33">"Chapter I; § 3: Field quantization and vacuum energy in the presence of boundaries"</a>. <i>Advances in the Casimir effect</i>. Oxford University Press. pp. 33 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-923874-3" title="Special:BookSources/978-0-19-923874-3"><bdi>978-0-19-923874-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+I%3B+%C2%A7+3%3A+Field+quantization+and+vacuum+energy+in+the+presence+of+boundaries&rft.btitle=Advances+in+the+Casimir+effect&rft.pages=33+%27%27ff%27%27&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=978-0-19-923874-3&rft.au=Michael+Bordag&rft.au=Galina+Leonidovna+Klimchitskaya&rft.au=Umar+Mohideen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCqE1f_s5PgYC%26pg%3DPA33&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span> Reviewed in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLamoreaux2010" class="citation journal cs1">Lamoreaux, Steve K. (2010). "Advances in the Casimir Effect Advances in the Casimir Effect, M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko Oxford U. Press, New York, 2009. $150.00 (749 pp.). ISBN 978-0-19-923874-3". <i>Physics Today</i>. <b>63</b> (8): 50–51. <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/2010PhT....63h..50B">2010PhT....63h..50B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.3480079">10.1063/1.3480079</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Advances+in+the+Casimir+Effect+Advances+in+the+Casimir+Effect%2C+M.+Bordag%2C+G.+L.+Klimchitskaya%2C+U.+Mohideen%2C+and+V.+M.+Mostepanenko+Oxford+U.+Press%2C+New+York%2C+2009.+%24150.00+%28749+pp.%29.+ISBN+978-0-19-923874-3&rft.volume=63&rft.issue=8&rft.pages=50-51&rft.date=2010&rft_id=info%3Adoi%2F10.1063%2F1.3480079&rft_id=info%3Abibcode%2F2010PhT....63h..50B&rft.aulast=Lamoreaux&rft.aufirst=Steve+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" 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="CITEREFGriffithsHo2001" class="citation journal cs1">Griffiths, D. J.; Ho, E. (2001). "Classical Casimir effect for beads on a string". <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a></i>. <b>69</b> (11): 1173. <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/2001AmJPh..69.1173G">2001AmJPh..69.1173G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1396620">10.1119/1.1396620</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Classical+Casimir+effect+for+beads+on+a+string&rft.volume=69&rft.issue=11&rft.pages=1173&rft.date=2001&rft_id=info%3Adoi%2F10.1119%2F1.1396620&rft_id=info%3Abibcode%2F2001AmJPh..69.1173G&rft.aulast=Griffiths&rft.aufirst=D.+J.&rft.au=Ho%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" 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="CITEREFCooke1998" class="citation journal cs1">Cooke, J. H. (1998). "Casimir force on a loaded string". <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a></i>. <b>66</b> (7): 569–572. <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/1998AmJPh..66..569C">1998AmJPh..66..569C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.18907">10.1119/1.18907</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Casimir+force+on+a+loaded+string&rft.volume=66&rft.issue=7&rft.pages=569-572&rft.date=1998&rft_id=info%3Adoi%2F10.1119%2F1.18907&rft_id=info%3Abibcode%2F1998AmJPh..66..569C&rft.aulast=Cooke&rft.aufirst=J.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDenardoPudaLarraza2009" class="citation journal cs1">Denardo, B. C.; Puda, J. J.; Larraza, A. S. (2009). <a rel="nofollow" class="external text" href="https://zenodo.org/record/883891">"A water wave analog of the Casimir effect"</a>. <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a></i>. <b>77</b> (12): 1095. <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/2009AmJPh..77.1095D">2009AmJPh..77.1095D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.3211416">10.1119/1.3211416</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=A+water+wave+analog+of+the+Casimir+effect&rft.volume=77&rft.issue=12&rft.pages=1095&rft.date=2009&rft_id=info%3Adoi%2F10.1119%2F1.3211416&rft_id=info%3Abibcode%2F2009AmJPh..77.1095D&rft.aulast=Denardo&rft.aufirst=B.+C.&rft.au=Puda%2C+J.+J.&rft.au=Larraza%2C+A.+S.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F883891&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarrazaDenardo1998" class="citation journal cs1">Larraza, A. S.; Denardo, B. (1998). "An acoustic Casimir effect". <i><a href="/wiki/Physics_Letters_A" class="mw-redirect" title="Physics Letters A">Physics Letters A</a></i>. <b>248</b> (2–4): 151. <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/1998PhLA..248..151L">1998PhLA..248..151L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0375-9601%2898%2900652-5">10.1016/S0375-9601(98)00652-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+A&rft.atitle=An+acoustic+Casimir+effect&rft.volume=248&rft.issue=2%E2%80%934&rft.pages=151&rft.date=1998&rft_id=info%3Adoi%2F10.1016%2FS0375-9601%2898%2900652-5&rft_id=info%3Abibcode%2F1998PhLA..248..151L&rft.aulast=Larraza&rft.aufirst=A.+S.&rft.au=Denardo%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Astrid Lambrecht, Serge Reynaud and Cyriaque Genet (2007) "<a rel="nofollow" class="external text" href="http://www.fisica.unipa.it/~cewqo2007/Archive/presentations/Genet.pdf">Casimir In The Nanoworld</a>" <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091122143950/http://www.fisica.unipa.it/~cewqo2007/Archive/presentations/Genet.pdf">Archived</a> 22 November 2009 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGenetIntravaiaLambrechtReynaud2004" class="citation journal cs1">Genet, C.; Intravaia, F.; Lambrecht, A.; Reynaud, S. (2004). <a rel="nofollow" class="external text" href="http://aflb.ensmp.fr/AFLB-291/aflb291p227.pdf">"Electromagnetic vacuum fluctuations, Casimir and Van der Waals forces"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Annales_de_la_Fondation_Louis_de_Broglie" class="mw-redirect" title="Annales de la Fondation Louis de Broglie">Annales de la Fondation Louis de Broglie</a></i>. <b>29</b> (1–2): 311–328. <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/quant-ph/0302072">quant-ph/0302072</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/2003quant.ph..2072G">2003quant.ph..2072G</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161003065907/http://aflb.ensmp.fr/AFLB-291/aflb291p227.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 3 October 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+de+la+Fondation+Louis+de+Broglie&rft.atitle=Electromagnetic+vacuum+fluctuations%2C+Casimir+and+Van+der+Waals+forces&rft.volume=29&rft.issue=1%E2%80%932&rft.pages=311-328&rft.date=2004&rft_id=info%3Aarxiv%2Fquant-ph%2F0302072&rft_id=info%3Abibcode%2F2003quant.ph..2072G&rft.aulast=Genet&rft.aufirst=C.&rft.au=Intravaia%2C+F.&rft.au=Lambrecht%2C+A.&rft.au=Reynaud%2C+S.&rft_id=http%3A%2F%2Faflb.ensmp.fr%2FAFLB-291%2Faflb291p227.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://focus.aps.org/story/v2/st28">The Force of Empty Space</a>, <i><a href="/wiki/Physical_Review_Focus" title="Physical Review Focus">Physical Review Focus</a></i>, 3 December 1998</span> </li> <li id="cite_note-Physics_World-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-Physics_World_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Physics_World_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLambrecht2002" class="citation web cs1">Lambrecht, A. (1 September 2002). <a rel="nofollow" class="external text" href="https://physicsworld.com/a/the-casimir-effect-a-force-from-nothing/">"The Casimir effect: a force from nothing"</a>. <i><a href="/wiki/Physics_World" title="Physics World">Physics World</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">17 July</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+World&rft.atitle=The+Casimir+effect%3A+a+force+from+nothing&rft.date=2002-09-01&rft.aulast=Lambrecht&rft.aufirst=A.&rft_id=https%3A%2F%2Fphysicsworld.com%2Fa%2Fthe-casimir-effect-a-force-from-nothing%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080129093425/http://www.aip.org/pnu/1996/split/pnu300-3.htm">"American Institute of Physics News Note 1996"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.aip.org/pnu/1996/split/pnu300-3.htm">the original</a> on 29 January 2008<span class="reference-accessdate">. Retrieved <span class="nowrap">28 February</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=American+Institute+of+Physics+News+Note+1996&rft_id=http%3A%2F%2Fwww.aip.org%2Fpnu%2F1996%2Fsplit%2Fpnu300-3.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-:0-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaffe2005" class="citation journal cs1"><a href="/wiki/Robert_Jaffe_(physicist)" title="Robert Jaffe (physicist)">Jaffe, R.</a> (2005). "Casimir effect and the quantum vacuum". <i><a href="/wiki/Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>72</b> (2): 021301. <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/hep-th/0503158">hep-th/0503158</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/2005PhRvD..72b1301J">2005PhRvD..72b1301J</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.72.021301">10.1103/PhysRevD.72.021301</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:13171179">13171179</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=Casimir+effect+and+the+quantum+vacuum&rft.volume=72&rft.issue=2&rft.pages=021301&rft.date=2005&rft_id=info%3Aarxiv%2Fhep-th%2F0503158&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13171179%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.72.021301&rft_id=info%3Abibcode%2F2005PhRvD..72b1301J&rft.aulast=Jaffe&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-:1-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCasimir1948" class="citation journal cs1">Casimir, H. B. G. (1948). <a rel="nofollow" class="external text" href="https://www.dwc.knaw.nl/DL/publications/PU00018547.pdf">"On the attraction between two perfectly conducting plates"</a> <span class="cs1-format">(PDF)</span>. <i>Proc. Kon. Ned. Akad. Wet</i>. <b>51</b>: 793. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130418012405/http://www.dwc.knaw.nl/DL/publications/PU00018547.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 18 April 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Kon.+Ned.+Akad.+Wet.&rft.atitle=On+the+attraction+between+two+perfectly+conducting+plates&rft.volume=51&rft.pages=793&rft.date=1948&rft.aulast=Casimir&rft.aufirst=H.+B.+G.&rft_id=https%3A%2F%2Fwww.dwc.knaw.nl%2FDL%2Fpublications%2FPU00018547.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuLiuXieWang2015" class="citation journal cs1">Du, Z. Z.; Liu, H. M.; Xie, Y. L.; Wang, Q. H.; Liu, J.-M. (7 December 2015). "Spin Casimir effect in noncollinear quantum antiferromagnets: Torque equilibrium spin wave approach". <i>Physical Review B</i>. <b>92</b> (21): 214409. <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/1506.05211">1506.05211</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/2015arXiv150605211D">2015arXiv150605211D</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%2FPhysRevB.92.214409">10.1103/PhysRevB.92.214409</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1098-0121">1098-0121</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:118348464">118348464</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+B&rft.atitle=Spin+Casimir+effect+in+noncollinear+quantum+antiferromagnets%3A+Torque+equilibrium+spin+wave+approach&rft.volume=92&rft.issue=21&rft.pages=214409&rft.date=2015-12-07&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118348464%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2015arXiv150605211D&rft_id=info%3Aarxiv%2F1506.05211&rft.issn=1098-0121&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.92.214409&rft.aulast=Du&rft.aufirst=Z.+Z.&rft.au=Liu%2C+H.+M.&rft.au=Xie%2C+Y.+L.&rft.au=Wang%2C+Q.+H.&rft.au=Liu%2C+J.-M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-Rugh-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rugh_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSE_Rugh,_H_ZinkernagelZinkernagel2002" class="citation journal cs1">SE Rugh, H Zinkernagel; Zinkernagel (2002). <a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/398/">"The quantum vacuum and the cosmological constant problem"</a>. <i>Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics</i>. <b>33</b> (4): 663–705. <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/hep-th/0012253">hep-th/0012253</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/2002SHPMP..33..663R">2002SHPMP..33..663R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS1355-2198%2802%2900033-3">10.1016/S1355-2198(02)00033-3</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:9007190">9007190</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studies+in+History+and+Philosophy+of+Science+Part+B%3A+Studies+in+History+and+Philosophy+of+Modern+Physics&rft.atitle=The+quantum+vacuum+and+the+cosmological+constant+problem&rft.volume=33&rft.issue=4&rft.pages=663-705&rft.date=2002&rft_id=info%3Aarxiv%2Fhep-th%2F0012253&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9007190%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2FS1355-2198%2802%2900033-3&rft_id=info%3Abibcode%2F2002SHPMP..33..663R&rft.au=SE+Rugh%2C+H+Zinkernagel&rft.au=Zinkernagel&rft_id=http%3A%2F%2Fphilsci-archive.pitt.edu%2F398%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-Bianchi-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bianchi_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBianchiRovelli2010" class="citation arxiv cs1">Bianchi, Eugenio; Rovelli, Carlo (2010). "Why all these prejudices against a constant?". <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/1002.3966">1002.3966</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/astro-ph.CO">astro-ph.CO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Why+all+these+prejudices+against+a+constant%3F&rft.date=2010&rft_id=info%3Aarxiv%2F1002.3966&rft.aulast=Bianchi&rft.aufirst=Eugenio&rft.au=Rovelli%2C+Carlo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwingerDeRaadMilton1978" class="citation journal cs1">Schwinger, Julian; DeRaad, Lester L.; Milton, Kimball A. (1978). "Casimir effect in dielectrics". <i>Annals of Physics</i>. <b>115</b> (1): 1–23. <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/1978AnPhy.115....1S">1978AnPhy.115....1S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0003-4916%2878%2990172-0">10.1016/0003-4916(78)90172-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Physics&rft.atitle=Casimir+effect+in+dielectrics&rft.volume=115&rft.issue=1&rft.pages=1-23&rft.date=1978&rft_id=info%3Adoi%2F10.1016%2F0003-4916%2878%2990172-0&rft_id=info%3Abibcode%2F1978AnPhy.115....1S&rft.aulast=Schwinger&rft.aufirst=Julian&rft.au=DeRaad%2C+Lester+L.&rft.au=Milton%2C+Kimball+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNikolic2016" class="citation journal cs1">Nikolic, Hrvoje (2016). "Proof that Casimir force does not originate from vacuum energy". <i>Physics Letters B</i>. <b>761</b>: 197–202. <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/1605.04143">1605.04143</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/2016PhLB..761..197N">2016PhLB..761..197N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physletb.2016.08.036">10.1016/j.physletb.2016.08.036</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:119265677">119265677</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+B&rft.atitle=Proof+that+Casimir+force+does+not+originate+from+vacuum+energy&rft.volume=761&rft.pages=197-202&rft.date=2016&rft_id=info%3Aarxiv%2F1605.04143&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119265677%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.physletb.2016.08.036&rft_id=info%3Abibcode%2F2016PhLB..761..197N&rft.aulast=Nikolic&rft.aufirst=Hrvoje&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNikolic2017" class="citation journal cs1">Nikolic, Hrvoje (2017). "Is zero-point energy physical? A toy model for Casimir-like effect". <i>Annals of Physics</i>. <b>383</b>: 181–195. <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/1702.03291">1702.03291</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/2017AnPhy.383..181N">2017AnPhy.383..181N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aop.2017.05.013">10.1016/j.aop.2017.05.013</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:118883930">118883930</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Physics&rft.atitle=Is+zero-point+energy+physical%3F+A+toy+model+for+Casimir-like+effect&rft.volume=383&rft.pages=181-195&rft.date=2017&rft_id=info%3Aarxiv%2F1702.03291&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118883930%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.aop.2017.05.013&rft_id=info%3Abibcode%2F2017AnPhy.383..181N&rft.aulast=Nikolic&rft.aufirst=Hrvoje&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">For a brief summary, see the introduction in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPassanteSpagnolo2007" class="citation journal cs1">Passante, R.; Spagnolo, S. (2007). "Casimir–Polder interatomic potential between two atoms at finite temperature and in the presence of boundary conditions". <i>Physical Review A</i>. <b>76</b> (4): 042112. <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/0708.2240">0708.2240</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/2007PhRvA..76d2112P">2007PhRvA..76d2112P</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.76.042112">10.1103/PhysRevA.76.042112</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:119651683">119651683</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=Casimir%E2%80%93Polder+interatomic+potential+between+two+atoms+at+finite+temperature+and+in+the+presence+of+boundary+conditions&rft.volume=76&rft.issue=4&rft.pages=042112&rft.date=2007&rft_id=info%3Aarxiv%2F0708.2240&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119651683%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.76.042112&rft_id=info%3Abibcode%2F2007PhRvA..76d2112P&rft.aulast=Passante&rft.aufirst=R.&rft.au=Spagnolo%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRuggieroVillani1977" class="citation journal cs1">Ruggiero, Zimerman; Villani (1977). <a rel="nofollow" class="external text" href="http://www.sbfisica.org.br/bjp/download/v07/v07a43.pdf">"Application of Analytic Regularization to the Casimir Forces"</a> <span class="cs1-format">(PDF)</span>. <i>Revista Brasileira de Física</i>. <b>7</b> (3). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150403165408/http://www.sbfisica.org.br/bjp/download/v07/v07a43.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 3 April 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Revista+Brasileira+de+F%C3%ADsica&rft.atitle=Application+of+Analytic+Regularization+to+the+Casimir+Forces&rft.volume=7&rft.issue=3&rft.date=1977&rft.aulast=Ruggiero&rft.aufirst=Zimerman&rft.au=Villani&rft_id=http%3A%2F%2Fwww.sbfisica.org.br%2Fbjp%2Fdownload%2Fv07%2Fv07a43.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDzyaloshinskiiKats2004" class="citation journal cs1">Dzyaloshinskii, I E; Kats, E I (2004). "Casimir forces in modulated systems". <i>Journal of Physics: Condensed Matter</i>. <b>16</b> (32): 5659. <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/cond-mat/0408348">cond-mat/0408348</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/2004JPCM...16.5659D">2004JPCM...16.5659D</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%2F0953-8984%2F16%2F32%2F003">10.1088/0953-8984/16/32/003</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:250897415">250897415</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+Physics%3A+Condensed+Matter&rft.atitle=Casimir+forces+in+modulated+systems&rft.volume=16&rft.issue=32&rft.pages=5659&rft.date=2004&rft_id=info%3Aarxiv%2Fcond-mat%2F0408348&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250897415%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0953-8984%2F16%2F32%2F003&rft_id=info%3Abibcode%2F2004JPCM...16.5659D&rft.aulast=Dzyaloshinskii&rft.aufirst=I+E&rft.au=Kats%2C+E+I&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">V. A. Parsegian, <i>Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists</i> (Cambridge Univ. Press, 2006).</span> </li> <li id="cite_note-Rodriguez11-review-31"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rodriguez11-review_31-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rodriguez11-review_31-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Rodriguez11-review_31-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRodriguezCapassoJohnson2011" class="citation journal cs1">Rodriguez, A. W.; Capasso, F.; Johnson, Steven G. (2011). "The Casimir effect in microstructured geometries". <i>Nature Photonics</i>. <b>5</b> (4): 211–221. <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/2011NaPho...5..211R">2011NaPho...5..211R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphoton.2011.39">10.1038/nphoton.2011.39</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Photonics&rft.atitle=The+Casimir+effect+in+microstructured+geometries&rft.volume=5&rft.issue=4&rft.pages=211-221&rft.date=2011&rft_id=info%3Adoi%2F10.1038%2Fnphoton.2011.39&rft_id=info%3Abibcode%2F2011NaPho...5..211R&rft.aulast=Rodriguez&rft.aufirst=A.+W.&rft.au=Capasso%2C+F.&rft.au=Johnson%2C+Steven+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span> Review article.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">B. V. Derjaguin, I. I. Abrikosova, and E. M. Lifshitz, <i>Quarterly Reviews, Chemical Society</i>, vol. 10, 295–329 (1956).</span> </li> <li id="cite_note-Reid2013-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-Reid2013_33-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReidWhiteJohnson2011" class="citation journal cs1">Reid, M. T. H.; White, J.; Johnson, S. G. (2011). "Computation of Casimir interactions between arbitrary three-dimensional objects with arbitrary material properties". <i>Physical Review A</i>. <b>84</b> (1): 010503(R). <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/1010.5539">1010.5539</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/2011PhRvA..84a0503R">2011PhRvA..84a0503R</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.84.010503">10.1103/PhysRevA.84.010503</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:197461628">197461628</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=Computation+of+Casimir+interactions+between+arbitrary+three-dimensional+objects+with+arbitrary+material+properties&rft.volume=84&rft.issue=1&rft.pages=010503%28R%29&rft.date=2011&rft_id=info%3Aarxiv%2F1010.5539&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A197461628%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.84.010503&rft_id=info%3Abibcode%2F2011PhRvA..84a0503R&rft.aulast=Reid&rft.aufirst=M.+T.+H.&rft.au=White%2C+J.&rft.au=Johnson%2C+S.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSparnaay1957" class="citation journal cs1">Sparnaay, M. J. (1957). "Attractive Forces between Flat Plates". <i>Nature</i>. <b>180</b> (4581): 334–335. <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/1957Natur.180..334S">1957Natur.180..334S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F180334b0">10.1038/180334b0</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:4263111">4263111</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Attractive+Forces+between+Flat+Plates&rft.volume=180&rft.issue=4581&rft.pages=334-335&rft.date=1957&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4263111%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F180334b0&rft_id=info%3Abibcode%2F1957Natur.180..334S&rft.aulast=Sparnaay&rft.aufirst=M.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSparnaay1958" class="citation journal cs1">Sparnaay, M (1958). "Measurements of attractive forces between flat plates". <i>Physica</i>. <b>24</b> (6–10): 751–764. <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/1958Phy....24..751S">1958Phy....24..751S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0031-8914%2858%2980090-7">10.1016/S0031-8914(58)80090-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica&rft.atitle=Measurements+of+attractive+forces+between+flat+plates&rft.volume=24&rft.issue=6%E2%80%9310&rft.pages=751-764&rft.date=1958&rft_id=info%3Adoi%2F10.1016%2FS0031-8914%2858%2980090-7&rft_id=info%3Abibcode%2F1958Phy....24..751S&rft.aulast=Sparnaay&rft.aufirst=M&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMohideenRoy1998" class="citation journal cs1">Mohideen, U.; Roy, Anushree (1998). "Precision Measurement of the Casimir Force from 0.1 to 0.9 µm". <i>Physical Review Letters</i>. <b>81</b> (21): 4549–4552. <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/physics/9805038">physics/9805038</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/1998PhRvL..81.4549M">1998PhRvL..81.4549M</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%2FPhysRevLett.81.4549">10.1103/PhysRevLett.81.4549</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:56132451">56132451</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+Letters&rft.atitle=Precision+Measurement+of+the+Casimir+Force+from+0.1+to+0.9+%C2%B5m&rft.volume=81&rft.issue=21&rft.pages=4549-4552&rft.date=1998&rft_id=info%3Aarxiv%2Fphysics%2F9805038&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A56132451%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.81.4549&rft_id=info%3Abibcode%2F1998PhRvL..81.4549M&rft.aulast=Mohideen&rft.aufirst=U.&rft.au=Roy%2C+Anushree&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBressiCarugnoOnofrioRuoso2002" class="citation journal cs1">Bressi, G.; Carugno, G.; Onofrio, R.; Ruoso, G. (2002). "Measurement of the Casimir Force between Parallel Metallic Surfaces". <i>Physical Review Letters</i>. <b>88</b> (4): 041804. <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/quant-ph/0203002">quant-ph/0203002</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/2002PhRvL..88d1804B">2002PhRvL..88d1804B</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%2FPhysRevLett.88.041804">10.1103/PhysRevLett.88.041804</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11801108">11801108</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:43354557">43354557</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+Letters&rft.atitle=Measurement+of+the+Casimir+Force+between+Parallel+Metallic+Surfaces&rft.volume=88&rft.issue=4&rft.pages=041804&rft.date=2002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A43354557%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2002PhRvL..88d1804B&rft_id=info%3Aarxiv%2Fquant-ph%2F0203002&rft_id=info%3Apmid%2F11801108&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.88.041804&rft.aulast=Bressi&rft.aufirst=G.&rft.au=Carugno%2C+G.&rft.au=Onofrio%2C+R.&rft.au=Ruoso%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlimchitskayaMohideenMostepanenko2009" class="citation journal cs1">Klimchitskaya, G. L.; Mohideen, U.; Mostepanenko, V. M. (21 December 2009). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.81.1827">"The Casimir force between real materials: Experiment and theory"</a>. <i>Reviews of Modern Physics</i>. <b>81</b> (4): 1827–1885. <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/0902.4022">0902.4022</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/2009RvMP...81.1827K">2009RvMP...81.1827K</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.81.1827">10.1103/RevModPhys.81.1827</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0034-6861">0034-6861</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=The+Casimir+force+between+real+materials%3A+Experiment+and+theory&rft.volume=81&rft.issue=4&rft.pages=1827-1885&rft.date=2009-12-21&rft_id=info%3Aarxiv%2F0902.4022&rft.issn=0034-6861&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.81.1827&rft_id=info%3Abibcode%2F2009RvMP...81.1827K&rft.aulast=Klimchitskaya&rft.aufirst=G.+L.&rft.au=Mohideen%2C+U.&rft.au=Mostepanenko%2C+V.+M.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FRevModPhys.81.1827&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZaoMarcetRodriguezReid2013" class="citation journal cs1">Zao, J.; Marcet, Z.; Rodriguez, A. W.; Reid, M. T. H.; McCauley, A. P.; Kravchenko, I. I.; Lu, T.; Bao, Y.; Johnson, S. G.; Chan, H. B.; et al. (14 May 2013). "Casimir forces on a silicon micromechanical chip". <i>Nature Communications</i>. <b>4</b>: 1845. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1207.6163">1207.6163</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/2013NatCo...4.1845Z">2013NatCo...4.1845Z</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fncomms2842">10.1038/ncomms2842</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23673630">23673630</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:46359798">46359798</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Communications&rft.atitle=Casimir+forces+on+a+silicon+micromechanical+chip&rft.volume=4&rft.pages=1845&rft.date=2013-05-14&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A46359798%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2013NatCo...4.1845Z&rft_id=info%3Aarxiv%2F1207.6163&rft_id=info%3Apmid%2F23673630&rft_id=info%3Adoi%2F10.1038%2Fncomms2842&rft.aulast=Zao&rft.aufirst=J.&rft.au=Marcet%2C+Z.&rft.au=Rodriguez%2C+A.+W.&rft.au=Reid%2C+M.+T.+H.&rft.au=McCauley%2C+A.+P.&rft.au=Kravchenko%2C+I.+I.&rft.au=Lu%2C+T.&rft.au=Bao%2C+Y.&rft.au=Johnson%2C+S.+G.&rft.au=Chan%2C+H.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuWangNgNikolic2017" class="citation journal cs1">Lu, T.; Wang, Mingkang; Ng, C. Y.; Nikolic, M.; Chan, C. T.; Rodriguez, Alejandro; Chan, H. B.; et al. (9 January 2017). "Measurement of non-monotonic Casimir forces between silicon nanostructures". <i>Nature Photonics</i>. <b>11</b> (2): 97–101. <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/1701.02351">1701.02351</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/2017NaPho..11...97T">2017NaPho..11...97T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphoton.2016.254">10.1038/nphoton.2016.254</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:119327017">119327017</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Photonics&rft.atitle=Measurement+of+non-monotonic+Casimir+forces+between+silicon+nanostructures&rft.volume=11&rft.issue=2&rft.pages=97-101&rft.date=2017-01-09&rft_id=info%3Aarxiv%2F1701.02351&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119327017%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2Fnphoton.2016.254&rft_id=info%3Abibcode%2F2017NaPho..11...97T&rft.aulast=Lu&rft.aufirst=T.&rft.au=Wang%2C+Mingkang&rft.au=Ng%2C+C.+Y.&rft.au=Nikolic%2C+M.&rft.au=Chan%2C+C.+T.&rft.au=Rodriguez%2C+Alejandro&rft.au=Chan%2C+H.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWangTangNgMessina2021" class="citation journal cs1">Wang, Mingkang; Tang, L.; Ng, C. Y.; Messina, Riccardo; Guizal, Brahim; Crosse, J. A.; Antezza, Mauro; Chan, C. T.; Chan, H. B.; et al. (26 January 2021). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7838308">"Strong geometry dependence of the Casimir force between interpenetrated rectangular gratings"</a>. <i>Nature Communications</i>. <b>12</b> (1): 600. <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/2009.02187">2009.02187</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/2021NatCo..12..600W">2021NatCo..12..600W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41467-021-20891-4">10.1038/s41467-021-20891-4</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7838308">7838308</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/33500401">33500401</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Communications&rft.atitle=Strong+geometry+dependence+of+the+Casimir+force+between+interpenetrated+rectangular+gratings&rft.volume=12&rft.issue=1&rft.pages=600&rft.date=2021-01-26&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7838308%23id-name%3DPMC&rft_id=info%3Abibcode%2F2021NatCo..12..600W&rft_id=info%3Aarxiv%2F2009.02187&rft_id=info%3Apmid%2F33500401&rft_id=info%3Adoi%2F10.1038%2Fs41467-021-20891-4&rft.aulast=Wang&rft.aufirst=Mingkang&rft.au=Tang%2C+L.&rft.au=Ng%2C+C.+Y.&rft.au=Messina%2C+Riccardo&rft.au=Guizal%2C+Brahim&rft.au=Crosse%2C+J.+A.&rft.au=Antezza%2C+Mauro&rft.au=Chan%2C+C.+T.&rft.au=Chan%2C+H.+B.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7838308&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAjdariDuplantierHonePeliti1992" class="citation journal cs1">Ajdari, A.; Duplantier, B.; Hone, D.; Peliti, L.; Prost, J. (March 1992). <a rel="nofollow" class="external text" href="https://jp2.journaldephysique.org/10.1051/jp2:1992145/pdf">"<span class="cs1-kern-left"></span>"Pseudo-Casimir" effect in liquid crystals"</a>. <i>Journal de Physique II</i>. <b>2</b> (3): 487–501. <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/1992JPhy2...2..487A">1992JPhy2...2..487A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1051%2Fjp2%3A1992145">10.1051/jp2:1992145</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:55236741">55236741</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+Physique+II&rft.atitle=%22Pseudo-Casimir%22+effect+in+liquid+crystals&rft.volume=2&rft.issue=3&rft.pages=487-501&rft.date=1992-03&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A55236741%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1051%2Fjp2%3A1992145&rft_id=info%3Abibcode%2F1992JPhy2...2..487A&rft.aulast=Ajdari&rft.aufirst=A.&rft.au=Duplantier%2C+B.&rft.au=Hone%2C+D.&rft.au=Peliti%2C+L.&rft.au=Prost%2C+J.&rft_id=https%3A%2F%2Fjp2.journaldephysique.org%2F10.1051%2Fjp2%3A1992145%2Fpdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFullingDavies1976" class="citation journal cs1">Fulling, S. A.; Davies, P. C. W. (1976). "Radiation from a Moving Mirror in Two Dimensional Space-Time: Conformal Anomaly". <i>Proceedings of the Royal Society A</i>. <b>348</b> (1654): 393. <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/1976RSPSA.348..393F">1976RSPSA.348..393F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1976.0045">10.1098/rspa.1976.0045</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:122176090">122176090</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A&rft.atitle=Radiation+from+a+Moving+Mirror+in+Two+Dimensional+Space-Time%3A+Conformal+Anomaly&rft.volume=348&rft.issue=1654&rft.pages=393&rft.date=1976&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122176090%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1976.0045&rft_id=info%3Abibcode%2F1976RSPSA.348..393F&rft.aulast=Fulling&rft.aufirst=S.+A.&rft.au=Davies%2C+P.+C.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation magazine cs1"><a rel="nofollow" class="external text" href="http://www.technologyreview.com/blog/arxiv/26813/">"First Observation of the Dynamical Casimir Effect"</a>. <i>Technology Review</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Technology+Review&rft.atitle=First+Observation+of+the+Dynamical+Casimir+Effect&rft_id=http%3A%2F%2Fwww.technologyreview.com%2Fblog%2Farxiv%2F26813%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilsonJohanssonPourkabirianSimoen2011" class="citation journal cs1">Wilson, C. M.; Johansson, G.; Pourkabirian, A.; Simoen, M.; Johansson, J. R.; Duty, T.; Nori, F.; Delsing, P. (2011). "Observation of the Dynamical Casimir Effect in a Superconducting Circuit". <i>Nature</i>. <b>479</b> (7373): 376–379. <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/1105.4714">1105.4714</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/2011Natur.479..376W">2011Natur.479..376W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature10561">10.1038/nature10561</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22094697">22094697</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:219735">219735</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Observation+of+the+Dynamical+Casimir+Effect+in+a+Superconducting+Circuit&rft.volume=479&rft.issue=7373&rft.pages=376-379&rft.date=2011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A219735%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2011Natur.479..376W&rft_id=info%3Aarxiv%2F1105.4714&rft_id=info%3Apmid%2F22094697&rft_id=info%3Adoi%2F10.1038%2Fnature10561&rft.aulast=Wilson&rft.aufirst=C.+M.&rft.au=Johansson%2C+G.&rft.au=Pourkabirian%2C+A.&rft.au=Simoen%2C+M.&rft.au=Johansson%2C+J.+R.&rft.au=Duty%2C+T.&rft.au=Nori%2C+F.&rft.au=Delsing%2C+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation magazine cs1"><a rel="nofollow" class="external text" href="https://www.pnas.org/content/110/11/4234">"Dynamical Casimir effect in a Josephson metamaterial"</a>. <i>Proceedings of the National Academy of Sciences of the United States of America</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences+of+the+United+States+of+America&rft.atitle=Dynamical+Casimir+effect+in+a+Josephson+metamaterial&rft_id=https%3A%2F%2Fwww.pnas.org%2Fcontent%2F110%2F11%2F4234&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVezzoliMussotWesterbergKudlinski2019" class="citation journal cs1">Vezzoli, S.; Mussot, A.; Westerberg, N.; Kudlinski, A.; Saleh, H. D.; Prain, A.; Biancalana, F.; Lantz, E.; Faccio, D. (2019). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/s42005-019-0183-z">"Optical analogue of the dynamical Casimir effect in a dispersion-oscillating fibre"</a>. <i>Communications Physics</i>. <b>2</b> (1): 84. <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/1811.04262">1811.04262</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/2019CmPhy...2...84V">2019CmPhy...2...84V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs42005-019-0183-z">10.1038/s42005-019-0183-z</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:53691352">53691352</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+Physics&rft.atitle=Optical+analogue+of+the+dynamical+Casimir+effect+in+a+dispersion-oscillating+fibre&rft.volume=2&rft.issue=1&rft.pages=84&rft.date=2019&rft_id=info%3Aarxiv%2F1811.04262&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53691352%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2Fs42005-019-0183-z&rft_id=info%3Abibcode%2F2019CmPhy...2...84V&rft.aulast=Vezzoli&rft.aufirst=S.&rft.au=Mussot%2C+A.&rft.au=Westerberg%2C+N.&rft.au=Kudlinski%2C+A.&rft.au=Saleh%2C+H.+D.&rft.au=Prain%2C+A.&rft.au=Biancalana%2C+F.&rft.au=Lantz%2C+E.&rft.au=Faccio%2C+D.&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2Fs42005-019-0183-z&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilczekLinderGood2020" class="citation journal cs1">Wilczek, F.; Linder, E. V.; Good, M.R.R. (2020). "Moving mirror model for quasithermal radiation fields". <i><a href="/wiki/Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>101</b> (2): 025012. <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/2020PhRvD.101b5012G">2020PhRvD.101b5012G</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.101.025012">10.1103/PhysRevD.101.025012</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F125524">1721.1/125524</a></span>. <a href="/wiki/OSTI_(identifier)" class="mw-redirect" title="OSTI (identifier)">OSTI</a> <a rel="nofollow" class="external text" href="https://www.osti.gov/biblio/1635008">1635008</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:213899274">213899274</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=Moving+mirror+model+for+quasithermal+radiation+fields&rft.volume=101&rft.issue=2&rft.pages=025012&rft.date=2020&rft_id=info%3Ahdl%2F1721.1%2F125524&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A213899274%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.osti.gov%2Fbiblio%2F1635008%23id-name%3DOSTI&rft_id=info%3Abibcode%2F2020PhRvD.101b5012G&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.101.025012&rft.aulast=Wilczek&rft.aufirst=F.&rft.au=Linder%2C+E.+V.&rft.au=Good%2C+M.R.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirrellDavies1982" class="citation book cs1">Birrell, N. D.; Davies, P. C. W. (1982). <a rel="nofollow" class="external text" href="https://doi.org/10.1017/CBO9780511622632"><i>Quantum Fields in Curved Space</i></a>. Cambridge Monographs on Mathematical Physics. Cambridge University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511622632">10.1017/CBO9780511622632</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-23385-9" title="Special:BookSources/978-0-521-23385-9"><bdi>978-0-521-23385-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Fields+in+Curved+Space&rft.series=Cambridge+Monographs+on+Mathematical+Physics&rft.pub=Cambridge+University+Press&rft.date=1982&rft_id=info%3Adoi%2F10.1017%2FCBO9780511622632&rft.isbn=978-0-521-23385-9&rft.aulast=Birrell&rft.aufirst=N.+D.&rft.au=Davies%2C+P.+C.+W.&rft_id=https%3A%2F%2Fdoi.org%2F10.1017%2FCBO9780511622632&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-DLP-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-DLP_50-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDzyaloshinskiiLifshitzPitaevskii1961" class="citation journal cs1">Dzyaloshinskii, I.E.; Lifshitz, E.M.; Pitaevskii, L.P. (1961). "The general theory of van der Waals forces". <i>Advances in Physics</i>. <b>10</b> (38): 165. <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/1961AdPhy..10..165D">1961AdPhy..10..165D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00018736100101281">10.1080/00018736100101281</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Physics&rft.atitle=The+general+theory+of+van+der+Waals+forces&rft.volume=10&rft.issue=38&rft.pages=165&rft.date=1961&rft_id=info%3Adoi%2F10.1080%2F00018736100101281&rft_id=info%3Abibcode%2F1961AdPhy..10..165D&rft.aulast=Dzyaloshinskii&rft.aufirst=I.E.&rft.au=Lifshitz%2C+E.M.&rft.au=Pitaevskii%2C+L.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMundayCapassoParsegian2009" class="citation journal cs1">Munday, J.N.; Capasso, F.; Parsegian, V.A. (2009). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4169270">"Measured long-range repulsive Casimir–Lifshitz forces"</a>. <i>Nature</i>. <b>457</b> (7226): 170–3. <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/2009Natur.457..170M">2009Natur.457..170M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature07610">10.1038/nature07610</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4169270">4169270</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19129843">19129843</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Measured+long-range+repulsive+Casimir%E2%80%93Lifshitz+forces&rft.volume=457&rft.issue=7226&rft.pages=170-3&rft.date=2009&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4169270%23id-name%3DPMC&rft_id=info%3Apmid%2F19129843&rft_id=info%3Adoi%2F10.1038%2Fnature07610&rft_id=info%3Abibcode%2F2009Natur.457..170M&rft.aulast=Munday&rft.aufirst=J.N.&rft.au=Capasso%2C+F.&rft.au=Parsegian%2C+V.A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4169270&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHighfield2007" class="citation news cs1">Highfield, Roger (6 August 2007). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080513213336/http://www.telegraph.co.uk/news/1559579/Physicists-have-%27solved%27-mystery-of-levitation.html">"Physicists have 'solved' mystery of levitation"</a>. <i>The Daily Telegraph</i>. London. Archived from <a rel="nofollow" class="external text" href="https://www.telegraph.co.uk/news/1559579/Physicists-have-%27solved%27-mystery-of-levitation.html">the original</a> on 13 May 2008<span class="reference-accessdate">. Retrieved <span class="nowrap">28 April</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Daily+Telegraph&rft.atitle=Physicists+have+%27solved%27+mystery+of+levitation&rft.date=2007-08-06&rft.aulast=Highfield&rft.aufirst=Roger&rft_id=https%3A%2F%2Fwww.telegraph.co.uk%2Fnews%2F1559579%2FPhysicists-have-%2527solved%2527-mystery-of-levitation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeonhardtPhilbin2007" class="citation journal cs1">Leonhardt, Ulf; Philbin, Thomas G. (August 2007). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F9%2F8%2F254">"Quantum levitation by left-handed metamaterials"</a>. <i><a href="/wiki/New_Journal_of_Physics" title="New Journal of Physics">New Journal of Physics</a></i>. <b>9</b> (8). <a href="/wiki/IOP_Publishing" title="IOP Publishing">IOP Publishing</a> and <a href="/wiki/German_Physical_Society" title="German Physical Society">German Physical Society</a>: 254. <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/quant-ph/0608115">quant-ph/0608115</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/2007NJPh....9..254L">2007NJPh....9..254L</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/10.1088%2F1367-2630%2F9%2F8%2F254">10.1088/1367-2630/9/8/254</a></span>.</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=Quantum+levitation+by+left-handed+metamaterials&rft.volume=9&rft.issue=8&rft.pages=254&rft.date=2007-08&rft_id=info%3Aarxiv%2Fquant-ph%2F0608115&rft_id=info%3Adoi%2F10.1088%2F1367-2630%2F9%2F8%2F254&rft_id=info%3Abibcode%2F2007NJPh....9..254L&rft.aulast=Leonhardt&rft.aufirst=Ulf&rft.au=Philbin%2C+Thomas+G.&rft_id=https%3A%2F%2Fdoi.org%2F10.1088%252F1367-2630%252F9%252F8%252F254&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiltonAbaloParasharPourtolami2012" class="citation journal cs1">Milton, K. A.; Abalo, E. K.; Parashar, Prachi; Pourtolami, Nima; Brevik, Iver; Ellingsen, Simen A. (2012). "Repulsive Casimir and Casimir–Polder Forces". <i>J. Phys. A</i>. <b>45</b> (37): 4006. <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/1202.6415">1202.6415</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/2012JPhA...45K4006M">2012JPhA...45K4006M</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%2F1751-8113%2F45%2F37%2F374006">10.1088/1751-8113/45/37/374006</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:118364958">118364958</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Phys.+A&rft.atitle=Repulsive+Casimir+and+Casimir%E2%80%93Polder+Forces&rft.volume=45&rft.issue=37&rft.pages=4006&rft.date=2012&rft_id=info%3Aarxiv%2F1202.6415&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118364958%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1751-8113%2F45%2F37%2F374006&rft_id=info%3Abibcode%2F2012JPhA...45K4006M&rft.aulast=Milton&rft.aufirst=K.+A.&rft.au=Abalo%2C+E.+K.&rft.au=Parashar%2C+Prachi&rft.au=Pourtolami%2C+Nima&rft.au=Brevik%2C+Iver&rft.au=Ellingsen%2C+Simen+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJiangWilczek2019" class="citation journal cs1">Jiang, Qing-Dong; Wilczek, Frank (4 March 2019). "Chiral Casimir forces: Repulsive, enhanced, tunable". <i>Physical Review B</i>. <b>99</b> (12): 125403. <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/1805.07994">1805.07994</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/2019PhRvB..99l5403J">2019PhRvB..99l5403J</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%2FPhysRevB.99.125403">10.1103/PhysRevB.99.125403</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:67802144">67802144</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+B&rft.atitle=Chiral+Casimir+forces%3A+Repulsive%2C+enhanced%2C+tunable&rft.volume=99&rft.issue=12&rft.pages=125403&rft.date=2019-03-04&rft_id=info%3Aarxiv%2F1805.07994&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A67802144%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.99.125403&rft_id=info%3Abibcode%2F2019PhRvB..99l5403J&rft.aulast=Jiang&rft.aufirst=Qing-Dong&rft.au=Wilczek%2C+Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyer1968" class="citation journal cs1">Boyer, Timothy H. (25 October 1968). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRev.174.1764">"Quantum Electromagnetic Zero-Point Energy of a Conducting Spherical Shell and the Casimir Model for a Charged Particle"</a>. <i>Physical Review</i>. <b>174</b> (5): 1764–1776. <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/1968PhRv..174.1764B">1968PhRv..174.1764B</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%2FPhysRev.174.1764">10.1103/PhysRev.174.1764</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&rft.atitle=Quantum+Electromagnetic+Zero-Point+Energy+of+a+Conducting+Spherical+Shell+and+the+Casimir+Model+for+a+Charged+Particle&rft.volume=174&rft.issue=5&rft.pages=1764-1776&rft.date=1968-10-25&rft_id=info%3Adoi%2F10.1103%2FPhysRev.174.1764&rft_id=info%3Abibcode%2F1968PhRv..174.1764B&rft.aulast=Boyer&rft.aufirst=Timothy+H.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRev.174.1764&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSanderson2009" class="citation journal cs1">Sanderson, Katharine (7 January 2009). <a rel="nofollow" class="external text" href="http://www.nature.com/articles/news.2009.4">"Quantum force gets repulsive"</a>. <i>Nature</i>: news.2009.4. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnews.2009.4">10.1038/news.2009.4</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0028-0836">0028-0836</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Quantum+force+gets+repulsive&rft.pages=news.2009.4&rft.date=2009-01-07&rft_id=info%3Adoi%2F10.1038%2Fnews.2009.4&rft.issn=0028-0836&rft.aulast=Sanderson&rft.aufirst=Katharine&rft_id=http%3A%2F%2Fwww.nature.com%2Farticles%2Fnews.2009.4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCapassoMundayIannuzziChan2007" class="citation journal cs1">Capasso, F.; Munday, J.N.; Iannuzzi, D.; Chan, H.B. (2007). "Casimir forces and quantum electrodynamical torques: physics and nanomechanics". <i>IEEE Journal of Selected Topics in Quantum Electronics</i>. <b>13</b> (2): 400. <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/2007IJSTQ..13..400C">2007IJSTQ..13..400C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FJSTQE.2007.893082">10.1109/JSTQE.2007.893082</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:32996610">32996610</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Journal+of+Selected+Topics+in+Quantum+Electronics&rft.atitle=Casimir+forces+and+quantum+electrodynamical+torques%3A+physics+and+nanomechanics&rft.volume=13&rft.issue=2&rft.pages=400&rft.date=2007&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A32996610%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FJSTQE.2007.893082&rft_id=info%3Abibcode%2F2007IJSTQ..13..400C&rft.aulast=Capasso&rft.aufirst=F.&rft.au=Munday%2C+J.N.&rft.au=Iannuzzi%2C+D.&rft.au=Chan%2C+H.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerryWalliserMacLay1995" class="citation journal cs1">Serry, F.M.; Walliser, D.; MacLay, G.J. (1995). <a rel="nofollow" class="external text" href="http://www.quantumfields.com/IEEEJMEMSACO.pdf">"The anharmonic Casimir oscillator (ACO)-the Casimir effect in a model microelectromechanical system"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Journal_of_Microelectromechanical_Systems" title="Journal of Microelectromechanical Systems">Journal of Microelectromechanical Systems</a></i>. <b>4</b> (4): 193. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F84.475546">10.1109/84.475546</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060313160526/http://www.quantumfields.com/IEEEJMEMSACO.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 13 March 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Microelectromechanical+Systems&rft.atitle=The+anharmonic+Casimir+oscillator+%28ACO%29-the+Casimir+effect+in+a+model+microelectromechanical+system&rft.volume=4&rft.issue=4&rft.pages=193&rft.date=1995&rft_id=info%3Adoi%2F10.1109%2F84.475546&rft.aulast=Serry&rft.aufirst=F.M.&rft.au=Walliser%2C+D.&rft.au=MacLay%2C+G.J.&rft_id=http%3A%2F%2Fwww.quantumfields.com%2FIEEEJMEMSACO.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerryWalliserMaclay1995" class="citation journal cs1">Serry, F. M.; Walliser, D.; Maclay, G. J. (1995). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060313160526/http://www.quantumfields.com/IEEEJMEMSACO.pdf">"The anharmonic Casimir oscillator (ACO)-the Casimir effect in a model microelectromechanical system"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Journal_of_Microelectromechanical_Systems" title="Journal of Microelectromechanical Systems">Journal of Microelectromechanical Systems</a></i>. <b>4</b> (4): 193–205. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F84.475546">10.1109/84.475546</a>. Archived from <a rel="nofollow" class="external text" href="http://www.quantumfields.com/IEEEJMEMSACO.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 13 March 2006<span class="reference-accessdate">. Retrieved <span class="nowrap">24 October</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Microelectromechanical+Systems&rft.atitle=The+anharmonic+Casimir+oscillator+%28ACO%29-the+Casimir+effect+in+a+model+microelectromechanical+system&rft.volume=4&rft.issue=4&rft.pages=193-205&rft.date=1995&rft_id=info%3Adoi%2F10.1109%2F84.475546&rft.aulast=Serry&rft.aufirst=F.+M.&rft.au=Walliser%2C+D.&rft.au=Maclay%2C+G.+J.&rft_id=http%3A%2F%2Fwww.quantumfields.com%2FIEEEJMEMSACO.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerryWalliserMaclay1998" class="citation journal cs1">Serry, F. Michael; Walliser, Dirk; Maclay, G. Jordan (1998). <a rel="nofollow" class="external text" href="https://www.mit.edu/~kardar/research/seminars/Casimir/SerryMEMS.pdf">"The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS)"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Applied Physics</i>. <b>84</b> (5): 2501–2506. <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/1998JAP....84.2501S">1998JAP....84.2501S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.368410">10.1063/1.368410</a><span class="reference-accessdate">. Retrieved <span class="nowrap">24 October</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Applied+Physics&rft.atitle=The+role+of+the+casimir+effect+in+the+static+deflection+and+stiction+of+membrane+strips+in+microelectromechanical+systems+%28MEMS%29&rft.volume=84&rft.issue=5&rft.pages=2501-2506&rft.date=1998&rft_id=info%3Adoi%2F10.1063%2F1.368410&rft_id=info%3Abibcode%2F1998JAP....84.2501S&rft.aulast=Serry&rft.aufirst=F.+Michael&rft.au=Walliser%2C+Dirk&rft.au=Maclay%2C+G.+Jordan&rft_id=https%3A%2F%2Fwww.mit.edu%2F~kardar%2Fresearch%2Fseminars%2FCasimir%2FSerryMEMS.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBordagKlimchitskayaMohideenMostepanenko2009" class="citation book cs1">Bordag, M; Klimchitskaya, G. L.; Mohideen, U.; Mostepanenko, V. M. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CqE1f_s5PgYC"><i>Advances in the Casimir Effect</i></a>. Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-923874-3" title="Special:BookSources/978-0-19-923874-3"><bdi>978-0-19-923874-3</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2009279136">2009279136</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/319209483">319209483</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advances+in+the+Casimir+Effect&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=2009&rft_id=info%3Aoclcnum%2F319209483&rft_id=info%3Alccn%2F2009279136&rft.isbn=978-0-19-923874-3&rft.aulast=Bordag&rft.aufirst=M&rft.au=Klimchitskaya%2C+G.+L.&rft.au=Mohideen%2C+U.&rft.au=Mostepanenko%2C+V.+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCqE1f_s5PgYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChan_et_al.2001" class="citation journal cs1">Chan, H. B.; Aksyuk, V. A.; Kleiman, R. N.; Bishop, D. J.; Capasso, F. (2001). <a rel="nofollow" class="external text" href="https://www.mit.edu/~kardar/research/seminars/Casimir/Science-Capasso.pdf">"Quantum Mechanical Actuation of Microelectromechanical Systems by the Casimir Force"</a> <span class="cs1-format">(PDF)</span>. <i>Science</i>. <b>291</b> (5510): 1941–1944. <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/2001Sci...291.1941C">2001Sci...291.1941C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1057984">10.1126/science.1057984</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11239149">11239149</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:17072357">17072357</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Quantum+Mechanical+Actuation+of+Microelectromechanical+Systems+by+the+Casimir+Force&rft.volume=291&rft.issue=5510&rft.pages=1941-1944&rft.date=2001&rft_id=info%3Adoi%2F10.1126%2Fscience.1057984&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17072357%23id-name%3DS2CID&rft_id=info%3Apmid%2F11239149&rft_id=info%3Abibcode%2F2001Sci...291.1941C&rft.aulast=Chan&rft.aufirst=H.+B.&rft.au=Aksyuk%2C+V.+A.&rft.au=Kleiman%2C+R.+N.&rft.au=Bishop%2C+D.+J.&rft.au=Capasso%2C+F.&rft_id=https%3A%2F%2Fwww.mit.edu%2F~kardar%2Fresearch%2Fseminars%2FCasimir%2FScience-Capasso.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEverettRoman,_Thomas2012" class="citation book cs1">Everett, Allen; Roman, Thomas (2012). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780226224985/page/167"><i>Time Travel and Warp Drives</i></a></span>. University of Chicago Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780226224985/page/167">167</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-22498-5" title="Special:BookSources/978-0-226-22498-5"><bdi>978-0-226-22498-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time+Travel+and+Warp+Drives&rft.pages=167&rft.pub=University+of+Chicago+Press&rft.date=2012&rft.isbn=978-0-226-22498-5&rft.aulast=Everett&rft.aufirst=Allen&rft.au=Roman%2C+Thomas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_9780226224985%2Fpage%2F167&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSopovaFord2002" class="citation journal cs1">Sopova, V.; Ford, L. H. (2002). "The Energy Density in the Casimir Effect". <i><a href="/wiki/Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>66</b> (4): 045026. <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/quant-ph/0204125">quant-ph/0204125</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/2002PhRvD..66d5026S">2002PhRvD..66d5026S</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.66.045026">10.1103/PhysRevD.66.045026</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:10649139">10649139</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=The+Energy+Density+in+the+Casimir+Effect&rft.volume=66&rft.issue=4&rft.pages=045026&rft.date=2002&rft_id=info%3Aarxiv%2Fquant-ph%2F0204125&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10649139%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.66.045026&rft_id=info%3Abibcode%2F2002PhRvD..66d5026S&rft.aulast=Sopova&rft.aufirst=V.&rft.au=Ford%2C+L.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFordRoman1995" class="citation journal cs1">Ford, L. H.; Roman, Thomas A. (1995). "Averaged Energy Conditions and Quantum Inequalities". <i><a href="/wiki/Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>51</b> (8): 4277–4286. <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/gr-qc/9410043">gr-qc/9410043</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/1995PhRvD..51.4277F">1995PhRvD..51.4277F</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.51.4277">10.1103/PhysRevD.51.4277</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10018903">10018903</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:7413835">7413835</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=Averaged+Energy+Conditions+and+Quantum+Inequalities&rft.volume=51&rft.issue=8&rft.pages=4277-4286&rft.date=1995&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7413835%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1995PhRvD..51.4277F&rft_id=info%3Aarxiv%2Fgr-qc%2F9410043&rft_id=info%3Apmid%2F10018903&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.51.4277&rft.aulast=Ford&rft.aufirst=L.+H.&rft.au=Roman%2C+Thomas+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120210233225/http://www.hawking.org.uk/space-and-time-warps.html">"Space and Time Warps"</a>. Hawking.org.uk. Archived from <a rel="nofollow" class="external text" href="http://www.hawking.org.uk/space-and-time-warps.html">the original</a> on 10 February 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">11 November</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Space+and+Time+Warps&rft.pub=Hawking.org.uk&rft_id=http%3A%2F%2Fwww.hawking.org.uk%2Fspace-and-time-warps.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> <li id="cite_note-time_travel-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-time_travel_68-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorrisThorneYurtsever1988" class="citation journal cs1">Morris, Michael; Thorne, Kip; Yurtsever, Ulvi (1988). <a rel="nofollow" class="external text" href="http://authors.library.caltech.edu/9262/1/MORprl88.pdf">"Wormholes, Time Machines, and the Weak Energy Condition"</a> <span class="cs1-format">(PDF)</span>. <i>Physical Review Letters</i>. <b>61</b> (13): 1446–1449. <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/1988PhRvL..61.1446M">1988PhRvL..61.1446M</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%2FPhysRevLett.61.1446">10.1103/PhysRevLett.61.1446</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10038800">10038800</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110709100524/http://authors.library.caltech.edu/9262/1/MORprl88.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 9 July 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Wormholes%2C+Time+Machines%2C+and+the+Weak+Energy+Condition&rft.volume=61&rft.issue=13&rft.pages=1446-1449&rft.date=1988&rft_id=info%3Apmid%2F10038800&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.61.1446&rft_id=info%3Abibcode%2F1988PhRvL..61.1446M&rft.aulast=Morris&rft.aufirst=Michael&rft.au=Thorne%2C+Kip&rft.au=Yurtsever%2C+Ulvi&rft_id=http%3A%2F%2Fauthors.library.caltech.edu%2F9262%2F1%2FMORprl88.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=17" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Introductory_readings">Introductory readings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=18" title="Edit section: Introductory readings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/Quantum/casimir.html">Casimir effect description</a> from <a href="/wiki/University_of_California,_Riverside" title="University of California, Riverside">University of California, Riverside</a>'s version of the <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/index.html">Usenet physics FAQ</a>.</li> <li>A. Lambrecht, <a rel="nofollow" class="external text" href="https://physicsworld.com/a/the-casimir-effect-a-force-from-nothing/">The Casimir effect: a force from nothing</a>, <i>Physics World</i>, September 2002.</li> <li><a rel="nofollow" class="external text" href="https://apod.nasa.gov/apod/ap061217.html">NASA Astronomy Picture of the Day: Casimir effect (17 December 2006)</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimpsonLeonhardt2015" class="citation book cs1">Simpson, W. M. R; Leonhardt, U. (2015). <i>Forces of the Quantum Vacuum: An introduction to Casimir physics</i>. <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4632-90-4" title="Special:BookSources/978-981-4632-90-4"><bdi>978-981-4632-90-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Forces+of+the+Quantum+Vacuum%3A+An+introduction+to+Casimir+physics&rft.pub=World+Scientific&rft.date=2015&rft.isbn=978-981-4632-90-4&rft.aulast=Simpson&rft.aufirst=W.+M.+R&rft.au=Leonhardt%2C+U.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Papers,_books_and_lectures"><span id="Papers.2C_books_and_lectures"></span>Papers, books and lectures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=19" title="Edit section: Papers, books and lectures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCasimirPolder1948" class="citation journal cs1"><a href="/wiki/Hendrik_Casimir" title="Hendrik Casimir">Casimir, H. B. G.</a>; <a href="/wiki/Dirk_Polder" title="Dirk Polder">Polder, D.</a> (1948). "The Influence of Retardation on the London-van der Waals Forces". <i><a href="/wiki/Physical_Review" title="Physical Review">Physical Review</a></i>. <b>73</b> (4): 360–372. <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/1948PhRv...73..360C">1948PhRv...73..360C</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%2FPhysRev.73.360">10.1103/PhysRev.73.360</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&rft.atitle=The+Influence+of+Retardation+on+the+London-van+der+Waals+Forces&rft.volume=73&rft.issue=4&rft.pages=360-372&rft.date=1948&rft_id=info%3Adoi%2F10.1103%2FPhysRev.73.360&rft_id=info%3Abibcode%2F1948PhRv...73..360C&rft.aulast=Casimir&rft.aufirst=H.+B.+G.&rft.au=Polder%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCasimir1948" class="citation journal cs1">Casimir, H. B. G. (1948). <a rel="nofollow" class="external text" href="http://www.dwc.knaw.nl/DL/publications/PU00018547.pdf">"On the attraction between two perfectly conducting plates"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen</i>. <b>B51</b>: 793–795.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Koninklijke+Nederlandse+Akademie+van+Wetenschappen&rft.atitle=On+the+attraction+between+two+perfectly+conducting+plates&rft.volume=B51&rft.pages=793-795&rft.date=1948&rft.aulast=Casimir&rft.aufirst=H.+B.+G.&rft_id=http%3A%2F%2Fwww.dwc.knaw.nl%2FDL%2Fpublications%2FPU00018547.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLamoreaux1997" class="citation journal cs1">Lamoreaux, S. K. (1997). "Demonstration of the Casimir Force in the 0.6 to 6 μm Range". <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>78</b> (1): 5–8. <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/1997PhRvL..78....5L">1997PhRvL..78....5L</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%2FPhysRevLett.78.5">10.1103/PhysRevLett.78.5</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:25323874">25323874</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+Letters&rft.atitle=Demonstration+of+the+Casimir+Force+in+the+0.6+to+6+%CE%BCm+Range&rft.volume=78&rft.issue=1&rft.pages=5-8&rft.date=1997&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A25323874%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.78.5&rft_id=info%3Abibcode%2F1997PhRvL..78....5L&rft.aulast=Lamoreaux&rft.aufirst=S.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBordagMohideenMostepanenko2001" class="citation journal cs1">Bordag, M.; Mohideen, U.; Mostepanenko, V. M. (October 2001). "New developments in the Casimir effect". <i>Physics Reports</i>. <b>353</b> (1–3): 1–205. <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/quant-ph/0106045">quant-ph/0106045</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/2001PhR...353....1B">2001PhR...353....1B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0370-1573%2801%2900015-1">10.1016/S0370-1573(01)00015-1</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:119352552">119352552</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Reports&rft.atitle=New+developments+in+the+Casimir+effect&rft.volume=353&rft.issue=1%E2%80%933&rft.pages=1-205&rft.date=2001-10&rft_id=info%3Aarxiv%2Fquant-ph%2F0106045&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119352552%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2FS0370-1573%2801%2900015-1&rft_id=info%3Abibcode%2F2001PhR...353....1B&rft.aulast=Bordag&rft.aufirst=M.&rft.au=Mohideen%2C+U.&rft.au=Mostepanenko%2C+V.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilton2001" class="citation book cs1">Milton, K. A. (2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/casimireffectphy0000milt"><i>The Casimir Effect: Physical Manifestations of Zero-point Energy</i></a></span> (Reprint ed.). <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-4397-5" title="Special:BookSources/978-981-02-4397-5"><bdi>978-981-02-4397-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Casimir+Effect%3A+Physical+Manifestations+of+Zero-point+Energy&rft.edition=Reprint&rft.pub=World+Scientific&rft.date=2001&rft.isbn=978-981-02-4397-5&rft.aulast=Milton&rft.aufirst=K.+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcasimireffectphy0000milt&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDalvitMilonniRobertsDa_Rosa2011" class="citation book cs1">Dalvit, Diego; Milonni, Peter; Roberts, David; Da Rosa, Felipe (2011). Dalvit, Diego; <a href="/wiki/Peter_W._Milonni" title="Peter W. Milonni">Milonni, Peter W.</a>; Roberts, David; da Rosa, Felipe (eds.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eqQRBwAAQBAJ"><i>Casimir Physics</i></a>. Lecture Notes in Physics. Vol. 834. <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/2011LNP...834.....D">2011LNP...834.....D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-20288-9">10.1007/978-3-642-20288-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-20287-2" title="Special:BookSources/978-3-642-20287-2"><bdi>978-3-642-20287-2</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0075-8450">0075-8450</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/844922239">844922239</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Casimir+Physics&rft.series=Lecture+Notes+in+Physics&rft.date=2011&rft_id=info%3Abibcode%2F2011LNP...834.....D&rft_id=info%3Aoclcnum%2F844922239&rft.issn=0075-8450&rft_id=info%3Adoi%2F10.1007%2F978-3-642-20288-9&rft.isbn=978-3-642-20287-2&rft.aulast=Dalvit&rft.aufirst=Diego&rft.au=Milonni%2C+Peter&rft.au=Roberts%2C+David&rft.au=Da+Rosa%2C+Felipe&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeqQRBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBressiCarugnoOnofrioRuoso2002" class="citation journal cs1">Bressi, G.; Carugno, G.; Onofrio, R.; Ruoso, G. (2002). "Measurement of the Casimir Force between Parallel Metallic Surfaces". <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>88</b> (4): 041804. <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/quant-ph/0203002">quant-ph/0203002</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/2002PhRvL..88d1804B">2002PhRvL..88d1804B</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%2FPhysRevLett.88.041804">10.1103/PhysRevLett.88.041804</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11801108">11801108</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:43354557">43354557</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+Letters&rft.atitle=Measurement+of+the+Casimir+Force+between+Parallel+Metallic+Surfaces&rft.volume=88&rft.issue=4&rft.pages=041804&rft.date=2002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A43354557%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2002PhRvL..88d1804B&rft_id=info%3Aarxiv%2Fquant-ph%2F0203002&rft_id=info%3Apmid%2F11801108&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.88.041804&rft.aulast=Bressi&rft.aufirst=G.&rft.au=Carugno%2C+G.&rft.au=Onofrio%2C+R.&rft.au=Ruoso%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKennethKlichMannRevzen2002" class="citation journal cs1">Kenneth, O.; Klich, I.; Mann, A.; Revzen, M. (2002). "Repulsive Casimir Forces". <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>89</b> (3): 033001. <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/quant-ph/0202114">quant-ph/0202114</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/2002PhRvL..89c3001K">2002PhRvL..89c3001K</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%2FPhysRevLett.89.033001">10.1103/PhysRevLett.89.033001</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12144387">12144387</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:20903628">20903628</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+Letters&rft.atitle=Repulsive+Casimir+Forces&rft.volume=89&rft.issue=3&rft.pages=033001&rft.date=2002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A20903628%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2002PhRvL..89c3001K&rft_id=info%3Aarxiv%2Fquant-ph%2F0202114&rft_id=info%3Apmid%2F12144387&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.89.033001&rft.aulast=Kenneth&rft.aufirst=O.&rft.au=Klich%2C+I.&rft.au=Mann%2C+A.&rft.au=Revzen%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarrow2005" class="citation web cs1"><a href="/wiki/John_D._Barrow" title="John D. Barrow">Barrow, J. D.</a> (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070930185128/http://www.gresham.ac.uk/event.asp?PageId=4&EventId=258">"Much Ado About Nothing"</a>. <i>Lecture at <a href="/wiki/Gresham_College" title="Gresham College">Gresham College</a></i>. Archived from <a rel="nofollow" class="external text" href="http://www.gresham.ac.uk/event.asp?PageId=4&EventId=258">the original</a> on 30 September 2007.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Lecture+at+Gresham+College&rft.atitle=Much+Ado+About+Nothing&rft.date=2005&rft.aulast=Barrow&rft.aufirst=J.+D.&rft_id=http%3A%2F%2Fwww.gresham.ac.uk%2Fevent.asp%3FPageId%3D4%26EventId%3D258&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span> (Includes discussion of French naval analogy.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarrow2000" class="citation book cs1">Barrow, J. D. (2000). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/bookofnothingvac0000barr"><i>The Book of Nothing: Vacuums, Voids, and the Latest Ideas About the Origins of the Universe</i></a></span>. <a href="/wiki/Pantheon_Books" title="Pantheon Books">Pantheon Books</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-09-928845-9" title="Special:BookSources/978-0-09-928845-9"><bdi>978-0-09-928845-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Book+of+Nothing%3A+Vacuums%2C+Voids%2C+and+the+Latest+Ideas+About+the+Origins+of+the+Universe&rft.pub=Pantheon+Books&rft.date=2000&rft.isbn=978-0-09-928845-9&rft.aulast=Barrow&rft.aufirst=J.+D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbookofnothingvac0000barr&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span> (Also includes discussion of French naval analogy.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDownling1989" class="citation journal cs1"><a href="/wiki/Jonathan_Dowling" title="Jonathan Dowling">Downling, J. P.</a> (1989). "The Mathematics of the Casimir Effect". <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>. <b>62</b> (5): 324–331. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0025570X.1989.11977464">10.1080/0025570X.1989.11977464</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=The+Mathematics+of+the+Casimir+Effect&rft.volume=62&rft.issue=5&rft.pages=324-331&rft.date=1989&rft_id=info%3Adoi%2F10.1080%2F0025570X.1989.11977464&rft.aulast=Downling&rft.aufirst=J.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li> <li>Patent No. PCT/RU2011/000847 Author Urmatskih.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Temperature_dependence">Temperature dependence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=20" title="Edit section: Temperature dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070210090006/http://www.nist.gov/public_affairs/newsfromnist_casimir-polder.htm">Measurements Recast Usual View of Elusive Force</a> from <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNesterenkoLambiaseScarpetta2005" class="citation journal cs1">Nesterenko, V. V.; Lambiase, G.; Scarpetta, G. (2005). "Calculation of the Casimir energy at zero and finite temperature: Some recent results". <i><a href="/wiki/Rivista_del_Nuovo_Cimento" class="mw-redirect" title="Rivista del Nuovo Cimento">Rivista del Nuovo Cimento</a></i>. <b>27</b> (6): 1–74. <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/hep-th/0503100">hep-th/0503100</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/2004NCimR..27f...1N">2004NCimR..27f...1N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1393%2Fncr%2Fi2005-10002-2">10.1393/ncr/i2005-10002-2</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:14693485">14693485</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rivista+del+Nuovo+Cimento&rft.atitle=Calculation+of+the+Casimir+energy+at+zero+and+finite+temperature%3A+Some+recent+results&rft.volume=27&rft.issue=6&rft.pages=1-74&rft.date=2005&rft_id=info%3Aarxiv%2Fhep-th%2F0503100&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14693485%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1393%2Fncr%2Fi2005-10002-2&rft_id=info%3Abibcode%2F2004NCimR..27f...1N&rft.aulast=Nesterenko&rft.aufirst=V.+V.&rft.au=Lambiase%2C+G.&rft.au=Scarpetta%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACasimir+effect" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Casimir_effect&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://xstructure.inr.ac.ru/x-bin/theme3.py?level=3&index1=313011">Casimir effect article search</a> on arxiv.org</li> <li>G. Lang, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040604161012/http://www.casimir.rl.ac.uk/default.htm">The Casimir Force</a> web site, 2002</li> <li>J. Babb, <a rel="nofollow" class="external text" href="http://www.cfa.harvard.edu/~babb/casimir-bib.html">bibliography on the Casimir Effect</a> web site, 2009</li> <li>H. Nikolic, <a rel="nofollow" class="external text" href="http://thphys.irb.hr/wiki/main/images/2/2c/Casimir.pdf">The origin of Casimir effect; Vacuum energy or van der Waals force?</a> presentation slides, 2018</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="Quantum_mechanics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible expanded 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:Quantum_mechanics_topics" title="Template:Quantum mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</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/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</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/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</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/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</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/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</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/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</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/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</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/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</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/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</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/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a class="mw-selflink selflink">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</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" 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