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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Schrödinger's_equation"> <div class="vector-toc-text"> 2.3 Schrödinger's equation </div> </a> <ul id="toc-Schrödinger's_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_the_Born_rule" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_Born_rule"> <div class="vector-toc-text"> 2.4 Relation to the Born rule </div> </a> <ul id="toc-Relation_to_the_Born_rule-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_conditional_wavefunction_of_a_subsystem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_conditional_wavefunction_of_a_subsystem"> <div class="vector-toc-text"> 2.5 The conditional wavefunction of a subsystem </div> </a> <ul id="toc-The_conditional_wavefunction_of_a_subsystem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Extensions"> <div class="vector-toc-text"> 3 Extensions </div> </a> <button aria-controls="toc-Extensions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Extensions subsection </button> <ul id="toc-Extensions-sublist" class="vector-toc-list"> <li id="toc-Relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativity"> <div class="vector-toc-text"> 3.1 Relativity </div> </a> <ul id="toc-Relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spin"> <div class="vector-toc-text"> 3.2 Spin </div> </a> <ul id="toc-Spin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stochastic_electrodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stochastic_electrodynamics"> <div class="vector-toc-text"> 3.3 Stochastic electrodynamics </div> </a> <ul id="toc-Stochastic_electrodynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_field_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_field_theory"> <div class="vector-toc-text"> 3.4 Quantum field theory </div> </a> <ul id="toc-Quantum_field_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curved_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Curved_space"> <div class="vector-toc-text"> 3.5 Curved space </div> </a> <ul id="toc-Curved_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exploiting_nonlocality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exploiting_nonlocality"> <div class="vector-toc-text"> 3.6 Exploiting nonlocality </div> </a> <ul id="toc-Exploiting_nonlocality-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Results" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Results"> <div class="vector-toc-text"> 4 Results </div> </a> <button aria-controls="toc-Results-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Results subsection </button> <ul id="toc-Results-sublist" class="vector-toc-list"> <li id="toc-Measuring_spin_and_polarization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measuring_spin_and_polarization"> <div class="vector-toc-text"> 4.1 Measuring spin and polarization </div> </a> <ul id="toc-Measuring_spin_and_polarization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measurements,_the_quantum_formalism,_and_observer_independence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measurements,_the_quantum_formalism,_and_observer_independence"> <div class="vector-toc-text"> 4.2 Measurements, the quantum formalism, and observer independence </div> </a> <ul id="toc-Measurements,_the_quantum_formalism,_and_observer_independence-sublist" class="vector-toc-list"> <li id="toc-Collapse_of_the_wavefunction" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Collapse_of_the_wavefunction"> <div class="vector-toc-text"> 4.2.1 Collapse of the wavefunction </div> </a> <ul id="toc-Collapse_of_the_wavefunction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operators_as_observables" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Operators_as_observables"> <div class="vector-toc-text"> 4.2.2 Operators as observables </div> </a> <ul id="toc-Operators_as_observables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hidden_variables" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Hidden_variables"> <div class="vector-toc-text"> 4.2.3 Hidden variables </div> </a> <ul id="toc-Hidden_variables-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Different_predictions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Different_predictions"> <div class="vector-toc-text"> 4.3 Different predictions </div> </a> <ul id="toc-Different_predictions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heisenberg's_uncertainty_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heisenberg's_uncertainty_principle"> <div class="vector-toc-text"> 4.4 Heisenberg's uncertainty principle </div> </a> <ul id="toc-Heisenberg's_uncertainty_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_entanglement,_Einstein–Podolsky–Rosen_paradox,_Bell's_theorem,_and_nonlocality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_entanglement,_Einstein–Podolsky–Rosen_paradox,_Bell's_theorem,_and_nonlocality"> <div class="vector-toc-text"> 4.5 Quantum entanglement, Einstein–Podolsky–Rosen paradox, Bell's theorem, and nonlocality </div> </a> <ul id="toc-Quantum_entanglement,_Einstein–Podolsky–Rosen_paradox,_Bell's_theorem,_and_nonlocality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classical_limit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_limit"> <div class="vector-toc-text"> 4.6 Classical limit </div> </a> <ul id="toc-Classical_limit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_trajectory_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_trajectory_method"> <div class="vector-toc-text"> 4.7 Quantum trajectory method </div> </a> <ul id="toc-Quantum_trajectory_method-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Similarities_with_the_many-worlds_interpretation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Similarities_with_the_many-worlds_interpretation"> <div class="vector-toc-text"> 5 Similarities with the many-worlds interpretation </div> </a> <ul id="toc-Similarities_with_the_many-worlds_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Occam's-razor_criticism" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Occam's-razor_criticism"> <div class="vector-toc-text"> 6 Occam's-razor criticism </div> </a> <ul id="toc-Occam's-razor_criticism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Derivations"> <div class="vector-toc-text"> 7 Derivations </div> </a> <ul id="toc-Derivations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 8 History </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Pilot-wave_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pilot-wave_theory"> <div class="vector-toc-text"> 8.1 Pilot-wave theory </div> </a> <ul id="toc-Pilot-wave_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bohmian_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bohmian_mechanics"> <div class="vector-toc-text"> 8.2 Bohmian mechanics </div> </a> <ul id="toc-Bohmian_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Causal_interpretation_and_ontological_interpretation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Causal_interpretation_and_ontological_interpretation"> <div class="vector-toc-text"> 8.3 Causal interpretation and ontological interpretation </div> </a> <ul id="toc-Causal_interpretation_and_ontological_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hydrodynamic_quantum_analogs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hydrodynamic_quantum_analogs"> <div class="vector-toc-text"> 8.4 Hydrodynamic quantum analogs </div> </a> <ul id="toc-Hydrodynamic_quantum_analogs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Surrealistic_trajectories" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Surrealistic_trajectories"> <div class="vector-toc-text"> 9 Surrealistic trajectories </div> </a> <ul id="toc-Surrealistic_trajectories-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 10 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 11 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 12 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> 13 Sources </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 14 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 15 External links </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 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Available in 19 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-19" aria-hidden="true" > 19 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%AF%D9%8A_%D8%A8%D8%B1%D9%88%D9%8A-%D8%A8%D9%88%D9%85" title="نظرية دي بروي-بوم – Arabic" lang="ar" hreflang="ar" data-title="نظرية دي بروي-بوم" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_Broglie-Bohm" title="Teoria de Broglie-Bohm – Catalan" lang="ca" hreflang="ca" data-title="Teoria de Broglie-Bohm" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Pilotb%C3%B8lge-teorien" title="Pilotbølge-teorien – Danish" lang="da" hreflang="da" data-title="Pilotbølge-teorien" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/De-Broglie-Bohm-Theorie" title="De-Broglie-Bohm-Theorie – German" lang="de" hreflang="de" data-title="De-Broglie-Bohm-Theorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Interpretaci%C3%B3n_de_Bohm" title="Interpretación de Bohm – Spanish" lang="es" hreflang="es" data-title="Interpretación de Bohm" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%D8%A8%D9%88%D9%87%D9%85%DB%8C" title="مکانیک بوهمی – Persian" lang="fa" hreflang="fa" data-title="مکانیک بوهمی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_de_De_Broglie-Bohm" title="Théorie de De Broglie-Bohm – French" lang="fr" hreflang="fr" data-title="Théorie de De Broglie-Bohm" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%93%9C_%EB%B8%8C%EB%A1%9C%EC%9D%B4-%EB%B4%84_%EC%9D%B4%EB%A1%A0" title="드 브로이-봄 이론 – Korean" lang="ko" hreflang="ko" data-title="드 브로이-봄 이론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Interpretazione_di_Bohm" title="Interpretazione di Bohm – Italian" lang="it" hreflang="it" data-title="Interpretazione di Bohm" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%90%D7%95%D7%A8%D7%99%D7%99%D7%AA_%D7%93%D7%94_%D7%91%D7%A8%D7%95%D7%99%D7%99-%D7%91%D7%95%D7%94%D7%9D" title="תאוריית דה ברויי-בוהם – Hebrew" lang="he" hreflang="he" data-title="תאוריית דה ברויי-בוהם" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theoria_undarum_gubernatoriarum_De_Broglie-Bohm" title="Theoria undarum gubernatoriarum De Broglie-Bohm – Latin" lang="la" hreflang="la" data-title="Theoria undarum gubernatoriarum De Broglie-Bohm" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9C%E3%83%BC%E3%83%A0%E8%A7%A3%E9%87%88" title="ボーム解釈 – Japanese" lang="ja" hreflang="ja" data-title="ボーム解釈" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A1%E0%A9%80_%E0%A8%AC%E0%A9%8D%E0%A8%B0%E0%A9%8B%E0%A8%97%E0%A8%B2%E0%A8%BE%E0%A8%87-%E0%A8%AC%E0%A9%8B%E0%A8%B9%E0%A8%AE_%E0%A8%A5%E0%A8%BF%E0%A8%8A%E0%A8%B0%E0%A9%80" title="ਡੀ ਬ੍ਰੋਗਲਾਇ-ਬੋਹਮ ਥਿਊਰੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਡੀ ਬ੍ਰੋਗਲਾਇ-ਬੋਹਮ ਥਿਊਰੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_de_Broglie%E2%80%99a-Bohma" title="Teoria de Broglie’a-Bohma – Polish" lang="pl" hreflang="pl" data-title="Teoria de Broglie’a-Bohma" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Interpreta%C3%A7%C3%A3o_de_Bohm" title="Interpretação de Bohm – Portuguese" lang="pt" hreflang="pt" data-title="Interpretação de Bohm" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B4%D0%B5_%D0%91%D1%80%D0%BE%D0%B9%D0%BB%D1%8F_%E2%80%94_%D0%91%D0%BE%D0%BC%D0%B0" title="Теория де Бройля — Бома – Russian" lang="ru" hreflang="ru" data-title="Теория де Бройля — Бома" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Pilot_dalga_teorisi" title="Pilot dalga teorisi – Turkish" lang="tr" hreflang="tr" data-title="Pilot dalga teorisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%B4%D0%B5_%D0%91%D1%80%D0%BE%D0%B9%D0%BB%D1%8F_%E2%80%94_%D0%91%D0%BE%D0%BC%D0%B0" title="Теорія де Бройля — Бома – Ukrainian" lang="uk" hreflang="uk" data-title="Теорія де Бройля — Бома" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BE%B7%E5%B8%83%E7%BD%97%E6%84%8F-%E7%8E%BB%E5%A7%86%E7%90%86%E8%AE%BA" title="德布罗意-玻姆理论 – Chinese" lang="zh" hreflang="zh" data-title="德布罗意-玻姆理论" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q899444#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div 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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"><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><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 }"><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"><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 class="mw-selflink selflink">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> The de Broglie–Bohm theory<a href="#cite_note-1">[a]</a> is an <a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">interpretation</a> of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> which postulates that, in addition to the <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</a>, an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all particles is defined by a <a href="#Guiding_equation">guiding equation</a>. The evolution of the wave function over time is given by the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>. The theory is named after <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a> (1892–1987) and <a href="/wiki/David_Bohm" title="David Bohm">David Bohm</a> (1917–1992). The theory is <a href="/wiki/Deterministic_system" title="Deterministic system">deterministic</a><a href="#cite_note-2">[1]</a> and explicitly <a href="/wiki/Principle_of_locality" title="Principle of locality">nonlocal</a>: the velocity of any one particle depends on the value of the guiding equation, which depends on the configuration of all the particles under consideration. Measurements are a particular case of quantum processes described by the theory—for which it yields the same quantum predictions as other interpretations of quantum mechanics. The theory does not have a "<a href="/wiki/Measurement_problem" title="Measurement problem">measurement problem</a>", due to the fact that the particles have a definite configuration at all times. The <a href="/wiki/Born_rule" title="Born rule">Born rule</a> in de Broglie–Bohm theory is not a postulate. Rather, in this theory, the link between the probability density and the wave function has the status of a theorem, a result of a separate postulate, the "<a href="/wiki/Quantum_equilibrium_hypothesis" class="mw-redirect" title="Quantum equilibrium hypothesis">quantum equilibrium hypothesis</a>", which is additional to the basic principles governing the wave function. There are several equivalent <a href="#Derivations">mathematical formulations</a> of the theory. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=1" title="Edit section: Overview">edit</a>]</div> De Broglie–Bohm theory is based on the following postulates: <ul><li>There is a configuration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> of the universe, described by coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9b1525f5a653e19f9fd37fd2701a768e171632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.168ex; height:3.009ex;" alt="{\displaystyle q^{k}}">, which is an element of the configuration space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}">. The configuration space is different for different versions of pilot-wave theory. For example, this may be the space of positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} _{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b5e8415b63b5600e16dd0001ae83bbe85885d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.097ex; height:2.676ex;" alt="{\displaystyle \mathbf {Q} _{k}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> particles, or, in case of field theory, the space of field configurations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546b660b2f3cfb5f34be7b3ed8371d54f5c74227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.524ex; height:2.843ex;" alt="{\displaystyle \phi (x)}">. The configuration evolves (for spin=0) according to the guiding equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{k}{\frac {dq^{k}}{dt}}(t)=\hbar \nabla _{k}\operatorname {Im} \ln \psi (q,t)=\hbar \operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(q,t)={\frac {m_{k}\mathbf {j} _{k}}{\psi ^{*}\psi }}=\operatorname {Re} \left({\frac {\mathbf {\hat {P}} _{k}\Psi }{\Psi }}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>Im</mi> <mo>⁡</mo> <mi>ln</mi> <mo>⁡</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>ψ</mi> </mrow> <mi>ψ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ψ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">P</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi mathvariant="normal">Ψ</mi> </mrow> <mi mathvariant="normal">Ψ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{k}{\frac {dq^{k}}{dt}}(t)=\hbar \nabla _{k}\operatorname {Im} \ln \psi (q,t)=\hbar \operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(q,t)={\frac {m_{k}\mathbf {j} _{k}}{\psi ^{*}\psi }}=\operatorname {Re} \left({\frac {\mathbf {\hat {P}} _{k}\Psi }{\Psi }}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd3ad407a1da576a55f7df29d776900c04eb0b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:75.383ex; height:7.509ex;" alt="{\displaystyle m_{k}{\frac {dq^{k}}{dt}}(t)=\hbar \nabla _{k}\operatorname {Im} \ln \psi (q,t)=\hbar \operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(q,t)={\frac {m_{k}\mathbf {j} _{k}}{\psi ^{*}\psi }}=\operatorname {Re} \left({\frac {\mathbf {\hat {P}} _{k}\Psi }{\Psi }}\right),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {j} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5d874dbb32b8b33e83ca521f592808387a486e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.164ex; width:0.98ex; height:2.509ex;" alt="{\displaystyle \mathbf {j} }"> is the <a href="/wiki/Probability_current" title="Probability current">probability current</a> or probability flux, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {P}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">P</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {P}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc3943cde360004c44c69724f23ec5001088451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {P}} }"> is the <a href="/wiki/Momentum_operator" title="Momentum operator">momentum operator</a>. Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (q,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (q,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae63ac01548619f4c3c78988c4a0c54d1c73f3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.266ex; height:2.843ex;" alt="{\displaystyle \psi (q,t)}"> is the standard complex-valued wavefunction from quantum theory, which evolves according to <a href="/wiki/Schr%C3%B6dinger%27s_equation" class="mw-redirect" title="Schrödinger's equation">Schrödinger's equation</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (q,t)=-\sum _{i=1}^{N}{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}\psi (q,t)+V(q)\psi (q,t).}"> <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 mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (q,t)=-\sum _{i=1}^{N}{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}\psi (q,t)+V(q)\psi (q,t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff5ab65bd1afc1b87cc0e143f6baf8be10fceb76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.917ex; height:7.343ex;" alt="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (q,t)=-\sum _{i=1}^{N}{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}\psi (q,t)+V(q)\psi (q,t).}">This completes the specification of the theory for any quantum theory with Hamilton operator of type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H=\sum {\frac {1}{2m_{i}}}{\hat {p}}_{i}^{2}+V({\hat {q}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle H=\sum {\frac {1}{2m_{i}}}{\hat {p}}_{i}^{2}+V({\hat {q}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05853a1f365298882626b8a36226925852f46322" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:21.956ex; height:3.843ex;" alt="{\textstyle H=\sum {\frac {1}{2m_{i}}}{\hat {p}}_{i}^{2}+V({\hat {q}})}">.</li> <li>The configuration is distributed according to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi (q,t)|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi (q,t)|^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70c45eb3cadd021b2b94b8ffe8b74ad2ab4888a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.613ex; height:3.343ex;" alt="{\displaystyle |\psi (q,t)|^{2}}"> at some moment of time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">, and this consequently holds for all times. Such a state is named quantum equilibrium. With quantum equilibrium, this theory agrees with the results of standard quantum mechanics.</li></ul> Even though this latter relation is frequently presented as an axiom of the theory, Bohm presented it as derivable from statistical-mechanical arguments in the original papers of 1952. This argument was further supported by the work of Bohm in 1953 and was substantiated by Vigier and Bohm's paper of 1954, in which they introduced stochastic fluid fluctuations that drive a process of asymptotic relaxation from <a href="/wiki/Quantum_non-equilibrium" title="Quantum non-equilibrium">quantum non-equilibrium</a> to quantum equilibrium (ρ → |ψ|2).<a href="#cite_note-3">[2]</a> <div class="mw-heading mw-heading3"><h3 id="Double-slit_experiment">Double-slit experiment</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=2" title="Edit section: Double-slit experiment">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Doppelspalt.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Doppelspalt.svg/220px-Doppelspalt.svg.png" decoding="async" width="220" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Doppelspalt.svg/330px-Doppelspalt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Doppelspalt.svg/440px-Doppelspalt.svg.png 2x" data-file-width="249" data-file-height="225" /></a><figcaption>The Bohmian trajectories for an electron going through the two-slit experiment. A similar pattern was also extrapolated from <a href="/wiki/Weak_measurement" title="Weak measurement">weak measurements</a> of single photons.<a href="#cite_note-4">[3]</a></figcaption></figure> The <a href="/wiki/Double-slit_experiment" title="Double-slit experiment">double-slit experiment</a> is an illustration of <a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</a>. In it, a beam of particles (such as electrons) travels through a barrier that has two slits. If a detector screen is on the side beyond the barrier, the pattern of detected particles shows interference fringes characteristic of waves arriving at the screen from two sources (the two slits); however, the interference pattern is made up of individual dots corresponding to particles that had arrived on the screen. The system seems to exhibit the behaviour of both waves (interference patterns) and particles (dots on the screen). If this experiment is modified so that one slit is closed, no interference pattern is observed. Thus, the state of both slits affects the final results. It can also be arranged to have a minimally invasive detector at one of the slits to detect which slit the particle went through. When that is done, the interference pattern disappears.<a href="#cite_note-5">[4]</a> In de Broglie–Bohm theory, the wavefunction is defined at both slits, but each particle has a well-defined trajectory that passes through exactly one of the slits. The final position of the particle on the detector screen and the slit through which the particle passes is determined by the initial position of the particle. Such initial position is not knowable or controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. In Bohm's 1952 papers he used the wavefunction to construct a <a href="/wiki/Quantum_potential" title="Quantum potential">quantum potential</a> that, when included in Newton's equations, gave the trajectories of the particles streaming through the two slits. In effect the wavefunction interferes with itself and guides the particles by the quantum potential in such a way that the particles avoid the regions in which the interference is destructive and are attracted to the regions in which the interference is constructive, resulting in the interference pattern on the detector screen. To explain the behavior when the particle is detected to go through one slit, one needs to appreciate the role of the conditional wavefunction and how it results in the collapse of the wavefunction; this is explained below. The basic idea is that the environment registering the detection effectively separates the two wave packets in configuration space. <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=3" title="Edit section: Theory">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="The_pilot_wave">The pilot wave</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=4" title="Edit section: The pilot wave">edit</a>]</div> The de Broglie–Bohm theory describes a pilot wave <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (q,t)\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (q,t)\in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b2a581123491b6a7dd9be1090ae3a33a4003f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.784ex; height:2.843ex;" alt="{\displaystyle \psi (q,t)\in \mathbb {C} }"> in a <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)"> configuration space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> and trajectories <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(t)\in Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(t)\in Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af5124a7bc57d22d5717abbf1b3e943466c52b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.397ex; height:2.843ex;" alt="{\displaystyle q(t)\in Q}"> of particles as in classical mechanics but defined by non-Newtonian mechanics.<a href="#cite_note-6">[5]</a> At every moment of time there exists not only a wavefunction, but also a well-defined configuration of the whole universe (i.e., the system as defined by the boundary conditions used in solving the Schrödinger equation). The de Broglie–Bohm theory works on particle positions and trajectories like <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> but the dynamics are different. In classical mechanics, the accelerations of the particles are imparted directly by forces, which exist in physical three-dimensional space. In de Broglie–Bohm theory, the quantum "field exerts a new kind of "quantum-mechanical" force".<a href="#cite_note-7">[6]</a>: 76  Bohm hypothesized that each particle has a "complex and subtle inner structure" that provides the capacity to react to the information provided by the wavefunction by the quantum potential.<a href="#cite_note-8">[7]</a> Also, unlike in classical mechanics, physical properties (e.g., mass, charge) are spread out over the wavefunction in de Broglie–Bohm theory, not localized at the position of the particle.<a href="#cite_note-9">[8]</a><a href="#cite_note-10">[9]</a> The wavefunction itself, and not the particles, determines the dynamical evolution of the system: the particles do not act back onto the wave function. As Bohm and Hiley worded it, "the Schrödinger equation for the quantum field does not have sources, nor does it have any other way by which the field could be directly affected by the condition of the particles [...] the quantum theory can be understood completely in terms of the assumption that the quantum field has no sources or other forms of dependence on the particles".<a href="#cite_note-11">[10]</a> P. Holland considers this lack of reciprocal action of particles and wave function to be one "[a]mong the many nonclassical properties exhibited by this theory".<a href="#cite_note-12">[11]</a> Holland later called this a merely apparent lack of back reaction, due to the incompleteness of the description.<a href="#cite_note-13">[12]</a> In what follows below, the setup for one particle moving in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> is given followed by the setup for N particles moving in 3 dimensions. In the first instance, configuration space and real space are the same, while in the second, real space is still <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">, but configuration space becomes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c256f1c3b38d079269116d276130a4999b0d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.192ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3N}}">. While the particle positions themselves are in real space, the velocity field and wavefunction are on configuration space, which is how particles are entangled with each other in this theory. <a href="#Extensions">Extensions</a> to this theory include spin and more complicated configuration spaces. We use variations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d0144479d6f47c30ad82a65d458966ccbe928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }"> for particle positions, while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> represents the complex-valued wavefunction on configuration space. <div class="mw-heading mw-heading3"><h3 id="Guiding_equation">Guiding equation</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=5" title="Edit section: Guiding equation">edit</a>]</div> For a spinless single particle moving in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">, the particle's velocity is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {Q} }{dt}}(t)={\frac {\hbar }{m}}\operatorname {Im} \left({\frac {\nabla \psi }{\psi }}\right)(\mathbf {Q} ,t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mi>m</mi> </mfrac> </mrow> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mi>ψ</mi> </mrow> <mi>ψ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {Q} }{dt}}(t)={\frac {\hbar }{m}}\operatorname {Im} \left({\frac {\nabla \psi }{\psi }}\right)(\mathbf {Q} ,t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9227e93adf534410d81d26c92b240576ca87631a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.89ex; height:6.176ex;" alt="{\displaystyle {\frac {d\mathbf {Q} }{dt}}(t)={\frac {\hbar }{m}}\operatorname {Im} \left({\frac {\nabla \psi }{\psi }}\right)(\mathbf {Q} ,t).}"></dd></dl> For many particles labeled <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} _{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b5e8415b63b5600e16dd0001ae83bbe85885d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.097ex; height:2.676ex;" alt="{\displaystyle \mathbf {Q} _{k}}"> for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-th particle their velocities are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {Q} _{k}}{dt}}(t)={\frac {\hbar }{m_{k}}}\operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(\mathbf {Q} _{1},\mathbf {Q} _{2},\ldots ,\mathbf {Q} _{N},t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mfrac> </mrow> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>ψ</mi> </mrow> <mi>ψ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {Q} _{k}}{dt}}(t)={\frac {\hbar }{m_{k}}}\operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(\mathbf {Q} _{1},\mathbf {Q} _{2},\ldots ,\mathbf {Q} _{N},t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9cf5fb6bb023de3dcfe4a7a56a772093c7d7c07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.184ex; height:6.176ex;" alt="{\displaystyle {\frac {d\mathbf {Q} _{k}}{dt}}(t)={\frac {\hbar }{m_{k}}}\operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(\mathbf {Q} _{1},\mathbf {Q} _{2},\ldots ,\mathbf {Q} _{N},t).}"></dd></dl> The main fact to notice is that this velocity field depends on the actual positions of all of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> particles in the universe. As explained below, in most experimental situations, the influence of all of those particles can be encapsulated into an effective wavefunction for a subsystem of the universe. <div class="mw-heading mw-heading3"><h3 id="Schrödinger's_equation">Schrödinger's equation</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=6" title="Edit section: Schrödinger's equation">edit</a>]</div> The one-particle Schrödinger equation governs the time evolution of a complex-valued wavefunction on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">. The equation represents a quantized version of the total energy of a classical system evolving under a real-valued potential function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi .}"> <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 mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>+</mo> <mi>V</mi> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8337e22518371695afec421ceb3e7b819da2840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.852ex; height:5.843ex;" alt="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi .}"></dd></dl> For many particles, the equation is the same except that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> are now on configuration space, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c256f1c3b38d079269116d276130a4999b0d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.192ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3N}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}\nabla _{k}^{2}\psi +V\psi .}"> <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 mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>ψ</mi> <mo>+</mo> <mi>V</mi> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}\nabla _{k}^{2}\psi +V\psi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a9fe79b4f86de29aa0a0aef07e31c2d8c27e59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.105ex; height:7.343ex;" alt="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}\nabla _{k}^{2}\psi +V\psi .}"></dd></dl> This is the same wavefunction as in conventional quantum mechanics. <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_Born_rule">Relation to the Born rule</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=7" title="Edit section: Relation to the Born rule">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Born_rule" title="Born rule">Born rule</a></div> In Bohm's original papers,<a href="#cite_note-:0-14">[13]</a> he discusses how de Broglie–Bohm theory results in the usual measurement results of quantum mechanics. The main idea is that this is true if the positions of the particles satisfy the statistical distribution given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1766f8c8e0a96326d9379b65a63900b3be22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.861ex; height:3.343ex;" alt="{\displaystyle |\psi |^{2}}">. And that distribution is guaranteed to be true for all time by the guiding equation if the initial distribution of the particles satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1766f8c8e0a96326d9379b65a63900b3be22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.861ex; height:3.343ex;" alt="{\displaystyle |\psi |^{2}}">. For a given experiment, one can postulate this as being true and verify it experimentally. But, as argued by Dürr et al.,<a href="#cite_note-dgz92-15">[14]</a> one needs to argue that this distribution for subsystems is typical. The authors argue that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1766f8c8e0a96326d9379b65a63900b3be22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.861ex; height:3.343ex;" alt="{\displaystyle |\psi |^{2}}">, by virtue of its equivariance under the dynamical evolution of the system, is the appropriate measure of typicality for <a href="/wiki/Initial_condition" title="Initial condition">initial conditions</a> of the positions of the particles. The authors then prove that the vast majority of possible initial configurations will give rise to statistics obeying the Born rule (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1766f8c8e0a96326d9379b65a63900b3be22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.861ex; height:3.343ex;" alt="{\displaystyle |\psi |^{2}}">) for measurement outcomes. In summary, in a universe governed by the de Broglie–Bohm dynamics, Born rule behavior is typical. The situation is thus analogous to the situation in classical statistical physics. A low-<a href="/wiki/Entropy" title="Entropy">entropy</a> initial condition will, with overwhelmingly high probability, evolve into a higher-entropy state: behavior consistent with the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a> is typical. There are anomalous initial conditions that would give rise to violations of the second law; however in the absence of some very detailed evidence supporting the realization of one of those conditions, it would be quite unreasonable to expect anything but the actually observed uniform increase of entropy. Similarly in the de Broglie–Bohm theory, there are anomalous initial conditions that would produce measurement statistics in violation of the Born rule (conflicting the predictions of standard quantum theory), but the typicality theorem shows that absent some specific reason to believe one of those special initial conditions was in fact realized, the Born rule behavior is what one should expect. It is in this qualified sense that the Born rule is, for the de Broglie–Bohm theory, a <a href="/wiki/Theorem" title="Theorem">theorem</a> rather than (as in ordinary quantum theory) an additional <a href="/wiki/Postulate" class="mw-redirect" title="Postulate">postulate</a>. It can also be shown that a distribution of particles which is not distributed according to the Born rule (that is, a distribution "out of quantum equilibrium") and evolving under the de Broglie–Bohm dynamics is overwhelmingly likely to evolve dynamically into a state distributed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1766f8c8e0a96326d9379b65a63900b3be22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.861ex; height:3.343ex;" alt="{\displaystyle |\psi |^{2}}">.<a href="#cite_note-16">[15]</a> <div class="mw-heading mw-heading3"><h3 id="The_conditional_wavefunction_of_a_subsystem">The conditional wavefunction of a subsystem</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=8" title="Edit section: The conditional wavefunction of a subsystem">edit</a>]</div> In the formulation of the de Broglie–Bohm theory, there is only a wavefunction for the entire universe (which always evolves by the Schrödinger equation). Here, the "universe" is simply the system limited by the same boundary conditions used to solve the Schrödinger equation. However, once the theory is formulated, it is convenient to introduce a notion of wavefunction also for subsystems of the universe. Let us write the wavefunction of the universe as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768c126040a32b1715209c9c8fa21ada56b417bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.634ex; height:3.176ex;" alt="{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{\text{I}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7fdac89c862bb052d03d995c0e3e6fe28c77efb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.905ex; height:3.009ex;" alt="{\displaystyle q^{\text{I}}}"> denotes the configuration variables associated to some subsystem (I) of the universe, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{\text{II}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{\text{II}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e0508438d6211ba35ada56f5ccb4595919cb12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.499ex; height:3.009ex;" alt="{\displaystyle q^{\text{II}}}"> denotes the remaining configuration variables. Denote respectively by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{\text{I}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{\text{I}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c485395b930f3dab78a57f700a4eb3e74d0c8bfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.313ex; height:3.176ex;" alt="{\displaystyle Q^{\text{I}}(t)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{\text{II}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{\text{II}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3424998e3e2ae67151b9e5ebae071a12f8aefe45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.907ex; height:3.176ex;" alt="{\displaystyle Q^{\text{II}}(t)}"> the actual configuration of subsystem (I) and of the rest of the universe. For simplicity, we consider here only the spinless case. The conditional wavefunction of subsystem (I) is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\text{I}}(t,q^{\text{I}})=\psi (t,q^{\text{I}},Q^{\text{II}}(t)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo>,</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\text{I}}(t,q^{\text{I}})=\psi (t,q^{\text{I}},Q^{\text{II}}(t)).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d337d6187f75f2026e4b3fa392ed701539efa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.715ex; height:3.176ex;" alt="{\displaystyle \psi ^{\text{I}}(t,q^{\text{I}})=\psi (t,q^{\text{I}},Q^{\text{II}}(t)).}"></dd></dl> It follows immediately from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(t)=(Q^{\text{I}}(t),Q^{\text{II}}(t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(t)=(Q^{\text{I}}(t),Q^{\text{II}}(t))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee86e277ae0194cca243b47a896e976893242487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.649ex; height:3.176ex;" alt="{\displaystyle Q(t)=(Q^{\text{I}}(t),Q^{\text{II}}(t))}"> satisfies the guiding equation that also the configuration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{\text{I}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{\text{I}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c485395b930f3dab78a57f700a4eb3e74d0c8bfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.313ex; height:3.176ex;" alt="{\displaystyle Q^{\text{I}}(t)}"> satisfies a guiding equation identical to the one presented in the formulation of the theory, with the universal wavefunction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> replaced with the conditional wavefunction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\text{I}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68e660788eb97125222cafc8fb3510caa04f13a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.339ex; height:3.009ex;" alt="{\displaystyle \psi ^{\text{I}}}">. Also, the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b7ab35402f0f501cbc361f5309fe64fd678cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.487ex; height:2.843ex;" alt="{\displaystyle Q(t)}"> is random with <a href="/wiki/Probability_density_function" title="Probability density function">probability density</a> given by the <a href="/wiki/Square_modulus" class="mw-redirect" title="Square modulus">square modulus</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa2cc7a52d0855168c0ab45a438e875eed456c7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.843ex; height:2.843ex;" alt="{\displaystyle \psi (t,\cdot )}"> implies that the <a href="/wiki/Conditional_probability_density_function" class="mw-redirect" title="Conditional probability density function">conditional probability density</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{\text{I}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{\text{I}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c485395b930f3dab78a57f700a4eb3e74d0c8bfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.313ex; height:3.176ex;" alt="{\displaystyle Q^{\text{I}}(t)}"> given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{\text{II}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{\text{II}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3424998e3e2ae67151b9e5ebae071a12f8aefe45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.907ex; height:3.176ex;" alt="{\displaystyle Q^{\text{II}}(t)}"> is given by the square modulus of the (normalized) conditional wavefunction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\text{I}}(t,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\text{I}}(t,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a58b689c1945bf773922f500d1da35e9d43a8b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.669ex; height:3.176ex;" alt="{\displaystyle \psi ^{\text{I}}(t,\cdot )}"> (in the terminology of Dürr et al.<a href="#cite_note-17">[16]</a> this fact is called the fundamental conditional probability formula). Unlike the universal wavefunction, the conditional wavefunction of a subsystem does not always evolve by the Schrödinger equation, but in many situations it does. For instance, if the universal wavefunction factors as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})=\psi ^{\text{I}}(t,q^{\text{I}})\psi ^{\text{II}}(t,q^{\text{II}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">)</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})=\psi ^{\text{I}}(t,q^{\text{I}})\psi ^{\text{II}}(t,q^{\text{II}}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a08e9236cd83291b8eded67ddff602cd7bf5c573" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.422ex; height:3.176ex;" alt="{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})=\psi ^{\text{I}}(t,q^{\text{I}})\psi ^{\text{II}}(t,q^{\text{II}}),}"></dd></dl> then the conditional wavefunction of subsystem (I) is (up to an irrelevant scalar factor) equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\text{I}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68e660788eb97125222cafc8fb3510caa04f13a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.339ex; height:3.009ex;" alt="{\displaystyle \psi ^{\text{I}}}"> (this is what standard quantum theory would regard as the wavefunction of subsystem (I)). If, in addition, the Hamiltonian does not contain an interaction term between subsystems (I) and (II), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\text{I}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68e660788eb97125222cafc8fb3510caa04f13a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.339ex; height:3.009ex;" alt="{\displaystyle \psi ^{\text{I}}}"> does satisfy a Schrödinger equation. More generally, assume that the universal wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> can be written in the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})=\psi ^{\text{I}}(t,q^{\text{I}})\psi ^{\text{II}}(t,q^{\text{II}})+\phi (t,q^{\text{I}},q^{\text{II}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo stretchy="false">)</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})=\psi ^{\text{I}}(t,q^{\text{I}})\psi ^{\text{II}}(t,q^{\text{II}})+\phi (t,q^{\text{I}},q^{\text{II}}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4685a5afe89c2f0cc041e40dbbdb9a1f6d8da46f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.769ex; height:3.176ex;" alt="{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})=\psi ^{\text{I}}(t,q^{\text{I}})\psi ^{\text{II}}(t,q^{\text{II}})+\phi (t,q^{\text{I}},q^{\text{II}}),}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> solves Schrödinger equation and, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (t,q^{\text{I}},Q^{\text{II}}(t))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> <mo>,</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (t,q^{\text{I}},Q^{\text{II}}(t))=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/784f844f3f395d690f921fc8dc41feeed71eee68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.175ex; height:3.176ex;" alt="{\displaystyle \phi (t,q^{\text{I}},Q^{\text{II}}(t))=0}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{\text{I}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7fdac89c862bb052d03d995c0e3e6fe28c77efb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.905ex; height:3.009ex;" alt="{\displaystyle q^{\text{I}}}">. Then, again, the conditional wavefunction of subsystem (I) is (up to an irrelevant scalar factor) equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\text{I}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68e660788eb97125222cafc8fb3510caa04f13a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.339ex; height:3.009ex;" alt="{\displaystyle \psi ^{\text{I}}}">, and if the Hamiltonian does not contain an interaction term between subsystems (I) and (II), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\text{I}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68e660788eb97125222cafc8fb3510caa04f13a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.339ex; height:3.009ex;" alt="{\displaystyle \psi ^{\text{I}}}"> satisfies a Schrödinger equation. The fact that the conditional wavefunction of a subsystem does not always evolve by the Schrödinger equation is related to the fact that the usual collapse rule of standard quantum theory emerges from the Bohmian formalism when one considers conditional wavefunctions of subsystems. <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=9" title="Edit section: Extensions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Relativity">Relativity</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=10" title="Edit section: Relativity">edit</a>]</div> Pilot-wave theory is explicitly nonlocal, which is in ostensible conflict with <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. Various extensions of "Bohm-like" mechanics exist that attempt to resolve this problem. Bohm himself in 1953 presented an extension of the theory satisfying the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> for a single particle. However, this was not extensible to the many-particle case because it used an absolute time.<a href="#cite_note-18">[17]</a> A renewed interest in constructing <a href="/wiki/Lorentz_scalar" title="Lorentz scalar">Lorentz-invariant</a> extensions of Bohmian theory arose in the 1990s; see Bohm and Hiley: The Undivided Universe<a href="#cite_note-19">[18]</a><a href="#cite_note-20">[19]</a> and references therein. Another approach is given by Dürr et al.,<a href="#cite_note-21">[20]</a> who use Bohm–Dirac models and a Lorentz-invariant foliation of space-time. Thus, Dürr et al. (1999) showed that it is possible to formally restore Lorentz invariance for the Bohm–Dirac theory by introducing additional structure. This approach still requires a <a href="/wiki/Foliation" title="Foliation">foliation</a> of space-time. While this is in conflict with the standard interpretation of relativity, the preferred foliation, if unobservable, does not lead to any empirical conflicts with relativity. In 2013, Dürr et al. suggested that the required foliation could be covariantly determined by the wavefunction.<a href="#cite_note-22">[21]</a> The relation between nonlocality and preferred foliation can be better understood as follows. In de Broglie–Bohm theory, nonlocality manifests as the fact that the velocity and acceleration of one particle depends on the instantaneous positions of all other particles. On the other hand, in the theory of relativity the concept of instantaneousness does not have an invariant meaning. Thus, to define particle trajectories, one needs an additional rule that defines which space-time points should be considered instantaneous. The simplest way to achieve this is to introduce a preferred foliation of space-time by hand, such that each hypersurface of the foliation defines a hypersurface of equal time. Initially, it had been considered impossible to set out a description of photon trajectories in the de Broglie–Bohm theory in view of the difficulties of describing bosons relativistically.<a href="#cite_note-ghose-1996-23">[22]</a> In 1996, <a href="/wiki/Partha_Ghose" title="Partha Ghose">Partha Ghose</a> presented a relativistic quantum-mechanical description of spin-0 and spin-1 bosons starting from the <a href="/wiki/Duffin%E2%80%93Kemmer%E2%80%93Petiau_equation" class="mw-redirect" title="Duffin–Kemmer–Petiau equation">Duffin–Kemmer–Petiau equation</a>, setting out Bohmian trajectories for massive bosons and for massless bosons (and therefore <a href="/wiki/Photon" title="Photon">photons</a>).<a href="#cite_note-ghose-1996-23">[22]</a> In 2001, <a href="/wiki/Jean-Pierre_Vigier" title="Jean-Pierre Vigier">Jean-Pierre Vigier</a> emphasized the importance of deriving a well-defined description of light in terms of particle trajectories in the framework of either the Bohmian mechanics or the Nelson stochastic mechanics.<a href="#cite_note-24">[23]</a> The same year, Ghose worked out Bohmian photon trajectories for specific cases.<a href="#cite_note-25">[24]</a> Subsequent <a href="/wiki/Weak_measurement" title="Weak measurement">weak-measurement</a> experiments yielded trajectories that coincide with the predicted trajectories.<a href="#cite_note-26">[25]</a><a href="#cite_note-27">[26]</a> The significance of these experimental findings is controversial.<a href="#cite_note-28">[27]</a> Chris Dewdney and G. Horton have proposed a relativistically covariant, wave-functional formulation of Bohm's quantum field theory<a href="#cite_note-29">[28]</a><a href="#cite_note-30">[29]</a> and have extended it to a form that allows the inclusion of gravity.<a href="#cite_note-31">[30]</a> Nikolić has proposed a Lorentz-covariant formulation of the Bohmian interpretation of many-particle wavefunctions.<a href="#cite_note-32">[31]</a> He has developed a generalized relativistic-invariant probabilistic interpretation of quantum theory,<a href="#cite_note-nikolicqft-33">[32]</a><a href="#cite_note-34">[33]</a><a href="#cite_note-35">[34]</a> in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1766f8c8e0a96326d9379b65a63900b3be22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.861ex; height:3.343ex;" alt="{\displaystyle |\psi |^{2}}"> is no longer a probability density in space, but a probability density in space-time. He uses this generalized probabilistic interpretation to formulate a relativistic-covariant version of de Broglie–Bohm theory without introducing a preferred foliation of space-time. His work also covers the extension of the Bohmian interpretation to a quantization of fields and strings.<a href="#cite_note-36">[35]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Quantum_potential#Relativistic_and_field-theoretic_extensions" title="Quantum potential">Quantum potential § Relativistic and field-theoretic extensions</a></div> Roderick I. Sutherland at the University in Sydney has a Lagrangian formalism for the pilot wave and its <a href="https://en.wiktionary.org/wiki/beable" class="extiw" title="wikt:beable">beables</a>. It draws on <a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Yakir Aharonov</a>'s retrocasual weak measurements to explain many-particle entanglement in a special relativistic way without the need for configuration space. The basic idea was already published by <a href="/wiki/Olivier_Costa_de_Beauregard" title="Olivier Costa de Beauregard">Costa de Beauregard</a> in the 1950s and is also used by <a href="/wiki/John_G._Cramer" title="John G. Cramer">John Cramer</a> in his transactional interpretation except the beables that exist between the von Neumann strong projection operator measurements. Sutherland's Lagrangian includes two-way action-reaction between pilot wave and beables. Therefore, it is a post-quantum non-statistical theory with final boundary conditions that violate the no-signal theorems of quantum theory. Just as special relativity is a limiting case of general relativity when the spacetime curvature vanishes, so, too is statistical no-entanglement signaling quantum theory with the Born rule a limiting case of the post-quantum action-reaction Lagrangian when the reaction is set to zero and the final boundary condition is integrated out.<a href="#cite_note-37">[36]</a> <div class="mw-heading mw-heading3"><h3 id="Spin">Spin</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=11" title="Edit section: Spin">edit</a>]</div> To incorporate <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>, the wavefunction becomes complex-vector-valued. The value space is called spin space; for a <a href="/wiki/Spin-1/2" title="Spin-1/2">spin-1/2</a> particle, spin space can be taken to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{2}}">. The guiding equation is modified by taking <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner products</a> in spin space to reduce the complex vectors to complex numbers. The Schrödinger equation is modified by adding a <a href="/wiki/Pauli_equation" title="Pauli equation">Pauli spin term</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d\mathbf {Q} _{k}}{dt}}(t)&={\frac {\hbar }{m_{k}}}\operatorname {Im} \left({\frac {(\psi ,D_{k}\psi )}{(\psi ,\psi )}}\right)(\mathbf {Q} _{1},\ldots ,\mathbf {Q} _{N},t),\\i\hbar {\frac {\partial }{\partial t}}\psi &=\left(-\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}D_{k}^{2}+V-\sum _{k=1}^{N}\mu _{k}{\frac {\mathbf {S} _{k}}{\hbar s_{k}}}\cdot \mathbf {B} (\mathbf {q} _{k})\right)\psi ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mfrac> </mrow> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo>,</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>V</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mi class="MJX-variant">ℏ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d\mathbf {Q} _{k}}{dt}}(t)&={\frac {\hbar }{m_{k}}}\operatorname {Im} \left({\frac {(\psi ,D_{k}\psi )}{(\psi ,\psi )}}\right)(\mathbf {Q} _{1},\ldots ,\mathbf {Q} _{N},t),\\i\hbar {\frac {\partial }{\partial t}}\psi &=\left(-\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}D_{k}^{2}+V-\sum _{k=1}^{N}\mu _{k}{\frac {\mathbf {S} _{k}}{\hbar s_{k}}}\cdot \mathbf {B} (\mathbf {q} _{k})\right)\psi ,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36e096c44b9fda59d9004a802686606ad2254c77" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:57.865ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {d\mathbf {Q} _{k}}{dt}}(t)&={\frac {\hbar }{m_{k}}}\operatorname {Im} \left({\frac {(\psi ,D_{k}\psi )}{(\psi ,\psi )}}\right)(\mathbf {Q} _{1},\ldots ,\mathbf {Q} _{N},t),\\i\hbar {\frac {\partial }{\partial t}}\psi &=\left(-\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}D_{k}^{2}+V-\sum _{k=1}^{N}\mu _{k}{\frac {\mathbf {S} _{k}}{\hbar s_{k}}}\cdot \mathbf {B} (\mathbf {q} _{k})\right)\psi ,\end{aligned}}}"> where <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{k},e_{k},\mu _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{k},e_{k},\mu _{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7b661b0ce81e6775472ae0eb3c9c00d14c2ad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.86ex; height:2.176ex;" alt="{\displaystyle m_{k},e_{k},\mu _{k}}"> — the mass, charge and <a href="/wiki/Magnetic_moment" title="Magnetic moment">magnetic moment</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">–th particle</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} _{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09491d73286bdeeeea46ccc7c69882a9c846a964" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.509ex;" alt="{\displaystyle \mathbf {S} _{k}}"> — the appropriate <a href="/wiki/Spin_(physics)#Operator" title="Spin (physics)">spin operator</a> acting in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">–th particle's spin space</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f159343172781e7666dbc88280c91f34117c30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.009ex;" alt="{\displaystyle s_{k}}"> — <a href="/wiki/Spin_quantum_number" title="Spin quantum number">spin quantum number</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">–th particle (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}=1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{k}=1/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf719c9fcbd0794d2c6ca3b300c71424c81d0082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.765ex; height:2.843ex;" alt="{\displaystyle s_{k}=1/2}"> for electron)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"> is <a href="/wiki/Vector_potential" title="Vector potential">vector potential</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/436aaad562d4e3626ff807cadb0185658e4e6c51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.795ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }"> is the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle D_{k}=\nabla _{k}-{\frac {ie_{k}}{\hbar }}\mathbf {A} (\mathbf {q} _{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle D_{k}=\nabla _{k}-{\frac {ie_{k}}{\hbar }}\mathbf {A} (\mathbf {q} _{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf3131b6215031a20cb07307a071634e409e5844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.334ex; height:4.009ex;" alt="{\textstyle D_{k}=\nabla _{k}-{\frac {ie_{k}}{\hbar }}\mathbf {A} (\mathbf {q} _{k})}"> is the covariant derivative, involving the vector potential, ascribed to the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">–th particle (in <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a>)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> — the wavefunction defined on the multidimensional configuration space; e.g. a system consisting of two spin-1/2 particles and one spin-1 particle has a wavefunction of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi :\mathbb {R} ^{9}\times \mathbb {R} \to \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}\otimes \mathbb {C} ^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⊗</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⊗</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi :\mathbb {R} ^{9}\times \mathbb {R} \to \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}\otimes \mathbb {C} ^{3},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d38aea01cd5b4378d149f95b477ff05d4e2d4c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.84ex; height:3.009ex;" alt="{\displaystyle \psi :\mathbb {R} ^{9}\times \mathbb {R} \to \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}\otimes \mathbb {C} ^{3},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }"> is a <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a>, so this spin space is 12-dimensional</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot ,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cdot ,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc515c912925128800226dd0b017be508069e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle (\cdot ,\cdot )}"> is the <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> in spin space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{d}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc165f5f6c2360e365b6693209c45fe805a0781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{d}}">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\phi ,\psi )=\sum _{s=1}^{d}\phi _{s}^{*}\psi _{s}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </munderover> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi ,\psi )=\sum _{s=1}^{d}\phi _{s}^{*}\psi _{s}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/719c2528fe40930f605da18877fd7d9f27bfcafc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.185ex; height:7.343ex;" alt="{\displaystyle (\phi ,\psi )=\sum _{s=1}^{d}\phi _{s}^{*}\psi _{s}.}"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Stochastic_electrodynamics">Stochastic electrodynamics</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=12" title="Edit section: Stochastic electrodynamics">edit</a>]</div> <a href="/wiki/Stochastic_electrodynamics" title="Stochastic electrodynamics">Stochastic electrodynamics</a> (SED) is an extension of the de Broglie–Bohm interpretation of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, with the electromagnetic <a href="/wiki/Zero-point_energy" title="Zero-point energy">zero-point field</a> (ZPF) playing a central role as the guiding <a href="/wiki/Pilot-wave" class="mw-redirect" title="Pilot-wave">pilot-wave</a>. Modern approaches to SED, like those proposed by the group around late Gerhard Grössing, among others, consider wave and particle-like quantum effects as well-coordinated emergent systems. These emergent systems are the result of speculated and calculated sub-quantum interactions with the zero-point field.<a href="#cite_note-38">[37]</a><a href="#cite_note-39">[38]</a><a href="#cite_note-40">[39]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_field_theory">Quantum field theory</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=13" title="Edit section: Quantum field theory">edit</a>]</div> In Dürr et al.,<a href="#cite_note-dgtz04-41">[40]</a><a href="#cite_note-42">[41]</a> the authors describe an extension of de Broglie–Bohm theory for handling <a href="/wiki/Creation_and_annihilation_operators" title="Creation and annihilation operators">creation and annihilation operators</a>, which they refer to as "Bell-type quantum field theories". The basic idea is that configuration space becomes the (disjoint) space of all possible configurations of any number of particles. For part of the time, the system evolves deterministically under the guiding equation with a fixed number of particles. But under a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a>, particles may be created and annihilated. The distribution of creation events is dictated by the wavefunction. The wavefunction itself is evolving at all times over the full multi-particle configuration space. Hrvoje Nikolić<a href="#cite_note-nikolicqft-33">[32]</a> introduces a purely deterministic de Broglie–Bohm theory of particle creation and destruction, according to which particle trajectories are continuous, but particle detectors behave as if particles have been created or destroyed even when a true creation or destruction of particles does not take place. <div class="mw-heading mw-heading3"><h3 id="Curved_space">Curved space</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=14" title="Edit section: Curved space">edit</a>]</div> To extend de Broglie–Bohm theory to curved space (<a href="/wiki/Riemannian_manifolds" class="mw-redirect" title="Riemannian manifolds">Riemannian manifolds</a> in mathematical parlance), one simply notes that all of the elements of these equations make sense, such as <a href="/wiki/Gradient" title="Gradient">gradients</a> and <a href="/wiki/Laplacian" class="mw-redirect" title="Laplacian">Laplacians</a>. Thus, we use equations that have the same form as above. Topological and <a href="/wiki/Boundary_conditions" class="mw-redirect" title="Boundary conditions">boundary conditions</a> may apply in supplementing the evolution of Schrödinger's equation. For a de Broglie–Bohm theory on curved space with spin, the spin space becomes a <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> over configuration space, and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space.<a href="#cite_note-43">[42]</a> The field equations for the de Broglie–Bohm theory in the relativistic case with spin can also be given for curved space-times with torsion.<a href="#cite_note-44">[43]</a><a href="#cite_note-45">[44]</a> In a general spacetime with curvature and torsion, the guiding equation for the <a href="/wiki/Four-velocity" title="Four-velocity">four-velocity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1794f924463f261d905efba7f417274435899f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.129ex; height:2.676ex;" alt="{\displaystyle u^{i}}"> of an elementary <a href="/wiki/Fermion" title="Fermion">fermion</a> particle is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{i}={\frac {e_{\mu }^{i}{\bar {\psi }}\gamma ^{\mu }\psi }{{\bar {\psi }}\psi }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ψ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>ψ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{i}={\frac {e_{\mu }^{i}{\bar {\psi }}\gamma ^{\mu }\psi }{{\bar {\psi }}\psi }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c35b4491cfbc652b62c2d71c9b5d4c884fe298ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.63ex; height:6.843ex;" alt="{\displaystyle u^{i}={\frac {e_{\mu }^{i}{\bar {\psi }}\gamma ^{\mu }\psi }{{\bar {\psi }}\psi }},}">where the wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> is a <a href="/wiki/Dirac_spinor" title="Dirac spinor">spinor</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890adebe2730294079a81b7bf08b2fe0f2c59909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.596ex; height:2.843ex;" alt="{\displaystyle {\bar {\psi }}}"> is the corresponding <a href="/wiki/Dirac_adjoint" title="Dirac adjoint">adjoint</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{\mu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9ad28fd0224d333e10919fe473cf40e52ac6c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.503ex; height:2.843ex;" alt="{\displaystyle \gamma ^{\mu }}"> are the <a href="/wiki/Gamma_matrices" title="Gamma matrices">Dirac matrices</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\mu }^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{\mu }^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fa225b66b3257f59f530d3871ee93a2fabb1cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.307ex; height:3.176ex;" alt="{\displaystyle e_{\mu }^{i}}"> is a <a href="/wiki/Tetrad_formalism" title="Tetrad formalism">tetrad</a>.<a href="#cite_note-FG-46">[45]</a> If the wave function propagates according to the <a href="/wiki/Dirac_equation_in_curved_spacetime" title="Dirac equation in curved spacetime">curved</a> Dirac equation, then the particle moves according to the <a href="/wiki/Mathisson-Papapetrou-Dixon_equations" class="mw-redirect" title="Mathisson-Papapetrou-Dixon equations">Mathisson-Papapetrou equations</a> of motion, which are an extension of the <a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">geodesic equation</a>. This relativistic wave-particle duality follows from the <a href="/wiki/Conservation_law" title="Conservation law">conservation laws</a> for the <a href="/wiki/Spin_tensor" title="Spin tensor">spin tensor</a> and <a href="/wiki/Stress-energy_tensor" class="mw-redirect" title="Stress-energy tensor">energy-momentum tensor</a>,<a href="#cite_note-FG-46">[45]</a> and also from the covariant <a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg picture</a> equation of motion.<a href="#cite_note-47">[46]</a> <div class="mw-heading mw-heading3"><h3 id="Exploiting_nonlocality">Exploiting nonlocality</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=15" title="Edit section: Exploiting nonlocality">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quantum_non-equilibrium" title="Quantum non-equilibrium">Quantum non-equilibrium</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Quantum_Theory_is_a_special_case_of_a_wider_physics.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Quantum_Theory_is_a_special_case_of_a_wider_physics.svg/286px-Quantum_Theory_is_a_special_case_of_a_wider_physics.svg.png" decoding="async" width="286" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Quantum_Theory_is_a_special_case_of_a_wider_physics.svg/429px-Quantum_Theory_is_a_special_case_of_a_wider_physics.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Quantum_Theory_is_a_special_case_of_a_wider_physics.svg/572px-Quantum_Theory_is_a_special_case_of_a_wider_physics.svg.png 2x" data-file-width="750" data-file-height="400" /></a><figcaption>Diagram made by <a href="/wiki/Antony_Valentini" title="Antony Valentini">Antony Valentini</a> in a lecture about the De Broglie–Bohm theory. Valentini argues quantum theory is a special equilibrium case of a wider physics and that it may be possible to observe and exploit <a href="/wiki/Quantum_non-equilibrium" title="Quantum non-equilibrium">quantum non-equilibrium</a><a href="#cite_note-48">[47]</a></figcaption></figure> <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">De Broglie</a> and Bohm's causal interpretation of quantum mechanics was later extended by Bohm, Vigier, Hiley, Valentini and others to include stochastic properties. Bohm and other physicists, including Valentini, view the Born rule linking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> to the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =R^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =R^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d510dfb50093f9115982f0b7b9c08916c936dba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.119ex; height:3.176ex;" alt="{\displaystyle \rho =R^{2}}"> as representing not a basic law, but a result of a system having reached quantum equilibrium during the course of the time development under the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>. It can be shown that, once an equilibrium has been reached, the system remains in such equilibrium over the course of its further evolution: this follows from the <a href="/wiki/Continuity_equation#Quantum_mechanics" title="Continuity equation">continuity equation</a> associated with the Schrödinger evolution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }">.<a href="#cite_note-49">[48]</a> It is less straightforward to demonstrate whether and how such an equilibrium is reached in the first place. <a href="/wiki/Antony_Valentini" title="Antony Valentini">Antony Valentini</a><a href="#cite_note-50">[49]</a> has extended de Broglie–Bohm theory to include signal nonlocality that would allow entanglement to be used as a stand-alone communication channel without a secondary classical "key" signal to "unlock" the message encoded in the entanglement. This violates orthodox quantum theory but has the virtue of making the parallel universes of the <a href="/wiki/Chaotic_inflation_theory" class="mw-redirect" title="Chaotic inflation theory">chaotic inflation theory</a> observable in principle. Unlike de Broglie–Bohm theory, Valentini's theory the wavefunction evolution also depends on the ontological variables. This introduces an instability, a feedback loop that pushes the hidden variables out of "sub-quantal heat death". The resulting theory becomes nonlinear and non-unitary. Valentini argues that the laws of quantum mechanics are <a href="/wiki/Emergence" title="Emergence">emergent</a> and form a "quantum equilibrium" that is analogous to thermal equilibrium in classical dynamics, such that other "<a href="/wiki/Quantum_non-equilibrium" title="Quantum non-equilibrium">quantum non-equilibrium</a>" distributions may in principle be observed and exploited, for which the statistical predictions of quantum theory are violated. It is controversially argued that quantum theory is merely a special case of a much wider nonlinear physics, a physics in which non-local (<a href="/wiki/Faster-than-light" title="Faster-than-light">superluminal</a>) signalling is possible, and in which the uncertainty principle can be violated.<a href="#cite_note-Valentini2009-51">[50]</a><a href="#cite_note-52">[51]</a> <div class="mw-heading mw-heading2"><h2 id="Results">Results</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=16" title="Edit section: Results">edit</a>]</div> Below are some highlights of the results that arise out of an analysis of de Broglie–Bohm theory. Experimental results agree with all of quantum mechanics' standard predictions insofar as it has them. But while standard quantum mechanics is limited to discussing the results of "measurements", de Broglie–Bohm theory governs the dynamics of a system without the intervention of outside observers (p. 117 in Bell<a href="#cite_note-bell-53">[52]</a>). The basis for agreement with standard quantum mechanics is that the particles are distributed according to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1766f8c8e0a96326d9379b65a63900b3be22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.861ex; height:3.343ex;" alt="{\displaystyle |\psi |^{2}}">. This is a statement of observer ignorance: the initial positions are represented by a statistical distribution so deterministic trajectories will result in a statistical distribution.<a href="#cite_note-dgz92-15">[14]</a> <div class="mw-heading mw-heading3"><h3 id="Measuring_spin_and_polarization">Measuring spin and polarization</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=17" title="Edit section: Measuring spin and polarization">edit</a>]</div> According to ordinary quantum theory, it is not possible to measure the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> or <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> of a particle directly; instead, the component in one direction is measured; the outcome from a single particle may be 1, meaning that the particle is aligned with the measuring apparatus, or −1, meaning that it is aligned the opposite way. An ensemble of particles prepared by a polarizer to be in state 1 will all measure polarized in state 1 in a subsequent apparatus. A polarized ensemble sent through a polarizer set at angle to the first pass will result in some values of 1 and some of −1 with a probability that depends on the relative alignment. For a full explanation of this, see the <a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach experiment</a>. In de Broglie–Bohm theory, the results of a spin experiment cannot be analyzed without some knowledge of the experimental setup. It is possible<a href="#cite_note-54">[53]</a> to modify the setup so that the trajectory of the particle is unaffected, but that the particle with one setup registers as spin-up, while in the other setup it registers as spin-down. Thus, for the de Broglie–Bohm theory, the particle's spin is not an intrinsic property of the particle; instead spin is, so to speak, in the wavefunction of the particle in relation to the particular device being used to measure the spin. This is an illustration of what is sometimes referred to as contextuality and is related to naive realism about operators.<a href="#cite_note-55">[54]</a> Interpretationally, measurement results are a deterministic property of the system and its environment, which includes information about the experimental setup including the context of co-measured observables; in no sense does the system itself possess the property being measured, as would have been the case in classical physics. <div class="mw-heading mw-heading3"><h3 id="Measurements,_the_quantum_formalism,_and_observer_independence">Measurements, the quantum formalism, and observer independence</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=18" title="Edit section: Measurements, the quantum formalism, and observer independence">edit</a>]</div> De Broglie–Bohm theory gives the almost results as (non-relativisitic) quantum mechanics. It treats the wavefunction as a fundamental object in the theory, as the wavefunction describes how the particles move. This means that no experiment can distinguish between the two theories. This section outlines the ideas as to how the standard quantum formalism arises out of quantum mechanics.<a href="#cite_note-:0-14">[13]</a><a href="#cite_note-dgz92-15">[14]</a> <div class="mw-heading mw-heading4"><h4 id="Collapse_of_the_wavefunction">Collapse of the wavefunction</h4>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=19" title="Edit section: Collapse of the wavefunction">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/De_Broglie%E2%80%93Bohm_theory" title="Special:EditPage/De Broglie–Bohm theory">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (September 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> De Broglie–Bohm theory is a theory that applies primarily to the whole universe. That is, there is a single wavefunction governing the motion of all of the particles in the universe according to the guiding equation. Theoretically, the motion of one particle depends on the positions of all of the other particles in the universe. In some situations, such as in experimental systems, we can represent the system itself in terms of a de Broglie–Bohm theory in which the wavefunction of the system is obtained by conditioning on the environment of the system. Thus, the system can be analyzed with Schrödinger's equation and the guiding equation, with an initial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1766f8c8e0a96326d9379b65a63900b3be22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.861ex; height:3.343ex;" alt="{\displaystyle |\psi |^{2}}"> distribution for the particles in the system (see the section on <a href="#The_conditional_wavefunction_of_a_subsystem">the conditional wavefunction of a subsystem</a> for details). It requires a special setup for the conditional wavefunction of a system to obey a quantum evolution. When a system interacts with its environment, such as through a measurement, the conditional wavefunction of the system evolves in a different way. The evolution of the universal wavefunction can become such that the wavefunction of the system appears to be in a superposition of distinct states. But if the environment has recorded the results of the experiment, then using the actual Bohmian configuration of the environment to condition on, the conditional wavefunction collapses to just one alternative, the one corresponding with the measurement results. <a href="/wiki/Wavefunction_collapse" class="mw-redirect" title="Wavefunction collapse">Collapse</a> of the universal wavefunction never occurs in de Broglie–Bohm theory. Its entire evolution is governed by Schrödinger's equation, and the particles' evolutions are governed by the guiding equation. Collapse only occurs in a <a href="/wiki/Phenomenology_(physics)" title="Phenomenology (physics)">phenomenological</a> way for systems that seem to follow their own Schrödinger's equation. As this is an effective description of the system, it is a matter of choice as to what to define the experimental system to include, and this will affect when "collapse" occurs. <div class="mw-heading mw-heading4"><h4 id="Operators_as_observables">Operators as observables</h4>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=20" title="Edit section: Operators as observables">edit</a>]</div> In the standard quantum formalism, measuring observables is generally thought of as measuring operators on the Hilbert space. For example, measuring position is considered to be a measurement of the position operator. This relationship between physical measurements and Hilbert space operators is, for standard quantum mechanics, an additional axiom of the theory. The de Broglie–Bohm theory, by contrast, requires no such measurement axioms (and measurement as such is not a dynamically distinct or special sub-category of physical processes in the theory). In particular, the usual operators-as-observables formalism is, for de Broglie–Bohm theory, a theorem.<a href="#cite_note-56">[55]</a> A major point of the analysis is that many of the measurements of the observables do not correspond to properties of the particles; they are (as in the case of spin discussed above) measurements of the wavefunction. In the history of de Broglie–Bohm theory, the proponents have often had to deal with claims that this theory is impossible. Such arguments are generally based on inappropriate analysis of operators as observables. If one believes that spin measurements are indeed measuring the spin of a particle that existed prior to the measurement, then one does reach contradictions. De Broglie–Bohm theory deals with this by noting that spin is not a feature of the particle, but rather that of the wavefunction. As such, it only has a definite outcome once the experimental apparatus is chosen. Once that is taken into account, the impossibility theorems become irrelevant. There are also objections to this theory based on what it says about particular situations usually involving eigenstates of an operator. For example, the ground state of hydrogen is a real wavefunction. According to the guiding equation, this means that the electron is at rest when in this state. Nevertheless, it is distributed according to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1766f8c8e0a96326d9379b65a63900b3be22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.861ex; height:3.343ex;" alt="{\displaystyle |\psi |^{2}}">, and no contradiction to experimental results is possible to detect. Operators as observables leads many to believe that many operators are equivalent. De Broglie–Bohm theory, from this perspective, chooses the position observable as a favored observable rather than, say, the momentum observable. Again, the link to the position observable is a consequence of the dynamics. The motivation for de Broglie–Bohm theory is to describe a system of particles. This implies that the goal of the theory is to describe the positions of those particles at all times. Other observables do not have this compelling ontological status. Having definite positions explains having definite results such as flashes on a detector screen. Other observables would not lead to that conclusion, but there need not be any problem in defining a mathematical theory for other observables; see Hyman et al.<a href="#cite_note-57">[56]</a> for an exploration of the fact that a probability density and probability current can be defined for any set of commuting operators. <div class="mw-heading mw-heading4"><h4 id="Hidden_variables">Hidden variables</h4>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=21" title="Edit section: Hidden variables">edit</a>]</div> De Broglie–Bohm theory is often referred to as a "hidden-variable" theory. Bohm used this description in his original papers on the subject, writing: "From the point of view of the <a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">usual interpretation</a>, these additional elements or parameters [permitting a detailed causal and continuous description of all processes] could be called 'hidden' variables." Bohm and Hiley later stated that they found Bohm's choice of the term "hidden variables" to be too restrictive. In particular, they argued that a particle is not actually hidden but rather "is what is most directly manifested in an observation [though] its properties cannot be observed with arbitrary precision (within the limits set by <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>)".<a href="#cite_note-58">[57]</a> However, others nevertheless treat the term "hidden variable" as a suitable description.<a href="#cite_note-59">[58]</a> Generalized particle trajectories can be extrapolated from numerous weak measurements on an ensemble of equally prepared systems, and such trajectories coincide with the de Broglie–Bohm trajectories. In particular, an experiment with two entangled photons, in which a set of Bohmian trajectories for one of the photons was determined using weak measurements and postselection, can be understood in terms of a nonlocal connection between that photon's trajectory and the other photon's polarization.<a href="#cite_note-60">[59]</a><a href="#cite_note-newscientist.com-61">[60]</a> However, not only the De Broglie–Bohm interpretation, but also many other interpretations of quantum mechanics that do not include such trajectories are consistent with such experimental evidence. <div class="mw-heading mw-heading3"><h3 id="Different_predictions">Different predictions</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=22" title="Edit section: Different predictions">edit</a>]</div> A specialized version of the <a href="/wiki/Double_slit_experiment" class="mw-redirect" title="Double slit experiment">double slit experiment</a> has been devised to test characteristics of the trajectory predictions.<a href="#cite_note-62">[61]</a> Experimental realization of this concept disagreed with the Bohm predictions.<a href="#cite_note-63">[62]</a> where they differed from standard quantum mechanics. These conclusions have been the subject of debate.<a href="#cite_note-64">[63]</a><a href="#cite_note-65">[64]</a> <div class="mw-heading mw-heading3"><h3 id="Heisenberg's_uncertainty_principle">Heisenberg's uncertainty principle</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=23" title="Edit section: Heisenberg's uncertainty principle">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/De_Broglie%E2%80%93Bohm_theory" title="Special:EditPage/De Broglie–Bohm theory">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (September 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> The Heisenberg's uncertainty principle states that when two complementary measurements are made, there is a limit to the product of their accuracy. As an example, if one measures the position with an accuracy of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"> and the momentum with an accuracy of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0758c326125ad3d8b96e515c7fd69164ec587b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.105ex; height:2.509ex;" alt="{\displaystyle \Delta p}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x\Delta p\gtrsim h.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mi mathvariant="normal">Δ</mi> <mi>p</mi> <mo>≳</mo> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x\Delta p\gtrsim h.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d938bd3080a161e9848e1bff0fc54bbe6d374f9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.455ex; height:2.843ex;" alt="{\displaystyle \Delta x\Delta p\gtrsim h.}"> In de Broglie–Bohm theory, there is always a matter of fact about the position and momentum of a particle. Each particle has a well-defined trajectory, as well as a wavefunction. Observers have limited knowledge as to what this trajectory is (and thus of the position and momentum). It is the lack of knowledge of the particle's trajectory that accounts for the uncertainty relation. What one can know about a particle at any given time is described by the wavefunction. Since the uncertainty relation can be derived from the wavefunction in other interpretations of quantum mechanics, it can be likewise derived (in the <a href="/wiki/Epistemology" title="Epistemology">epistemic</a> sense mentioned above) on the de Broglie–Bohm theory. To put the statement differently, the particles' positions are only known statistically. As in <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, successive observations of the particles' positions refine the experimenter's knowledge of the particles' <a href="/wiki/Initial_conditions" class="mw-redirect" title="Initial conditions">initial conditions</a>. Thus, with succeeding observations, the initial conditions become more and more restricted. This formalism is consistent with the normal use of the Schrödinger equation. For the derivation of the uncertainty relation, see <a href="/wiki/Heisenberg_uncertainty_principle" class="mw-redirect" title="Heisenberg uncertainty principle">Heisenberg uncertainty principle</a>, noting that this article describes the principle from the viewpoint of the <a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a>. <div class="mw-heading mw-heading3"><h3 id="Quantum_entanglement,_Einstein–Podolsky–Rosen_paradox,_Bell's_theorem,_and_nonlocality">Quantum entanglement, Einstein–Podolsky–Rosen paradox, Bell's theorem, and nonlocality</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=24" title="Edit section: Quantum entanglement, Einstein–Podolsky–Rosen paradox, Bell's theorem, and nonlocality">edit</a>]</div> De Broglie–Bohm theory highlighted the issue of <a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">nonlocality</a>: it inspired <a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">John Stewart Bell</a> to prove his now-famous <a href="/wiki/Bell%27s_theorem" title="Bell's theorem">theorem</a>,<a href="#cite_note-66">[65]</a> which in turn led to the <a href="/wiki/Bell_test_experiments" class="mw-redirect" title="Bell test experiments">Bell test experiments</a>. In the <a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">Einstein–Podolsky–Rosen paradox</a>, the authors describe a thought experiment that one could perform on a pair of particles that have interacted, the results of which they interpreted as indicating that quantum mechanics is an incomplete theory.<a href="#cite_note-67">[66]</a> Decades later <a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">John Bell</a> proved <a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's theorem</a> (see p. 14 in Bell<a href="#cite_note-bell-53">[52]</a>), in which he showed that, if they are to agree with the empirical predictions of quantum mechanics, all such "hidden-variable" completions of quantum mechanics must either be nonlocal (as the Bohm interpretation is) or give up the assumption that experiments produce unique results (see <a href="/wiki/Counterfactual_definiteness" title="Counterfactual definiteness">counterfactual definiteness</a> and <a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">many-worlds interpretation</a>). In particular, Bell proved that any local theory with unique results must make empirical predictions satisfying a statistical constraint called "Bell's inequality". <a href="/wiki/Alain_Aspect" title="Alain Aspect">Alain Aspect</a> performed a series of <a href="/wiki/Bell_test_experiments" class="mw-redirect" title="Bell test experiments">Bell test experiments</a> that test Bell's inequality using an EPR-type setup. Aspect's results show experimentally that Bell's inequality is in fact violated, meaning that the relevant quantum-mechanical predictions are correct. In these Bell test experiments, entangled pairs of particles are created; the particles are separated, traveling to remote measuring apparatus. The orientation of the measuring apparatus can be changed while the particles are in flight, demonstrating the apparent nonlocality of the effect. The de Broglie–Bohm theory makes the same (empirically correct) predictions for the Bell test experiments as ordinary quantum mechanics. It is able to do this because it is manifestly nonlocal. It is often criticized or rejected based on this; Bell's attitude was: "It is a merit of the de Broglie–Bohm version to bring this [nonlocality] out so explicitly that it cannot be ignored."<a href="#cite_note-68">[67]</a> The de Broglie–Bohm theory describes the physics in the Bell test experiments as follows: to understand the evolution of the particles, we need to set up a wave equation for both particles; the orientation of the apparatus affects the wavefunction. The particles in the experiment follow the guidance of the wavefunction. It is the wavefunction that carries the faster-than-light effect of changing the orientation of the apparatus. <a href="/wiki/Tim_Maudlin#Philosophical_work" title="Tim Maudlin">Maudlin</a> provides an analysis of exactly what kind of nonlocality is present and how it is compatible with relativity.<a href="#cite_note-69">[68]</a> Bell has shown that the nonlocality does not allow <a href="/wiki/Superluminal_communication" title="Superluminal communication">superluminal communication</a>. Maudlin has shown this in greater detail. <div class="mw-heading mw-heading3"><h3 id="Classical_limit">Classical limit</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=25" title="Edit section: Classical limit">edit</a>]</div> Bohm's formulation of de Broglie–Bohm theory in a classical-looking version has the merits that the emergence of classical behavior seems to follow immediately for any situation in which the quantum potential is negligible, as noted by Bohm in 1952. Modern methods of <a href="/wiki/Decoherence" class="mw-redirect" title="Decoherence">decoherence</a> are relevant to an analysis of this limit. See Allori et al.<a href="#cite_note-70">[69]</a> for steps towards a rigorous analysis. <div class="mw-heading mw-heading3"><h3 id="Quantum_trajectory_method">Quantum trajectory method</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=26" title="Edit section: Quantum trajectory method">edit</a>]</div> Work by <a href="/wiki/Robert_E._Wyatt" title="Robert E. Wyatt">Robert E. Wyatt</a> in the early 2000s attempted to use the Bohm "particles" as an adaptive mesh that follows the actual trajectory of a quantum state in time and space. In the "quantum trajectory" method, one samples the quantum wavefunction with a mesh of quadrature points. One then evolves the quadrature points in time according to the Bohm equations of motion. At each time step, one then re-synthesizes the wavefunction from the points, recomputes the quantum forces, and continues the calculation. (QuickTime movies of this for H + H2 reactive scattering can be found on the <a rel="nofollow" class="external text" href="http://research.cm.utexas.edu/rwyatt/movies/qtm/index.html">Wyatt group web-site</a> at UT Austin.) This approach has been adapted, extended, and used by a number of researchers in the chemical physics community as a way to compute semi-classical and quasi-classical molecular dynamics. A 2007 issue of <a href="/wiki/The_Journal_of_Physical_Chemistry_A" title="The Journal of Physical Chemistry A">The Journal of Physical Chemistry A</a> was dedicated to Prof. Wyatt and his work on "computational Bohmian dynamics".<a href="#cite_note-71">[70]</a> <a href="/wiki/Eric_R._Bittner" title="Eric R. Bittner">Eric R. Bittner</a>'s group<a href="#cite_note-h523-72">[71]</a> at the <a href="/wiki/University_of_Houston" title="University of Houston">University of Houston</a> has advanced a statistical variant of this approach that uses Bayesian sampling technique to sample the quantum density and compute the quantum potential on a structureless mesh of points. This technique was recently used to estimate quantum effects in the heat capacity of small clusters Nen for n ≈ 100. There remain difficulties using the Bohmian approach, mostly associated with the formation of singularities in the quantum potential due to nodes in the quantum wavefunction. In general, nodes forming due to interference effects lead to the case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{-1}\nabla ^{2}R\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{-1}\nabla ^{2}R\to \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b18b623fce7661225b2df367270953033936e887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.436ex; height:2.676ex;" alt="{\displaystyle R^{-1}\nabla ^{2}R\to \infty .}"> This results in an infinite force on the sample particles forcing them to move away from the node and often crossing the path of other sample points (which violates single-valuedness). Various schemes have been developed to overcome this; however, no general solution has yet emerged. These methods, as does Bohm's Hamilton–Jacobi formulation, do not apply to situations in which the full dynamics of spin need to be taken into account. The properties of trajectories in the de Broglie–Bohm theory differ significantly from the <a href="/wiki/Method_of_quantum_characteristics" title="Method of quantum characteristics">Moyal quantum trajectories</a> as well as the <a href="/wiki/Quantum_stochastic_calculus#Quantum_trajectories" title="Quantum stochastic calculus">quantum trajectories</a> from the unraveling of an open quantum system. <div class="mw-heading mw-heading2"><h2 id="Similarities_with_the_many-worlds_interpretation">Similarities with the many-worlds interpretation</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=27" title="Edit section: Similarities with the many-worlds interpretation">edit</a>]</div> Kim Joris Boström has proposed a non-relativistic quantum mechanical theory that combines elements of de Broglie-Bohm mechanics and <a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a>'s many-worlds. In particular, the unreal many-worlds interpretation of Hawking and Weinberg is similar to the Bohmian concept of unreal empty branch worlds: <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">The second issue with Bohmian mechanics may, at first sight, appear rather harmless, but which on a closer look develops considerable destructive power: the issue of empty branches. These are the components of the post-measurement state that do not guide any particles because they do not have the actual configuration q in their support. At first sight, the empty branches do not appear problematic but on the contrary very helpful as they enable the theory to explain unique outcomes of measurements. Also, they seem to explain why there is an effective "collapse of the wavefunction", as in ordinary quantum mechanics. On a closer view, though, one must admit that these empty branches do not actually disappear. As the wavefunction is taken to describe a really existing field, all their branches really exist and will evolve forever by the Schrödinger dynamics, no matter how many of them will become empty in the course of the evolution. Every branch of the global wavefunction potentially describes a complete world which is, according to Bohm's ontology, only a possible world that would be the actual world if only it were filled with particles, and which is in every respect identical to a corresponding world in Everett's theory. Only one branch at a time is occupied by particles, thereby representing the actual world, while all other branches, though really existing as part of a really existing wavefunction, are empty and thus contain some sort of "zombie worlds" with planets, oceans, trees, cities, cars and people who talk like us and behave like us, but who do not actually exist. Now, if the Everettian theory may be accused of ontological extravagance, then Bohmian mechanics could be accused of ontological wastefulness. On top of the ontology of empty branches comes the additional ontology of particle positions that are, on account of the quantum equilibrium hypothesis, forever unknown to the observer. Yet, the actual configuration is never needed for the calculation of the statistical predictions in experimental reality, for these can be obtained by mere wavefunction algebra. From this perspective, Bohmian mechanics may appear as a wasteful and redundant theory. I think it is considerations like these that are the biggest obstacle in the way of a general acceptance of Bohmian mechanics.<a href="#cite_note-73">[72]</a></blockquote> Many authors have expressed critical views of de Broglie–Bohm theory by comparing it to Everett's many-worlds approach. Many (but not all) proponents of de Broglie–Bohm theory (such as Bohm and Bell) interpret the universal wavefunction as physically real. According to some supporters of Everett's theory, if the (never collapsing) wavefunction is taken to be physically real, then it is natural to interpret the theory as having the same many worlds as Everett's theory. In the Everettian view the role of the Bohmian particle is to act as a "pointer", tagging, or selecting, just one branch of the <a href="/wiki/Universal_wavefunction" title="Universal wavefunction">universal wavefunction</a> (the assumption that this branch indicates which wave packet determines the observed result of a given experiment is called the "result assumption"<a href="#cite_note-BrownWallace-74">[73]</a>); the other branches are designated "empty" and implicitly assumed by Bohm to be devoid of conscious observers.<a href="#cite_note-BrownWallace-74">[73]</a> <a href="/wiki/H._Dieter_Zeh" title="H. Dieter Zeh">H. Dieter Zeh</a> comments on these "empty" branches:<a href="#cite_note-75">[74]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">It is usually overlooked that Bohm's theory contains the same "many worlds" of dynamically separate branches as the Everett interpretation (now regarded as "empty" wave components), since it is based on precisely the same ... <a href="/wiki/Universal_wavefunction" title="Universal wavefunction">global wave function</a> ...</blockquote> <a href="/wiki/David_Deutsch" title="David Deutsch">David Deutsch</a> has expressed the same point more "acerbically":<a href="#cite_note-BrownWallace-74">[73]</a><a href="#cite_note-76">[75]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">Pilot-wave theories are parallel-universe theories in a state of chronic denial.</blockquote> This conclusion has been challenged by Detlef Dürr and Justin Lazarovici: <blockquote>The Bohmian, of course, cannot accept this argument. For her, it is decidedly the particle configuration in three-dimensional space and not the wave function on the abstract configuration space that constitutes a world (or rather, the world). Instead, she will accuse the Everettian of not having local beables (in Bell's sense) in her theory, that is, the ontological variables that refer to localized entities in three-dimensional space or four-dimensional spacetime. The many worlds of her theory thus merely appear as a grotesque consequence of this omission.<a href="#cite_note-77">[76]</a></blockquote> <div class="mw-heading mw-heading2"><h2 id="Occam's-razor_criticism">Occam's-razor criticism</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=28" title="Edit section: Occam's-razor criticism">edit</a>]</div> Both <a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Hugh Everett III</a> and Bohm treated the wavefunction as a <a href="/wiki/Scientific_realism" title="Scientific realism">physically real</a> <a href="/wiki/Field_(physics)" title="Field (physics)">field</a>. Everett's <a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">many-worlds interpretation</a> is an attempt to demonstrate that the wavefunction alone is sufficient to account for all our observations. When we see the particle detectors flash or hear the click of a <a href="/wiki/Geiger_counter" title="Geiger counter">Geiger counter</a>, Everett's theory interprets this as our wavefunction responding to changes in the detector's wavefunction, which is responding in turn to the passage of another wavefunction (which we think of as a "particle", but is actually just another <a href="/wiki/Wave_packet" title="Wave packet">wave packet</a>).<a href="#cite_note-BrownWallace-74">[73]</a> No particle (in the Bohm sense of having a defined position and velocity) exists according to that theory. For this reason Everett sometimes referred to his own <a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">many-worlds approach</a> as the "pure wave theory". Of Bohm's 1952 approach, Everett said:<a href="#cite_note-78">[77]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">Our main criticism of this view is on the grounds of simplicity – if one desires to hold the view that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> is a real field, then the associated particle is superfluous, since, as we have endeavored to illustrate, the pure wave theory is itself satisfactory.</blockquote> In the Everettian view, then, the Bohm particles are superfluous entities, similar to, and equally as unnecessary as, for example, the <a href="/wiki/Luminiferous_ether" class="mw-redirect" title="Luminiferous ether">luminiferous ether</a>, which was found to be unnecessary in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. This argument is sometimes called the "redundancy argument", since the superfluous particles are redundant in the sense of <a href="/wiki/Occam%27s_razor" title="Occam's razor">Occam's razor</a>.<a href="#cite_note-79">[78]</a> According to <a href="/wiki/Harvey_Brown_(philosopher)" class="mw-redirect" title="Harvey Brown (philosopher)">Brown</a> & Wallace,<a href="#cite_note-BrownWallace-74">[73]</a> the de Broglie–Bohm particles play no role in the solution of the measurement problem. For these authors,<a href="#cite_note-BrownWallace-74">[73]</a> the "result assumption" (see above) is inconsistent with the view that there is no measurement problem in the predictable outcome (i.e. single-outcome) case. They also say<a href="#cite_note-BrownWallace-74">[73]</a> that a standard <a href="/wiki/Tacit_assumption" title="Tacit assumption">tacit assumption</a> of de Broglie–Bohm theory (that an observer becomes aware of configurations of particles of ordinary objects by means of correlations between such configurations and the configuration of the particles in the observer's brain) is unreasonable. This conclusion has been challenged by <a href="/wiki/Antony_Valentini" title="Antony Valentini">Valentini</a>,<a href="#cite_note-80">[79]</a> who argues that the entirety of such objections arises from a failure to interpret de Broglie–Bohm theory on its own terms. According to <a href="/wiki/Peter_R._Holland" title="Peter R. Holland">Peter R. Holland</a>, in a wider Hamiltonian framework, theories can be formulated in which particles do act back on the wave function.<a href="#cite_note-81">[80]</a> <div class="mw-heading mw-heading2"><h2 id="Derivations">Derivations</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=29" title="Edit section: Derivations">edit</a>]</div> De Broglie–Bohm theory has been derived many times and in many ways. Below are six derivations, all of which are very different and lead to different ways of understanding and extending this theory. <ul><li><a href="/wiki/Schr%C3%B6dinger_equation#Derivation" title="Schrödinger equation">Schrödinger's equation</a> can be derived by using <a href="/wiki/Photoelectric_effect" title="Photoelectric effect">Einstein's light quanta hypothesis</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\hbar \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\hbar \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb16565f02349106457258633097e0d0414a8e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.626ex; height:2.176ex;" alt="{\displaystyle E=\hbar \omega }"> and <a href="/wiki/Matter_wave" title="Matter wave">de Broglie's hypothesis</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =\hbar \mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\hbar \mathbf {k} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e73faf974069648fc56ca99c29c35950e6819c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.301ex; height:2.509ex;" alt="{\displaystyle \mathbf {p} =\hbar \mathbf {k} }">.</li></ul> <dl><dd>The guiding equation can be derived in a similar fashion. We assume a plane wave: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {x} ,t)=Ae^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {x} ,t)=Ae^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/304e6917b9cfd5d622f892a36e126d3aee93cb5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.958ex; height:3.343ex;" alt="{\displaystyle \psi (\mathbf {x} ,t)=Ae^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}}">. Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\mathbf {k} =\nabla \psi /\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\mathbf {k} =\nabla \psi /\psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69096b3f1f12e1abf374f5c0c5ce07a3da37e070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.437ex; height:2.843ex;" alt="{\displaystyle i\mathbf {k} =\nabla \psi /\psi }">. Assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =m\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m\mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a271a96e7b925fd39686375167c76d406e87c813" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.035ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} =m\mathbf {v} }"> for the particle's actual velocity, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} ={\frac {\hbar }{m}}\operatorname {Im} \left({\frac {\nabla \psi }{\psi }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mi>m</mi> </mfrac> </mrow> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mi>ψ</mi> </mrow> <mi>ψ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\frac {\hbar }{m}}\operatorname {Im} \left({\frac {\nabla \psi }{\psi }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/560b1d2cdebfbfe22ac3103a145650b95f3cf9c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.255ex; height:6.176ex;" alt="{\displaystyle \mathbf {v} ={\frac {\hbar }{m}}\operatorname {Im} \left({\frac {\nabla \psi }{\psi }}\right)}">. Thus, we have the guiding equation.</dd> <dd>Notice that this derivation does not use Schrödinger's equation.</dd></dl> <ul><li>Preserving the density under the time evolution is another method of derivation. This is the method that Bell cites. It is this method that generalizes to many possible alternative theories. The starting point is the <a href="/wiki/Continuity_equation" title="Continuity equation">continuity equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\partial \rho }{\partial t}}=\nabla \cdot (\rho v^{\psi })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\partial \rho }{\partial t}}=\nabla \cdot (\rho v^{\psi })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/410e4ca9324f22fcd035b44f3e623b31a9283ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.319ex; height:5.676ex;" alt="{\displaystyle -{\frac {\partial \rho }{\partial t}}=\nabla \cdot (\rho v^{\psi })}"> [<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>] for the density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =|\psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =|\psi |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3102dd83fdcdaa7cb4ff7c44b81562624d1efc54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.161ex; height:3.343ex;" alt="{\displaystyle \rho =|\psi |^{2}}">. This equation describes a probability flow along a current. We take the velocity field associated with this current as the velocity field whose integral curves yield the motion of the particle.</li> <li>A method applicable for particles without spin is to do a polar decomposition of the wavefunction and transform Schrödinger's equation into two coupled equations: the <a href="/wiki/Continuity_equation" title="Continuity equation">continuity equation</a> from above and the <a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a>. This is the method used by Bohm in 1952. The decomposition and equations are as follows:</li></ul> <dl><dd>Decomposition: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {x} ,t)=R(\mathbf {x} ,t)e^{iS(\mathbf {x} ,t)/\hbar }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {x} ,t)=R(\mathbf {x} ,t)e^{iS(\mathbf {x} ,t)/\hbar }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7580f43f2d7ef549ed9aceeaff93f5af1bc81d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.227ex; height:3.343ex;" alt="{\displaystyle \psi (\mathbf {x} ,t)=R(\mathbf {x} ,t)e^{iS(\mathbf {x} ,t)/\hbar }.}"> Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}(\mathbf {x} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}(\mathbf {x} ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fa16b3e8a4634fab23612c1ad96a5b67abd192c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:3.176ex;" alt="{\displaystyle R^{2}(\mathbf {x} ,t)}"> corresponds to the probability density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\mathbf {x} ,t)=|\psi (\mathbf {x} ,t)|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {x} ,t)=|\psi (\mathbf {x} ,t)|^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08a1b7a54cbf719de50bcc3f7ae8b98e9ee2a47f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.349ex; height:3.343ex;" alt="{\displaystyle \rho (\mathbf {x} ,t)=|\psi (\mathbf {x} ,t)|^{2}}">.</dd></dl> <dl><dd>Continuity equation: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\partial \rho (\mathbf {x} ,t)}{\partial t}}=\nabla \cdot \left(\rho (\mathbf {x} ,t){\frac {\nabla S(\mathbf {x} ,t)}{m}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\partial \rho (\mathbf {x} ,t)}{\partial t}}=\nabla \cdot \left(\rho (\mathbf {x} ,t){\frac {\nabla S(\mathbf {x} ,t)}{m}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8efd1c8ffdf4e6ed4a24947e4f72593f50bb1acc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.054ex; height:6.343ex;" alt="{\displaystyle -{\frac {\partial \rho (\mathbf {x} ,t)}{\partial t}}=\nabla \cdot \left(\rho (\mathbf {x} ,t){\frac {\nabla S(\mathbf {x} ,t)}{m}}\right)}">.</dd></dl> <dl><dd>Hamilton–Jacobi equation: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial S(\mathbf {x} ,t)}{\partial t}}=-\left[{\frac {1}{2m}}(\nabla S(\mathbf {x} ,t))^{2}+V-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R(\mathbf {x} ,t)}{R(\mathbf {x} ,t)}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>V</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial S(\mathbf {x} ,t)}{\partial t}}=-\left[{\frac {1}{2m}}(\nabla S(\mathbf {x} ,t))^{2}+V-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R(\mathbf {x} ,t)}{R(\mathbf {x} ,t)}}\right].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d074f127c5c63c03adb9c44a9937cb2aaa5b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.408ex; height:7.509ex;" alt="{\displaystyle {\frac {\partial S(\mathbf {x} ,t)}{\partial t}}=-\left[{\frac {1}{2m}}(\nabla S(\mathbf {x} ,t))^{2}+V-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R(\mathbf {x} ,t)}{R(\mathbf {x} ,t)}}\right].}"></dd></dl> <dl><dd>The Hamilton–Jacobi equation is the equation derived from a Newtonian system with potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R}{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> </mrow> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R}{R}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f39d1e81d58033ee4855c2f3c75a26608b4eff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.257ex; height:5.843ex;" alt="{\displaystyle V-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R}{R}}}"> and velocity field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\nabla S}{m}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mi>S</mi> </mrow> <mi>m</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\nabla S}{m}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe965a756faa542d8b733156ffe749da1e9cc414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.918ex; height:5.343ex;" alt="{\displaystyle {\frac {\nabla S}{m}}.}"> The potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> is the classical potential that appears in Schrödinger's equation, and the other term involving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> is the <a href="/wiki/Quantum_potential" title="Quantum potential">quantum potential</a>, terminology introduced by Bohm.</dd></dl> <dl><dd>This leads to viewing the quantum theory as particles moving under the classical force modified by a quantum force. However, unlike standard <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>, the initial velocity field is already specified by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\nabla S}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mi>S</mi> </mrow> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\nabla S}{m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d785e4b5cb66a5b5ea2b3796c59cb5613dcc292f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.271ex; height:5.343ex;" alt="{\displaystyle {\frac {\nabla S}{m}}}">, which is a symptom of this being a first-order theory, not a second-order theory.</dd></dl> <ul><li>A fourth derivation was given by Dürr et al.<a href="#cite_note-dgz92-15">[14]</a> In their derivation, they derive the velocity field by demanding the appropriate transformation properties given by the various symmetries that Schrödinger's equation satisfies, once the wavefunction is suitably transformed. The guiding equation is what emerges from that analysis.</li> <li>A fifth derivation, given by Dürr et al.<a href="#cite_note-dgtz04-41">[40]</a> is appropriate for generalization to quantum field theory and the Dirac equation. The idea is that a velocity field can also be understood as a first-order differential operator acting on functions. Thus, if we know how it acts on functions, we know what it is. Then given the Hamiltonian operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}">, the equation to satisfy for all functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> (with associated multiplication operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}">) is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v(f))(q)=\operatorname {Re} {\frac {\left(\psi ,{\frac {i}{\hbar }}[H,{\hat {f}}]\psi \right)}{(\psi ,\psi )}}(q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>(</mo> <mrow> <mi>ψ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>H</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">]</mo> <mi>ψ</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v(f))(q)=\operatorname {Re} {\frac {\left(\psi ,{\frac {i}{\hbar }}[H,{\hat {f}}]\psi \right)}{(\psi ,\psi )}}(q)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a152b01c38974ca0aae94ace66f0c754bc0197a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.533ex; height:8.343ex;" alt="{\displaystyle (v(f))(q)=\operatorname {Re} {\frac {\left(\psi ,{\frac {i}{\hbar }}[H,{\hat {f}}]\psi \right)}{(\psi ,\psi )}}(q)}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v,w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v,w)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1ba70e3b2d314bc63be62afe1e5c3a80b95834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.635ex; height:2.843ex;" alt="{\displaystyle (v,w)}"> is the local Hermitian inner product on the value space of the wavefunction.</li></ul> <dl><dd>This formulation allows for stochastic theories such as the creation and annihilation of particles.</dd></dl> <ul><li>A further derivation has been given by Peter R. Holland, on which he bases his quantum-physics textbook The Quantum Theory of Motion.<a href="#cite_note-82">[81]</a> It is based on three basic postulates and an additional fourth postulate that links the wavefunction to measurement probabilities: <ol><li>A physical system consists in a spatiotemporally propagating wave and a point particle guided by it.</li> <li>The wave is described mathematically by a solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> to Schrödinger's wave equation.</li> <li>The particle motion is described by a solution to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\dot {x}} (t)=[\nabla S(\mathbf {x} (t),t))]/m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">x</mi> <mo mathvariant="bold">˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi mathvariant="normal">∇</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\dot {x}} (t)=[\nabla S(\mathbf {x} (t),t))]/m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11edf93808e16578b12ee0236405b6b464153e71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.737ex; height:2.843ex;" alt="{\displaystyle \mathbf {\dot {x}} (t)=[\nabla S(\mathbf {x} (t),t))]/m}"> in dependence on initial condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (t=0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t=0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5083bba6954eed4b98c03e7a78280f2779fd38ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.321ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t=0)}">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> the phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }">.<div class="paragraphbreak" style="margin-top:0.5em"></div>The fourth postulate is subsidiary yet consistent with the first three:</li> <li>The probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\mathbf {x} (t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {x} (t))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c47a5db22900a501f2438f33ca1e69fb2e1e1546" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.071ex; height:2.843ex;" alt="{\displaystyle \rho (\mathbf {x} (t))}"> to find the particle in the differential volume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{3}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{3}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9c17db5d994411d2f06ba6818f2fc930e7c4ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.602ex; height:2.676ex;" alt="{\displaystyle d^{3}x}"> at time t equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi (\mathbf {x} (t))|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi (\mathbf {x} (t))|^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6c58cb0923ba34344041e07693170ce2e1706f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.73ex; height:3.343ex;" alt="{\displaystyle |\psi (\mathbf {x} (t))|^{2}}">.</li></ol></li></ul> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=30" title="Edit section: History">edit</a>]</div> The theory was historically developed in the 1920s by de Broglie, who, in 1927, was persuaded to abandon it in favour of the then-mainstream Copenhagen interpretation. David Bohm, dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's pilot-wave theory in 1952. Bohm's suggestions were not then widely received, partly due to reasons unrelated to their content, such as Bohm's youthful <a href="/wiki/Communist" class="mw-redirect" title="Communist">communist</a> affiliations.<a href="#cite_note-83">[82]</a> The de Broglie–Bohm theory was widely deemed unacceptable by mainstream theorists, mostly because of its explicit non-locality. On the theory, <a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">John Stewart Bell</a>, author of the 1964 <a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's theorem</a> wrote in 1982: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">Bohm showed explicitly how parameters could indeed be introduced, into nonrelativistic wave mechanics, with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary reference to the "observer", could be eliminated. ...<div class="paragraphbreak" style="margin-top:0.5em"></div>But why then had Born not told me of this "pilot wave"? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing "impossibility" proofs, after 1952, and as recently as 1978?... Why is the pilot wave picture ignored in text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show us that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?<a href="#cite_note-84">[83]</a></blockquote> Since the 1990s, there has been renewed interest in formulating extensions to de Broglie–Bohm theory, attempting to reconcile it with <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> and <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, besides other features such as <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> or curved spatial geometries.<a href="#cite_note-85">[84]</a> De Broglie–Bohm theory has a history of different formulations and names. In this section, each stage is given a name and a main reference. <div class="mw-heading mw-heading3"><h3 id="Pilot-wave_theory">Pilot-wave theory</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=31" title="Edit section: Pilot-wave theory">edit</a>]</div> <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a> presented his <a href="/wiki/Pilot_wave_theory" title="Pilot wave theory">pilot wave theory</a> at the 1927 Solvay Conference,<a href="#cite_note-86">[85]</a> after close collaboration with Schrödinger,[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] who developed his wave equation for de Broglie's theory.[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>] At the end of the presentation, <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a> pointed out that it was not compatible with a semi-classical technique Fermi had previously adopted in the case of inelastic scattering. Contrary to a popular legend, de Broglie actually gave the correct rebuttal that the particular technique could not be generalized for Pauli's purpose, although the audience might have been lost in the technical details and de Broglie's mild manner left the impression that Pauli's objection was valid. He was eventually persuaded to abandon this theory nonetheless because he was "discouraged by criticisms which [it] roused".<a href="#cite_note-87">[86]</a> De Broglie's theory already applies to multiple spin-less particles, but lacks an adequate theory of measurement as no one understood <a href="/wiki/Quantum_decoherence" title="Quantum decoherence">quantum decoherence</a> at the time. An analysis of de Broglie's presentation is given in Bacciagaluppi et al.[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>]<a href="#cite_note-88">[87]</a><a href="#cite_note-89">[88]</a> Also, in 1932 <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a> published a paper,<a href="#cite_note-90">[89]</a> that was widely (and erroneously, as shown by <a href="/wiki/Jeffrey_Bub" title="Jeffrey Bub">Jeffrey Bub</a><a href="#cite_note-91">[90]</a>) believed to prove that all hidden-variable theories are impossible. This sealed the fate of de Broglie's theory for the next two decades. In 1926, <a href="/wiki/Erwin_Madelung" title="Erwin Madelung">Erwin Madelung</a> had developed a hydrodynamic version of <a href="/wiki/Schr%C3%B6dinger%27s_equation" class="mw-redirect" title="Schrödinger's equation">Schrödinger's equation</a>, which is incorrectly[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] considered as a basis for the density current derivation of the de Broglie–Bohm theory.<a href="#cite_note-92">[91]</a> The <a href="/wiki/Madelung_equations" title="Madelung equations">Madelung equations</a>, being quantum <a href="/wiki/Euler_equations_(fluid_dynamics)" title="Euler equations (fluid dynamics)">Euler equations (fluid dynamics)</a>, differ philosophically from the de Broglie–Bohm mechanics<a href="#cite_note-93">[92]</a> and are the basis of the <a href="/wiki/Stochastic_interpretation" class="mw-redirect" title="Stochastic interpretation">stochastic interpretation</a> of quantum mechanics. <a href="/wiki/Peter_R._Holland" title="Peter R. Holland">Peter R. Holland</a> has pointed out that, earlier in 1927, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a> had actually submitted a preprint with a similar proposal but, not convinced, had withdrawn it before publication.<a href="#cite_note-94">[93]</a> According to Holland, failure to appreciate key points of the de Broglie–Bohm theory has led to confusion, the key point being "that the trajectories of a many-body quantum system are correlated not because the particles exert a direct force on one another (à la Coulomb) but because all are acted upon by an entity – mathematically described by the wavefunction or functions of it – that lies beyond them".<a href="#cite_note-95">[94]</a> This entity is the <a href="/wiki/Quantum_potential" title="Quantum potential">quantum potential</a>. After publishing a popular textbook on Quantum Mechanics that adhered entirely to the Copenhagen orthodoxy, Bohm was persuaded by Einstein to take a critical look at von Neumann's theorem. The result was 'A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I and II' [Bohm 1952]. It was an independent origination of the pilot wave theory, and extended it to incorporate a consistent theory of measurement, and to address a criticism of Pauli that de Broglie did not properly respond to; it is taken to be deterministic (though Bohm hinted in the original papers that there should be disturbances to this, in the way <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> disturbs Newtonian mechanics). This stage is known as the de Broglie–Bohm Theory in Bell's work [Bell 1987] and is the basis for 'The Quantum Theory of Motion' [Holland 1993]. This stage applies to multiple particles, and is deterministic. The de Broglie–Bohm theory is an example of a <a href="/wiki/Hidden-variables_theory" class="mw-redirect" title="Hidden-variables theory">hidden-variables theory</a>. Bohm originally hoped that hidden variables could provide a <a href="/wiki/Principle_of_locality" title="Principle of locality">local</a>, <a href="/wiki/Causal" class="mw-redirect" title="Causal">causal</a>, <a href="/wiki/Objectivity_(philosophy)" class="mw-redirect" title="Objectivity (philosophy)">objective</a> description that would resolve or eliminate many of the paradoxes of quantum mechanics, such as <a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a>, the <a href="/wiki/Measurement_problem" title="Measurement problem">measurement problem</a> and the collapse of the wavefunction. However, <a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's theorem</a> complicates this hope, as it demonstrates that there can be no local hidden-variable theory that is compatible with the predictions of quantum mechanics. The Bohmian interpretation is <a href="/wiki/Causal" class="mw-redirect" title="Causal">causal</a> but not <a href="/wiki/Principle_of_locality" title="Principle of locality">local</a>. Bohm's paper was largely ignored or panned by other physicists. <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, who had suggested that Bohm search for a realist alternative to the prevailing <a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen approach</a>, did not consider Bohm's interpretation to be a satisfactory answer to the quantum nonlocality question, calling it "too cheap",<a href="#cite_note-96">[95]</a> while <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> considered it a "superfluous 'ideological superstructure' ".<a href="#cite_note-97">[96]</a> <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a>, who had been unconvinced by de Broglie in 1927, conceded to Bohm as follows: <blockquote>I just received your long letter of 20th November, and I also have studied more thoroughly the details of your paper. I do not see any longer the possibility of any logical contradiction as long as your results agree completely with those of the usual wave mechanics and as long as no means is given to measure the values of your hidden parameters both in the measuring apparatus and in the observe [sic] system. As far as the whole matter stands now, your 'extra wave-mechanical predictions' are still a check, which cannot be cashed.<a href="#cite_note-98">[97]</a></blockquote> He subsequently described Bohm's theory as "artificial metaphysics".<a href="#cite_note-99">[98]</a> According to physicist <a href="/wiki/Max_Dresden" title="Max Dresden">Max Dresden</a>, when Bohm's theory was presented at the <a href="/wiki/Institute_for_Advanced_Study" title="Institute for Advanced Study">Institute for Advanced Study</a> in Princeton, many of the objections were <a href="/wiki/Ad_hominem" title="Ad hominem">ad hominem</a>, focusing on Bohm's sympathy with communists as exemplified by his refusal to give testimony to the <a href="/wiki/House_Un-American_Activities_Committee" title="House Un-American Activities Committee">House Un-American Activities Committee</a>.<a href="#cite_note-100">[99]</a> In 1979, Chris Philippidis, Chris Dewdney and <a href="/wiki/Basil_Hiley" title="Basil Hiley">Basil Hiley</a> were the first to perform numeric computations on the basis of the quantum potential to deduce ensembles of particle trajectories.<a href="#cite_note-101">[100]</a><a href="#cite_note-102">[101]</a> Their work renewed the interests of physicists in the Bohm interpretation of quantum physics.<a href="#cite_note-103">[102]</a> Eventually <a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">John Bell</a> began to defend the theory. In "Speakable and Unspeakable in Quantum Mechanics" [Bell 1987], several of the papers refer to hidden-variables theories (which include Bohm's). The trajectories of the Bohm model that would result for particular experimental arrangements were termed "surreal" by some.<a href="#cite_note-ESSW_1992-104">[103]</a><a href="#cite_note-105">[104]</a> Still in 2016, mathematical physicist <a href="/wiki/Sheldon_Goldstein" title="Sheldon Goldstein">Sheldon Goldstein</a> said of Bohm's theory: "There was a time when you couldn't even talk about it because it was heretical. It probably still is the kiss of death for a physics career to be actually working on Bohm, but maybe that's changing."<a href="#cite_note-newscientist.com-61">[60]</a> <div class="mw-heading mw-heading3"><h3 id="Bohmian_mechanics">Bohmian mechanics</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=32" title="Edit section: Bohmian mechanics">edit</a>]</div> Bohmian mechanics is the same theory, but with an emphasis on the notion of current flow, which is determined on the basis of the <a href="/wiki/Quantum_equilibrium_hypothesis" class="mw-redirect" title="Quantum equilibrium hypothesis">quantum equilibrium hypothesis</a> that the probability follows the Born rule.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] The term "Bohmian mechanics" is also often used to include most of the further extensions past the spin-less version of Bohm.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] While de Broglie–Bohm theory has <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangians</a> and <a href="/wiki/Hamilton-Jacobi_equations" class="mw-redirect" title="Hamilton-Jacobi equations">Hamilton-Jacobi equations</a> as a primary focus and backdrop, with the icon of the <a href="/wiki/Quantum_potential" title="Quantum potential">quantum potential</a>, Bohmian mechanics considers the <a href="/wiki/Continuity_equation" title="Continuity equation">continuity equation</a> as primary and has the guiding equation as its icon. They are mathematically equivalent in so far as the Hamilton-Jacobi formulation applies, i.e., spin-less particles. All of non-relativistic quantum mechanics can be fully accounted for in this theory. Recent studies have used this formalism to compute the evolution of many-body quantum systems, with a considerable increase in speed as compared to other quantum-based methods.<a href="#cite_note-106">[105]</a> <div class="mw-heading mw-heading3"><h3 id="Causal_interpretation_and_ontological_interpretation">Causal interpretation and ontological interpretation</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=33" title="Edit section: Causal interpretation and ontological interpretation">edit</a>]</div> Bohm developed his original ideas, calling them the Causal Interpretation. Later he felt that causal sounded too much like deterministic and preferred to call his theory the Ontological Interpretation. The main reference is "The Undivided Universe" (Bohm, Hiley 1993). This stage covers work by Bohm and in collaboration with <a href="/wiki/Jean-Pierre_Vigier" title="Jean-Pierre Vigier">Jean-Pierre Vigier</a> and Basil Hiley. Bohm is clear that this theory is non-deterministic (the work with Hiley includes a stochastic theory). As such, this theory is not strictly speaking a formulation of de Broglie–Bohm theory, but it deserves mention here because the term "Bohm Interpretation" is ambiguous between this theory and de Broglie–Bohm theory. In 1996 <a href="/wiki/Philosopher_of_science" class="mw-redirect" title="Philosopher of science">philosopher of science</a> <a href="/wiki/Arthur_Fine" title="Arthur Fine">Arthur Fine</a> gave an in-depth analysis of possible interpretations of Bohm's model of 1952.<a href="#cite_note-107">[106]</a> William Simpson has suggested a <a href="/wiki/Hylomorphism" title="Hylomorphism">hylomorphic</a> interpretation of Bohmian mechanics, in which the cosmos is an Aristotelian substance composed of material particles and a substantial form. The wave function is assigned a dispositional role in choreographing the trajectories of the particles.<a href="#cite_note-108">[107]</a> <div class="mw-heading mw-heading3"><h3 id="Hydrodynamic_quantum_analogs">Hydrodynamic quantum analogs</h3>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=34" title="Edit section: Hydrodynamic quantum analogs">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hydrodynamic_quantum_analogs" title="Hydrodynamic quantum analogs">Hydrodynamic quantum analogs</a></div> Experiments on hydrodynamical analogs of quantum mechanics beginning with the work of Couder and Fort (2006)<a href="#cite_note-109">[108]</a><a href="#cite_note-110">[109]</a> have purported to show that macroscopic classical pilot-waves can exhibit characteristics previously thought to be restricted to the quantum realm. Hydrodynamic pilot-wave analogs have been claimed to duplicate the double slit experiment, tunneling, quantized orbits, and numerous other quantum phenomena which have led to a resurgence in interest in pilot wave theories.<a href="#cite_note-111">[110]</a><a href="#cite_note-112">[111]</a><a href="#cite_note-113">[112]</a> The analogs have been compared to the <a href="/wiki/Faraday_wave" title="Faraday wave">Faraday wave</a>.<a href="#cite_note-114">[113]</a> These results have been disputed: experiments fail to reproduce aspects of the double-slit experiments.<a href="#cite_note-115">[114]</a><a href="#cite_note-116">[115]</a> High precision measurements in the tunneling case point to a different origin of the unpredictable crossing: rather than initial position uncertainty or environmental noise, interactions at the barrier seem to be involved.<a href="#cite_note-117">[116]</a> Another classical analog has been reported in surface gravity waves.<a href="#cite_note-118">[117]</a> <table class="wikitable" style="text-align: center;"> <caption>A comparison by Bush (2015)<a href="#cite_note-119">[118]</a> among the walking droplet system, de Broglie's double-solution pilot-wave theory<a href="#cite_note-120">[119]</a><a href="#cite_note-121">[120]</a> and its extension to SED<a href="#cite_note-122">[121]</a><a href="#cite_note-123">[122]</a> </caption> <tbody><tr> <th> </th> <th>Hydrodynamic walkers </th> <th>de Broglie </th> <th>SED pilot wave </th></tr> <tr> <th>Driving </th> <td>bath vibration</td> <td>internal clock</td> <td>vacuum fluctuations </td></tr> <tr> <th>Spectrum </th> <td>monochromatic</td> <td>monochromatic</td> <td>broad </td></tr> <tr> <th>Trigger </th> <td>bouncing</td> <td><a href="/wiki/Zitterbewegung" title="Zitterbewegung">zitterbewegung</a></td> <td>zitterbewegung </td></tr> <tr> <th>Trigger frequency </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{F}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8413a2d5ea20f5c563fffbb0d2d639ace881842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.909ex; height:2.009ex;" alt="{\displaystyle \omega _{F}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{c}=mc^{2}/\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{c}=mc^{2}/\hbar }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9036180576a16090f4d11e07f56f37a40a9a867f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.059ex; height:3.176ex;" alt="{\displaystyle \omega _{c}=mc^{2}/\hbar }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{c}=mc^{2}/\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{c}=mc^{2}/\hbar }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9036180576a16090f4d11e07f56f37a40a9a867f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.059ex; height:3.176ex;" alt="{\displaystyle \omega _{c}=mc^{2}/\hbar }"> </td></tr> <tr> <th>Energetics </th> <td>GPE <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"> wave</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mc^{2}\leftrightarrow \hbar \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">↔</mo> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mc^{2}\leftrightarrow \hbar \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27bdb3c872b01975bac5cd5ce06b85b7651d59e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.468ex; height:2.676ex;" alt="{\displaystyle mc^{2}\leftrightarrow \hbar \omega }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mc^{2}\leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mc^{2}\leftrightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52c414b30fe35407ea0d7ede80ca3f52d3b1012a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.07ex; height:2.676ex;" alt="{\displaystyle mc^{2}\leftrightarrow }"> EM </td></tr> <tr> <th>Resonance </th> <td>droplet-wave</td> <td>harmony of phases</td> <td>unspecified </td></tr> <tr> <th>Dispersion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb1537a1f4d4185fe800d4682abd78bdab60402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.466ex; height:2.843ex;" alt="{\displaystyle \omega (k)}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{F}^{2}\approx \sigma k^{3}/\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>≈</mo> <mi>σ</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{F}^{2}\approx \sigma k^{3}/\rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5299dfb19bf69450dddee0fbc77a68fb75334e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.967ex; height:3.343ex;" alt="{\displaystyle \omega _{F}^{2}\approx \sigma k^{3}/\rho }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{2}\approx \omega _{c}^{2}+c^{2}k^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≈</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}\approx \omega _{c}^{2}+c^{2}k^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed42604d9d7ea33715594ae5d7f1eae248d85bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.266ex; height:3.009ex;" alt="{\displaystyle \omega ^{2}\approx \omega _{c}^{2}+c^{2}k^{2}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =ck}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>c</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =ck}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a121709f0c11324e6983899f0606e33ce4e7072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.762ex; height:2.176ex;" alt="{\displaystyle \omega =ck}"> </td></tr> <tr> <th>Carrier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{F}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7f7401356632e7b37f1622bdde1d5a33801f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle \lambda _{F}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{dB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{dB}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cadc88dd1b0e862ae47ccee950a8e72159061db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.695ex; height:2.509ex;" alt="{\displaystyle \lambda _{dB}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89ec57820b5836734173272deaf187341a85917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.299ex; height:2.509ex;" alt="{\displaystyle \lambda _{c}}"> </td></tr> <tr> <th>Statistical <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{F}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7f7401356632e7b37f1622bdde1d5a33801f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle \lambda _{F}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{dB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{dB}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cadc88dd1b0e862ae47ccee950a8e72159061db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.695ex; height:2.509ex;" alt="{\displaystyle \lambda _{dB}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{dB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{dB}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cadc88dd1b0e862ae47ccee950a8e72159061db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.695ex; height:2.509ex;" alt="{\displaystyle \lambda _{dB}}"> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Surrealistic_trajectories">Surrealistic trajectories</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=35" title="Edit section: Surrealistic trajectories">edit</a>]</div> In 1992, Englert, Scully, Sussman, and Walther proposed experiments that would show particles taking paths that differ from the Bohm trajectories. <a href="#cite_note-ESSW_1992-104">[103]</a> They described the Bohm trajectories as "surrealistic"; their proposal was later referred to as ESSW after the last names of the authors.<a href="#cite_note-MahlerRozema-124">[123]</a> In 2016, Mahler et al. verified the ESSW predictions. However they propose the surealistic effect is a consequence the nonlocality inherent in Bohm's theory.<a href="#cite_note-MahlerRozema-124">[123]</a><a href="#cite_note-125">[124]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=36" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Madelung_equations" title="Madelung equations">Madelung equations</a></li> <li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local hidden-variable theory</a></li> <li><a href="/wiki/Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a></li> <li><a href="/wiki/Fluid_analogs_in_quantum_mechanics" class="mw-redirect" title="Fluid analogs in quantum mechanics">Fluid analogs in quantum mechanics</a></li> <li><a href="/wiki/Probability_current" title="Probability current">Probability current</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=37" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> Also known as the <a href="/wiki/Pilot_wave_theory" title="Pilot wave theory">pilot wave theory</a>, Bohmian mechanics, Bohm's interpretation, and the causal interpretation. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=38" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBohm1952" class="citation journal cs1">Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of 'Hidden Variables' I". Physical Review. 85 (2): 166–179. <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/1952PhRv...85..166B">1952PhRv...85..166B</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.85.166">10.1103/PhysRev.85.166</a>. <q>In contrast to the usual interpretation, this alternative interpretation permits us to conceive of each individual system as being in a precisely definable state, whose changes with time are determined by definite laws, analogous to (but not identical with) the classical equations of motion. Quantum-mechanical probabilities are regarded (like their counterparts in classical statistical mechanics) as only a practical necessity and not as an inherent lack of complete determination in the properties of matter at the quantum level.</q></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=A+Suggested+Interpretation+of+the+Quantum+Theory+in+Terms+of+%27Hidden+Variables%27+I&rft.volume=85&rft.issue=2&rft.pages=166-179&rft.date=1952&rft_id=info%3Adoi%2F10.1103%2FPhysRev.85.166&rft_id=info%3Abibcode%2F1952PhRv...85..166B&rft.aulast=Bohm&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> Publications of D. Bohm in 1952 and 1953 and of J.-P. Vigier in 1954 as cited in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntony_ValentiniHans_Westman2005" class="citation journal cs1">Antony Valentini; Hans Westman (2005). "Dynamical origin of quantum probabilities". Proc. R. Soc. A. 461 (2053): 253–272. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0403034">quant-ph/0403034</a>. <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/2005RSPSA.461..253V">2005RSPSA.461..253V</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.252.849">10.1.1.252.849</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.2004.1394">10.1098/rspa.2004.1394</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:6589887">6589887</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+R.+Soc.+A&rft.atitle=Dynamical+origin+of+quantum+probabilities&rft.volume=461&rft.issue=2053&rft.pages=253-272&rft.date=2005&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6589887%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2005RSPSA.461..253V&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.252.849%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1098%2Frspa.2004.1394&rft_id=info%3Aarxiv%2Fquant-ph%2F0403034&rft.au=Antony+Valentini&rft.au=Hans+Westman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> <a rel="nofollow" class="external text" href="http://rspa.royalsocietypublishing.org/content/461/2053/253.full.pdf#page=3">p. 254</a>. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKocsisBravermanRavetsStevens2011" class="citation journal cs1">Kocsis, Sacha; Braverman, Boris; Ravets, Sylvain; Stevens, Martin J.; Mirin, Richard P.; Shalm, L. Krister; Steinberg, Aephraim M. (3 June 2011). <a rel="nofollow" class="external text" href="https://www.science.org/doi/10.1126/science.1202218">"Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer"</a>. Science. 332 (6034): 1170–1173. <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/2011Sci...332.1170K">2011Sci...332.1170K</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.1202218">10.1126/science.1202218</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/0036-8075">0036-8075</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/21636767">21636767</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:27351467">27351467</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=Observing+the+Average+Trajectories+of+Single+Photons+in+a+Two-Slit+Interferometer&rft.volume=332&rft.issue=6034&rft.pages=1170-1173&rft.date=2011-06-03&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A27351467%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2011Sci...332.1170K&rft.issn=0036-8075&rft_id=info%3Adoi%2F10.1126%2Fscience.1202218&rft_id=info%3Apmid%2F21636767&rft.aulast=Kocsis&rft.aufirst=Sacha&rft.au=Braverman%2C+Boris&rft.au=Ravets%2C+Sylvain&rft.au=Stevens%2C+Martin+J.&rft.au=Mirin%2C+Richard+P.&rft.au=Shalm%2C+L.+Krister&rft.au=Steinberg%2C+Aephraim+M.&rft_id=https%3A%2F%2Fwww.science.org%2Fdoi%2F10.1126%2Fscience.1202218&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZeilinger1999" class="citation journal cs1">Zeilinger, Anton (1 March 1999). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.71.S288">"Experiment and the foundations of quantum physics"</a>. 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European Journal of Physics. 25 (6): 765–769. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0404128">quant-ph/0404128</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%2F0143-0807%2F25%2F6%2F008">10.1088/0143-0807/25/6/008</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/0143-0807">0143-0807</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Physics&rft.atitle=How+to+teach+quantum+mechanics&rft.volume=25&rft.issue=6&rft.pages=765-769&rft.date=2004-11-01&rft_id=info%3Aarxiv%2Fquant-ph%2F0404128&rft.issn=0143-0807&rft_id=info%3Adoi%2F10.1088%2F0143-0807%2F25%2F6%2F008&rft.aulast=Passon&rft.aufirst=Oliver&rft_id=https%3A%2F%2Fiopscience.iop.org%2Farticle%2F10.1088%2F0143-0807%2F25%2F6%2F008&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBohm1957" class="citation book cs1">Bohm, David (1957). Causality and Chance in Modern Physics. Routledge & Kegan Paul and D. Van Nostrand. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8122-1002-6" title="Special:BookSources/978-0-8122-1002-6"><bdi>978-0-8122-1002-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Causality+and+Chance+in+Modern+Physics&rft.pub=Routledge+%26+Kegan+Paul+and+D.+Van+Nostrand&rft.date=1957&rft.isbn=978-0-8122-1002-6&rft.aulast=Bohm&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> D. Bohm and B. Hiley: The undivided universe: An ontological interpretation of quantum theory, p. 37. </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> H. R. Brown, C. Dewdney and G. Horton: "Bohm particles and their detection in the light of neutron interferometry", Foundations of Physics, 1995, Volume 25, Number 2, pp. 329–347. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> J. Anandan, "The Quantum Measurement Problem and the Possible Role of the Gravitational Field", Foundations of Physics, March 1999, Volume 29, Issue 3, pp. 333–348. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBohmHiley1995" class="citation book cs1">Bohm, David; Hiley, Basil J. (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vt9XKjc4WAQC&pg=PA24">The undivided universe: an ontological interpretation of quantum theory</a>. Routledge. p. 24. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-12185-9" title="Special:BookSources/978-0-415-12185-9"><bdi>978-0-415-12185-9</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+undivided+universe%3A+an+ontological+interpretation+of+quantum+theory&rft.pages=24&rft.pub=Routledge&rft.date=1995&rft.isbn=978-0-415-12185-9&rft.aulast=Bohm&rft.aufirst=David&rft.au=Hiley%2C+Basil+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dvt9XKjc4WAQC%26pg%3DPA24&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHolland1995" class="citation book cs1">Holland, Peter R. (26 January 1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BsEfVBzToRMC&pg=PA26">The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics</a>. Cambridge University Press. p. 26. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-48543-2" title="Special:BookSources/978-0-521-48543-2"><bdi>978-0-521-48543-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Quantum+Theory+of+Motion%3A+An+Account+of+the+de+Broglie-Bohm+Causal+Interpretation+of+Quantum+Mechanics&rft.pages=26&rft.pub=Cambridge+University+Press&rft.date=1995-01-26&rft.isbn=978-0-521-48543-2&rft.aulast=Holland&rft.aufirst=Peter+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBsEfVBzToRMC%26pg%3DPA26&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHolland2001" class="citation journal cs1">Holland, P. (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111110140052/http://users.ox.ac.uk/~gree0579/index_files/NuovoCimento2.pdf#page=31">"Hamiltonian theory of wave and particle in quantum mechanics II: Hamilton-Jacobi theory and particle back-reaction"</a> (PDF). Nuovo Cimento B. 116 (10): 1143–1172. <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/2001NCimB.116.1143H">2001NCimB.116.1143H</a>. Archived from <a rel="nofollow" class="external text" href="http://users.ox.ac.uk/~gree0579/index_files/NuovoCimento2.pdf#page=31">the original</a> (PDF) on 10 November 2011. Retrieved 1 August 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuovo+Cimento+B&rft.atitle=Hamiltonian+theory+of+wave+and+particle+in+quantum+mechanics+II%3A+Hamilton-Jacobi+theory+and+particle+back-reaction&rft.volume=116&rft.issue=10&rft.pages=1143-1172&rft.date=2001&rft_id=info%3Abibcode%2F2001NCimB.116.1143H&rft.aulast=Holland&rft.aufirst=P.&rft_id=http%3A%2F%2Fusers.ox.ac.uk%2F~gree0579%2Findex_files%2FNuovoCimento2.pdf%23page%3D31&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-:0-14">^ <a href="#cite_ref-:0_14-0">a</a> <a href="#cite_ref-:0_14-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBohm1952" class="citation journal cs1">Bohm, David (15 January 1952). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRev.85.166">"A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I"</a>. Physical Review. 85 (2): 166–179. <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.85.166">10.1103/PhysRev.85.166</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=A+Suggested+Interpretation+of+the+Quantum+Theory+in+Terms+of+%22Hidden%22+Variables.+I&rft.volume=85&rft.issue=2&rft.pages=166-179&rft.date=1952-01-15&rft_id=info%3Adoi%2F10.1103%2FPhysRev.85.166&rft.issn=0031-899X&rft.aulast=Bohm&rft.aufirst=David&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRev.85.166&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-dgz92-15">^ <a href="#cite_ref-dgz92_15-0">a</a> <a href="#cite_ref-dgz92_15-1">b</a> <a href="#cite_ref-dgz92_15-2">c</a> <a href="#cite_ref-dgz92_15-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDürrGoldsteinZanghì1992" class="citation journal cs1">Dürr, D.; Goldstein, S.; Zanghì, N. (1992). "Quantum Equilibrium and the Origin of Absolute Uncertainty". Journal of Statistical Physics. 67 (5–6): 843–907. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0308039">quant-ph/0308039</a>. <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/1992JSP....67..843D">1992JSP....67..843D</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%2FBF01049004">10.1007/BF01049004</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:15749334">15749334</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+Statistical+Physics&rft.atitle=Quantum+Equilibrium+and+the+Origin+of+Absolute+Uncertainty&rft.volume=67&rft.issue=5%E2%80%936&rft.pages=843-907&rft.date=1992&rft_id=info%3Aarxiv%2Fquant-ph%2F0308039&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15749334%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01049004&rft_id=info%3Abibcode%2F1992JSP....67..843D&rft.aulast=D%C3%BCrr&rft.aufirst=D.&rft.au=Goldstein%2C+S.&rft.au=Zangh%C3%AC%2C+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTowlerRussellValentini2012" class="citation journal cs1">Towler, M. D.; Russell, N. J.; Valentini, A. (2012). "Timescales for dynamical relaxation to the Born rule". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 468 (2140): 990. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1103.1589">1103.1589</a>. <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/2012RSPSA.468..990T">2012RSPSA.468..990T</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.2011.0598">10.1098/rspa.2011.0598</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:119178440">119178440</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%3A+Mathematical%2C+Physical+and+Engineering+Sciences&rft.atitle=Timescales+for+dynamical+relaxation+to+the+Born+rule&rft.volume=468&rft.issue=2140&rft.pages=990&rft.date=2012&rft_id=info%3Aarxiv%2F1103.1589&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119178440%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.2011.0598&rft_id=info%3Abibcode%2F2012RSPSA.468..990T&rft.aulast=Towler&rft.aufirst=M.+D.&rft.au=Russell%2C+N.+J.&rft.au=Valentini%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988">. A video of the electron density in a 2D box evolving under this process is available <a rel="nofollow" class="external text" href="http://www.tcm.phy.cam.ac.uk/~mdt26/raw_movie.gif">here</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303230023/http://www.tcm.phy.cam.ac.uk/~mdt26/raw_movie.gif">Archived</a> 3 March 2016 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDürrGoldsteinZanghí2003" class="citation journal cs1">Dürr, Detlef; Goldstein, Sheldon; Zanghí, Nino (2003). "Quantum Equilibrium and the Origin of Absolute Uncertainty". Journal of Statistical Physics. 67 (5–6): 843–907. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0308039">quant-ph/0308039</a>. <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/1992JSP....67..843D">1992JSP....67..843D</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%2FBF01049004">10.1007/BF01049004</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:15749334">15749334</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+Statistical+Physics&rft.atitle=Quantum+Equilibrium+and+the+Origin+of+Absolute+Uncertainty&rft.volume=67&rft.issue=5%E2%80%936&rft.pages=843-907&rft.date=2003&rft_id=info%3Aarxiv%2Fquant-ph%2F0308039&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15749334%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01049004&rft_id=info%3Abibcode%2F1992JSP....67..843D&rft.aulast=D%C3%BCrr&rft.aufirst=Detlef&rft.au=Goldstein%2C+Sheldon&rft.au=Zangh%C3%AD%2C+Nino&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPasson2006" class="citation journal cs1">Passon, Oliver (2006). "What you always wanted to know about Bohmian mechanics but were afraid to ask". Physics and Philosophy. 3 (2006). <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0611032">quant-ph/0611032</a>. <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/2006quant.ph.11032P">2006quant.ph.11032P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.17877%2FDE290R-14213">10.17877/DE290R-14213</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2003%2F23108">2003/23108</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:45526627">45526627</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+and+Philosophy&rft.atitle=What+you+always+wanted+to+know+about+Bohmian+mechanics+but+were+afraid+to+ask&rft.volume=3&rft.issue=2006&rft.date=2006&rft_id=info%3Ahdl%2F2003%2F23108&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A45526627%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2006quant.ph.11032P&rft_id=info%3Aarxiv%2Fquant-ph%2F0611032&rft_id=info%3Adoi%2F10.17877%2FDE290R-14213&rft.aulast=Passon&rft.aufirst=Oliver&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNikolic2004" class="citation journal cs1">Nikolic, H. (2004). "Bohmian particle trajectories in relativistic bosonic quantum field theory". Foundations of Physics Letters. 17 (4): 363–380. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0208185">quant-ph/0208185</a>. <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/2004FoPhL..17..363N">2004FoPhL..17..363N</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.253.838">10.1.1.253.838</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FB%3AFOPL.0000035670.31755.0a">10.1023/B:FOPL.0000035670.31755.0a</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:1927035">1927035</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics+Letters&rft.atitle=Bohmian+particle+trajectories+in+relativistic+bosonic+quantum+field+theory&rft.volume=17&rft.issue=4&rft.pages=363-380&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1927035%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2004FoPhL..17..363N&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.253.838%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1023%2FB%3AFOPL.0000035670.31755.0a&rft_id=info%3Aarxiv%2Fquant-ph%2F0208185&rft.aulast=Nikolic&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNikolic2005" class="citation journal cs1">Nikolic, H. (2005). "Bohmian particle trajectories in relativistic fermionic quantum field theory". Foundations of Physics Letters. 18 (2): 123–138. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0302152">quant-ph/0302152</a>. <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/2005FoPhL..18..123N">2005FoPhL..18..123N</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%2Fs10702-005-3957-3">10.1007/s10702-005-3957-3</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:15304186">15304186</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics+Letters&rft.atitle=Bohmian+particle+trajectories+in+relativistic+fermionic+quantum+field+theory&rft.volume=18&rft.issue=2&rft.pages=123-138&rft.date=2005&rft_id=info%3Aarxiv%2Fquant-ph%2F0302152&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15304186%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10702-005-3957-3&rft_id=info%3Abibcode%2F2005FoPhL..18..123N&rft.aulast=Nikolic&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDürrGoldsteinMünch-BerndlZanghì1999" class="citation journal cs1">Dürr, D.; Goldstein, S.; Münch-Berndl, K.; Zanghì, N. 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Physical Review A. 60 (4): 2729–2736. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9801070">quant-ph/9801070</a>. <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/1999PhRvA..60.2729D">1999PhRvA..60.2729D</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.60.2729">10.1103/physreva.60.2729</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:52562586">52562586</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=Hypersurface+Bohm%E2%80%93Dirac+Models&rft.volume=60&rft.issue=4&rft.pages=2729-2736&rft.date=1999&rft_id=info%3Aarxiv%2Fquant-ph%2F9801070&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A52562586%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2Fphysreva.60.2729&rft_id=info%3Abibcode%2F1999PhRvA..60.2729D&rft.aulast=D%C3%BCrr&rft.aufirst=D.&rft.au=Goldstein%2C+S.&rft.au=M%C3%BCnch-Berndl%2C+K.&rft.au=Zangh%C3%AC%2C+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDürrGoldsteinNorsenStruyve2014" class="citation journal cs1">Dürr, Detlef; Goldstein, Sheldon; Norsen, Travis; Struyve, Ward; Zanghì, Nino (2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3896068">"Can Bohmian mechanics be made relativistic?"</a>. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 470 (2162): 20130699. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1307.1714">1307.1714</a>. <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/2013RSPSA.47030699D">2013RSPSA.47030699D</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.2013.0699">10.1098/rspa.2013.0699</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3896068">3896068</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/24511259">24511259</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%3A+Mathematical%2C+Physical+and+Engineering+Sciences&rft.atitle=Can+Bohmian+mechanics+be+made+relativistic%3F&rft.volume=470&rft.issue=2162&rft.pages=20130699&rft.date=2014&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3896068%23id-name%3DPMC&rft_id=info%3Abibcode%2F2013RSPSA.47030699D&rft_id=info%3Aarxiv%2F1307.1714&rft_id=info%3Apmid%2F24511259&rft_id=info%3Adoi%2F10.1098%2Frspa.2013.0699&rft.aulast=D%C3%BCrr&rft.aufirst=Detlef&rft.au=Goldstein%2C+Sheldon&rft.au=Norsen%2C+Travis&rft.au=Struyve%2C+Ward&rft.au=Zangh%C3%AC%2C+Nino&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3896068&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-ghose-1996-23">^ <a href="#cite_ref-ghose-1996_23-0">a</a> <a href="#cite_ref-ghose-1996_23-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGhose1996" class="citation journal cs1">Ghose, Partha (1996). 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Retrieved 18 March 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Physical+Chemistry+A&rft.atitle=The+Short+Story+of+My+Life+and+My+Career+in+Quantum+Propagation&rft.volume=111&rft.issue=41&rft.pages=10171-10185&rft.date=2007-10-11&rft_id=info%3Apmid%2F17927265&rft_id=info%3Adoi%2F10.1021%2Fjp079540%2B&rft_id=info%3Abibcode%2F2007JPCA..11110171.&rft.aulast=Wyatt&rft.aufirst=Robert&rft_id=https%3A%2F%2Fpubs.acs.org%2Fdoi%2Ffull%2F10.1021%2Fjp079540%252B&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-h523-72"><a href="#cite_ref-h523_72-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://web.archive.org/web/20210805161220/http://k2.chem.uh.edu/">"Bittner Group Webpage"</a>. k2.chem.uh.edu. 10 March 2021. Archived from <a rel="nofollow" class="external text" href="http://k2.chem.uh.edu/">the original</a> on 5 August 2021. Retrieved 10 July 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=k2.chem.uh.edu&rft.atitle=Bittner+Group+Webpage&rft.date=2021-03-10&rft_id=http%3A%2F%2Fk2.chem.uh.edu%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFValentiniWestman2012" class="citation arxiv cs1">Valentini, Antony; Westman, Hans (2012). "Combining Bohm and Everett: Axiomatics for a Standalone Quantum Mechanics". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1208.5632">1208.5632</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/quant-ph">quant-ph</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=Combining+Bohm+and+Everett%3A+Axiomatics+for+a+Standalone+Quantum+Mechanics&rft.date=2012&rft_id=info%3Aarxiv%2F1208.5632&rft.aulast=Valentini&rft.aufirst=Antony&rft.au=Westman%2C+Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-BrownWallace-74">^ <a href="#cite_ref-BrownWallace_74-0">a</a> <a href="#cite_ref-BrownWallace_74-1">b</a> <a href="#cite_ref-BrownWallace_74-2">c</a> <a href="#cite_ref-BrownWallace_74-3">d</a> <a href="#cite_ref-BrownWallace_74-4">e</a> <a href="#cite_ref-BrownWallace_74-5">f</a> <a href="#cite_ref-BrownWallace_74-6">g</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrownWallace2005" class="citation journal cs1"><a href="/wiki/Harvey_Brown_(philosopher)" class="mw-redirect" title="Harvey Brown (philosopher)">Brown, Harvey R.</a>; Wallace, David (2005). <a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/archive/00001659/01/Cushing.pdf">"Solving the measurement problem: de Broglie–Bohm loses out to Everett"</a> (PDF). Foundations of Physics. 35 (4): 517–540. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0403094">quant-ph/0403094</a>. <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/2005FoPh...35..517B">2005FoPh...35..517B</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%2Fs10701-004-2009-3">10.1007/s10701-004-2009-3</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:412240">412240</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=Solving+the+measurement+problem%3A+de+Broglie%E2%80%93Bohm+loses+out+to+Everett&rft.volume=35&rft.issue=4&rft.pages=517-540&rft.date=2005&rft_id=info%3Aarxiv%2Fquant-ph%2F0403094&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A412240%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10701-004-2009-3&rft_id=info%3Abibcode%2F2005FoPh...35..517B&rft.aulast=Brown&rft.aufirst=Harvey+R.&rft.au=Wallace%2C+David&rft_id=http%3A%2F%2Fphilsci-archive.pitt.edu%2Farchive%2F00001659%2F01%2FCushing.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> Abstract: "The quantum theory of de Broglie and Bohm solves the measurement problem, but the hypothetical corpuscles play no role in the argument. The solution finds a more natural home in the Everett interpretation." </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <a href="/wiki/Daniel_Dennett" title="Daniel Dennett">Daniel Dennett</a> (2000). With a little help from my friends. In D. Ross, A. Brook, and D. Thompson (Eds.), Dennett's Philosophy: a comprehensive assessment. MIT Press/Bradford, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-68117-X" title="Special:BookSources/0-262-68117-X">0-262-68117-X</a>. </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeutsch1996" class="citation journal cs1"><a href="/wiki/David_Deutsch" title="David Deutsch">Deutsch, David</a> (1996). "Comment on Lockwood". British Journal for the Philosophy of Science. 47 (2): 222–228. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbjps%2F47.2.222">10.1093/bjps/47.2.222</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=British+Journal+for+the+Philosophy+of+Science&rft.atitle=Comment+on+Lockwood&rft.volume=47&rft.issue=2&rft.pages=222-228&rft.date=1996&rft_id=info%3Adoi%2F10.1093%2Fbjps%2F47.2.222&rft.aulast=Deutsch&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDürrLazarovici2022" class="citation book cs1"><a href="/w/index.php?title=Detlef_D%C3%BCrr_and_Justin_Lazarovici&action=edit&redlink=1" class="new" title="Detlef Dürr and Justin Lazarovici (page does not exist)">Dürr, Detlef</a>; Lazarovici, Justin (2022). Understanding Quantum Mechanics: The World According to Modern Quantum Foundations. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-40067-5" title="Special:BookSources/978-3-030-40067-5"><bdi>978-3-030-40067-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Quantum+Mechanics%3A+The+World+According+to+Modern+Quantum+Foundations&rft.pub=Springer&rft.date=2022&rft.isbn=978-3-030-40067-5&rft.aulast=D%C3%BCrr&rft.aufirst=Detlef&rft.au=Lazarovici%2C+Justin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> See section VI of Everett's dissertation <a rel="nofollow" class="external text" href="https://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf">Theory of the Universal Wavefunction</a>, pp. 3–140 of <a href="/wiki/Bryce_Seligman_DeWitt" class="mw-redirect" title="Bryce Seligman DeWitt">Bryce Seligman DeWitt</a>, <a href="/w/index.php?title=R._Neill_Graham&action=edit&redlink=1" class="new" title="R. Neill Graham (page does not exist)">R. Neill Graham</a>, eds, The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a> (1973), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08131-X" title="Special:BookSources/0-691-08131-X">0-691-08131-X</a>. </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCallender" class="citation report cs1"><a href="/wiki/Craig_Callender" title="Craig Callender">Callender, Craig</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100612191323/http://philosophy.ucsd.edu/faculty/ccallender/The%20Redundancy%20Argument%20Against%20Bohmian%20Mechanics.doc">The Redundancy Argument Against Bohmian Mechanics</a> (Report). Archived from <a rel="nofollow" class="external text" href="http://philosophy.ucsd.edu/faculty/ccallender/The%20Redundancy%20Argument%20Against%20Bohmian%20Mechanics.doc">the original</a> on 12 June 2010. Retrieved 23 November 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=The+Redundancy+Argument+Against+Bohmian+Mechanics&rft.aulast=Callender&rft.aufirst=Craig&rft_id=http%3A%2F%2Fphilosophy.ucsd.edu%2Ffaculty%2Fccallender%2FThe%2520Redundancy%2520Argument%2520Against%2520Bohmian%2520Mechanics.doc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFValentini2010" class="citation book cs1">Valentini, Antony (2010). "De Broglie-Bohm Pilot-Wave Theory: Many Worlds in Denial?". In Saunders, Simon; Barrett, Jon; Kent, Adrian (eds.). Many Worlds? Everett, Quantum Theory, and Reality. Vol. 2010. Oxford University Press. pp. 476–509. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0811.0810">0811.0810</a>. <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/2008arXiv0811.0810V">2008arXiv0811.0810V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Facprof%3Aoso%2F9780199560561.003.0019">10.1093/acprof:oso/9780199560561.003.0019</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-956056-1" title="Special:BookSources/978-0-19-956056-1"><bdi>978-0-19-956056-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=De+Broglie-Bohm+Pilot-Wave+Theory%3A+Many+Worlds+in+Denial%3F&rft.btitle=Many+Worlds%3F+Everett%2C+Quantum+Theory%2C+and+Reality&rft.pages=476-509&rft.pub=Oxford+University+Press&rft.date=2010&rft_id=info%3Aarxiv%2F0811.0810&rft_id=info%3Adoi%2F10.1093%2Facprof%3Aoso%2F9780199560561.003.0019&rft_id=info%3Abibcode%2F2008arXiv0811.0810V&rft.isbn=978-0-19-956056-1&rft.aulast=Valentini&rft.aufirst=Antony&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHolland2001" class="citation journal cs1">Holland, Peter (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111110140052/http://users.ox.ac.uk/~gree0579/index_files/NuovoCimento2.pdf">"Hamiltonian Theory of Wave and Particle in Quantum Mechanics I, II"</a> (PDF). Nuovo Cimento B. 116: 1043, 1143. Archived from <a rel="nofollow" class="external text" href="http://users.ox.ac.uk/~gree0579/index_files/NuovoCimento2.pdf">the original</a> (PDF) on 10 November 2011. Retrieved 17 July 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuovo+Cimento+B&rft.atitle=Hamiltonian+Theory+of+Wave+and+Particle+in+Quantum+Mechanics+I%2C+II&rft.volume=116&rft.pages=1043%2C+1143&rft.date=2001&rft.aulast=Holland&rft.aufirst=Peter&rft_id=http%3A%2F%2Fusers.ox.ac.uk%2F~gree0579%2Findex_files%2FNuovoCimento2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> Peter R. Holland: The quantum theory of motion, Cambridge University Press, 1993 (re-printed 2000, transferred to digital printing 2004), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-48543-6" title="Special:BookSources/0-521-48543-6">0-521-48543-6</a>, p. 66 ff. </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> F. David Peat, Infinite Potential: The Life and Times of David Bohm (1997), p. 133. James T. Cushing, Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony (1994) discusses "the hegemony of the Copenhagen interpretation of quantum mechanics" over theories like Bohmian mechanics as an example of how the acceptance of scientific theories may be guided by social aspects. </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBell1982" class="citation journal cs1">Bell, J. S. (1 October 1982). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/BF01889272">"On the impossible pilot wave"</a>. Foundations of Physics. 12 (10): 989–999. <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/1982FoPh...12..989B">1982FoPh...12..989B</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%2FBF01889272">10.1007/BF01889272</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/1572-9516">1572-9516</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:120592799">120592799</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=On+the+impossible+pilot+wave&rft.volume=12&rft.issue=10&rft.pages=989-999&rft.date=1982-10-01&rft_id=info%3Adoi%2F10.1007%2FBF01889272&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120592799%23id-name%3DS2CID&rft.issn=1572-9516&rft_id=info%3Abibcode%2F1982FoPh...12..989B&rft.aulast=Bell&rft.aufirst=J.+S.&rft_id=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2FBF01889272&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> David Bohm and Basil J. Hiley, The Undivided Universe – An Ontological Interpretation of Quantum Theory appeared after Bohm's death, in 1993; <a rel="nofollow" class="external text" href="http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/files/bhr.pdf">reviewed</a> by Sheldon Goldstein in Physics Today (1994). J. Cushing, A. Fine, S. 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London: Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-12185-9" title="Special:BookSources/978-0-415-12185-9"><bdi>978-0-415-12185-9</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+Undivided+Universe%3A+An+ontological+interpretation+of+quantum+theory&rft.place=London&rft.pub=Routledge&rft.date=1993&rft.isbn=978-0-415-12185-9&rft.aulast=Bohm&rft.aufirst=David&rft.au=B.J.+Hiley&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDürrSheldon_GoldsteinRoderich_TumulkaNino_Zanghì2004" class="citation journal cs1">Dürr, Detlef; Sheldon Goldstein; Roderich Tumulka; Nino Zanghì (December 2004). <a rel="nofollow" class="external text" href="http://www.math.rutgers.edu/~oldstein/papers/bohmech.pdf">"Bohmian Mechanics"</a> (PDF). Physical Review Letters. 93 (9): 090402. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0303156">quant-ph/0303156</a>. <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/2004PhRvL..93i0402D">2004PhRvL..93i0402D</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.8444">10.1.1.8.8444</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.93.090402">10.1103/PhysRevLett.93.090402</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/0031-9007">0031-9007</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/15447078">15447078</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:8720296">8720296</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=Bohmian+Mechanics&rft.volume=93&rft.issue=9&rft.pages=090402&rft.date=2004-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8720296%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.93.090402&rft_id=info%3Abibcode%2F2004PhRvL..93i0402D&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.8.8444%23id-name%3DCiteSeerX&rft.issn=0031-9007&rft_id=info%3Apmid%2F15447078&rft_id=info%3Aarxiv%2Fquant-ph%2F0303156&rft.aulast=D%C3%BCrr&rft.aufirst=Detlef&rft.au=Sheldon+Goldstein&rft.au=Roderich+Tumulka&rft.au=Nino+Zangh%C3%AC&rft_id=http%3A%2F%2Fwww.math.rutgers.edu%2F~oldstein%2Fpapers%2Fbohmech.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldstein2001" class="citation journal cs1">Goldstein, Sheldon (2001). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/qm-bohm/">"Bohmian Mechanics"</a>. <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Stanford+Encyclopedia+of+Philosophy&rft.atitle=Bohmian+Mechanics&rft.date=2001&rft.aulast=Goldstein&rft.aufirst=Sheldon&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fqm-bohm%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2004" class="citation journal cs1">Hall, Michael J. W. (2004). "Incompleteness of trajectory-based interpretations of quantum mechanics". Journal of Physics A: Mathematical and General. 37 (40): 9549–9556. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0406054">quant-ph/0406054</a>. <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/2004JPhA...37.9549H">2004JPhA...37.9549H</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.252.5757">10.1.1.252.5757</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%2F0305-4470%2F37%2F40%2F015">10.1088/0305-4470/37/40/015</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:15196269">15196269</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+A%3A+Mathematical+and+General&rft.atitle=Incompleteness+of+trajectory-based+interpretations+of+quantum+mechanics&rft.volume=37&rft.issue=40&rft.pages=9549-9556&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15196269%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2004JPhA...37.9549H&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.252.5757%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1088%2F0305-4470%2F37%2F40%2F015&rft_id=info%3Aarxiv%2Fquant-ph%2F0406054&rft.aulast=Hall&rft.aufirst=Michael+J.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> (Demonstrates incompleteness of the Bohm interpretation in the face of fractal, differentiable-nowhere wavefunctions.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHolland1993" class="citation book cs1">Holland, Peter R. (1993). The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-48543-2" title="Special:BookSources/978-0-521-48543-2"><bdi>978-0-521-48543-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Quantum+Theory+of+Motion%3A+An+Account+of+the+de+Broglie%E2%80%93Bohm+Causal+Interpretation+of+Quantum+Mechanics&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1993&rft.isbn=978-0-521-48543-2&rft.aulast=Holland&rft.aufirst=Peter+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNikolic2005" class="citation journal cs1">Nikolic, H. (2005). "Relativistic quantum mechanics and the Bohmian interpretation". Foundations of Physics Letters. 18 (6): 549–561. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0406173">quant-ph/0406173</a>. <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/2005FoPhL..18..549N">2005FoPhL..18..549N</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.252.6803">10.1.1.252.6803</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%2Fs10702-005-1128-1">10.1007/s10702-005-1128-1</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:14006204">14006204</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics+Letters&rft.atitle=Relativistic+quantum+mechanics+and+the+Bohmian+interpretation&rft.volume=18&rft.issue=6&rft.pages=549-561&rft.date=2005&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14006204%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2005FoPhL..18..549N&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.252.6803%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1007%2Fs10702-005-1128-1&rft_id=info%3Aarxiv%2Fquant-ph%2F0406173&rft.aulast=Nikolic&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPasson2004" class="citation arxiv cs1">Passon, Oliver (2004). "Why isn't every physicist a Bohmian?". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0412119">quant-ph/0412119</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+isn%27t+every+physicist+a+Bohmian%3F&rft.date=2004&rft_id=info%3Aarxiv%2Fquant-ph%2F0412119&rft.aulast=Passon&rft.aufirst=Oliver&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSanzF._Borondo2007" class="citation journal cs1">Sanz, A. S.; F. Borondo (2007). "A Bohmian view on quantum decoherence". European Physical Journal D. 44 (2): 319–326. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0310096">quant-ph/0310096</a>. <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/2007EPJD...44..319S">2007EPJD...44..319S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1140%2Fepjd%2Fe2007-00191-8">10.1140/epjd/e2007-00191-8</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:18449109">18449109</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Physical+Journal+D&rft.atitle=A+Bohmian+view+on+quantum+decoherence&rft.volume=44&rft.issue=2&rft.pages=319-326&rft.date=2007&rft_id=info%3Aarxiv%2Fquant-ph%2F0310096&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18449109%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1140%2Fepjd%2Fe2007-00191-8&rft_id=info%3Abibcode%2F2007EPJD...44..319S&rft.aulast=Sanz&rft.aufirst=A.+S.&rft.au=F.+Borondo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSanz2005" class="citation journal cs1">Sanz, A. S. (2005). "A Bohmian approach to quantum fractals". Journal of Physics A: Mathematical and General. 38 (26): 6037–6049. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0412050">quant-ph/0412050</a>. <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/2005JPhA...38.6037S">2005JPhA...38.6037S</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%2F0305-4470%2F38%2F26%2F013">10.1088/0305-4470/38/26/013</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:17633797">17633797</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+A%3A+Mathematical+and+General&rft.atitle=A+Bohmian+approach+to+quantum+fractals&rft.volume=38&rft.issue=26&rft.pages=6037-6049&rft.date=2005&rft_id=info%3Aarxiv%2Fquant-ph%2F0412050&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17633797%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0305-4470%2F38%2F26%2F013&rft_id=info%3Abibcode%2F2005JPhA...38.6037S&rft.aulast=Sanz&rft.aufirst=A.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"> (Describes a Bohmian resolution to the dilemma posed by non-differentiable wavefunctions.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilverman1993" class="citation book cs1">Silverman, Mark P. (1993). <a rel="nofollow" class="external text" href="https://archive.org/details/andyetitmoves00mark">And Yet It Moves: Strange Systems and Subtle Questions in Physics</a>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-44631-0" title="Special:BookSources/978-0-521-44631-0"><bdi>978-0-521-44631-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=And+Yet+It+Moves%3A+Strange+Systems+and+Subtle+Questions+in+Physics&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1993&rft.isbn=978-0-521-44631-0&rft.aulast=Silverman&rft.aufirst=Mark+P.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fandyetitmoves00mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStreater2003" class="citation web cs1"><a href="/wiki/Ray_Streater" title="Ray Streater">Streater, Ray F.</a> (2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060613191814/http://www.mth.kcl.ac.uk/~streater/lostcauses.html#XI">"Bohmian mechanics is a 'lost cause'"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.mth.kcl.ac.uk/~streater/lostcauses.html#XI">the original</a> on 13 June 2006. Retrieved 25 June 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Bohmian+mechanics+is+a+%27lost+cause%27&rft.date=2003&rft.aulast=Streater&rft.aufirst=Ray+F.&rft_id=http%3A%2F%2Fwww.mth.kcl.ac.uk%2F~streater%2Flostcauses.html%23XI&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFValentiniHans_Westman2005" class="citation journal cs1"><a href="/wiki/Antony_Valentini" title="Antony Valentini">Valentini, Antony</a>; Hans Westman (2005). "Dynamical Origin of Quantum Probabilities". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 461 (2053): 253–272. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0403034">quant-ph/0403034</a>. <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/2005RSPSA.461..253V">2005RSPSA.461..253V</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.252.849">10.1.1.252.849</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.2004.1394">10.1098/rspa.2004.1394</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:6589887">6589887</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%3A+Mathematical%2C+Physical+and+Engineering+Sciences&rft.atitle=Dynamical+Origin+of+Quantum+Probabilities&rft.volume=461&rft.issue=2053&rft.pages=253-272&rft.date=2005&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6589887%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2005RSPSA.461..253V&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.252.849%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1098%2Frspa.2004.1394&rft_id=info%3Aarxiv%2Fquant-ph%2F0403034&rft.aulast=Valentini&rft.aufirst=Antony&rft.au=Hans+Westman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=-139823">Bohmian mechanics on arxiv.org</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=40" title="Edit section: Further reading">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col div-col-small" style="column-width: 30em;"> <ul><li><a href="/wiki/John_S._Bell" class="mw-redirect" title="John S. Bell">John S. Bell</a>: Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, Cambridge University Press, 2004, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-81862-1" title="Special:BookSources/0-521-81862-1">0-521-81862-1</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">David Bohm</a>, <a href="/wiki/Basil_Hiley" title="Basil Hiley">Basil Hiley</a>: The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge Chapman & Hall, 1993, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-415-06588-7" title="Special:BookSources/0-415-06588-7">0-415-06588-7</a></li> <li>Detlef Dürr, Sheldon Goldstein, Nino Zanghì: Quantum Physics Without Quantum Philosophy, Springer, 2012, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-30690-7" title="Special:BookSources/978-3-642-30690-7">978-3-642-30690-7</a></li> <li>Detlef Dürr, Stefan Teufel: Bohmian Mechanics: The Physics and Mathematics of Quantum Theory, Springer, 2009, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-89343-1" title="Special:BookSources/978-3-540-89343-1">978-3-540-89343-1</a></li> <li><a href="/wiki/Peter_R._Holland" title="Peter R. Holland">Peter R. Holland</a>: The quantum theory of motion, Cambridge University Press, 1993 (re-printed 2000, transferred to digital printing 2004), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-48543-6" title="Special:BookSources/0-521-48543-6">0-521-48543-6</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&section=41" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <a href="https://en.wikiquote.org/wiki/Special:Search/De_Broglie%E2%80%93Bohm_theory" class="extiw" title="q:Special:Search/De Broglie–Bohm theory">De Broglie–Bohm theory</a>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <a href="https://en.wikiversity.org/wiki/Making_sense_of_quantum_mechanics" class="extiw" title="v:Making sense of quantum mechanics"> Making sense of quantum mechanics</a></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col div-col-small" style="column-width: 30em;"> <ul><li><a rel="nofollow" class="external text" href="http://www.annualreviews.org/doi/abs/10.1146/annurev-fluid-010814-014506">"Pilot-Wave Hydrodynamics"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210318163615/https://www.annualreviews.org/doi/abs/10.1146/annurev-fluid-010814-014506">Archived</a> 18 March 2021 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Bush, J. W. M., Annual Review of Fluid Mechanics, 2015</li> <li><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/qm-bohm">"Bohmian Mechanics" (Stanford Encyclopedia of Philosophy)</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'Dowd2016" class="citation web cs1"><a href="/wiki/Matt_O%27Dowd_(astrophysicist)" title="Matt O'Dowd (astrophysicist)">O'Dowd, Matt</a> (30 November 2016). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=RlXdsyctD50">"Pilot Wave Theory and Quantum Realism"</a>. <a href="/wiki/PBS_Space_Time" class="mw-redirect" title="PBS Space Time">PBS Space Time</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211211/RlXdsyctD50">Archived</a> from the original on 11 December 2021 – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PBS+Space+Time&rft.atitle=Pilot+Wave+Theory+and+Quantum+Realism&rft.date=2016-11-30&rft.aulast=O%27Dowd&rft.aufirst=Matt&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DRlXdsyctD50&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.youtube.com/playlist?list=PL7LbfRoKBR5OpRjt8toBOmzqGjH7zaM1m">"Videos answering frequently asked questions about Bohmian Mechanics"</a> – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Videos+answering+frequently+asked+questions+about+Bohmian+Mechanics&rft_id=https%3A%2F%2Fwww.youtube.com%2Fplaylist%3Flist%3DPL7LbfRoKBR5OpRjt8toBOmzqGjH7zaM1m&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Broglie%E2%80%93Bohm+theory" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.bohmian-mechanics.net/">"Bohmian-Mechanics.net"</a>, the homepage of the international research network on Bohmian Mechanics that was started by D. Dürr, S. Goldstein and N. Zanghì.</li> <li><a rel="nofollow" class="external text" href="http://www.mathematik.uni-muenchen.de/~bohmmech/">Workgroup Bohmian Mechanics at LMU Munich (D. Dürr)</a></li> <li><a rel="nofollow" class="external text" href="http://bohm-mechanics.uibk.ac.at/">Bohmian Mechanics Group at University of Innsbruck (G. Grübl)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141125213942/http://bohm-mechanics.uibk.ac.at/">Archived</a> 25 November 2014 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html">"Pilot waves, Bohmian metaphysics, and the foundations of quantum mechanics"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160410173517/http://www.tcm.phy.cam.ac.uk/%7Emdt26/pilot_waves.html">Archived</a> 10 April 2016 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, lecture course on de Broglie-Bohm theory by <a href="/wiki/Mike_Towler" class="mw-redirect" title="Mike Towler">Mike Towler</a>, Cambridge University.</li> <li><a rel="nofollow" class="external text" href="http://www.vallico.net/tti/deBB_10/conference.html">"21st-century directions in de Broglie-Bohm theory and beyond"</a>, August 2010 international conference on de Broglie-Bohm theory. Site contains slides for all the talks – the latest cutting-edge deBB research.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110626194505/http://www.aip.org.au/Congress2010/Abstracts/Monday%206%20Dec%20-%20Orals/Session_3E/Kocsis_Observing_the_Trajectories.pdf">"Observing the Trajectories of a Single Photon Using Weak Measurement"</a></li> <li><a rel="nofollow" class="external text" href="https://www.physicsforums.com/blog.php?b=3077">"Bohmian trajectories are no longer 'hidden variables'"</a></li> <li><a rel="nofollow" class="external text" href="http://dbohm.com/">The David Bohm Society</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=W2Xb2GFK2yc">De Broglie–Bohm theory inspired visualization of atomic orbitals.</a></li></ul> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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.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 autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template: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 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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 class="mw-selflink selflink">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 href="/wiki/Casimir_effect" title="Casimir effect">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> 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