Path integral formulation

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class="vector-toc-numb">2</span> <span>Classical limit</span> </div> </a> <ul id="toc-Classical_limit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Feynman's_interpretation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Feynman's_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Feynman's interpretation</span> </div> </a> <ul id="toc-Feynman's_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Path_integral_in_quantum_mechanics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Path_integral_in_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Path integral in quantum mechanics</span> </div> </a> <button aria-controls="toc-Path_integral_in_quantum_mechanics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Path integral in quantum mechanics subsection</span> </button> <ul id="toc-Path_integral_in_quantum_mechanics-sublist" class="vector-toc-list"> <li id="toc-Time-slicing_derivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time-slicing_derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Time-slicing derivation</span> </div> </a> <ul id="toc-Time-slicing_derivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Path_integral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Path_integral"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Path integral</span> </div> </a> <ul id="toc-Path_integral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Free_particle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free_particle"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Free particle</span> </div> </a> <ul id="toc-Free_particle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_harmonic_oscillator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_harmonic_oscillator"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Simple harmonic oscillator</span> </div> </a> <ul id="toc-Simple_harmonic_oscillator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coulomb_potential" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coulomb_potential"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Coulomb potential</span> </div> </a> <ul id="toc-Coulomb_potential-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Schrödinger_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Schrödinger_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>The Schrödinger equation</span> </div> </a> <ul id="toc-The_Schrödinger_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equations_of_motion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equations_of_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Equations of motion</span> </div> </a> <ul id="toc-Equations_of_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stationary-phase_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stationary-phase_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Stationary-phase approximation</span> </div> </a> <ul id="toc-Stationary-phase_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Canonical_commutation_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Canonical_commutation_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.9</span> <span>Canonical commutation relations</span> </div> </a> <ul id="toc-Canonical_commutation_relations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Particle_in_curved_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Particle_in_curved_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.10</span> <span>Particle in curved space</span> </div> </a> <ul id="toc-Particle_in_curved_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measure-theoretic_factors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measure-theoretic_factors"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.11</span> <span>Measure-theoretic factors</span> </div> </a> <ul id="toc-Measure-theoretic_factors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expectation_values_and_matrix_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expectation_values_and_matrix_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.12</span> <span>Expectation values and matrix elements</span> </div> </a> <ul id="toc-Expectation_values_and_matrix_elements-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Euclidean_path_integrals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Euclidean_path_integrals"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Euclidean path integrals</span> </div> </a> <button aria-controls="toc-Euclidean_path_integrals-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Euclidean path integrals subsection</span> </button> <ul id="toc-Euclidean_path_integrals-sublist" class="vector-toc-list"> <li id="toc-Wick_rotation_and_the_Feynman–Kac_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wick_rotation_and_the_Feynman–Kac_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Wick rotation and the Feynman–Kac formula</span> </div> </a> <ul id="toc-Wick_rotation_and_the_Feynman–Kac_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Path_integral_and_the_partition_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Path_integral_and_the_partition_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Path integral and the partition function</span> </div> </a> <ul id="toc-Path_integral_and_the_partition_function-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_field_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Quantum_field_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Quantum field theory</span> </div> </a> <button aria-controls="toc-Quantum_field_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Quantum field theory subsection</span> </button> <ul id="toc-Quantum_field_theory-sublist" class="vector-toc-list"> <li id="toc-Propagator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propagator"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Propagator</span> </div> </a> <ul id="toc-Propagator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Functionals_of_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Functionals_of_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Functionals of fields</span> </div> </a> <ul id="toc-Functionals_of_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expectation_values" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expectation_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Expectation values</span> </div> </a> <ul id="toc-Expectation_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_a_probability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_probability"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>As a probability</span> </div> </a> <ul id="toc-As_a_probability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Schwinger–Dyson_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Schwinger–Dyson_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Schwinger–Dyson equations</span> </div> </a> <ul id="toc-Schwinger–Dyson_equations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Localization" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Localization"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Localization</span> </div> </a> <button aria-controls="toc-Localization-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Localization subsection</span> </button> <ul id="toc-Localization-sublist" class="vector-toc-list"> <li id="toc-Ward–Takahashi_identities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ward–Takahashi_identities"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Ward–Takahashi identities</span> </div> </a> <ul id="toc-Ward–Takahashi_identities-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Caveats" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Caveats"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Caveats</span> </div> </a> <button aria-controls="toc-Caveats-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Caveats subsection</span> </button> <ul id="toc-Caveats-sublist" class="vector-toc-list"> <li id="toc-Need_for_regulators_and_renormalization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Need_for_regulators_and_renormalization"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Need for regulators and renormalization</span> </div> </a> <ul id="toc-Need_for_regulators_and_renormalization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordering_prescription" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordering_prescription"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Ordering prescription</span> </div> </a> <ul id="toc-Ordering_prescription-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Path_integral_in_quantum-mechanical_interpretation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Path_integral_in_quantum-mechanical_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Path integral in quantum-mechanical interpretation</span> </div> </a> <ul id="toc-Path_integral_in_quantum-mechanical_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_gravity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Quantum_gravity"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Quantum gravity</span> </div> </a> <ul id="toc-Quantum_gravity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_tunneling" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Quantum_tunneling"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Quantum tunneling</span> </div> </a> <ul id="toc-Quantum_tunneling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Remarks" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Remarks"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Remarks</span> </div> </a> <ul id="toc-Remarks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-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"> <span class="vector-toc-numb">16</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Path integral formulation</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 22 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-22" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">22 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B5%D9%8A%D8%BA%D8%A9_%D8%AA%D9%83%D8%A7%D9%85%D9%84_%D8%A7%D9%84%D9%85%D8%B3%D8%A7%D8%B1" title="صيغة تكامل المسار – Arabic" lang="ar" hreflang="ar" data-title="صيغة تكامل المسار" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BE%E0%A4%A5_%E0%A4%87%E0%A4%82%E0%A4%9F%E0%A5%80%E0%A4%97%E0%A5%8D%E0%A4%B0%E0%A4%B2_%E0%A4%AB%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AE%E0%A5%81%E0%A4%B2%E0%A5%87%E0%A4%B6%E0%A4%A8" title="पाथ इंटीग्रल फार्मुलेशन – Bhojpuri" lang="bh" hreflang="bh" data-title="पाथ इंटीग्रल फार्मुलेशन" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Formulaci%C3%B3_de_la_integral_de_camins" title="Formulació de la integral de camins – Catalan" lang="ca" hreflang="ca" data-title="Formulació de la integral de camins" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Pfadintegral" title="Pfadintegral – German" lang="de" hreflang="de" data-title="Pfadintegral" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Integral_de_caminos_(mec%C3%A1nica_cu%C3%A1ntica)" title="Integral de caminos (mecánica cuántica) – Spanish" lang="es" hreflang="es" data-title="Integral de caminos (mecánica cuántica)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B1%D9%85%D9%88%D9%84%E2%80%8C%D8%A8%D9%86%D8%AF%DB%8C_%D8%A7%D9%86%D8%AA%DA%AF%D8%B1%D8%A7%D9%84_%D9%85%D8%B3%DB%8C%D8%B1" title="فرمول‌بندی انتگرال مسیر – Persian" lang="fa" hreflang="fa" data-title="فرمول‌بندی انتگرال مسیر" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Int%C3%A9grale_de_chemin" title="Intégrale de chemin – French" lang="fr" hreflang="fr" data-title="Intégrale de chemin" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B2%BD%EB%A1%9C_%EC%A0%81%EB%B6%84_%EA%B3%B5%EC%8B%9D%ED%99%94" title="경로 적분 공식화 – Korean" lang="ko" hreflang="ko" data-title="경로 적분 공식화" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%BE%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1%D5%B5%D5%AB_%D5%B1%D6%87%D5%A1%D5%AF%D5%A5%D6%80%D5%BA%D5%B8%D6%82%D5%B4%D5%B6_%D5%A8%D5%BD%D5%BF_%D5%B0%D5%A5%D5%BF%D5%A1%D5%A3%D5%AE%D5%A5%D6%80%D5%B8%D5%BE_%D5%AB%D5%B6%D5%BF%D5%A5%D5%A3%D6%80%D5%A1%D5%AC%D5%B6%D5%A5%D6%80%D5%AB" title="Քվանտային մեխանիկայի ձևակերպումն ըստ հետագծերով ինտեգրալների – Armenian" lang="hy" hreflang="hy" data-title="Քվանտային մեխանիկայի ձևակերպումն ըստ հետագծերով ինտեգրալների" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Rumus_integral_lintasan" title="Rumus integral lintasan – Indonesian" lang="id" hreflang="id" data-title="Rumus integral lintasan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Integrale_sui_cammini" title="Integrale sui cammini – Italian" lang="it" hreflang="it" data-title="Integrale sui cammini" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99_%D7%9E%D7%A1%D7%9C%D7%95%D7%9C" title="אינטגרלי מסלול – Hebrew" lang="he" hreflang="he" data-title="אינטגרלי מסלול" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Padintegraal" title="Padintegraal – Dutch" lang="nl" hreflang="nl" data-title="Padintegraal" 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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">This article is about a formulation of quantum mechanics. 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:19.0em;"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_number" title="Quantum number">Quantum number</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">State</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell's inequality</a></li> <li><a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li></ul> </div> <ul><li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a> <ul><li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><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 class="mw-selflink selflink">Sum-over-histories (path integral)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced topics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Blackett</a></li> <li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">Bohm</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Davisson</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Hans_Kramers" title="Hans Kramers">Kramers</a></li> <li><a href="/wiki/Willis_Lamb" title="Willis Lamb">Lamb</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Isidor_Isaac_Rabi" title="Isidor Isaac Rabi">Rabi</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Michelle_Simmons" title="Michelle Simmons">Simmons</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics" title="Template:Quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics" title="Template talk:Quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics" title="Special:EditPage/Template:Quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>path integral formulation</b> is a description in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> that generalizes the <a href="/wiki/Stationary_action_principle" class="mw-redirect" title="Stationary action principle">stationary action principle</a> of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or <a href="/wiki/Functional_integral" class="mw-redirect" title="Functional integral">functional integral</a>, over an infinity of quantum-mechanically possible trajectories to compute a <a href="/wiki/Probability_amplitude" title="Probability amplitude">quantum amplitude</a>. </p><p>This formulation has proven crucial to the subsequent development of <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>, because manifest <a href="/wiki/Lorentz_covariance" title="Lorentz covariance">Lorentz covariance</a> (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of <a href="/wiki/Canonical_quantization" title="Canonical quantization">canonical quantization</a>. Unlike previous methods, the path integral allows one to easily change <a href="/wiki/Coordinates" class="mw-redirect" title="Coordinates">coordinates</a> between very different <a href="/wiki/Canonical_coordinates" title="Canonical coordinates">canonical</a> descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> of a theory, which naturally enters the path integrals (for interactions of a certain type, these are <i>coordinate space</i> or <i>Feynman path integrals</i>), than the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>. Possible downsides of the approach include that <a href="/wiki/Unitarity" class="mw-redirect" title="Unitarity">unitarity</a> (this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the <a href="/wiki/S-matrix" title="S-matrix">S-matrix</a> is obscure in the formulation. The path-integral approach has proven to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by <i>deriving</i> either approach from the other, problems associated with one or the other approach (as exemplified by Lorentz covariance or unitarity) go away.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The path integral also relates quantum and <a href="/wiki/Stochastic" title="Stochastic">stochastic</a> processes, and this provided the basis for the grand synthesis of the 1970s, which unified <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> with the <a href="/wiki/Statistical_field_theory" title="Statistical field theory">statistical field theory</a> of a fluctuating field near a <a href="/wiki/Second-order_phase_transition" class="mw-redirect" title="Second-order phase transition">second-order phase transition</a>. The <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> is a <a href="/wiki/Diffusion_equation" title="Diffusion equation">diffusion equation</a> with an imaginary diffusion constant, and the path integral is an <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> of a method for summing up all possible <a href="/wiki/Random_walk" title="Random walk">random walks</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The path integral has impacted a wide array of sciences, including <a href="/wiki/Polymer_physics" title="Polymer physics">polymer physics</a>, quantum field theory, <a href="/wiki/String_theory" title="String theory">string theory</a> and <a href="/wiki/Cosmology" title="Cosmology">cosmology</a>. In physics, it is a foundation for <a href="/wiki/Lattice_gauge_theory" title="Lattice gauge theory">lattice gauge theory</a> and <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>.<sup id="cite_ref-:02_3-0" class="reference"><a href="#cite_note-:02-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It has been called the "most powerful formula in physics",<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> with <a href="/wiki/Stephen_Wolfram" title="Stephen Wolfram">Stephen Wolfram</a> also declaring it to be the "fundamental mathematical construct of modern quantum mechanics and quantum field theory".<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The basic idea of the path integral formulation can be traced back to <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a>, who introduced the <a href="/wiki/Wiener_integral" class="mw-redirect" title="Wiener integral">Wiener integral</a> for solving problems in diffusion and <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This idea was extended to the use of the <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> in quantum mechanics by <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, whose 1933 paper gave birth to path integral formulation.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_9-0" class="reference"><a href="#cite_note-:0-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:02_3-1" class="reference"><a href="#cite_note-:02-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The complete method was developed in 1948 by <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a>.<sup id="cite_ref-FOOTNOTEFeynman1948_10-0" class="reference"><a href="#cite_note-FOOTNOTEFeynman1948-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Some preliminaries were worked out earlier in his doctoral work under the supervision of <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a>. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the <a href="/wiki/Wheeler%E2%80%93Feynman_absorber_theory" title="Wheeler–Feynman absorber theory">Wheeler–Feynman absorber theory</a> using a <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> (rather than a <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>) as a starting point. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Feynman_paths.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Feynman_paths.png/220px-Feynman_paths.png" decoding="async" width="220" height="331" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/bf/Feynman_paths.png 1.5x" data-file-width="253" data-file-height="381" /></a><figcaption>These are five of the infinitely many paths available for a particle to move from point A at time t to point B at time t’(>t).</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Quantum_action_principle">Quantum action principle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=1" title="Edit section: Quantum action principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In quantum mechanics, as in classical mechanics, the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> is the generator of time translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator (multiplied by the negative <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, <span class="texhtml">−<i>i</i></span>). For states with a definite energy, this is a statement of the <a href="/wiki/De_Broglie_relation" class="mw-redirect" title="De Broglie relation">de Broglie relation</a> between frequency and energy, and the general relation is consistent with that plus the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</a>. </p><p>The Hamiltonian in classical mechanics is derived from a <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a>, which is a more fundamental quantity in the context of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. The Hamiltonian indicates how to march forward in time, but the time is different in different <a href="/wiki/Frame_of_reference" title="Frame of reference">reference frames</a>. The Lagrangian is a <a href="/wiki/Lorentz_scalar" title="Lorentz scalar">Lorentz scalar</a>, while the Hamiltonian is the time component of a <a href="/wiki/Four-vector" title="Four-vector">four-vector</a>. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics. </p><p>The Hamiltonian is a function of the position and momentum at one time, and it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a <a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre transformation</a>, and the condition that determines the classical equations of motion (the <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equations</a>) is that the <a href="/wiki/Action_(physics)" title="Action (physics)">action</a> has an extremum. </p><p>In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with <a href="/wiki/Discretization" title="Discretization">discretization</a> in time, the Legendre transform becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon H=p(t){\big (}q(t+\varepsilon )-q(t){\big )}-\varepsilon L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mi>H</mi> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>ε</mi> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon H=p(t){\big (}q(t+\varepsilon )-q(t){\big )}-\varepsilon L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/264d03a4cbb7ef8a3e5e6b85f2d52180b05a202f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.902ex; height:3.176ex;" alt="{\displaystyle \varepsilon H=p(t){\big (}q(t+\varepsilon )-q(t){\big )}-\varepsilon L}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {\partial L}{\partial {\dot {q}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {\partial L}{\partial {\dot {q}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f194505359be178a7c4e1ca3891ec0cdd2dbadff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:8.741ex; height:5.843ex;" alt="{\displaystyle p={\frac {\partial L}{\partial {\dot {q}}}},}"></span></dd></dl> <p>where the partial derivative with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399dc6b6e91a780c89824ccc26b4453b289e4387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.377ex; height:2.509ex;" alt="{\displaystyle {\dot {q}}}"></span> holds <span class="texhtml"><i>q</i>(<i>t</i> + <i>ε</i>)</span> fixed. The inverse Legendre transform is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon L=\varepsilon p{\dot {q}}-\varepsilon H,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mi>L</mi> <mo>=</mo> <mi>ε</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>ε</mi> <mi>H</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon L=\varepsilon p{\dot {q}}-\varepsilon H,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/933c8d1eb7249e784f1ea6812caeac3804101104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.029ex; height:2.509ex;" alt="{\displaystyle \varepsilon L=\varepsilon p{\dot {q}}-\varepsilon H,}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {q}}={\frac {\partial H}{\partial p}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q}}={\frac {\partial H}{\partial p}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9fd910fdf7e268174de84c82bedfa60b3d3aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.34ex; height:5.843ex;" alt="{\displaystyle {\dot {q}}={\frac {\partial H}{\partial p}},}"></span></dd></dl> <p>and the partial derivative now is with respect to <span class="texhtml mvar" style="font-style:italic;">p</span> at fixed <span class="texhtml mvar" style="font-style:italic;">q</span>. </p><p>In quantum mechanics, the state is a <a href="/wiki/Quantum_superposition" title="Quantum superposition">superposition of different states</a> with different values of <span class="texhtml mvar" style="font-style:italic;">q</span>, or different values of <span class="texhtml mvar" style="font-style:italic;">p</span>, and the quantities <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">q</span> can be interpreted as noncommuting operators. The operator <span class="texhtml mvar" style="font-style:italic;">p</span> is only definite on states that are indefinite with respect to <span class="texhtml mvar" style="font-style:italic;">q</span>. So consider two states separated in time and act with the operator corresponding to the Lagrangian: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i{\big [}p{\big (}q(t+\varepsilon )-q(t){\big )}-\varepsilon H(p,q){\big ]}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>ε</mi> <mi>H</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i{\big [}p{\big (}q(t+\varepsilon )-q(t){\big )}-\varepsilon H(p,q){\big ]}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612a1518d97cd3ba738de130868d805c22b46aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.277ex; height:3.843ex;" alt="{\displaystyle e^{i{\big [}p{\big (}q(t+\varepsilon )-q(t){\big )}-\varepsilon H(p,q){\big ]}}.}"></span></dd></dl> <p>If the multiplications implicit in this formula are reinterpreted as <i>matrix</i> multiplications, the first factor is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-ipq(t)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>p</mi> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-ipq(t)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d368d626d04992055c9db70aaba5b2afdcedb7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.265ex; height:3.176ex;" alt="{\displaystyle e^{-ipq(t)},}"></span></dd></dl> <p>and if this is also interpreted as a matrix multiplication, the sum over all states integrates over all <span class="texhtml"><i>q</i>(<i>t</i>)</span>, and so it takes the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> in <span class="texhtml"><i>q</i>(<i>t</i>)</span> to change basis to <span class="texhtml"><i>p</i>(<i>t</i>)</span>. That is the action on the Hilbert space – <em>change basis to <span class="texhtml mvar" style="font-style:italic;">p</span> at time <span class="texhtml mvar" style="font-style:italic;">t</span></em>. </p><p>Next comes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-i\varepsilon H(p,q)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ε</mi> <mi>H</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-i\varepsilon H(p,q)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eabb4e30ff29bab18d63c6c4f1c1db9ef57ba061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.354ex; height:3.176ex;" alt="{\displaystyle e^{-i\varepsilon H(p,q)},}"></span></dd></dl> <p>or <em>evolve an infinitesimal time into the future</em>. </p><p>Finally, the last factor in this interpretation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ipq(t+\varepsilon )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>p</mi> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ipq(t+\varepsilon )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc318645e8610a08c3412766f8041f01cf1130aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.031ex; height:3.176ex;" alt="{\displaystyle e^{ipq(t+\varepsilon )},}"></span></dd></dl> <p>which means <em>change basis back to <span class="texhtml mvar" style="font-style:italic;">q</span> at a later time</em>. </p><p>This is not very different from just ordinary time evolution: the <span class="texhtml mvar" style="font-style:italic;">H</span> factor contains all the dynamical information – it pushes the state forward in time. The first part and the last part are just Fourier transforms to change to a pure <span class="texhtml mvar" style="font-style:italic;">q</span> basis from an intermediate <span class="texhtml mvar" style="font-style:italic;">p</span> basis. </p><p>Another way of saying this is that since the Hamiltonian is naturally a function of <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">q</span>, exponentiating this quantity and changing basis from <span class="texhtml mvar" style="font-style:italic;">p</span> to <span class="texhtml mvar" style="font-style:italic;">q</span> at each step allows the matrix element of <span class="texhtml mvar" style="font-style:italic;">H</span> to be expressed as a simple function along each path. This function is the quantum analog of the classical action. This observation is due to <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Dirac further noted that one could square the time-evolution operator in the <span class="texhtml mvar" style="font-style:italic;">S</span> representation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\varepsilon S},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ε</mi> <mi>S</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\varepsilon S},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d21d41d68fb7b5fadf4f5531ff0212ca685916d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.356ex; height:3.009ex;" alt="{\displaystyle e^{i\varepsilon S},}"></span></dd></dl> <p>and this gives the time-evolution operator between time <span class="texhtml mvar" style="font-style:italic;">t</span> and time <span class="texhtml"><i>t</i> + 2<i>ε</i></span>. While in the <span class="texhtml mvar" style="font-style:italic;">H</span> representation the quantity that is being summed over the intermediate states is an obscure matrix element, in the <span class="texhtml mvar" style="font-style:italic;">S</span> representation it is reinterpreted as a quantity associated to the path. In the limit that one takes a large power of this operator, one reconstructs the full quantum evolution between two states, the early one with a fixed value of <span class="texhtml"><i>q</i>(0)</span> and the later one with a fixed value of <span class="texhtml"><i>q</i>(<i>t</i>)</span>. The result is a sum over paths with a phase, which is the quantum action. </p> <div class="mw-heading mw-heading2"><h2 id="Classical_limit">Classical limit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=2" title="Edit section: Classical limit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Crucially, Dirac identified the effect of the classical limit on the quantum form of the action principle: </p> <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"><p>...we see that the integrand in (11) must be of the form <span class="texhtml"><i>e</i><sup><i>iF</i>/<i>h</i></sup></span>, where <span class="texhtml mvar" style="font-style:italic;">F</span> is a function of <span class="texhtml"><i>q</i><sub><i>T</i></sub>, <i>q</i><sub>1</sub>, <i>q</i><sub>2</sub>, … <i>q</i><sub><i>m</i></sub>, <i>q</i><sub><i>t</i></sub></span>, which remains finite as <span class="texhtml mvar" style="font-style:italic;">h</span> tends to zero. Let us now picture one of the intermediate <span class="texhtml mvar" style="font-style:italic;">q</span>s, say <span class="texhtml mvar" style="font-style:italic;">q<sub>k</sub></span>, as varying continuously while the other ones are fixed. Owing to the smallness of <span class="texhtml mvar" style="font-style:italic;">h</span>, we shall then in general have <i>F</i>/<i>h</i> varying extremely rapidly. This means that <span class="texhtml"><i>e</i><sup><i>iF</i>/<i>h</i></sup></span> will vary periodically with a very high frequency about the value zero, as a result of which its integral will be practically zero. The only important part in the domain of integration of <span class="texhtml mvar" style="font-style:italic;">q<sub>k</sub></span> is thus that for which a comparatively large variation in <span class="texhtml mvar" style="font-style:italic;">q<sub>k</sub></span> produces only a very small variation in <span class="texhtml mvar" style="font-style:italic;">F</span>. This part is the neighbourhood of a point for which <span class="texhtml mvar" style="font-style:italic;">F</span> is stationary with respect to small variations in <span class="texhtml mvar" style="font-style:italic;">q<sub>k</sub></span>. We can apply this argument to each of the variables of integration ... and obtain the result that the only important part in the domain of integration is that for which <span class="texhtml mvar" style="font-style:italic;">F</span> is stationary for small variations in all intermediate <span class="texhtml mvar" style="font-style:italic;">q</span>s. ... We see that <span class="texhtml mvar" style="font-style:italic;">F</span> has for its classical analogue <span class="texhtml"><span style="position:relative; top:0.2em"><span style="font-style:italic; margin-right:0.3em;">∫</span><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>t</i></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>T</i></sub></span></span></span> <i>L dt</i></span>, which is just the action function, which classical mechanics requires to be stationary for small variations in all the intermediate <span class="texhtml mvar" style="font-style:italic;">q</span>s. This shows the way in which equation (11) goes over into classical results when <span class="texhtml mvar" style="font-style:italic;">h</span> becomes extremely small. </p><div class="templatequotecite">— <cite>Dirac (1933), p. 69</cite></div></blockquote> <p>That is, in the limit of action that is large compared to the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a> <span class="texhtml mvar" style="font-style:italic;">ħ</span> – the classical limit – the path integral is dominated by solutions that are in the neighborhood of <a href="/wiki/Stationary_point" title="Stationary point">stationary points</a> of the action. The classical path arises naturally in the classical limit. </p> <div class="mw-heading mw-heading2"><h2 id="Feynman's_interpretation"><span id="Feynman.27s_interpretation"></span>Feynman's interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=3" title="Edit section: Feynman's interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dirac's work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schrödinger equation or the <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relations</a> from this rule. This was done by Feynman. </p><p>Feynman showed that Dirac's quantum action was, for most cases of interest, simply equal to the classical action, appropriately discretized. This means that the classical action is the phase acquired by quantum evolution between two fixed endpoints. He proposed to recover all of quantum mechanics from the following postulates: </p> <ol><li>The <a href="/wiki/Probability" title="Probability">probability</a> for an event is given by the <a href="/wiki/Squared_modulus" class="mw-redirect" title="Squared modulus">squared modulus</a> of a complex number called the "probability amplitude".</li> <li>The <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitude</a> is given by adding together the contributions of all paths in configuration space.</li> <li>The contribution of a path is proportional to <span class="texhtml"><i>e</i><sup><i>iS</i>/<i>ħ</i></sup></span>, where <span class="texhtml mvar" style="font-style:italic;">S</span> is the <a href="/wiki/Action_(physics)" title="Action (physics)">action</a> given by the <a href="/wiki/Time_integral" class="mw-redirect" title="Time integral">time integral</a> of the <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> along the path.</li></ol> <p>In order to find the overall probability amplitude for a given process, then, one adds up, or <a href="/wiki/Integral" title="Integral">integrates</a>, the amplitude of the 3rd postulate over the space of <i>all</i> possible paths of the system in between the initial and final states, including those that are absurd by classical standards. In calculating the probability amplitude for a single particle to go from one space-time coordinate to another, it is correct to include paths in which the particle describes elaborate <a href="/wiki/Curlicues" class="mw-redirect" title="Curlicues">curlicues</a>, curves in which the particle shoots off into outer space and flies back again, and so forth. The <b>path integral</b> assigns to all these amplitudes <i>equal weight</i> but varying <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a>, or argument of the <a href="/wiki/Complex_number" title="Complex number">complex number</a>. Contributions from paths wildly different from the classical trajectory may be suppressed by <a href="/wiki/Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a> (see below). </p><p>Feynman showed that this formulation of quantum mechanics is equivalent to the <a href="/wiki/Quantization_(physics)" title="Quantization (physics)">canonical approach to quantum mechanics</a> when the Hamiltonian is at most quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> for the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> corresponding to the given action. </p><p>The path integral formulation of quantum field theory represents the <a href="/wiki/Transition_amplitude" class="mw-redirect" title="Transition amplitude">transition amplitude</a> (corresponding to the classical <a href="/wiki/Correlation_function" title="Correlation function">correlation function</a>) as a weighted sum of all possible histories of the system from the initial to the final state. A <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a> is a graphical representation of a <a href="/wiki/Perturbative" class="mw-redirect" title="Perturbative">perturbative</a> contribution to the transition amplitude. </p> <div class="mw-heading mw-heading2"><h2 id="Path_integral_in_quantum_mechanics">Path integral in quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=4" title="Edit section: Path integral in quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Time-slicing_derivation">Time-slicing derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=5" title="Edit section: Time-slicing derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Relation_between_Schr%C3%B6dinger%27s_equation_and_the_path_integral_formulation_of_quantum_mechanics" title="Relation between Schrödinger's equation and the path integral formulation of quantum mechanics">Relation between Schrödinger's equation and the path integral formulation of quantum mechanics</a></div> <p>One common approach to deriving the path integral formula is to divide the time interval into small pieces. Once this is done, the <a href="/wiki/Lie_product_formula" title="Lie product formula">Trotter product formula</a> tells us that the noncommutativity of the kinetic and potential energy operators can be ignored. </p><p>For a particle in a smooth potential, the path integral is approximated by <a href="/wiki/Zigzag" title="Zigzag">zigzag</a> paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position <span class="texhtml mvar" style="font-style:italic;">x<sub>a</sub></span> at time <span class="texhtml mvar" style="font-style:italic;">t<sub>a</sub></span> to <span class="texhtml mvar" style="font-style:italic;">x<sub>b</sub></span> at time <span class="texhtml mvar" style="font-style:italic;">t<sub>b</sub></span>, the time sequence </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{a}=t_{0}<t_{1}<\cdots <t_{n-1}<t_{n}<t_{n+1}=t_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{a}=t_{0}<t_{1}<\cdots <t_{n-1}<t_{n}<t_{n+1}=t_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc66411b53e57c20b745801e739b17a8d123f260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:42.294ex; height:2.343ex;" alt="{\displaystyle t_{a}=t_{0}<t_{1}<\cdots <t_{n-1}<t_{n}<t_{n+1}=t_{b}}"></span></dd></dl> <p>can be divided up into <span class="texhtml"><i>n</i> + 1</span> smaller segments <span class="texhtml"><i>t<sub>j</sub></i> − <i>t</i><sub><i>j</i> − 1</sub></span>, where <span class="texhtml"><i>j</i> = 1, ..., <i>n</i> + 1</span>, of fixed duration </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =\Delta t={\frac {t_{b}-t_{a}}{n+1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =\Delta t={\frac {t_{b}-t_{a}}{n+1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7391ab86e4a569a83ee168941cef95207afd73f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.098ex; height:5.343ex;" alt="{\displaystyle \varepsilon =\Delta t={\frac {t_{b}-t_{a}}{n+1}}.}"></span></dd></dl> <p>This process is called <i>time-slicing</i>. </p><p>An approximation for the path integral can be computed as proportional to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{-\infty }^{+\infty }\cdots \int \limits _{-\infty }^{+\infty }\exp \left({\frac {i}{\hbar }}\int _{t_{a}}^{t_{b}}L{\big (}x(t),v(t){\big )}\,dt\right)\,dx_{0}\,\cdots \,dx_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo>⋯</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>⋯</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{-\infty }^{+\infty }\cdots \int \limits _{-\infty }^{+\infty }\exp \left({\frac {i}{\hbar }}\int _{t_{a}}^{t_{b}}L{\big (}x(t),v(t){\big )}\,dt\right)\,dx_{0}\,\cdots \,dx_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec431e4e3ffa22961e53de6101441d42ac3db7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.447ex; width:52.878ex; height:9.009ex;" alt="{\displaystyle \int \limits _{-\infty }^{+\infty }\cdots \int \limits _{-\infty }^{+\infty }\exp \left({\frac {i}{\hbar }}\int _{t_{a}}^{t_{b}}L{\big (}x(t),v(t){\big )}\,dt\right)\,dx_{0}\,\cdots \,dx_{n},}"></span></dd></dl> <p>where <span class="texhtml"><i>L</i>(<i>x</i>, <i>v</i>)</span> is the Lagrangian of the one-dimensional system with position variable <span class="texhtml"><i>x</i>(<i>t</i>)</span> and velocity <span class="texhtml"><i>v</i> = <i>ẋ</i>(<i>t</i>)</span> considered (see below), and <span class="texhtml mvar" style="font-style:italic;">dx<sub>j</sub></span> corresponds to the position at the <span class="texhtml mvar" style="font-style:italic;">j</span>th time step, if the time integral is approximated by a sum of <span class="texhtml mvar" style="font-style:italic;">n</span> terms.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup> </p><p>In the limit <span class="texhtml mvar" style="font-style:italic;"><i>n</i> → ∞</span>, this becomes a <a href="/wiki/Functional_integral" class="mw-redirect" title="Functional integral">functional integral</a>, which, apart from a nonessential factor, is directly the product of the probability amplitudes <span class="texhtml"><span class="nowrap">⟨<i>x<sub>b</sub></i>, <i>t<sub>b</sub></i>|<i>x<sub>a</sub></i>, <i>t<sub>a</sub></i>⟩</span></span> (more precisely, since one must work with a continuous spectrum, the respective densities) to find the quantum mechanical particle at <span class="texhtml mvar" style="font-style:italic;">t<sub>a</sub></span> in the initial state <span class="texhtml mvar" style="font-style:italic;">x<sub>a</sub></span> and at <span class="texhtml mvar" style="font-style:italic;">t<sub>b</sub></span> in the final state <span class="texhtml mvar" style="font-style:italic;">x<sub>b</sub></span>. </p><p>Actually <span class="texhtml mvar" style="font-style:italic;">L</span> is the classical <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> of the one-dimensional system considered, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x,{\dot {x}})=T-V={\frac {1}{2}}m|{\dot {x}}|^{2}-V(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo>−</mo> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(x,{\dot {x}})=T-V={\frac {1}{2}}m|{\dot {x}}|^{2}-V(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c7332b04b925cdb293f48a48cd5e607ee0243d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.029ex; height:5.176ex;" alt="{\displaystyle L(x,{\dot {x}})=T-V={\frac {1}{2}}m|{\dot {x}}|^{2}-V(x)}"></span></dd></dl> <p>and the abovementioned "zigzagging" corresponds to the appearance of the terms </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \left({\frac {i}{\hbar }}\varepsilon \sum _{j=1}^{n+1}L\left({\tilde {x}}_{j},{\frac {x_{j}-x_{j-1}}{\varepsilon }},j\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mi>ε</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mi>ε</mi> </mfrac> </mrow> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left({\frac {i}{\hbar }}\varepsilon \sum _{j=1}^{n+1}L\left({\tilde {x}}_{j},{\frac {x_{j}-x_{j-1}}{\varepsilon }},j\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0d6855863d5d1052006afed70f08dd1964646b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:35.501ex; height:7.676ex;" alt="{\displaystyle \exp \left({\frac {i}{\hbar }}\varepsilon \sum _{j=1}^{n+1}L\left({\tilde {x}}_{j},{\frac {x_{j}-x_{j-1}}{\varepsilon }},j\right)\right)}"></span></dd></dl> <p>in the <a href="/wiki/Riemann_sum" title="Riemann sum">Riemann sum</a> approximating the time integral, which are finally integrated over <span class="texhtml"><i>x</i><sub>1</sub></span> to <span class="texhtml mvar" style="font-style:italic;">x<sub>n</sub></span> with the integration measure <span class="texhtml"><i>dx</i><sub>1</sub>...<i>dx<sub>n</sub></i></span>, <span class="texhtml mvar" style="font-style:italic;">x̃<sub>j</sub></span> is an arbitrary value of the interval corresponding to <span class="texhtml mvar" style="font-style:italic;">j</span>, e.g. its center, <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>x<sub>j</sub></i> + <i>x</i><sub><i>j</i>−1</sub></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>. </p><p>Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute. </p> <div class="mw-heading mw-heading3"><h3 id="Path_integral">Path integral</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=6" title="Edit section: Path integral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In terms of the wave function in the position representation, the path integral formula reads as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}{\mathcal {D}}\mathbf {x} \,e^{iS[\mathbf {x} ,{\dot {\mathbf {x} }}]}\psi _{0}(\mathbf {x} (t))\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Z</mi> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mspace width="thinmathspace" /> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">]</mo> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}{\mathcal {D}}\mathbf {x} \,e^{iS[\mathbf {x} ,{\dot {\mathbf {x} }}]}\psi _{0}(\mathbf {x} (t))\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63847726e284004a822be0b4d8210a8a713392c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.482ex; height:6.009ex;" alt="{\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}{\mathcal {D}}\mathbf {x} \,e^{iS[\mathbf {x} ,{\dot {\mathbf {x} }}]}\psi _{0}(\mathbf {x} (t))\,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}\mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33025fa4f5b581baa561370b3ca0733ff7eca552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}\mathbf {x} }"></span> denotes integration over all paths <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (0)=x}"> <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> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (0)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e8de1f51758515b495f6ce024b6813d0d8f771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.811ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (0)=x}"></span> and where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> is a normalization factor. Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <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></span><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}"></span> is the action, given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S[\mathbf {x} ,{\dot {\mathbf {x} }}]=\int dt\,L(\mathbf {x} (t),{\dot {\mathbf {x} }}(t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi>L</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[\mathbf {x} ,{\dot {\mathbf {x} }}]=\int dt\,L(\mathbf {x} (t),{\dot {\mathbf {x} }}(t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5011bdd160e07f4c3de4e918534275113f529f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.316ex; height:5.676ex;" alt="{\displaystyle S[\mathbf {x} ,{\dot {\mathbf {x} }}]=\int dt\,L(\mathbf {x} (t),{\dot {\mathbf {x} }}(t))}"></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Path_integral_example.webm/220px--Path_integral_example.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="315" data-durationhint="14" data-mwtitle="Path_integral_example.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Path_integral_example.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/ed/Path_integral_example.webm/Path_integral_example.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="336" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/ed/Path_integral_example.webm/Path_integral_example.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="504" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/ed/Path_integral_example.webm/Path_integral_example.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="756" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/e/ed/Path_integral_example.webm" type="video/webm; codecs="vp8"" data-width="800" data-height="1144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/ed/Path_integral_example.webm/Path_integral_example.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="100" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/ed/Path_integral_example.webm/Path_integral_example.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="168" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/ed/Path_integral_example.webm/Path_integral_example.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="252" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/ed/Path_integral_example.webm/Path_integral_example.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="252" data-height="360" /></video></span><figcaption>The diagram shows the contribution to the path integral of a free particle for a set of paths, eventually drawing a <a href="/wiki/Cornu_Spiral" class="mw-redirect" title="Cornu Spiral">Cornu Spiral</a>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Free_particle">Free particle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=7" title="Edit section: Free particle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The path integral representation gives the quantum amplitude to go from point <span class="texhtml mvar" style="font-style:italic;">x</span> to point <span class="texhtml mvar" style="font-style:italic;">y</span> as an integral over all paths. For a free-particle action (for simplicity let <span class="texhtml"><i>m</i> = 1</span>, <span class="texhtml"><i>ħ</i> = 1</span>) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\int {\frac {{\dot {x}}^{2}}{2}}\,\mathrm {d} t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int {\frac {{\dot {x}}^{2}}{2}}\,\mathrm {d} t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9fb837ae9c23c36cfef1e4ec12e710f6b94340d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.565ex; height:6.176ex;" alt="{\displaystyle S=\int {\frac {{\dot {x}}^{2}}{2}}\,\mathrm {d} t,}"></span></dd></dl> <p>the integral can be evaluated explicitly. </p><p>To do this, it is convenient to start without the factor <span class="texhtml mvar" style="font-style:italic;">i</span> in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions. The amplitude (or Kernel) reads: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x-y;T)=\int _{x(0)=x}^{x(T)=y}\exp \left(-\int _{0}^{T}{\frac {{\dot {x}}^{2}}{2}}\,\mathrm {d} t\right)\,{\mathcal {D}}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mrow> </msubsup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x-y;T)=\int _{x(0)=x}^{x(T)=y}\exp \left(-\int _{0}^{T}{\frac {{\dot {x}}^{2}}{2}}\,\mathrm {d} t\right)\,{\mathcal {D}}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38578413b0f2f2753766bfc26b09934031bbb5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.829ex; height:6.676ex;" alt="{\displaystyle K(x-y;T)=\int _{x(0)=x}^{x(T)=y}\exp \left(-\int _{0}^{T}{\frac {{\dot {x}}^{2}}{2}}\,\mathrm {d} t\right)\,{\mathcal {D}}x.}"></span></dd></dl> <p>Splitting the integral into time slices: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x-y;T)=\int _{x(0)=x}^{x(T)=y}\prod _{t}\exp \left(-{\tfrac {1}{2}}\left({\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}\right)^{2}\varepsilon \right)\,{\mathcal {D}}x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mrow> </msubsup> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </munder> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>ε</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ε</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x-y;T)=\int _{x(0)=x}^{x(T)=y}\prod _{t}\exp \left(-{\tfrac {1}{2}}\left({\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}\right)^{2}\varepsilon \right)\,{\mathcal {D}}x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3f654e2576e32a0faa3449d36d75adb88450ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.854ex; height:7.509ex;" alt="{\displaystyle K(x-y;T)=\int _{x(0)=x}^{x(T)=y}\prod _{t}\exp \left(-{\tfrac {1}{2}}\left({\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}\right)^{2}\varepsilon \right)\,{\mathcal {D}}x,}"></span></dd></dl> <p>where the <span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">D</span> is interpreted as a finite collection of integrations at each integer multiple of <span class="texhtml mvar" style="font-style:italic;">ε</span>. Each factor in the product is a Gaussian as a function of <span class="texhtml"><i>x</i>(<i>t</i> + <i>ε</i>)</span> centered at <span class="texhtml"><i>x</i>(<i>t</i>)</span> with variance <span class="texhtml mvar" style="font-style:italic;">ε</span>. The multiple integrals are a repeated <a href="/wiki/Convolution" title="Convolution">convolution</a> of this Gaussian <span class="texhtml mvar" style="font-style:italic;">G<sub>ε</sub></span> with copies of itself at adjacent times: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x-y;T)=G_{\varepsilon }*G_{\varepsilon }*\cdots *G_{\varepsilon },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo>∗</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo>∗</mo> <mo>⋯</mo> <mo>∗</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x-y;T)=G_{\varepsilon }*G_{\varepsilon }*\cdots *G_{\varepsilon },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c8c68d1e9618f8116c22c29f9acfa9153905f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.399ex; height:2.843ex;" alt="{\displaystyle K(x-y;T)=G_{\varepsilon }*G_{\varepsilon }*\cdots *G_{\varepsilon },}"></span></dd></dl> <p>where the number of convolutions is <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>T</i></span><span class="sr-only">/</span><span class="den"><i>ε</i></span></span>⁠</span></span>. The result is easy to evaluate by taking the Fourier transform of both sides, so that the convolutions become multiplications: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {K}}(p;T)={\tilde {G}}_{\varepsilon }(p)^{T/\varepsilon }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>K</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ε</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {K}}(p;T)={\tilde {G}}_{\varepsilon }(p)^{T/\varepsilon }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aac132a0745408f024632036b6773989720595ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.241ex; height:3.343ex;" alt="{\displaystyle {\tilde {K}}(p;T)={\tilde {G}}_{\varepsilon }(p)^{T/\varepsilon }.}"></span></dd></dl> <p>The Fourier transform of the Gaussian <span class="texhtml mvar" style="font-style:italic;">G</span> is another Gaussian of reciprocal variance: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {G}}_{\varepsilon }(p)=e^{-{\frac {\varepsilon p^{2}}{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ε</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {G}}_{\varepsilon }(p)=e^{-{\frac {\varepsilon p^{2}}{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf988d3d6bd952118eb59c5e78af4d8368af7a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.074ex; height:4.843ex;" alt="{\displaystyle {\tilde {G}}_{\varepsilon }(p)=e^{-{\frac {\varepsilon p^{2}}{2}}},}"></span></dd></dl> <p>and the result is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {K}}(p;T)=e^{-{\frac {Tp^{2}}{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>K</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {K}}(p;T)=e^{-{\frac {Tp^{2}}{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3ccb7fd1cb9adec985603774502f27838e3c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.302ex; height:4.843ex;" alt="{\displaystyle {\tilde {K}}(p;T)=e^{-{\frac {Tp^{2}}{2}}}.}"></span></dd></dl> <p>The Fourier transform gives <span class="texhtml mvar" style="font-style:italic;">K</span>, and it is a Gaussian again with reciprocal variance: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x-y;T)\propto e^{-{\frac {(x-y)^{2}}{2T}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x-y;T)\propto e^{-{\frac {(x-y)^{2}}{2T}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493c1573d059984180078e6989a3989646e81f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.351ex; height:4.843ex;" alt="{\displaystyle K(x-y;T)\propto e^{-{\frac {(x-y)^{2}}{2T}}}.}"></span></dd></dl> <p>The proportionality constant is not really determined by the time-slicing approach, only the ratio of values for different endpoint choices is determined. The proportionality constant should be chosen to ensure that between each two time slices the time evolution is quantum-mechanically unitary, but a more illuminating way to fix the normalization is to consider the path integral as a description of a stochastic process. </p><p>The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path. The probability is the product over each segment of the probability of selecting that segment, so that each segment is probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>, which can be interpreted as the first historical evaluation of a statistical path integral. </p><p>The probability interpretation gives a natural normalization choice. The path integral should be defined so that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int K(x-y;T)\,dy=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int K(x-y;T)\,dy=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9323946bd94dc3ddabe233f052832a85cc49190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.118ex; height:5.676ex;" alt="{\displaystyle \int K(x-y;T)\,dy=1.}"></span></dd></dl> <p>This condition normalizes the Gaussian and produces a kernel that obeys the diffusion equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}K(x;T)={\frac {\nabla ^{2}}{2}}K.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mi>K</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}K(x;T)={\frac {\nabla ^{2}}{2}}K.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e43adca3cbe8b03da6ad8b1ccf431b920e4580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.404ex; height:5.843ex;" alt="{\displaystyle {\frac {d}{dt}}K(x;T)={\frac {\nabla ^{2}}{2}}K.}"></span></dd></dl> <p>For oscillatory path integrals, ones with an <span class="texhtml mvar" style="font-style:italic;">i</span> in the numerator, the time slicing produces convolved Gaussians, just as before. Now, however, the convolution product is marginally singular, since it requires careful limits to evaluate the oscillating integrals. To make the factors well defined, the easiest way is to add a small imaginary part to the time increment <span class="texhtml mvar" style="font-style:italic;">ε</span>. This is closely related to <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a>. Then the same convolution argument as before gives the propagation kernel: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x-y;T)\propto e^{\frac {i(x-y)^{2}}{2T}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </mfrac> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x-y;T)\propto e^{\frac {i(x-y)^{2}}{2T}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eabc7cbf68857f6f5f3e03c803fada0f0b2ab76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.533ex; height:4.843ex;" alt="{\displaystyle K(x-y;T)\propto e^{\frac {i(x-y)^{2}}{2T}},}"></span></dd></dl> <p>which, with the same normalization as before (not the sum-squares normalization – this function has a divergent norm), obeys a free Schrödinger equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}K(x;T)=i{\frac {\nabla ^{2}}{2}}K.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mi>K</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}K(x;T)=i{\frac {\nabla ^{2}}{2}}K.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c8580e44630077fc72dbcfdf95a93366603644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.207ex; height:5.843ex;" alt="{\displaystyle {\frac {d}{dt}}K(x;T)=i{\frac {\nabla ^{2}}{2}}K.}"></span></dd></dl> <p>This means that any superposition of <span class="texhtml mvar" style="font-style:italic;">K</span>s will also obey the same equation, by linearity. Defining </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{t}(y)=\int \psi _{0}(x)K(x-y;t)\,dx=\int \psi _{0}(x)\int _{x(0)=x}^{x(t)=y}e^{iS}\,{\mathcal {D}}x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>;</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{t}(y)=\int \psi _{0}(x)K(x-y;t)\,dx=\int \psi _{0}(x)\int _{x(0)=x}^{x(t)=y}e^{iS}\,{\mathcal {D}}x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6074bf6efee143c85cf1ce3486c5fc2226e7637e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.56ex; height:6.676ex;" alt="{\displaystyle \psi _{t}(y)=\int \psi _{0}(x)K(x-y;t)\,dx=\int \psi _{0}(x)\int _{x(0)=x}^{x(t)=y}e^{iS}\,{\mathcal {D}}x,}"></span></dd></dl> <p>then <span class="texhtml mvar" style="font-style:italic;">ψ<sub>t</sub></span> obeys the free Schrödinger equation just as <span class="texhtml mvar" style="font-style:italic;">K</span> does: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\frac {\partial }{\partial t}}\psi _{t}=-{\frac {\nabla ^{2}}{2}}\psi _{t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i{\frac {\partial }{\partial t}}\psi _{t}=-{\frac {\nabla ^{2}}{2}}\psi _{t}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7ebed051995d7cd520bbd75d5bd06c2892dd0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.854ex; height:5.843ex;" alt="{\displaystyle i{\frac {\partial }{\partial t}}\psi _{t}=-{\frac {\nabla ^{2}}{2}}\psi _{t}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Simple_harmonic_oscillator">Simple harmonic oscillator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=8" title="Edit section: Simple harmonic oscillator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Propagator#Basic_examples:_propagator_of_free_particle_and_harmonic_oscillator" title="Propagator">Propagator § Basic examples: propagator of free particle and harmonic oscillator</a>, and <a href="/wiki/Mehler_kernel" title="Mehler kernel">Mehler kernel</a></div> <p>The Lagrangian for the simple harmonic oscillator is<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af505239ef251e25561abfed6571b112e81993b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.854ex; height:3.509ex;" alt="{\displaystyle {\mathcal {L}}={\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}.}"></span></dd></dl> <p>Write its trajectory <span class="texhtml"><i>x</i>(<i>t</i>)</span> as the classical trajectory plus some perturbation, <span class="texhtml"><i>x</i>(<i>t</i>) = <i>x</i><sub>c</sub>(<i>t</i>) + <i>δx</i>(<i>t</i>)</span> and the action as <span class="texhtml"><i>S</i> = <i>S</i><sub>c</sub> + <i>δS</i></span>. The classical trajectory can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{c}}(t)=x_{i}{\frac {\sin \omega (t_{f}-t)}{\sin \omega (t_{f}-t_{i})}}+x_{f}{\frac {\sin \omega (t-t_{i})}{\sin \omega (t_{f}-t_{i})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{c}}(t)=x_{i}{\frac {\sin \omega (t_{f}-t)}{\sin \omega (t_{f}-t_{i})}}+x_{f}{\frac {\sin \omega (t-t_{i})}{\sin \omega (t_{f}-t_{i})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8dab27acfaf09dab338e4c264fcabcf394749a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.701ex; height:6.509ex;" alt="{\displaystyle x_{\text{c}}(t)=x_{i}{\frac {\sin \omega (t_{f}-t)}{\sin \omega (t_{f}-t_{i})}}+x_{f}{\frac {\sin \omega (t-t_{i})}{\sin \omega (t_{f}-t_{i})}}.}"></span></dd></dl> <p>This trajectory yields the classical action </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S_{\text{c}}&=\int _{t_{i}}^{t_{f}}{\mathcal {L}}\,dt=\int _{t_{i}}^{t_{f}}\left({\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}\right)\,dt\\[6pt]&={\frac {1}{2}}m\omega \left({\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega (t_{f}-t_{i})-2x_{i}x_{f}}{\sin \omega (t_{f}-t_{i})}}\right)~.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mi>ω</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{\text{c}}&=\int _{t_{i}}^{t_{f}}{\mathcal {L}}\,dt=\int _{t_{i}}^{t_{f}}\left({\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}\right)\,dt\\[6pt]&={\frac {1}{2}}m\omega \left({\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega (t_{f}-t_{i})-2x_{i}x_{f}}{\sin \omega (t_{f}-t_{i})}}\right)~.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045fbb738649f09823736e127c36f5082f118e84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:49.935ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}S_{\text{c}}&=\int _{t_{i}}^{t_{f}}{\mathcal {L}}\,dt=\int _{t_{i}}^{t_{f}}\left({\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}\right)\,dt\\[6pt]&={\frac {1}{2}}m\omega \left({\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega (t_{f}-t_{i})-2x_{i}x_{f}}{\sin \omega (t_{f}-t_{i})}}\right)~.\end{aligned}}}"></span></dd></dl> <p>Next, expand the deviation from the classical path as a Fourier series, and calculate the contribution to the action <span class="texhtml mvar" style="font-style:italic;">δS</span>, which gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=S_{\text{c}}+\sum _{n=1}^{\infty }{\tfrac {1}{2}}a_{n}^{2}{\frac {m}{2}}\left({\frac {(n\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=S_{\text{c}}+\sum _{n=1}^{\infty }{\tfrac {1}{2}}a_{n}^{2}{\frac {m}{2}}\left({\frac {(n\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbbd1c86b39b0779d4e07ba8a4320ed4c6ce69e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.55ex; height:7.509ex;" alt="{\displaystyle S=S_{\text{c}}+\sum _{n=1}^{\infty }{\tfrac {1}{2}}a_{n}^{2}{\frac {m}{2}}\left({\frac {(n\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right).}"></span></dd></dl> <p>This means that the propagator is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}K(x_{f},t_{f};x_{i},t_{i})&=Qe^{\frac {iS_{\text{c}}}{\hbar }}\prod _{j=1}^{\infty }{\frac {j\pi }{\sqrt {2}}}\int da_{j}\exp {\left({\frac {i}{2\hbar }}a_{j}^{2}{\frac {m}{2}}\left({\frac {(j\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right)\right)}\\[6pt]&=e^{\frac {iS_{\text{c}}}{\hbar }}Q\prod _{j=1}^{\infty }\left(1-\left({\frac {\omega (t_{f}-t_{i})}{j\pi }}\right)^{2}\right)^{-{\frac {1}{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>Q</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>j</mi> <mi>π</mi> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>∫</mo> <mi>d</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mn>2</mn> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </msup> <mi>Q</mi> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>j</mi> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K(x_{f},t_{f};x_{i},t_{i})&=Qe^{\frac {iS_{\text{c}}}{\hbar }}\prod _{j=1}^{\infty }{\frac {j\pi }{\sqrt {2}}}\int da_{j}\exp {\left({\frac {i}{2\hbar }}a_{j}^{2}{\frac {m}{2}}\left({\frac {(j\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right)\right)}\\[6pt]&=e^{\frac {iS_{\text{c}}}{\hbar }}Q\prod _{j=1}^{\infty }\left(1-\left({\frac {\omega (t_{f}-t_{i})}{j\pi }}\right)^{2}\right)^{-{\frac {1}{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4182856aaa82017672eff0d426a858a535e27d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.211ex; margin-bottom: -0.293ex; width:80.473ex; height:18.176ex;" alt="{\displaystyle {\begin{aligned}K(x_{f},t_{f};x_{i},t_{i})&=Qe^{\frac {iS_{\text{c}}}{\hbar }}\prod _{j=1}^{\infty }{\frac {j\pi }{\sqrt {2}}}\int da_{j}\exp {\left({\frac {i}{2\hbar }}a_{j}^{2}{\frac {m}{2}}\left({\frac {(j\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right)\right)}\\[6pt]&=e^{\frac {iS_{\text{c}}}{\hbar }}Q\prod _{j=1}^{\infty }\left(1-\left({\frac {\omega (t_{f}-t_{i})}{j\pi }}\right)^{2}\right)^{-{\frac {1}{2}}}\end{aligned}}}"></span></dd></dl> <p>for some normalization </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q={\sqrt {\frac {m}{2\pi i\hbar (t_{f}-t_{i})}}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>m</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\sqrt {\frac {m}{2\pi i\hbar (t_{f}-t_{i})}}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcef5b4fde23d8ddbe7d93c4202e576cea16c256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:22.192ex; height:7.509ex;" alt="{\displaystyle Q={\sqrt {\frac {m}{2\pi i\hbar (t_{f}-t_{i})}}}~.}"></span></dd></dl> <p>Using the infinite-product representation of the <a href="/wiki/Sinc_function" title="Sinc function">sinc function</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{j=1}^{\infty }\left(1-{\frac {x^{2}}{j^{2}}}\right)={\frac {\sin \pi x}{\pi x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>π</mi> <mi>x</mi> </mrow> <mrow> <mi>π</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{j=1}^{\infty }\left(1-{\frac {x^{2}}{j^{2}}}\right)={\frac {\sin \pi x}{\pi x}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f224b21501534e5f2f34bf4abae9e9b9a4d390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.486ex; height:7.176ex;" alt="{\displaystyle \prod _{j=1}^{\infty }\left(1-{\frac {x^{2}}{j^{2}}}\right)={\frac {\sin \pi x}{\pi x}},}"></span></dd></dl> <p>the propagator can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=Qe^{\frac {iS_{\text{c}}}{\hbar }}{\sqrt {\frac {\omega (t_{f}-t_{i})}{\sin \omega (t_{f}-t_{i})}}}=e^{\frac {iS_{c}}{\hbar }}{\sqrt {\frac {m\omega }{2\pi i\hbar \sin \omega (t_{f}-t_{i})}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Q</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>m</mi> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=Qe^{\frac {iS_{\text{c}}}{\hbar }}{\sqrt {\frac {\omega (t_{f}-t_{i})}{\sin \omega (t_{f}-t_{i})}}}=e^{\frac {iS_{c}}{\hbar }}{\sqrt {\frac {m\omega }{2\pi i\hbar \sin \omega (t_{f}-t_{i})}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e636c22db862102eebb96605ddf07c6dea45d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:69.385ex; height:8.009ex;" alt="{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=Qe^{\frac {iS_{\text{c}}}{\hbar }}{\sqrt {\frac {\omega (t_{f}-t_{i})}{\sin \omega (t_{f}-t_{i})}}}=e^{\frac {iS_{c}}{\hbar }}{\sqrt {\frac {m\omega }{2\pi i\hbar \sin \omega (t_{f}-t_{i})}}}.}"></span></dd></dl> <p>Let <span class="texhtml"><i>T</i> = <i>t<sub>f</sub></i> − <i>t<sub>i</sub></i></span>. One may write this propagator in terms of energy eigenstates as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}K(x_{f},t_{f};x_{i},t_{i})&=\left({\frac {m\omega }{2\pi i\hbar \sin \omega T}}\right)^{\frac {1}{2}}\exp {\left({\frac {i}{\hbar }}{\tfrac {1}{2}}m\omega {\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega T-2x_{i}x_{f}}{\sin \omega T}}\right)}\\[6pt]&=\sum _{n=0}^{\infty }\exp {\left(-{\frac {iE_{n}T}{\hbar }}\right)}\psi _{n}(x_{f})\psi _{n}(x_{i})^{*}~.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>T</mi> <mo>−</mo> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>T</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K(x_{f},t_{f};x_{i},t_{i})&=\left({\frac {m\omega }{2\pi i\hbar \sin \omega T}}\right)^{\frac {1}{2}}\exp {\left({\frac {i}{\hbar }}{\tfrac {1}{2}}m\omega {\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega T-2x_{i}x_{f}}{\sin \omega T}}\right)}\\[6pt]&=\sum _{n=0}^{\infty }\exp {\left(-{\frac {iE_{n}T}{\hbar }}\right)}\psi _{n}(x_{f})\psi _{n}(x_{i})^{*}~.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f63c46394c38005de72cb1af4bec51b97aec548e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:76.896ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}K(x_{f},t_{f};x_{i},t_{i})&=\left({\frac {m\omega }{2\pi i\hbar \sin \omega T}}\right)^{\frac {1}{2}}\exp {\left({\frac {i}{\hbar }}{\tfrac {1}{2}}m\omega {\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega T-2x_{i}x_{f}}{\sin \omega T}}\right)}\\[6pt]&=\sum _{n=0}^{\infty }\exp {\left(-{\frac {iE_{n}T}{\hbar }}\right)}\psi _{n}(x_{f})\psi _{n}(x_{i})^{*}~.\end{aligned}}}"></span></dd></dl> <p>Using the identities <span class="texhtml"><i>i</i> sin <i>ωT</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>e</i><sup><i>iωT</i></sup> (1 − <i>e</i><sup>−2<i>iωT</i></sup>)</span> and <span class="texhtml">cos <i>ωT</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>e</i><sup><i>iωT</i></sup> (1 + <i>e</i><sup>−2<i>iωT</i></sup>)</span>, this amounts to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega T}{2}}\left(1-e^{-2i\omega T}\right)^{-{\frac {1}{2}}}\exp {\left(-{\frac {m\omega }{2\hbar }}\left(\left(x_{i}^{2}+x_{f}^{2}\right){\frac {1+e^{-2i\omega T}}{1-e^{-2i\omega T}}}-{\frac {4x_{i}x_{f}e^{-i\omega T}}{1-e^{-2i\omega T}}}\right)\right)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>ω</mi> </mrow> <mrow> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>T</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mi>ω</mi> <mi>T</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mi>ω</mi> <mi>T</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mi>ω</mi> <mi>T</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>T</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mi>ω</mi> <mi>T</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega T}{2}}\left(1-e^{-2i\omega T}\right)^{-{\frac {1}{2}}}\exp {\left(-{\frac {m\omega }{2\hbar }}\left(\left(x_{i}^{2}+x_{f}^{2}\right){\frac {1+e^{-2i\omega T}}{1-e^{-2i\omega T}}}-{\frac {4x_{i}x_{f}e^{-i\omega T}}{1-e^{-2i\omega T}}}\right)\right)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/102427365f73601f7911a15e5c8c712cddd954c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:103.334ex; height:7.509ex;" alt="{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega T}{2}}\left(1-e^{-2i\omega T}\right)^{-{\frac {1}{2}}}\exp {\left(-{\frac {m\omega }{2\hbar }}\left(\left(x_{i}^{2}+x_{f}^{2}\right){\frac {1+e^{-2i\omega T}}{1-e^{-2i\omega T}}}-{\frac {4x_{i}x_{f}e^{-i\omega T}}{1-e^{-2i\omega T}}}\right)\right)}.}"></span></dd></dl> <p>One may absorb all terms after the first <span class="texhtml"><i>e</i><sup>−<i>iωT</i>/2</sup></span> into <span class="texhtml"><i>R</i>(<i>T</i>)</span>, thereby obtaining </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega T}{2}}\cdot R(T).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>ω</mi> </mrow> <mrow> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>T</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>⋅</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega T}{2}}\cdot R(T).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0231e8ba479c20a38882eb90dcaf0096519fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.075ex; height:6.009ex;" alt="{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega T}{2}}\cdot R(T).}"></span></dd></dl> <p>One may finally expand <span class="texhtml"><i>R</i>(<i>T</i>)</span> in powers of <span class="texhtml"><i>e</i><sup>−<i>iωT</i></sup></span>: All terms in this expansion get multiplied by the <span class="texhtml"><i>e</i><sup>−<i>iωT</i>/2</sup></span> factor in the front, yielding terms of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\frac {-i\omega T}{2}}e^{-in\omega T}=e^{-i\omega T\left({\frac {1}{2}}+n\right)}\quad {\text{for }}n=0,1,2,\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>T</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>ω</mi> <mi>T</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>T</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\frac {-i\omega T}{2}}e^{-in\omega T}=e^{-i\omega T\left({\frac {1}{2}}+n\right)}\quad {\text{for }}n=0,1,2,\ldots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8800550034bc52cdd80fdaaae4e641c014b95ade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.446ex; height:4.176ex;" alt="{\displaystyle e^{\frac {-i\omega T}{2}}e^{-in\omega T}=e^{-i\omega T\left({\frac {1}{2}}+n\right)}\quad {\text{for }}n=0,1,2,\ldots .}"></span></dd></dl> <p>Comparison to the above eigenstate expansion yields the standard energy spectrum for the simple harmonic oscillator, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}=\left(n+{\tfrac {1}{2}}\right)\hbar \omega ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}=\left(n+{\tfrac {1}{2}}\right)\hbar \omega ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a67ef15c25fe3750bc13569f005e8b2388a246d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.422ex; height:3.509ex;" alt="{\displaystyle E_{n}=\left(n+{\tfrac {1}{2}}\right)\hbar \omega ~.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Coulomb_potential">Coulomb potential</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=9" title="Edit section: Coulomb potential"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Feynman's time-sliced approximation does not, however, exist for the most important quantum-mechanical path integrals of atoms, due to the singularity of the <a href="/wiki/Coulomb_potential" class="mw-redirect" title="Coulomb potential">Coulomb potential</a> <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>e</i><sup>2</sup></span><span class="sr-only">/</span><span class="den"><i>r</i></span></span>⁠</span></span> at the origin. Only after replacing the time <span class="texhtml mvar" style="font-style:italic;">t</span> by another path-dependent pseudo-time parameter </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\int {\frac {dt}{r(t)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\int {\frac {dt}{r(t)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd49ea87f7a7f93569c19d5dfcab81ee5eecfe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.303ex; height:6.176ex;" alt="{\displaystyle s=\int {\frac {dt}{r(t)}}}"></span></dd></dl> <p>the singularity is removed and a time-sliced approximation exists, which is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by <a href="/wiki/%C4%B0smail_Hakk%C4%B1_Duru" title="İsmail Hakkı Duru">İsmail Hakkı Duru</a> and <a href="/wiki/Hagen_Kleinert" title="Hagen Kleinert">Hagen Kleinert</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The combination of a path-dependent time transformation and a coordinate transformation is an important tool to solve many path integrals and is called generically the <a href="/wiki/Duru%E2%80%93Kleinert_transformation" title="Duru–Kleinert transformation">Duru–Kleinert transformation</a>. </p> <div class="mw-heading mw-heading3"><h3 id="The_Schrödinger_equation"><span id="The_Schr.C3.B6dinger_equation"></span>The Schrödinger equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=10" title="Edit section: The Schrödinger equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Relation_between_Schr%C3%B6dinger%27s_equation_and_the_path_integral_formulation_of_quantum_mechanics" title="Relation between Schrödinger's equation and the path integral formulation of quantum mechanics">Relation between Schrödinger's equation and the path integral formulation of quantum mechanics</a></div> <p>The path integral reproduces the Schrödinger equation for the initial and final state even when a potential is present. This is easiest to see by taking a path-integral over infinitesimally separated times. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (y;t+\varepsilon )=\int _{-\infty }^{\infty }\psi (x;t)\int _{x(t)=x}^{x(t+\varepsilon )=y}e^{i\int _{t}^{t+\varepsilon }{\bigl (}{\frac {1}{2}}{\dot {x}}^{2}-V(x){\bigr )}dt}Dx(t)\,dx\qquad (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>;</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>+</mo> <mi>ε</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mi>d</mi> <mi>t</mi> </mrow> </msup> <mi>D</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (y;t+\varepsilon )=\int _{-\infty }^{\infty }\psi (x;t)\int _{x(t)=x}^{x(t+\varepsilon )=y}e^{i\int _{t}^{t+\varepsilon }{\bigl (}{\frac {1}{2}}{\dot {x}}^{2}-V(x){\bigr )}dt}Dx(t)\,dx\qquad (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88aef6037796cb5a34144b114fa77f287b13428d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:69.103ex; height:6.676ex;" alt="{\displaystyle \psi (y;t+\varepsilon )=\int _{-\infty }^{\infty }\psi (x;t)\int _{x(t)=x}^{x(t+\varepsilon )=y}e^{i\int _{t}^{t+\varepsilon }{\bigl (}{\frac {1}{2}}{\dot {x}}^{2}-V(x){\bigr )}dt}Dx(t)\,dx\qquad (1)}"></span></dd></dl> <p>Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of <span class="texhtml mvar" style="font-style:italic;">ẋ</span>, the path integral has most weight for <span class="texhtml mvar" style="font-style:italic;">y</span> close to <span class="texhtml mvar" style="font-style:italic;">x</span>. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. (This separation of the kinetic and potential energy terms in the exponent is essentially the <a href="/wiki/Lie_product_formula" title="Lie product formula">Trotter product formula</a>.) The exponential of the action is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-i\varepsilon V(x)}e^{i{\frac {{\dot {x}}^{2}}{2}}\varepsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ε</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mi>ε</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-i\varepsilon V(x)}e^{i{\frac {{\dot {x}}^{2}}{2}}\varepsilon }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0bbb2a67794e00226a656c2031f1e78554830ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.477ex; height:4.509ex;" alt="{\displaystyle e^{-i\varepsilon V(x)}e^{i{\frac {{\dot {x}}^{2}}{2}}\varepsilon }}"></span></dd></dl> <p>The first term rotates the phase of <span class="texhtml"><i>ψ</i>(<i>x</i>)</span> locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to <span class="texhtml mvar" style="font-style:italic;">i</span> times a diffusion process. To lowest order in <span class="texhtml mvar" style="font-style:italic;">ε</span> they are additive; in any case one has with (1): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (y;t+\varepsilon )\approx \int \psi (x;t)e^{-i\varepsilon V(x)}e^{\frac {i(x-y)^{2}}{2\varepsilon }}\,dx\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>;</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mo>∫</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ε</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>ε</mi> </mrow> </mfrac> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (y;t+\varepsilon )\approx \int \psi (x;t)e^{-i\varepsilon V(x)}e^{\frac {i(x-y)^{2}}{2\varepsilon }}\,dx\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b561b8044049c91d0fef8d840548e78ec2e16f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.775ex; height:6.343ex;" alt="{\displaystyle \psi (y;t+\varepsilon )\approx \int \psi (x;t)e^{-i\varepsilon V(x)}e^{\frac {i(x-y)^{2}}{2\varepsilon }}\,dx\,.}"></span></dd></dl> <p>As mentioned, the spread in <span class="texhtml mvar" style="font-style:italic;">ψ</span> is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase that slowly varies from point to point from the potential: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \psi }{\partial t}}=i\cdot \left({\tfrac {1}{2}}\nabla ^{2}-V(x)\right)\psi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>i</mi> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \psi }{\partial t}}=i\cdot \left({\tfrac {1}{2}}\nabla ^{2}-V(x)\right)\psi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d96c015c4233b89f3279e11b712efefe68af35e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.079ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \psi }{\partial t}}=i\cdot \left({\tfrac {1}{2}}\nabla ^{2}-V(x)\right)\psi \,}"></span></dd></dl> <p>and this is the Schrödinger equation. The normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment. </p> <div class="mw-heading mw-heading3"><h3 id="Equations_of_motion">Equations of motion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=11" title="Edit section: Equations of motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the states obey the Schrödinger equation, the path integral must reproduce the Heisenberg equations of motion for the averages of <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">ẋ</span> variables, but it is instructive to see this directly. The direct approach shows that the expectation values calculated from the path integral reproduce the usual ones of quantum mechanics. </p><p>Start by considering the path integral with some fixed initial state </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \psi _{0}(x)\int _{x(0)=x}e^{iS(x,{\dot {x}})}\,Dx\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>D</mi> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \psi _{0}(x)\int _{x(0)=x}e^{iS(x,{\dot {x}})}\,Dx\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c7b92e4e1623d083c4372de223e8a01e6cfe34c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.441ex; height:6.009ex;" alt="{\displaystyle \int \psi _{0}(x)\int _{x(0)=x}e^{iS(x,{\dot {x}})}\,Dx\,}"></span></dd></dl> <p>Now <span class="texhtml mvar" style="font-style:italic;"><i>x</i>(<i>t</i>)</span> at each separate time is a separate integration variable. So it is legitimate to change variables in the integral by shifting: <span class="texhtml"><i>x</i>(<i>t</i>) = <i>u</i>(<i>t</i>) + <i>ε</i>(<i>t</i>)</span> where <span class="texhtml"><i>ε</i>(<i>t</i>)</span> is a different shift at each time but <span class="texhtml"><i>ε</i>(0) = <i>ε</i>(<i>T</i>) = 0</span>, since the endpoints are not integrated: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}e^{iS(u+\varepsilon ,{\dot {u}}+{\dot {\varepsilon }})}\,Du\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>ε</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>D</mi> <mi>u</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}e^{iS(u+\varepsilon ,{\dot {u}}+{\dot {\varepsilon }})}\,Du\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49348010ca1e3cce2ffd5505c661cbe5de83078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.779ex; height:6.009ex;" alt="{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}e^{iS(u+\varepsilon ,{\dot {u}}+{\dot {\varepsilon }})}\,Du\,}"></span></dd></dl> <p>The change in the integral from the shift is, to first infinitesimal order in <span class="texhtml mvar" style="font-style:italic;">ε</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}\left(\int {\frac {\partial S}{\partial u}}\varepsilon +{\frac {\partial S}{\partial {\dot {u}}}}{\dot {\varepsilon }}\,dt\right)e^{iS}\,Du\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mi>ε</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>D</mi> <mi>u</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}\left(\int {\frac {\partial S}{\partial u}}\varepsilon +{\frac {\partial S}{\partial {\dot {u}}}}{\dot {\varepsilon }}\,dt\right)e^{iS}\,Du\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21031da9d3c96e070170fe590cd434e9ee7ba980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.236ex; height:6.343ex;" alt="{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}\left(\int {\frac {\partial S}{\partial u}}\varepsilon +{\frac {\partial S}{\partial {\dot {u}}}}{\dot {\varepsilon }}\,dt\right)e^{iS}\,Du\,}"></span></dd></dl> <p>which, integrating by parts in <span class="texhtml mvar" style="font-style:italic;">t</span>, gives: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}-\left(\int \left({\frac {d}{dt}}{\frac {\partial S}{\partial {\dot {u}}}}-{\frac {\partial S}{\partial u}}\right)\varepsilon (t)\,dt\right)e^{iS}\,Du\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </msub> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>∫</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>D</mi> <mi>u</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}-\left(\int \left({\frac {d}{dt}}{\frac {\partial S}{\partial {\dot {u}}}}-{\frac {\partial S}{\partial u}}\right)\varepsilon (t)\,dt\right)e^{iS}\,Du\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48d352044a014d2ca0b39001299cc2590f6d4797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:54.424ex; height:6.343ex;" alt="{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}-\left(\int \left({\frac {d}{dt}}{\frac {\partial S}{\partial {\dot {u}}}}-{\frac {\partial S}{\partial u}}\right)\varepsilon (t)\,dt\right)e^{iS}\,Du\,}"></span></dd></dl> <p>But this was just a shift of integration variables, which doesn't change the value of the integral for any choice of <span class="texhtml mvar" style="font-style:italic;"><i>ε</i>(<i>t</i>)</span>. The conclusion is that this first order variation is zero for an arbitrary initial state and at any arbitrary point in time: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle \psi _{0}\left|{\frac {\delta S}{\delta x}}(t)\right|\psi _{0}\right\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>S</mi> </mrow> <mrow> <mi>δ</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle \psi _{0}\left|{\frac {\delta S}{\delta x}}(t)\right|\psi _{0}\right\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3deb73b105bbf0afd7c88762e5b44d91834eaa38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.983ex; height:6.176ex;" alt="{\displaystyle \left\langle \psi _{0}\left|{\frac {\delta S}{\delta x}}(t)\right|\psi _{0}\right\rangle =0}"></span></dd></dl> <p>this is the Heisenberg equation of motion. </p><p>If the action contains terms that multiply <span class="texhtml mvar" style="font-style:italic;">ẋ</span> and <span class="texhtml mvar" style="font-style:italic;">x</span>, at the same moment in time, the manipulations above are only heuristic, because the multiplication rules for these quantities is just as noncommuting in the path integral as it is in the operator formalism. </p> <div class="mw-heading mw-heading3"><h3 id="Stationary-phase_approximation">Stationary-phase approximation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=12" title="Edit section: Stationary-phase approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the variation in the action exceeds <span class="texhtml mvar" style="font-style:italic;">ħ</span> by many orders of magnitude, we typically have destructive interference other than in the vicinity of those trajectories satisfying the <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</a>, which is now reinterpreted as the condition for constructive interference. This can be shown using the method of stationary phase applied to the propagator. As <span class="texhtml mvar" style="font-style:italic;">ħ</span> decreases, the exponential in the integral oscillates rapidly in the complex domain for any change in the action. Thus, in the limit that <span class="texhtml mvar" style="font-style:italic;">ħ</span> goes to zero, only points where the classical action does not vary contribute to the propagator. </p> <div class="mw-heading mw-heading3"><h3 id="Canonical_commutation_relations">Canonical commutation relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=13" title="Edit section: Canonical commutation relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The formulation of the path integral does not make it clear at first sight that the quantities <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">p</span> do not commute. In the path integral, these are just integration variables and they have no obvious ordering. Feynman discovered that the non-commutativity is still present.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>To see this, consider the simplest path integral, the brownian walk. This is not yet quantum mechanics, so in the path-integral the action is not multiplied by <span class="texhtml mvar" style="font-style:italic;">i</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\int \left({\frac {dx}{dt}}\right)^{2}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>∫</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int \left({\frac {dx}{dt}}\right)^{2}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf7cef442e5f87a67c185ff90262fe15d68e635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.478ex; height:6.509ex;" alt="{\displaystyle S=\int \left({\frac {dx}{dt}}\right)^{2}\,dt}"></span></dd></dl> <p>The quantity <span class="texhtml mvar" style="font-style:italic;"><i>x</i>(<i>t</i>)</span> is fluctuating, and the derivative is defined as the limit of a discrete difference. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dx}{dt}}={\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dt}}={\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86c75fe6e962a2b610fb1a7f2cf24914ea559d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.038ex; height:5.843ex;" alt="{\displaystyle {\frac {dx}{dt}}={\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}"></span></dd></dl> <p>The distance that a random walk moves is proportional to <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>t</i></span></span></span>, so that: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t+\varepsilon )-x(t)\approx {\sqrt {\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ε</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t+\varepsilon )-x(t)\approx {\sqrt {\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256a6a50c22d21c2a65e876ed25a7a2230fa48fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.839ex; height:3.009ex;" alt="{\displaystyle x(t+\varepsilon )-x(t)\approx {\sqrt {\varepsilon }}}"></span></dd></dl> <p>This shows that the random walk is not differentiable, since the ratio that defines the derivative diverges with probability one. </p><p>The quantity <span class="texhtml mvar" style="font-style:italic;">xẋ</span> is ambiguous, with two possible meanings: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1]=x{\frac {dx}{dt}}=x(t){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1]=x{\frac {dx}{dt}}=x(t){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8f251860d68e9e7f988d44fd900c0a91e48e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.901ex; height:5.843ex;" alt="{\displaystyle [1]=x{\frac {dx}{dt}}=x(t){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [2]=x{\frac {dx}{dt}}=x(t+\varepsilon ){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [2]=x{\frac {dx}{dt}}=x(t+\varepsilon ){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/299c595d865a660a326c75e3c627e2077126c765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.825ex; height:5.843ex;" alt="{\displaystyle [2]=x{\frac {dx}{dt}}=x(t+\varepsilon ){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}"></span></dd></dl> <p>In elementary calculus, the two are only different by an amount that goes to 0 as <span class="texhtml mvar" style="font-style:italic;">ε</span> goes to 0. But in this case, the difference between the two is not 0: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [2]-[1]={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}\approx {\frac {\varepsilon }{\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo>−</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>ε</mi> </mfrac> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [2]-[1]={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}\approx {\frac {\varepsilon }{\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c987304e52bfa1dadac2aab1c402035017a524c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.611ex; height:6.509ex;" alt="{\displaystyle [2]-[1]={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}\approx {\frac {\varepsilon }{\varepsilon }}}"></span></dd></dl> <p>Let </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2571d78c6aeff6c1801bbf532ccdfb122dca9bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.768ex; height:6.509ex;" alt="{\displaystyle f(t)={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}}"></span></dd></dl> <p>Then <span class="texhtml"><i>f</i>(<i>t</i>)</span> is a rapidly fluctuating statistical quantity, whose average value is 1, i.e. a normalized "Gaussian process". The fluctuations of such a quantity can be described by a statistical Lagrangian </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}=(f(t)-1)^{2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=(f(t)-1)^{2}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9c0546fc3bf0a7796a025b7e6e26134e4f5ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.53ex; height:3.176ex;" alt="{\displaystyle {\mathcal {L}}=(f(t)-1)^{2}\,,}"></span></dd></dl> <p>and the equations of motion for <span class="texhtml mvar" style="font-style:italic;">f</span> derived from extremizing the action <span class="texhtml mvar" style="font-style:italic;">S</span> corresponding to <span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">L</span> just set it equal to 1. In physics, such a quantity is "equal to 1 as an operator identity". In mathematics, it "weakly converges to 1". In either case, it is 1 in any expectation value, or when averaged over any interval, or for all practical purpose. </p><p>Defining the time order to <i>be</i> the operator order: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,{\dot {x}}]=x{\frac {dx}{dt}}-{\frac {dx}{dt}}x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,{\dot {x}}]=x{\frac {dx}{dt}}-{\frac {dx}{dt}}x=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a27062611afdbf93bdf30d31f8a003633c25c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.61ex; height:5.509ex;" alt="{\displaystyle [x,{\dot {x}}]=x{\frac {dx}{dt}}-{\frac {dx}{dt}}x=1}"></span></dd></dl> <p>This is called the <a href="/wiki/It%C5%8D_lemma" class="mw-redirect" title="Itō lemma">Itō lemma</a> in <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">stochastic calculus</a>, and the (euclideanized) canonical commutation relations in physics. </p><p>For a general statistical action, a similar argument shows that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[x,{\frac {\partial S}{\partial {\dot {x}}}}\right]=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[x,{\frac {\partial S}{\partial {\dot {x}}}}\right]=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c068c9f75efe46dc8da5b8154c2c6b524f81baff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.733ex; height:6.176ex;" alt="{\displaystyle \left[x,{\frac {\partial S}{\partial {\dot {x}}}}\right]=1}"></span></dd></dl> <p>and in quantum mechanics, the extra imaginary unit in the action converts this to the canonical commutation relation, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,p]=i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,p]=i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68322d6a3131d29b89c3b525d5380b219a220821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.728ex; height:2.843ex;" alt="{\displaystyle [x,p]=i}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Particle_in_curved_space">Particle in curved space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=14" title="Edit section: Particle in curved space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a particle in curved space the kinetic term depends on the position, and the above time slicing cannot be applied, this being a manifestation of the notorious <a href="/w/index.php?title=Operator_ordering_problem&action=edit&redlink=1" class="new" title="Operator ordering problem (page does not exist)">operator ordering problem</a> in Schrödinger quantum mechanics. One may, however, solve this problem by transforming the time-sliced flat-space path integral to curved space using a multivalued coordinate transformation (<a href="/w/index.php?title=Nonholonomic_mapping&action=edit&redlink=1" class="new" title="Nonholonomic mapping (page does not exist)">nonholonomic mapping</a> explained <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/~kleinert/b5/psfiles/pthic10.pdf">here</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Measure-theoretic_factors">Measure-theoretic factors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=15" title="Edit section: Measure-theoretic factors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sometimes (e.g. a particle moving in curved space) we also have measure-theoretic factors in the functional integral: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \mu [x]e^{iS[x]}\,{\mathcal {D}}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>μ</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \mu [x]e^{iS[x]}\,{\mathcal {D}}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/751ff6cb46202a7a93d1d0735da68ec9883b1f94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.56ex; height:5.676ex;" alt="{\displaystyle \int \mu [x]e^{iS[x]}\,{\mathcal {D}}x.}"></span></dd></dl> <p>This factor is needed to restore unitarity. </p><p>For instance, if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\int \left({\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x)\right)\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int \left({\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x)\right)\,dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebe8910a82658939b6e55525577a0f73f1f056c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.029ex; height:5.676ex;" alt="{\displaystyle S=\int \left({\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x)\right)\,dt,}"></span></dd></dl> <p>then it means that each spatial slice is multiplied by the measure <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>g</i></span></span></span>. This measure cannot be expressed as a functional multiplying the <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">D</span><i>x</i></span> measure because they belong to entirely different classes. </p> <div class="mw-heading mw-heading3"><h3 id="Expectation_values_and_matrix_elements">Expectation values and matrix elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=16" title="Edit section: Expectation values and matrix elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Matrix elements of the kind <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x_{f}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t-t')}F({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t')}|x_{i}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x_{f}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t-t')}F({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t')}|x_{i}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac6db587024451bf2f30ce1ccb0ffd83ed22e6af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.539ex; height:4.176ex;" alt="{\displaystyle \langle x_{f}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t-t')}F({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t')}|x_{i}\rangle }"></span> take the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{x(0)=x_{i}}^{x(t)=x_{f}}{\mathcal {D}}[x]F(x(t'))e^{{\frac {i}{\hbar }}\int dtL(x(t),{\dot {x}}(t))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo>∫</mo> <mi>d</mi> <mi>t</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{x(0)=x_{i}}^{x(t)=x_{f}}{\mathcal {D}}[x]F(x(t'))e^{{\frac {i}{\hbar }}\int dtL(x(t),{\dot {x}}(t))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d451d6c7235252420f0e3fa8adb5960e61ec43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.263ex; height:6.676ex;" alt="{\displaystyle \int _{x(0)=x_{i}}^{x(t)=x_{f}}{\mathcal {D}}[x]F(x(t'))e^{{\frac {i}{\hbar }}\int dtL(x(t),{\dot {x}}(t))}}"></span>.</dd></dl> <p>This generalizes to multiple operators, for example </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x_{f}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t-t_{1})}F_{1}({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{1}-t_{2})}F_{2}({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{2})}|x_{i}\rangle =\int _{x(0)=x_{i}}^{x(t)=x_{f}}{\mathcal {D}}[x]F_{1}(x(t_{1}))F_{2}(x(t_{2}))e^{{\frac {i}{\hbar }}\int dtL(x(t),{\dot {x}}(t))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo>∫</mo> <mi>d</mi> <mi>t</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x_{f}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t-t_{1})}F_{1}({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{1}-t_{2})}F_{2}({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{2})}|x_{i}\rangle =\int _{x(0)=x_{i}}^{x(t)=x_{f}}{\mathcal {D}}[x]F_{1}(x(t_{1}))F_{2}(x(t_{2}))e^{{\frac {i}{\hbar }}\int dtL(x(t),{\dot {x}}(t))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b8e00db6a2ef338d85b7542a7e8abe3ac8cf16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:98.612ex; height:6.676ex;" alt="{\displaystyle \langle x_{f}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t-t_{1})}F_{1}({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{1}-t_{2})}F_{2}({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{2})}|x_{i}\rangle =\int _{x(0)=x_{i}}^{x(t)=x_{f}}{\mathcal {D}}[x]F_{1}(x(t_{1}))F_{2}(x(t_{2}))e^{{\frac {i}{\hbar }}\int dtL(x(t),{\dot {x}}(t))}}"></span>,</dd></dl> <p>and to the general vacuum expectation value (in the large time limit) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle F\rangle ={\frac {\int {\mathcal {D}}[\phi ]F(\phi )e^{{\frac {i}{\hbar }}S[\phi ]}}{\int {\mathcal {D}}[\phi ]e^{{\frac {i}{\hbar }}S[\phi ]}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>F</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mi>S</mi> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> </mrow> </msup> </mrow> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mi>S</mi> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle F\rangle ={\frac {\int {\mathcal {D}}[\phi ]F(\phi )e^{{\frac {i}{\hbar }}S[\phi ]}}{\int {\mathcal {D}}[\phi ]e^{{\frac {i}{\hbar }}S[\phi ]}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5cde79b9d454bfe5f02829e19906c80cdcaa392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:24.553ex; height:9.176ex;" alt="{\displaystyle \langle F\rangle ={\frac {\int {\mathcal {D}}[\phi ]F(\phi )e^{{\frac {i}{\hbar }}S[\phi ]}}{\int {\mathcal {D}}[\phi ]e^{{\frac {i}{\hbar }}S[\phi ]}}}}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Euclidean_path_integrals">Euclidean path integrals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=17" title="Edit section: Euclidean path integrals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is very common in path integrals to perform a <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a> from real to imaginary times. In the setting of quantum field theory, the Wick rotation changes the geometry of space-time from Lorentzian to Euclidean; as a result, Wick-rotated path integrals are often called Euclidean path integrals. </p> <div class="mw-heading mw-heading3"><h3 id="Wick_rotation_and_the_Feynman–Kac_formula"><span id="Wick_rotation_and_the_Feynman.E2.80.93Kac_formula"></span>Wick rotation and the Feynman–Kac formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=18" title="Edit section: Wick rotation and the Feynman–Kac formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we replace <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -it}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -it}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829b87bc6da6307747ee22a6efc715157f21b573" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.45ex; height:2.343ex;" alt="{\displaystyle -it}"></span>, the time-evolution operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-it{\hat {H}}/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>t</mi> <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>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-it{\hat {H}}/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/176cc8ba3c6734526faaead85cb4003e107444f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.96ex; height:3.176ex;" alt="{\displaystyle e^{-it{\hat {H}}/\hbar }}"></span> is replaced by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-t{\hat {H}}/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> <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>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-t{\hat {H}}/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625411e43e274eb530b976882ae3e16332c30d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.393ex; height:3.176ex;" alt="{\displaystyle e^{-t{\hat {H}}/\hbar }}"></span>. (This change is known as a <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a>.) If we repeat the derivation of the path-integral formula in this setting, we obtain<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}e^{-S_{\mathrm {Euclidean} }(\mathbf {x} ,{\dot {\mathbf {x} }})/\hbar }\psi _{0}(\mathbf {x} (t))\,{\mathcal {D}}\mathbf {x} \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Z</mi> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}e^{-S_{\mathrm {Euclidean} }(\mathbf {x} ,{\dot {\mathbf {x} }})/\hbar }\psi _{0}(\mathbf {x} (t))\,{\mathcal {D}}\mathbf {x} \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b04cb2c0a23d3694406e5e7b6e6fddd1376576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.145ex; height:6.009ex;" alt="{\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}e^{-S_{\mathrm {Euclidean} }(\mathbf {x} ,{\dot {\mathbf {x} }})/\hbar }\psi _{0}(\mathbf {x} (t))\,{\mathcal {D}}\mathbf {x} \,}"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\mathrm {Euclidean} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\mathrm {Euclidean} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9983ab7b93675f3a9e3d918b0768566babc737d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.715ex; height:2.509ex;" alt="{\displaystyle S_{\mathrm {Euclidean} }}"></span> is the Euclidean action, given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\mathrm {Euclidean} }(\mathbf {x} ,{\dot {\mathbf {x} }})=\int \left[{\frac {m}{2}}|{\dot {\mathbf {x} }}(t)|^{2}+V(\mathbf {x} (t))\right]\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</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> <mo>+</mo> <mi>V</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> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\mathrm {Euclidean} }(\mathbf {x} ,{\dot {\mathbf {x} }})=\int \left[{\frac {m}{2}}|{\dot {\mathbf {x} }}(t)|^{2}+V(\mathbf {x} (t))\right]\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa1dbe41729019d2dd493ceaec5fa7c73a2553c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.865ex; height:5.676ex;" alt="{\displaystyle S_{\mathrm {Euclidean} }(\mathbf {x} ,{\dot {\mathbf {x} }})=\int \left[{\frac {m}{2}}|{\dot {\mathbf {x} }}(t)|^{2}+V(\mathbf {x} (t))\right]\,dt}"></span>.</dd></dl> <p>Note the sign change between this and the normal action, where the potential energy term is negative. (The term <i>Euclidean</i> is from the context of quantum field theory, where the change from real to imaginary time changes the space-time geometry from Lorentzian to Euclidean.) </p><p>Now, the contribution of the kinetic energy to the path integral is as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{Z}}\int _{\mathbf {x} (0)=x}f(\mathbf {x} )e^{-{\frac {m}{2}}\int |{\dot {\mathbf {x} }}|^{2}dt}\,{\mathcal {D}}\mathbf {x} \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Z</mi> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{Z}}\int _{\mathbf {x} (0)=x}f(\mathbf {x} )e^{-{\frac {m}{2}}\int |{\dot {\mathbf {x} }}|^{2}dt}\,{\mathcal {D}}\mathbf {x} \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4d6e54b162dadf9965680e1efd29d96fab8afa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.185ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{Z}}\int _{\mathbf {x} (0)=x}f(\mathbf {x} )e^{-{\frac {m}{2}}\int |{\dot {\mathbf {x} }}|^{2}dt}\,{\mathcal {D}}\mathbf {x} \,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {x} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e41ea95e6949bf4cef6426116364ba87e0fdcd60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.499ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {x} )}"></span> includes all the remaining dependence of the integrand on the path. This integral has a rigorous mathematical interpretation as integration against the <a href="/wiki/Wiener_process" title="Wiener process">Wiener measure</a>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535a0e2c3feb2a293ed4deedde308b7c1b74d6f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \mu _{x}}"></span>. The Wiener measure, constructed by <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> gives a rigorous foundation to <a href="/wiki/Brownian_motion#Einstein.27s_theory" title="Brownian motion">Einstein's mathematical model of Brownian motion</a>. The subscript <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> indicates that the measure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535a0e2c3feb2a293ed4deedde308b7c1b74d6f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \mu _{x}}"></span> is supported on paths <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (0)=x}"> <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> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (0)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e8de1f51758515b495f6ce024b6813d0d8f771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.811ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (0)=x}"></span>. </p><p>We then have a rigorous version of the Feynman path integral, known as the <a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a>:<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x,t)=\int e^{-\int V(\mathbf {x} (t))\,dt/\hbar }\,\psi _{0}(\mathbf {x} (t))\,d\mu _{x}(\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>∫</mo> <mi>V</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> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x,t)=\int e^{-\int V(\mathbf {x} (t))\,dt/\hbar }\,\psi _{0}(\mathbf {x} (t))\,d\mu _{x}(\mathbf {x} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f990b22f876c66b18e5e17282b60485a5bcf1e3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.797ex; height:5.676ex;" alt="{\displaystyle \psi (x,t)=\int e^{-\int V(\mathbf {x} (t))\,dt/\hbar }\,\psi _{0}(\mathbf {x} (t))\,d\mu _{x}(\mathbf {x} )}"></span>,</dd></dl> <p>where now <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710fe5889ab03540462927622f3da789e03271d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.526ex; height:2.843ex;" alt="{\displaystyle \psi (x,t)}"></span> satisfies the Wick-rotated version of the Schrödinger equation, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar {\frac {\partial }{\partial t}}\psi (x,t)=-{\hat {H}}\psi (x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar {\frac {\partial }{\partial t}}\psi (x,t)=-{\hat {H}}\psi (x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2667eb3e404028885df487bd970b83160357b3de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.322ex; height:5.509ex;" alt="{\displaystyle \hbar {\frac {\partial }{\partial t}}\psi (x,t)=-{\hat {H}}\psi (x,t)}"></span>.</dd></dl> <p>Although the Wick-rotated Schrödinger equation does not have a direct physical meaning, interesting properties of the Schrödinger operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb06de5217295d7fbdbf68fb9c5309a513fc99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}"></span> can be extracted by studying it.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Much of the study of quantum field theories from the path-integral perspective, in both the mathematics and physics literatures, is done in the Euclidean setting, that is, after a Wick rotation. In particular, there are various results showing that if a Euclidean field theory with suitable properties can be constructed, one can then undo the Wick rotation to recover the physical, Lorentzian theory.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> On the other hand, it is much more difficult to give a meaning to path integrals (even Euclidean path integrals) in quantum field theory than in quantum mechanics.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Path_integral_and_the_partition_function">Path integral and the partition function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=19" title="Edit section: Path integral and the partition function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The path integral is just the generalization of the integral above to all quantum mechanical problems— </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\int e^{\frac {i{\mathcal {S}}[\mathbf {x} ]}{\hbar }}\,{\mathcal {D}}\mathbf {x} \quad {\text{where }}{\mathcal {S}}[\mathbf {x} ]=\int _{0}^{t_{f}}L[\mathbf {x} (t),{\dot {\mathbf {x} }}(t)]\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">]</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>where </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> </msubsup> <mi>L</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\int e^{\frac {i{\mathcal {S}}[\mathbf {x} ]}{\hbar }}\,{\mathcal {D}}\mathbf {x} \quad {\text{where }}{\mathcal {S}}[\mathbf {x} ]=\int _{0}^{t_{f}}L[\mathbf {x} (t),{\dot {\mathbf {x} }}(t)]\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4081ea7aee92bd8923f1300d76a3ab2e2f0dfd72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.165ex; height:6.176ex;" alt="{\displaystyle Z=\int e^{\frac {i{\mathcal {S}}[\mathbf {x} ]}{\hbar }}\,{\mathcal {D}}\mathbf {x} \quad {\text{where }}{\mathcal {S}}[\mathbf {x} ]=\int _{0}^{t_{f}}L[\mathbf {x} (t),{\dot {\mathbf {x} }}(t)]\,dt}"></span></dd></dl> <p>is the <a href="/wiki/Action_(physics)" title="Action (physics)">action</a> of the classical problem in which one investigates the path starting at time <span class="texhtml"><i>t</i> = 0</span> and ending at time <span class="texhtml"><i>t</i> = t<sub>f</sub></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}\mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33025fa4f5b581baa561370b3ca0733ff7eca552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}\mathbf {x} }"></span> denotes the integration measure over all paths. In the classical limit, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}[\mathbf {x} ]\gg \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">]</mo> <mo>≫</mo> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}[\mathbf {x} ]\gg \hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63becd74c8345f71aa038ebbe0c1dd5a173d8c37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.117ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}[\mathbf {x} ]\gg \hbar }"></span>, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.<sup id="cite_ref-Feynman-Hibbs_21-0" class="reference"><a href="#cite_note-Feynman-Hibbs-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>The connection with <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> follows. Considering only paths that begin and end in the same configuration, perform the <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a> <span class="texhtml"><i>it</i> = <i>ħβ</i></span>, i.e., make time imaginary, and integrate over all possible beginning-ending configurations. The Wick-rotated path integral—described in the previous subsection, with the ordinary action replaced by its "Euclidean" counterpart—now resembles the <a href="/wiki/Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a> of statistical mechanics defined in a <a href="/wiki/Canonical_ensemble" title="Canonical ensemble">canonical ensemble</a> with inverse temperature proportional to imaginary time, <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>T</i></span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">i<i>k</i><sub>B</sub><i>t</i></span><span class="sr-only">/</span><span class="den"><i>ħ</i></span></span>⁠</span></span>. Strictly speaking, though, this is the partition function for a <a href="/wiki/Statistical_field_theory" title="Statistical field theory">statistical field theory</a>. </p><p>Clearly, such a deep analogy between quantum mechanics and statistical mechanics cannot be dependent on the formulation. In the canonical formulation, one sees that the unitary evolution operator of a state is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha ;t\rangle =e^{-{\frac {iHt}{\hbar }}}|\alpha ;0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo>;</mo> <mi>t</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>H</mi> <mi>t</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo>;</mo> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha ;t\rangle =e^{-{\frac {iHt}{\hbar }}}|\alpha ;0\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23225f7f1ed784d5c26e324a73c40f6d4814d77b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.804ex; height:4.176ex;" alt="{\displaystyle |\alpha ;t\rangle =e^{-{\frac {iHt}{\hbar }}}|\alpha ;0\rangle }"></span></dd></dl> <p>where the state <span class="texhtml mvar" style="font-style:italic;">α</span> is evolved from time <span class="texhtml"><i>t</i> = 0</span>. If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in (imaginary) time <span class="texhtml mvar" style="font-style:italic;">iβ</span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\operatorname {Tr} \left[e^{-H\beta }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>Tr</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>H</mi> <mi>β</mi> </mrow> </msup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\operatorname {Tr} \left[e^{-H\beta }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c392558011f101562fa6ffd0c6d9c53a22f830ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.303ex; height:3.343ex;" alt="{\displaystyle Z=\operatorname {Tr} \left[e^{-H\beta }\right]}"></span></dd></dl> <p>which is precisely the partition function of statistical mechanics for the same system at the temperature quoted earlier. One aspect of this equivalence was also known to <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> who remarked that the equation named after him looked like the <a href="/wiki/Diffusion_equation" title="Diffusion equation">diffusion equation</a> after Wick rotation. Note, however, that the Euclidean path integral is actually in the form of a <i>classical</i> statistical mechanics model. </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_field_theory">Quantum field theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=20" title="Edit section: Quantum field theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert-image" typeof="mw:File/Frameless"><a href="/wiki/Feynman_diagram" title="Feynman diagram"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/211px-Feynmann_Diagram_Gluon_Radiation.svg.png" decoding="async" width="211" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/317px-Feynmann_Diagram_Gluon_Radiation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/422px-Feynmann_Diagram_Gluon_Radiation.svg.png 2x" data-file-width="400" data-file-height="250" /></a></span><div class="sidebar-caption"><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></div></td></tr><tr><td class="sidebar-above"> <a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Field_(physics)" title="Field (physics)">Field theory</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">Weak force</a></li> <li><a href="/wiki/Strong_force" class="mw-redirect" title="Strong force">Strong force</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">Symmetries</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">C-symmetry</a></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">P-symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a></li> <li><a href="/wiki/Lorentz_symmetry" class="mw-redirect" title="Lorentz symmetry">Lorentz symmetry</a></li> <li><a href="/wiki/Poincar%C3%A9_symmetry" class="mw-redirect" title="Poincaré symmetry">Poincaré symmetry</a></li> <li><a href="/wiki/Gauge_symmetry_(mathematics)" title="Gauge symmetry (mathematics)">Gauge symmetry</a></li> <li><a href="/wiki/Explicit_symmetry_breaking" title="Explicit symmetry breaking">Explicit symmetry breaking</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Noether_charge" class="mw-redirect" title="Noether charge">Noether charge</a></li> <li><a href="/wiki/Topological_charge" class="mw-redirect" title="Topological charge">Topological charge</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Tools</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomaly</a></li> <li><a href="/wiki/Background_field_method" title="Background field method">Background field method</a></li> <li><a href="/wiki/BRST_quantization" title="BRST quantization">BRST quantization</a></li> <li><a href="/wiki/Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">Correlation function</a></li> <li><a href="/wiki/Crossing_(physics)" title="Crossing (physics)">Crossing</a></li> <li><a href="/wiki/Effective_action" title="Effective action">Effective action</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective field theory</a></li> <li><a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">Expectation value</a></li> <li><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></li> <li><a href="/wiki/Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li> <li><a href="/wiki/LSZ_reduction_formula" title="LSZ reduction formula">LSZ reduction formula</a></li> <li><a href="/wiki/Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">Partition function</a></li> <li><a href="/wiki/Path_Integral_Formulation" class="mw-redirect" title="Path Integral Formulation">Path Integral Formulation</a></li> <li><a href="/wiki/Propagator_(Quantum_Theory)" class="mw-redirect" title="Propagator (Quantum Theory)">Propagator</a></li> <li><a href="/wiki/Quantization_(physics)" title="Quantization (physics)">Quantization</a></li> <li><a href="/wiki/Regularization_(physics)" title="Regularization (physics)">Regularization</a></li> <li><a href="/wiki/Renormalization" title="Renormalization">Renormalization</a></li> <li><a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">Vacuum state</a></li> <li><a href="/wiki/Wick%27s_theorem" title="Wick's theorem">Wick's theorem</a></li> <li><a href="/wiki/Wightman_axioms" title="Wightman axioms">Wightman axioms</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a></li> <li><a href="/wiki/Proca_action" title="Proca action">Proca equations</a></li> <li><a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></li> <li><a href="/wiki/Bargmann%E2%80%93Wigner_equations" title="Bargmann–Wigner equations">Bargmann–Wigner equations</a></li> <li><a href="/wiki/Schwinger-Dyson_equation" class="mw-redirect" title="Schwinger-Dyson equation">Schwinger-Dyson equation</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group equation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Incomplete theories</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></li> <li><a href="/wiki/Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Stephen_Louis_Adler" class="mw-redirect" title="Stephen Louis Adler">Adler</a></li> <li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li> <li><a href="/wiki/Alexey_Andreevich_Anselm" class="mw-redirect" title="Alexey Andreevich Anselm">Anselm</a></li> <li><a href="/wiki/Valentine_Bargmann" title="Valentine Bargmann">Bargmann</a></li> <li><a href="/wiki/Carlo_Becchi" title="Carlo Becchi">Becchi</a></li> <li><a href="/wiki/Alexander_Belavin" title="Alexander Belavin">Belavin</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Felix_Berezin" title="Felix Berezin">Berezin</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/James_Bjorken" title="James Bjorken">Bjorken</a></li> <li><a href="/wiki/Konrad_Bleuler" title="Konrad Bleuler">Bleuer</a></li> <li><a href="/wiki/Nikolay_Bogolyubov" title="Nikolay Bogolyubov">Bogoliubov</a></li> <li><a href="/wiki/Stanley_Brodsky" title="Stanley Brodsky">Brodsky</a></li> <li><a href="/wiki/Robert_Brout" title="Robert Brout">Brout</a></li> <li><a href="/wiki/Detlev_Buchholz" title="Detlev Buchholz">Buchholz</a></li> <li><a href="/wiki/Freddy_Cachazo" title="Freddy Cachazo">Cachazo</a></li> <li><a href="/wiki/Curtis_Callan" title="Curtis Callan">Callan</a></li> <li><a href="/wiki/John_Cardy" title="John Cardy">Cardy</a></li> <li><a href="/wiki/Sidney_Coleman" title="Sidney Coleman">Coleman</a></li> <li><a href="/wiki/Alain_Connes" title="Alain Connes">Connes</a></li> <li><a href="/wiki/Roger_Dashen" title="Roger Dashen">Dashen</a></li> <li><a href="/wiki/Bryce_DeWitt" title="Bryce DeWitt">DeWitt</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Sergio_Doplicher" title="Sergio Doplicher">Doplicher</a></li> <li><a href="/wiki/Freeman_Dyson" title="Freeman Dyson">Dyson</a></li> <li><a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">Englert</a></li> <li><a href="/wiki/Ludvig_Faddeev" title="Ludvig Faddeev">Faddeev</a></li> <li><a href="/wiki/Victor_Sergeevich_Fadin" class="mw-redirect" title="Victor Sergeevich Fadin">Fadin</a></li> <li><a href="/wiki/Pierre_Fayet" title="Pierre Fayet">Fayet</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Markus_Fierz" title="Markus Fierz">Fierz</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Paul_Frampton" title="Paul Frampton">Frampton</a></li> <li><a href="/wiki/Harald_Fritzsch" title="Harald Fritzsch">Fritzsch</a></li> <li><a href="/wiki/J%C3%BCrg_Fr%C3%B6hlich" title="Jürg Fröhlich">Fröhlich</a></li> <li><a href="/wiki/Klaus_Fredenhagen" title="Klaus Fredenhagen">Fredenhagen</a></li> <li><a href="/wiki/Wendell_H._Furry" title="Wendell H. Furry">Furry</a></li> <li><a href="/wiki/Sheldon_Glashow" title="Sheldon Glashow">Glashow</a></li> <li><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann</a></li> <li><a href="/wiki/James_Glimm" title="James Glimm">Glimm</a></li> <li><a href="/wiki/Jeffrey_Goldstone" title="Jeffrey Goldstone">Goldstone</a></li> <li><a href="/wiki/Vladimir_Gribov" title="Vladimir Gribov">Gribov</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Suraj_N._Gupta" title="Suraj N. Gupta">Gupta</a></li> <li><a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Guralnik</a></li> <li><a href="/wiki/Rudolf_Haag" title="Rudolf Haag">Haag</a></li> <li><a href="/wiki/C._R._Hagen" title="C. R. Hagen">Hagen</a></li> <li><a href="/wiki/Moo-Young_Han" title="Moo-Young Han">Han</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/Klaus_Hepp" title="Klaus Hepp">Hepp</a></li> <li><a href="/wiki/Peter_Higgs" title="Peter Higgs">Higgs</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li> <li><a href="/wiki/Dmitri_Ivanenko" title="Dmitri Ivanenko">Ivanenko</a></li> <li><a href="/wiki/Roman_Jackiw" title="Roman Jackiw">Jackiw</a></li> <li><a href="/wiki/Arthur_Jaffe" title="Arthur Jaffe">Jaffe</a></li> <li><a href="/wiki/Giovanni_Jona-Lasinio" title="Giovanni Jona-Lasinio">Jona-Lasinio</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Res_Jost" title="Res Jost">Jost</a></li> <li><a href="/wiki/Gunnar_K%C3%A4ll%C3%A9n" title="Gunnar Källén">Källén</a></li> <li><a href="/wiki/Henry_Way_Kendall" title="Henry Way Kendall">Kendall</a></li> <li><a href="/wiki/Toichiro_Kinoshita" title="Toichiro Kinoshita">Kinoshita</a></li> <li><a href="/wiki/Kim_Jihn-eui" title="Kim Jihn-eui">Kim</a></li> <li><a href="/wiki/Igor_R._Klebanov" class="mw-redirect" title="Igor R. Klebanov">Klebanov</a></li> <li><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li> <li><a href="/wiki/Dirk_Kreimer" title="Dirk Kreimer">Kreimer</a></li> <li><a href="/wiki/Eduard_A._Kuraev" title="Eduard A. Kuraev">Kuraev</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Benjamin_W._Lee" title="Benjamin W. Lee">Lee</a></li> <li><a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Lee</a></li> <li><a href="/wiki/Harry_Lehmann" title="Harry Lehmann">Lehmann</a></li> <li><a href="/wiki/Heinrich_Leutwyler" title="Heinrich Leutwyler">Leutwyler</a></li> <li><a href="/wiki/Lev_Lipatov" title="Lev Lipatov">Lipatov</a></li> <li><a href="/wiki/Jan_%C5%81opusza%C5%84ski_(physicist)" title="Jan Łopuszański (physicist)">Łopuszański</a></li> <li><a href="/wiki/Francis_E._Low" title="Francis E. Low">Low</a></li> <li><a href="/wiki/Gerhart_L%C3%BCders" title="Gerhart Lüders">Lüders</a></li> <li><a href="/wiki/Luciano_Maiani" title="Luciano Maiani">Maiani</a></li> <li><a href="/wiki/Ettore_Majorana" title="Ettore Majorana">Majorana</a></li> <li><a href="/wiki/Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li> <li><a href="/wiki/Takeo_Matsubara" title="Takeo Matsubara">Matsubara</a></li> <li><a href="/wiki/Alexander_Arkadyevich_Migdal" title="Alexander Arkadyevich Migdal">Migdal</a></li> <li><a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Mills</a></li> <li><a href="/wiki/Christian_M%C3%B8ller" title="Christian Møller">Møller</a></li> <li><a href="/wiki/Mark_Naimark" title="Mark Naimark">Naimark</a></li> <li><a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Nambu</a></li> <li><a href="/wiki/Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li> <li><a href="/wiki/Kazuhiko_Nishijima" title="Kazuhiko Nishijima">Nishijima</a></li> <li><a href="/wiki/Reinhard_Oehme" title="Reinhard Oehme">Oehme</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Hugh_Osborn" title="Hugh Osborn">Osborn</a></li> <li><a href="/wiki/Konrad_Osterwalder" title="Konrad Osterwalder">Osterwalder</a></li> <li><a href="/wiki/Giorgio_Parisi" title="Giorgio Parisi">Parisi</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Roberto_Peccei" title="Roberto Peccei">Peccei</a></li> <li><a href="/wiki/Michael_Peskin" title="Michael Peskin">Peskin</a></li> <li><a href="/wiki/Jan_Christoph_Plefka" title="Jan Christoph Plefka">Plefka</a></li> <li><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li> <li><a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li> <li><a href="/wiki/Isaak_Pomeranchuk" title="Isaak Pomeranchuk">Pomeranchuk</a></li> <li><a href="/wiki/Victor_Popov" title="Victor Popov">Popov</a></li> <li><a href="/wiki/Alexandru_Proca" title="Alexandru Proca">Proca</a></li> <li><a href="/wiki/Helen_Quinn" title="Helen Quinn">Quinn</a></li> <li><a href="/wiki/Alain_Rouet" title="Alain Rouet">Rouet</a></li> <li><a href="/wiki/Valery_Rubakov" title="Valery Rubakov">Rubakov</a></li> <li><a href="/wiki/David_Ruelle" title="David Ruelle">Ruelle</a></li> <li><a href="/wiki/Jun_John_Sakurai" class="mw-redirect" title="Jun John Sakurai">Sakurai</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Robert_Schrader" title="Robert Schrader">Schrader</a></li> <li><a href="/wiki/Albert_Schwarz" title="Albert Schwarz">Schwarz</a></li> <li><a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Schwinger</a></li> <li><a href="/wiki/Irving_Segal" title="Irving Segal">Segal</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Gordon_Walter_Semenoff" title="Gordon Walter Semenoff">Semenoff</a></li> <li><a href="/wiki/Mikhail_Shifman" title="Mikhail Shifman">Shifman</a></li> <li><a href="/wiki/Dmitry_Shirkov" title="Dmitry Shirkov">Shirkov</a></li> <li><a href="/wiki/Tony_Skyrme" title="Tony Skyrme">Skyrme</a></li> <li><a href="/wiki/Charles_M._Sommerfield" title="Charles M. Sommerfield">Sommerfield</a></li> <li><a href="/wiki/Raymond_Stora" title="Raymond Stora">Stora</a></li> <li><a href="/wiki/Ernst_Stueckelberg" title="Ernst Stueckelberg">Stueckelberg</a></li> <li><a href="/wiki/George_Sudarshan" class="mw-redirect" title="George Sudarshan">Sudarshan</a></li> <li><a href="/wiki/Kurt_Symanzik" title="Kurt Symanzik">Symanzik</a></li> <li><a href="/wiki/Yasushi_Takahashi" title="Yasushi Takahashi">Takahashi</a></li> <li><a href="/wiki/Walter_Thirring" title="Walter Thirring">Thirring</a></li> <li><a href="/wiki/Shin%27ichir%C5%8D_Tomonaga" title="Shin'ichirō Tomonaga">Tomonaga</a></li> <li><a href="/wiki/Igor_Tyutin" title="Igor Tyutin">Tyutin</a></li> <li><a href="/wiki/Arkady_Vainshtein" title="Arkady Vainshtein">Vainshtein</a></li> <li><a href="/wiki/Martinus_Veltman" class="mw-redirect" title="Martinus Veltman">Veltman</a></li> <li><a href="/wiki/Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano</a></li> <li><a href="/wiki/Miguel_%C3%81ngel_Virasoro_(physicist)" title="Miguel Ángel Virasoro (physicist)">Virasoro</a></li> <li><a href="/wiki/John_Clive_Ward" title="John Clive Ward">Ward</a></li> <li><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg</a></li> <li><a href="/wiki/Victor_Weisskopf" title="Victor Weisskopf">Weisskopf</a></li> <li><a href="/wiki/Gregor_Wentzel" title="Gregor Wentzel">Wentzel</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Christof_Wetterich" title="Christof Wetterich">Wetterich</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Gian_Carlo_Wick" title="Gian Carlo Wick">Wick</a></li> <li><a href="/wiki/Arthur_Wightman" title="Arthur Wightman">Wightman</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek</a></li> <li><a href="/wiki/Kenneth_G._Wilson" title="Kenneth G. Wilson">Wilson</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang</a></li> <li><a href="/wiki/Hideki_Yukawa" title="Hideki Yukawa">Yukawa</a></li> <li><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Anthony_Zee" title="Anthony Zee">Zee</a></li> <li><a href="/wiki/Wolfhart_Zimmermann" title="Wolfhart Zimmermann">Zimmermann</a></li> <li><a href="/wiki/Jean_Zinn-Justin" title="Jean Zinn-Justin">Zinn-Justin</a></li> <li><a href="/wiki/Jean-Bernard_Zuber" title="Jean-Bernard Zuber">Zuber</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li></ul> <p><br /> </p> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_field_theory" title="Template:Quantum field theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_field_theory" title="Template talk:Quantum field theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_field_theory" title="Special:EditPage/Template:Quantum field theory"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>Both the Schrödinger and Heisenberg approaches to quantum mechanics single out time and are not in the spirit of relativity. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\varphi (x),\partial _{t}\varphi (y)]=i\delta ^{3}(x-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\varphi (x),\partial _{t}\varphi (y)]=i\delta ^{3}(x-y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0dd61e9a1a73f935161139e2d89278803d7d99f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.676ex; height:3.176ex;" alt="{\displaystyle [\varphi (x),\partial _{t}\varphi (y)]=i\delta ^{3}(x-y)}"></span></dd></dl> <p>for two simultaneous spatial positions <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>, and this is not a relativistically invariant concept. The results of a calculation <i>are</i> covariant, but the symmetry is not apparent in intermediate stages. If naive field-theory calculations did not produce infinite answers in the <a href="/wiki/Continuum_limit" title="Continuum limit">continuum limit</a>, this would not have been such a big problem – it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry. This makes it difficult to extract the physical predictions, which require a <a href="/wiki/Renormalization" title="Renormalization">careful limiting procedure</a>. </p><p>The problem of lost symmetry also appears in classical mechanics, where the Hamiltonian formulation also superficially singles out time. The Lagrangian formulation makes the relativistic invariance apparent. In the same way, the path integral is manifestly relativistic. It reproduces the Schrödinger equation, the Heisenberg equations of motion, and the canonical commutation relations and shows that they are compatible with relativity. It extends the Heisenberg-type operator algebra to <a href="/wiki/Operator_product_expansion" title="Operator product expansion">operator product rules</a>, which are new relations difficult to see in the old formalism. </p><p>Further, different choices of canonical variables lead to very different-seeming formulations of the same theory. The transformations between the variables can be very complicated, but the path integral makes them into reasonably straightforward changes of integration variables. For these reasons, the Feynman path integral has made earlier formalisms largely obsolete. </p><p>The price of a path integral representation is that the unitarity of a theory is no longer self-evident, but it can be proven by changing variables to some canonical representation. The path integral itself also deals with larger mathematical spaces than is usual, which requires more careful mathematics, not all of which has been fully worked out. The path integral historically was not immediately accepted, partly because it took many years to incorporate fermions properly. This required physicists to invent an entirely new mathematical object – the <a href="/wiki/Grassmann_variable" class="mw-redirect" title="Grassmann variable">Grassmann variable</a> – which also allowed changes of variables to be done naturally, as well as allowing <a href="/wiki/Faddeev%E2%80%93Popov_ghost" title="Faddeev–Popov ghost">constrained quantization</a>. </p><p>The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time. This makes some naive identities <a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">fail</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Propagator">Propagator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=21" title="Edit section: Propagator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In relativistic theories, there is both a particle and field representation for every theory. The field representation is a sum over all field configurations, and the particle representation is a sum over different particle paths. </p><p>The nonrelativistic formulation is traditionally given in terms of particle paths, not fields. There, the path integral in the usual variables, with fixed boundary conditions, gives the probability amplitude for a particle to go from point <span class="texhtml mvar" style="font-style:italic;">x</span> to point <span class="texhtml mvar" style="font-style:italic;">y</span> in time <span class="texhtml mvar" style="font-style:italic;">T</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x,y;T)=\langle y;T\mid x;0\rangle =\int _{x(0)=x}^{x(T)=y}e^{iS[x]}\,Dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo>∣</mo> <mi>x</mi> <mo>;</mo> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>D</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x,y;T)=\langle y;T\mid x;0\rangle =\int _{x(0)=x}^{x(T)=y}e^{iS[x]}\,Dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13a2490ba92f4528eac85157f3af098d2b480c11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.002ex; height:6.676ex;" alt="{\displaystyle K(x,y;T)=\langle y;T\mid x;0\rangle =\int _{x(0)=x}^{x(T)=y}e^{iS[x]}\,Dx.}"></span></dd></dl> <p>This is called the <a href="/wiki/Propagator" title="Propagator">propagator</a>. To obtain the final state at <span class="texhtml"><i>y</i></span> we simply apply <span class="texhtml"><i>K</i>(<i>x</i>,<i>y</i>; <i>T</i>)</span> to the initial state and integrate over <span class="texhtml"><i>x</i></span> resulting in: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{T}(y)=\int _{x}\psi _{0}(x)K(x,y;T)\,dx=\int ^{x(T)=y}\psi _{0}(x(0))e^{iS[x]}\,Dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>D</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{T}(y)=\int _{x}\psi _{0}(x)K(x,y;T)\,dx=\int ^{x(T)=y}\psi _{0}(x(0))e^{iS[x]}\,Dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99e0c051b9f404b3a46ec79d4913270c62448ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:59.94ex; height:6.343ex;" alt="{\displaystyle \psi _{T}(y)=\int _{x}\psi _{0}(x)K(x,y;T)\,dx=\int ^{x(T)=y}\psi _{0}(x(0))e^{iS[x]}\,Dx.}"></span></dd></dl> <p>For a spatially homogeneous system, where <span class="texhtml"><i>K</i>(<i>x</i>, <i>y</i>)</span> is only a function of <span class="texhtml">(<i>x</i> − <i>y</i>)</span>, the integral is a <a href="/wiki/Convolution" title="Convolution">convolution</a>, the final state is the initial state convolved with the propagator: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{T}=\psi _{0}*K(;T).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∗</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{T}=\psi _{0}*K(;T).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8a62f0d839d382329f84a624bdf4f166fcb8a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.955ex; height:2.843ex;" alt="{\displaystyle \psi _{T}=\psi _{0}*K(;T).}"></span></dd></dl> <p>For a free particle of mass <span class="texhtml mvar" style="font-style:italic;">m</span>, the propagator can be evaluated either explicitly from the path integral or by noting that the Schrödinger equation is a diffusion equation in imaginary time, and the solution must be a normalized Gaussian: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x,y;T)\propto e^{\frac {im(x-y)^{2}}{2T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </mfrac> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x,y;T)\propto e^{\frac {im(x-y)^{2}}{2T}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a5bf4d04ff665e526f86bba5c11fab45b6e855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.898ex; height:4.843ex;" alt="{\displaystyle K(x,y;T)\propto e^{\frac {im(x-y)^{2}}{2T}}.}"></span></dd></dl> <p>Taking the Fourier transform in <span class="texhtml">(<i>x</i> − <i>y</i>)</span> produces another Gaussian: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(p;T)=e^{\frac {iTp^{2}}{2m}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>T</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(p;T)=e^{\frac {iTp^{2}}{2m}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb67f065d8be714481716c93d85416487aba929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.484ex; height:4.843ex;" alt="{\displaystyle K(p;T)=e^{\frac {iTp^{2}}{2m}},}"></span></dd></dl> <p>and in <span class="texhtml mvar" style="font-style:italic;">p</span>-space the proportionality factor here is constant in time, as will be verified in a moment. The Fourier transform in time, extending <span class="texhtml"><i>K</i>(<i>p</i>; <i>T</i>)</span> to be zero for negative times, gives Green's function, or the frequency-space propagator: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\text{F}}(p,E)={\frac {-i}{E-{\frac {{\vec {p}}^{2}}{2m}}+i\varepsilon }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> </mrow> <mrow> <mi>E</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <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"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mi>ε</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\text{F}}(p,E)={\frac {-i}{E-{\frac {{\vec {p}}^{2}}{2m}}+i\varepsilon }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8300a0649b3aeb8ecf383e6933204c742e400b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:25.945ex; height:8.009ex;" alt="{\displaystyle G_{\text{F}}(p,E)={\frac {-i}{E-{\frac {{\vec {p}}^{2}}{2m}}+i\varepsilon }},}"></span></dd></dl> <p>which is the reciprocal of the operator that annihilates the wavefunction in the Schrödinger equation, which wouldn't have come out right if the proportionality factor weren't constant in the <span class="texhtml mvar" style="font-style:italic;">p</span>-space representation. </p><p>The infinitesimal term in the denominator is a small positive number, which guarantees that the inverse Fourier transform in <span class="texhtml mvar" style="font-style:italic;">E</span> will be nonzero only for future times. For past times, the inverse Fourier transform contour closes toward values of <span class="texhtml mvar" style="font-style:italic;">E</span> where there is no singularity. This guarantees that <span class="texhtml mvar" style="font-style:italic;">K</span> propagates the particle into the future and is the reason for the subscript "F" on <span class="texhtml mvar" style="font-style:italic;">G</span>. The infinitesimal term can be interpreted as an infinitesimal rotation toward imaginary time. </p><p>It is also possible to reexpress the nonrelativistic time evolution in terms of propagators going toward the past, since the Schrödinger equation is time-reversible. The past propagator is the same as the future propagator except for the obvious difference that it vanishes in the future, and in the Gaussian <span class="texhtml mvar" style="font-style:italic;">t</span> is replaced by <span class="texhtml">−<i>t</i></span>. In this case, the interpretation is that these are the quantities to convolve the final wavefunction so as to get the initial wavefunction: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\text{B}}(p,E)={\frac {-i}{-E-{\frac {i{\vec {p}}^{2}}{2m}}+i\varepsilon }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> </mrow> <mrow> <mo>−</mo> <mi>E</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msup> <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"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mi>ε</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\text{B}}(p,E)={\frac {-i}{-E-{\frac {i{\vec {p}}^{2}}{2m}}+i\varepsilon }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89bdb63a8d50f2f7ff78eb22aaca9e9c39afebad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:27.972ex; height:8.009ex;" alt="{\displaystyle G_{\text{B}}(p,E)={\frac {-i}{-E-{\frac {i{\vec {p}}^{2}}{2m}}+i\varepsilon }}.}"></span></dd></dl> <p>Given the nearly identical only change is the sign of <span class="texhtml mvar" style="font-style:italic;">E</span> and <span class="texhtml mvar" style="font-style:italic;">ε</span>, the parameter <span class="texhtml mvar" style="font-style:italic;">E</span> in Green's function can either be the energy if the paths are going toward the future, or the negative of the energy if the paths are going toward the past. </p><p>For a nonrelativistic theory, the time as measured along the path of a moving particle and the time as measured by an outside observer are the same. In relativity, this is no longer true. For a relativistic theory the propagator should be defined as the sum over all paths that travel between two points in a fixed proper time, as measured along the path (these paths describe the trajectory of a particle in space and in time): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x-y,\mathrm {T} )=\int _{x(0)=x}^{x(\mathrm {T} )=y}e^{i\int _{0}^{\mathrm {T} }{\sqrt {{\dot {x}}^{2}-\alpha }}\,d\tau }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>α</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x-y,\mathrm {T} )=\int _{x(0)=x}^{x(\mathrm {T} )=y}e^{i\int _{0}^{\mathrm {T} }{\sqrt {{\dot {x}}^{2}-\alpha }}\,d\tau }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/915e63173a04cfab5bc79a311de9f12ad190fa2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.947ex; height:6.676ex;" alt="{\displaystyle K(x-y,\mathrm {T} )=\int _{x(0)=x}^{x(\mathrm {T} )=y}e^{i\int _{0}^{\mathrm {T} }{\sqrt {{\dot {x}}^{2}-\alpha }}\,d\tau }.}"></span></dd></dl> <p>The integral above is not trivial to interpret because of the square root. Fortunately, there is a heuristic trick. The sum is over the relativistic arc length of the path of an oscillating quantity, and like the nonrelativistic path integral should be interpreted as slightly rotated into imaginary time. The function <span class="texhtml"><i>K</i>(<i>x</i> − <i>y</i>, <i>τ</i>)</span> can be evaluated when the sum is over paths in Euclidean space: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x-y,\mathrm {T} )=e^{-\alpha \mathrm {T} }\int _{x(0)=x}^{x(\mathrm {T} )=y}e^{-L}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>L</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x-y,\mathrm {T} )=e^{-\alpha \mathrm {T} }\int _{x(0)=x}^{x(\mathrm {T} )=y}e^{-L}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0f6ed0273aeed1380948335be0575df2be6293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.177ex; height:6.676ex;" alt="{\displaystyle K(x-y,\mathrm {T} )=e^{-\alpha \mathrm {T} }\int _{x(0)=x}^{x(\mathrm {T} )=y}e^{-L}.}"></span></dd></dl> <p>This describes a sum over all paths of length <span class="texhtml">Τ</span> of the exponential of minus the length. This can be given a probability interpretation. The sum over all paths is a probability average over a path constructed step by step. The total number of steps is proportional to <span class="texhtml">Τ</span>, and each step is less likely the longer it is. By the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>, the result of many independent steps is a Gaussian of variance proportional to <span class="texhtml">Τ</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x-y,\mathrm {T} )=e^{-\alpha \mathrm {T} }e^{-{\frac {(x-y)^{2}}{\mathrm {T} }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x-y,\mathrm {T} )=e^{-\alpha \mathrm {T} }e^{-{\frac {(x-y)^{2}}{\mathrm {T} }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/675e0a92d99721723e9a93dd77121a4e0a07b718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.225ex; height:4.843ex;" alt="{\displaystyle K(x-y,\mathrm {T} )=e^{-\alpha \mathrm {T} }e^{-{\frac {(x-y)^{2}}{\mathrm {T} }}}.}"></span></dd></dl> <p>The usual definition of the relativistic propagator only asks for the amplitude is to travel from <span class="texhtml mvar" style="font-style:italic;">x</span> to <span class="texhtml mvar" style="font-style:italic;">y</span>, after summing over all the possible proper times it could take: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x-y)=\int _{0}^{\infty }K(x-y,\mathrm {T} )W(\mathrm {T} )\,d\mathrm {T} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">)</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x-y)=\int _{0}^{\infty }K(x-y,\mathrm {T} )W(\mathrm {T} )\,d\mathrm {T} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e375b40bae2a1c5171599987a09b1de670eb3dc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.789ex; height:5.843ex;" alt="{\displaystyle K(x-y)=\int _{0}^{\infty }K(x-y,\mathrm {T} )W(\mathrm {T} )\,d\mathrm {T} ,}"></span></dd></dl> <p>where <span class="texhtml"><i>W</i>(Τ)</span> is a weight factor, the relative importance of paths of different proper time. By the translation symmetry in proper time, this weight can only be an exponential factor and can be absorbed into the constant <span class="texhtml mvar" style="font-style:italic;">α</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x-y)=\int _{0}^{\infty }e^{-{\frac {(x-y)^{2}}{\mathrm {T} }}-\alpha \mathrm {T} }\,d\mathrm {T} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x-y)=\int _{0}^{\infty }e^{-{\frac {(x-y)^{2}}{\mathrm {T} }}-\alpha \mathrm {T} }\,d\mathrm {T} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9487d8f6961816d2ff54d7684db67b077386399f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.205ex; height:6.343ex;" alt="{\displaystyle K(x-y)=\int _{0}^{\infty }e^{-{\frac {(x-y)^{2}}{\mathrm {T} }}-\alpha \mathrm {T} }\,d\mathrm {T} .}"></span></dd></dl> <p>This is the <a href="/wiki/Feynman_diagram#Schwinger_representation" title="Feynman diagram">Schwinger representation</a>. Taking a Fourier transform over the variable <span class="texhtml">(<i>x</i> − <i>y</i>)</span> can be done for each value of <span class="texhtml">Τ</span> separately, and because each separate <span class="texhtml">Τ</span> contribution is a Gaussian, gives whose Fourier transform is another Gaussian with reciprocal width. So in <span class="texhtml mvar" style="font-style:italic;">p</span>-space, the propagator can be reexpressed simply: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(p)=\int _{0}^{\infty }e^{-\mathrm {T} p^{2}-\mathrm {T} \alpha }\,d\mathrm {T} ={\frac {1}{p^{2}+\alpha }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mi>α</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(p)=\int _{0}^{\infty }e^{-\mathrm {T} p^{2}-\mathrm {T} \alpha }\,d\mathrm {T} ={\frac {1}{p^{2}+\alpha }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0592d1f4026e3e314b9bd0073056fe7f829f5caa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.24ex; height:6.009ex;" alt="{\displaystyle K(p)=\int _{0}^{\infty }e^{-\mathrm {T} p^{2}-\mathrm {T} \alpha }\,d\mathrm {T} ={\frac {1}{p^{2}+\alpha }},}"></span></dd></dl> <p>which is the Euclidean propagator for a scalar particle. Rotating <span class="texhtml"><i>p</i><sub>0</sub></span> to be imaginary gives the usual relativistic propagator, up to a factor of <span class="texhtml">−<i>i</i></span> and an ambiguity, which will be clarified below: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(p)={\frac {i}{p_{0}^{2}-{\vec {p}}^{2}-m^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <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"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(p)={\frac {i}{p_{0}^{2}-{\vec {p}}^{2}-m^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a6988472056f0d1ce0cd3cdc72425f25a09cbbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.005ex; height:6.676ex;" alt="{\displaystyle K(p)={\frac {i}{p_{0}^{2}-{\vec {p}}^{2}-m^{2}}}.}"></span></dd></dl> <p>This expression can be interpreted in the nonrelativistic limit, where it is convenient to split it by <a href="/wiki/Partial_fractions" class="mw-redirect" title="Partial fractions">partial fractions</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2p_{0}K(p)={\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}}}+{\frac {i}{p_{0}+{\sqrt {{\vec {p}}^{2}+m^{2}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>K</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <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"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <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"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2p_{0}K(p)={\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}}}+{\frac {i}{p_{0}+{\sqrt {{\vec {p}}^{2}+m^{2}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b33c414bec47189dd9db8e311dac5909febfed2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:48.093ex; height:8.009ex;" alt="{\displaystyle 2p_{0}K(p)={\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}}}+{\frac {i}{p_{0}+{\sqrt {{\vec {p}}^{2}+m^{2}}}}}.}"></span></dd></dl> <p>For states where one nonrelativistic particle is present, the initial wavefunction has a frequency distribution concentrated near <span class="texhtml"><i>p</i><sub>0</sub> = <i>m</i></span>. When convolving with the propagator, which in <span class="texhtml mvar" style="font-style:italic;">p</span> space just means multiplying by the propagator, the second term is suppressed and the first term is enhanced. For frequencies near <span class="texhtml"><i>p</i><sub>0</sub> = <i>m</i></span>, the dominant first term has the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2mK_{\text{NR}}(p)={\frac {i}{(p_{0}-m)-{\frac {{\vec {p}}^{2}}{2m}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>m</mi> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NR</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <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"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2mK_{\text{NR}}(p)={\frac {i}{(p_{0}-m)-{\frac {{\vec {p}}^{2}}{2m}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296ae343851f1dd8a5d237496c94ab5ea3245803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:30.265ex; height:8.009ex;" alt="{\displaystyle 2mK_{\text{NR}}(p)={\frac {i}{(p_{0}-m)-{\frac {{\vec {p}}^{2}}{2m}}}}.}"></span></dd></dl> <p>This is the expression for the nonrelativistic <a href="/wiki/Green%27s_function" title="Green's function">Green's function</a> of a free Schrödinger particle. </p><p>The second term has a nonrelativistic limit also, but this limit is concentrated on frequencies that are negative. The second pole is dominated by contributions from paths where the proper time and the coordinate time are ticking in an opposite sense, which means that the second term is to be interpreted as the antiparticle. The nonrelativistic analysis shows that with this form the antiparticle still has positive energy. </p><p>The proper way to express this mathematically is that, adding a small suppression factor in proper time, the limit where <span class="texhtml"><i>t</i> → −∞</span> of the first term must vanish, while the <span class="texhtml"><i>t</i> → +∞</span> limit of the second term must vanish. In the Fourier transform, this means shifting the pole in <span class="texhtml"><i>p</i><sub>0</sub></span> slightly, so that the inverse Fourier transform will pick up a small decay factor in one of the time directions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(p)={\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}+i\varepsilon }}+{\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}-i\varepsilon }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <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"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>i</mi> <mi>ε</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <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"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>i</mi> <mi>ε</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(p)={\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}+i\varepsilon }}+{\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}-i\varepsilon }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de9899361bfc2d8478d48e60d2409a267077a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:54.16ex; height:8.009ex;" alt="{\displaystyle K(p)={\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}+i\varepsilon }}+{\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}-i\varepsilon }}.}"></span></dd></dl> <p>Without these terms, the pole contribution could not be unambiguously evaluated when taking the inverse Fourier transform of <span class="texhtml"><i>p</i><sub>0</sub></span>. The terms can be recombined: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(p)={\frac {i}{p^{2}-m^{2}+i\varepsilon }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mi>ε</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(p)={\frac {i}{p^{2}-m^{2}+i\varepsilon }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba78f8a18e484438945f39ea35e67d14fd8df3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.511ex; height:5.843ex;" alt="{\displaystyle K(p)={\frac {i}{p^{2}-m^{2}+i\varepsilon }},}"></span></dd></dl> <p>which when factored, produces opposite-sign infinitesimal terms in each factor. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. The <span class="texhtml mvar" style="font-style:italic;">ε</span> term introduces a small imaginary part to the <span class="texhtml"><i>α</i> = <i>m</i><sup>2</sup></span>, which in the Minkowski version is a small exponential suppression of long paths. </p><p>So in the relativistic case, the Feynman path-integral representation of the propagator includes paths going backwards in time, which describe antiparticles. The paths that contribute to the relativistic propagator go forward and backwards in time, and the <a href="/wiki/Feynman%E2%80%93Stueckelberg_interpretation" class="mw-redirect" title="Feynman–Stueckelberg interpretation">interpretation</a> of this is that the amplitude for a free particle to travel between two points includes amplitudes for the particle to fluctuate into an antiparticle, travel back in time, then forward again. </p><p>Unlike the nonrelativistic case, it is impossible to produce a relativistic theory of local particle propagation without including antiparticles. All local differential operators have inverses that are nonzero outside the light cone, meaning that it is impossible to keep a particle from travelling faster than light. Such a particle cannot have a Green's function that is only nonzero in the future in a relativistically invariant theory. </p> <div class="mw-heading mw-heading3"><h3 id="Functionals_of_fields">Functionals of fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=22" title="Edit section: Functionals of fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>However, the path integral formulation is also extremely important in <i>direct</i> application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a <a href="/wiki/Field_(physics)" title="Field (physics)">field</a> over all space. The action is referred to technically as a <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functional</a> of the field: <span class="texhtml"><i>S</i>[<i>ϕ</i>]</span>, where the field <span class="texhtml"><i>ϕ</i>(<i>x<sup>μ</sup></i>)</span> is itself a function of space and time, and the square brackets are a reminder that the action depends on all the field's values everywhere, not just some particular value. <i>One</i> such given function <span class="texhtml"><i>ϕ</i>(<i>x<sup>μ</sup></i>)</span> of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> is called a <i>field configuration</i>. In principle, one integrates Feynman's amplitude over the class of all possible field configurations. </p><p>Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort (not yet entirely successful) has been made toward making these <a href="/wiki/Functional_integral" class="mw-redirect" title="Functional integral">functional integrals</a> mathematically precise. </p><p>Such a functional integral is extremely similar to the <a href="/wiki/Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a> in <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>. Indeed, it is sometimes <i>called</i> a <a href="/wiki/Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">partition function</a>, and the two are essentially mathematically identical except for the factor of <span class="texhtml mvar" style="font-style:italic;">i</span> in the exponent in Feynman's postulate 3. <a href="/wiki/Analytic_continuation" title="Analytic continuation">Analytically continuing</a> the integral to an imaginary time variable (called a <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a>) makes the functional integral even more like a statistical partition function and also tames some of the mathematical difficulties of working with these integrals. </p> <div class="mw-heading mw-heading3"><h3 id="Expectation_values">Expectation values</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=23" title="Edit section: Expectation values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, if the <a href="/wiki/Action_(physics)" title="Action (physics)">action</a> is given by the <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functional</a> <span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">S</span> of field configurations (which only depends locally on the fields), then the <a href="/wiki/Time-ordered" class="mw-redirect" title="Time-ordered">time-ordered</a> <a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">vacuum expectation value</a> of <a href="/w/index.php?title=Polynomially_bounded&action=edit&redlink=1" class="new" title="Polynomially bounded (page does not exist)">polynomially bounded</a> functional <span class="texhtml mvar" style="font-style:italic;">F</span>, <span class="texhtml"><span class="nowrap">⟨<i>F</i>⟩</span></span>, is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle F\rangle ={\frac {\int {\mathcal {D}}\varphi F[\varphi ]e^{i{\mathcal {S}}[\varphi ]}}{\int {\mathcal {D}}\varphi e^{i{\mathcal {S}}[\varphi ]}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>F</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>φ</mi> <mi>F</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> </msup> </mrow> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>φ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle F\rangle ={\frac {\int {\mathcal {D}}\varphi F[\varphi ]e^{i{\mathcal {S}}[\varphi ]}}{\int {\mathcal {D}}\varphi e^{i{\mathcal {S}}[\varphi ]}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8dd3d8f7f56934dd5dc162ad33111913bac8a96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.731ex; height:7.009ex;" alt="{\displaystyle \langle F\rangle ={\frac {\int {\mathcal {D}}\varphi F[\varphi ]e^{i{\mathcal {S}}[\varphi ]}}{\int {\mathcal {D}}\varphi e^{i{\mathcal {S}}[\varphi ]}}}.}"></span></dd></dl> <p>The symbol <span class="texhtml">∫<span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">D</span><i>ϕ</i></span> here is a concise way to represent the infinite-dimensional integral over all possible field configurations on all of space-time. As stated above, the unadorned path integral in the denominator ensures proper normalization. </p> <div class="mw-heading mw-heading3"><h3 id="As_a_probability">As a probability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=24" title="Edit section: As a probability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Strictly speaking, the only question that can be asked in physics is: <i>What fraction of states satisfying condition <span class="texhtml mvar" style="font-style:italic;">A</span> also satisfy condition <span class="texhtml mvar" style="font-style:italic;">B</span>?</i> The answer to this is a number between 0 and 1, which can be interpreted as a <a href="/wiki/Conditional_probability" title="Conditional probability">conditional probability</a>, written as <span class="texhtml">P(<i>B</i>|<i>A</i>)</span>. In terms of path integration, since <span class="texhtml">P(<i>B</i>|<i>A</i>) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">P(<i>A</i>∩<i>B</i>) </span><span class="sr-only">/</span><span class="den"> P(<i>A</i>)</span></span>⁠</span></span>, this means </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} (B\mid A)={\frac {\sum _{F\subset A\cap B}\left|\int {\mathcal {D}}\varphi O_{\text{in}}[\varphi ]e^{i{\mathcal {S}}[\varphi ]}F[\varphi ]\right|^{2}}{\sum _{F\subset A}\left|\int {\mathcal {D}}\varphi O_{\text{in}}[\varphi ]e^{i{\mathcal {S}}[\varphi ]}F[\varphi ]\right|^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∣</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mo>⊂</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mrow> </munder> <msup> <mrow> <mo>|</mo> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>φ</mi> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> </msup> <mi>F</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mo>⊂</mo> <mi>A</mi> </mrow> </munder> <msup> <mrow> <mo>|</mo> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>φ</mi> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> </msup> <mi>F</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (B\mid A)={\frac {\sum _{F\subset A\cap B}\left|\int {\mathcal {D}}\varphi O_{\text{in}}[\varphi ]e^{i{\mathcal {S}}[\varphi ]}F[\varphi ]\right|^{2}}{\sum _{F\subset A}\left|\int {\mathcal {D}}\varphi O_{\text{in}}[\varphi ]e^{i{\mathcal {S}}[\varphi ]}F[\varphi ]\right|^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae50ce50d99fed4b30b60f1abd8bd23bfc21227a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.715ex; height:8.176ex;" alt="{\displaystyle \operatorname {P} (B\mid A)={\frac {\sum _{F\subset A\cap B}\left|\int {\mathcal {D}}\varphi O_{\text{in}}[\varphi ]e^{i{\mathcal {S}}[\varphi ]}F[\varphi ]\right|^{2}}{\sum _{F\subset A}\left|\int {\mathcal {D}}\varphi O_{\text{in}}[\varphi ]e^{i{\mathcal {S}}[\varphi ]}F[\varphi ]\right|^{2}}},}"></span></dd></dl> <p>where the functional <span class="texhtml"><i>O</i><sub>in</sub>[<i>ϕ</i>]</span> is the superposition of all incoming states that could lead to the states we are interested in. In particular, this could be a state corresponding to the state of the Universe just after the <a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a>, although for actual calculation this can be simplified using heuristic methods. Since this expression is a quotient of path integrals, it is naturally normalised. </p> <div class="mw-heading mw-heading3"><h3 id="Schwinger–Dyson_equations"><span id="Schwinger.E2.80.93Dyson_equations"></span>Schwinger–Dyson equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=25" title="Edit section: Schwinger–Dyson equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Schwinger%E2%80%93Dyson_equation" title="Schwinger–Dyson equation">Schwinger–Dyson equation</a></div> <p>Since this formulation of quantum mechanics is analogous to classical action principle, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case. </p><p>In the language of functional analysis, we can write the <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equations</a> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta {\mathcal {S}}[\varphi ]}{\delta \varphi }}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>δ</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta {\mathcal {S}}[\varphi ]}{\delta \varphi }}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb4de731adf904b88deade6b70258828ed4f09b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.452ex; height:6.343ex;" alt="{\displaystyle {\frac {\delta {\mathcal {S}}[\varphi ]}{\delta \varphi }}=0}"></span></dd></dl> <p>(the left-hand side is a <a href="/wiki/Functional_derivative" title="Functional derivative">functional derivative</a>; the equation means that the action is stationary under small changes in the field configuration). The quantum analogues of these equations are called the <a href="/wiki/Schwinger%E2%80%93Dyson_equation" title="Schwinger–Dyson equation">Schwinger–Dyson equations</a>. </p><p>If the <a href="/w/index.php?title=Functional_measure&action=edit&redlink=1" class="new" title="Functional measure (page does not exist)">functional measure</a> <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">D</span><i>ϕ</i></span> turns out to be <a href="/wiki/Translational_symmetry" title="Translational symmetry">translationally invariant</a> (we'll assume this for the rest of this article, although this does not hold for, let's say <a href="/wiki/Nonlinear_sigma_model" class="mw-redirect" title="Nonlinear sigma model">nonlinear sigma models</a>), and if we assume that after a <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i{\mathcal {S}}[\varphi ]},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i{\mathcal {S}}[\varphi ]},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72853bae20ab0e1b75f33011ada66f826ecab088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.575ex; height:3.176ex;" alt="{\displaystyle e^{i{\mathcal {S}}[\varphi ]},}"></span></dd></dl> <p>which now becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-H[\varphi ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>H</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-H[\varphi ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569838b0b4ff05dc6e23bc07778f9f962fcfd243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.043ex; height:2.843ex;" alt="{\displaystyle e^{-H[\varphi ]}}"></span></dd></dl> <p>for some <span class="texhtml mvar" style="font-style:italic;">H</span>, it goes to zero faster than a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of any <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> for large values of <span class="texhtml mvar" style="font-style:italic;">φ</span>, then we can <a href="/wiki/Integration_by_parts" title="Integration by parts">integrate by parts</a> (after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger–Dyson equations for the expectation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle {\frac {\delta F[\varphi ]}{\delta \varphi }}\right\rangle =-i\left\langle F[\varphi ]{\frac {\delta {\mathcal {S}}[\varphi ]}{\delta \varphi }}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>F</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>δ</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mrow> <mo>⟨</mo> <mrow> <mi>F</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>δ</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle {\frac {\delta F[\varphi ]}{\delta \varphi }}\right\rangle =-i\left\langle F[\varphi ]{\frac {\delta {\mathcal {S}}[\varphi ]}{\delta \varphi }}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab0605db250f78b7a72d8c9a8be5ea981e98938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.253ex; height:6.343ex;" alt="{\displaystyle \left\langle {\frac {\delta F[\varphi ]}{\delta \varphi }}\right\rangle =-i\left\langle F[\varphi ]{\frac {\delta {\mathcal {S}}[\varphi ]}{\delta \varphi }}\right\rangle }"></span></dd></dl> <p>for any polynomially-bounded functional <span class="texhtml mvar" style="font-style:italic;">F</span>. In the <a href="/wiki/DeWitt_notation" title="DeWitt notation">deWitt notation</a> this looks like<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle F_{,i}\right\rangle =-i\left\langle F{\mathcal {S}}_{,i}\right\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mrow> <mo>⟨</mo> <mrow> <mi>F</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle F_{,i}\right\rangle =-i\left\langle F{\mathcal {S}}_{,i}\right\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae0792fb8c9f9b1c6dc017e0f140733ccf111157" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.907ex; height:3.009ex;" alt="{\displaystyle \left\langle F_{,i}\right\rangle =-i\left\langle F{\mathcal {S}}_{,i}\right\rangle .}"></span></dd></dl> <p>These equations are the analog of the <a href="/wiki/On-shell" class="mw-redirect" title="On-shell">on-shell</a> EL equations. The time ordering is taken before the time derivatives inside the <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">S</span><sub>,<i>i</i></sub></span>. </p><p>If <span class="texhtml mvar" style="font-style:italic;">J</span> (called the <a href="/wiki/Source_field" title="Source field">source field</a>) is an element of the <a href="/wiki/Dual_space" title="Dual space">dual space</a> of the field configurations (which has at least an <a href="/w/index.php?title=Affine_structure&action=edit&redlink=1" class="new" title="Affine structure (page does not exist)">affine structure</a> because of the assumption of the <a href="/wiki/Translational_invariance" class="mw-redirect" title="Translational invariance">translational invariance</a> for the functional measure), then the <a href="/wiki/Generating_functional" class="mw-redirect" title="Generating functional">generating functional</a> <span class="texhtml mvar" style="font-style:italic;">Z</span> of the source fields is <b>defined</b> to be </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z[J]=\int {\mathcal {D}}\varphi e^{i\left({\mathcal {S}}[\varphi ]+\langle J,\varphi \rangle \right)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>φ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>J</mi> <mo>,</mo> <mi>φ</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z[J]=\int {\mathcal {D}}\varphi e^{i\left({\mathcal {S}}[\varphi ]+\langle J,\varphi \rangle \right)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c0bb94dfefb303272fe0566a3d021312ac35966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.422ex; height:5.676ex;" alt="{\displaystyle Z[J]=\int {\mathcal {D}}\varphi e^{i\left({\mathcal {S}}[\varphi ]+\langle J,\varphi \rangle \right)}.}"></span></dd></dl> <p>Note that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta ^{n}Z}{\delta J(x_{1})\cdots \delta J(x_{n})}}[J]=i^{n}\,Z[J]\,\left\langle \varphi (x_{1})\cdots \varphi (x_{n})\right\rangle _{J},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>Z</mi> </mrow> <mrow> <mi>δ</mi> <mi>J</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mi>δ</mi> <mi>J</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <msub> <mrow> <mo>⟨</mo> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta ^{n}Z}{\delta J(x_{1})\cdots \delta J(x_{n})}}[J]=i^{n}\,Z[J]\,\left\langle \varphi (x_{1})\cdots \varphi (x_{n})\right\rangle _{J},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51ad1e13e298e27ca37cea282dfb558ada113de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.845ex; height:6.176ex;" alt="{\displaystyle {\frac {\delta ^{n}Z}{\delta J(x_{1})\cdots \delta J(x_{n})}}[J]=i^{n}\,Z[J]\,\left\langle \varphi (x_{1})\cdots \varphi (x_{n})\right\rangle _{J},}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z^{,i_{1}\cdots i_{n}}[J]=i^{n}Z[J]\left\langle \varphi ^{i_{1}}\cdots \varphi ^{i_{n}}\right\rangle _{J},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <msub> <mrow> <mo>⟨</mo> <mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⋯</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z^{,i_{1}\cdots i_{n}}[J]=i^{n}Z[J]\left\langle \varphi ^{i_{1}}\cdots \varphi ^{i_{n}}\right\rangle _{J},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b4fbd7299210e2c476c87216723801b4cdefbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.633ex; height:3.509ex;" alt="{\displaystyle Z^{,i_{1}\cdots i_{n}}[J]=i^{n}Z[J]\left\langle \varphi ^{i_{1}}\cdots \varphi ^{i_{n}}\right\rangle _{J},}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle F\rangle _{J}={\frac {\int {\mathcal {D}}\varphi F[\varphi ]e^{i\left({\mathcal {S}}[\varphi ]+\langle J,\varphi \rangle \right)}}{\int {\mathcal {D}}\varphi e^{i\left({\mathcal {S}}[\varphi ]+\langle J,\varphi \rangle \right)}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>F</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>φ</mi> <mi>F</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>J</mi> <mo>,</mo> <mi>φ</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>φ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>J</mi> <mo>,</mo> <mi>φ</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle F\rangle _{J}={\frac {\int {\mathcal {D}}\varphi F[\varphi ]e^{i\left({\mathcal {S}}[\varphi ]+\langle J,\varphi \rangle \right)}}{\int {\mathcal {D}}\varphi e^{i\left({\mathcal {S}}[\varphi ]+\langle J,\varphi \rangle \right)}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5557cbd2cdb86590e7c666b51770fcc80d6ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.414ex; height:7.009ex;" alt="{\displaystyle \langle F\rangle _{J}={\frac {\int {\mathcal {D}}\varphi F[\varphi ]e^{i\left({\mathcal {S}}[\varphi ]+\langle J,\varphi \rangle \right)}}{\int {\mathcal {D}}\varphi e^{i\left({\mathcal {S}}[\varphi ]+\langle J,\varphi \rangle \right)}}}.}"></span></dd></dl> <p>Basically, if <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">D</span><i>φ</i> <i>e</i><sup><i>i</i><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">S</span>[<i>φ</i>]</sup></span> is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of <a href="/wiki/Quantum_field_theory" title="Quantum field theory">QFT</a>, unlike its Wick-rotated <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> analogue, because we have <a href="/wiki/Time_ordering" class="mw-redirect" title="Time ordering">time ordering</a> complications here!), then <span class="texhtml"><span class="nowrap">⟨<i>φ</i>(<i>x</i><sub>1</sub>) ... <i>φ</i>(<i>x<sub>n</sub></i>)⟩</span></span> are its <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a>, and <span class="texhtml mvar" style="font-style:italic;">Z</span> is its <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>. </p><p>If <span class="texhtml mvar" style="font-style:italic;">F</span> is a functional of <span class="texhtml mvar" style="font-style:italic;">φ</span>, then for an <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operator</a> <span class="texhtml mvar" style="font-style:italic;">K</span>, <span class="texhtml"><i>F</i>[<i>K</i>]</span> is defined to be the operator that substitutes <span class="texhtml mvar" style="font-style:italic;">K</span> for <span class="texhtml mvar" style="font-style:italic;">φ</span>. For example, if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[\varphi ]={\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}\varphi (x_{1})\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}\varphi (x_{n}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[\varphi ]={\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}\varphi (x_{1})\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}\varphi (x_{n}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfaa9b6dbc2a4f700fedbf508daffb46857b8c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:34.33ex; height:7.176ex;" alt="{\displaystyle F[\varphi ]={\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}\varphi (x_{1})\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}\varphi (x_{n}),}"></span></dd></dl> <p>and <span class="texhtml mvar" style="font-style:italic;">G</span> is a functional of <span class="texhtml mvar" style="font-style:italic;">J</span>, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\left[-i{\frac {\delta }{\delta J}}\right]G[J]=(-i)^{n}{\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}{\frac {\delta }{\delta J(x_{1})}}\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}{\frac {\delta }{\delta J(x_{n})}}G[J].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>J</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mi>G</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>J</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>J</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>G</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\left[-i{\frac {\delta }{\delta J}}\right]G[J]=(-i)^{n}{\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}{\frac {\delta }{\delta J(x_{1})}}\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}{\frac {\delta }{\delta J(x_{n})}}G[J].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b809c494c78617b22f3a7eb51229916366f3bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:59.206ex; height:7.176ex;" alt="{\displaystyle F\left[-i{\frac {\delta }{\delta J}}\right]G[J]=(-i)^{n}{\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}{\frac {\delta }{\delta J(x_{1})}}\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}{\frac {\delta }{\delta J(x_{n})}}G[J].}"></span></dd></dl> <p>Then, from the properties of the <a href="/wiki/Functional_integral" class="mw-redirect" title="Functional integral">functional integrals</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}[\varphi ]+J(x)\right\rangle _{J}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>⟨</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> </mrow> <mrow> <mi>δ</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}[\varphi ]+J(x)\right\rangle _{J}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacbb7505e8ede2b17344cd5da8021c52f05e1d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.828ex; height:6.343ex;" alt="{\displaystyle \left\langle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}[\varphi ]+J(x)\right\rangle _{J}=0}"></span></dd></dl> <p>we get the "master" Schwinger–Dyson equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Z[J]=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> </mrow> <mrow> <mi>δ</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>J</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Z[J]=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c943233dae4175a4d8c19052414ab8cc7a58bd31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.989ex; height:6.343ex;" alt="{\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Z[J]=0,}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{,i}[-i\partial ]Z+J_{i}Z=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mo>−</mo> <mi>i</mi> <mi mathvariant="normal">∂</mi> <mo stretchy="false">]</mo> <mi>Z</mi> <mo>+</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>Z</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{,i}[-i\partial ]Z+J_{i}Z=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f80fc729f6339c28c610ea01f6ebbc39c954e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.087ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}_{,i}[-i\partial ]Z+J_{i}Z=0.}"></span></dd></dl> <p>If the functional measure is not translationally invariant, it might be possible to express it as the product <span class="texhtml"><i>M</i>[<i>φ</i>] <span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">D</span><i>φ</i></span>, where <span class="texhtml mvar" style="font-style:italic;">M</span> is a functional and <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">D</span><i>φ</i></span> is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the <a href="/w/index.php?title=Target_space&action=edit&redlink=1" class="new" title="Target space (page does not exist)">target space</a> is diffeomorphic to <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. However, if the <a href="/wiki/Target_manifold" class="mw-redirect" title="Target manifold">target manifold</a> is some topologically nontrivial space, the concept of a translation does not even make any sense. </p><p>In that case, we would have to replace the <span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">S</span> in this equation by another functional </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathcal {S}}}={\mathcal {S}}-i\ln M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathcal {S}}}={\mathcal {S}}-i\ln M.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f9efd4db21031b657e2debac7315327f7d6d60f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.741ex; height:3.009ex;" alt="{\displaystyle {\hat {\mathcal {S}}}={\mathcal {S}}-i\ln M.}"></span></dd></dl> <p>If we expand this equation as a <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> about <i>J</i> = 0, we get the entire set of Schwinger–Dyson equations. </p> <div class="mw-heading mw-heading2"><h2 id="Localization">Localization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=26" title="Edit section: Localization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The path integrals are usually thought of as being the sum of all paths through an infinite space–time. However, in <a href="/wiki/Local_quantum_field_theory" class="mw-redirect" title="Local quantum field theory">local quantum field theory</a> we would restrict everything to lie within a finite <i>causally complete</i> region, for example inside a double light-cone. This gives a more mathematically precise and physically rigorous definition of quantum field theory. </p> <div class="mw-heading mw-heading3"><h3 id="Ward–Takahashi_identities"><span id="Ward.E2.80.93Takahashi_identities"></span>Ward–Takahashi identities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=27" title="Edit section: Ward–Takahashi identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ward%E2%80%93Takahashi_identity" title="Ward–Takahashi identity">Ward–Takahashi identity</a></div> <p>Now how about the <a href="/wiki/On_shell" class="mw-redirect" title="On shell">on shell</a> <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> for the classical case? Does it have a quantum analog as well? Yes, but with a caveat. The functional measure would have to be invariant under the one parameter group of symmetry transformation as well. </p><p>Let's just assume for simplicity here that the symmetry in question is local (not local in the sense of a <a href="/wiki/Gauge_symmetry" class="mw-redirect" title="Gauge symmetry">gauge symmetry</a>, but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question). Let's also assume that the action is local in the sense that it is the integral over spacetime of a <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a>, and that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q[{\mathcal {L}}(x)]=\partial _{\mu }f^{\mu }(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q[{\mathcal {L}}(x)]=\partial _{\mu }f^{\mu }(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/783dfeb077d75c74001b16901af455cdcee9739a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.114ex; height:3.009ex;" alt="{\displaystyle Q[{\mathcal {L}}(x)]=\partial _{\mu }f^{\mu }(x)}"></span></dd></dl> <p>for some function <span class="texhtml mvar" style="font-style:italic;">f</span> where <span class="texhtml mvar" style="font-style:italic;">f</span> only depends locally on <span class="texhtml mvar" style="font-style:italic;">φ</span> (and possibly the spacetime position). </p><p>If we don't assume any special boundary conditions, this would not be a "true" symmetry in the true sense of the term in general unless <span class="texhtml"><i>f</i> = 0</span> or something. Here, <span class="texhtml mvar" style="font-style:italic;">Q</span> is a <a href="/wiki/Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivation</a> that generates the one parameter group in question. We could have <a href="/wiki/Antiderivation" class="mw-redirect" title="Antiderivation">antiderivations</a> as well, such as <a href="/wiki/BRST_quantization" title="BRST quantization">BRST</a> and <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a>. </p><p>Let's also assume </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\mathcal {D}}\varphi \,Q[F][\varphi ]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>φ</mi> <mspace width="thinmathspace" /> <mi>Q</mi> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\mathcal {D}}\varphi \,Q[F][\varphi ]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a938d847265651ecf1e7da4508e8bae8bdac5961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.227ex; height:5.676ex;" alt="{\displaystyle \int {\mathcal {D}}\varphi \,Q[F][\varphi ]=0}"></span></dd></dl> <p>for any polynomially-bounded functional <span class="texhtml mvar" style="font-style:italic;">F</span>. This property is called the invariance of the measure, and this does not hold in general. (See <i><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">anomaly (physics)</a></i> for more details.) </p><p>Then, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\mathcal {D}}\varphi \,Q\left[Fe^{iS}\right][\varphi ]=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>φ</mi> <mspace width="thinmathspace" /> <mi>Q</mi> <mrow> <mo>[</mo> <mrow> <mi>F</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\mathcal {D}}\varphi \,Q\left[Fe^{iS}\right][\varphi ]=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dfd63c0fd16df415605732e39e93ac92b70969c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.237ex; height:5.676ex;" alt="{\displaystyle \int {\mathcal {D}}\varphi \,Q\left[Fe^{iS}\right][\varphi ]=0,}"></span></dd></dl> <p>which implies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Q[F]\rangle +i\left\langle F\int _{\partial V}f^{\mu }\,ds_{\mu }\right\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>Q</mi> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mi>i</mi> <mrow> <mo>⟨</mo> <mrow> <mi>F</mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </msub> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle Q[F]\rangle +i\left\langle F\int _{\partial V}f^{\mu }\,ds_{\mu }\right\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc0bdaaad1763e314c753caf7a59ec37fb6dba32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.155ex; height:6.176ex;" alt="{\displaystyle \langle Q[F]\rangle +i\left\langle F\int _{\partial V}f^{\mu }\,ds_{\mu }\right\rangle =0}"></span></dd></dl> <p>where the integral is over the boundary. This is the quantum analog of Noether's theorem. </p><p>Now, let's assume even further that <span class="texhtml mvar" style="font-style:italic;">Q</span> is a local integral </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\int d^{d}x\,q(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\int d^{d}x\,q(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19b038cc44cc47e1c4374749e7bc703aaf08746c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.753ex; height:5.676ex;" alt="{\displaystyle Q=\int d^{d}x\,q(x)}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)[\varphi (y)]=\delta ^{(d)}(X-y)Q[\varphi (y)]\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(x)[\varphi (y)]=\delta ^{(d)}(X-y)Q[\varphi (y)]\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/025974d6d70aca94e21a0324d78ff850a5d7741a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.3ex; height:3.343ex;" alt="{\displaystyle q(x)[\varphi (y)]=\delta ^{(d)}(X-y)Q[\varphi (y)]\,}"></span></dd></dl> <p>so that\ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)[S]=\partial _{\mu }j^{\mu }(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(x)[S]=\partial _{\mu }j^{\mu }(x)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e24e0482aaf028b21c9706779595151eba65129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.265ex; height:3.009ex;" alt="{\displaystyle q(x)[S]=\partial _{\mu }j^{\mu }(x)\,}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{\mu }(x)=f^{\mu }(x)-{\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}(x)Q[\varphi ]\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j^{\mu }(x)=f^{\mu }(x)-{\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}(x)Q[\varphi ]\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0d2f4b2a1a3bf0c05e419d2faf62fb3b6d2301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.027ex; width:34.692ex; height:6.176ex;" alt="{\displaystyle j^{\mu }(x)=f^{\mu }(x)-{\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}(x)Q[\varphi ]\,}"></span></dd></dl> <p>(this is assuming the Lagrangian only depends on <span class="texhtml mvar" style="font-style:italic;">φ</span> and its first partial derivatives! More general Lagrangians would require a modification to this definition!). We're not insisting that <span class="texhtml"><i>q</i>(<i>x</i>)</span> is the generator of a symmetry (i.e. we are <i>not</i> insisting upon the <a href="/wiki/Gauge_principle" title="Gauge principle">gauge principle</a>), but just that <span class="texhtml mvar" style="font-style:italic;">Q</span> is. And we also assume the even stronger assumption that the functional measure is locally invariant: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\mathcal {D}}\varphi \,q(x)[F][\varphi ]=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mi>φ</mi> <mspace width="thinmathspace" /> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\mathcal {D}}\varphi \,q(x)[F][\varphi ]=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab340dbb8a6c70857e63053f5126bfcb8e5efd23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.244ex; height:5.676ex;" alt="{\displaystyle \int {\mathcal {D}}\varphi \,q(x)[F][\varphi ]=0.}"></span></dd></dl> <p>Then, we would have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle q(x)[F]\rangle +i\langle Fq(x)[S]\rangle =\langle q(x)[F]\rangle +i\left\langle F\partial _{\mu }j^{\mu }(x)\right\rangle =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mi>i</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>F</mi> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mi>i</mi> <mrow> <mo>⟨</mo> <mrow> <mi>F</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle q(x)[F]\rangle +i\langle Fq(x)[S]\rangle =\langle q(x)[F]\rangle +i\left\langle F\partial _{\mu }j^{\mu }(x)\right\rangle =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd258e91dd9d104cbda7c100b7a3051c3a182699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.664ex; height:3.009ex;" alt="{\displaystyle \langle q(x)[F]\rangle +i\langle Fq(x)[S]\rangle =\langle q(x)[F]\rangle +i\left\langle F\partial _{\mu }j^{\mu }(x)\right\rangle =0.}"></span></dd></dl> <p>Alternatively, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)[S]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Q[\varphi (x)]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]=\partial _{\mu }j^{\mu }(x)\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Q[\varphi (x)]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mo stretchy="false">]</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>J</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>J</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>J</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>J</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(x)[S]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Q[\varphi (x)]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]=\partial _{\mu }j^{\mu }(x)\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Q[\varphi (x)]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5d0cb54c839fe82d7a5030dfb1455dc0b610bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:107.835ex; height:6.176ex;" alt="{\displaystyle q(x)[S]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Q[\varphi (x)]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]=\partial _{\mu }j^{\mu }(x)\left[-i{\frac {\delta }{\delta J}}\right]Z[J]+J(x)Q[\varphi (x)]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]=0.}"></span></dd></dl> <p>The above two equations are the Ward–Takahashi identities. </p><p>Now for the case where <span class="texhtml"><i>f</i> = 0</span>, we can forget about all the boundary conditions and locality assumptions. We'd simply have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle Q[F]\right\rangle =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <mi>Q</mi> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">]</mo> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle Q[F]\right\rangle =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afdc3e73929181acdd9426370f11dc3d12db08cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.59ex; height:2.843ex;" alt="{\displaystyle \left\langle Q[F]\right\rangle =0.}"></span></dd></dl> <p>Alternatively, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int d^{d}x\,J(x)Q[\varphi (x)]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>J</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int d^{d}x\,J(x)Q[\varphi (x)]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda2802a1efe3f726404ca358543487fb48fd119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.558ex; height:6.176ex;" alt="{\displaystyle \int d^{d}x\,J(x)Q[\varphi (x)]\left[-i{\frac {\delta }{\delta J}}\right]Z[J]=0.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Caveats">Caveats</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=28" title="Edit section: Caveats"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Need_for_regulators_and_renormalization">Need for regulators and renormalization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=29" title="Edit section: Need for regulators and renormalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Path integrals as they are defined here require the introduction of <a href="/wiki/Regularization_(physics)" title="Regularization (physics)">regulators</a>. Changing the scale of the regulator leads to the <a href="/wiki/Renormalization_group" title="Renormalization group">renormalization group</a>. In fact, renormalization is the major obstruction to making path integrals well-defined. </p> <div class="mw-heading mw-heading3"><h3 id="Ordering_prescription">Ordering prescription</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=30" title="Edit section: Ordering prescription"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Regardless of whether one works in configuration space or phase space, when equating the <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">operator formalism</a> and the path integral formulation, an ordering prescription is required to resolve the ambiguity in the correspondence between non-commutative operators and the commutative functions that appear in path integrands. For example, the operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <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"> <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">{\displaystyle {\frac {1}{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41a032688cbdb4fe6c687449a43888fe8e03f916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.121ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}"></span> can be translated back as either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qp-{\frac {i\hbar }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mi>p</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qp-{\frac {i\hbar }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/643225cca3dd9112a8f2cd04c28c3d654edfb2ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.024ex; height:5.343ex;" alt="{\displaystyle qp-{\frac {i\hbar }{2}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qp+{\frac {i\hbar }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qp+{\frac {i\hbar }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e78e37a424c84a56adc6d6675560bcb183de66bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.024ex; height:5.343ex;" alt="{\displaystyle qp+{\frac {i\hbar }{2}}}"></span>, or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qp}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qp}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/068c7d66b3d8851a83610e06d5370d113ac387af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.009ex;" alt="{\displaystyle qp}"></span> depending on whether one chooses the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {q}}{\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>q</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {q}}{\hat {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bb1d840482dd19c737085c98b31ed24aac554b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.737ex; height:2.509ex;" alt="{\displaystyle {\hat {q}}{\hat {p}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}{\hat {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}{\hat {q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d87a17961aa867cb559c093a28e17fea2e174d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.826ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}{\hat {q}}}"></span>, or Weyl ordering prescription; conversely, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qp}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qp}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/068c7d66b3d8851a83610e06d5370d113ac387af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.009ex;" alt="{\displaystyle qp}"></span> can be translated to either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {q}}{\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>q</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {q}}{\hat {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bb1d840482dd19c737085c98b31ed24aac554b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.737ex; height:2.509ex;" alt="{\displaystyle {\hat {q}}{\hat {p}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}{\hat {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}{\hat {q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d87a17961aa867cb559c093a28e17fea2e174d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.826ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}{\hat {q}}}"></span>, or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <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"> <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">{\displaystyle {\frac {1}{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41a032688cbdb4fe6c687449a43888fe8e03f916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.121ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}"></span> for the same respective choice of ordering prescription. </p> <div class="mw-heading mw-heading2"><h2 id="Path_integral_in_quantum-mechanical_interpretation">Path integral in quantum-mechanical interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=31" title="Edit section: Path integral in quantum-mechanical interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In one <a href="/wiki/Interpretation_of_quantum_mechanics" class="mw-redirect" title="Interpretation of quantum mechanics">interpretation of quantum mechanics</a>, the "sum over histories" interpretation, the path integral is taken to be fundamental, and reality is viewed as a single indistinguishable "class" of paths that all share the same events.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> For this interpretation, it is crucial to understand what exactly an event is. The sum-over-histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> claim the interpretation explains the <a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">Einstein–Podolsky–Rosen paradox</a> without resorting to <a href="/wiki/Action_at_a_distance" title="Action at a distance">nonlocality</a>. </p><p>Some<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag possibly uses too-vague attribution or weasel words. (August 2014)">who?</span></a></i>]</sup> advocates of interpretations of quantum mechanics emphasizing <a href="/wiki/Decoherence" class="mw-redirect" title="Decoherence">decoherence</a> have attempted to make more rigorous the notion of extracting a classical-like "coarse-grained" history from the space of all possible histories. </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_gravity">Quantum gravity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=32" title="Edit section: Quantum gravity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Whereas in quantum mechanics the path integral formulation is fully equivalent to other formulations, it may be that it can be extended to quantum gravity, which would make it different from the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> model. Feynman had some success in this direction, and his work has been extended by <a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a> and others.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Approaches that use this method include <a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">causal dynamical triangulations</a> and <a href="/wiki/Spinfoam" class="mw-redirect" title="Spinfoam">spinfoam</a> models. </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_tunneling">Quantum tunneling</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=33" title="Edit section: Quantum tunneling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Quantum tunnelling</a> can be modeled by using the path integral formation to determine the action of the trajectory through a potential barrier. Using the <a href="/wiki/WKB_approximation" title="WKB approximation">WKB approximation</a>, the tunneling rate (<span class="texhtml">Γ</span>) can be determined to be of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =A_{\mathrm {o} }\exp \left(-{\frac {S_{\mathrm {eff} }}{\hbar }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma =A_{\mathrm {o} }\exp \left(-{\frac {S_{\mathrm {eff} }}{\hbar }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e021a7ab98970144e8ea030e9c5466ce53d43ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.964ex; height:6.176ex;" alt="{\displaystyle \Gamma =A_{\mathrm {o} }\exp \left(-{\frac {S_{\mathrm {eff} }}{\hbar }}\right)}"></span></dd></dl> <p>with the effective action <span class="texhtml"><i>S</i><sub>eff</sub></span> and pre-exponential factor <span class="texhtml"><i>A</i><sub>o</sub></span>. This form is specifically useful in a <a href="/wiki/Dissipative_system" title="Dissipative system">dissipative system</a>, in which the systems and surroundings must be modeled together. Using the <a href="/wiki/Langevin_equation" title="Langevin equation">Langevin equation</a> to model <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a>, the path integral formation can be used to determine an effective action and pre-exponential model to see the effect of dissipation on tunnelling.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> From this model, tunneling rates of macroscopic systems (at finite temperatures) can be predicted. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=34" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Static_forces_and_virtual-particle_exchange" title="Static forces and virtual-particle exchange">Static forces and virtual-particle exchange</a></li> <li><a href="/wiki/Feynman_checkerboard" title="Feynman checkerboard">Feynman checkerboard</a></li> <li><a href="/wiki/Berezin_integral" title="Berezin integral">Berezin integral</a></li> <li><a href="/wiki/Propagator" title="Propagator">Propagators</a></li> <li><a href="/wiki/Wheeler%E2%80%93Feynman_absorber_theory" title="Wheeler–Feynman absorber theory">Wheeler–Feynman absorber theory</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Path_integrals_in_polymer_science" title="Path integrals in polymer science">Path integrals in polymer science</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Remarks">Remarks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=35" title="Edit section: Remarks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">For a simplified, step-by-step derivation of the above relation, see <a rel="nofollow" class="external text" href="http://www.quantumfieldtheory.info/website_Chap18.pdf">Path Integrals in Quantum Theories: A Pedagogic 1st Step</a>.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">For a brief account of the origins of these difficulties, see <a href="#CITEREFHall2013">Hall 2013</a>, Section 20.6.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=36" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeinberg2002">Weinberg 2002</a>, Chapter 9.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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 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M.</a> (1933). <a rel="nofollow" class="external text" href="http://www.hep.anl.gov/czachos/soysoy/Dirac33.pdf">"The Lagrangian in Quantum Mechanics"</a> <span class="cs1-format">(PDF)</span>. <i>Physikalische Zeitschrift der Sowjetunion</i>. <b>3</b>: <span class="nowrap">64–</span>72.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physikalische+Zeitschrift+der+Sowjetunion&rft.atitle=The+Lagrangian+in+Quantum+Mechanics&rft.volume=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E64-%3C%2Fspan%3E72&rft.date=1933&rft.aulast=Dirac&rft.aufirst=Paul+A.+M.&rft_id=http%3A%2F%2Fwww.hep.anl.gov%2Fczachos%2Fsoysoy%2FDirac33.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuruKleinert1979" class="citation journal cs1"><a href="/wiki/%C4%B0smail_Hakk%C4%B1_Duru" title="İsmail Hakkı Duru">Duru, İ. H.</a>; <a href="/wiki/Hagen_Kleinert" title="Hagen Kleinert">Kleinert, Hagen</a> (1979). <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/~kleinert/kleiner_re65/65.pdf">"Solution of the path integral for the H-atom"</a> <span class="cs1-format">(PDF)</span>. <i>Physics Letters</i>. <b>84B</b> (2): <span class="nowrap">185–</span>188. <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/1979PhLB...84..185D">1979PhLB...84..185D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2879%2990280-6">10.1016/0370-2693(79)90280-6</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2007-11-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters&rft.atitle=Solution+of+the+path+integral+for+the+H-atom&rft.volume=84B&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E185-%3C%2Fspan%3E188&rft.date=1979&rft_id=info%3Adoi%2F10.1016%2F0370-2693%2879%2990280-6&rft_id=info%3Abibcode%2F1979PhLB...84..185D&rft.aulast=Duru&rft.aufirst=%C4%B0.+H.&rft.au=Kleinert%2C+Hagen&rft_id=http%3A%2F%2Fwww.physik.fu-berlin.de%2F~kleinert%2Fkleiner_re65%2F65.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEtingof2002" class="citation web cs1"><a href="/wiki/Pavel_Etingof" title="Pavel Etingof">Etingof, P.</a> (2002). <a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/mathematics/18-238-geometry-and-quantum-field-theory-fall-2002/index.htm">"Geometry and Quantum Field Theory"</a>. MIT OpenCourseWare.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Geometry+and+Quantum+Field+Theory&rft.pub=MIT+OpenCourseWare&rft.date=2002&rft.aulast=Etingof&rft.aufirst=P.&rft_id=http%3A%2F%2Focw.mit.edu%2Fcourses%2Fmathematics%2F18-238-geometry-and-quantum-field-theory-fall-2002%2Findex.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span> <small>This course, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals.</small></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman2005" class="citation book cs1"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman, R. P.</a> (2005) [1942/1948]. Brown, L. M. (ed.). <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/910611"><i>Feynman's Thesis — A New Approach to Quantum Theory</i></a>. World Scientific. <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/2005ftna.book.....B">2005ftna.book.....B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F5852">10.1142/5852</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-256-366-8" title="Special:BookSources/978-981-256-366-8"><bdi>978-981-256-366-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Feynman%27s+Thesis+%E2%80%94+A+New+Approach+to+Quantum+Theory&rft.pub=World+Scientific&rft.date=2005&rft_id=info%3Adoi%2F10.1142%2F5852&rft_id=info%3Abibcode%2F2005ftna.book.....B&rft.isbn=978-981-256-366-8&rft.aulast=Feynman&rft.aufirst=R.+P.&rft_id=https%3A%2F%2Fcds.cern.ch%2Frecord%2F910611&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span> <small>The 1942 thesis. Also includes Dirac's 1933 paper and Feynman's 1948 publication.</small></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman1948" class="citation journal cs1">Feynman, R. P. (1948). <a rel="nofollow" class="external text" href="https://authors.library.caltech.edu/47756/1/FEYrmp48.pdf">"Space-Time Approach to Non-Relativistic Quantum Mechanics"</a> <span class="cs1-format">(PDF)</span>. <i>Reviews of Modern Physics</i>. <b>20</b> (2): <span class="nowrap">367–</span>387. <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/1948RvMP...20..367F">1948RvMP...20..367F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.20.367">10.1103/RevModPhys.20.367</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Space-Time+Approach+to+Non-Relativistic+Quantum+Mechanics&rft.volume=20&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E367-%3C%2Fspan%3E387&rft.date=1948&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.20.367&rft_id=info%3Abibcode%2F1948RvMP...20..367F&rft.aulast=Feynman&rft.aufirst=R.+P.&rft_id=https%3A%2F%2Fauthors.library.caltech.edu%2F47756%2F1%2FFEYrmp48.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynmanHibbs1965" class="citation book cs1">Feynman, R. P.; Hibbs, A. R. (1965). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quantummechanics0000feyn"><i>Quantum Mechanics and Path Integrals</i></a></span>. New York: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-020650-2" title="Special:BookSources/978-0-07-020650-2"><bdi>978-0-07-020650-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics+and+Path+Integrals&rft.place=New+York&rft.pub=McGraw-Hill&rft.date=1965&rft.isbn=978-0-07-020650-2&rft.aulast=Feynman&rft.aufirst=R.+P.&rft.au=Hibbs%2C+A.+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantummechanics0000feyn&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span> <small>The historical reference written by the inventor of the path integral formulation himself and one of his students.</small></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynmanHibbsStyer2010" class="citation book cs1">Feynman, R. P.; <a href="/wiki/Albert_Hibbs" title="Albert Hibbs">Hibbs, A. R.</a>; <a href="/wiki/Daniel_F._Styer" title="Daniel F. Styer">Styer, D. F.</a> (2010). <i>Quantum Mechanics and Path Integrals</i>. Mineola, NY: Dover Publications. pp. <span class="nowrap">29–</span>31. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47722-0" title="Special:BookSources/978-0-486-47722-0"><bdi>978-0-486-47722-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics+and+Path+Integrals&rft.place=Mineola%2C+NY&rft.pages=%3Cspan+class%3D%22nowrap%22%3E29-%3C%2Fspan%3E31&rft.pub=Dover+Publications&rft.date=2010&rft.isbn=978-0-486-47722-0&rft.aulast=Feynman&rft.aufirst=R.+P.&rft.au=Hibbs%2C+A.+R.&rft.au=Styer%2C+D.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGell-Mann1993" class="citation book cs1"><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann, Murray</a> (1993). "Most of the Good Stuff". In Brown, Laurie M.; Rigden, John S. (eds.). <i>Memories Of Richard Feynman</i>. American Institute of Physics. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0883188705" title="Special:BookSources/978-0883188705"><bdi>978-0883188705</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Most+of+the+Good+Stuff&rft.btitle=Memories+Of+Richard+Feynman&rft.pub=American+Institute+of+Physics&rft.date=1993&rft.isbn=978-0883188705&rft.aulast=Gell-Mann&rft.aufirst=Murray&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlimmJaffe1981" class="citation book cs1">Glimm, J. & Jaffe, A. (1981). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quantumphysicsfu0000glim"><i>Quantum Physics: A Functional Integral Point of View</i></a></span>. New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90562-4" title="Special:BookSources/978-0-387-90562-4"><bdi>978-0-387-90562-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Physics%3A+A+Functional+Integral+Point+of+View&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1981&rft.isbn=978-0-387-90562-4&rft.aulast=Glimm&rft.aufirst=J.&rft.au=Jaffe%2C+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumphysicsfu0000glim&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGroscheSteiner1998" class="citation book cs1">Grosche, Christian & Steiner, Frank (1998). <i>Handbook of Feynman Path Integrals</i>. Springer Tracts in Modern Physics 145. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-57135-3" title="Special:BookSources/978-3-540-57135-3"><bdi>978-3-540-57135-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Feynman+Path+Integrals&rft.series=Springer+Tracts+in+Modern+Physics+145&rft.pub=Springer-Verlag&rft.date=1998&rft.isbn=978-3-540-57135-3&rft.aulast=Grosche&rft.aufirst=Christian&rft.au=Steiner%2C+Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrosche1992" class="citation arxiv cs1">Grosche, Christian (1992). "An Introduction into the Feynman Path Integral". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9302097">hep-th/9302097</a></span>.</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=An+Introduction+into+the+Feynman+Path+Integral&rft.date=1992&rft_id=info%3Aarxiv%2Fhep-th%2F9302097&rft.aulast=Grosche&rft.aufirst=Christian&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2013" class="citation book cs1">Hall, Brian C. (2013). <i>Quantum Theory for Mathematicians</i>. Graduate Texts in Mathematics. Vol. 267. Springer. <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/2013qtm..book.....H">2013qtm..book.....H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4614-7116-5">10.1007/978-1-4614-7116-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-7115-8" title="Special:BookSources/978-1-4614-7115-8"><bdi>978-1-4614-7115-8</bdi></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:117837329">117837329</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory+for+Mathematicians&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2013&rft_id=info%3Adoi%2F10.1007%2F978-1-4614-7116-5&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A117837329%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2013qtm..book.....H&rft.isbn=978-1-4614-7115-8&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFInomataKuratsujiGerry1992" class="citation book cs1">Inomata, Akira; Kuratsuji, Hiroshi; Gerry, Christopher (1992). <i>Path Integrals and Coherent States of SU(2) and SU(1,1)</i>. Singapore: World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-0656-7" title="Special:BookSources/978-981-02-0656-7"><bdi>978-981-02-0656-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Path+Integrals+and+Coherent+States+of+SU%282%29+and+SU%281%2C1%29&rft.place=Singapore&rft.pub=World+Scientific&rft.date=1992&rft.isbn=978-981-02-0656-7&rft.aulast=Inomata&rft.aufirst=Akira&rft.au=Kuratsuji%2C+Hiroshi&rft.au=Gerry%2C+Christopher&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJankePelster2008" class="citation book cs1">Janke, W.; Pelster, Axel, eds. (2008). <i>Path Integrals--New Trends And Perspectives</i>. Proceedings Of The 9Th International Conference. World Scientific Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-283-726-4" title="Special:BookSources/978-981-283-726-4"><bdi>978-981-283-726-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Path+Integrals--New+Trends+And+Perspectives&rft.series=Proceedings+Of+The+9Th+International+Conference&rft.pub=World+Scientific+Publishing&rft.date=2008&rft.isbn=978-981-283-726-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnsonLapidus2002" class="citation book cs1">Johnson, Gerald W.; Lapidus, Michel L. (2002). <i>The Feynman Integral and Feynman's Operational Calculus</i>. Oxford Mathematical Monographs. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-851572-2" title="Special:BookSources/978-0-19-851572-2"><bdi>978-0-19-851572-2</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+Feynman+Integral+and+Feynman%27s+Operational+Calculus&rft.series=Oxford+Mathematical+Monographs&rft.pub=Oxford+University+Press&rft.date=2002&rft.isbn=978-0-19-851572-2&rft.aulast=Johnson&rft.aufirst=Gerald+W.&rft.au=Lapidus%2C+Michel+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlauder2010" class="citation book cs1"><a href="/wiki/John_R._Klauder" title="John R. Klauder">Klauder, John R.</a> (2010). <i>A Modern Approach to Functional Integration</i>. New York: Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4790-2" title="Special:BookSources/978-0-8176-4790-2"><bdi>978-0-8176-4790-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Modern+Approach+to+Functional+Integration&rft.place=New+York&rft.pub=Birkh%C3%A4user&rft.date=2010&rft.isbn=978-0-8176-4790-2&rft.aulast=Klauder&rft.aufirst=John+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleinert2004" class="citation book cs1"><a href="/wiki/Hagen_Kleinert" title="Hagen Kleinert">Kleinert, Hagen</a> (2004). <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/~kleinert/b5"><i>Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets</i></a> (4th ed.). Singapore: World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-238-107-1" title="Special:BookSources/978-981-238-107-1"><bdi>978-981-238-107-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Path+Integrals+in+Quantum+Mechanics%2C+Statistics%2C+Polymer+Physics%2C+and+Financial+Markets&rft.place=Singapore&rft.edition=4th&rft.pub=World+Scientific&rft.date=2004&rft.isbn=978-981-238-107-1&rft.aulast=Kleinert&rft.aufirst=Hagen&rft_id=http%3A%2F%2Fwww.physik.fu-berlin.de%2F~kleinert%2Fb5&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMacKenzie2000" class="citation arxiv cs1">MacKenzie, Richard (2000). "Path Integral Methods and Applications". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0004090">quant-ph/0004090</a></span>.</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=Path+Integral+Methods+and+Applications&rft.date=2000&rft_id=info%3Aarxiv%2Fquant-ph%2F0004090&rft.aulast=MacKenzie&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMazzucchi2009" class="citation book cs1">Mazzucchi, S. (2009). <i>Mathematical Feynman path integrals and their applications</i>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-283-690-8" title="Special:BookSources/978-981-283-690-8"><bdi>978-981-283-690-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Feynman+path+integrals+and+their+applications&rft.pub=World+Scientific&rft.date=2009&rft.isbn=978-981-283-690-8&rft.aulast=Mazzucchi&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMüller-Kirsten2012" class="citation book cs1">Müller-Kirsten, Harald J. W. (2012). <i>Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral</i> (2nd ed.). Singapore: World Scientific.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantum+Mechanics%3A+Schr%C3%B6dinger+Equation+and+Path+Integral&rft.place=Singapore&rft.edition=2nd&rft.pub=World+Scientific&rft.date=2012&rft.aulast=M%C3%BCller-Kirsten&rft.aufirst=Harald+J.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRivers1987" class="citation book cs1">Rivers, R. J. (1987). <i>Path Integrals Methods in Quantum Field Theory</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-25979-8" title="Special:BookSources/978-0-521-25979-8"><bdi>978-0-521-25979-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Path+Integrals+Methods+in+Quantum+Field+Theory&rft.pub=Cambridge+University+Press&rft.date=1987&rft.isbn=978-0-521-25979-8&rft.aulast=Rivers&rft.aufirst=R.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyder1985" class="citation book cs1">Ryder, Lewis H. (1985). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quantumfieldtheo0000ryde"><i>Quantum Field Theory</i></a></span>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-33859-2" title="Special:BookSources/978-0-521-33859-2"><bdi>978-0-521-33859-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory&rft.pub=Cambridge+University+Press&rft.date=1985&rft.isbn=978-0-521-33859-2&rft.aulast=Ryder&rft.aufirst=Lewis+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumfieldtheo0000ryde&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span> Highly readable textbook; introduction to relativistic QFT for particle physics.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchulman1981" class="citation book cs1">Schulman, L S. (1981). <i>Techniques & Applications of Path Integration</i>. New York: John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-44528-1" title="Special:BookSources/978-0-486-44528-1"><bdi>978-0-486-44528-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Techniques+%26+Applications+of+Path+Integration&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=1981&rft.isbn=978-0-486-44528-1&rft.aulast=Schulman&rft.aufirst=L+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimon1979" class="citation book cs1"><a href="/wiki/Barry_Simon" title="Barry Simon">Simon, B.</a> (1979). <i>Functional Integration and Quantum Physics</i>. New York: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-6941-3" title="Special:BookSources/978-0-8218-6941-3"><bdi>978-0-8218-6941-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Integration+and+Quantum+Physics&rft.place=New+York&rft.pub=Academic+Press&rft.date=1979&rft.isbn=978-0-8218-6941-3&rft.aulast=Simon&rft.aufirst=B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSinhaSorkin1991" class="citation journal cs1">Sinha, Sukanya; Sorkin, Rafael D. (1991). <a rel="nofollow" class="external text" href="https://www.perimeterinstitute.ca/personal/rsorkin/some.papers/63.eprb.pdf">"A Sum-over-histories Account of an EPR(B) Experiment"</a> <span class="cs1-format">(PDF)</span>. <i>Foundations of Physics Letters</i>. <b>4</b> (4): <span class="nowrap">303–</span>335. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991FoPhL...4..303S">1991FoPhL...4..303S</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%2FBF00665892">10.1007/BF00665892</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:121370426">121370426</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=A+Sum-over-histories+Account+of+an+EPR%28B%29+Experiment&rft.volume=4&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E303-%3C%2Fspan%3E335&rft.date=1991&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121370426%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00665892&rft_id=info%3Abibcode%2F1991FoPhL...4..303S&rft.aulast=Sinha&rft.aufirst=Sukanya&rft.au=Sorkin%2C+Rafael+D.&rft_id=https%3A%2F%2Fwww.perimeterinstitute.ca%2Fpersonal%2Frsorkin%2Fsome.papers%2F63.eprb.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTomé1998" class="citation book cs1"><a href="/wiki/Wolfgang_A._Tom%C3%A9" title="Wolfgang A. Tomé">Tomé, W. A.</a> (1998). <i>Path Integrals on Group Manifolds</i>. Singapore: World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-3355-6" title="Special:BookSources/978-981-02-3355-6"><bdi>978-981-02-3355-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Path+Integrals+on+Group+Manifolds&rft.place=Singapore&rft.pub=World+Scientific&rft.date=1998&rft.isbn=978-981-02-3355-6&rft.aulast=Tom%C3%A9&rft.aufirst=W.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span> Discusses the definition of Path Integrals for systems whose kinematical variables are the generators of a real separable, connected Lie group with irreducible, square integrable representations.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Vleck1928" class="citation journal cs1"><a href="/wiki/John_Hasbrouck_Van_Vleck" title="John Hasbrouck Van Vleck">Van Vleck, J. H.</a> (1928). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1085402">"The correspondence principle in the statistical interpretation of quantum mechanics"</a>. <i>Proceedings of the National Academy of Sciences of the United States of America</i>. <b>14</b> (2): <span class="nowrap">178–</span>188. <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/1928PNAS...14..178V">1928PNAS...14..178V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.14.2.178">10.1073/pnas.14.2.178</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1085402">1085402</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16577107">16577107</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+National+Academy+of+Sciences+of+the+United+States+of+America&rft.atitle=The+correspondence+principle+in+the+statistical+interpretation+of+quantum+mechanics&rft.volume=14&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E178-%3C%2Fspan%3E188&rft.date=1928&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1085402%23id-name%3DPMC&rft_id=info%3Apmid%2F16577107&rft_id=info%3Adoi%2F10.1073%2Fpnas.14.2.178&rft_id=info%3Abibcode%2F1928PNAS...14..178V&rft.aulast=Van+Vleck&rft.aufirst=J.+H.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1085402&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg2002" class="citation cs2"><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg, S.</a> (2002) [1995], <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quantumtheoryoff00stev"><i>Foundations</i></a></span>, The Quantum Theory of Fields, vol. 1, Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55001-7" title="Special:BookSources/978-0-521-55001-7"><bdi>978-0-521-55001-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations&rft.place=Cambridge&rft.series=The+Quantum+Theory+of+Fields&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-0-521-55001-7&rft.aulast=Weinberg&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumtheoryoff00stev&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZee2010" class="citation book cs1"><a href="/wiki/Anthony_Zee" title="Anthony Zee">Zee, A.</a> (2010-02-21). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780691140346"><i>Quantum Field Theory in a Nutshell</i></a></span> (Second ed.). Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-14034-6" title="Special:BookSources/978-0-691-14034-6"><bdi>978-0-691-14034-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory+in+a+Nutshell&rft.edition=Second&rft.pub=Princeton+University+Press&rft.date=2010-02-21&rft.isbn=978-0-691-14034-6&rft.aulast=Zee&rft.aufirst=A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_9780691140346&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span> A great introduction to Path Integrals (Chapter 1) and QFT in general.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZinn_Justin2004" class="citation book cs1"><a href="/wiki/Jean_Zinn-Justin" title="Jean Zinn-Justin">Zinn Justin, J.</a> (2004). <i>Path Integrals in Quantum Mechanics</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-856674-8" title="Special:BookSources/978-0-19-856674-8"><bdi>978-0-19-856674-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Path+Integrals+in+Quantum+Mechanics&rft.pub=Oxford+University+Press&rft.date=2004&rft.isbn=978-0-19-856674-8&rft.aulast=Zinn+Justin&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeshmukh2023" class="citation book cs1"><a href="/wiki/Pranawachandra_Deshmukh" title="Pranawachandra Deshmukh">Deshmukh, P. C.</a> (2023). <i>Quantum Mechanics Formalism, Methodologies, and Applications</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1316512258" title="Special:BookSources/978-1316512258"><bdi>978-1316512258</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics+Formalism%2C+Methodologies%2C+and+Applications&rft.pub=Cambridge+University+Press&rft.date=2023&rft.isbn=978-1316512258&rft.aulast=Deshmukh&rft.aufirst=P.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APath+integral+formulation" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Path_integral_formulation&action=edit&section=38" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><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" /></a></span></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Path_integral_formulation" class="extiw" title="q:Special:Search/Path integral formulation">Path integral formulation</a></b></i>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/Path_integral">Path integral on Scholarpedia</a></li> <li><a rel="nofollow" class="external text" href="http://www.quantumfieldtheory.info/website_Chap18.pdf">Path Integrals in Quantum Theories: A Pedagogic 1st Step</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=QTjmLBzAdAA">A mathematically rigorous approach to perturbative path integrals</a> via animation on YouTube</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=vSFRN-ymfgE">Feynman's Infinite Quantum Paths</a> | PBS Space Time. 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(Video, 15:48)</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_mechanics332" style="padding:3px"><table class="nowraplinks hlist mw-collapsible expanded navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics332" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a class="mw-selflink selflink">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a 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> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Quantum_mechanics" title="Category:Quantum mechanics">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Richard_Feynman43" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed 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:Richard_Feynman" title="Template:Richard Feynman"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Richard_Feynman" title="Template talk:Richard Feynman"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Richard_Feynman" title="Special:EditPage/Template:Richard Feynman"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Richard_Feynman43" style="font-size:114%;margin:0 4em"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Career</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Wheeler%E2%80%93Feynman_absorber_theory" title="Wheeler–Feynman absorber theory">Wheeler–Feynman absorber theory</a></li> <li><a href="/wiki/Bethe%E2%80%93Feynman_formula" title="Bethe–Feynman formula">Bethe–Feynman formula</a></li> <li><a href="/wiki/Hellmann%E2%80%93Feynman_theorem" title="Hellmann–Feynman theorem">Hellmann–Feynman theorem</a></li> <li><a href="/wiki/Feynman_slash_notation" title="Feynman slash notation">Feynman slash notation</a></li> <li><a href="/wiki/Feynman_parametrization" title="Feynman parametrization">Feynman parametrization</a></li> <li><a class="mw-selflink selflink">Path integral formulation</a></li> <li><a href="/wiki/Parton_(particle_physics)" title="Parton (particle physics)">Parton model</a></li> <li><a href="/wiki/Sticky_bead_argument" title="Sticky bead argument">Sticky bead argument</a></li> <li><a href="/wiki/One-electron_universe" title="One-electron universe">One-electron universe</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automaton</a></li> <li><a href="/wiki/Rogers_Commission_Report" title="Rogers Commission Report">Rogers Commission Report</a></li> <li><a href="/wiki/Feynman_checkerboard" title="Feynman checkerboard">Feynman checkerboard</a></li> <li><a href="/wiki/Feynman_sprinkler" title="Feynman sprinkler">Feynman sprinkler</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Works</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>"<a href="/wiki/There%27s_Plenty_of_Room_at_the_Bottom" title="There's Plenty of Room at the Bottom">There's Plenty of Room at the Bottom</a>" <span style="font-size:85%;">(1959)</span></li> <li><i><a href="/wiki/The_Feynman_Lectures_on_Physics" title="The Feynman Lectures on Physics">The Feynman Lectures on Physics</a></i> <span style="font-size:85%;">(1964)</span></li> <li><i><a href="/wiki/The_Character_of_Physical_Law" title="The Character of Physical Law">The Character of Physical Law</a></i> <span style="font-size:85%;">(1965)</span></li> <li><i><a href="/wiki/QED:_The_Strange_Theory_of_Light_and_Matter" title="QED: The Strange Theory of Light and Matter">QED: The Strange Theory of Light and Matter</a></i> <span style="font-size:85%;">(1985)</span></li> <li><i><a href="/wiki/Surely_You%27re_Joking,_Mr._Feynman!" title="Surely You're Joking, Mr. Feynman!">Surely You're Joking, Mr. Feynman!</a></i> <span style="font-size:85%;">(1985)</span></li> <li><i><a href="/wiki/What_Do_You_Care_What_Other_People_Think%3F" title="What Do You Care What Other People Think?">What Do You Care What Other People Think?</a></i> <span style="font-size:85%;">(1988)</span></li> <li><i><a href="/wiki/Feynman%27s_Lost_Lecture" title="Feynman's Lost Lecture">Feynman's Lost Lecture: The Motion of Planets Around the Sun</a></i> <span style="font-size:85%;">(1997)</span></li> <li><i><a href="/wiki/The_Meaning_of_It_All" title="The Meaning of It All">The Meaning of It All</a></i> <span style="font-size:85%;">(1999)</span></li> <li><i><a href="/wiki/The_Pleasure_of_Finding_Things_Out" title="The Pleasure of Finding Things Out">The Pleasure of Finding Things Out</a></i> <span style="font-size:85%;">(1999)</span></li> <li><i><a href="/wiki/Perfectly_Reasonable_Deviations_from_the_Beaten_Track" title="Perfectly Reasonable Deviations from the Beaten Track">Perfectly Reasonable Deviations from the Beaten Track</a></i> <span style="font-size:85%;">(2005)</span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Family</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Joan_Feynman" title="Joan Feynman">Joan Feynman</a> <span style="font-size:85%;">(sister)</span></li> <li><a href="/wiki/Charles_Hirshberg" title="Charles Hirshberg">Charles Hirshberg</a> <span style="font-size:85%;">(nephew)</span></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 hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_things_named_after_Richard_Feynman" title="List of things named after Richard Feynman">Namesakes</a></li> <li><i><a href="/wiki/Quantum_Man:_Richard_Feynman%27s_Life_in_Science" title="Quantum Man: Richard Feynman's Life in Science">Quantum Man: Richard Feynman's Life in Science</a></i></li> <li><i><a href="/wiki/Tuva_or_Bust!" title="Tuva or Bust!">Tuva or Bust!</a></i></li> <li><a href="/wiki/Feynman_Prize_in_Nanotechnology" title="Feynman Prize in Nanotechnology">Feynman Prize in Nanotechnology</a></li> <li><a href="/wiki/Infinity_(1996_film)" title="Infinity (1996 film)"><i>Infinity</i> <small>(1996 film)</small></a></li> <li><a href="/wiki/QED_(play)" title="QED (play)"><i>QED</i> <span style="font-size:85%;">(2001 play)</span></a></li> <li><a href="/wiki/The_Challenger_Disaster" title="The Challenger Disaster"><i>The Challenger Disaster</i> <small>(2013 film)</small></a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Path_integral_formulation&oldid=1261078587">https://en.wikipedia.org/w/index.php?title=Path_integral_formulation&oldid=1261078587</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Concepts_in_physics" title="Category:Concepts in physics">Concepts in physics</a></li><li><a href="/wiki/Category:Statistical_mechanics" title="Category:Statistical mechanics">Statistical mechanics</a></li><li><a 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[\"CITEREFInomataKuratsujiGerry1992\"] = 1,\n [\"CITEREFJankePelster2008\"] = 1,\n [\"CITEREFJohnsonLapidus2002\"] = 1,\n [\"CITEREFKlauder2010\"] = 1,\n [\"CITEREFKleinert2004\"] = 1,\n [\"CITEREFMacKenzie2000\"] = 1,\n [\"CITEREFMazzucchi2009\"] = 1,\n [\"CITEREFMüller-Kirsten2012\"] = 1,\n [\"CITEREFPössel2006\"] = 1,\n [\"CITEREFRivers1987\"] = 1,\n [\"CITEREFRyder1985\"] = 1,\n [\"CITEREFSchulman1981\"] = 1,\n [\"CITEREFSimon1979\"] = 1,\n [\"CITEREFSinhaSorkin1991\"] = 1,\n [\"CITEREFTomé1998\"] = 1,\n [\"CITEREFVan_Vleck1928\"] = 1,\n [\"CITEREFVinokur2015\"] = 1,\n [\"CITEREFWeinberg2002\"] = 1,\n [\"CITEREFWolfram2020\"] = 1,\n [\"CITEREFWood2023\"] = 1,\n [\"CITEREFZee2010\"] = 1,\n [\"CITEREFZinn-Justin2009\"] = 1,\n [\"CITEREFZinn_Justin2004\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 2,\n [\"=\"] = 24,\n [\"About\"] = 1,\n [\"Angbr\"] = 2,\n [\"Blockquote\"] = 1,\n [\"Bra-ket\"] = 1,\n [\"Citation\"] = 1,\n [\"Cite arXiv\"] = 4,\n [\"Cite book\"] = 25,\n [\"Cite journal\"] = 9,\n [\"Cite web\"] = 6,\n [\"Colend\"] = 1,\n [\"Cols\"] = 1,\n [\"Divcol\"] = 1,\n [\"Divcol end\"] = 1,\n [\"Harvnb\"] = 16,\n [\"Intmath\"] = 1,\n [\"Main\"] = 4,\n [\"Math\"] = 100,\n [\"Mathcal\"] = 12,\n [\"Mvar\"] = 146,\n [\"Quantum field theory\"] = 1,\n [\"Quantum mechanics\"] = 1,\n [\"Quantum mechanics topics\"] = 1,\n [\"Reflist\"] = 2,\n [\"Richard Feynman\"] = 1,\n [\"See also\"] = 1,\n [\"Sfn\"] = 1,\n [\"Sfrac\"] = 8,\n [\"Short description\"] = 1,\n [\"Sqrt\"] = 2,\n [\"Who\"] = 1,\n [\"Wikiquote\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-549fbf847c-4kpp4","timestamp":"20250224161332","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Path integral 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Path integral formulation - Wikipedia