Quantum field theory

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id="toc-Condensed-matter-physics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Principles" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Principles"> <div class="vector-toc-text"> 2 Principles </div> </a> <button aria-controls="toc-Principles-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Principles subsection </button> <ul id="toc-Principles-sublist" class="vector-toc-list"> <li id="toc-Classical_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_fields"> <div class="vector-toc-text"> 2.1 Classical fields </div> </a> <ul id="toc-Classical_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Canonical_quantization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Canonical_quantization"> <div class="vector-toc-text"> 2.2 Canonical quantization </div> </a> <ul id="toc-Canonical_quantization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Path_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Path_integrals"> <div class="vector-toc-text"> 2.3 Path integrals </div> </a> <ul id="toc-Path_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Two-point_correlation_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two-point_correlation_function"> <div class="vector-toc-text"> 2.4 Two-point correlation function </div> </a> <ul id="toc-Two-point_correlation_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Feynman_diagram" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Feynman_diagram"> <div class="vector-toc-text"> 2.5 Feynman diagram </div> </a> <ul id="toc-Feynman_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Renormalization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Renormalization"> <div class="vector-toc-text"> 2.6 Renormalization </div> </a> <ul id="toc-Renormalization-sublist" class="vector-toc-list"> <li id="toc-Renormalization_group" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Renormalization_group"> <div class="vector-toc-text"> 2.6.1 Renormalization group </div> </a> <ul id="toc-Renormalization_group-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_theories"> <div class="vector-toc-text"> 2.7 Other theories </div> </a> <ul id="toc-Other_theories-sublist" class="vector-toc-list"> <li id="toc-Gauge_symmetry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Gauge_symmetry"> <div class="vector-toc-text"> 2.7.1 Gauge symmetry </div> </a> <ul id="toc-Gauge_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spontaneous_symmetry-breaking" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Spontaneous_symmetry-breaking"> <div class="vector-toc-text"> 2.7.2 Spontaneous symmetry-breaking </div> </a> <ul id="toc-Spontaneous_symmetry-breaking-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Supersymmetry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Supersymmetry"> <div class="vector-toc-text"> 2.7.3 Supersymmetry </div> </a> <ul id="toc-Supersymmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_spacetimes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Other_spacetimes"> <div class="vector-toc-text"> 2.7.4 Other spacetimes </div> </a> <ul id="toc-Other_spacetimes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_quantum_field_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Topological_quantum_field_theory"> <div class="vector-toc-text"> 2.7.5 Topological quantum field theory </div> </a> <ul id="toc-Topological_quantum_field_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Perturbative_and_non-perturbative_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Perturbative_and_non-perturbative_methods"> <div class="vector-toc-text"> 2.8 Perturbative and non-perturbative methods </div> </a> <ul id="toc-Perturbative_and_non-perturbative_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_rigor" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematical_rigor"> <div class="vector-toc-text"> 3 Mathematical rigor </div> </a> <ul id="toc-Mathematical_rigor-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"> 4 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 5 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 6 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 7 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </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">Quantum field theory</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 64 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-64" aria-hidden="true" > 64 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Quantenfeldtheorie" title="Quantenfeldtheorie – Alemannic" lang="gsw" hreflang="gsw" data-title="Quantenfeldtheorie" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%AD%D9%82%D9%84_%D8%A7%D9%84%D9%83%D9%85%D9%88%D9%85%D9%8A" title="نظرية الحقل الكمومي – Arabic" lang="ar" hreflang="ar" data-title="نظرية الحقل الكمومي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_cu%C3%A1ntica_de_campos" title="Teoría cuántica de campos – Asturian" lang="ast" hreflang="ast" data-title="Teoría cuántica de campos" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85_%D8%AA%D8%A6%D9%88%D8%B1%DB%8C%D8%B3%DB%8C" title="کوانتوم تئوریسی – South Azerbaijani" lang="azb" hreflang="azb" data-title="کوانتوم تئوریسی" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="কোয়ান্টাম ক্ষেত্র তত্ত্ব – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম ক্ষেত্র তত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D0%B0%D1%8F_%D1%82%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%BF%D0%BE%D0%BB%D1%8F" title="Квантавая тэорыя поля – Belarusian" lang="be" hreflang="be" data-title="Квантавая тэорыя поля" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D0%B0%D1%8F_%D1%82%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%BF%D0%BE%D0%BB%D1%8F" title="Квантавая тэорыя поля – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Квантавая тэорыя поля" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%BF%D0%BE%D0%BB%D0%B5%D1%82%D0%BE" title="Квантова теория на полето – Bulgarian" lang="bg" hreflang="bg" data-title="Квантова теория на полето" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_qu%C3%A0ntica_de_camps" title="Teoria quàntica de camps – Catalan" lang="ca" hreflang="ca" data-title="Teoria quàntica de camps" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvantov%C3%A1_teorie_pole" title="Kvantová teorie pole – Czech" lang="cs" hreflang="cs" data-title="Kvantová teorie pole" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvantefeltteori" title="Kvantefeltteori – Danish" lang="da" hreflang="da" data-title="Kvantefeltteori" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quantenfeldtheorie" title="Quantenfeldtheorie – German" lang="de" hreflang="de" data-title="Quantenfeldtheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kvantv%C3%A4ljateooria" title="Kvantväljateooria – Estonian" lang="et" hreflang="et" data-title="Kvantväljateooria" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B2%CE%B1%CE%BD%CF%84%CE%B9%CE%BA%CE%AE_%CE%B8%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CF%80%CE%B5%CE%B4%CE%AF%CE%BF%CF%85" title="Κβαντική θεωρία πεδίου – Greek" lang="el" hreflang="el" data-title="Κβαντική θεωρία πεδίου" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_cu%C3%A1ntica_de_campos" title="Teoría cuántica de campos – Spanish" lang="es" hreflang="es" data-title="Teoría cuántica de campos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvantuma_kampa_teorio" title="Kvantuma kampa teorio – Esperanto" lang="eo" hreflang="eo" data-title="Kvantuma kampa teorio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eremu-teoria_kuantiko" title="Eremu-teoria kuantiko – Basque" lang="eu" hreflang="eu" data-title="Eremu-teoria kuantiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D9%85%DB%8C%D8%AF%D8%A7%D9%86%E2%80%8C%D9%87%D8%A7%DB%8C_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C" title="نظریه میدان‌های کوانتومی – Persian" lang="fa" hreflang="fa" data-title="نظریه میدان‌های کوانتومی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_quantique_des_champs" title="Théorie quantique des champs – French" lang="fr" hreflang="fr" data-title="Théorie quantique des champs" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/R%C3%A9imsetheoiric_chandamach" title="Réimsetheoiric chandamach – Irish" lang="ga" hreflang="ga" data-title="Réimsetheoiric chandamach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_cu%C3%A1ntica_de_campos" title="Teoría cuántica de campos – Galician" lang="gl" hreflang="gl" data-title="Teoría cuántica de campos" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%96%91%EC%9E%90%EC%9E%A5%EB%A1%A0" title="양자장론 – Korean" lang="ko" hreflang="ko" data-title="양자장론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%A1%D5%B7%D5%BF%D5%AB_%D6%84%D5%BE%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Դաշտի քվանտային տեսություն – Armenian" lang="hy" hreflang="hy" data-title="Դաշտի քվանտային տեսություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BE_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="प्रमात्रा क्षेत्र सिद्धान्त – Hindi" lang="hi" hreflang="hi" data-title="प्रमात्रा क्षेत्र सिद्धान्त" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kvantna_teorija_polja" title="Kvantna teorija polja – Croatian" lang="hr" hreflang="hr" data-title="Kvantna teorija polja" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_medan_kuantum" title="Teori medan kuantum – Indonesian" lang="id" hreflang="id" data-title="Teori medan kuantum" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_quantistica_dei_campi" title="Teoria quantistica dei campi – Italian" lang="it" hreflang="it" data-title="Teoria quantistica dei campi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A9%D7%93%D7%95%D7%AA_%D7%94%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99%D7%AA" title="תורת השדות הקוונטית – Hebrew" lang="he" hreflang="he" data-title="תורת השדות הקוונטית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%82%D1%8B%D2%9B_%D3%A9%D1%80%D1%96%D1%81_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Кванттық өріс теориясы – Kazakh" lang="kk" hreflang="kk" data-title="Кванттық өріс теориясы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%82%D1%8B%D0%BA_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F" title="Кванттык теория – Kyrgyz" lang="ky" hreflang="ky" data-title="Кванттык теория" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theoria_camporum_quanticorum" title="Theoria camporum quanticorum – Latin" lang="la" hreflang="la" data-title="Theoria camporum quanticorum" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kvantin%C4%97_lauko_teorija" title="Kvantinė lauko teorija – Lithuanian" lang="lt" hreflang="lt" data-title="Kvantinė lauko teorija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kvantumt%C3%A9relm%C3%A9let" title="Kvantumtérelmélet – Hungarian" lang="hu" hreflang="hu" data-title="Kvantumtérelmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_medan_kuantum" title="Teori medan kuantum – Malay" lang="ms" hreflang="ms" data-title="Teori medan kuantum" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwantumveldentheorie" title="Kwantumveldentheorie – Dutch" lang="nl" hreflang="nl" data-title="Kwantumveldentheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A0%B4%E3%81%AE%E9%87%8F%E5%AD%90%E8%AB%96" title="場の量子論 – Japanese" lang="ja" hreflang="ja" data-title="場の量子論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kvantefeltteori" title="Kvantefeltteori – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvantefeltteori" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kvantefeltteori" title="Kvantefeltteori – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kvantefeltteori" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Teoria_quantica_dei_camps" title="Teoria quantica dei camps – Occitan" lang="oc" hreflang="oc" data-title="Teoria quantica dei camps" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Maydon_kvant_nazariyasi" title="Maydon kvant nazariyasi – Uzbek" lang="uz" hreflang="uz" data-title="Maydon kvant nazariyasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%81%E0%A8%86%E0%A8%82%E0%A8%9F%E0%A8%AE_%E0%A8%AB%E0%A9%80%E0%A8%B2%E0%A8%A1_%E0%A8%A5%E0%A8%BF%E0%A8%8A%E0%A8%B0%E0%A9%80" title="ਕੁਆਂਟਮ ਫੀਲਡ ਥਿਊਰੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੁਆਂਟਮ ਫੀਲਡ ਥਿਊਰੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D9%B9%D9%85_%D9%81%DB%8C%D9%84%DA%88_%D8%AA%DA%BE%DB%8C%D9%88%D8%B1%DB%8C" title="کوانٹم فیلڈ تھیوری – Western Punjabi" lang="pnb" hreflang="pnb" data-title="کوانٹم فیلڈ تھیوری" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%DA%A9%D9%88%D8%A7%D9%86%D9%BC%D9%88%D9%85%D9%8A_%D8%B3%D8%A7%D8%AD%DB%90_%D9%86%D8%B8%D8%B1%DB%8C%D9%87" title="د کوانټومي ساحې نظریه – Pashto" lang="ps" hreflang="ps" data-title="د کوانټومي ساحې نظریه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kwantowa_teoria_pola" title="Kwantowa teoria pola – Polish" lang="pl" hreflang="pl" data-title="Kwantowa teoria pola" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_qu%C3%A2ntica_de_campos" title="Teoria quântica de campos – Portuguese" lang="pt" hreflang="pt" data-title="Teoria quântica de campos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_cuantic%C4%83_a_c%C3%A2mpurilor" title="Teoria cuantică a câmpurilor – Romanian" lang="ro" hreflang="ro" data-title="Teoria cuantică a câmpurilor" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BF%D0%BE%D0%BB%D1%8F" title="Квантовая теория поля – Russian" lang="ru" hreflang="ru" data-title="Квантовая теория поля" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kvantov%C3%A1_te%C3%B3ria_po%C4%BEa" title="Kvantová teória poľa – Slovak" lang="sk" hreflang="sk" data-title="Kvantová teória poľa" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvantna_teorija_polja" title="Kvantna teorija polja – Slovenian" lang="sl" hreflang="sl" data-title="Kvantna teorija polja" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%DB%8C%DB%8C_%D8%A8%D9%88%D8%A7%D8%B1%DB%8C_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%DB%86%D9%85%DB%8C" title="تیۆریی بواری کوانتۆمی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="تیۆریی بواری کوانتۆمی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BF%D0%BE%D1%99%D0%B0" title="Квантна теорија поља – Serbian" lang="sr" hreflang="sr" data-title="Квантна теорија поља" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kvantna_teorija_polja" title="Kvantna teorija polja – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Kvantna teorija polja" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kvanttikentt%C3%A4teoria" title="Kvanttikenttäteoria – Finnish" lang="fi" hreflang="fi" data-title="Kvanttikenttäteoria" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kvantf%C3%A4ltteori" title="Kvantfältteori – Swedish" lang="sv" hreflang="sv" data-title="Kvantfältteori" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teoryang_quantum_field" title="Teoryang quantum field – Tagalog" lang="tl" hreflang="tl" data-title="Teoryang quantum field" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%81%E0%AE%B5%E0%AE%BE%E0%AE%A3%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D_%E0%AE%AA%E0%AF%81%E0%AE%B2%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="குவாண்டம் புலக்கோட்பாடு – Tamil" lang="ta" hreflang="ta" data-title="குவாண்டம் புலக்கோட்பாடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D1%8B%D1%80%D0%BD%D1%8B%D2%A3_%D0%BA%D0%B2%D0%B0%D0%BD%D1%82_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D0%B5" title="Кырның квант теориясе – Tatar" lang="tt" hreflang="tt" data-title="Кырның квант теориясе" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%AA%E0%B8%99%E0%B8%B2%E0%B8%A1%E0%B8%84%E0%B8%A7%E0%B8%AD%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A1" title="ทฤษฎีสนามควอนตัม – Thai" lang="th" hreflang="th" data-title="ทฤษฎีสนามควอนตัม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kuantum_alan_teorisi" title="Kuantum alan teorisi – Turkish" lang="tr" hreflang="tr" data-title="Kuantum alan teorisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%BF%D0%BE%D0%BB%D1%8F" title="Квантова теорія поля – Ukrainian" lang="uk" hreflang="uk" data-title="Квантова теорія поля" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_tr%C6%B0%E1%BB%9Dng_l%C6%B0%E1%BB%A3ng_t%E1%BB%AD" title="Lý thuyết trường lượng tử – Vietnamese" lang="vi" hreflang="vi" data-title="Lý thuyết trường lượng tử" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E5%9C%BA%E8%AE%BA" title="量子场论 – Wu" lang="wuu" hreflang="wuu" data-title="量子场论" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E5%A0%B4%E8%AB%96" title="量子場論 – 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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"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a class="mw-selflink selflink">Quantum field theory</a></th></tr><tr><td class="sidebar-image"><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><div class="sidebar-caption"><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></div></td></tr><tr><td class="sidebar-above"> <a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Field_(physics)" title="Field (physics)">Field theory</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">Weak force</a></li> <li><a href="/wiki/Strong_force" class="mw-redirect" title="Strong force">Strong force</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">Symmetries</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">C-symmetry</a></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">P-symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a></li> <li><a href="/wiki/Lorentz_symmetry" class="mw-redirect" title="Lorentz symmetry">Lorentz symmetry</a></li> <li><a href="/wiki/Poincar%C3%A9_symmetry" class="mw-redirect" title="Poincaré symmetry">Poincaré symmetry</a></li> <li><a href="/wiki/Gauge_symmetry_(mathematics)" title="Gauge symmetry (mathematics)">Gauge symmetry</a></li> <li><a href="/wiki/Explicit_symmetry_breaking" title="Explicit symmetry breaking">Explicit symmetry breaking</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Noether_charge" class="mw-redirect" title="Noether charge">Noether charge</a></li> <li><a href="/wiki/Topological_charge" class="mw-redirect" title="Topological charge">Topological charge</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Tools</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomaly</a></li> <li><a href="/wiki/Background_field_method" title="Background field method">Background field method</a></li> <li><a href="/wiki/BRST_quantization" title="BRST quantization">BRST quantization</a></li> <li><a href="/wiki/Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">Correlation function</a></li> <li><a href="/wiki/Crossing_(physics)" title="Crossing (physics)">Crossing</a></li> <li><a href="/wiki/Effective_action" title="Effective action">Effective action</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective field theory</a></li> <li><a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">Expectation value</a></li> <li><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></li> <li><a href="/wiki/Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li> <li><a href="/wiki/LSZ_reduction_formula" title="LSZ reduction formula">LSZ reduction formula</a></li> <li><a href="/wiki/Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">Partition function</a></li> <li><a href="/wiki/Path_Integral_Formulation" class="mw-redirect" title="Path Integral Formulation">Path Integral Formulation</a></li> <li><a href="/wiki/Propagator_(Quantum_Theory)" class="mw-redirect" title="Propagator (Quantum Theory)">Propagator</a></li> <li><a href="/wiki/Quantization_(physics)" title="Quantization (physics)">Quantization</a></li> <li><a href="/wiki/Regularization_(physics)" title="Regularization (physics)">Regularization</a></li> <li><a href="/wiki/Renormalization" title="Renormalization">Renormalization</a></li> <li><a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">Vacuum state</a></li> <li><a href="/wiki/Wick%27s_theorem" title="Wick's theorem">Wick's theorem</a></li> <li><a href="/w/index.php?title=Wightman_Axioms&action=edit&redlink=1" class="new" title="Wightman Axioms (page does not exist)">Wightman Axioms</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a></li> <li><a href="/wiki/Proca_action" title="Proca action">Proca equations</a></li> <li><a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></li> <li><a href="/wiki/Bargmann%E2%80%93Wigner_equations" title="Bargmann–Wigner equations">Bargmann–Wigner equations</a></li> <li><a href="/wiki/Schwinger-Dyson_equation" class="mw-redirect" title="Schwinger-Dyson equation">Schwinger-Dyson equation</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group equation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Incomplete theories</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></li> <li><a href="/wiki/Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Stephen_Louis_Adler" class="mw-redirect" title="Stephen Louis Adler">Adler</a></li> <li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li> <li><a href="/wiki/Alexey_Andreevich_Anselm" class="mw-redirect" title="Alexey Andreevich Anselm">Anselm</a></li> <li><a href="/wiki/Valentine_Bargmann" title="Valentine Bargmann">Bargmann</a></li> <li><a href="/wiki/Carlo_Becchi" title="Carlo Becchi">Becchi</a></li> <li><a href="/wiki/Alexander_Belavin" title="Alexander Belavin">Belavin</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Felix_Berezin" title="Felix Berezin">Berezin</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/James_Bjorken" title="James Bjorken">Bjorken</a></li> <li><a href="/wiki/Konrad_Bleuler" title="Konrad Bleuler">Bleuer</a></li> <li><a href="/wiki/Nikolay_Bogolyubov" title="Nikolay Bogolyubov">Bogoliubov</a></li> <li><a href="/wiki/Stanley_Brodsky" title="Stanley Brodsky">Brodsky</a></li> <li><a href="/wiki/Robert_Brout" title="Robert Brout">Brout</a></li> <li><a href="/wiki/Detlev_Buchholz" title="Detlev Buchholz">Buchholz</a></li> <li><a href="/wiki/Freddy_Cachazo" title="Freddy Cachazo">Cachazo</a></li> <li><a href="/wiki/Curtis_Callan" title="Curtis Callan">Callan</a></li> <li><a href="/wiki/Sidney_Coleman" title="Sidney Coleman">Coleman</a></li> <li><a href="/wiki/Alain_Connes" title="Alain Connes">Connes</a></li> <li><a href="/wiki/Roger_Dashen" title="Roger Dashen">Dashen</a></li> <li><a href="/wiki/Bryce_DeWitt" title="Bryce DeWitt">DeWitt</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Sergio_Doplicher" title="Sergio Doplicher">Doplicher</a></li> <li><a href="/wiki/Freeman_Dyson" title="Freeman Dyson">Dyson</a></li> <li><a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">Englert</a></li> <li><a href="/wiki/Ludvig_Faddeev" title="Ludvig Faddeev">Faddeev</a></li> <li><a href="/wiki/Victor_Sergeevich_Fadin" class="mw-redirect" title="Victor Sergeevich Fadin">Fadin</a></li> <li><a href="/wiki/Pierre_Fayet" title="Pierre Fayet">Fayet</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Markus_Fierz" title="Markus Fierz">Fierz</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Paul_Frampton" title="Paul Frampton">Frampton</a></li> <li><a href="/wiki/Harald_Fritzsch" title="Harald Fritzsch">Fritzsch</a></li> <li><a href="/wiki/J%C3%BCrg_Fr%C3%B6hlich" title="Jürg Fröhlich">Fröhlich</a></li> <li><a href="/wiki/Klaus_Fredenhagen" title="Klaus Fredenhagen">Fredenhagen</a></li> <li><a href="/wiki/Wendell_H._Furry" title="Wendell H. Furry">Furry</a></li> <li><a href="/wiki/Sheldon_Glashow" title="Sheldon Glashow">Glashow</a></li> <li><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gelfand</a></li> <li><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann</a></li> <li><a href="/wiki/James_Glimm" title="James Glimm">Glimm</a></li> <li><a href="/wiki/Jeffrey_Goldstone" title="Jeffrey Goldstone">Goldstone</a></li> <li><a href="/wiki/Vladimir_Gribov" title="Vladimir Gribov">Gribov</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Suraj_N._Gupta" title="Suraj N. Gupta">Gupta</a></li> <li><a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Guralnik</a></li> <li><a href="/wiki/Rudolf_Haag" title="Rudolf Haag">Haag</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/Klaus_Hepp" title="Klaus Hepp">Hepp</a></li> <li><a href="/wiki/Peter_Higgs" title="Peter Higgs">Higgs</a></li> <li><a href="/wiki/C._R._Hagen" title="C. R. Hagen">Hagen</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li> <li><a href="/wiki/Dmitri_Ivanenko" title="Dmitri Ivanenko">Ivanenko</a></li> <li><a href="/wiki/Roman_Jackiw" title="Roman Jackiw">Jackiw</a></li> <li><a href="/wiki/Arthur_Jaffe" title="Arthur Jaffe">Jaffe</a></li> <li><a href="/wiki/Giovanni_Jona-Lasinio" title="Giovanni Jona-Lasinio">Jona-Lasinio</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Res_Jost" title="Res Jost">Jost</a></li> <li><a href="/wiki/Gunnar_K%C3%A4ll%C3%A9n" title="Gunnar Källén">Källén</a></li> <li><a href="/wiki/Henry_Way_Kendall" title="Henry Way Kendall">Kendall</a></li> <li><a href="/wiki/Toichiro_Kinoshita" title="Toichiro Kinoshita">Kinoshita</a></li> <li><a href="/wiki/Igor_R._Klebanov" class="mw-redirect" title="Igor R. Klebanov">Klebanov</a></li> <li><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li> <li><a href="/wiki/Dirk_Kreimer" title="Dirk Kreimer">Kreimer</a></li> <li><a href="/wiki/Eduard_A._Kuraev" title="Eduard A. Kuraev">Kuraev</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Lee</a></li> <li><a href="/wiki/Harry_Lehmann" title="Harry Lehmann">Lehmann</a></li> <li><a href="/wiki/Heinrich_Leutwyler" title="Heinrich Leutwyler">Leutwyler</a></li> <li><a href="/wiki/Lev_Lipatov" title="Lev Lipatov">Lipatov</a></li> <li><a href="/wiki/Jan_%C5%81opusza%C5%84ski_(physicist)" title="Jan Łopuszański (physicist)">Łopuszański</a></li> <li><a href="/wiki/Francis_E._Low" title="Francis E. Low">Low</a></li> <li><a href="/wiki/Gerhart_L%C3%BCders" title="Gerhart Lüders">Lüders</a></li> <li><a href="/wiki/Luciano_Maiani" title="Luciano Maiani">Maiani</a></li> <li><a href="/wiki/Ettore_Majorana" title="Ettore Majorana">Majorana</a></li> <li><a href="/wiki/Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li> <li><a href="/wiki/Alexander_Arkadyevich_Migdal" title="Alexander Arkadyevich Migdal">Migdal</a></li> <li><a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Mills</a></li> <li><a href="/wiki/Christian_M%C3%B8ller" title="Christian Møller">Møller</a></li> <li><a href="/wiki/Mark_Naimark" title="Mark Naimark">Naimark</a></li> <li><a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Nambu</a></li> <li><a href="/wiki/Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li> <li><a href="/wiki/Kazuhiko_Nishijima" title="Kazuhiko Nishijima">Nishijima</a></li> <li><a href="/wiki/Reinhard_Oehme" title="Reinhard Oehme">Oehme</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Konrad_Osterwalder" title="Konrad Osterwalder">Osterwalder</a></li> <li><a href="/wiki/Giorgio_Parisi" title="Giorgio Parisi">Parisi</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Michael_Peskin" title="Michael Peskin">Peskin</a></li> <li><a href="/wiki/Jan_Christoph_Plefka" title="Jan Christoph Plefka">Plefka</a></li> <li><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li> <li><a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li> <li><a href="/wiki/Isaak_Pomeranchuk" title="Isaak Pomeranchuk">Pomeranchuk</a></li> <li><a href="/wiki/Victor_Popov" title="Victor Popov">Popov</a></li> <li><a href="/wiki/Alexandru_Proca" title="Alexandru Proca">Proca</a></li> <li><a href="/wiki/Valery_Rubakov" title="Valery Rubakov">Rubakov</a></li> <li><a href="/wiki/David_Ruelle" title="David Ruelle">Ruelle</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Robert_Schrader" title="Robert Schrader">Schrader</a></li> <li><a href="/wiki/Albert_Schwarz" title="Albert Schwarz">Schwarz</a></li> <li><a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Schwinger</a></li> <li><a href="/wiki/Irving_Segal" title="Irving Segal">Segal</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Gordon_Walter_Semenoff" title="Gordon Walter Semenoff">Semenoff</a></li> <li><a href="/wiki/Mikhail_Shifman" title="Mikhail Shifman">Shifman</a></li> <li><a href="/wiki/Dmitry_Shirkov" title="Dmitry Shirkov">Shirkov</a></li> <li><a href="/wiki/Tony_Skyrme" title="Tony Skyrme">Skyrme</a></li> <li><a href="/wiki/Charles_M._Sommerfield" title="Charles M. Sommerfield">Sommerfield</a></li> <li><a href="/wiki/Raymond_Stora" title="Raymond Stora">Stora</a></li> <li><a href="/wiki/Ernst_Stueckelberg" title="Ernst Stueckelberg">Stueckelberg</a></li> <li><a href="/wiki/George_Sudarshan" class="mw-redirect" title="George Sudarshan">Sudarshan</a></li> <li><a href="/wiki/Kurt_Symanzik" title="Kurt Symanzik">Symanzik</a></li> <li><a href="/wiki/Walter_Thirring" title="Walter Thirring">Thirring</a></li> <li><a href="/wiki/Shin%27ichir%C5%8D_Tomonaga" title="Shin'ichirō Tomonaga">Tomonaga</a></li> <li><a href="/wiki/Igor_Tyutin" title="Igor Tyutin">Tyutin</a></li> <li><a href="/wiki/Arkady_Vainshtein" title="Arkady Vainshtein">Vainshtein</a></li> <li><a href="/wiki/Martinus_Veltman" class="mw-redirect" title="Martinus Veltman">Veltman</a></li> <li><a href="/wiki/Miguel_%C3%81ngel_Virasoro_(physicist)" title="Miguel Ángel Virasoro (physicist)">Virasoro</a></li> <li><a href="/wiki/John_Clive_Ward" title="John Clive Ward">Ward</a></li> <li><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg</a></li> <li><a href="/wiki/Victor_Weisskopf" title="Victor Weisskopf">Weisskopf</a></li> <li><a href="/wiki/Gregor_Wentzel" title="Gregor Wentzel">Wentzel</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Christof_Wetterich" title="Christof Wetterich">Wetterich</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Gian_Carlo_Wick" title="Gian Carlo Wick">Wick</a></li> <li><a href="/wiki/Arthur_Wightman" title="Arthur Wightman">Wightman</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek</a></li> <li><a href="/wiki/Kenneth_G._Wilson" title="Kenneth G. Wilson">Wilson</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang</a></li> <li><a href="/wiki/Hideki_Yukawa" title="Hideki Yukawa">Yukawa</a></li> <li><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Anthony_Zee" title="Anthony Zee">Zee</a></li> <li><a href="/wiki/Wolfhart_Zimmermann" title="Wolfhart Zimmermann">Zimmermann</a></li> <li><a href="/wiki/Jean_Zinn-Justin" title="Jean Zinn-Justin">Zinn-Justin</a></li> <li><a href="/wiki/Jean-Bernard_Zuber" title="Jean-Bernard Zuber">Zuber</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><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_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> In <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>, quantum field theory (QFT) is a theoretical framework that combines <a href="/wiki/Classical_field_theory" title="Classical field theory">classical field theory</a>, <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, and <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>.<a href="#cite_note-peskin-1">[1]</a>: xi  QFT is used in <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a> to construct <a href="/wiki/Physical_model" class="mw-redirect" title="Physical model">physical models</a> of <a href="/wiki/Subatomic_particle" title="Subatomic particle">subatomic particles</a> and in <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a> to construct models of <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticles</a>. The current <a href="/wiki/Standard_model_of_particle_physics" class="mw-redirect" title="Standard model of particle physics">standard model of particle physics</a> is based on quantum field theory. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=1" title="Edit section: History">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History of quantum field theory</a></div> Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between <a href="/wiki/Light" title="Light">light</a> and <a href="/wiki/Electrons" class="mw-redirect" title="Electrons">electrons</a>, culminating in the first quantum field theory—<a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the <a href="/wiki/Renormalization" title="Renormalization">renormalization</a> procedure. A second major barrier came with QFT's apparent inability to describe the <a href="/wiki/Weak_interaction" title="Weak interaction">weak</a> and <a href="/wiki/Strong_interaction" title="Strong interaction">strong interactions</a>, to the point where some theorists called for the abandonment of the field theoretic approach. The development of <a href="/wiki/Gauge_theory" title="Gauge theory">gauge theory</a> and the completion of the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> in the 1970s led to a renaissance of quantum field theory. <div class="mw-heading mw-heading3"><h3 id="Theoretical_background">Theoretical background</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=2" title="Edit section: Theoretical background">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Magnet0873.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Magnet0873.png/200px-Magnet0873.png" decoding="async" width="200" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Magnet0873.png/300px-Magnet0873.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Magnet0873.png/400px-Magnet0873.png 2x" data-file-width="444" data-file-height="298" /></a><figcaption><a href="/wiki/Magnetic_field_lines" class="mw-redirect" title="Magnetic field lines">Magnetic field lines</a> visualized using <a href="/wiki/Iron_filings" title="Iron filings">iron filings</a>. When a piece of paper is sprinkled with iron filings and placed above a bar magnet, the filings align according to the direction of the magnetic field, forming arcs allowing viewers to clearly see the poles of the magnet and to see the magnetic field generated.</figcaption></figure> Quantum field theory results from the combination of <a href="/wiki/Classical_field_theory" title="Classical field theory">classical field theory</a>, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, and <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>.<a href="#cite_note-peskin-1">[1]</a>: xi  A brief overview of these theoretical precursors follows. The earliest successful classical field theory is one that emerged from <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a>, despite the complete absence of the concept of fields from his 1687 treatise <a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Philosophiæ Naturalis Principia Mathematica</a>. The force of gravity as described by Isaac Newton is an "<a href="/wiki/Action_at_a_distance" title="Action at a distance">action at a distance</a>"—its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with <a href="/wiki/Richard_Bentley" title="Richard Bentley">Richard Bentley</a>, however, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact".<a href="#cite_note-Hobson-2">[2]</a>: 4  It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields—a numerical quantity (a <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> in the case of <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a>) assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered merely a mathematical trick.<a href="#cite_note-weinberg-3">[3]</a>: 18  Fields began to take on an existence of their own with the development of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a> in the 19th century. <a href="/wiki/Michael_Faraday" title="Michael Faraday">Michael Faraday</a> coined the English term "field" in 1845. He introduced fields as properties of space (even when it is devoid of matter) having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines of force". This description of fields remains to this day.<a href="#cite_note-Hobson-2">[2]</a><a href="#cite_note-Heilbron2003-4">[4]</a>: 301 <a href="#cite_note-Thomson1893-5">[5]</a>: 2  The theory of <a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">classical electromagnetism</a> was completed in 1864 with <a href="/wiki/Maxwell%27s_equation" class="mw-redirect" title="Maxwell's equation">Maxwell's equations</a>, which described the relationship between the <a href="/wiki/Electric_field" title="Electric field">electric field</a>, the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>, <a href="/wiki/Electric_current" title="Electric current">electric current</a>, and <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>. Maxwell's equations implied the existence of <a href="/wiki/Electromagnetic_waves" class="mw-redirect" title="Electromagnetic waves">electromagnetic waves</a>, a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. Action-at-a-distance was thus conclusively refuted.<a href="#cite_note-Hobson-2">[2]</a>: 19  Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in <a href="/wiki/Emission_spectrum" title="Emission spectrum">atomic spectra</a>, nor for the distribution of <a href="/wiki/Blackbody_radiation" class="mw-redirect" title="Blackbody radiation">blackbody radiation</a> in different wavelengths.<a href="#cite_note-weisskopf-6">[6]</a> <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a>'s study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>, as tiny <a href="/wiki/Oscillator" class="mw-redirect" title="Oscillator">oscillators</a> with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillators</a>. This process of restricting energies to discrete values is called quantization.<a href="#cite_note-Heisenberg1999-7">[7]</a>: Ch.2  Building on this idea, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> proposed in 1905 an explanation for the <a href="/wiki/Photoelectric_effect" title="Photoelectric effect">photoelectric effect</a>, that light is composed of individual packets of energy called <a href="/wiki/Photon" title="Photon">photons</a> (the quanta of light). This implied that the electromagnetic radiation, while being waves in the classical electromagnetic field, also exists in the form of particles.<a href="#cite_note-weisskopf-6">[6]</a> In 1913, <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> introduced the <a href="/wiki/Bohr_model" title="Bohr model">Bohr model</a> of atomic structure, wherein <a href="/wiki/Electrons" class="mw-redirect" title="Electrons">electrons</a> within atoms can only take on a series of discrete, rather than continuous, energies. This is another example of quantization. The Bohr model successfully explained the discrete nature of atomic spectral lines. In 1924, <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a> proposed the hypothesis of <a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</a>, that microscopic particles exhibit both wave-like and particle-like properties under different circumstances.<a href="#cite_note-weisskopf-6">[6]</a> Uniting these scattered ideas, a coherent discipline, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, was formulated between 1925 and 1926, with important contributions from <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a>, <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a>, <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="/wiki/Max_Born" title="Max Born">Max Born</a>, <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>, <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, and <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a>.<a href="#cite_note-weinberg-3">[3]</a>: 22–23  In the same year as his paper on the photoelectric effect, Einstein published his theory of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, built on Maxwell's electromagnetism. New rules, called <a href="/wiki/Lorentz_transformations" class="mw-redirect" title="Lorentz transformations">Lorentz transformations</a>, were given for the way time and space coordinates of an event change under changes in the observer's velocity, and the distinction between time and space was blurred.<a href="#cite_note-weinberg-3">[3]</a>: 19  It was proposed that all physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations. Two difficulties remained. Observationally, the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> underlying quantum mechanics could explain the <a href="/wiki/Stimulated_emission" title="Stimulated emission">stimulated emission</a> of radiation from atoms, where an electron emits a new photon under the action of an external electromagnetic field, but it was unable to explain <a href="/wiki/Spontaneous_emission" title="Spontaneous emission">spontaneous emission</a>, where an electron spontaneously decreases in energy and emits a photon even without the action of an external electromagnetic field. Theoretically, the Schrödinger equation could not describe photons and was inconsistent with the principles of special relativity—it treats time as an ordinary number while promoting spatial coordinates to <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a>.<a href="#cite_note-weisskopf-6">[6]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_electrodynamics">Quantum electrodynamics</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=3" title="Edit section: Quantum electrodynamics">edit</a>]</div> Quantum field theory naturally began with the study of electromagnetic interactions, as the electromagnetic field was the only known classical field as of the 1920s.<a href="#cite_note-shifman-8">[8]</a>: 1  Through the works of Born, Heisenberg, and <a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a> in 1925–1926, a quantum theory of the free electromagnetic field (one with no interactions with matter) was developed via <a href="/wiki/Canonical_quantization" title="Canonical quantization">canonical quantization</a> by treating the electromagnetic field as a set of <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillators</a>.<a href="#cite_note-shifman-8">[8]</a>: 1  With the exclusion of interactions, however, such a theory was yet incapable of making quantitative predictions about the real world.<a href="#cite_note-weinberg-3">[3]</a>: 22  In his seminal 1927 paper The quantum theory of the emission and absorption of radiation, Dirac coined the term <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> (QED), a theory that adds upon the terms describing the free electromagnetic field an additional interaction term between electric <a href="/wiki/Current_density" title="Current density">current density</a> and the <a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">electromagnetic vector potential</a>. Using first-order <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a>, he successfully explained the phenomenon of spontaneous emission. According to the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a> in quantum mechanics, quantum harmonic oscillators cannot remain stationary, but they have a non-zero minimum energy and must always be oscillating, even in the lowest energy state (the <a href="/wiki/Ground_state" title="Ground state">ground state</a>). Therefore, even in a perfect <a href="/wiki/Vacuum" title="Vacuum">vacuum</a>, there remains an oscillating electromagnetic field having <a href="/wiki/Zero-point_energy" title="Zero-point energy">zero-point energy</a>. It is this <a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">quantum fluctuation</a> of electromagnetic fields in the vacuum that "stimulates" the spontaneous emission of radiation by electrons in atoms. Dirac's theory was hugely successful in explaining both the emission and absorption of radiation by atoms; by applying second-order perturbation theory, it was able to account for the <a href="/wiki/Scattering" title="Scattering">scattering</a> of photons, <a href="/wiki/Resonance_fluorescence" title="Resonance fluorescence">resonance fluorescence</a> and non-relativistic <a href="/wiki/Compton_scattering" title="Compton scattering">Compton scattering</a>. Nonetheless, the application of higher-order perturbation theory was plagued with problematic infinities in calculations.<a href="#cite_note-weisskopf-6">[6]</a>: 71  In 1928, Dirac wrote down a <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a> that described relativistic electrons: the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>. It had the following important consequences: the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> of an electron is 1/2; the electron <a href="/wiki/G-factor_(physics)" title="G-factor (physics)">g-factor</a> is 2; it led to the correct Sommerfeld formula for the <a href="/wiki/Fine_structure" title="Fine structure">fine structure</a> of the <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a>; and it could be used to derive the <a href="/wiki/Klein%E2%80%93Nishina_formula" title="Klein–Nishina formula">Klein–Nishina formula</a> for relativistic Compton scattering. Although the results were fruitful, the theory also apparently implied the existence of negative energy states, which would cause atoms to be unstable, since they could always decay to lower energy states by the emission of radiation.<a href="#cite_note-weisskopf-6">[6]</a>: 71–72  The prevailing view at the time was that the world was composed of two very different ingredients: material particles (such as electrons) and <a href="/wiki/Field_(physics)#Quantum_fields" title="Field (physics)">quantum fields</a> (such as photons). Material particles were considered to be eternal, with their physical state described by the probabilities of finding each particle in any given region of space or range of velocities. On the other hand, photons were considered merely the <a href="/wiki/Excited_state" title="Excited state">excited states</a> of the underlying quantized electromagnetic field, and could be freely created or destroyed. It was between 1928 and 1930 that Jordan, <a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Eugene Wigner</a>, Heisenberg, Pauli, and <a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Enrico Fermi</a> discovered that material particles could also be seen as excited states of quantum fields. Just as photons are excited states of the quantized electromagnetic field, so each type of particle had its corresponding quantum field: an electron field, a proton field, etc. Given enough energy, it would now be possible to create material particles. Building on this idea, Fermi proposed in 1932 an explanation for <a href="/wiki/Beta_decay" title="Beta decay">beta decay</a> known as <a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi's interaction</a>. <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">Atomic nuclei</a> do not contain electrons per se, but in the process of decay, an electron is created out of the surrounding electron field, analogous to the photon created from the surrounding electromagnetic field in the radiative decay of an excited atom.<a href="#cite_note-weinberg-3">[3]</a>: 22–23  It was realized in 1929 by Dirac and others that negative energy states implied by the Dirac equation could be removed by assuming the existence of particles with the same mass as electrons but opposite electric charge. This not only ensured the stability of atoms, but it was also the first proposal of the existence of <a href="/wiki/Antimatter" title="Antimatter">antimatter</a>. Indeed, the evidence for <a href="/wiki/Positron" title="Positron">positrons</a> was discovered in 1932 by <a href="/wiki/Carl_David_Anderson" title="Carl David Anderson">Carl David Anderson</a> in <a href="/wiki/Cosmic_ray" title="Cosmic ray">cosmic rays</a>. With enough energy, such as by absorbing a photon, an electron-positron pair could be created, a process called <a href="/wiki/Pair_production" title="Pair production">pair production</a>; the reverse process, annihilation, could also occur with the emission of a photon. This showed that particle numbers need not be fixed during an interaction. Historically, however, positrons were at first thought of as "holes" in an infinite electron sea, rather than a new kind of particle, and this theory was referred to as the <a href="/wiki/Dirac_hole_theory" title="Dirac hole theory">Dirac hole theory</a>.<a href="#cite_note-weisskopf-6">[6]</a>: 72 <a href="#cite_note-weinberg-3">[3]</a>: 23  QFT naturally incorporated antiparticles in its formalism.<a href="#cite_note-weinberg-3">[3]</a>: 24  <div class="mw-heading mw-heading3"><h3 id="Infinities_and_renormalization">Infinities and renormalization</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=4" title="Edit section: Infinities and renormalization">edit</a>]</div> <a href="/wiki/Robert_Oppenheimer" class="mw-redirect" title="Robert Oppenheimer">Robert Oppenheimer</a> showed in 1930 that higher-order perturbative calculations in QED always resulted in infinite quantities, such as the electron <a href="/wiki/Self-energy" title="Self-energy">self-energy</a> and the vacuum zero-point energy of the electron and photon fields,<a href="#cite_note-weisskopf-6">[6]</a> suggesting that the computational methods at the time could not properly deal with interactions involving photons with extremely high momenta.<a href="#cite_note-weinberg-3">[3]</a>: 25  It was not until 20 years later that a systematic approach to remove such infinities was developed. A series of papers was published between 1934 and 1938 by <a href="/wiki/Ernst_Stueckelberg" title="Ernst Stueckelberg">Ernst Stueckelberg</a> that established a relativistically invariant formulation of QFT. In 1947, Stueckelberg also independently developed a complete renormalization procedure. Such achievements were not understood and recognized by the theoretical community.<a href="#cite_note-weisskopf-6">[6]</a> Faced with these infinities, <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a> and Heisenberg proposed, in 1937 and 1943 respectively, to supplant the problematic QFT with the so-called <a href="/wiki/S-matrix_theory" title="S-matrix theory">S-matrix theory</a>. Since the specific details of microscopic interactions are inaccessible to observations, the theory should only attempt to describe the relationships between a small number of <a href="/wiki/Observable" title="Observable">observables</a> (e.g. the energy of an atom) in an interaction, rather than be concerned with the microscopic minutiae of the interaction. In 1945, <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a> and Wheeler daringly suggested abandoning QFT altogether and proposed <a href="/wiki/Action-at-a-distance" class="mw-redirect" title="Action-at-a-distance">action-at-a-distance</a> as the mechanism of particle interactions.<a href="#cite_note-weinberg-3">[3]</a>: 26  In 1947, <a href="/wiki/Willis_Lamb" title="Willis Lamb">Willis Lamb</a> and <a href="/wiki/Robert_Retherford" title="Robert Retherford">Robert Retherford</a> measured the minute difference in the 2S1/2 and 2P1/2 energy levels of the hydrogen atom, also called the <a href="/wiki/Lamb_shift" title="Lamb shift">Lamb shift</a>. By ignoring the contribution of photons whose energy exceeds the electron mass, <a href="/wiki/Hans_Bethe" title="Hans Bethe">Hans Bethe</a> successfully estimated the numerical value of the Lamb shift.<a href="#cite_note-weisskopf-6">[6]</a><a href="#cite_note-weinberg-3">[3]</a>: 28  Subsequently, <a href="/wiki/Norman_Myles_Kroll" title="Norman Myles Kroll">Norman Myles Kroll</a>, Lamb, <a href="/wiki/James_Bruce_French" title="James Bruce French">James Bruce French</a>, and <a href="/wiki/Victor_Weisskopf" title="Victor Weisskopf">Victor Weisskopf</a> again confirmed this value using an approach in which infinities cancelled other infinities to result in finite quantities. However, this method was clumsy and unreliable and could not be generalized to other calculations.<a href="#cite_note-weisskopf-6">[6]</a> The breakthrough eventually came around 1950 when a more robust method for eliminating infinities was developed by <a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Julian Schwinger</a>, <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a>, <a href="/wiki/Freeman_Dyson" title="Freeman Dyson">Freeman Dyson</a>, and <a href="/wiki/Shinichiro_Tomonaga" class="mw-redirect" title="Shinichiro Tomonaga">Shinichiro Tomonaga</a>. The main idea is to replace the calculated values of mass and charge, infinite though they may be, by their finite measured values. This systematic computational procedure is known as <a href="/wiki/Renormalization" title="Renormalization">renormalization</a> and can be applied to arbitrary order in perturbation theory.<a href="#cite_note-weisskopf-6">[6]</a> As Tomonaga said in his Nobel lecture:<blockquote>Since those parts of the modified mass and charge due to field reactions [become infinite], it is impossible to calculate them by the theory. However, the mass and charge observed in experiments are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite. On the other hand, the mass and charge appearing in the theory are… the values modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental values for them phenomenologically... This procedure is called the renormalization of mass and charge… After long, laborious calculations, less skillful than Schwinger's, we obtained a result... which was in agreement with [the] Americans'.<a href="#cite_note-9">[9]</a></blockquote> By applying the renormalization procedure, calculations were finally made to explain the electron's <a href="/wiki/Anomalous_magnetic_moment" class="mw-redirect" title="Anomalous magnetic moment">anomalous magnetic moment</a> (the deviation of the electron <a href="/wiki/G-factor_(physics)" title="G-factor (physics)">g-factor</a> from 2) and <a href="/wiki/Vacuum_polarization" title="Vacuum polarization">vacuum polarization</a>. These results agreed with experimental measurements to a remarkable degree, thus marking the end of a "war against infinities".<a href="#cite_note-weisskopf-6">[6]</a> At the same time, Feynman introduced the <a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integral formulation</a> of quantum mechanics and <a href="/wiki/Feynman_diagrams" class="mw-redirect" title="Feynman diagrams">Feynman diagrams</a>.<a href="#cite_note-shifman-8">[8]</a>: 2  The latter can be used to visually and intuitively organize and to help compute terms in the perturbative expansion. Each diagram can be interpreted as paths of particles in an interaction, with each vertex and line having a corresponding mathematical expression, and the product of these expressions gives the <a href="/wiki/Scattering_amplitude" title="Scattering amplitude">scattering amplitude</a> of the interaction represented by the diagram.<a href="#cite_note-peskin-1">[1]</a>: 5  It was with the invention of the renormalization procedure and Feynman diagrams that QFT finally arose as a complete theoretical framework.<a href="#cite_note-shifman-8">[8]</a>: 2  <div class="mw-heading mw-heading3"><h3 id="Non-renormalizability">Non-renormalizability</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=5" title="Edit section: Non-renormalizability">edit</a>]</div> Given the tremendous success of QED, many theorists believed, in the few years after 1949, that QFT could soon provide an understanding of all microscopic phenomena, not only the interactions between photons, electrons, and positrons. Contrary to this optimism, QFT entered yet another period of depression that lasted for almost two decades.<a href="#cite_note-weinberg-3">[3]</a>: 30  The first obstacle was the limited applicability of the renormalization procedure. In perturbative calculations in QED, all infinite quantities could be eliminated by redefining a small (finite) number of physical quantities (namely the mass and charge of the electron). Dyson proved in 1949 that this is only possible for a small class of theories called "renormalizable theories", of which QED is an example. However, most theories, including the <a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi theory</a> of the <a href="/wiki/Weak_interaction" title="Weak interaction">weak interaction</a>, are "non-renormalizable". Any perturbative calculation in these theories beyond the first order would result in infinities that could not be removed by redefining a finite number of physical quantities.<a href="#cite_note-weinberg-3">[3]</a>: 30  The second major problem stemmed from the limited validity of the Feynman diagram method, which is based on a series expansion in perturbation theory. In order for the series to converge and low-order calculations to be a good approximation, the <a href="/wiki/Coupling_constant" title="Coupling constant">coupling constant</a>, in which the series is expanded, must be a sufficiently small number. The coupling constant in QED is the <a href="/wiki/Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a> α ≈ 1/137, which is small enough that only the simplest, lowest order, Feynman diagrams need to be considered in realistic calculations. In contrast, the coupling constant in the <a href="/wiki/Strong_interaction" title="Strong interaction">strong interaction</a> is roughly of the order of one, making complicated, higher order, Feynman diagrams just as important as simple ones. There was thus no way of deriving reliable quantitative predictions for the strong interaction using perturbative QFT methods.<a href="#cite_note-weinberg-3">[3]</a>: 31  With these difficulties looming, many theorists began to turn away from QFT. Some focused on <a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">symmetry</a> principles and <a href="/wiki/Conservation_law" title="Conservation law">conservation laws</a>, while others picked up the old S-matrix theory of Wheeler and Heisenberg. QFT was used heuristically as guiding principles, but not as a basis for quantitative calculations.<a href="#cite_note-weinberg-3">[3]</a>: 31  <div class="mw-heading mw-heading3"><h3 id="Source_theory">Source theory</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=6" title="Edit section: Source theory">edit</a>]</div> Schwinger, however, took a different route. For more than a decade he and his students had been nearly the only exponents of field theory,<a href="#cite_note-MiltonMehra-10">[10]</a>: 454  but in 1951<a href="#cite_note-11">[11]</a><a href="#cite_note-12">[12]</a> he found a way around the problem of the infinities with a new method using external sources as currents coupled to gauge fields.<a href="#cite_note-13">[13]</a> Motivated by the former findings, Schwinger kept pursuing this approach in order to "quantumly" generalize the <a href="/wiki/Lagrangian_mechanics#Lagrange_multipliers_and_constraints" title="Lagrangian mechanics">classical process</a> of coupling external forces to the configuration space parameters known as Lagrange multipliers. He summarized his <a href="/wiki/Source_field" title="Source field">source theory</a> in 1966<a href="#cite_note-14">[14]</a> then expanded the theory's applications to quantum electrodynamics in his three volume-set titled: Particles, Sources, and Fields.<a href="#cite_note-Perseus_Books-15">[15]</a><a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a> Developments in pion physics, in which the new viewpoint was most successfully applied, convinced him of the great advantages of mathematical simplicity and conceptual clarity that its use bestowed.<a href="#cite_note-Perseus_Books-15">[15]</a> In source theory there are no divergences, and no renormalization. It may be regarded as the calculational tool of field theory, but it is more general.<a href="#cite_note-18">[18]</a> Using source theory, Schwinger was able to calculate the anomalous magnetic moment of the electron, which he had done in 1947, but this time with no ‘distracting remarks’ about infinite quantities.<a href="#cite_note-MiltonMehra-10">[10]</a>: 467  Schwinger also applied source theory to his QFT theory of gravity, and was able to reproduce all four of Einstein's classic results: gravitational red shift, deflection and slowing of light by gravity, and the perihelion precession of Mercury.<a href="#cite_note-19">[19]</a> The neglect of source theory by the physics community was a major disappointment for Schwinger:<blockquote>The lack of appreciation of these facts by others was depressing, but understandable. -J. Schwinger<a href="#cite_note-Perseus_Books-15">[15]</a></blockquote>See "<a href="/wiki/Julian_Schwinger#Career" title="Julian Schwinger">the shoes incident</a>" between J. Schwinger and <a href="/wiki/Steven_Weinberg" title="Steven Weinberg">S. Weinberg</a>.<a href="#cite_note-MiltonMehra-10">[10]</a> <div class="mw-heading mw-heading3"><h3 id="Standard_model">Standard model</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=7" title="Edit section: Standard model">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Standard_Model_of_Elementary_Particles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/300px-Standard_Model_of_Elementary_Particles.svg.png" decoding="async" width="300" height="287" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/450px-Standard_Model_of_Elementary_Particles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/600px-Standard_Model_of_Elementary_Particles.svg.png 2x" data-file-width="1390" data-file-height="1330" /></a><figcaption><a href="/wiki/Elementary_particles" class="mw-redirect" title="Elementary particles">Elementary particles</a> of the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>: six types of <a href="/wiki/Quark" title="Quark">quarks</a>, six types of <a href="/wiki/Lepton" title="Lepton">leptons</a>, four types of <a href="/wiki/Gauge_boson" title="Gauge boson">gauge bosons</a> that carry <a href="/wiki/Fundamental_interaction" title="Fundamental interaction">fundamental interactions</a>, as well as the <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a>, which endow elementary particles with mass.</figcaption></figure> In 1954, <a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang Chen-Ning</a> and <a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Robert Mills</a> generalized the <a href="/wiki/Gauge_theory" title="Gauge theory">local symmetry</a> of QED, leading to <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">non-Abelian gauge theories</a> (also known as Yang–Mills theories), which are based on more complicated local <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry groups</a>.<a href="#cite_note-thooft-20">[20]</a>: 5  In QED, (electrically) charged particles interact via the exchange of photons, while in non-Abelian gauge theory, particles carrying a new type of "<a href="/wiki/Charge_(physics)" title="Charge (physics)">charge</a>" interact via the exchange of massless <a href="/wiki/Gauge_boson" title="Gauge boson">gauge bosons</a>. Unlike photons, these gauge bosons themselves carry charge.<a href="#cite_note-weinberg-3">[3]</a>: 32 <a href="#cite_note-21">[21]</a> <a href="/wiki/Sheldon_Glashow" title="Sheldon Glashow">Sheldon Glashow</a> developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in 1960. In 1964, <a href="/wiki/Abdus_Salam" title="Abdus Salam">Abdus Salam</a> and <a href="/wiki/John_Clive_Ward" title="John Clive Ward">John Clive Ward</a> arrived at the same theory through a different path. This theory, nevertheless, was non-renormalizable.<a href="#cite_note-coleman-22">[22]</a> <a href="/wiki/Peter_Higgs" title="Peter Higgs">Peter Higgs</a>, <a href="/wiki/Robert_Brout" title="Robert Brout">Robert Brout</a>, <a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">François Englert</a>, <a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Gerald Guralnik</a>, <a href="/wiki/C._R._Hagen" title="C. R. Hagen">Carl Hagen</a>, and <a href="/wiki/T._W._B._Kibble" class="mw-redirect" title="T. W. B. Kibble">Tom Kibble</a> proposed in their famous <a href="/wiki/1964_PRL_symmetry_breaking_papers" title="1964 PRL symmetry breaking papers">Physical Review Letters papers</a> that the gauge symmetry in Yang–Mills theories could be broken by a mechanism called <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneous symmetry breaking</a>, through which originally massless gauge bosons could acquire mass.<a href="#cite_note-thooft-20">[20]</a>: 5–6  By combining the earlier theory of Glashow, Salam, and Ward with the idea of spontaneous symmetry breaking, <a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Steven Weinberg</a> wrote down in 1967 a theory describing <a href="/wiki/Electroweak_interaction" title="Electroweak interaction">electroweak interactions</a> between all <a href="/wiki/Lepton" title="Lepton">leptons</a> and the effects of the <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a>. His theory was at first mostly ignored,<a href="#cite_note-coleman-22">[22]</a><a href="#cite_note-thooft-20">[20]</a>: 6  until it was brought back to light in 1971 by <a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">Gerard 't Hooft</a>'s proof that non-Abelian gauge theories are renormalizable. The electroweak theory of Weinberg and Salam was extended from leptons to <a href="/wiki/Quark" title="Quark">quarks</a> in 1970 by Glashow, <a href="/wiki/John_Iliopoulos" title="John Iliopoulos">John Iliopoulos</a>, and <a href="/wiki/Luciano_Maiani" title="Luciano Maiani">Luciano Maiani</a>, marking its completion.<a href="#cite_note-coleman-22">[22]</a> <a href="/wiki/Harald_Fritzsch" title="Harald Fritzsch">Harald Fritzsch</a>, <a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Murray Gell-Mann</a>, and <a href="/wiki/Heinrich_Leutwyler" title="Heinrich Leutwyler">Heinrich Leutwyler</a> discovered in 1971 that certain phenomena involving the <a href="/wiki/Strong_interaction" title="Strong interaction">strong interaction</a> could also be explained by non-Abelian gauge theory. <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a> (QCD) was born. In 1973, <a href="/wiki/David_Gross" title="David Gross">David Gross</a>, <a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Frank Wilczek</a>, and <a href="/wiki/Hugh_David_Politzer" title="Hugh David Politzer">Hugh David Politzer</a> showed that non-Abelian gauge theories are "<a href="/wiki/Asymptotic_freedom" title="Asymptotic freedom">asymptotically free</a>", meaning that under renormalization, the coupling constant of the strong interaction decreases as the interaction energy increases. (Similar discoveries had been made numerous times previously, but they had been largely ignored.) <a href="#cite_note-thooft-20">[20]</a>: 11  Therefore, at least in high-energy interactions, the coupling constant in QCD becomes sufficiently small to warrant a perturbative series expansion, making quantitative predictions for the strong interaction possible.<a href="#cite_note-weinberg-3">[3]</a>: 32  These theoretical breakthroughs brought about a renaissance in QFT. The full theory, which includes the electroweak theory and chromodynamics, is referred to today as the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of elementary particles.<a href="#cite_note-23">[23]</a> The Standard Model successfully describes all <a href="/wiki/Fundamental_interaction" title="Fundamental interaction">fundamental interactions</a> except <a href="/wiki/Gravity" title="Gravity">gravity</a>, and its many predictions have been met with remarkable experimental confirmation in subsequent decades.<a href="#cite_note-shifman-8">[8]</a>: 3  The <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a>, central to the mechanism of spontaneous symmetry breaking, was finally detected in 2012 at <a href="/wiki/CERN" title="CERN">CERN</a>, marking the complete verification of the existence of all constituents of the Standard Model.<a href="#cite_note-24">[24]</a> <div class="mw-heading mw-heading3"><h3 id="Other_developments">Other developments</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=8" title="Edit section: Other developments">edit</a>]</div> The 1970s saw the development of non-perturbative methods in non-Abelian gauge theories. The <a href="/wiki/%27t_Hooft%E2%80%93Polyakov_monopole" title="'t Hooft–Polyakov monopole">'t Hooft–Polyakov monopole</a> was discovered theoretically by 't Hooft and <a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Alexander Polyakov</a>, <a href="/wiki/Flux_tube" title="Flux tube">flux tubes</a> by <a href="/wiki/Holger_Bech_Nielsen" title="Holger Bech Nielsen">Holger Bech Nielsen</a> and <a href="/w/index.php?title=Poul_Olesen&action=edit&redlink=1" class="new" title="Poul Olesen (page does not exist)">Poul Olesen</a>, and <a href="/wiki/Instanton" title="Instanton">instantons</a> by Polyakov and coauthors. These objects are inaccessible through perturbation theory.<a href="#cite_note-shifman-8">[8]</a>: 4  <a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a> also appeared in the same period. The first supersymmetric QFT in four dimensions was built by <a href="/wiki/Yuri_Golfand" title="Yuri Golfand">Yuri Golfand</a> and <a href="/w/index.php?title=Evgeny_Likhtman&action=edit&redlink=1" class="new" title="Evgeny Likhtman (page does not exist)">Evgeny Likhtman</a> in 1970, but their result failed to garner widespread interest due to the <a href="/wiki/Iron_Curtain" title="Iron Curtain">Iron Curtain</a>. Supersymmetry only took off in the theoretical community after the work of <a href="/wiki/Julius_Wess" title="Julius Wess">Julius Wess</a> and <a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Bruno Zumino</a> in 1973.<a href="#cite_note-shifman-8">[8]</a>: 7  Among the four fundamental interactions, gravity remains the only one that lacks a consistent QFT description. Various attempts at a theory of <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a> led to the development of <a href="/wiki/String_theory" title="String theory">string theory</a>,<a href="#cite_note-shifman-8">[8]</a>: 6  itself a type of two-dimensional QFT with <a href="/wiki/Conformal_symmetry" title="Conformal symmetry">conformal symmetry</a>.<a href="#cite_note-polchinski1-25">[25]</a> <a href="/wiki/Jo%C3%ABl_Scherk" title="Joël Scherk">Joël Scherk</a> and <a href="/wiki/John_Henry_Schwarz" title="John Henry Schwarz">John Schwarz</a> first proposed in 1974 that string theory could be the quantum theory of gravity.<a href="#cite_note-26">[26]</a> <div class="mw-heading mw-heading3"><h3 id="Condensed-matter-physics">Condensed-matter-physics</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=9" title="Edit section: Condensed-matter-physics">edit</a>]</div> Although quantum field theory arose from the study of interactions between elementary particles, it has been successfully applied to other physical systems, particularly to <a href="/wiki/Many-body_system" class="mw-redirect" title="Many-body system">many-body systems</a> in <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>. Historically, the Higgs mechanism of spontaneous symmetry breaking was a result of <a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Yoichiro Nambu</a>'s application of <a href="/wiki/Superconductor" class="mw-redirect" title="Superconductor">superconductor</a> theory to elementary particles, while the concept of renormalization came out of the study of second-order <a href="/wiki/Phase_transition" title="Phase transition">phase transitions</a> in matter.<a href="#cite_note-27">[27]</a> Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticle</a>—<a href="/wiki/Phonon" title="Phonon">phonons</a>. Lev Landau claimed that low-energy excitations in many condensed matter systems could be described in terms of interactions between a set of quasiparticles. The Feynman diagram method of QFT was naturally well suited to the analysis of various phenomena in condensed matter systems.<a href="#cite_note-wilczek-28">[28]</a> Gauge theory is used to describe the quantization of <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> in superconductors, the <a href="/wiki/Resistivity" class="mw-redirect" title="Resistivity">resistivity</a> in the <a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">quantum Hall effect</a>, as well as the relation between frequency and voltage in the AC <a href="/wiki/Josephson_effect" title="Josephson effect">Josephson effect</a>.<a href="#cite_note-wilczek-28">[28]</a> <div class="mw-heading mw-heading2"><h2 id="Principles">Principles</h2>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=10" title="Edit section: Principles">edit</a>]</div> For simplicity, <a href="/wiki/Natural_units" title="Natural units">natural units</a> are used in the following sections, in which the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a> ħ and the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> c are both set to one. <div class="mw-heading mw-heading3"><h3 id="Classical_fields">Classical fields</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=11" title="Edit section: Classical fields">edit</a>]</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/Classical_field_theory" title="Classical field theory">Classical field theory</a></div> A classical <a href="/wiki/Field_(physics)" title="Field (physics)">field</a> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of spatial and time coordinates.<a href="#cite_note-tong1-29">[29]</a> Examples include the <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a> in <a href="/wiki/Newtonian_gravity" class="mw-redirect" title="Newtonian gravity">Newtonian gravity</a> g(x, t) and the <a href="/wiki/Electric_field" title="Electric field">electric field</a> E(x, t) and <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> B(x, t) in <a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">classical electromagnetism</a>. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many <a href="/wiki/Degrees_of_freedom_(mechanics)" title="Degrees of freedom (mechanics)">degrees of freedom</a>.<a href="#cite_note-tong1-29">[29]</a><a href="#cite_note-30">[30]</a> Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the <a href="/wiki/Photoelectric_effect" title="Photoelectric effect">photoelectric effect</a> are best explained by discrete particles (<a href="/wiki/Photon" title="Photon">photons</a>), rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields. <a href="/wiki/Canonical_quantization" title="Canonical quantization">Canonical quantization</a> and <a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integrals</a> are two common formulations of QFT.<a href="#cite_note-zee-31">[31]</a>: 61  To motivate the fundamentals of QFT, an overview of classical field theory follows. The simplest classical field is a real <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> — a <a href="/wiki/Real_number" title="Real number">real number</a> at every point in space that changes in time. It is denoted as ϕ(x, t), where x is the position vector, and t is the time. Suppose the <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> of the field, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">, is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\int d^{3}x\,{\mathcal {L}}=\int d^{3}x\,\left[{\frac {1}{2}}{\dot {\phi }}^{2}-{\frac {1}{2}}(\nabla \phi )^{2}-{\frac {1}{2}}m^{2}\phi ^{2}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow> <mo>[</mo> <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>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\int d^{3}x\,{\mathcal {L}}=\int d^{3}x\,\left[{\frac {1}{2}}{\dot {\phi }}^{2}-{\frac {1}{2}}(\nabla \phi )^{2}-{\frac {1}{2}}m^{2}\phi ^{2}\right],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c9ce91f7eb79d2c06f42cc431ab9bbca59ca89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.317ex; height:6.176ex;" alt="{\displaystyle L=\int d^{3}x\,{\mathcal {L}}=\int d^{3}x\,\left[{\frac {1}{2}}{\dot {\phi }}^{2}-{\frac {1}{2}}(\nabla \phi )^{2}-{\frac {1}{2}}m^{2}\phi ^{2}\right],}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"> is the Lagrangian density, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\phi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0446aa46e762e6b105ed6cd084731c4a37b8a3e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.467ex; height:3.009ex;" alt="{\displaystyle {\dot {\phi }}}"> is the time-derivative of the field, ∇ is the gradient operator, and m is a real parameter (the "mass" of the field). Applying the <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</a> on the Lagrangian:<a href="#cite_note-peskin-1">[1]</a>: 16  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial t}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial t)}}\right]+\sum _{i=1}^{3}{\frac {\partial }{\partial x^{i}}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial x^{i})}}\right]-{\frac {\partial {\mathcal {L}}}{\partial \phi }}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial t)}}\right]+\sum _{i=1}^{3}{\frac {\partial }{\partial x^{i}}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial x^{i})}}\right]-{\frac {\partial {\mathcal {L}}}{\partial \phi }}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b60bbe90a93512487bc5805c9f9c6753483396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.314ex; height:7.176ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial t)}}\right]+\sum _{i=1}^{3}{\frac {\partial }{\partial x^{i}}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial x^{i})}}\right]-{\frac {\partial {\mathcal {L}}}{\partial \phi }}=0,}"></dd></dl> we obtain the <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a> for the field, which describe the way it varies in time and space: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}+m^{2}\right)\phi =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mi mathvariant="normal">∇</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> </mrow> <mo>)</mo> </mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}+m^{2}\right)\phi =0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31df19e0122c3dd2417717d3addd13bdf2bc6dad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.915ex; height:6.343ex;" alt="{\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}+m^{2}\right)\phi =0.}"></dd></dl> This is known as the <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a>.<a href="#cite_note-peskin-1">[1]</a>: 17  The Klein–Gordon equation is a <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a>, so its solutions can be expressed as a sum of <a href="/wiki/Normal_mode" title="Normal mode">normal modes</a> (obtained via <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>) as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left(a_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+a_{\mathbf {p} }^{*}e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>p</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </msup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mi>t</mi> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left(a_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+a_{\mathbf {p} }^{*}e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306fbef48f739372d29ee11687a0907b6ba5761f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.833ex; height:7.009ex;" alt="{\displaystyle \phi (\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left(a_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+a_{\mathbf {p} }^{*}e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right),}"></dd></dl> where a is a <a href="/wiki/Complex_number" title="Complex number">complex number</a> (normalized by convention), * denotes <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>, and ωp is the frequency of the normal mode: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\mathbf {p} }={\sqrt {|\mathbf {p} |^{2}+m^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathbf {p} }={\sqrt {|\mathbf {p} |^{2}+m^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5c7ab2cc77dd3509d2679a70e31f7d5210ec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:18.566ex; height:4.843ex;" alt="{\displaystyle \omega _{\mathbf {p} }={\sqrt {|\mathbf {p} |^{2}+m^{2}}}.}"></dd></dl> Thus each normal mode corresponding to a single p can be seen as a classical <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a> with frequency ωp.<a href="#cite_note-peskin-1">[1]</a>: 21,26  <div class="mw-heading mw-heading3"><h3 id="Canonical_quantization">Canonical quantization</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=12" title="Edit section: Canonical quantization">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Canonical_quantization" title="Canonical quantization">Canonical quantization</a></div> The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a>. The displacement of a classical harmonic oscillator is described by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)={\frac {1}{\sqrt {2\omega }}}ae^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}a^{*}e^{i\omega t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>ω</mi> </msqrt> </mfrac> </mrow> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>ω</mi> </msqrt> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)={\frac {1}{\sqrt {2\omega }}}ae^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}a^{*}e^{i\omega t},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee75bd3f42d37ef8780c248c8072208b0ab0884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.116ex; height:6.176ex;" alt="{\displaystyle x(t)={\frac {1}{\sqrt {2\omega }}}ae^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}a^{*}e^{i\omega t},}"></dd></dl> where a is a complex number (normalized by convention), and ω is the oscillator's frequency. Note that x is the displacement of a particle in simple harmonic motion from the equilibrium position, not to be confused with the spatial label x of a quantum field. For a quantum harmonic oscillator, x(t) is promoted to a <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/355c897c5f2c1bf80dd12138be16e8ec2f211eb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle {\hat {x}}(t)}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}(t)={\frac {1}{\sqrt {2\omega }}}{\hat {a}}e^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}{\hat {a}}^{\dagger }e^{i\omega t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>ω</mi> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>ω</mi> </msqrt> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}(t)={\frac {1}{\sqrt {2\omega }}}{\hat {a}}e^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}{\hat {a}}^{\dagger }e^{i\omega t}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4e44d98a630c2f359a9b8f04286f20abcc18de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.024ex; height:6.176ex;" alt="{\displaystyle {\hat {x}}(t)={\frac {1}{\sqrt {2\omega }}}{\hat {a}}e^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}{\hat {a}}^{\dagger }e^{i\omega t}.}"></dd></dl> Complex numbers a and a* are replaced by the <a href="/wiki/Annihilation_operator" class="mw-redirect" title="Annihilation operator">annihilation operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233a5bda7c263f804b049be11c03d12e3d65103a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.176ex;" alt="{\displaystyle {\hat {a}}}"> and the <a href="/wiki/Creation_operator" class="mw-redirect" title="Creation operator">creation operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {a}}^{\dagger }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b262475ca337b6f7f24ebb322576d5bb4daec96a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.192ex; height:2.843ex;" alt="{\displaystyle {\hat {a}}^{\dagger }}">, respectively, where † denotes <a href="/wiki/Hermitian_conjugation" class="mw-redirect" title="Hermitian conjugation">Hermitian conjugation</a>. The <a href="/wiki/Commutation_relation" class="mw-redirect" title="Commutation relation">commutation relation</a> between the two is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\hat {a}},{\hat {a}}^{\dagger }\right]=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\hat {a}},{\hat {a}}^{\dagger }\right]=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac6a25fa663fd271e485103c5f5510df6bfe657e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.558ex; height:4.843ex;" alt="{\displaystyle \left[{\hat {a}},{\hat {a}}^{\dagger }\right]=1.}"></dd></dl> The <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> of the simple harmonic oscillator can be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\hbar \omega .}"> <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> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\hbar \omega .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e12d17462173e7817850cd19fb034314ff946b80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.574ex; height:5.176ex;" alt="{\displaystyle {\hat {H}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\hbar \omega .}"></dd></dl> The <a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">vacuum state</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }">, which is the lowest energy state, is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}|0\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {a}}|0\rangle =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fae897c43431cddcff81a415da3cf2f80281d4b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.205ex; height:2.843ex;" alt="{\displaystyle {\hat {a}}|0\rangle =0}"></dd></dl> and has energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\hbar \omega .}"> <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> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\hbar \omega .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f35b3c27690e000543692954a6009a5ab30c923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.398ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\hbar \omega .}"> One can easily check that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\hat {H}},{\hat {a}}^{\dagger }]=\hbar \omega {\hat {a}}^{\dagger },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\hat {H}},{\hat {a}}^{\dagger }]=\hbar \omega {\hat {a}}^{\dagger },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd37c8e5f013ca40c74d1bdf24dca7b06db2883b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.273ex; height:3.343ex;" alt="{\displaystyle [{\hat {H}},{\hat {a}}^{\dagger }]=\hbar \omega {\hat {a}}^{\dagger },}"> which implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {a}}^{\dagger }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b262475ca337b6f7f24ebb322576d5bb4daec96a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.192ex; height:2.843ex;" alt="{\displaystyle {\hat {a}}^{\dagger }}"> increases the energy of the simple harmonic oscillator by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257e7f4e184cd5ca0743d3e3cc9b0f0f025dce11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.752ex; height:2.176ex;" alt="{\displaystyle \hbar \omega }">. For example, the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}^{\dagger }|0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {a}}^{\dagger }|0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc5956feb86d5eb3f95c4eebac246185465c562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.343ex;" alt="{\displaystyle {\hat {a}}^{\dagger }|0\rangle }"> is an eigenstate of energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\hbar \omega /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\hbar \omega /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68c57490690604672dab53305f597ff09e514ab8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:2.843ex;" alt="{\displaystyle 3\hbar \omega /2}">. Any energy eigenstate state of a single harmonic oscillator can be obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> by successively applying the creation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {a}}^{\dagger }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b262475ca337b6f7f24ebb322576d5bb4daec96a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.192ex; height:2.843ex;" alt="{\displaystyle {\hat {a}}^{\dagger }}">:<a href="#cite_note-peskin-1">[1]</a>: 20  and any state of the system can be expressed as a linear combination of the states <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n\rangle \propto \left({\hat {a}}^{\dagger }\right)^{n}|0\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∝</mo> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |n\rangle \propto \left({\hat {a}}^{\dagger }\right)^{n}|0\rangle .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75da0a02a4496b2ec5383643523ed217385bb9a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.591ex; height:4.843ex;" alt="{\displaystyle |n\rangle \propto \left({\hat {a}}^{\dagger }\right)^{n}|0\rangle .}"></dd></dl> A similar procedure can be applied to the real scalar field ϕ, by promoting it to a quantum field operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\phi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f81c80ffc527c401c28130f592ec69f5f83d5ad5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.467ex; height:3.176ex;" alt="{\displaystyle {\hat {\phi }}}">, while the annihilation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}_{\mathbf {p} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {a}}_{\mathbf {p} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c53518307a01430e40f28258bf5e9e21e78b17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.512ex; height:2.843ex;" alt="{\displaystyle {\hat {a}}_{\mathbf {p} }}">, the creation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98b89f9276bbf2494fbcceb271bef33088a58314" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.512ex; height:3.676ex;" alt="{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}"> and the angular frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\mathbf {p} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathbf {p} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9b69f2fe1d8ab6cb6c702c4ef22e2251eb1d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.728ex; height:2.343ex;" alt="{\displaystyle \omega _{\mathbf {p} }}">are now for a particular p: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\phi }}(\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left({\hat {a}}_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+{\hat {a}}_{\mathbf {p} }^{\dagger }e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>p</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </msup> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mi>t</mi> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\phi }}(\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left({\hat {a}}_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+{\hat {a}}_{\mathbf {p} }^{\dagger }e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03216657bfa9100539f9fe5b033c07a43607d5a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.56ex; height:7.009ex;" alt="{\displaystyle {\hat {\phi }}(\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left({\hat {a}}_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+{\hat {a}}_{\mathbf {p} }^{\dagger }e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right).}"></dd></dl> Their commutation relations are:<a href="#cite_note-peskin-1">[1]</a>: 21  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=(2\pi )^{3}\delta (\mathbf {p} -\mathbf {q} ),\quad \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }\right]=\left[{\hat {a}}_{\mathbf {p} }^{\dagger },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mo>,</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=(2\pi )^{3}\delta (\mathbf {p} -\mathbf {q} ),\quad \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }\right]=\left[{\hat {a}}_{\mathbf {p} }^{\dagger },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e717bcbc4eacc629e354ccb1313c85c8d7d1f030" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.13ex; height:4.843ex;" alt="{\displaystyle \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=(2\pi )^{3}\delta (\mathbf {p} -\mathbf {q} ),\quad \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }\right]=\left[{\hat {a}}_{\mathbf {p} }^{\dagger },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=0,}"></dd></dl> where δ is the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. The vacuum state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}_{\mathbf {p} }|0\rangle =0,\quad {\text{for all }}\mathbf {p} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {a}}_{\mathbf {p} }|0\rangle =0,\quad {\text{for all }}\mathbf {p} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2d04822804455ddeba3229328d6fee8da50cec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.379ex; height:3.009ex;" alt="{\displaystyle {\hat {a}}_{\mathbf {p} }|0\rangle =0,\quad {\text{for all }}\mathbf {p} .}"></dd></dl> Any quantum state of the field can be obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> by successively applying creation operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98b89f9276bbf2494fbcceb271bef33088a58314" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.512ex; height:3.676ex;" alt="{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}"> (or by a linear combination of such states), e.g. <a href="#cite_note-peskin-1">[1]</a>: 22  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\hat {a}}_{\mathbf {p} _{3}}^{\dagger }\right)^{3}{\hat {a}}_{\mathbf {p} _{2}}^{\dagger }\left({\hat {a}}_{\mathbf {p} _{1}}^{\dagger }\right)^{2}|0\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\hat {a}}_{\mathbf {p} _{3}}^{\dagger }\right)^{3}{\hat {a}}_{\mathbf {p} _{2}}^{\dagger }\left({\hat {a}}_{\mathbf {p} _{1}}^{\dagger }\right)^{2}|0\rangle .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df04bec97043dad717b06f168ee296d113c7d2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.052ex; height:5.176ex;" alt="{\displaystyle \left({\hat {a}}_{\mathbf {p} _{3}}^{\dagger }\right)^{3}{\hat {a}}_{\mathbf {p} _{2}}^{\dagger }\left({\hat {a}}_{\mathbf {p} _{1}}^{\dagger }\right)^{2}|0\rangle .}"></dd></dl> While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a <a href="/wiki/Fock_space" title="Fock space">Fock space</a>, which can account for the fact that particle numbers are not fixed in relativistic quantum systems.<a href="#cite_note-32">[32]</a> The process of quantizing an arbitrary number of particles instead of a single particle is often also called <a href="/wiki/Second_quantization" title="Second quantization">second quantization</a>.<a href="#cite_note-peskin-1">[1]</a>: 19  The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields, <a href="/wiki/Dirac_field" class="mw-redirect" title="Dirac field">Dirac fields</a>,<a href="#cite_note-peskin-1">[1]</a>: 52  <a href="/wiki/Vector_field" title="Vector field">vector fields</a> (e.g. the electromagnetic field), and even <a href="/wiki/String_theory" title="String theory">strings</a>.<a href="#cite_note-33">[33]</a> However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a> would be necessary. The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a <a href="/wiki/Quartic_interaction" title="Quartic interaction">quartic interaction</a> term could be introduced to the Lagrangian of the real scalar field:<a href="#cite_note-peskin-1">[1]</a>: 77  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi )\left(\partial ^{\mu }\phi \right)-{\frac {1}{2}}m^{2}\phi ^{2}-{\frac {\lambda }{4!}}\phi ^{4},}"> <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"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <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> <mo>(</mo> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi )\left(\partial ^{\mu }\phi \right)-{\frac {1}{2}}m^{2}\phi ^{2}-{\frac {\lambda }{4!}}\phi ^{4},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27c42c384c0207e1610ef64374d2266576215dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.447ex; height:5.509ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi )\left(\partial ^{\mu }\phi \right)-{\frac {1}{2}}m^{2}\phi ^{2}-{\frac {\lambda }{4!}}\phi ^{4},}"></dd></dl> where μ is a spacetime index, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{0}=\partial /\partial t,\ \partial _{1}=\partial /\partial x^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mo>,</mo> <mtext> </mtext> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{0}=\partial /\partial t,\ \partial _{1}=\partial /\partial x^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daa8a221ba5a51a5dc021de4e86d0f453e8a5e89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.21ex; height:3.176ex;" alt="{\displaystyle \partial _{0}=\partial /\partial t,\ \partial _{1}=\partial /\partial x^{1}}">, etc. The summation over the index μ has been omitted following the <a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a>. If the parameter λ is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory. <div class="mw-heading mw-heading3"><h3 id="Path_integrals">Path integrals</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=13" title="Edit section: Path integrals">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></div> The <a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integral formulation</a> of QFT is concerned with the direct computation of the <a href="/wiki/Scattering_amplitude" title="Scattering amplitude">scattering amplitude</a> of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitude</a> for a system to evolve from some initial state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi _{I}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ϕ</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 |\phi _{I}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60c79158ab415eec95711cd6c086b4a4abb0bfd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.998ex; height:2.843ex;" alt="{\displaystyle |\phi _{I}\rangle }"> at time t = 0 to some final state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi _{F}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi _{F}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c42baf7504488abdc90f9d8e7b0f3dc5e16b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.4ex; height:2.843ex;" alt="{\displaystyle |\phi _{F}\rangle }"> at t = T, the total time T is divided into N small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let H be the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> (i.e. <a href="/wiki/Time_evolution_operator" class="mw-redirect" title="Time evolution operator">generator of time evolution</a>), then<a href="#cite_note-zee-31">[31]</a>: 10  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int d\phi _{1}\int d\phi _{2}\cdots \int d\phi _{N-1}\,\langle \phi _{F}|e^{-iHT/N}|\phi _{N-1}\rangle \cdots \langle \phi _{2}|e^{-iHT/N}|\phi _{1}\rangle \langle \phi _{1}|e^{-iHT/N}|\phi _{I}\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</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> <mi>i</mi> <mi>H</mi> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∫</mo> <mi>d</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <mo>∫</mo> <mi>d</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</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> <mi>i</mi> <mi>H</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>⋯</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>H</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>H</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int d\phi _{1}\int d\phi _{2}\cdots \int d\phi _{N-1}\,\langle \phi _{F}|e^{-iHT/N}|\phi _{N-1}\rangle \cdots \langle \phi _{2}|e^{-iHT/N}|\phi _{1}\rangle \langle \phi _{1}|e^{-iHT/N}|\phi _{I}\rangle .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eabc789bdce84345b9c34b04c418542c9657455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:98.414ex; height:5.676ex;" alt="{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int d\phi _{1}\int d\phi _{2}\cdots \int d\phi _{N-1}\,\langle \phi _{F}|e^{-iHT/N}|\phi _{N-1}\rangle \cdots \langle \phi _{2}|e^{-iHT/N}|\phi _{1}\rangle \langle \phi _{1}|e^{-iHT/N}|\phi _{I}\rangle .}"></dd></dl> Taking the limit N → ∞, the above product of integrals becomes the Feynman path integral:<a href="#cite_note-peskin-1">[1]</a>: 282 <a href="#cite_note-zee-31">[31]</a>: 12  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int {\mathcal {D}}\phi (t)\,\exp \left\{i\int _{0}^{T}dt\,L\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</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> <mi>i</mi> <mi>H</mi> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo fence="false" 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> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>{</mo> <mrow> <mi>i</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi>L</mi> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int {\mathcal {D}}\phi (t)\,\exp \left\{i\int _{0}^{T}dt\,L\right\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f4d3d74408b0009728df0de45d2d8ec6b5e562c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.596ex; height:6.343ex;" alt="{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int {\mathcal {D}}\phi (t)\,\exp \left\{i\int _{0}^{T}dt\,L\right\},}"></dd></dl> where L is the Lagrangian involving ϕ and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian H via <a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre transformation</a>. The initial and final conditions of the path integral are respectively <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (0)=\phi _{I},\quad \phi (T)=\phi _{F}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (0)=\phi _{I},\quad \phi (T)=\phi _{F}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/176a710894eca43b61fcc5d8900bfbb768885ea7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.683ex; height:2.843ex;" alt="{\displaystyle \phi (0)=\phi _{I},\quad \phi (T)=\phi _{F}.}"></dd></dl> In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand. <div class="mw-heading mw-heading3"><h3 id="Two-point_correlation_function">Two-point correlation function</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=14" title="Edit section: Two-point correlation function">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">Correlation function (quantum field theory)</a></div> In calculations, one often encounters expression like<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \quad {\text{or}}\quad \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em" /> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ω</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \quad {\text{or}}\quad \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7b053d4520f1d14bacbfc3e6ba26e9e4d05969e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.278ex; height:2.843ex;" alt="{\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \quad {\text{or}}\quad \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle }">in the free or interacting theory, respectively. Here, <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><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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> are position <a href="/wiki/Four-vector" title="Four-vector">four-vectors</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is the <a href="/wiki/Time_ordering" class="mw-redirect" title="Time ordering">time ordering</a> operator that shuffles its operands so the time-components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1871ffeb57c11624b375dbb7157d5887c706eb87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{0}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd56a211bc86164ec0847c63e676c95a484c548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.215ex; height:3.009ex;" alt="{\displaystyle y^{0}}"> increase from right to left, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Omega \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ω</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Omega \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353415bb622cc3448da08826b5d4f779c17cba95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.23ex; height:2.843ex;" alt="{\displaystyle |\Omega \rangle }"> is the ground state (vacuum state) of the interacting theory, different from the free ground state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }">. This expression represents the probability amplitude for the field to propagate from y to x, and goes by multiple names, like the two-point <a href="/wiki/Propagator" title="Propagator">propagator</a>, two-point <a href="/wiki/Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">correlation function</a>, two-point <a href="/wiki/Green%27s_function" title="Green's function">Green's function</a> or two-point function for short.<a href="#cite_note-peskin-1">[1]</a>: 82  The free two-point function, also known as the <a href="/wiki/Feynman_propagator" class="mw-redirect" title="Feynman propagator">Feynman propagator</a>, can be found for the real scalar field by either canonical quantization or path integrals to be<a href="#cite_note-peskin-1">[1]</a>: 31,288 <a href="#cite_note-zee-31">[31]</a>: 23  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \equiv D_{F}(x-y)=\lim _{\epsilon \to 0}\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i}{p_{\mu }p^{\mu }-m^{2}+i\epsilon }}e^{-ip_{\mu }(x^{\mu }-y^{\mu })}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>≡</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>p</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </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> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \equiv D_{F}(x-y)=\lim _{\epsilon \to 0}\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i}{p_{\mu }p^{\mu }-m^{2}+i\epsilon }}e^{-ip_{\mu }(x^{\mu }-y^{\mu })}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3409dcaa02262501a6bb39d0e5a715f3778336ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:75.379ex; height:6.676ex;" alt="{\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \equiv D_{F}(x-y)=\lim _{\epsilon \to 0}\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i}{p_{\mu }p^{\mu }-m^{2}+i\epsilon }}e^{-ip_{\mu }(x^{\mu }-y^{\mu })}.}"></dd></dl> In an interacting theory, where the Lagrangian or Hamiltonian contains terms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{I}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{I}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1d8adcca80705f291227712d1c272b8c60a55c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.293ex; height:2.843ex;" alt="{\displaystyle L_{I}(t)}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{I}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{I}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34fa5cb5a2eaeda404215c59c40d177f944c9fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.641ex; height:2.843ex;" alt="{\displaystyle H_{I}(t)}"> that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the free two-point function. In canonical quantization, the two-point correlation function can be written as:<a href="#cite_note-peskin-1">[1]</a>: 87  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\left\langle 0\left|T\left\{\phi _{I}(x)\phi _{I}(y)\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}\right|0\right\rangle }{\left\langle 0\left|T\left\{\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}\right|0\right\rangle }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ω</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>⟨</mo> <mrow> <mn>0</mn> <mrow> <mo>|</mo> <mrow> <mi>T</mi> <mrow> <mo>{</mo> <mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>i</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mn>0</mn> </mrow> <mo>⟩</mo> </mrow> <mrow> <mo>⟨</mo> <mrow> <mn>0</mn> <mrow> <mo>|</mo> <mrow> <mi>T</mi> <mrow> <mo>{</mo> <mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>i</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mn>0</mn> </mrow> <mo>⟩</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\left\langle 0\left|T\left\{\phi _{I}(x)\phi _{I}(y)\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}\right|0\right\rangle }{\left\langle 0\left|T\left\{\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}\right|0\right\rangle }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5da3c13010e11cdcfa46460fe31c93cda1ea9e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:78.043ex; height:10.176ex;" alt="{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\left\langle 0\left|T\left\{\phi _{I}(x)\phi _{I}(y)\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}\right|0\right\rangle }{\left\langle 0\left|T\left\{\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}\right|0\right\rangle }},}"></dd></dl> where ε is an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> number and ϕI is the field operator under the free theory. Here, the <a href="/wiki/Exponential_function" title="Exponential function">exponential</a> should be understood as its <a href="/wiki/Power_series" title="Power series">power series</a> expansion. For example, in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3f7052b477f76bbe009fc399bd9bf28b08510b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.44ex; height:3.009ex;" alt="{\displaystyle \phi ^{4}}">-theory, the interacting term of the Hamiltonian is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H_{I}(t)=\int d^{3}x\,{\frac {\lambda }{4!}}\phi _{I}(x)^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle H_{I}(t)=\int d^{3}x\,{\frac {\lambda }{4!}}\phi _{I}(x)^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f28adb88ceac9816c1dcacf8838646196b2f2e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:23.289ex; height:3.843ex;" alt="{\textstyle H_{I}(t)=\int d^{3}x\,{\frac {\lambda }{4!}}\phi _{I}(x)^{4}}">,<a href="#cite_note-peskin-1">[1]</a>: 84  and the expansion of the two-point correlator in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> becomes<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle ={\frac {\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(x)\phi _{I}(y)\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }{\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ω</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="true" scriptlevel="0"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋯</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> <mstyle displaystyle="true" scriptlevel="0"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋯</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle ={\frac {\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(x)\phi _{I}(y)\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }{\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f913b383a859ad2f6a2c18175dbd70af56f157" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:92.411ex; height:14.509ex;" alt="{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle ={\frac {\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(x)\phi _{I}(y)\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }{\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }}.}">This perturbation expansion expresses the interacting two-point function in terms of quantities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 0|\cdots |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle 0|\cdots |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0394cb7d7e65f9822fe96e240efae92ffe1cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.925ex; height:2.843ex;" alt="{\displaystyle \langle 0|\cdots |0\rangle }"> that are evaluated in the free theory. In the path integral formulation, the two-point correlation function can be written<a href="#cite_note-peskin-1">[1]</a>: 284  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\int {\mathcal {D}}\phi \,\phi (x)\phi (y)\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}{\int {\mathcal {D}}\phi \,\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ω</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> </munder> <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> <mspace width="thinmathspace" /> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>i</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>z</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> </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> <mspace width="thinmathspace" /> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>i</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>z</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\int {\mathcal {D}}\phi \,\phi (x)\phi (y)\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}{\int {\mathcal {D}}\phi \,\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a53cf57879d423ddf41fcff180629b0917de9fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:64.393ex; height:10.176ex;" alt="{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\int {\mathcal {D}}\phi \,\phi (x)\phi (y)\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}{\int {\mathcal {D}}\phi \,\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"> is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in λ, reducing the interacting two-point function to quantities in the free theory. <a href="/wiki/Wick%27s_theorem" title="Wick's theorem">Wick's theorem</a> further reduce any n-point correlation function in the free theory to a sum of products of two-point correlation functions. For example, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\langle 0|T\{\phi (x_{1})\phi (x_{2})\phi (x_{3})\phi (x_{4})\}|0\rangle &=\langle 0|T\{\phi (x_{1})\phi (x_{2})\}|0\rangle \langle 0|T\{\phi (x_{3})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{3})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{4})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{3})\}|0\rangle .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <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> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <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> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <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> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <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> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle 0|T\{\phi (x_{1})\phi (x_{2})\phi (x_{3})\phi (x_{4})\}|0\rangle &=\langle 0|T\{\phi (x_{1})\phi (x_{2})\}|0\rangle \langle 0|T\{\phi (x_{3})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{3})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{4})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{3})\}|0\rangle .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b095c661a237453fc0fbb80a67fd603878b1f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:77.035ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\langle 0|T\{\phi (x_{1})\phi (x_{2})\phi (x_{3})\phi (x_{4})\}|0\rangle &=\langle 0|T\{\phi (x_{1})\phi (x_{2})\}|0\rangle \langle 0|T\{\phi (x_{3})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{3})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{4})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{3})\}|0\rangle .\end{aligned}}}"></dd></dl> Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory.<a href="#cite_note-peskin-1">[1]</a>: 90  This makes the Feynman propagator one of the most important quantities in quantum field theory. <div class="mw-heading mw-heading3"><h3 id="Feynman_diagram">Feynman diagram</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=15" title="Edit section: Feynman diagram">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></div> Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a>. For example, the λ1 term in the two-point correlation function in the ϕ4 theory is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {-i\lambda }{4!}}\int d^{4}z\,\langle 0|T\{\phi (x)\phi (y)\phi (z)\phi (z)\phi (z)\phi (z)\}|0\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> <mi>λ</mi> </mrow> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>z</mi> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {-i\lambda }{4!}}\int d^{4}z\,\langle 0|T\{\phi (x)\phi (y)\phi (z)\phi (z)\phi (z)\phi (z)\}|0\rangle .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2273463abbd76f517963e9a16b8db601d8b6e15e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.559ex; height:5.843ex;" alt="{\displaystyle {\frac {-i\lambda }{4!}}\int d^{4}z\,\langle 0|T\{\phi (x)\phi (y)\phi (z)\phi (z)\phi (z)\phi (z)\}|0\rangle .}"></dd></dl> After applying Wick's theorem, one of the terms is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12\cdot {\frac {-i\lambda }{4!}}\int d^{4}z\,D_{F}(x-z)D_{F}(y-z)D_{F}(z-z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>i</mi> <mi>λ</mi> </mrow> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>z</mi> <mspace width="thinmathspace" /> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12\cdot {\frac {-i\lambda }{4!}}\int d^{4}z\,D_{F}(x-z)D_{F}(y-z)D_{F}(z-z).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63d1372db86ab2b7aabd2a97de91d57fd19ae541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.117ex; height:5.843ex;" alt="{\displaystyle 12\cdot {\frac {-i\lambda }{4!}}\int d^{4}z\,D_{F}(x-z)D_{F}(y-z)D_{F}(z-z).}"></dd></dl> This term can instead be obtained from the Feynman diagram <dl><dd><a href="/wiki/File:Phi-4_one-loop.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Phi-4_one-loop.svg/200px-Phi-4_one-loop.svg.png" decoding="async" width="200" height="92" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Phi-4_one-loop.svg/300px-Phi-4_one-loop.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Phi-4_one-loop.svg/400px-Phi-4_one-loop.svg.png 2x" data-file-width="512" data-file-height="235" /></a>.</dd></dl> The diagram consists of <ul><li>external vertices connected with one edge and represented by dots (here labeled <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><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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">).</li> <li>internal vertices connected with four edges and represented by dots (here labeled <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">).</li> <li>edges connecting the vertices and represented by lines.</li></ul> Every vertex corresponds to a single <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> field factor at the corresponding point in spacetime, while the edges correspond to the propagators between the spacetime points. The term in the perturbation series corresponding to the diagram is obtained by writing down the expression that follows from the so-called Feynman rules: <ol><li>For every internal vertex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6e920bac39ad09fff4efef16254595091a1025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}">, write down a factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -i\lambda \int d^{4}z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi>λ</mi> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -i\lambda \int d^{4}z_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61baf6595e8c2db314df6d3f59bb2e3daa208921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.311ex; height:3.176ex;" alt="{\textstyle -i\lambda \int d^{4}z_{i}}">.</li> <li>For every edge that connects two vertices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6e920bac39ad09fff4efef16254595091a1025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412a06424b2eeb1f51d963bc33fb3bd5c3df5f49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.991ex; height:2.343ex;" alt="{\displaystyle z_{j}}">, write down a factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{F}(z_{i}-z_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{F}(z_{i}-z_{j})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2347cf567e1db2b1595c4b85f1c358a1d205061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.909ex; height:3.009ex;" alt="{\displaystyle D_{F}(z_{i}-z_{j})}">.</li> <li>Divide by the symmetry factor of the diagram.</li></ol> With the symmetry factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}">, following these rules yields exactly the expression above. By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space.<a href="#cite_note-peskin-1">[1]</a>: 91–94  In order to compute the n-point correlation function to the k-th order, list all valid Feynman diagrams with n external points and k or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Omega |T\{\phi (x_{1})\cdots \phi (x_{n})\}|\Omega \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo fence="false" stretchy="false">{</mo> <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> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ω</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Omega |T\{\phi (x_{1})\cdots \phi (x_{n})\}|\Omega \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc73e4e939e28688f1d5499fa0333bdb5d13f54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.239ex; height:2.843ex;" alt="{\displaystyle \langle \Omega |T\{\phi (x_{1})\cdots \phi (x_{n})\}|\Omega \rangle }"></dd></dl> is equal to the sum of (expressions corresponding to) all connected diagrams with n external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the ϕ4 interaction theory discussed above, every vertex must have four legs.<a href="#cite_note-peskin-1">[1]</a>: 98  In realistic applications, the scattering amplitude of a certain interaction or the <a href="/wiki/Decay_rate" class="mw-redirect" title="Decay rate">decay rate</a> of a particle can be computed from the <a href="/wiki/S-matrix" title="S-matrix">S-matrix</a>, which itself can be found using the Feynman diagram method.<a href="#cite_note-peskin-1">[1]</a>: 102–115  Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing n loops are referred to as n-loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction.<a href="#cite_note-zee-31">[31]</a>: 44  Lines whose end points are vertices can be thought of as the propagation of <a href="/wiki/Virtual_particle" title="Virtual particle">virtual particles</a>.<a href="#cite_note-peskin-1">[1]</a>: 31  <div class="mw-heading mw-heading3"><h3 id="Renormalization">Renormalization</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=16" title="Edit section: Renormalization">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Renormalization" title="Renormalization">Renormalization</a></div> Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The <a href="/wiki/Renormalisation" class="mw-redirect" title="Renormalisation">renormalisation</a> procedure is a systematic process for removing such infinities. Parameters appearing in the Lagrangian, such as the mass m and the coupling constant λ, have no physical meaning — m, λ, and the field strength ϕ are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off Λ, obtain expressions for the physical quantities, and then take the limit Λ → ∞. This is an example of <a href="/wiki/Regularization_(physics)" title="Regularization (physics)">regularization</a>, a class of methods to treat divergences in QFT, with Λ being the regulator. The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of ϕ4 theory, the field strength is first redefined: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =Z^{1/2}\phi _{r},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =Z^{1/2}\phi _{r},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa51647d9e14f72d08511c3d7df73bb9b4f14e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.896ex; height:3.176ex;" alt="{\displaystyle \phi =Z^{1/2}\phi _{r},}"></dd></dl> where ϕ is the bare field, ϕr is the renormalized field, and Z is a constant to be determined. The Lagrangian density becomes: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}m_{r}^{2}\phi _{r}^{2}-{\frac {\lambda _{r}}{4!}}\phi _{r}^{4}+{\frac {1}{2}}\delta _{Z}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}\delta _{m}\phi _{r}^{2}-{\frac {\delta _{\lambda }}{4!}}\phi _{r}^{4},}"> <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"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}m_{r}^{2}\phi _{r}^{2}-{\frac {\lambda _{r}}{4!}}\phi _{r}^{4}+{\frac {1}{2}}\delta _{Z}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}\delta _{m}\phi _{r}^{2}-{\frac {\delta _{\lambda }}{4!}}\phi _{r}^{4},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b7d9af31c7bda750a2df3ca12efd8ef17e1456" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:78.506ex; height:5.509ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}m_{r}^{2}\phi _{r}^{2}-{\frac {\lambda _{r}}{4!}}\phi _{r}^{4}+{\frac {1}{2}}\delta _{Z}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}\delta _{m}\phi _{r}^{2}-{\frac {\delta _{\lambda }}{4!}}\phi _{r}^{4},}"></dd></dl> where mr and λr are the experimentally measurable, renormalized, mass and coupling constant, respectively, and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{Z}=Z-1,\quad \delta _{m}=m^{2}Z-m_{r}^{2},\quad \delta _{\lambda }=\lambda Z^{2}-\lambda _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> <mo>=</mo> <mi>Z</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>Z</mi> <mo>−</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <mspace width="1em" /> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo>=</mo> <mi>λ</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{Z}=Z-1,\quad \delta _{m}=m^{2}Z-m_{r}^{2},\quad \delta _{\lambda }=\lambda Z^{2}-\lambda _{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa7aecb97908769affb9eefd1a9102bb1a89c4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:49.072ex; height:3.009ex;" alt="{\displaystyle \delta _{Z}=Z-1,\quad \delta _{m}=m^{2}Z-m_{r}^{2},\quad \delta _{\lambda }=\lambda Z^{2}-\lambda _{r}}"></dd></dl> are constants to be determined. The first three terms are the ϕ4 Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularization scheme (such as the cut-off regularization introduced above or <a href="/wiki/Dimensional_regularization" title="Dimensional regularization">dimensional regularization</a>); call the regulator Λ. Compute Feynman diagrams, in which divergent terms will depend on Λ. Then, define δZ, δm, and δλ such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit Λ → ∞ is taken. In this way, meaningful finite quantities are obtained.<a href="#cite_note-peskin-1">[1]</a>: 323–326  It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of elementary particles is a renormalizable QFT,<a href="#cite_note-peskin-1">[1]</a>: 719–727  while <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a> is non-renormalizable.<a href="#cite_note-peskin-1">[1]</a>: 798 <a href="#cite_note-zee-31">[31]</a>: 421  <div class="mw-heading mw-heading4"><h4 id="Renormalization_group">Renormalization group</h4>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=17" title="Edit section: Renormalization group">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group</a></div> The <a href="/wiki/Renormalization_group" title="Renormalization group">renormalization group</a>, developed by <a href="/wiki/Kenneth_G._Wilson" title="Kenneth G. Wilson">Kenneth Wilson</a>, is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales.<a href="#cite_note-peskin-1">[1]</a>: 393  The way in which each parameter changes with scale is described by its <a href="/wiki/Beta_function_(physics)" title="Beta function (physics)">β function</a>.<a href="#cite_note-peskin-1">[1]</a>: 417  Correlation functions, which underlie quantitative physical predictions, change with scale according to the <a href="/wiki/Callan%E2%80%93Symanzik_equation" title="Callan–Symanzik equation">Callan–Symanzik equation</a>.<a href="#cite_note-peskin-1">[1]</a>: 410–411  As an example, the coupling constant in QED, namely the <a href="/wiki/Elementary_charge" title="Elementary charge">elementary charge</a> e, has the following β function: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta (e)\equiv {\frac {1}{\Lambda }}{\frac {de}{d\Lambda }}={\frac {e^{3}}{12\pi ^{2}}}+O{\mathord {\left(e^{5}\right)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi mathvariant="normal">Λ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>e</mi> </mrow> <mrow> <mi>d</mi> <mi mathvariant="normal">Λ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>12</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta (e)\equiv {\frac {1}{\Lambda }}{\frac {de}{d\Lambda }}={\frac {e^{3}}{12\pi ^{2}}}+O{\mathord {\left(e^{5}\right)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5bba61e7ce55fb8efd0c0c55d06542f3470cb88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.614ex; height:6.009ex;" alt="{\displaystyle \beta (e)\equiv {\frac {1}{\Lambda }}{\frac {de}{d\Lambda }}={\frac {e^{3}}{12\pi ^{2}}}+O{\mathord {\left(e^{5}\right)}},}"></dd></dl> where Λ is the energy scale under which the measurement of e is performed. This <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> implies that the observed elementary charge increases as the scale increases.<a href="#cite_note-34">[34]</a> The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant.<a href="#cite_note-peskin-1">[1]</a>: 420  The coupling constant g in <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>, a non-Abelian gauge theory based on the symmetry group <a href="/wiki/Special_unitary_group" title="Special unitary group">SU(3)</a>, has the following β function: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta (g)\equiv {\frac {1}{\Lambda }}{\frac {dg}{d\Lambda }}={\frac {g^{3}}{16\pi ^{2}}}\left(-11+{\frac {2}{3}}N_{f}\right)+O{\mathord {\left(g^{5}\right)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi mathvariant="normal">Λ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>g</mi> </mrow> <mrow> <mi>d</mi> <mi mathvariant="normal">Λ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>16</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>11</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta (g)\equiv {\frac {1}{\Lambda }}{\frac {dg}{d\Lambda }}={\frac {g^{3}}{16\pi ^{2}}}\left(-11+{\frac {2}{3}}N_{f}\right)+O{\mathord {\left(g^{5}\right)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15bc256bf294a6051d72c59eb99c1bc41ab75bd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.464ex; height:6.343ex;" alt="{\displaystyle \beta (g)\equiv {\frac {1}{\Lambda }}{\frac {dg}{d\Lambda }}={\frac {g^{3}}{16\pi ^{2}}}\left(-11+{\frac {2}{3}}N_{f}\right)+O{\mathord {\left(g^{5}\right)}},}"></dd></dl> where Nf is the number of <a href="/wiki/Quark" title="Quark">quark</a> <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">flavours</a>. In the case where Nf ≤ 16 (the Standard Model has Nf = 6), the coupling constant g decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as <a href="/wiki/Asymptotic_freedom" title="Asymptotic freedom">asymptotic freedom</a>.<a href="#cite_note-peskin-1">[1]</a>: 531  <a href="/wiki/Conformal_field_theories" class="mw-redirect" title="Conformal field theories">Conformal field theories</a> (CFTs) are special QFTs that admit <a href="/wiki/Conformal_symmetry" title="Conformal symmetry">conformal symmetry</a>. They are insensitive to changes in the scale, as all their coupling constants have vanishing β function. (The converse is not true, however — the vanishing of all β functions does not imply conformal symmetry of the theory.)<a href="#cite_note-35">[35]</a> Examples include <a href="/wiki/String_theory" title="String theory">string theory</a><a href="#cite_note-polchinski1-25">[25]</a> and <a href="/wiki/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 supersymmetric Yang–Mills theory</a>.<a href="#cite_note-36">[36]</a> According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off Λ, i.e. that the theory is no longer valid at energies higher than Λ, and all degrees of freedom above the scale Λ are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable <a href="/wiki/Effective_field_theory" title="Effective field theory">effective field theory</a>.<a href="#cite_note-peskin-1">[1]</a>: 402–403  The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them.<a href="#cite_note-shifman-8">[8]</a>: 2  According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off Λ from calculations in such a theory merely indicates that new physical phenomena appear at scales above Λ, where a new theory is necessary.<a href="#cite_note-zee-31">[31]</a>: 156  <div class="mw-heading mw-heading3"><h3 id="Other_theories">Other theories</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=18" title="Edit section: Other theories">edit</a>]</div> The quantization and renormalization procedures outlined in the preceding sections are performed for the free theory and <a href="/wiki/Quartic_interaction" title="Quartic interaction">ϕ4 theory</a> of the real scalar field. A similar process can be done for other types of fields, including the <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex</a> scalar field, the <a href="/wiki/Vector_field" title="Vector field">vector field</a>, and the <a href="/wiki/Dirac_field" class="mw-redirect" title="Dirac field">Dirac field</a>, as well as other types of interaction terms, including the electromagnetic interaction and the <a href="/wiki/Yukawa_interaction" title="Yukawa interaction">Yukawa interaction</a>. As an example, <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> contains a Dirac field ψ representing the <a href="/wiki/Electron" title="Electron">electron</a> field and a vector field Aμ representing the electromagnetic field (<a href="/wiki/Photon" title="Photon">photon</a> field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical <a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">electromagnetic four-potential</a>, rather than the classical electric and magnetic fields.) The full QED Lagrangian density is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-e{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ψ</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-e{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2282f6dbffe9874803719aa011f6056fc0d4e37a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.526ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-e{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },}"></dd></dl> where γμ are <a href="/wiki/Dirac_matrices" class="mw-redirect" title="Dirac matrices">Dirac matrices</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dafa5ba70b83e82cde041f28d40f2d367580457f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.504ex; height:3.176ex;" alt="{\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77fbf0c39a13f9706357f83b041774749598ded1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.137ex; height:2.843ex;" alt="{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}"> is the <a href="/wiki/Electromagnetic_field_strength" class="mw-redirect" title="Electromagnetic field strength">electromagnetic field strength</a>. The parameters in this theory are the (bare) electron mass m and the (bare) <a href="/wiki/Elementary_charge" title="Elementary charge">elementary charge</a> e. The first and second terms in the Lagrangian density correspond to the free Dirac field and free vector fields, respectively. The last term describes the interaction between the electron and photon fields, which is treated as a perturbation from the free theories.<a href="#cite_note-peskin-1">[1]</a>: 78  <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Electron-positron-annihilation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Electron-positron-annihilation.svg/220px-Electron-positron-annihilation.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Electron-positron-annihilation.svg/330px-Electron-positron-annihilation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Electron-positron-annihilation.svg/440px-Electron-positron-annihilation.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption></figcaption></figure> Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an <a href="/wiki/Off-shell" class="mw-redirect" title="Off-shell">off-shell</a> photon, and then decaying into a new pair of electron and positron. Time runs from left to right. Arrows pointing forward in time represent the propagation of electrons, while those pointing backward in time represent the propagation of positrons. A wavy line represents the propagation of a photon. Each vertex in QED Feynman diagrams must have an incoming and an outgoing fermion (positron/electron) leg as well as a photon leg. <div class="mw-heading mw-heading4"><h4 id="Gauge_symmetry">Gauge symmetry</h4>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=19" title="Edit section: Gauge symmetry">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></div> If the following transformation to the fields is performed at every spacetime point x (a local transformation), then the QED Lagrangian remains unchanged, or invariant: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)\to e^{i\alpha (x)}\psi (x),\quad A_{\mu }(x)\to A_{\mu }(x)+ie^{-1}e^{-i\alpha (x)}\partial _{\mu }e^{i\alpha (x)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)\to e^{i\alpha (x)}\psi (x),\quad A_{\mu }(x)\to A_{\mu }(x)+ie^{-1}e^{-i\alpha (x)}\partial _{\mu }e^{i\alpha (x)},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eecf0aee23661310cefb2869f2b10c24bf968fec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.006ex; height:3.509ex;" alt="{\displaystyle \psi (x)\to e^{i\alpha (x)}\psi (x),\quad A_{\mu }(x)\to A_{\mu }(x)+ie^{-1}e^{-i\alpha (x)}\partial _{\mu }e^{i\alpha (x)},}"></dd></dl> where α(x) is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the <a href="/wiki/Action_(physics)" title="Action (physics)">action</a>) is invariant under a certain local transformation, then the transformation is referred to as a <a href="/wiki/Gauge_symmetry" class="mw-redirect" title="Gauge symmetry">gauge symmetry</a> of the theory.<a href="#cite_note-peskin-1">[1]</a>: 482–483  Gauge symmetries form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\alpha (x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\alpha (x)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/743e1ae90f3e5c27b0b056cd3f8b447fc4eff121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.155ex; height:2.843ex;" alt="{\displaystyle e^{i\alpha (x)}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\alpha '(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msup> <mi>α</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\alpha '(x)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef394e671080c6b0c0ab22997c971eb047d3c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.686ex; height:2.843ex;" alt="{\displaystyle e^{i\alpha '(x)}}"> is yet another symmetry transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i[\alpha (x)+\alpha '(x)]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">[</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>α</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i[\alpha (x)+\alpha '(x)]}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/171b083b739b394140c10da874c3e4c7d23fe601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.151ex; height:2.843ex;" alt="{\displaystyle e^{i[\alpha (x)+\alpha '(x)]}}">. For any α(x), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\alpha (x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\alpha (x)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/743e1ae90f3e5c27b0b056cd3f8b447fc4eff121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.155ex; height:2.843ex;" alt="{\displaystyle e^{i\alpha (x)}}"> is an element of the <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> group, thus QED is said to have U(1) gauge symmetry.<a href="#cite_note-peskin-1">[1]</a>: 496  The photon field Aμ may be referred to as the U(1) <a href="/wiki/Gauge_boson" title="Gauge boson">gauge boson</a>. U(1) is an <a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a>, meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on <a href="/wiki/Non-Abelian_group" class="mw-redirect" title="Non-Abelian group">non-Abelian groups</a>, giving rise to <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">non-Abelian gauge theories</a> (also known as Yang–Mills theories).<a href="#cite_note-peskin-1">[1]</a>: 489  <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a>, which describes the strong interaction, is a non-Abelian gauge theory with an <a href="/wiki/Special_unitary_group" title="Special unitary group">SU(3)</a> gauge symmetry. It contains three Dirac fields ψi, i = 1,2,3 representing <a href="/wiki/Quark" title="Quark">quark</a> fields as well as eight vector fields Aa,μ, a = 1,...,8 representing <a href="/wiki/Gluon" title="Gluon">gluon</a> fields, which are the SU(3) gauge bosons.<a href="#cite_note-peskin-1">[1]</a>: 547  The QCD Lagrangian density is:<a href="#cite_note-peskin-1">[1]</a>: 490–491  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}=i{\bar {\psi }}^{i}\gamma ^{\mu }(D_{\mu })^{ij}\psi ^{j}-{\frac {1}{4}}F_{\mu \nu }^{a}F^{a,\mu \nu }-m{\bar {\psi }}^{i}\psi ^{i},}"> <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> <mi>i</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=i{\bar {\psi }}^{i}\gamma ^{\mu }(D_{\mu })^{ij}\psi ^{j}-{\frac {1}{4}}F_{\mu \nu }^{a}F^{a,\mu \nu }-m{\bar {\psi }}^{i}\psi ^{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c0805046b3dc9149a58c16fd67fb5c23af534c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.161ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}=i{\bar {\psi }}^{i}\gamma ^{\mu }(D_{\mu })^{ij}\psi ^{j}-{\frac {1}{4}}F_{\mu \nu }^{a}F^{a,\mu \nu }-m{\bar {\psi }}^{i}\psi ^{i},}"></dd></dl> where Dμ is the gauge <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }=\partial _{\mu }-igA_{\mu }^{a}t^{a},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <mi>g</mi> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }=\partial _{\mu }-igA_{\mu }^{a}t^{a},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/429452ed8f9b403b7597eb775dd1aa506ff1008c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.018ex; height:3.009ex;" alt="{\displaystyle D_{\mu }=\partial _{\mu }-igA_{\mu }^{a}t^{a},}"></dd></dl> where g is the coupling constant, ta are the eight <a href="/wiki/Lie_algebra" title="Lie algebra">generators</a> of SU(3) in the <a href="/wiki/Fundamental_representation" title="Fundamental representation">fundamental representation</a> (3×3 matrices), <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>+</mo> <mi>g</mi> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9174c7b5dc59ed20be48d38c292db9a6ff8a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.393ex; height:3.343ex;" alt="{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c},}"></dd></dl> and fabc are the <a href="/wiki/Structure_constants" title="Structure constants">structure constants</a> of SU(3). Repeated indices i,j,a are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{i}(x)\to U^{ij}(x)\psi ^{j}(x),\quad A_{\mu }^{a}(x)t^{a}\to U(x)\left[A_{\mu }^{a}(x)t^{a}+ig^{-1}\partial _{\mu }\right]U^{\dagger }(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>+</mo> <mi>i</mi> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{i}(x)\to U^{ij}(x)\psi ^{j}(x),\quad A_{\mu }^{a}(x)t^{a}\to U(x)\left[A_{\mu }^{a}(x)t^{a}+ig^{-1}\partial _{\mu }\right]U^{\dagger }(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1ca05167f65c0cba1f6243946464914e11ab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:67.926ex; height:3.343ex;" alt="{\displaystyle \psi ^{i}(x)\to U^{ij}(x)\psi ^{j}(x),\quad A_{\mu }^{a}(x)t^{a}\to U(x)\left[A_{\mu }^{a}(x)t^{a}+ig^{-1}\partial _{\mu }\right]U^{\dagger }(x),}"></dd></dl> where U(x) is an element of SU(3) at every spacetime point x: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x)=e^{i\alpha (x)^{a}t^{a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(x)=e^{i\alpha (x)^{a}t^{a}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4e62102001aa04cd813a6e8cc0bec2f964bb4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.156ex; height:3.343ex;" alt="{\displaystyle U(x)=e^{i\alpha (x)^{a}t^{a}}.}"></dd></dl> The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called <a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">anomaly</a>. For instance, in the path integral formulation, despite the invariance of the Lagrangian density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}[\phi ,\partial _{\mu }\phi ]}"> <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 stretchy="false">[</mo> <mi>ϕ</mi> <mo>,</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}[\phi ,\partial _{\mu }\phi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/562e89f35d6535c06ecb8c3cc70aa8da84920ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.16ex; height:3.009ex;" alt="{\displaystyle {\mathcal {L}}[\phi ,\partial _{\mu }\phi ]}"> under a certain local transformation of the fields, the <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int {\mathcal {D}}\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" 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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int {\mathcal {D}}\phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d02aa00ebe1ebfe5cdfdb54a1b608fb7967e9660" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.982ex; height:3.176ex;" alt="{\textstyle \int {\mathcal {D}}\phi }"> of the path integral may change.<a href="#cite_note-zee-31">[31]</a>: 243  For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group SU(3) × SU(2) × U(1), in which all anomalies exactly cancel.<a href="#cite_note-peskin-1">[1]</a>: 705–707  The theoretical foundation of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, the <a href="/wiki/Equivalence_principle" title="Equivalence principle">equivalence principle</a>, can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>.<a href="#cite_note-37">[37]</a> <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> states that every continuous symmetry, i.e. the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding <a href="/wiki/Conservation_law" title="Conservation law">conservation law</a>.<a href="#cite_note-peskin-1">[1]</a>: 17–18 <a href="#cite_note-zee-31">[31]</a>: 73  For example, the U(1) symmetry of QED implies <a href="/wiki/Charge_conservation" title="Charge conservation">charge conservation</a>.<a href="#cite_note-38">[38]</a> Gauge-transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field Aμ, being a <a href="/wiki/Four-vector" title="Four-vector">four-vector</a>, has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the <a href="/wiki/Photon_polarization" title="Photon polarization">polarization</a>. The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing Aμ can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description.<a href="#cite_note-zee-31">[31]</a>: 168  To account for the gauge redundancy in the path integral formulation, one must perform the so-called <a href="/wiki/Faddeev%E2%80%93Popov_ghost" title="Faddeev–Popov ghost">Faddeev–Popov</a> <a href="/wiki/Gauge_fixing" title="Gauge fixing">gauge fixing</a> procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally.<a href="#cite_note-peskin-1">[1]</a>: 512–515  A more rigorous generalization of the Faddeev–Popov procedure is given by <a href="/wiki/BRST_quantization" title="BRST quantization">BRST quantization</a>.<a href="#cite_note-peskin-1">[1]</a>: 517  <div class="mw-heading mw-heading4"><h4 id="Spontaneous_symmetry-breaking">Spontaneous symmetry-breaking</h4>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=20" title="Edit section: Spontaneous symmetry-breaking">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></div> <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a> is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it.<a href="#cite_note-peskin-1">[1]</a>: 347  To illustrate the mechanism, consider a linear <a href="/wiki/Sigma_model" title="Sigma model">sigma model</a> containing N real scalar fields, described by the Lagrangian density: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\phi ^{i}\right)\left(\partial ^{\mu }\phi ^{i}\right)+{\frac {1}{2}}\mu ^{2}\phi ^{i}\phi ^{i}-{\frac {\lambda }{4}}\left(\phi ^{i}\phi ^{i}\right)^{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"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mn>4</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\phi ^{i}\right)\left(\partial ^{\mu }\phi ^{i}\right)+{\frac {1}{2}}\mu ^{2}\phi ^{i}\phi ^{i}-{\frac {\lambda }{4}}\left(\phi ^{i}\phi ^{i}\right)^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2624f26f917374e36966e8251d3c0897037849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.027ex; height:5.343ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\phi ^{i}\right)\left(\partial ^{\mu }\phi ^{i}\right)+{\frac {1}{2}}\mu ^{2}\phi ^{i}\phi ^{i}-{\frac {\lambda }{4}}\left(\phi ^{i}\phi ^{i}\right)^{2},}"></dd></dl> where μ and λ are real parameters. The theory admits an <a href="/wiki/Orthogonal_group" title="Orthogonal group">O(N)</a> global symmetry: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{i}\to R^{ij}\phi ^{j},\quad R\in \mathrm {O} (N).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>R</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ^{i}\to R^{ij}\phi ^{j},\quad R\in \mathrm {O} (N).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb7d83f04c827e608f477a88bd80597526bd2f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.625ex; height:3.176ex;" alt="{\displaystyle \phi ^{i}\to R^{ij}\phi ^{j},\quad R\in \mathrm {O} (N).}"></dd></dl> The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field ϕ0 satisfying <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{0}^{i}\phi _{0}^{i}={\frac {\mu ^{2}}{\lambda }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>λ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{0}^{i}\phi _{0}^{i}={\frac {\mu ^{2}}{\lambda }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/443d1f0831d71b6aef5faa858343c4a66ad6b772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.917ex; height:5.843ex;" alt="{\displaystyle \phi _{0}^{i}\phi _{0}^{i}={\frac {\mu ^{2}}{\lambda }}.}"></dd></dl> Without loss of generality, let the ground state be in the N-th direction: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{0}^{i}=\left(0,\cdots ,0,{\frac {\mu }{\sqrt {\lambda }}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <msqrt> <mi>λ</mi> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{0}^{i}=\left(0,\cdots ,0,{\frac {\mu }{\sqrt {\lambda }}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f8a52bc3a4e8f8dd835fbc3f68f846bae6a270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.658ex; height:6.509ex;" alt="{\displaystyle \phi _{0}^{i}=\left(0,\cdots ,0,{\frac {\mu }{\sqrt {\lambda }}}\right).}"></dd></dl> The original N fields can be rewritten as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{i}(x)=\left(\pi ^{1}(x),\cdots ,\pi ^{N-1}(x),{\frac {\mu }{\sqrt {\lambda }}}+\sigma (x)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <msqrt> <mi>λ</mi> </msqrt> </mfrac> </mrow> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ^{i}(x)=\left(\pi ^{1}(x),\cdots ,\pi ^{N-1}(x),{\frac {\mu }{\sqrt {\lambda }}}+\sigma (x)\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de55d6a4ef042cf48a829b30cfc627b4b3839e6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:44.319ex; height:6.509ex;" alt="{\displaystyle \phi ^{i}(x)=\left(\pi ^{1}(x),\cdots ,\pi ^{N-1}(x),{\frac {\mu }{\sqrt {\lambda }}}+\sigma (x)\right),}"></dd></dl> and the original Lagrangian density as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\pi ^{k}\right)\left(\partial ^{\mu }\pi ^{k}\right)+{\frac {1}{2}}\left(\partial _{\mu }\sigma \right)\left(\partial ^{\mu }\sigma \right)-{\frac {1}{2}}\left(2\mu ^{2}\right)\sigma ^{2}-{\sqrt {\lambda }}\mu \sigma ^{3}-{\sqrt {\lambda }}\mu \pi ^{k}\pi ^{k}\sigma -{\frac {\lambda }{2}}\pi ^{k}\pi ^{k}\sigma ^{2}-{\frac {\lambda }{4}}\left(\pi ^{k}\pi ^{k}\right)^{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"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>σ</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>σ</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> </msqrt> </mrow> <mi>μ</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> </msqrt> </mrow> <mi>μ</mi> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>σ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mn>4</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\pi ^{k}\right)\left(\partial ^{\mu }\pi ^{k}\right)+{\frac {1}{2}}\left(\partial _{\mu }\sigma \right)\left(\partial ^{\mu }\sigma \right)-{\frac {1}{2}}\left(2\mu ^{2}\right)\sigma ^{2}-{\sqrt {\lambda }}\mu \sigma ^{3}-{\sqrt {\lambda }}\mu \pi ^{k}\pi ^{k}\sigma -{\frac {\lambda }{2}}\pi ^{k}\pi ^{k}\sigma ^{2}-{\frac {\lambda }{4}}\left(\pi ^{k}\pi ^{k}\right)^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc13b6116f5e4bf1ea9477f929b3d60b71f4339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:101.864ex; height:5.343ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\pi ^{k}\right)\left(\partial ^{\mu }\pi ^{k}\right)+{\frac {1}{2}}\left(\partial _{\mu }\sigma \right)\left(\partial ^{\mu }\sigma \right)-{\frac {1}{2}}\left(2\mu ^{2}\right)\sigma ^{2}-{\sqrt {\lambda }}\mu \sigma ^{3}-{\sqrt {\lambda }}\mu \pi ^{k}\pi ^{k}\sigma -{\frac {\lambda }{2}}\pi ^{k}\pi ^{k}\sigma ^{2}-{\frac {\lambda }{4}}\left(\pi ^{k}\pi ^{k}\right)^{2},}"></dd></dl> where k = 1, ..., N − 1. The original O(N) global symmetry is no longer manifest, leaving only the <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> O(N − 1). The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken.<a href="#cite_note-peskin-1">[1]</a>: 349–350  <a href="/wiki/Goldstone%27s_theorem" class="mw-redirect" title="Goldstone's theorem">Goldstone's theorem</a> states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example, O(N) has N(N − 1)/2 continuous symmetries (the dimension of its <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>), while O(N − 1) has (N − 1)(N − 2)/2. The number of broken symmetries is their difference, N − 1, which corresponds to the N − 1 massless fields πk.<a href="#cite_note-peskin-1">[1]</a>: 351  On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson.<a href="#cite_note-peskin-1">[1]</a>: 743–744  In the QFT of <a href="/wiki/Ferromagnetism" title="Ferromagnetism">ferromagnetism</a>, spontaneous symmetry breaking can explain the alignment of <a href="/wiki/Magnetic_dipole" title="Magnetic dipole">magnetic dipoles</a> at low temperatures.<a href="#cite_note-zee-31">[31]</a>: 199  In the Standard Model of elementary particles, the <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W and Z bosons</a>, which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a>, a process called the <a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a>.<a href="#cite_note-peskin-1">[1]</a>: 690  <div class="mw-heading mw-heading4"><h4 id="Supersymmetry">Supersymmetry</h4>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=21" title="Edit section: Supersymmetry">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></div> All experimentally known symmetries in nature relate <a href="/wiki/Boson" title="Boson">bosons</a> to bosons and <a href="/wiki/Fermion" title="Fermion">fermions</a> to fermions. Theorists have hypothesized the existence of a type of symmetry, called <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a>, that relates bosons and fermions.<a href="#cite_note-peskin-1">[1]</a>: 795 <a href="#cite_note-zee-31">[31]</a>: 443  The Standard Model obeys <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré symmetry</a>, whose generators are the spacetime <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a> Pμ and the <a href="/wiki/Lorentz_transformations" class="mw-redirect" title="Lorentz transformations">Lorentz transformations</a> Jμν.<a href="#cite_note-WeinbergQFT-39">[39]</a>: 58–60  In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators Qα, called <a href="/wiki/Supercharge" title="Supercharge">supercharges</a>, which themselves transform as <a href="/wiki/Weyl_fermion" class="mw-redirect" title="Weyl fermion">Weyl fermions</a>.<a href="#cite_note-peskin-1">[1]</a>: 795 <a href="#cite_note-zee-31">[31]</a>: 444  The symmetry group generated by all these generators is known as the <a href="/wiki/Super-Poincar%C3%A9_group" class="mw-redirect" title="Super-Poincaré group">super-Poincaré group</a>. In general there can be more than one set of supersymmetry generators, QαI, I = 1, ..., N, which generate the corresponding N = 1 supersymmetry, N = 2 supersymmetry, and so on.<a href="#cite_note-peskin-1">[1]</a>: 795 <a href="#cite_note-zee-31">[31]</a>: 450  Supersymmetry can also be constructed in other dimensions,<a href="#cite_note-40">[40]</a> most notably in (1+1) dimensions for its application in <a href="/wiki/Superstring_theory" title="Superstring theory">superstring theory</a>.<a href="#cite_note-41">[41]</a> The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group.<a href="#cite_note-zee-31">[31]</a>: 448  Examples of such theories include: <a href="/wiki/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">Minimal Supersymmetric Standard Model</a> (MSSM), <a href="/wiki/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 supersymmetric Yang–Mills theory</a>,<a href="#cite_note-zee-31">[31]</a>: 450  and superstring theory. In a supersymmetric theory, every fermion has a bosonic <a href="/wiki/Superpartner" title="Superpartner">superpartner</a> and vice versa.<a href="#cite_note-zee-31">[31]</a>: 444  If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called <a href="/wiki/Supergravity" title="Supergravity">supergravity</a>.<a href="#cite_note-NathArnowitt-42">[42]</a> Supersymmetry is a potential solution to many current problems in physics. For example, the <a href="/wiki/Hierarchy_problem" title="Hierarchy problem">hierarchy problem</a> of the Standard Model—why the mass of the Higgs boson is not radiatively corrected (under renormalization) to a very high scale such as the <a href="/wiki/Grand_Unified_Theory" title="Grand Unified Theory">grand unified scale</a> or the <a href="/wiki/Planck_mass" class="mw-redirect" title="Planck mass">Planck scale</a>—can be resolved by relating the <a href="/wiki/Higgs_field" class="mw-redirect" title="Higgs field">Higgs field</a> and its super-partner, the <a href="/wiki/Higgsino" title="Higgsino">Higgsino</a>. Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of <a href="/wiki/Dark_matter" title="Dark matter">dark matter</a>.<a href="#cite_note-peskin-1">[1]</a>: 796–797 <a href="#cite_note-43">[43]</a> Nevertheless, experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.<a href="#cite_note-peskin-1">[1]</a>: 797 <a href="#cite_note-zee-31">[31]</a>: 443  <div class="mw-heading mw-heading4"><h4 id="Other_spacetimes">Other spacetimes</h4>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=22" title="Edit section: Other spacetimes">edit</a>]</div> The ϕ4 theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT a priori imposes no restriction on the number of dimensions nor the geometry of spacetime. In <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>, QFT is used to describe <a href="/wiki/Two-dimensional_electron_gas" title="Two-dimensional electron gas">(2+1)-dimensional electron gases</a>.<a href="#cite_note-44">[44]</a> In <a href="/wiki/High-energy_physics" class="mw-redirect" title="High-energy physics">high-energy physics</a>, <a href="/wiki/String_theory" title="String theory">string theory</a> is a type of (1+1)-dimensional QFT,<a href="#cite_note-zee-31">[31]</a>: 452 <a href="#cite_note-polchinski1-25">[25]</a> while <a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a> uses gravity in <a href="/wiki/Extra_dimensions" title="Extra dimensions">extra dimensions</a> to produce gauge theories in lower dimensions.<a href="#cite_note-zee-31">[31]</a>: 428–429  In Minkowski space, the flat <a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metric</a> ημν is used to <a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">raise and lower</a> spacetime indices in the Lagrangian, e.g. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }A^{\mu }=\eta _{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ϕ</mi> <mo>=</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mi>ϕ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }A^{\mu }=\eta _{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d10af7a1162d0b8fe30e9ee41e4de2d7da8920b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.827ex; height:3.009ex;" alt="{\displaystyle A_{\mu }A^{\mu }=\eta _{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}"></dd></dl> where ημν is the inverse of ημν satisfying ημρηρν = δμν. For <a href="/wiki/Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">QFTs in curved spacetime</a> on the other hand, a general metric (such as the <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metric</a> describing a <a href="/wiki/Black_hole" title="Black hole">black hole</a>) is used: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }A^{\mu }=g_{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ϕ</mi> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mi>ϕ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }A^{\mu }=g_{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b353dc8b4cc4af9454b6f378a69495d39256194c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.725ex; height:3.009ex;" alt="{\displaystyle A_{\mu }A^{\mu }=g_{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}"></dd></dl> where gμν is the inverse of gμν. For a real scalar field, the Lagrangian density in a general spacetime background is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\sqrt {|g|}}\left({\frac {1}{2}}g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right),}"> <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"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mi>ϕ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\sqrt {|g|}}\left({\frac {1}{2}}g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ff0e92e1b18f615644c213274949efd5bab46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.832ex; height:6.176ex;" alt="{\displaystyle {\mathcal {L}}={\sqrt {|g|}}\left({\frac {1}{2}}g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right),}"></dd></dl> where g = det(gμν), and ∇μ denotes the <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a>.<a href="#cite_note-45">[45]</a> The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background. <div class="mw-heading mw-heading4"><h4 id="Topological_quantum_field_theory">Topological quantum field theory</h4>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=23" title="Edit section: Topological quantum field theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></div> The correlation functions and physical predictions of a QFT depend on the spacetime metric gμν. For a special class of QFTs called <a href="/wiki/Topological_quantum_field_theories" class="mw-redirect" title="Topological quantum field theories">topological quantum field theories</a> (TQFTs), all correlation functions are independent of continuous changes in the spacetime metric.<a href="#cite_note-46">[46]</a>: 36  QFTs in curved spacetime generally change according to the geometry (local structure) of the spacetime background, while TQFTs are invariant under spacetime <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a> but are sensitive to the <a href="/wiki/Topology" title="Topology">topology</a> (global structure) of spacetime. This means that all calculational results of TQFTs are <a href="/wiki/Topological_invariant" class="mw-redirect" title="Topological invariant">topological invariants</a> of the underlying spacetime. <a href="/wiki/Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons theory</a> is an example of TQFT and has been used to construct models of quantum gravity.<a href="#cite_note-47">[47]</a> Applications of TQFT include the <a href="/wiki/Fractional_quantum_Hall_effect" title="Fractional quantum Hall effect">fractional quantum Hall effect</a> and <a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">topological quantum computers</a>.<a href="#cite_note-48">[48]</a>: 1–5  The world line trajectory of fractionalized particles (known as <a href="/wiki/Anyons" class="mw-redirect" title="Anyons">anyons</a>) can form a link configuration in the spacetime,<a href="#cite_note-49">[49]</a> which relates the braiding statistics of anyons in physics to the link invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond.<a href="#cite_note-50">[50]</a> <div class="mw-heading mw-heading3"><h3 id="Perturbative_and_non-perturbative_methods">Perturbative and non-perturbative methods</h3>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=24" title="Edit section: Perturbative and non-perturbative methods">edit</a>]</div> Using <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a>, the total effect of a small interaction term can be approximated order by order by a series expansion in the number of <a href="/wiki/Virtual_particle" title="Virtual particle">virtual particles</a> participating in the interaction. Every term in the expansion may be understood as one possible way for (physical) particles to interact with each other via virtual particles, expressed visually using a <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a>. The <a href="/wiki/Electromagnetic_force" class="mw-redirect" title="Electromagnetic force">electromagnetic force</a> between two electrons in QED is represented (to first order in perturbation theory) by the propagation of a virtual photon. In a similar manner, the <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W and Z bosons</a> carry the weak interaction, while <a href="/wiki/Gluon" title="Gluon">gluons</a> carry the strong interaction. The interpretation of an interaction as a sum of intermediate states involving the exchange of various virtual particles only makes sense in the framework of perturbation theory. In contrast, non-perturbative methods in QFT treat the interacting Lagrangian as a whole without any series expansion. Instead of particles that carry interactions, these methods have spawned such concepts as <a href="/wiki/%27t_Hooft%E2%80%93Polyakov_monopole" title="'t Hooft–Polyakov monopole">'t Hooft–Polyakov monopole</a>, <a href="/wiki/Domain_wall" title="Domain wall">domain wall</a>, <a href="/wiki/Flux_tube" title="Flux tube">flux tube</a>, and <a href="/wiki/Instanton" title="Instanton">instanton</a>.<a href="#cite_note-shifman-8">[8]</a> Examples of QFTs that are completely solvable non-perturbatively include <a href="/wiki/Minimal_model_(physics)" title="Minimal model (physics)">minimal models</a> of <a href="/wiki/Conformal_field_theory" title="Conformal field theory">conformal field theory</a><a href="#cite_note-51">[51]</a> and the <a href="/wiki/Thirring_model" title="Thirring model">Thirring model</a>.<a href="#cite_note-52">[52]</a> <div class="mw-heading mw-heading2"><h2 id="Mathematical_rigor">Mathematical rigor</h2>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=25" title="Edit section: Mathematical rigor">edit</a>]</div> In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to <a href="/wiki/Haag%27s_theorem" title="Haag's theorem">Haag's theorem</a>, there does not exist a well-defined <a href="/wiki/Interaction_picture" title="Interaction picture">interaction picture</a> for QFT, which implies that <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a> of QFT, which underlies the entire <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a> method, is fundamentally ill-defined.<a href="#cite_note-53">[53]</a> However, perturbative quantum field theory, which only requires that quantities be computable as a formal power series without any convergence requirements, can be given a rigorous mathematical treatment. In particular, <a href="/wiki/Kevin_Costello" title="Kevin Costello">Kevin Costello</a>'s monograph Renormalization and Effective Field Theory<a href="#cite_note-costello-54">[54]</a> provides a rigorous formulation of perturbative renormalization that combines both the effective-field theory approaches of <a href="/wiki/Leo_Kadanoff" title="Leo Kadanoff">Kadanoff</a>, <a href="/wiki/Kenneth_G._Wilson" title="Kenneth G. Wilson">Wilson</a>, and <a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a>, together with the <a href="/wiki/Batalin-Vilkovisky" class="mw-redirect" title="Batalin-Vilkovisky">Batalin-Vilkovisky</a> approach to quantizing gauge theories. Furthermore, perturbative path-integral methods, typically understood as formal computational methods inspired from finite-dimensional integration theory,<a href="#cite_note-ren-55">[55]</a> can be given a sound mathematical interpretation from their finite-dimensional analogues.<a href="#cite_note-nguyen-56">[56]</a> Since the 1950s,<a href="#cite_note-buchholz-57">[57]</a> theoretical physicists and mathematicians have attempted to organize all QFTs into a set of <a href="/wiki/Axiom" title="Axiom">axioms</a>, in order to establish the existence of concrete models of relativistic QFT in a mathematically rigorous way and to study their properties. This line of study is called <a href="/wiki/Constructive_quantum_field_theory" title="Constructive quantum field theory">constructive quantum field theory</a>, a subfield of <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a>,<a href="#cite_note-summers-58">[58]</a>: 2  which has led to such results as <a href="/wiki/CPT_theorem" class="mw-redirect" title="CPT theorem">CPT theorem</a>, <a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin–statistics theorem</a>, and <a href="/wiki/Goldstone%27s_theorem" class="mw-redirect" title="Goldstone's theorem">Goldstone's theorem</a>,<a href="#cite_note-buchholz-57">[57]</a> and also to mathematically rigorous constructions of many interacting QFTs in two and three spacetime dimensions, e.g. two-dimensional scalar field theories with arbitrary polynomial interactions,<a href="#cite_note-Simon-59">[59]</a> the three-dimensional scalar field theories with a quartic interaction, etc.<a href="#cite_note-Glimm1987-60">[60]</a> Compared to ordinary QFT, <a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theory</a> and <a href="/wiki/Conformal_field_theory" title="Conformal field theory">conformal field theory</a> are better supported mathematically — both can be classified in the framework of <a href="/wiki/Representation_(mathematics)" title="Representation (mathematics)">representations</a> of <a href="/wiki/Cobordism" title="Cobordism">cobordisms</a>.<a href="#cite_note-61">[61]</a> <a href="/wiki/Algebraic_quantum_field_theory" title="Algebraic quantum field theory">Algebraic quantum field theory</a> is another approach to the axiomatization of QFT, in which the fundamental objects are local operators and the algebraic relations between them. Axiomatic systems following this approach include <a href="/wiki/Wightman_axioms" title="Wightman axioms">Wightman axioms</a> and <a href="/wiki/Haag%E2%80%93Kastler_axioms" class="mw-redirect" title="Haag–Kastler axioms">Haag–Kastler axioms</a>.<a href="#cite_note-summers-58">[58]</a>: 2–3  One way to construct theories satisfying Wightman axioms is to use <a href="/wiki/Osterwalder%E2%80%93Schrader_axioms" class="mw-redirect" title="Osterwalder–Schrader axioms">Osterwalder–Schrader axioms</a>, which give the necessary and sufficient conditions for a real time theory to be obtained from an <a href="/wiki/Imaginary_time" title="Imaginary time">imaginary time</a> theory by <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> (<a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a>).<a href="#cite_note-summers-58">[58]</a>: 10  <a href="/wiki/Yang%E2%80%93Mills_existence_and_mass_gap" title="Yang–Mills existence and mass gap">Yang–Mills existence and mass gap</a>, one of the <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a>, concerns the well-defined existence of <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theories</a> as set out by the above axioms. The full problem statement is as follows.<a href="#cite_note-62">[62]</a> <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"> Prove that for any <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Simple_group" title="Simple group">simple</a> <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a> G, a non-trivial quantum Yang–Mills theory exists on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"> and has a <a href="/wiki/Mass_gap" title="Mass gap">mass gap</a> Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in <a href="#CITEREFStreaterWightman1964">Streater & Wightman (1964)</a>, <a href="#CITEREFOsterwalderSchrader1973">Osterwalder & Schrader (1973)</a> and <a href="#CITEREFOsterwalderSchrader1975">Osterwalder & Schrader (1975)</a>. </blockquote> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=26" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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href="#cite_ref-peskin_1-0">a</a> <a href="#cite_ref-peskin_1-1">b</a> <a href="#cite_ref-peskin_1-2">c</a> <a href="#cite_ref-peskin_1-3">d</a> <a href="#cite_ref-peskin_1-4">e</a> <a href="#cite_ref-peskin_1-5">f</a> <a href="#cite_ref-peskin_1-6">g</a> <a href="#cite_ref-peskin_1-7">h</a> <a href="#cite_ref-peskin_1-8">i</a> <a href="#cite_ref-peskin_1-9">j</a> <a href="#cite_ref-peskin_1-10">k</a> <a href="#cite_ref-peskin_1-11">l</a> <a href="#cite_ref-peskin_1-12">m</a> <a href="#cite_ref-peskin_1-13">n</a> <a href="#cite_ref-peskin_1-14">o</a> <a href="#cite_ref-peskin_1-15">p</a> <a href="#cite_ref-peskin_1-16">q</a> <a href="#cite_ref-peskin_1-17">r</a> <a href="#cite_ref-peskin_1-18">s</a> <a href="#cite_ref-peskin_1-19">t</a> <a href="#cite_ref-peskin_1-20">u</a> <a href="#cite_ref-peskin_1-21">v</a> <a href="#cite_ref-peskin_1-22">w</a> <a href="#cite_ref-peskin_1-23">x</a> <a href="#cite_ref-peskin_1-24">y</a> <a href="#cite_ref-peskin_1-25">z</a> <a href="#cite_ref-peskin_1-26">aa</a> <a href="#cite_ref-peskin_1-27">ab</a> <a href="#cite_ref-peskin_1-28">ac</a> <a href="#cite_ref-peskin_1-29">ad</a> <a href="#cite_ref-peskin_1-30">ae</a> <a href="#cite_ref-peskin_1-31">af</a> <a href="#cite_ref-peskin_1-32">ag</a> <a href="#cite_ref-peskin_1-33">ah</a> <a href="#cite_ref-peskin_1-34">ai</a> <a href="#cite_ref-peskin_1-35">aj</a> <a href="#cite_ref-peskin_1-36">ak</a> <a href="#cite_ref-peskin_1-37">al</a> <a href="#cite_ref-peskin_1-38">am</a> <a href="#cite_ref-peskin_1-39">an</a> <a href="#cite_ref-peskin_1-40">ao</a> <a href="#cite_ref-peskin_1-41">ap</a> <a href="#cite_ref-peskin_1-42">aq</a> <a href="#cite_ref-peskin_1-43">ar</a> <a href="#cite_ref-peskin_1-44">as</a> <a href="#cite_ref-peskin_1-45">at</a> <a href="#cite_ref-peskin_1-46">au</a> <a href="#cite_ref-peskin_1-47">av</a> <a href="#cite_ref-peskin_1-48">aw</a> <a href="#cite_ref-peskin_1-49">ax</a> <a href="#cite_ref-peskin_1-50">ay</a> <a href="#cite_ref-peskin_1-51">az</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPeskinSchroeder1995" class="citation book cs1"><a href="/wiki/Michael_Peskin" title="Michael Peskin">Peskin, M.</a>; Schroeder, D. (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=i35LALN0GosC">An Introduction to Quantum Field Theory</a>. Westview Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-50397-5" title="Special:BookSources/978-0-201-50397-5"><bdi>978-0-201-50397-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Quantum+Field+Theory&rft.pub=Westview+Press&rft.date=1995&rft.isbn=978-0-201-50397-5&rft.aulast=Peskin&rft.aufirst=M.&rft.au=Schroeder%2C+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Di35LALN0GosC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-Hobson-2">^ <a href="#cite_ref-Hobson_2-0">a</a> <a href="#cite_ref-Hobson_2-1">b</a> <a href="#cite_ref-Hobson_2-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHobson2013" class="citation journal cs1">Hobson, Art (2013). "There are no particles, there are only fields". <a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a>. 81 (211): 211–223. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1204.4616">1204.4616</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013AmJPh..81..211H">2013AmJPh..81..211H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.4789885">10.1119/1.4789885</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:18254182">18254182</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=There+are+no+particles%2C+there+are+only+fields&rft.volume=81&rft.issue=211&rft.pages=211-223&rft.date=2013&rft_id=info%3Aarxiv%2F1204.4616&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18254182%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.4789885&rft_id=info%3Abibcode%2F2013AmJPh..81..211H&rft.aulast=Hobson&rft.aufirst=Art&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-weinberg-3">^ <a href="#cite_ref-weinberg_3-0">a</a> <a href="#cite_ref-weinberg_3-1">b</a> <a href="#cite_ref-weinberg_3-2">c</a> <a href="#cite_ref-weinberg_3-3">d</a> <a href="#cite_ref-weinberg_3-4">e</a> <a href="#cite_ref-weinberg_3-5">f</a> <a href="#cite_ref-weinberg_3-6">g</a> <a href="#cite_ref-weinberg_3-7">h</a> <a href="#cite_ref-weinberg_3-8">i</a> <a href="#cite_ref-weinberg_3-9">j</a> <a href="#cite_ref-weinberg_3-10">k</a> <a href="#cite_ref-weinberg_3-11">l</a> <a href="#cite_ref-weinberg_3-12">m</a> <a href="#cite_ref-weinberg_3-13">n</a> <a href="#cite_ref-weinberg_3-14">o</a> <a href="#cite_ref-weinberg_3-15">p</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg1977" class="citation journal cs1"><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg, Steven</a> (1977). "The Search for Unity: Notes for a History of Quantum Field Theory". Daedalus. 106 (4): 17–35. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/20024506">20024506</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Daedalus&rft.atitle=The+Search+for+Unity%3A+Notes+for+a+History+of+Quantum+Field+Theory&rft.volume=106&rft.issue=4&rft.pages=17-35&rft.date=1977&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F20024506%23id-name%3DJSTOR&rft.aulast=Weinberg&rft.aufirst=Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-Heilbron2003-4"><a href="#cite_ref-Heilbron2003_4-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeilbron2003" class="citation book cs1"><a href="/wiki/John_L._Heilbron" title="John L. Heilbron">Heilbron, J. L.</a>, ed. (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/oxfordcompaniont0000unse_s7n3">The Oxford companion to the history of modern science</a>. Oxford ; New York: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-511229-0" title="Special:BookSources/978-0-19-511229-0"><bdi>978-0-19-511229-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Oxford+companion+to+the+history+of+modern+science&rft.place=Oxford+%3B+New+York&rft.pub=Oxford+University+Press&rft.date=2003&rft.isbn=978-0-19-511229-0&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Foxfordcompaniont0000unse_s7n3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-Thomson1893-5"><a href="#cite_ref-Thomson1893_5-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomsonMaxwell1893" class="citation book cs1"><a href="/wiki/Joseph_John_Thomson" class="mw-redirect" title="Joseph John Thomson">Thomson, Joseph John</a>; Maxwell, James Clerk (1893). <a rel="nofollow" class="external text" href="https://archive.org/details/notesonrecentres00thom">Notes on recent researches in electricity and magnetism, intended as a sequel to Professor Clerk-Maxwell's 'Treatise on Electricity and Magnetism'</a>. <a href="/wiki/Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notes+on+recent+researches+in+electricity+and+magnetism%2C+intended+as+a+sequel+to+Professor+Clerk-Maxwell%27s+%27Treatise+on+Electricity+and+Magnetism%27&rft.pub=Clarendon+Press&rft.date=1893&rft.aulast=Thomson&rft.aufirst=Joseph+John&rft.au=Maxwell%2C+James+Clerk&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnotesonrecentres00thom&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-weisskopf-6">^ <a href="#cite_ref-weisskopf_6-0">a</a> <a href="#cite_ref-weisskopf_6-1">b</a> <a href="#cite_ref-weisskopf_6-2">c</a> <a href="#cite_ref-weisskopf_6-3">d</a> <a href="#cite_ref-weisskopf_6-4">e</a> <a href="#cite_ref-weisskopf_6-5">f</a> <a href="#cite_ref-weisskopf_6-6">g</a> <a href="#cite_ref-weisskopf_6-7">h</a> <a href="#cite_ref-weisskopf_6-8">i</a> <a href="#cite_ref-weisskopf_6-9">j</a> <a href="#cite_ref-weisskopf_6-10">k</a> <a href="#cite_ref-weisskopf_6-11">l</a> <a href="#cite_ref-weisskopf_6-12">m</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisskopf1981" class="citation journal cs1"><a href="/wiki/Victor_Weisskopf" title="Victor Weisskopf">Weisskopf, Victor</a> (November 1981). "The development of field theory in the last 50 years". <a href="/wiki/Physics_Today" title="Physics Today">Physics Today</a>. 34 (11): 69–85. <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/1981PhT....34k..69W">1981PhT....34k..69W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.2914365">10.1063/1.2914365</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=The+development+of+field+theory+in+the+last+50+years&rft.volume=34&rft.issue=11&rft.pages=69-85&rft.date=1981-11&rft_id=info%3Adoi%2F10.1063%2F1.2914365&rft_id=info%3Abibcode%2F1981PhT....34k..69W&rft.aulast=Weisskopf&rft.aufirst=Victor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-Heisenberg1999-7"><a href="#cite_ref-Heisenberg1999_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeisenberg1999" class="citation book cs1"><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg, Werner</a> (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/physics-and-philosophy-the-revolution-in-modern-scirnce-werner-heisenberg-f.-s.-c.-northrop">Physics and philosophy: the revolution in modern science</a>. Great minds series. Amherst, N.Y: <a href="/wiki/Prometheus_Books" title="Prometheus Books">Prometheus Books</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-57392-694-2" title="Special:BookSources/978-1-57392-694-2"><bdi>978-1-57392-694-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+and+philosophy%3A+the+revolution+in+modern+science&rft.place=Amherst%2C+N.Y&rft.series=Great+minds+series&rft.pub=Prometheus+Books&rft.date=1999&rft.isbn=978-1-57392-694-2&rft.aulast=Heisenberg&rft.aufirst=Werner&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysics-and-philosophy-the-revolution-in-modern-scirnce-werner-heisenberg-f.-s.-c.-northrop&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-shifman-8">^ <a href="#cite_ref-shifman_8-0">a</a> <a href="#cite_ref-shifman_8-1">b</a> <a href="#cite_ref-shifman_8-2">c</a> <a href="#cite_ref-shifman_8-3">d</a> <a href="#cite_ref-shifman_8-4">e</a> <a href="#cite_ref-shifman_8-5">f</a> <a href="#cite_ref-shifman_8-6">g</a> <a href="#cite_ref-shifman_8-7">h</a> <a href="#cite_ref-shifman_8-8">i</a> <a href="#cite_ref-shifman_8-9">j</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShifman2012" class="citation book cs1"><a href="/wiki/Mikhail_Shifman" title="Mikhail Shifman">Shifman, M.</a> (2012). Advanced Topics in Quantum Field Theory. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-19084-8" title="Special:BookSources/978-0-521-19084-8"><bdi>978-0-521-19084-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Topics+in+Quantum+Field+Theory&rft.pub=Cambridge+University+Press&rft.date=2012&rft.isbn=978-0-521-19084-8&rft.aulast=Shifman&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTomonaga1966" class="citation journal cs1">Tomonaga, Shinichiro (1966). <a rel="nofollow" class="external text" href="https://www.nobelprize.org/prizes/physics/1965/tomonaga/lecture/">"Development of Quantum Electrodynamics"</a>. Science. 154 (3751): 864–868. <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/1966Sci...154..864T">1966Sci...154..864T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.154.3751.864">10.1126/science.154.3751.864</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17744604">17744604</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Development+of+Quantum+Electrodynamics&rft.volume=154&rft.issue=3751&rft.pages=864-868&rft.date=1966&rft_id=info%3Apmid%2F17744604&rft_id=info%3Adoi%2F10.1126%2Fscience.154.3751.864&rft_id=info%3Abibcode%2F1966Sci...154..864T&rft.aulast=Tomonaga&rft.aufirst=Shinichiro&rft_id=https%3A%2F%2Fwww.nobelprize.org%2Fprizes%2Fphysics%2F1965%2Ftomonaga%2Flecture%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-MiltonMehra-10">^ <a href="#cite_ref-MiltonMehra_10-0">a</a> <a href="#cite_ref-MiltonMehra_10-1">b</a> <a href="#cite_ref-MiltonMehra_10-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiltonMehra2000" class="citation book cs1">Milton, K. A.; Mehra, Jagdish (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9SmZSN8F164C">Climbing the Mountain: The Scientific Biography of Julian Schwinger</a> (Repr ed.). Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850658-4" title="Special:BookSources/978-0-19-850658-4"><bdi>978-0-19-850658-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Climbing+the+Mountain%3A+The+Scientific+Biography+of+Julian+Schwinger&rft.place=Oxford&rft.edition=Repr&rft.pub=Oxford+University+Press&rft.date=2000&rft.isbn=978-0-19-850658-4&rft.aulast=Milton&rft.aufirst=K.+A.&rft.au=Mehra%2C+Jagdish&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9SmZSN8F164C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwinger1951" class="citation journal cs1">Schwinger, Julian (July 1951). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063400">"On the Green's functions of quantized fields. I"</a>. Proceedings of the National Academy of Sciences. 37 (7): 452–455. <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/1951PNAS...37..452S">1951PNAS...37..452S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.37.7.452">10.1073/pnas.37.7.452</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063400">1063400</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16578383">16578383</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&rft.atitle=On+the+Green%27s+functions+of+quantized+fields.+I&rft.volume=37&rft.issue=7&rft.pages=452-455&rft.date=1951-07&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1063400%23id-name%3DPMC&rft_id=info%3Abibcode%2F1951PNAS...37..452S&rft_id=info%3Apmid%2F16578383&rft_id=info%3Adoi%2F10.1073%2Fpnas.37.7.452&rft.issn=0027-8424&rft.aulast=Schwinger&rft.aufirst=Julian&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1063400&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwinger1951" class="citation journal cs1">Schwinger, Julian (July 1951). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063401">"On the Green's functions of quantized fields. II"</a>. Proceedings of the National Academy of Sciences. 37 (7): 455–459. <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/1951PNAS...37..455S">1951PNAS...37..455S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.37.7.455">10.1073/pnas.37.7.455</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063401">1063401</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16578384">16578384</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&rft.atitle=On+the+Green%27s+functions+of+quantized+fields.+II&rft.volume=37&rft.issue=7&rft.pages=455-459&rft.date=1951-07&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1063401%23id-name%3DPMC&rft_id=info%3Abibcode%2F1951PNAS...37..455S&rft_id=info%3Apmid%2F16578384&rft_id=info%3Adoi%2F10.1073%2Fpnas.37.7.455&rft.issn=0027-8424&rft.aulast=Schwinger&rft.aufirst=Julian&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1063401&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchweber2005" class="citation journal cs1">Schweber, Silvan S. (2005-05-31). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1142349">"The sources of Schwinger's Green's functions"</a>. Proceedings of the National Academy of Sciences. 102 (22): 7783–7788. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.0405167101">10.1073/pnas.0405167101</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1142349">1142349</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15930139">15930139</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&rft.atitle=The+sources+of+Schwinger%27s+Green%27s+functions&rft.volume=102&rft.issue=22&rft.pages=7783-7788&rft.date=2005-05-31&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1142349%23id-name%3DPMC&rft.issn=0027-8424&rft_id=info%3Apmid%2F15930139&rft_id=info%3Adoi%2F10.1073%2Fpnas.0405167101&rft.aulast=Schweber&rft.aufirst=Silvan+S.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1142349&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwinger1966" class="citation journal cs1">Schwinger, Julian (1966). "Particles and Sources". Phys Rev. 152 (4): 1219. <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/1966PhRv..152.1219S">1966PhRv..152.1219S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.152.1219">10.1103/PhysRev.152.1219</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys+Rev&rft.atitle=Particles+and+Sources&rft.volume=152&rft.issue=4&rft.pages=1219&rft.date=1966&rft_id=info%3Adoi%2F10.1103%2FPhysRev.152.1219&rft_id=info%3Abibcode%2F1966PhRv..152.1219S&rft.aulast=Schwinger&rft.aufirst=Julian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-Perseus_Books-15">^ <a href="#cite_ref-Perseus_Books_15-0">a</a> <a href="#cite_ref-Perseus_Books_15-1">b</a> <a href="#cite_ref-Perseus_Books_15-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwinger1998" class="citation book cs1">Schwinger, Julian (1998). Particles, Sources and Fields vol. 1. Reading, MA: Perseus Books. p. xi. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7382-0053-0" title="Special:BookSources/0-7382-0053-0"><bdi>0-7382-0053-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Particles%2C+Sources+and+Fields+vol.+1&rft.place=Reading%2C+MA&rft.pages=xi&rft.pub=Perseus+Books&rft.date=1998&rft.isbn=0-7382-0053-0&rft.aulast=Schwinger&rft.aufirst=Julian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwinger1998" class="citation book cs1">Schwinger, Julian (1998). Particles, sources, and fields. 2 (1. print ed.). Reading, Mass: Advanced Book Program, Perseus Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7382-0054-5" title="Special:BookSources/978-0-7382-0054-5"><bdi>978-0-7382-0054-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Particles%2C+sources%2C+and+fields.+2&rft.place=Reading%2C+Mass&rft.edition=1.+print&rft.pub=Advanced+Book+Program%2C+Perseus+Books&rft.date=1998&rft.isbn=978-0-7382-0054-5&rft.aulast=Schwinger&rft.aufirst=Julian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwinger1998" class="citation book cs1">Schwinger, Julian (1998). Particles, sources, and fields. 3 (1. print ed.). Reading, Mass: Advanced Book Program, Perseus Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7382-0055-2" title="Special:BookSources/978-0-7382-0055-2"><bdi>978-0-7382-0055-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Particles%2C+sources%2C+and+fields.+3&rft.place=Reading%2C+Mass&rft.edition=1.+print&rft.pub=Advanced+Book+Program%2C+Perseus+Books&rft.date=1998&rft.isbn=978-0-7382-0055-2&rft.aulast=Schwinger&rft.aufirst=Julian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC.R._Hagen1967" class="citation book cs1">C.R. Hagen; et al., eds. (1967). Proc of the 1967 Int. Conference on Particles and Fields. NY: Interscience. p. 128.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proc+of+the+1967+Int.+Conference+on+Particles+and+Fields&rft.place=NY&rft.pages=128&rft.pub=Interscience&rft.date=1967&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwinger1998" class="citation book cs1">Schwinger, Julian (1998). Particles, Sources and Fields vol. 1. Reading, MA: Perseus Bookks. pp. 82–85.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Particles%2C+Sources+and+Fields+vol.+1&rft.place=Reading%2C+MA&rft.pages=82-85&rft.pub=Perseus+Bookks&rft.date=1998&rft.aulast=Schwinger&rft.aufirst=Julian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-thooft-20">^ <a href="#cite_ref-thooft_20-0">a</a> <a href="#cite_ref-thooft_20-1">b</a> <a href="#cite_ref-thooft_20-2">c</a> <a href="#cite_ref-thooft_20-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREF't_Hooft2015" class="citation book cs1"><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft, Gerard</a> (2015-03-17). "The Evolution of Quantum Field Theory". The Standard Theory of Particle Physics. Advanced Series on Directions in High Energy Physics. Vol. 26. pp. 1–27. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1503.05007">1503.05007</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016stpp.conf....1T">2016stpp.conf....1T</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%2F9789814733519_0001">10.1142/9789814733519_0001</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4733-50-2" title="Special:BookSources/978-981-4733-50-2"><bdi>978-981-4733-50-2</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:119198452">119198452</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Evolution+of+Quantum+Field+Theory&rft.btitle=The+Standard+Theory+of+Particle+Physics&rft.series=Advanced+Series+on+Directions+in+High+Energy+Physics&rft.pages=1-27&rft.date=2015-03-17&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119198452%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2016stpp.conf....1T&rft_id=info%3Aarxiv%2F1503.05007&rft_id=info%3Adoi%2F10.1142%2F9789814733519_0001&rft.isbn=978-981-4733-50-2&rft.aulast=%27t+Hooft&rft.aufirst=Gerard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYangMills1954" class="citation journal cs1"><a href="/wiki/Chen-Ning_Yang" class="mw-redirect" title="Chen-Ning Yang">Yang, C. N.</a>; <a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Mills, R. L.</a> (1954-10-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.96.191">"Conservation of Isotopic Spin and Isotopic Gauge Invariance"</a>. <a href="/wiki/Physical_Review" title="Physical Review">Physical Review</a>. 96 (1): 191–195. <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/1954PhRv...96..191Y">1954PhRv...96..191Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.96.191">10.1103/PhysRev.96.191</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Conservation+of+Isotopic+Spin+and+Isotopic+Gauge+Invariance&rft.volume=96&rft.issue=1&rft.pages=191-195&rft.date=1954-10-01&rft_id=info%3Adoi%2F10.1103%2FPhysRev.96.191&rft_id=info%3Abibcode%2F1954PhRv...96..191Y&rft.aulast=Yang&rft.aufirst=C.+N.&rft.au=Mills%2C+R.+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRev.96.191&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-coleman-22">^ <a href="#cite_ref-coleman_22-0">a</a> <a href="#cite_ref-coleman_22-1">b</a> <a href="#cite_ref-coleman_22-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFColeman1979" class="citation journal cs1"><a href="/wiki/Sidney_Coleman" title="Sidney Coleman">Coleman, Sidney</a> (1979-12-14). "The 1979 Nobel Prize in Physics". <a href="/wiki/Science_(journal)" title="Science (journal)">Science</a>. 206 (4424): 1290–1292. <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/1979Sci...206.1290C">1979Sci...206.1290C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.206.4424.1290">10.1126/science.206.4424.1290</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1749117">1749117</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17799637">17799637</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=The+1979+Nobel+Prize+in+Physics&rft.volume=206&rft.issue=4424&rft.pages=1290-1292&rft.date=1979-12-14&rft_id=info%3Adoi%2F10.1126%2Fscience.206.4424.1290&rft_id=info%3Apmid%2F17799637&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1749117%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F1979Sci...206.1290C&rft.aulast=Coleman&rft.aufirst=Sidney&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSutton" class="citation web cs1"><a href="/wiki/Christine_Sutton" title="Christine Sutton">Sutton, Christine</a>. <a rel="nofollow" class="external text" href="https://www.britannica.com/science/Standard-Model">"Standard model"</a>. britannica.com. <a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a>. Retrieved 2018-08-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=britannica.com&rft.atitle=Standard+model&rft.aulast=Sutton&rft.aufirst=Christine&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2FStandard-Model&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKibble2014" class="citation arxiv cs1"><a href="/wiki/Tom_Kibble" title="Tom Kibble">Kibble, Tom W. B.</a> (2014-12-12). "The Standard Model of Particle Physics". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1412.4094">1412.4094</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/physics.hist-ph">physics.hist-ph</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=The+Standard+Model+of+Particle+Physics&rft.date=2014-12-12&rft_id=info%3Aarxiv%2F1412.4094&rft.aulast=Kibble&rft.aufirst=Tom+W.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-polchinski1-25">^ <a href="#cite_ref-polchinski1_25-0">a</a> <a href="#cite_ref-polchinski1_25-1">b</a> <a href="#cite_ref-polchinski1_25-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolchinski2005" class="citation book cs1"><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski, Joseph</a> (2005). String Theory. Vol. 1. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-67227-6" title="Special:BookSources/978-0-521-67227-6"><bdi>978-0-521-67227-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=String+Theory&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=978-0-521-67227-6&rft.aulast=Polchinski&rft.aufirst=Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwarz2012" class="citation arxiv cs1"><a href="/wiki/John_Henry_Schwarz" title="John Henry Schwarz">Schwarz, John H.</a> (2012-01-04). "The Early History of String Theory and Supersymmetry". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1201.0981">1201.0981</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/physics.hist-ph">physics.hist-ph</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=The+Early+History+of+String+Theory+and+Supersymmetry&rft.date=2012-01-04&rft_id=info%3Aarxiv%2F1201.0981&rft.aulast=Schwarz&rft.aufirst=John+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://science.energy.gov/~/media/hep/pdf/Reports/HEP-BES_Roundtable_Report.pdf">"Common Problems in Condensed Matter and High Energy Physics"</a> (PDF). science.energy.gov. Office of Science, <a href="/wiki/U.S._Department_of_Energy" class="mw-redirect" title="U.S. Department of Energy">U.S. Department of Energy</a>. 2015-02-02. Retrieved 2018-07-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=science.energy.gov&rft.atitle=Common+Problems+in+Condensed+Matter+and+High+Energy+Physics&rft.date=2015-02-02&rft_id=https%3A%2F%2Fscience.energy.gov%2F~%2Fmedia%2Fhep%2Fpdf%2FReports%2FHEP-BES_Roundtable_Report.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-wilczek-28">^ <a href="#cite_ref-wilczek_28-0">a</a> <a href="#cite_ref-wilczek_28-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilczek2016" class="citation journal cs1"><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek, Frank</a> (2016-04-19). "Particle Physics and Condensed Matter: The Saga Continues". Physica Scripta. 2016 (T168): 014003. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1604.05669">1604.05669</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016PhST..168a4003W">2016PhST..168a4003W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0031-8949%2FT168%2F1%2F014003">10.1088/0031-8949/T168/1/014003</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:118439678">118439678</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+Scripta&rft.atitle=Particle+Physics+and+Condensed+Matter%3A+The+Saga+Continues&rft.volume=2016&rft.issue=T168&rft.pages=014003&rft.date=2016-04-19&rft_id=info%3Aarxiv%2F1604.05669&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118439678%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0031-8949%2FT168%2F1%2F014003&rft_id=info%3Abibcode%2F2016PhST..168a4003W&rft.aulast=Wilczek&rft.aufirst=Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-tong1-29">^ <a href="#cite_ref-tong1_29-0">a</a> <a href="#cite_ref-tong1_29-1">b</a> <a href="#CITEREFTong2015">Tong 2015</a>, Chapter 1 </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> In fact, its number of degrees of freedom is uncountable, because the vector space dimension of the space of continuous (differentiable, real analytic) functions on even a finite dimensional Euclidean space is uncountable. On the other hand, subspaces (of these function spaces) that one typically considers, such as Hilbert spaces (e.g. the space of square integrable real valued functions) or separable Banach spaces (e.g. the space of continuous real-valued functions on a compact interval, with the uniform convergence norm), have denumerable (i. e. countably infinite) dimension in the category of Banach spaces (though still their Euclidean vector space dimension is uncountable), so in these restricted contexts, the number of degrees of freedom (interpreted now as the vector space dimension of a dense subspace rather than the vector space dimension of the function space of interest itself) is denumerable. </li> <li id="cite_note-zee-31">^ <a href="#cite_ref-zee_31-0">a</a> <a href="#cite_ref-zee_31-1">b</a> <a href="#cite_ref-zee_31-2">c</a> <a href="#cite_ref-zee_31-3">d</a> <a href="#cite_ref-zee_31-4">e</a> <a href="#cite_ref-zee_31-5">f</a> <a href="#cite_ref-zee_31-6">g</a> <a href="#cite_ref-zee_31-7">h</a> <a href="#cite_ref-zee_31-8">i</a> <a href="#cite_ref-zee_31-9">j</a> <a href="#cite_ref-zee_31-10">k</a> <a href="#cite_ref-zee_31-11">l</a> <a href="#cite_ref-zee_31-12">m</a> <a href="#cite_ref-zee_31-13">n</a> <a href="#cite_ref-zee_31-14">o</a> <a href="#cite_ref-zee_31-15">p</a> <a href="#cite_ref-zee_31-16">q</a> <a href="#cite_ref-zee_31-17">r</a> <a href="#cite_ref-zee_31-18">s</a> <a href="#cite_ref-zee_31-19">t</a> <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). <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780691140346">Quantum Field Theory in a Nutshell</a>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-01019-9" title="Special:BookSources/978-0-691-01019-9"><bdi>978-0-691-01019-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory+in+a+Nutshell&rft.pub=Princeton+University+Press&rft.date=2010&rft.isbn=978-0-691-01019-9&rft.aulast=Zee&rft.aufirst=A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_9780691140346&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFock1932" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock, V.</a> (1932-03-10). "Konfigurationsraum und zweite Quantelung". Zeitschrift für Physik (in German). 75 (9–10): 622–647. <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/1932ZPhy...75..622F">1932ZPhy...75..622F</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%2FBF01344458">10.1007/BF01344458</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:186238995">186238995</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=Konfigurationsraum+und+zweite+Quantelung&rft.volume=75&rft.issue=9%E2%80%9310&rft.pages=622-647&rft.date=1932-03-10&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186238995%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01344458&rft_id=info%3Abibcode%2F1932ZPhy...75..622F&rft.aulast=Fock&rft.aufirst=V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeckerBeckerSchwarz2007" class="citation book cs1">Becker, Katrin; <a href="/wiki/Melanie_Becker" title="Melanie Becker">Becker, Melanie</a>; <a href="/wiki/John_Henry_Schwarz" title="John Henry Schwarz">Schwarz, John H.</a> (2007). <a rel="nofollow" class="external text" href="https://archive.org/details/stringtheorymthe00beck_649">String Theory and M-Theory</a>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/stringtheorymthe00beck_649/page/n53">36</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-86069-7" title="Special:BookSources/978-0-521-86069-7"><bdi>978-0-521-86069-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=String+Theory+and+M-Theory&rft.pages=36&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-86069-7&rft.aulast=Becker&rft.aufirst=Katrin&rft.au=Becker%2C+Melanie&rft.au=Schwarz%2C+John+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstringtheorymthe00beck_649&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFujita2008" class="citation arxiv cs1">Fujita, Takehisa (2008-02-01). "Physics of Renormalization Group Equation in QED". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/0606101">hep-th/0606101</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Physics+of+Renormalization+Group+Equation+in+QED&rft.date=2008-02-01&rft_id=info%3Aarxiv%2Fhep-th%2F0606101&rft.aulast=Fujita&rft.aufirst=Takehisa&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAharonyGur-AriKlinghoffer2015" class="citation journal cs1">Aharony, Ofer; Gur-Ari, Guy; Klinghoffer, Nizan (2015-05-19). "The Holographic Dictionary for Beta Functions of Multi-trace Coupling Constants". Journal of High Energy Physics. 2015 (5): 31. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1501.06664">1501.06664</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015JHEP...05..031A">2015JHEP...05..031A</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%2FJHEP05%282015%29031">10.1007/JHEP05(2015)031</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:115167208">115167208</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+High+Energy+Physics&rft.atitle=The+Holographic+Dictionary+for+Beta+Functions+of+Multi-trace+Coupling+Constants&rft.volume=2015&rft.issue=5&rft.pages=31&rft.date=2015-05-19&rft_id=info%3Aarxiv%2F1501.06664&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A115167208%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FJHEP05%282015%29031&rft_id=info%3Abibcode%2F2015JHEP...05..031A&rft.aulast=Aharony&rft.aufirst=Ofer&rft.au=Gur-Ari%2C+Guy&rft.au=Klinghoffer%2C+Nizan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKovacs1999" class="citation arxiv cs1">Kovacs, Stefano (1999-08-26). "N = 4 supersymmetric Yang–Mills theory and the AdS/SCFT correspondence". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9908171">hep-th/9908171</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=%3Cspan+class%3D%22texhtml+%22+%3EN+%3D+4%3C%2Fspan%3E+supersymmetric+Yang%E2%80%93Mills+theory+and+the+AdS%2FSCFT+correspondence&rft.date=1999-08-26&rft_id=info%3Aarxiv%2Fhep-th%2F9908171&rft.aulast=Kovacs&rft.aufirst=Stefano&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> Veltman, M. J. G. (1976). Methods in Field Theory, Proceedings of the Les Houches Summer School, Les Houches, France, 1975. </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrading2002" class="citation journal cs1"><a href="/wiki/Katherine_Brading" title="Katherine Brading">Brading, Katherine A.</a> (March 2002). "Which symmetry? Noether, Weyl, and conservation of electric charge". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 33 (1): 3–22. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002SHPMP..33....3B">2002SHPMP..33....3B</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.569.106">10.1.1.569.106</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS1355-2198%2801%2900033-8">10.1016/S1355-2198(01)00033-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studies+in+History+and+Philosophy+of+Science+Part+B%3A+Studies+in+History+and+Philosophy+of+Modern+Physics&rft.atitle=Which+symmetry%3F+Noether%2C+Weyl%2C+and+conservation+of+electric+charge&rft.volume=33&rft.issue=1&rft.pages=3-22&rft.date=2002-03&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.569.106%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1016%2FS1355-2198%2801%2900033-8&rft_id=info%3Abibcode%2F2002SHPMP..33....3B&rft.aulast=Brading&rft.aufirst=Katherine+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-WeinbergQFT-39"><a href="#cite_ref-WeinbergQFT_39-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg1995" class="citation book cs1"><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg, Steven</a> (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumtheoryoff00stev">The Quantum Theory of Fields</a>. Cambridge University Press. <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=The+Quantum+Theory+of+Fields&rft.pub=Cambridge+University+Press&rft.date=1995&rft.isbn=978-0-521-55001-7&rft.aulast=Weinberg&rft.aufirst=Steven&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumtheoryoff00stev&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_WitLouis1998" class="citation arxiv cs1">de Wit, Bernard; Louis, Jan (1998-02-18). "Supersymmetry and Dualities in various dimensions". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9801132">hep-th/9801132</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Supersymmetry+and+Dualities+in+various+dimensions&rft.date=1998-02-18&rft_id=info%3Aarxiv%2Fhep-th%2F9801132&rft.aulast=de+Wit&rft.aufirst=Bernard&rft.au=Louis%2C+Jan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolchinski2005" class="citation book cs1"><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski, Joseph</a> (2005). String Theory. Vol. 2. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-67228-3" title="Special:BookSources/978-0-521-67228-3"><bdi>978-0-521-67228-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=String+Theory&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=978-0-521-67228-3&rft.aulast=Polchinski&rft.aufirst=Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-NathArnowitt-42"><a href="#cite_ref-NathArnowitt_42-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNathArnowitt1975" class="citation journal cs1">Nath, P.; Arnowitt, R. (1975). "Generalized Super-Gauge Symmetry as a New Framework for Unified Gauge Theories". Physics Letters B. 56 (2): 177. <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/1975PhLB...56..177N">1975PhLB...56..177N</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%2875%2990297-x">10.1016/0370-2693(75)90297-x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+B&rft.atitle=Generalized+Super-Gauge+Symmetry+as+a+New+Framework+for+Unified+Gauge+Theories&rft.volume=56&rft.issue=2&rft.pages=177&rft.date=1975&rft_id=info%3Adoi%2F10.1016%2F0370-2693%2875%2990297-x&rft_id=info%3Abibcode%2F1975PhLB...56..177N&rft.aulast=Nath&rft.aufirst=P.&rft.au=Arnowitt%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunoz2017" class="citation journal cs1">Munoz, Carlos (2017-01-18). "Models of Supersymmetry for Dark Matter". EPJ Web of Conferences. 136: 01002. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1701.05259">1701.05259</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017EPJWC.13601002M">2017EPJWC.13601002M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1051%2Fepjconf%2F201713601002">10.1051/epjconf/201713601002</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:55199323">55199323</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=EPJ+Web+of+Conferences&rft.atitle=Models+of+Supersymmetry+for+Dark+Matter&rft.volume=136&rft.pages=01002&rft.date=2017-01-18&rft_id=info%3Aarxiv%2F1701.05259&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A55199323%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1051%2Fepjconf%2F201713601002&rft_id=info%3Abibcode%2F2017EPJWC.13601002M&rft.aulast=Munoz&rft.aufirst=Carlos&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorandiSodanoTagliacozzoTognetti2000" class="citation book cs1">Morandi, G.; Sodano, P.; Tagliacozzo, A.; Tognetti, V. (2000). <a rel="nofollow" class="external text" href="https://www.springer.com/us/book/9783540671770">Field Theories for Low-Dimensional Condensed Matter Systems</a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-04273-1" title="Special:BookSources/978-3-662-04273-1"><bdi>978-3-662-04273-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Field+Theories+for+Low-Dimensional+Condensed+Matter+Systems&rft.pub=Springer&rft.date=2000&rft.isbn=978-3-662-04273-1&rft.aulast=Morandi&rft.aufirst=G.&rft.au=Sodano%2C+P.&rft.au=Tagliacozzo%2C+A.&rft.au=Tognetti%2C+V.&rft_id=https%3A%2F%2Fwww.springer.com%2Fus%2Fbook%2F9783540671770&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFParkerToms2009" class="citation book cs1">Parker, Leonard E.; Toms, David J. (2009). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumfieldtheo00park">Quantum Field Theory in Curved Spacetime</a>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/quantumfieldtheo00park/page/n58">43</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-87787-9" title="Special:BookSources/978-0-521-87787-9"><bdi>978-0-521-87787-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory+in+Curved+Spacetime&rft.pages=43&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0-521-87787-9&rft.aulast=Parker&rft.aufirst=Leonard+E.&rft.au=Toms%2C+David+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumfieldtheo00park&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIvancevicIvancevic2008" class="citation arxiv cs1">Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2008-12-11). "Undergraduate Lecture Notes in Topological Quantum Field Theory". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0810.0344v5">0810.0344v5</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math-th">math-th</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Undergraduate+Lecture+Notes+in+Topological+Quantum+Field+Theory&rft.date=2008-12-11&rft_id=info%3Aarxiv%2F0810.0344v5&rft.aulast=Ivancevic&rft.aufirst=Vladimir+G.&rft.au=Ivancevic%2C+Tijana+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarlip1998" class="citation book cs1"><a href="/wiki/Steve_Carlip" title="Steve Carlip">Carlip, Steven</a> (1998). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/books/quantum-gravity-in-21-dimensions/D2F727B6822014270F423D82501E674A">Quantum Gravity in 2+1 Dimensions</a>. Cambridge University Press. pp. 27–29. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2312.12596">2312.12596</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511564192">10.1017/CBO9780511564192</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780511564192" title="Special:BookSources/9780511564192"><bdi>9780511564192</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+Gravity+in+2%2B1+Dimensions&rft.pages=27-29&rft.pub=Cambridge+University+Press&rft.date=1998&rft_id=info%3Aarxiv%2F2312.12596&rft_id=info%3Adoi%2F10.1017%2FCBO9780511564192&rft.isbn=9780511564192&rft.aulast=Carlip&rft.aufirst=Steven&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fbooks%2Fquantum-gravity-in-21-dimensions%2FD2F727B6822014270F423D82501E674A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarquevilleRunkel2018" class="citation journal cs1">Carqueville, Nils; Runkel, Ingo (2018). "Introductory lectures on topological quantum field theory". Banach Center Publications. 114: 9–47. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1705.05734">1705.05734</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Fbc114-1">10.4064/bc114-1</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119166976">119166976</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Banach+Center+Publications&rft.atitle=Introductory+lectures+on+topological+quantum+field+theory&rft.volume=114&rft.pages=9-47&rft.date=2018&rft_id=info%3Aarxiv%2F1705.05734&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119166976%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4064%2Fbc114-1&rft.aulast=Carqueville&rft.aufirst=Nils&rft.au=Runkel%2C+Ingo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWitten1989" class="citation journal cs1"><a href="/wiki/Edward_Witten" title="Edward Witten">Witten, Edward</a> (1989). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.cmp/1104178138">"Quantum Field Theory and the Jones Polynomial"</a>. <a href="/wiki/Communications_in_Mathematical_Physics" title="Communications in Mathematical Physics">Communications in Mathematical Physics</a>. 121 (3): 351–399. <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/1989CMaPh.121..351W">1989CMaPh.121..351W</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%2FBF01217730">10.1007/BF01217730</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0990772">0990772</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:14951363">14951363</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Mathematical+Physics&rft.atitle=Quantum+Field+Theory+and+the+Jones+Polynomial&rft.volume=121&rft.issue=3&rft.pages=351-399&rft.date=1989&rft_id=info%3Adoi%2F10.1007%2FBF01217730&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0990772%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14951363%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1989CMaPh.121..351W&rft.aulast=Witten&rft.aufirst=Edward&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.cmp%2F1104178138&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPutrovWangYau2017" class="citation journal cs1">Putrov, Pavel; Wang, Juven; Yau, Shing-Tung (2017). "Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions". <a href="/wiki/Annals_of_Physics" title="Annals of Physics">Annals of Physics</a>. 384 (C): 254–287. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1612.09298">1612.09298</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017AnPhy.384..254P">2017AnPhy.384..254P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aop.2017.06.019">10.1016/j.aop.2017.06.019</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:119578849">119578849</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Physics&rft.atitle=Braiding+Statistics+and+Link+Invariants+of+Bosonic%2FFermionic+Topological+Quantum+Matter+in+2%2B1+and+3%2B1+dimensions&rft.volume=384&rft.issue=C&rft.pages=254-287&rft.date=2017&rft_id=info%3Aarxiv%2F1612.09298&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119578849%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.aop.2017.06.019&rft_id=info%3Abibcode%2F2017AnPhy.384..254P&rft.aulast=Putrov&rft.aufirst=Pavel&rft.au=Wang%2C+Juven&rft.au=Yau%2C+Shing-Tung&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDi_FrancescoMathieuSénéchal1997" class="citation book cs1">Di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5u7jBwAAQBAJ">Conformal Field Theory</a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-7475-9" title="Special:BookSources/978-1-4612-7475-9"><bdi>978-1-4612-7475-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Conformal+Field+Theory&rft.pub=Springer&rft.date=1997&rft.isbn=978-1-4612-7475-9&rft.aulast=Di+Francesco&rft.aufirst=Philippe&rft.au=Mathieu%2C+Pierre&rft.au=S%C3%A9n%C3%A9chal%2C+David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5u7jBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThirring1958" class="citation journal cs1"><a href="/wiki/Walter_Thirring" title="Walter Thirring">Thirring, W.</a> (1958). "A Soluble Relativistic Field Theory?". <a href="/wiki/Annals_of_Physics" title="Annals of Physics">Annals of Physics</a>. 3 (1): 91–112. <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/1958AnPhy...3...91T">1958AnPhy...3...91T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0003-4916%2858%2990015-0">10.1016/0003-4916(58)90015-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Physics&rft.atitle=A+Soluble+Relativistic+Field+Theory%3F&rft.volume=3&rft.issue=1&rft.pages=91-112&rft.date=1958&rft_id=info%3Adoi%2F10.1016%2F0003-4916%2858%2990015-0&rft_id=info%3Abibcode%2F1958AnPhy...3...91T&rft.aulast=Thirring&rft.aufirst=W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaag1955" class="citation journal cs1"><a href="/wiki/Rudolf_Haag" title="Rudolf Haag">Haag, Rudolf</a> (1955). <a rel="nofollow" class="external text" href="http://cdsweb.cern.ch/record/212242/files/p1.pdf">"On Quantum Field Theories"</a> (PDF). Dan Mat Fys Medd. 29 (12).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Dan+Mat+Fys+Medd&rft.atitle=On+Quantum+Field+Theories&rft.volume=29&rft.issue=12&rft.date=1955&rft.aulast=Haag&rft.aufirst=Rudolf&rft_id=http%3A%2F%2Fcdsweb.cern.ch%2Frecord%2F212242%2Ffiles%2Fp1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-costello-54"><a href="#cite_ref-costello_54-0">^</a> Kevin Costello, Renormalization and Effective Field Theory, Mathematical Surveys and Monographs Volume 170, American Mathematical Society, 2011, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-5288-0" title="Special:BookSources/978-0-8218-5288-0">978-0-8218-5288-0</a> </li> <li id="cite_note-ren-55"><a href="#cite_ref-ren_55-0">^</a> Gerald B. Folland, Quantum Field Theory: A Tourist Guide for Mathematicians, Mathematical Surveys and Monographs Volume 149, American Mathematical Society, 2008, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0821847058" title="Special:BookSources/0821847058">0821847058</a> | chapter=8 </li> <li id="cite_note-nguyen-56"><a href="#cite_ref-nguyen_56-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNguyen2016" class="citation journal cs1">Nguyen, Timothy (2016). "The perturbative approach to path integrals: A succinct mathematical treatment". J. Math. Phys. 57 (9): 092301. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1505.04809">1505.04809</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016JMP....57i2301N">2016JMP....57i2301N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.4962800">10.1063/1.4962800</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:54813572">54813572</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Math.+Phys.&rft.atitle=The+perturbative+approach+to+path+integrals%3A+A+succinct+mathematical+treatment&rft.volume=57&rft.issue=9&rft.pages=092301&rft.date=2016&rft_id=info%3Aarxiv%2F1505.04809&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A54813572%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1063%2F1.4962800&rft_id=info%3Abibcode%2F2016JMP....57i2301N&rft.aulast=Nguyen&rft.aufirst=Timothy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-buchholz-57">^ <a href="#cite_ref-buchholz_57-0">a</a> <a href="#cite_ref-buchholz_57-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuchholz2000" class="citation book cs1"><a href="/wiki/Detlev_Buchholz" title="Detlev Buchholz">Buchholz, Detlev</a> (2000). "Current Trends in Axiomatic Quantum Field Theory". Quantum Field Theory. Lecture Notes in Physics. Vol. 558. pp. 43–64. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9811233">hep-th/9811233</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000LNP...558...43B">2000LNP...558...43B</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%2F3-540-44482-3_4">10.1007/3-540-44482-3_4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-67972-1" title="Special:BookSources/978-3-540-67972-1"><bdi>978-3-540-67972-1</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:5052535">5052535</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Current+Trends+in+Axiomatic+Quantum+Field+Theory&rft.btitle=Quantum+Field+Theory&rft.series=Lecture+Notes+in+Physics&rft.pages=43-64&rft.date=2000&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5052535%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2000LNP...558...43B&rft_id=info%3Aarxiv%2Fhep-th%2F9811233&rft_id=info%3Adoi%2F10.1007%2F3-540-44482-3_4&rft.isbn=978-3-540-67972-1&rft.aulast=Buchholz&rft.aufirst=Detlev&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-summers-58">^ <a href="#cite_ref-summers_58-0">a</a> <a href="#cite_ref-summers_58-1">b</a> <a href="#cite_ref-summers_58-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSummers2016" class="citation arxiv cs1">Summers, Stephen J. (2016-03-31). "A Perspective on Constructive Quantum Field Theory". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1203.3991v2">1203.3991v2</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math-ph">math-ph</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+Perspective+on+Constructive+Quantum+Field+Theory&rft.date=2016-03-31&rft_id=info%3Aarxiv%2F1203.3991v2&rft.aulast=Summers&rft.aufirst=Stephen+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-Simon-59"><a href="#cite_ref-Simon_59-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimon1974" class="citation book cs1">Simon, Barry (1974). The P(phi)_2 Euclidean (quantum) field theory. Princeton, New Jersey: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08144-1" title="Special:BookSources/0-691-08144-1"><bdi>0-691-08144-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/905864308">905864308</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+P%28phi%29_2+Euclidean+%28quantum%29+field+theory&rft.place=Princeton%2C+New+Jersey&rft.pub=Princeton+University+Press&rft.date=1974&rft_id=info%3Aoclcnum%2F905864308&rft.isbn=0-691-08144-1&rft.aulast=Simon&rft.aufirst=Barry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-Glimm1987-60"><a href="#cite_ref-Glimm1987_60-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlimmJaffe1987" class="citation book cs1">Glimm, James; Jaffe, Arthur (1987). Quantum Physics : a Functional Integral Point of View. New York, NY: Springer New York. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-4728-9" title="Special:BookSources/978-1-4612-4728-9"><bdi>978-1-4612-4728-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/852790676">852790676</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%2C+NY&rft.pub=Springer+New+York&rft.date=1987&rft_id=info%3Aoclcnum%2F852790676&rft.isbn=978-1-4612-4728-9&rft.aulast=Glimm&rft.aufirst=James&rft.au=Jaffe%2C+Arthur&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSatiSchreiber2012" class="citation arxiv cs1">Sati, Hisham; <a href="/wiki/Urs_Schreiber" title="Urs Schreiber">Schreiber, Urs</a> (2012-01-06). "Survey of mathematical foundations of QFT and perturbative string theory". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1109.0955v2">1109.0955v2</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math-ph">math-ph</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Survey+of+mathematical+foundations+of+QFT+and+perturbative+string+theory&rft.date=2012-01-06&rft_id=info%3Aarxiv%2F1109.0955v2&rft.aulast=Sati&rft.aufirst=Hisham&rft.au=Schreiber%2C+Urs&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaffeWitten" class="citation web cs1"><a href="/wiki/Arthur_Jaffe" title="Arthur Jaffe">Jaffe, Arthur</a>; <a href="/wiki/Edward_Witten" title="Edward Witten">Witten, Edward</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150330003812/http://www.claymath.org/sites/default/files/yangmills.pdf">"Quantum Yang–Mills Theory"</a> (PDF). <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a>. Archived from <a rel="nofollow" class="external text" href="http://www.claymath.org/sites/default/files/yangmills.pdf">the original</a> (PDF) on 2015-03-30. Retrieved 2018-07-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Quantum+Yang%E2%80%93Mills+Theory&rft.pub=Clay+Mathematics+Institute&rft.aulast=Jaffe&rft.aufirst=Arthur&rft.au=Witten%2C+Edward&rft_id=http%3A%2F%2Fwww.claymath.org%2Fsites%2Fdefault%2Ffiles%2Fyangmills.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"> </li> </ol></div></div> <dl><dt>Bibliography</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStreaterWightman1964" class="citation book cs1">Streater, R.; Wightman, A. (1964). <a rel="nofollow" class="external text" href="https://archive.org/details/pctspinstatistic0000stre">PCT, Spin and Statistics and all That</a>. W. A. Benjamin.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=PCT%2C+Spin+and+Statistics+and+all+That&rft.pub=W.+A.+Benjamin&rft.date=1964&rft.aulast=Streater&rft.aufirst=R.&rft.au=Wightman%2C+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpctspinstatistic0000stre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOsterwalderSchrader1973" class="citation journal cs1">Osterwalder, K.; Schrader, R. (1973). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.cmp/1103858969">"Axioms for Euclidean Green's functions"</a>. <a href="/wiki/Communications_in_Mathematical_Physics" title="Communications in Mathematical Physics">Communications in Mathematical Physics</a>. 31 (2): 83–112. <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/1973CMaPh..31...83O">1973CMaPh..31...83O</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%2FBF01645738">10.1007/BF01645738</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:189829853">189829853</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Mathematical+Physics&rft.atitle=Axioms+for+Euclidean+Green%27s+functions&rft.volume=31&rft.issue=2&rft.pages=83-112&rft.date=1973&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A189829853%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01645738&rft_id=info%3Abibcode%2F1973CMaPh..31...83O&rft.aulast=Osterwalder&rft.aufirst=K.&rft.au=Schrader%2C+R.&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.cmp%2F1103858969&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOsterwalderSchrader1975" class="citation journal cs1">Osterwalder, K.; Schrader, R. (1975). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.cmp/1103899050">"Axioms for Euclidean Green's functions II"</a>. <a href="/wiki/Communications_in_Mathematical_Physics" title="Communications in Mathematical Physics">Communications in Mathematical Physics</a>. 42 (3): 281–305. <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/1975CMaPh..42..281O">1975CMaPh..42..281O</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%2FBF01608978">10.1007/BF01608978</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:119389461">119389461</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Mathematical+Physics&rft.atitle=Axioms+for+Euclidean+Green%27s+functions+II&rft.volume=42&rft.issue=3&rft.pages=281-305&rft.date=1975&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119389461%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01608978&rft_id=info%3Abibcode%2F1975CMaPh..42..281O&rft.aulast=Osterwalder&rft.aufirst=K.&rft.au=Schrader%2C+R.&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.cmp%2F1103899050&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=28" title="Edit section: Further reading">edit</a>]</div> <dl><dt>General readers</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPais1994" class="citation book cs1"><a href="/wiki/Abraham_Pais" title="Abraham Pais">Pais, A.</a> (1994) [1986]. Inward Bound: Of Matter and Forces in the Physical World (reprint ed.). Oxford, New York, Toronto: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0198519973" title="Special:BookSources/978-0198519973"><bdi>978-0198519973</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inward+Bound%3A+Of+Matter+and+Forces+in+the+Physical+World&rft.place=Oxford%2C+New+York%2C+Toronto&rft.edition=reprint&rft.pub=Oxford+University+Press&rft.date=1994&rft.isbn=978-0198519973&rft.aulast=Pais&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchweber1994" class="citation book cs1"><a href="/wiki/S._S._Schweber" class="mw-redirect" title="S. S. Schweber">Schweber, S. S.</a> (1994). <a rel="nofollow" class="external text" href="https://archive.org/details/qedmenwhomadeitd0000schw">QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga</a>. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780691033273" title="Special:BookSources/9780691033273"><bdi>9780691033273</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=QED+and+the+Men+Who+Made+It%3A+Dyson%2C+Feynman%2C+Schwinger%2C+and+Tomonaga&rft.pub=Princeton+University+Press&rft.date=1994&rft.isbn=9780691033273&rft.aulast=Schweber&rft.aufirst=S.+S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fqedmenwhomadeitd0000schw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman2001" class="citation book cs1"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman, R.P.</a> (2001) [1964]. <a rel="nofollow" class="external text" href="https://archive.org/details/characterofphysi0000feyn_u5j3">The Character of Physical Law</a>. <a href="/wiki/MIT_Press" title="MIT Press">MIT Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-56003-0" title="Special:BookSources/978-0-262-56003-0"><bdi>978-0-262-56003-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Character+of+Physical+Law&rft.pub=MIT+Press&rft.date=2001&rft.isbn=978-0-262-56003-0&rft.aulast=Feynman&rft.aufirst=R.P.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcharacterofphysi0000feyn_u5j3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman2006" class="citation book cs1">Feynman, R.P. (2006) [1985]. QED: The Strange Theory of Light and Matter. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12575-6" title="Special:BookSources/978-0-691-12575-6"><bdi>978-0-691-12575-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=QED%3A+The+Strange+Theory+of+Light+and+Matter&rft.pub=Princeton+University+Press&rft.date=2006&rft.isbn=978-0-691-12575-6&rft.aulast=Feynman&rft.aufirst=R.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGribbin1998" class="citation book cs1"><a href="/wiki/John_Gribbin" title="John Gribbin">Gribbin, J.</a> (1998). Q is for Quantum: Particle Physics from A to Z. <a href="/wiki/Weidenfeld_%26_Nicolson" title="Weidenfeld & Nicolson">Weidenfeld & Nicolson</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-297-81752-9" title="Special:BookSources/978-0-297-81752-9"><bdi>978-0-297-81752-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Q+is+for+Quantum%3A+Particle+Physics+from+A+to+Z&rft.pub=Weidenfeld+%26+Nicolson&rft.date=1998&rft.isbn=978-0-297-81752-9&rft.aulast=Gribbin&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li></ul> <dl><dt>Introductory texts</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcMahon2008" class="citation book cs1">McMahon, D. (2008). Quantum Field Theory. <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-154382-8" title="Special:BookSources/978-0-07-154382-8"><bdi>978-0-07-154382-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory&rft.pub=McGraw-Hill&rft.date=2008&rft.isbn=978-0-07-154382-8&rft.aulast=McMahon&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBogolyubovShirkov1982" class="citation book cs1"><a href="/wiki/Nikolay_Bogolyubov" title="Nikolay Bogolyubov">Bogolyubov, N.</a>; <a href="/wiki/Dmitry_Shirkov" title="Dmitry Shirkov">Shirkov, D.</a> (1982). Quantum Fields. <a href="/wiki/Benjamin_Cummings" title="Benjamin Cummings">Benjamin Cummings</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-0983-6" title="Special:BookSources/978-0-8053-0983-6"><bdi>978-0-8053-0983-6</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+Fields&rft.pub=Benjamin+Cummings&rft.date=1982&rft.isbn=978-0-8053-0983-6&rft.aulast=Bogolyubov&rft.aufirst=N.&rft.au=Shirkov%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrampton2000" class="citation book cs1"><a href="/wiki/Paul_Frampton" title="Paul Frampton">Frampton, P.H.</a> (2000). Gauge Field Theories. Frontiers in Physics (2nd ed.). <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Field+Theories&rft.series=Frontiers+in+Physics&rft.edition=2nd&rft.pub=Wiley&rft.date=2000&rft.aulast=Frampton&rft.aufirst=P.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988">; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrampton2008" class="citation book cs1">Frampton, Paul H. (22 September 2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AwhkM6hVj-wC">2008, 3rd edition</a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3527408351" title="Special:BookSources/978-3527408351"><bdi>978-3527408351</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=2008%2C+3rd+edition&rft.pub=John+Wiley+%26+Sons&rft.date=2008-09-22&rft.isbn=978-3527408351&rft.aulast=Frampton&rft.aufirst=Paul+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAwhkM6hVj-wC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreinerMüller2000" class="citation book cs1"><a href="/wiki/Walter_Greiner" title="Walter Greiner">Greiner, W.</a>; <a href="/wiki/Berndt_M%C3%BCller" title="Berndt Müller">Müller, B.</a> (2000). Gauge Theory of Weak Interactions. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-67672-0" title="Special:BookSources/978-3-540-67672-0"><bdi>978-3-540-67672-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Theory+of+Weak+Interactions&rft.pub=Springer&rft.date=2000&rft.isbn=978-3-540-67672-0&rft.aulast=Greiner&rft.aufirst=W.&rft.au=M%C3%BCller%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFItzyksonZuber1980" class="citation book cs1"><a href="/wiki/Claude_Itzykson" title="Claude Itzykson">Itzykson, C.</a>; <a href="/wiki/Jean-Bernard_Zuber" title="Jean-Bernard Zuber">Zuber, J.-B.</a> (1980). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumfieldtheo0000itzy">Quantum Field Theory</a>. <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-032071-0" title="Special:BookSources/978-0-07-032071-0"><bdi>978-0-07-032071-0</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+Field+Theory&rft.pub=McGraw-Hill&rft.date=1980&rft.isbn=978-0-07-032071-0&rft.aulast=Itzykson&rft.aufirst=C.&rft.au=Zuber%2C+J.-B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumfieldtheo0000itzy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKane1987" class="citation book cs1"><a href="/wiki/Gordon_L._Kane" title="Gordon L. Kane">Kane, G.L.</a> (1987). Modern Elementary Particle Physics. <a href="/wiki/Perseus_Books_Group" title="Perseus Books Group">Perseus Group</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-11749-3" title="Special:BookSources/978-0-201-11749-3"><bdi>978-0-201-11749-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Elementary+Particle+Physics&rft.pub=Perseus+Group&rft.date=1987&rft.isbn=978-0-201-11749-3&rft.aulast=Kane&rft.aufirst=G.L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleinertSchulte-Frohlinde2001" class="citation book cs1"><a href="/wiki/Hagen_Kleinert" title="Hagen Kleinert">Kleinert, H.</a>; Schulte-Frohlinde, Verena (2001). <a rel="nofollow" class="external text" href="http://users.physik.fu-berlin.de/~kleinert/re.html#B6">Critical Properties of φ4-Theories</a>. <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-4658-7" title="Special:BookSources/978-981-02-4658-7"><bdi>978-981-02-4658-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Critical+Properties+of+%26phi%3B%3Csup%3E4%3C%2Fsup%3E-Theories&rft.pub=World+Scientific&rft.date=2001&rft.isbn=978-981-02-4658-7&rft.aulast=Kleinert&rft.aufirst=H.&rft.au=Schulte-Frohlinde%2C+Verena&rft_id=http%3A%2F%2Fusers.physik.fu-berlin.de%2F~kleinert%2Fre.html%23B6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleinert2008" class="citation book cs1">Kleinert, H. (2008). <a rel="nofollow" class="external text" href="http://users.physik.fu-berlin.de/~kleinert/public_html/kleiner_reb11/psfiles/mvf.pdf">Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation</a> (PDF). World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-279-170-2" title="Special:BookSources/978-981-279-170-2"><bdi>978-981-279-170-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multivalued+Fields+in+Condensed+Matter%2C+Electrodynamics%2C+and+Gravitation&rft.pub=World+Scientific&rft.date=2008&rft.isbn=978-981-279-170-2&rft.aulast=Kleinert&rft.aufirst=H.&rft_id=http%3A%2F%2Fusers.physik.fu-berlin.de%2F~kleinert%2Fpublic_html%2Fkleiner_reb11%2Fpsfiles%2Fmvf.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLancasterBlundell2014" class="citation book cs1">Lancaster, Tom; Blundell, Stephen (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Y-0kAwAAQBAJ">Quantum field theory for the gifted amateur</a>. Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-969933-9" title="Special:BookSources/978-0-19-969933-9"><bdi>978-0-19-969933-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/859651399">859651399</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+for+the+gifted+amateur&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=2014&rft_id=info%3Aoclcnum%2F859651399&rft.isbn=978-0-19-969933-9&rft.aulast=Lancaster&rft.aufirst=Tom&rft.au=Blundell%2C+Stephen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DY-0kAwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLoudon1983" class="citation book cs1"><a href="/wiki/Rodney_Loudon" title="Rodney Loudon">Loudon, R.</a> (1983). The Quantum Theory of Light. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-851155-7" title="Special:BookSources/978-0-19-851155-7"><bdi>978-0-19-851155-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Quantum+Theory+of+Light&rft.pub=Oxford+University+Press&rft.date=1983&rft.isbn=978-0-19-851155-7&rft.aulast=Loudon&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMandlShaw1993" class="citation book cs1"><a href="/wiki/Franz_Mandl_(physicist)" title="Franz Mandl (physicist)">Mandl, F.</a>; Shaw, G. (1993). Quantum Field Theory. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-94186-6" title="Special:BookSources/978-0-471-94186-6"><bdi>978-0-471-94186-6</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&rft.pub=John+Wiley+%26+Sons&rft.date=1993&rft.isbn=978-0-471-94186-6&rft.aulast=Mandl&rft.aufirst=F.&rft.au=Shaw%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyder1985" class="citation book cs1"><a href="/wiki/Lewis_Ryder" title="Lewis Ryder">Ryder, L.H.</a> (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nnuW_kVJ500C">Quantum Field Theory</a>. <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/978-0-521-33859-2" 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=L.H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnnuW_kVJ500C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartz2014" class="citation book cs1">Schwartz, M.D. (2014). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180322014256/http://schwartzqft.com/">Quantum Field Theory and the Standard Model</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1107034730" title="Special:BookSources/978-1107034730"><bdi>978-1107034730</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://www.schwartzqft.com">the original</a> on 2018-03-22. Retrieved 2020-05-13.</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+and+the+Standard+Model&rft.pub=Cambridge+University+Press&rft.date=2014&rft.isbn=978-1107034730&rft.aulast=Schwartz&rft.aufirst=M.D.&rft_id=http%3A%2F%2Fwww.schwartzqft.com&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYnduráin1996" class="citation book cs1"><a href="/wiki/Francisco_Jos%C3%A9_Yndur%C3%A1in" title="Francisco José Ynduráin">Ynduráin, F.J.</a> (1996). Relativistic Quantum Mechanics and Introduction to Field Theory (1st ed.). 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/1996rqmi.book.....Y">1996rqmi.book.....Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-61057-8">10.1007/978-3-642-61057-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-60453-2" title="Special:BookSources/978-3-540-60453-2"><bdi>978-3-540-60453-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativistic+Quantum+Mechanics+and+Introduction+to+Field+Theory&rft.edition=1st&rft.pub=Springer&rft.date=1996&rft_id=info%3Adoi%2F10.1007%2F978-3-642-61057-8&rft_id=info%3Abibcode%2F1996rqmi.book.....Y&rft.isbn=978-3-540-60453-2&rft.aulast=Yndur%C3%A1in&rft.aufirst=F.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreinerReinhardt1996" class="citation book cs1"><a href="/wiki/Walter_Greiner" title="Walter Greiner">Greiner, W.</a>; Reinhardt, J. (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/fieldquantizatio0000grei">Field Quantization</a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-59179-5" title="Special:BookSources/978-3-540-59179-5"><bdi>978-3-540-59179-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Field+Quantization&rft.pub=Springer&rft.date=1996&rft.isbn=978-3-540-59179-5&rft.aulast=Greiner&rft.aufirst=W.&rft.au=Reinhardt%2C+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffieldquantizatio0000grei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeskinSchroeder1995" class="citation book cs1"><a href="/wiki/Michael_Peskin" title="Michael Peskin">Peskin, M.</a>; Schroeder, D. (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=i35LALN0GosC">An Introduction to Quantum Field Theory</a>. <a href="/wiki/Westview_Press" title="Westview Press">Westview Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-50397-5" title="Special:BookSources/978-0-201-50397-5"><bdi>978-0-201-50397-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Quantum+Field+Theory&rft.pub=Westview+Press&rft.date=1995&rft.isbn=978-0-201-50397-5&rft.aulast=Peskin&rft.aufirst=M.&rft.au=Schroeder%2C+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Di35LALN0GosC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScharf2014" class="citation book cs1">Scharf, Günter (2014) [1989]. Finite Quantum Electrodynamics: The Causal Approach (third ed.). Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0486492735" title="Special:BookSources/978-0486492735"><bdi>978-0486492735</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+Quantum+Electrodynamics%3A+The+Causal+Approach&rft.edition=third&rft.pub=Dover+Publications&rft.date=2014&rft.isbn=978-0486492735&rft.aulast=Scharf&rft.aufirst=G%C3%BCnter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSrednicki2007" class="citation book cs1">Srednicki, M. (2007). <a rel="nofollow" class="external text" href="http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=0521864496">Quantum Field Theory</a>. <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/978-0521-8644-97" title="Special:BookSources/978-0521-8644-97"><bdi>978-0521-8644-97</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=2007&rft.isbn=978-0521-8644-97&rft.aulast=Srednicki&rft.aufirst=M.&rft_id=http%3A%2F%2Fwww.cambridge.org%2Fus%2Fcatalogue%2Fcatalogue.asp%3Fisbn%3D0521864496&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTong2015" class="citation web cs1"><a href="/wiki/David_Tong_(physicist)" title="David Tong (physicist)">Tong, David</a> (2015). <a rel="nofollow" class="external text" href="http://www.damtp.cam.ac.uk/user/tong/qft.html">"Lectures on Quantum Field Theory"</a>. Retrieved 2016-02-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lectures+on+Quantum+Field+Theory&rft.date=2015&rft.aulast=Tong&rft.aufirst=David&rft_id=http%3A%2F%2Fwww.damtp.cam.ac.uk%2Fuser%2Ftong%2Fqft.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams2022" class="citation book cs1">Williams, A.G. (2022). Introduction to Quantum Field Theory: Classical Mechanics to Gauge Field Theories. <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/978-1108470902" title="Special:BookSources/978-1108470902"><bdi>978-1108470902</bdi></a>.</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+Field+Theory%3A+Classical+Mechanics+to+Gauge+Field+Theories&rft.pub=Cambridge+University+Press&rft.date=2022&rft.isbn=978-1108470902&rft.aulast=Williams&rft.aufirst=A.G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></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, Anthony</a> (2010). <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780691140346">Quantum Field Theory in a Nutshell</a> (2nd ed.). <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0691140346" title="Special:BookSources/978-0691140346"><bdi>978-0691140346</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=2nd&rft.pub=Princeton+University+Press&rft.date=2010&rft.isbn=978-0691140346&rft.aulast=Zee&rft.aufirst=Anthony&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_9780691140346&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li></ul> <dl><dt>Advanced texts</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown1994" class="citation book cs1"><a href="/wiki/Lowell_S._Brown" title="Lowell S. Brown">Brown, Lowell S.</a> (1994). Quantum Field Theory. <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/978-0-521-46946-3" title="Special:BookSources/978-0-521-46946-3"><bdi>978-0-521-46946-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=978-0-521-46946-3&rft.aulast=Brown&rft.aufirst=Lowell+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBogoliubovLogunovOksakTodorov1990" class="citation book cs1">Bogoliubov, N.; <a href="/wiki/Anatoly_Logunov" title="Anatoly Logunov">Logunov, A.A.</a>; Oksak, A.I.; Todorov, I.T. (1990). General Principles of Quantum Field Theory. <a href="/wiki/Kluwer_Academic_Publishers" class="mw-redirect" title="Kluwer Academic Publishers">Kluwer Academic Publishers</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-0540-8" title="Special:BookSources/978-0-7923-0540-8"><bdi>978-0-7923-0540-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Principles+of+Quantum+Field+Theory&rft.pub=Kluwer+Academic+Publishers&rft.date=1990&rft.isbn=978-0-7923-0540-8&rft.aulast=Bogoliubov&rft.aufirst=N.&rft.au=Logunov%2C+A.A.&rft.au=Oksak%2C+A.I.&rft.au=Todorov%2C+I.T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg1995" class="citation book cs1"><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg, S.</a> (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumtheoryoff00stev">The Quantum Theory of Fields</a>. Vol. 1. <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/978-0521550017" title="Special:BookSources/978-0521550017"><bdi>978-0521550017</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Quantum+Theory+of+Fields&rft.pub=Cambridge+University+Press&rft.date=1995&rft.isbn=978-0521550017&rft.aulast=Weinberg&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumtheoryoff00stev&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Quantum_field_theory&action=edit&section=29" title="Edit section: External links">edit</a>]</div> <ul><li><a href="/wiki/File:Commons-logo.svg" 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Quantum_field_theory">"Quantum field theory"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Quantum+field+theory&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DQuantum_field_theory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+field+theory" class="Z3988"></li> <li><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: "<a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/quantum-field-theory/">Quantum Field Theory</a>", by Meinard Kuhlmann.</li> <li>Siegel, Warren, 2005. <a rel="nofollow" class="external text" href="http://insti.physics.sunysb.edu/%7Esiegel/errata.html">Fields.</a> <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9912205">hep-th/9912205</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.nat.vu.nl/~mulders/QFT-0.pdf">Quantum Field Theory</a> by P. J. Mulders</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 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href="/wiki/Template_talk:Quantum_field_theories" title="Template talk:Quantum field theories"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_field_theories" title="Special:EditPage/Template:Quantum field theories"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_field_theories" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Quantum field theories</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</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/Algebraic_quantum_field_theory" title="Algebraic quantum field theory">Algebraic QFT</a></li> <li><a href="/wiki/Axiomatic_quantum_field_theory" title="Axiomatic quantum field theory">Axiomatic QFT</a></li> <li><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></li> <li><a href="/wiki/Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li> <li><a href="/wiki/Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative QFT</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">QFT in curved spacetime</a></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Thermal_quantum_field_theory" title="Thermal quantum field theory">Thermal QFT</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological QFT</a></li> <li><a href="/wiki/Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">Two-dimensional conformal field theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Models</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Regular</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%E2%80%93Infeld_model" title="Born–Infeld model">Born–Infeld</a></li> <li><a href="/wiki/Euler%E2%80%93Heisenberg_Lagrangian" title="Euler–Heisenberg Lagrangian">Euler–Heisenberg</a></li> <li><a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau</a></li> <li><a href="/wiki/Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma</a></li> <li><a href="/wiki/Proca_action" title="Proca action">Proca</a></li> <li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Quartic_interaction" title="Quartic interaction">Quartic interaction</a></li> <li><a href="/wiki/Scalar_electrodynamics" title="Scalar electrodynamics">Scalar electrodynamics</a></li> <li><a href="/wiki/Scalar_chromodynamics" title="Scalar chromodynamics">Scalar chromodynamics</a></li> <li><a href="/wiki/Soler_model" title="Soler model">Soler</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills</a></li> <li><a href="/wiki/Yang%E2%80%93Mills%E2%80%93Higgs_equations" title="Yang–Mills–Higgs equations">Yang–Mills–Higgs</a></li> <li><a href="/wiki/Yukawa_interaction" title="Yukawa interaction">Yukawa</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Low dimensional</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/Two-dimensional_Yang%E2%80%93Mills_theory" title="Two-dimensional Yang–Mills theory">2D Yang–Mills</a></li> <li><a href="/wiki/Bullough%E2%80%93Dodd_model" title="Bullough–Dodd model">Bullough–Dodd</a></li> <li><a href="/wiki/Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu</a></li> <li><a href="/wiki/Schwinger_model" title="Schwinger model">Schwinger</a></li> <li><a href="/wiki/Sine-Gordon_equation" title="Sine-Gordon equation">Sine-Gordon</a></li> <li><a href="/wiki/Thirring_model" title="Thirring model">Thirring</a></li> <li><a href="/wiki/Thirring%E2%80%93Wess_model" title="Thirring–Wess model">Thirring–Wess</a></li> <li><a href="/wiki/Toda_field_theory" title="Toda field theory">Toda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Conformal</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/Massless_free_scalar_bosons_in_two_dimensions" title="Massless free scalar bosons in two dimensions">2D free massless scalar</a></li> <li><a href="/wiki/Liouville_field_theory" title="Liouville field theory">Liouville</a></li> <li><a href="/wiki/Minimal_model_(physics)" title="Minimal model (physics)">Minimal</a></li> <li><a href="/wiki/Polyakov_action" title="Polyakov action">Polyakov</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supersymmetric</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/4D_N_%3D_1_global_supersymmetry" title="4D N = 1 global supersymmetry">4D N = 1</a></li> <li><a href="/wiki/N_%3D_1_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 1 supersymmetric Yang–Mills theory">N = 1 super Yang–Mills</a></li> <li><a href="/wiki/Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten</a></li> <li><a href="/wiki/Super_QCD" title="Super QCD">Super QCD</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino_model" title="Wess–Zumino model">Wess–Zumino</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Superconformal</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/6D_(2,0)_superconformal_field_theory" title="6D (2,0) superconformal field theory">6D (2,0)</a></li> <li><a href="/wiki/ABJM_superconformal_field_theory" title="ABJM superconformal field theory">ABJM</a></li> <li><a href="/wiki/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 super Yang–Mills</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supergravity</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/Pure_4D_N_%3D_1_supergravity" title="Pure 4D N = 1 supergravity">Pure 4D N = 1</a></li> <li><a href="/wiki/4D_N_%3D_1_supergravity" title="4D N = 1 supergravity">4D N = 1</a></li> <li><a href="/wiki/N_%3D_8_supergravity" title="N = 8 supergravity">4D N = 8</a></li> <li><a href="/wiki/Higher-dimensional_supergravity" title="Higher-dimensional supergravity">Higher dimensional</a></li> <li><a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I</a></li> <li><a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA</a></li> <li><a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB</a></li> <li><a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">11D</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Topological</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/BF_model" title="BF model">BF</a></li> <li><a href="/wiki/Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Particle theory</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/Chiral_model" title="Chiral model">Chiral</a></li> <li><a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi</a></li> <li><a href="/wiki/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">MSSM</a></li> <li><a href="/wiki/Nambu%E2%80%93Jona-Lasinio_model" title="Nambu–Jona-Lasinio model">Nambu–Jona-Lasinio</a></li> <li><a href="/wiki/Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li> <li><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></li> <li><a href="/wiki/Stueckelberg_action" title="Stueckelberg action">Stueckelberg</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Cosmic_string" title="Cosmic string">Cosmic string</a></li> <li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li> <li><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/Loop_quantum_cosmology" title="Loop quantum cosmology">Loop quantum cosmology</a></li> <li><a href="/wiki/On_shell_and_off_shell" title="On shell and off shell">On shell and off shell</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_foam" title="Quantum foam">Quantum foam</a></li> <li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuations</a> <ul><li><a href="/wiki/Template:Quantum_electrodynamics" title="Template:Quantum electrodynamics">links</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a> <ul><li><a href="/wiki/Template:Quantum_gravity" title="Template:Quantum gravity">links</a></li></ul></li> <li><a href="/wiki/Quantum_hadrodynamics" title="Quantum hadrodynamics">Quantum hadrodynamics</a></li> <li><a href="/wiki/Quantum_hydrodynamics" title="Quantum hydrodynamics">Quantum hydrodynamics</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a> <ul><li><a href="/wiki/Template:Quantum_information" title="Template:Quantum information">links</a></li></ul></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>See also: <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/23px-Symbol_template_class_pink.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/31px-Symbol_template_class_pink.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics">Template:Quantum mechanics topics</a></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="Quantum_mechanics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a 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 class="mw-selflink selflink">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><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" /> <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="Quantum_gravity" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_gravity" title="Template:Quantum gravity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_gravity" title="Template talk:Quantum gravity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_gravity" title="Special:EditPage/Template:Quantum gravity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_gravity" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Central concepts</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/AdS/CFT_correspondence" title="AdS/CFT correspondence">AdS/CFT correspondence</a></li> <li><a href="/wiki/Batalin%E2%80%93Vilkovisky_formalism" title="Batalin–Vilkovisky formalism">Batalin–Vilkovisky formalism</a></li> <li><a href="/wiki/CA-duality" title="CA-duality">CA-duality</a></li> <li><a href="/wiki/Causal_patch" title="Causal patch">Causal patch</a></li> <li><a href="/wiki/Faddeev%E2%80%93Popov_ghost" title="Faddeev–Popov ghost">Faddeev–Popov ghost</a></li> <li><a href="/wiki/Gravitational_anomaly" title="Gravitational anomaly">Gravitational anomaly</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Holographic_principle" title="Holographic principle">Holographic principle</a></li> <li><a href="/wiki/IR/UV_mixing" title="IR/UV mixing">IR/UV mixing</a></li> <li><a href="/wiki/Planck_units" title="Planck units">Planck units</a></li> <li><a href="/wiki/Quantum_foam" title="Quantum foam">Quantum foam</a></li> <li><a href="/wiki/Ryu%E2%80%93Takayanagi_conjecture" title="Ryu–Takayanagi conjecture">Ryu–Takayanagi conjecture</a></li> <li><a href="/wiki/Trans-Planckian_problem" title="Trans-Planckian problem">Trans-Planckian problem</a></li> <li><a href="/wiki/Weinberg%E2%80%93Witten_theorem" title="Weinberg–Witten theorem">Weinberg–Witten theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Toy_model" title="Toy model">Toy models</a></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/(2%2B1)-dimensional_topological_gravity" title="(2+1)-dimensional topological gravity">2+1D topological gravity</a></li> <li><a href="/wiki/CGHS_model" title="CGHS model">CGHS model</a></li> <li><a href="/wiki/Jackiw%E2%80%93Teitelboim_gravity" title="Jackiw–Teitelboim gravity">Jackiw–Teitelboim gravity</a></li> <li><a href="/wiki/Liouville_gravity" class="mw-redirect" title="Liouville gravity">Liouville gravity</a></li> <li><a href="/wiki/RST_model" title="RST model">RST model</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">Quantum field theory in curved spacetime</a></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/Bunch%E2%80%93Davies_vacuum" title="Bunch–Davies vacuum">Bunch–Davies vacuum</a></li> <li><a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a></li> <li><a href="/wiki/Semiclassical_gravity" title="Semiclassical gravity">Semiclassical gravity</a></li> <li><a href="/wiki/Unruh_effect" title="Unruh effect">Unruh effect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Black_hole" title="Black hole">Black holes</a></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/Black_hole_complementarity" title="Black hole complementarity">Black hole complementarity</a></li> <li><a href="/wiki/Black_hole_information_paradox" title="Black hole information paradox">Black hole information paradox</a></li> <li><a href="/wiki/Black_hole_thermodynamics" title="Black hole thermodynamics">Black-hole thermodynamics</a></li> <li><a href="/wiki/Bekenstein_bound" title="Bekenstein bound">Bekenstein bound</a></li> <li><a href="/wiki/Bousso%27s_holographic_bound" title="Bousso's holographic bound">Bousso's holographic bound</a></li> <li><a href="/wiki/Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">Cosmic censorship hypothesis</a></li> <li><a href="/wiki/ER_%3D_EPR" title="ER = EPR">ER = EPR</a></li> <li><a href="/wiki/Firewall_(physics)" title="Firewall (physics)">Firewall (physics)</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Gravitational singularity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Approaches</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/String_theory" title="String theory">String theory</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/Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</a></li> <li><a href="/wiki/M-theory" title="M-theory">M-theory</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Superstring_theory" title="Superstring theory">Superstring theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</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/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euclidean_quantum_gravity" title="Euclidean quantum gravity">Euclidean quantum gravity</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/Hartle%E2%80%93Hawking_state" title="Hartle–Hawking state">Hartle–Hawking state</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Others</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/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li> <li><a href="/wiki/Causal_sets" title="Causal sets">Causal sets</a></li> <li><a href="/wiki/Dual_graviton" title="Dual graviton">Dual graviton</a></li> <li><a href="/wiki/Group_field_theory" title="Group field theory">Group field theory</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Spin_foam" title="Spin foam">Spin foam</a></li> <li><a href="/wiki/Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a></li> <li><a href="/wiki/Twistor_theory" title="Twistor theory">Twistor theory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</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/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a> <ul><li><a href="/wiki/Eternal_inflation" title="Eternal inflation">Eternal inflation</a></li> <li><a href="/wiki/FRW/CFT_duality" title="FRW/CFT duality">FRW/CFT duality</a></li> <li><a href="/wiki/Multiverse" title="Multiverse">Multiverse</a></li></ul></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>See also: <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/23px-Symbol_template_class_pink.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/31px-Symbol_template_class_pink.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics">Template:Quantum mechanics topics</a></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="Major_branches_of_physics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Branches_of_physics" title="Template:Branches of physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Branches_of_physics" title="Template talk:Branches of physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Branches_of_physics" title="Special:EditPage/Template:Branches of physics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_branches_of_physics" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Branches_of_physics" title="Branches of physics">branches of physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisions</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/Basic_research" title="Basic research">Pure</a></li> <li><a href="/wiki/Applied_physics" title="Applied physics">Applied</a> <ul><li><a href="/wiki/Engineering_physics" title="Engineering physics">Engineering</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Approaches</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/Experimental_physics" title="Experimental physics">Experimental</a></li> <li><a href="/wiki/Theoretical_physics" title="Theoretical physics">Theoretical</a> <ul><li><a href="/wiki/Computational_physics" title="Computational physics">Computational</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Classical_physics" title="Classical physics">Classical</a></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/Classical_mechanics" title="Classical mechanics">Classical mechanics</a> <ul><li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newtonian</a></li> <li><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li></ul></li> <li><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></li> <li><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Classical electromagnetism</a></li> <li><a href="/wiki/Classical_optics" class="mw-redirect" title="Classical optics">Classical optics</a> <ul><li><a href="/wiki/Geometrical_optics" title="Geometrical optics">Ray</a></li> <li><a href="/wiki/Physical_optics" title="Physical optics">Wave</a></li></ul></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a> <ul><li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Modern_physics" title="Modern physics">Modern</a></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/Relativistic_mechanics" title="Relativistic mechanics">Relativistic mechanics</a> <ul><li><a href="/wiki/Special_relativity" title="Special relativity">Special</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General</a></li></ul></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li> <li><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Atomic,_molecular,_and_optical_physics" title="Atomic, molecular, and optical physics">Atomic, molecular, and optical physics</a> <ul><li><a href="/wiki/Atomic_physics" title="Atomic physics">Atomic</a></li> <li><a href="/wiki/Molecular_physics" title="Molecular physics">Molecular</a></li> <li><a href="/wiki/Optics#Modern_optics" title="Optics">Modern optics</a></li></ul></li> <li><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a> <ul><li><a href="/wiki/Solid-state_physics" title="Solid-state physics">Solid-state physics</a></li> <li><a href="/wiki/Crystallography" title="Crystallography">Crystallography</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Applied_and_interdisciplinary_physics" title="Category:Applied and interdisciplinary physics">Interdisciplinary</a></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/Astrophysics" title="Astrophysics">Astrophysics</a></li> <li><a href="/wiki/Atmospheric_physics" title="Atmospheric physics">Atmospheric physics</a></li> <li><a href="/wiki/Biophysics" title="Biophysics">Biophysics</a></li> <li><a href="/wiki/Chemical_physics" title="Chemical physics">Chemical physics</a></li> <li><a href="/wiki/Geophysics" title="Geophysics">Geophysics</a></li> <li><a href="/wiki/Materials_science" title="Materials science">Materials science</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Medical_physics" title="Medical physics">Medical physics</a></li> <li><a href="/wiki/Physical_oceanography" title="Physical oceanography">Ocean physics</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</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 hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_physics" title="History of physics">History of physics</a></li> <li><a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a></li> <li><a href="/wiki/Philosophy_of_physics" title="Philosophy of physics">Philosophy of physics</a></li> <li><a href="/wiki/Physics_education" title="Physics education">Physics education</a></li> <li><a href="/wiki/Timeline_of_fundamental_physics_discoveries" title="Timeline of fundamental physics discoveries">Timeline of physics discoveries</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="Standard_Model" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Standard_model_of_physics" title="Template:Standard model of physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Standard_model_of_physics" title="Template talk:Standard model of physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Standard_model_of_physics" title="Special:EditPage/Template:Standard model of physics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Standard_Model" style="font-size:114%;margin:0 4em"><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</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/Particle_physics" title="Particle physics">Particle physics</a> <ul><li><a href="/wiki/Fermion" title="Fermion">Fermions</a></li> <li><a href="/wiki/Gauge_boson" title="Gauge boson">Gauge boson</a></li> <li><a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a></li></ul></li> <li><a class="mw-selflink selflink">Quantum field theory</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Strong_interaction" title="Strong interaction">Strong interaction</a> <ul><li><a href="/wiki/Color_charge" title="Color charge">Color charge</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Quark_model" title="Quark model">Quark model</a></li></ul></li> <li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a> <ul><li><a href="/wiki/Weak_interaction" title="Weak interaction">Weak interaction</a></li> <li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi's interaction</a></li> <li><a href="/wiki/Weak_hypercharge" title="Weak hypercharge">Weak hypercharge</a></li> <li><a href="/wiki/Weak_isospin" title="Weak isospin">Weak isospin</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/150px-Standard_Model_of_Elementary_Particles.svg.png" decoding="async" width="150" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/225px-Standard_Model_of_Elementary_Particles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/300px-Standard_Model_of_Elementary_Particles.svg.png 2x" data-file-width="1390" data-file-height="1330" /></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constituents</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/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">CKM matrix</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li> <li><a href="/wiki/Mathematical_formulation_of_the_Standard_Model" title="Mathematical formulation of the Standard Model">Mathematical formulation of the Standard Model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Beyond the Standard Model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Evidence</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/Hierarchy_problem" title="Hierarchy problem">Hierarchy problem</a></li> <li><a href="/wiki/Dark_matter" title="Dark matter">Dark matter</a></li> <li><a href="/wiki/Cosmological_constant" title="Cosmological constant">Cosmological constant</a> <ul><li><a href="/wiki/Cosmological_constant_problem" title="Cosmological constant problem">problem</a></li></ul></li> <li><a href="/wiki/CP_violation" title="CP violation">Strong CP problem</a></li> <li><a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">Neutrino oscillation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</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/Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Grand_Unified_Theory" title="Grand Unified Theory">Grand Unified Theory</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</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/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">MSSM</a></li> <li><a href="/wiki/Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li> <li><a href="/wiki/Split_supersymmetry" title="Split supersymmetry">Split supersymmetry</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</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/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Superstring_theory" title="Superstring theory">Superstring theory</a></li> <li><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li> <li><a href="/wiki/Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</a></li> <li><a href="/wiki/Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a></li> <li><a href="/wiki/Twistor_theory" title="Twistor theory">Twistor theory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</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/Laboratori_Nazionali_del_Gran_Sasso" title="Laboratori Nazionali del Gran Sasso">Gran Sasso</a></li> <li><a href="/wiki/India-based_Neutrino_Observatory" title="India-based Neutrino Observatory">INO</a></li> <li><a href="/wiki/Large_Hadron_Collider" title="Large Hadron Collider">LHC</a></li> <li><a href="/wiki/Sudbury_Neutrino_Observatory" title="Sudbury Neutrino Observatory">SNO</a></li> <li><a href="/wiki/Super-Kamiokande" title="Super-Kamiokande">Super-K</a></li> <li><a href="/wiki/Tevatron" title="Tevatron">Tevatron</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><img alt="" 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Quantum field theory - Wikipedia