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<title>Quantum electrodynamics</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Quantum electrodynamics</span></span>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></th></tr><tr><td class="sidebar-image"><div class="sidebar-caption"><a href="Feynman_diagram" title="Feynman diagram">Feynman diagram</a></div></td></tr><tr><td class="sidebar-above">

<a href="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="Field_(physics)" title="Field (physics)">Field theory</a></li>

<li><a href="Electromagnetism" title="Electromagnetism">Electromagnetism</a></li>

<li><a href="Weak_force" class="mw-redirect" title="Weak force">Weak force</a></li>

<li><a href="Strong_force" class="mw-redirect" title="Strong force">Strong force</a></li>

<li><a href="Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li>

<li><a href="Special_relativity" title="Special relativity">Special relativity</a></li>

<li><a href="General_relativity" title="General relativity">General relativity</a></li>

<li><a href="Gauge_theory" title="Gauge theory">Gauge theory</a></li>

<li><a href="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="Symmetry_(physics)" title="Symmetry (physics)">Symmetries</a></div><div class="sidebar-list-content mw-collapsible-content">

<ul><li><a href="Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li>

<li><a href="Charge_conjugation" class="mw-redirect" title="Charge conjugation">C-symmetry</a></li>

<li><a href="Parity_(physics)" title="Parity (physics)">P-symmetry</a></li>

<li><a href="T-symmetry" title="T-symmetry">T-symmetry</a></li>

<li><a href="Lorentz_symmetry" class="mw-redirect" title="Lorentz symmetry">Lorentz symmetry</a></li>

<li><a href="Poincar%C3%A9_symmetry" class="mw-redirect" title="Poincaré symmetry">Poincaré symmetry</a></li>

<li><a href="Gauge_symmetry_(mathematics)" title="Gauge symmetry (mathematics)">Gauge symmetry</a></li>

<li><a href="Explicit_symmetry_breaking" title="Explicit symmetry breaking">Explicit symmetry breaking</a></li>

<li><a href="Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li>

<li><a href="Noether_charge" class="mw-redirect" title="Noether charge">Noether charge</a></li>

<li><a href="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="Anomaly_(physics)" title="Anomaly (physics)">Anomaly</a></li>

<li><a href="Background_field_method" title="Background field method">Background field method</a></li>

<li><a href="BRST_quantization" title="BRST quantization">BRST quantization</a></li>

<li><a href="Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">Correlation function</a></li>

<li><a href="Crossing_(physics)" title="Crossing (physics)">Crossing</a></li>

<li><a href="Effective_action" title="Effective action">Effective action</a></li>

<li><a href="Effective_field_theory" title="Effective field theory">Effective field theory</a></li>

<li><a href="Vacuum_expectation_value" title="Vacuum expectation value">Expectation value</a></li>

<li><a href="Feynman_diagram" title="Feynman diagram">Feynman diagram</a></li>

<li><a href="Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li>

<li><a href="LSZ_reduction_formula" title="LSZ reduction formula">LSZ reduction formula</a></li>

<li><a href="Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">Partition function</a></li>

<li><a href="Path_Integral_Formulation" class="mw-redirect" title="Path Integral Formulation">Path Integral Formulation</a></li>

<li><a href="Propagator_(Quantum_Theory)" class="mw-redirect" title="Propagator (Quantum Theory)">Propagator</a></li>

<li><a href="Quantization_(physics)" title="Quantization (physics)">Quantization</a></li>

<li><a href="Regularization_(physics)" title="Regularization (physics)">Regularization</a></li>

<li><a href="Renormalization" title="Renormalization">Renormalization</a></li>

<li><a href="Vacuum_state" class="mw-redirect" title="Vacuum state">Vacuum state</a></li>

<li><a href="Wick's_theorem" title="Wick's theorem">Wick's theorem</a></li>

<li><a href="Wightman_axioms" title="Wightman axioms">Wightman axioms</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content">

<ul><li><a href="Dirac_equation" title="Dirac equation">Dirac equation</a></li>

<li><a href="Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a></li>

<li><a href="Proca_action" title="Proca action">Proca equations</a></li>

<li><a href="Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></li>

<li><a href="Bargmann%E2%80%93Wigner_equations" title="Bargmann–Wigner equations">Bargmann–Wigner equations</a></li>

<li><a href="Schwinger-Dyson_equation" class="mw-redirect" title="Schwinger-Dyson equation">Schwinger-Dyson equation</a></li>

<li><a href="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="Standard_Model" title="Standard Model">Standard Model</a></div><div class="sidebar-list-content mw-collapsible-content">

<ul>

<li><a href="Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a></li>

<li><a href="Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li>

<li><a href="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="String_theory" title="String theory">String theory</a></li>

<li><a href="Supersymmetry" title="Supersymmetry">Supersymmetry</a></li>

<li><a href="Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li>

<li><a href="Theory_of_everything" title="Theory of everything">Theory of everything</a></li>

<li><a href="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="Stephen_Louis_Adler" class="mw-redirect" title="Stephen Louis Adler">Adler</a></li>

<li><a href="Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li>

<li><a href="Alexey_Andreevich_Anselm" class="mw-redirect" title="Alexey Andreevich Anselm">Anselm</a></li>

<li><a href="Valentine_Bargmann" title="Valentine Bargmann">Bargmann</a></li>

<li><a href="Carlo_Becchi" title="Carlo Becchi">Becchi</a></li>

<li><a href="Alexander_Belavin" title="Alexander Belavin">Belavin</a></li>

<li><a href="Felix_Berezin" title="Felix Berezin">Berezin</a></li>

<li><a href="Hans_Bethe" title="Hans Bethe">Bethe</a></li>

<li><a href="James_Bjorken" title="James Bjorken">Bjorken</a></li>

<li><a href="Konrad_Bleuler" title="Konrad Bleuler">Bleuer</a></li>

<li><a href="Nikolay_Bogolyubov" title="Nikolay Bogolyubov">Bogoliubov</a></li>

<li><a href="Stanley_Brodsky" title="Stanley Brodsky">Brodsky</a></li>

<li><a href="Robert_Brout" title="Robert Brout">Brout</a></li>

<li><a href="Detlev_Buchholz" title="Detlev Buchholz">Buchholz</a></li>

<li><a href="Freddy_Cachazo" title="Freddy Cachazo">Cachazo</a></li>

<li><a href="Curtis_Callan" title="Curtis Callan">Callan</a></li>

<li><a href="John_Cardy" title="John Cardy">Cardy</a></li>

<li><a href="Sidney_Coleman" title="Sidney Coleman">Coleman</a></li>

<li><a href="Alain_Connes" title="Alain Connes">Connes</a></li>

<li><a href="Roger_Dashen" title="Roger Dashen">Dashen</a></li>

<li><a href="Bryce_DeWitt" title="Bryce DeWitt">DeWitt</a></li>

<li><a href="Paul_Dirac" title="Paul Dirac">Dirac</a></li>

<li><a href="Sergio_Doplicher" title="Sergio Doplicher">Doplicher</a></li>

<li><a href="Freeman_Dyson" title="Freeman Dyson">Dyson</a></li>

<li><a href="Fran%C3%A7ois_Englert" title="François Englert">Englert</a></li>

<li><a href="Ludvig_Faddeev" title="Ludvig Faddeev">Faddeev</a></li>

<li><a href="Victor_Sergeevich_Fadin" class="mw-redirect" title="Victor Sergeevich Fadin">Fadin</a></li>

<li><a href="Pierre_Fayet" title="Pierre Fayet">Fayet</a></li>

<li><a href="Enrico_Fermi" title="Enrico Fermi">Fermi</a></li>

<li><a href="Richard_Feynman" title="Richard Feynman">Feynman</a></li>

<li><a href="Markus_Fierz" title="Markus Fierz">Fierz</a></li>

<li><a href="Paul_Frampton" title="Paul Frampton">Frampton</a></li>

<li><a href="Harald_Fritzsch" title="Harald Fritzsch">Fritzsch</a></li>

<li><a href="J%C3%BCrg_Fr%C3%B6hlich" title="Jürg Fröhlich">Fröhlich</a></li>

<li><a href="Klaus_Fredenhagen" title="Klaus Fredenhagen">Fredenhagen</a></li>

<li><a href="Wendell_H._Furry" title="Wendell H. Furry">Furry</a></li>

<li><a href="Sheldon_Glashow" title="Sheldon Glashow">Glashow</a></li>

<li><a href="James_Glimm" title="James Glimm">Glimm</a></li>

<li><a href="Jeffrey_Goldstone" title="Jeffrey Goldstone">Goldstone</a></li>

<li><a href="Vladimir_Gribov" title="Vladimir Gribov">Gribov</a></li>

<li><a href="David_Gross" title="David Gross">Gross</a></li>

<li><a href="Suraj_N._Gupta" title="Suraj N. Gupta">Gupta</a></li>

<li><a href="Gerald_Guralnik" title="Gerald Guralnik">Guralnik</a></li>

<li><a href="C._R._Hagen" title="C. R. Hagen">Hagen</a></li>

<li><a href="Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li>

<li><a href="Peter_Higgs" title="Peter Higgs">Higgs</a></li>

<li><a href="Gerard_'t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li>

<li><a href="John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li>

<li><a href="Dmitri_Ivanenko" title="Dmitri Ivanenko">Ivanenko</a></li>

<li><a href="Roman_Jackiw" title="Roman Jackiw">Jackiw</a></li>

<li><a href="Arthur_Jaffe" title="Arthur Jaffe">Jaffe</a></li>

<li><a href="Giovanni_Jona-Lasinio" title="Giovanni Jona-Lasinio">Jona-Lasinio</a></li>

<li><a href="Pascual_Jordan" title="Pascual Jordan">Jordan</a></li>

<li><a href="Gunnar_K%C3%A4ll%C3%A9n" title="Gunnar Källén">Källén</a></li>

<li><a href="Henry_Way_Kendall" title="Henry Way Kendall">Kendall</a></li>

<li><a href="Toichiro_Kinoshita" title="Toichiro Kinoshita">Kinoshita</a></li>

<li><a href="Igor_R._Klebanov" class="mw-redirect" title="Igor R. Klebanov">Klebanov</a></li>

<li><a href="Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li>

<li><a href="Dirk_Kreimer" title="Dirk Kreimer">Kreimer</a></li>

<li><a href="Eduard_A._Kuraev" title="Eduard A. Kuraev">Kuraev</a></li>

<li><a href="Lev_Landau" title="Lev Landau">Landau</a></li>

<li><a href="Harry_Lehmann" title="Harry Lehmann">Lehmann</a></li>

<li><a href="Heinrich_Leutwyler" title="Heinrich Leutwyler">Leutwyler</a></li>

<li><a href="Lev_Lipatov" title="Lev Lipatov">Lipatov</a></li>

<li><a href="Jan_%C5%81opusza%C5%84ski_(physicist)" title="Jan Łopuszański (physicist)">Łopuszański</a></li>

<li><a href="Gerhart_L%C3%BCders" title="Gerhart Lüders">Lüders</a></li>

<li><a href="Luciano_Maiani" title="Luciano Maiani">Maiani</a></li>

<li><a href="Ettore_Majorana" title="Ettore Majorana">Majorana</a></li>

<li><a href="Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li>

<li><a href="Takeo_Matsubara" title="Takeo Matsubara">Matsubara</a></li>

<li><a href="Alexander_Arkadyevich_Migdal" class="mw-redirect" title="Alexander Arkadyevich Migdal">Migdal</a></li>

<li><a href="Robert_Mills_(physicist)" title="Robert Mills (physicist)">Mills</a></li>

<li><a href="Christian_M%C3%B8ller" title="Christian Møller">Møller</a></li>

<li><a href="Mark_Naimark" title="Mark Naimark">Naimark</a></li>

<li><a href="Yoichiro_Nambu" title="Yoichiro Nambu">Nambu</a></li>

<li><a href="Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li>

<li><a href="Kazuhiko_Nishijima" title="Kazuhiko Nishijima">Nishijima</a></li>

<li><a href="Reinhard_Oehme" title="Reinhard Oehme">Oehme</a></li>

<li><a href="J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li>

<li><a href="Hugh_Osborn" title="Hugh Osborn">Osborn</a></li>

<li><a href="Konrad_Osterwalder" title="Konrad Osterwalder">Osterwalder</a></li>

<li><a href="Giorgio_Parisi" title="Giorgio Parisi">Parisi</a></li>

<li><a href="Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li>

<li><a href="Roberto_Peccei" title="Roberto Peccei">Peccei</a></li>

<li><a href="Michael_Peskin" title="Michael Peskin">Peskin</a></li>

<li><a href="Jan_Christoph_Plefka" title="Jan Christoph Plefka">Plefka</a></li>

<li><a href="Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li>

<li><a href="Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li>

<li><a href="Isaak_Pomeranchuk" title="Isaak Pomeranchuk">Pomeranchuk</a></li>

<li><a href="Victor_Popov" title="Victor Popov">Popov</a></li>

<li><a href="Alexandru_Proca" title="Alexandru Proca">Proca</a></li>

<li><a href="Helen_Quinn" title="Helen Quinn">Quinn</a></li>

<li><a href="Alain_Rouet" title="Alain Rouet">Rouet</a></li>

<li><a href="Valery_Rubakov" title="Valery Rubakov">Rubakov</a></li>

<li><a href="David_Ruelle" title="David Ruelle">Ruelle</a></li>

<li><a href="Jun_John_Sakurai" class="mw-redirect" title="Jun John Sakurai">Sakurai</a></li>

<li><a href="Abdus_Salam" title="Abdus Salam">Salam</a></li>

<li><a href="Robert_Schrader" title="Robert Schrader">Schrader</a></li>

<li><a href="Albert_Schwarz" title="Albert Schwarz">Schwarz</a></li>

<li><a href="Julian_Schwinger" title="Julian Schwinger">Schwinger</a></li>

<li><a href="Irving_Segal" title="Irving Segal">Segal</a></li>

<li><a href="Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li>

<li><a href="Gordon_Walter_Semenoff" title="Gordon Walter Semenoff">Semenoff</a></li>

<li><a href="Mikhail_Shifman" title="Mikhail Shifman">Shifman</a></li>

<li><a href="Dmitry_Shirkov" title="Dmitry Shirkov">Shirkov</a></li>

<li><a href="Tony_Skyrme" title="Tony Skyrme">Skyrme</a></li>

<li><a href="Charles_M._Sommerfield" title="Charles M. Sommerfield">Sommerfield</a></li>

<li><a href="Raymond_Stora" title="Raymond Stora">Stora</a></li>

<li><a href="Ernst_Stueckelberg" title="Ernst Stueckelberg">Stueckelberg</a></li>

<li><a href="George_Sudarshan" class="mw-redirect" title="George Sudarshan">Sudarshan</a></li>

<li><a href="Kurt_Symanzik" title="Kurt Symanzik">Symanzik</a></li>

<li><a href="Yasushi_Takahashi" title="Yasushi Takahashi">Takahashi</a></li>

<li><a href="Walter_Thirring" title="Walter Thirring">Thirring</a></li>

<li><a href="Shin'ichir%C5%8D_Tomonaga" title="Shin'ichirō Tomonaga">Tomonaga</a></li>

<li><a href="Igor_Tyutin" title="Igor Tyutin">Tyutin</a></li>

<li><a href="Arkady_Vainshtein" title="Arkady Vainshtein">Vainshtein</a></li>

<li><a href="Martinus_Veltman" class="mw-redirect" title="Martinus Veltman">Veltman</a></li>

<li><a href="Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano</a></li>

<li><a href="Miguel_%C3%81ngel_Virasoro_(physicist)" title="Miguel Ángel Virasoro (physicist)">Virasoro</a></li>

<li><a href="Steven_Weinberg" title="Steven Weinberg">Weinberg</a></li>

<li><a href="Victor_Weisskopf" title="Victor Weisskopf">Weisskopf</a></li>

<li><a href="Gregor_Wentzel" title="Gregor Wentzel">Wentzel</a></li>

<li><a href="Christof_Wetterich" title="Christof Wetterich">Wetterich</a></li>

<li><a href="Arthur_Wightman" title="Arthur Wightman">Wightman</a></li>

<li><a href="Eugene_Wigner" title="Eugene Wigner">Wigner</a></li>

<li><a href="Frank_Wilczek" title="Frank Wilczek">Wilczek</a></li>

<li><a href="Kenneth_G._Wilson" title="Kenneth G. Wilson">Wilson</a></li>

<li><a href="Edward_Witten" title="Edward Witten">Witten</a></li>

<li><a href="Hideki_Yukawa" title="Hideki Yukawa">Yukawa</a></li>

<li><a href="Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li>

<li><a href="Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li>

<li><a href="Wolfhart_Zimmermann" title="Wolfhart Zimmermann">Zimmermann</a></li>

<li><a href="Jean_Zinn-Justin" title="Jean Zinn-Justin">Zinn-Justin</a></li>

<li><a href="Jean-Bernard_Zuber" title="Jean-Bernard Zuber">Zuber</a></li>

<li><a href="Bruno_Zumino" title="Bruno Zumino">Zumino</a></li></ul>

<p><br>

</p>

</div></div></div></td>

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<p>In <a href="Particle_physics" title="Particle physics">particle physics</a>, <b>quantum electrodynamics</b> (<b>QED</b>) is the <a href="Theory_of_relativity" title="Theory of relativity">relativistic</a> <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a> of <a href="Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>.<sup id="cite_ref-feynman1_1-0" class="reference"><a href="#cite_note-feynman1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-feynbook_2-0" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In essence, it describes how <a href="Light" title="Light">light</a> and <a href="Matter" title="Matter">matter</a> interact and is the first theory where full agreement between <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and <a href="Special_relativity" title="Special relativity">special relativity</a> is achieved.<sup id="cite_ref-feynbook_2-1" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> QED mathematically describes all <a href="Phenomenon" title="Phenomenon">phenomena</a> involving <a href="Electric_charge" title="Electric charge">electrically charged</a> particles interacting by means of exchange of <a href="Photon" title="Photon">photons</a> and represents the <a href="Quantum_mechanics" title="Quantum mechanics">quantum</a> counterpart of <a href="Classical_electromagnetism" title="Classical electromagnetism">classical electromagnetism</a> giving a complete account of matter and light interaction.<sup id="cite_ref-feynbook_2-2" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>In technical terms, QED can be described as a <a href="Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a> of the electromagnetic <a href="Quantum_vacuum_state" title="Quantum vacuum state">quantum vacuum</a>. <a href="Richard_Feynman" title="Richard Feynman">Richard Feynman</a> called it "the jewel of physics" for its <a href="Precision_tests_of_QED" title="Precision tests of QED">extremely accurate predictions</a> of quantities like the <a href="Anomalous_magnetic_moment" class="mw-redirect" title="Anomalous magnetic moment">anomalous magnetic moment</a> of the electron and the <a href="Lamb_shift" title="Lamb shift">Lamb shift</a> of the <a href="Energy_level" title="Energy level">energy levels</a> of <a href="Hydrogen" title="Hydrogen">hydrogen</a>.<sup id="cite_ref-feynbook_2-3" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Ch1">: Ch1 </span></sup> It is the most precise and stringently tested theory in physics.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="History_of_quantum_mechanics" title="History of quantum mechanics">History of quantum mechanics</a> and <a href="History_of_quantum_field_theory" title="History of quantum field theory">History of quantum field theory</a></div>

<p>The first formulation of a <a href="Quantum_mechanics" title="Quantum mechanics">quantum theory</a> describing radiation and matter interaction is attributed to <a href="Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, who during the 1920s computed the coefficient of <a href="Spontaneous_emission" title="Spontaneous emission">spontaneous emission</a> of an <a href="Atom" title="Atom">atom</a>.<sup id="cite_ref-dirac_6-0" class="reference"><a href="#cite_note-dirac-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> He is credited with coining the term "quantum electrodynamics".<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>Dirac described the quantization of the <a href="Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> as an ensemble of <a href="Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillators</a> with the introduction of the concept of <a href="Creation_and_annihilation_operators" title="Creation and annihilation operators">creation and annihilation operators</a> of particles. In the following years, with contributions from <a href="Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a>, <a href="Eugene_Wigner" title="Eugene Wigner">Eugene Wigner</a>, <a href="Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a>, <a href="Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> and <a href="Enrico_Fermi" title="Enrico Fermi">Enrico Fermi</a>,<sup id="cite_ref-fermi_8-0" class="reference"><a href="#cite_note-fermi-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> physicists came to believe that, in principle, it was possible to perform any computation for any physical process involving photons and charged particles. However, further studies by <a href="Felix_Bloch" title="Felix Bloch">Felix Bloch</a> with <a href="Arnold_Nordsieck" title="Arnold Nordsieck">Arnold Nordsieck</a>,<sup id="cite_ref-bloch_9-0" class="reference"><a href="#cite_note-bloch-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and <a href="Victor_Weisskopf" title="Victor Weisskopf">Victor Weisskopf</a>,<sup id="cite_ref-weisskopf_10-0" class="reference"><a href="#cite_note-weisskopf-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> in 1937 and 1939, revealed that such computations were reliable only at a first order of <a href="Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a>, a problem already pointed out by <a href="Robert_Oppenheimer" class="mw-redirect" title="Robert Oppenheimer">Robert Oppenheimer</a>.<sup id="cite_ref-oppenheimer_11-0" class="reference"><a href="#cite_note-oppenheimer-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> At higher orders in the series infinities emerged, making such computations meaningless and casting doubt on the theory's internal consistency. This suggested that <a href="Special_relativity" title="Special relativity">special relativity</a> and <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> were fundamentally incompatible.

</p>

<p>Difficulties increased through the end of the 1940s. Improvements in <a href="Microwave" title="Microwave">microwave</a> technology made it possible to take more precise measurements of the shift of the levels of a <a href="Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a>,<sup id="cite_ref-lamb_12-0" class="reference"><a href="#cite_note-lamb-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> later known as the <a href="Lamb_shift" title="Lamb shift">Lamb shift</a> and <a href="Magnetic_moment" title="Magnetic moment">magnetic moment</a> of the electron.<sup id="cite_ref-foley_13-0" class="reference"><a href="#cite_note-foley-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> These experiments exposed discrepancies that the theory was unable to explain.

</p><p>A first indication of a possible solution was given by <a href="Hans_Bethe" title="Hans Bethe">Hans Bethe</a> in 1947.<sup id="cite_ref-bethe_14-0" class="reference"><a href="#cite_note-bethe-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-schweber_15-0" class="reference"><a href="#cite_note-schweber-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> He made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by <a href="Willis_Lamb" title="Willis Lamb">Willis Lamb</a> and <a href="Robert_Retherford" title="Robert Retherford">Robert Retherford</a>.<sup id="cite_ref-bethe_14-1" class="reference"><a href="#cite_note-bethe-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Despite limitations of the computation, agreement was excellent. The idea was simply to attach infinities to corrections of mass and charge that were actually fixed to a finite value by experiments. In this way, the infinities get absorbed in those constants and yield a finite result with good experimental agreement. This procedure was named <a href="Renormalization" title="Renormalization">renormalization</a>.

</p>

<p>Based on Bethe's intuition and fundamental papers on the subject by <a href="Shin'ichir%C5%8D_Tomonaga" title="Shin'ichirō Tomonaga">Shin'ichirō Tomonaga</a>,<sup id="cite_ref-tomonaga_16-0" class="reference"><a href="#cite_note-tomonaga-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <a href="Julian_Schwinger" title="Julian Schwinger">Julian Schwinger</a>,<sup id="cite_ref-schwinger1_17-0" class="reference"><a href="#cite_note-schwinger1-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-schwinger2_18-0" class="reference"><a href="#cite_note-schwinger2-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <a href="Richard_Feynman" title="Richard Feynman">Richard Feynman</a><sup id="cite_ref-feynman1_1-1" class="reference"><a href="#cite_note-feynman1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-feynman2_19-0" class="reference"><a href="#cite_note-feynman2-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-feynman3_20-0" class="reference"><a href="#cite_note-feynman3-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> and <a href="Freeman_Dyson" title="Freeman Dyson">Freeman Dyson</a>,<sup id="cite_ref-dyson1_21-0" class="reference"><a href="#cite_note-dyson1-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-dyson2_22-0" class="reference"><a href="#cite_note-dyson2-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> it was finally possible to produce fully <a href="Lorentz_covariance" title="Lorentz covariance">covariant</a> formulations that were finite at any order in a perturbation series of quantum electrodynamics. Tomonaga, Schwinger, and Feynman were jointly awarded the 1965 <a href="Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> for their work in this area.<sup id="cite_ref-nobel65_23-0" class="reference"><a href="#cite_note-nobel65-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Their contributions, and Dyson's, were about <a href="Lorentz_covariance" title="Lorentz covariance">covariant</a> and <a href="Gauge-invariant" class="mw-redirect" title="Gauge-invariant">gauge-invariant</a> formulations of quantum electrodynamics that allow computations of observables at any order of <a href="Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a>. Feynman's mathematical technique, based on his <a href="Feynman_diagram" title="Feynman diagram">diagrams</a>, initially seemed unlike the field-theoretic, <a href="Operator_(physics)" title="Operator (physics)">operator</a>-based approach of Schwinger and Tomonaga, but Dyson later showed that the two approaches were equivalent.<sup id="cite_ref-dyson1_21-1" class="reference"><a href="#cite_note-dyson1-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Renormalization, the need to attach a physical meaning at certain divergences appearing in the theory through <a href="Integral" title="Integral">integrals</a>, became one of the fundamental aspects of <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a> and is seen as a criterion for a theory's general acceptability. Even though renormalization works well in practice, Feynman was never entirely comfortable with its mathematical validity, referring to renormalization as a "shell game" and "hocus pocus".<sup id="cite_ref-feynbook_2-4" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 128">: 128 </span></sup>

</p><p>Neither Feynman nor Dirac were happy with that way to approach the observations made in theoretical physics, above all in quantum mechanics.<sup id="cite_ref-:1_24-0" class="reference"><a href="#cite_note-:1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>

</p><p>QED is the model and template for all subsequent quantum field theories. One such subsequent theory is <a href="Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>, which began in the early 1960s and attained its present form in the 1970s, developed by <a href="H._David_Politzer" class="mw-redirect" title="H. David Politzer">H. David Politzer</a>, <a href="Sidney_Coleman" title="Sidney Coleman">Sidney Coleman</a>, <a href="David_Gross" title="David Gross">David Gross</a> and <a href="Frank_Wilczek" title="Frank Wilczek">Frank Wilczek</a>. Building on Schwinger's pioneering work, <a href="Gerald_Guralnik" title="Gerald Guralnik">Gerald Guralnik</a>, <a href="C._R._Hagen" title="C. R. Hagen">Dick Hagen</a>, and <a href="Tom_W._B._Kibble" class="mw-redirect" title="Tom W. B. Kibble">Tom Kibble</a>,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> <a href="Peter_Higgs" title="Peter Higgs">Peter Higgs</a>, <a href="Jeffrey_Goldstone" title="Jeffrey Goldstone">Jeffrey Goldstone</a>, and others, <a href="Sheldon_Glashow" title="Sheldon Glashow">Sheldon Glashow</a>, <a href="Steven_Weinberg" title="Steven Weinberg">Steven Weinberg</a> and <a href="Abdus_Salam" title="Abdus Salam">Abdus Salam</a> independently showed how the <a href="Weak_nuclear_force" class="mw-redirect" title="Weak nuclear force">weak nuclear force</a> and quantum electrodynamics could be merged into a single <a href="Electroweak_force" class="mw-redirect" title="Electroweak force">electroweak force</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Feynman's_view_of_quantum_electrodynamics">Feynman's view of quantum electrodynamics</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Introduction">Introduction</h3></div>

<p>Near the end of his life, <a href="Richard_Feynman" title="Richard Feynman">Richard Feynman</a> gave a series of lectures on QED intended for the lay public. These lectures were transcribed and published as Feynman (1985), <i><a href="QED%3A_The_Strange_Theory_of_Light_and_Matter" title="QED: The Strange Theory of Light and Matter">QED: The Strange Theory of Light and Matter</a></i>,<sup id="cite_ref-feynbook_2-5" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> a classic non-mathematical exposition of QED from the point of view articulated below.

</p><p>The key components of Feynman's presentation of QED are three basic actions.<sup id="cite_ref-feynbook_2-6" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 85">: 85 </span></sup>

</p>

<dl><dd>A <a href="Photon" title="Photon">photon</a> goes from one place and time to another place and time.</dd>

<dd>An <a href="Electron" title="Electron">electron</a> goes from one place and time to another place and time.</dd>

<dd>An electron emits or absorbs a photon at a certain place and time.</dd></dl>

<p>These actions are represented in the form of visual shorthand by the three basic elements of <a href="Feynman_diagram" title="Feynman diagram">diagrams</a>: a wavy line for the photon, a straight line for the electron and a junction of two straight lines and a wavy one for a vertex representing emission or absorption of a photon by an electron. These can all be seen in the adjacent diagram.

</p><p>As well as the visual shorthand for the actions, Feynman introduces another kind of shorthand for the numerical quantities called <a class="mw-selflink-fragment" href="#Probability_amplitudes">probability amplitudes</a>. The probability is the square of the absolute value of total probability amplitude, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{probability}}=|f({\text{amplitude}})|^{2}}">

<mtext>probability</mtext>

</mrow>

</mrow>

<mtext>amplitude</mtext>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{probability}}=|f({\text{amplitude}})|^{2}}</annotation>

</semantics>

</math></span><img src="./1aec0b02d1975e9eb147582a3d0acde2ad67e8ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.928ex; height:3.343ex;" alt="{\displaystyle {\text{probability}}=|f({\text{amplitude}})|^{2}}" loading="lazy"></span>. If a photon moves from one place and time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> to another place and time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span>, the associated quantity is written in Feynman's shorthand as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A{\text{ to }}B)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A{\text{ to }}B)}</annotation>

</semantics>

</math></span><img src="./d8ce640fb38aa48636e45180c49b5813e86abdf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.29ex; height:2.843ex;" alt="{\displaystyle P(A{\text{ to }}B)}" loading="lazy"></span>, and it depends on only the momentum and polarization of the photon. The similar quantity for an electron moving from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> is written <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(C{\text{ to }}D)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(C{\text{ to }}D)}</annotation>

</semantics>

</math></span><img src="./15b2c882c772435c97991c1d7e21cfb220fc3c74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.504ex; height:2.843ex;" alt="{\displaystyle E(C{\text{ to }}D)}" loading="lazy"></span>. It depends on the momentum and polarization of the electron, in addition to a constant Feynman calls <i>n</i>, sometimes called the "bare" mass of the electron: it is related to, but not the same as, the measured electron mass. Finally, the quantity that tells us about the probability amplitude for an electron to emit or absorb a photon Feynman calls <i>j</i>, and is sometimes called the "bare" charge of the electron: it is a constant, and is related to, but not the same as, the measured <a href="Elementary_charge" title="Elementary charge">electron charge</a> <i>e</i>.<sup id="cite_ref-feynbook_2-7" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 91">: 91 </span></sup>

</p><p>QED is based on the assumption that complex interactions of many electrons and photons can be represented by fitting together a suitable collection of the above three building blocks and then using the probability amplitudes to calculate the probability of any such complex interaction. It turns out that the basic idea of QED can be communicated while assuming that the square of the total of the probability amplitudes mentioned above (<i>P</i>(<i>A</i> to <i>B</i>), <i>E</i>(<i>C</i> to <i>D</i>) and <i>j</i>) acts just like our everyday <a href="Probability" title="Probability">probability</a> (a simplification made in Feynman's book). Later on, this will be corrected to include specifically quantum-style mathematics, following Feynman.

</p><p>The basic rules of probability amplitudes that will be used are:<sup id="cite_ref-feynbook_2-8" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 93">: 93 </span></sup>

</p>

<div><ol style="list-style-type:lower-alpha"><li>If an event can occur via a number of <i>indistinguishable</i> alternative processes (a.k.a. "virtual" processes), then its probability amplitude is the <b>sum</b> of the probability amplitudes of the alternatives.</li><li>If a virtual process involves a number of independent or concomitant sub-processes, then the probability amplitude of the total (compound) process is the <b>product</b> of the probability amplitudes of the sub-processes.</li></ol></div>

<p>The indistinguishability criterion in (a) is very important: it means that there is <i>no observable feature present in the given system</i> that in any way "reveals" which alternative is taken. In such a case, one cannot observe which alternative actually takes place without changing the experimental setup in some way (e.g. by introducing a new apparatus into the system). Whenever one <i>is</i> able to observe which alternative takes place, one always finds that the <i>probability</i> of the event is the sum of the <i>probabilities</i> of the alternatives. Indeed, if this were not the case, the very term "alternatives" to describe these processes would be inappropriate. What (a) says is that once the <i>physical means</i> for observing which alternative occurred is <i>removed</i>, one cannot still say that the event is occurring through "exactly one of the alternatives" in the sense of adding probabilities; one must add the amplitudes instead.<sup id="cite_ref-feynbook_2-9" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 82">: 82 </span></sup>

</p><p>Similarly, the independence criterion in (b) is very important: it only applies to processes which are not "entangled".

</p>

<div class="mw-heading mw-heading3"><h3 id="Basic_constructions">Basic constructions</h3></div>

<p>Suppose we start with one electron at a certain place and time (this place and time being given the arbitrary label <i>A</i>) and a photon at another place and time (given the label <i>B</i>). A typical question from a physical standpoint is: "What is the probability of finding an electron at <i>C</i> (another place and a later time) and a photon at <i>D</i> (yet another place and time)?". The simplest process to achieve this end is for the electron to move from <i>A</i> to <i>C</i> (an elementary action) and for the photon to move from <i>B</i> to <i>D</i> (another elementary action). From a knowledge of the probability amplitudes of each of these sub-processes – <i>E</i>(<i>A</i> to <i>C</i>) and <i>P</i>(<i>B</i> to <i>D</i>) – we would expect to calculate the probability amplitude of both happening together by multiplying them, using rule b) above. This gives a simple estimated overall probability amplitude, which is squared to give an estimated probability.

</p>

<p>But there are other ways in which the result could come about. The electron might move to a place and time <i>E</i>, where it absorbs the photon; then move on before emitting another photon at <i>F</i>; then move on to <i>C</i>, where it is detected, while the new photon moves on to <i>D</i>. The probability of this complex process can again be calculated by knowing the probability amplitudes of each of the individual actions: three electron actions, two photon actions and two vertexes – one emission and one absorption. We would expect to find the total probability amplitude by multiplying the probability amplitudes of each of the actions, for any chosen positions of <i>E</i> and <i>F</i>. We then, using rule a) above, have to add up all these probability amplitudes for all the alternatives for <i>E</i> and <i>F</i>. (This is not elementary in practice and involves <a href="Integral" title="Integral">integration</a>.) But there is another possibility, which is that the electron first moves to <i>G</i>, where it emits a photon, which goes on to <i>D</i>, while the electron moves on to <i>H</i>, where it absorbs the first photon, before moving on to <i>C</i>. Again, we can calculate the probability amplitude of these possibilities (for all points <i>G</i> and <i>H</i>). We then have a better estimation for the total probability amplitude by adding the probability amplitudes of these two possibilities to our original simple estimate. Incidentally, the name given to this process of a photon interacting with an electron in this way is <a href="Compton_scattering" title="Compton scattering">Compton scattering</a>.

</p><p>An <i>infinite number</i> of other intermediate "virtual" processes exist in which photons are absorbed or emitted. For each of these processes, a Feynman diagram could be drawn describing it. This implies a complex computation for the resulting probability amplitudes, but provided it is the case that the more complicated the diagram, the less it contributes to the result, it is only a matter of time and effort to find as accurate an answer as one wants to the original question. This is the basic approach of QED. To calculate the probability of <i>any</i> interactive process between electrons and photons, it is a matter of first noting, with Feynman diagrams, all the possible ways in which the process can be constructed from the three basic elements. Each diagram involves some calculation involving definite rules to find the associated probability amplitude.

</p><p>That basic scaffolding remains when one moves to a quantum description, but some conceptual changes are needed. One is that whereas we might expect in our everyday life that there would be some constraints on the points to which a particle can move, that is <i>not</i> true in full quantum electrodynamics. There is a nonzero probability amplitude of an electron at <i>A</i>, or a photon at <i>B</i>, moving as a basic action to <i>any other place and time in the universe</i>. That includes places that could only be reached at speeds greater than that of light and also <i>earlier times</i>. (An electron moving backwards in time can be viewed as a <a href="Positron" title="Positron">positron</a> moving forward in time.)<sup id="cite_ref-feynbook_2-10" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 89, 98–99">: 89, 98–99 </span></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Probability_amplitudes">Probability amplitudes</h3></div>

<p><a href="Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> introduces an important change in the way probabilities are computed. Probabilities are still represented by the usual real numbers we use for probabilities in our everyday world, but probabilities are computed as the <a href="Square_modulus" class="mw-redirect" title="Square modulus">square modulus</a> of <a href="Probability_amplitude" title="Probability amplitude">probability amplitudes</a>, which are <a href="Complex_number" title="Complex number">complex numbers</a>.

</p><p>Feynman avoids exposing the reader to the mathematics of complex numbers by using a simple but accurate representation of them as arrows on a piece of paper or screen. (These must not be confused with the arrows of Feynman diagrams, which are simplified representations in two dimensions of a relationship between points in three dimensions of space and one of time.) The amplitude arrows are fundamental to the description of the world given by quantum theory. They are related to our everyday ideas of probability by the simple rule that the probability of an event is the <i>square</i> of the length of the corresponding amplitude arrow. So, for a given process, if two probability amplitudes, <b>v</b> and <b>w</b>, are involved, the probability of the process will be given either by

</p>

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<annotation encoding="application/x-tex">{\displaystyle P=|\mathbf {v} +\mathbf {w} |^{2}}</annotation>

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</math></span><img src="./3e923cc9d1c6b59f2184133a5227640a811c3909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.374ex; height:3.343ex;" alt="{\displaystyle P=|\mathbf {v} +\mathbf {w} |^{2}}" loading="lazy"></span></dd></dl>

<p>or

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<annotation encoding="application/x-tex">{\displaystyle P=|\mathbf {v} \,\mathbf {w} |^{2}.}</annotation>

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</math></span><img src="./65b36ecdebf263ca4aff40f1870bdb946e021495.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.568ex; height:3.343ex;" alt="{\displaystyle P=|\mathbf {v} \,\mathbf {w} |^{2}.}" loading="lazy"></span></dd></dl>

<p>The rules as regards adding or multiplying, however, are the same as above. But where you would expect to add or multiply probabilities, instead you add or multiply probability amplitudes that now are complex numbers.

</p>

<p>Addition and multiplication are common operations in the theory of complex numbers and are given in the figures. The sum is found as follows. Let the start of the second arrow be at the end of the first. The sum is then a third arrow that goes directly from the beginning of the first to the end of the second. The product of two arrows is an arrow whose length is the product of the two lengths. The direction of the product is found by adding the angles that each of the two have been turned through relative to a reference direction: that gives the angle that the product is turned relative to the reference direction.

</p><p>That change, from probabilities to probability amplitudes, complicates the mathematics without changing the basic approach. But that change is still not quite enough because it fails to take into account the fact that both photons and electrons can be polarized, which is to say that their orientations in space and time have to be taken into account. Therefore, <i>P</i>(<i>A</i> to <i>B</i>) consists of 16 complex numbers, or probability amplitude arrows.<sup id="cite_ref-feynbook_2-11" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 120–121">: 120–121 </span></sup> There are also some minor changes to do with the quantity <i>j</i>, which may have to be rotated by a multiple of 90° for some polarizations, which is only of interest for the detailed bookkeeping.

</p><p>Associated with the fact that the electron can be polarized is another small necessary detail, which is connected with the fact that an electron is a <a href="Fermion" title="Fermion">fermion</a> and obeys <a href="Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">Fermi–Dirac statistics</a>. The basic rule is that if we have the probability amplitude for a given complex process involving more than one electron, then when we include (as we always must) the complementary Feynman diagram in which we exchange two electron events, the resulting amplitude is the reverse – the negative – of the first. The simplest case would be two electrons starting at <i>A</i> and <i>B</i> ending at <i>C</i> and <i>D</i>. The amplitude would be calculated as the "difference", <span class="nowrap"><i>E</i>(<i>A</i> to <i>D</i>) × <i>E</i>(<i>B</i> to <i>C</i>) − <i>E</i>(<i>A</i> to <i>C</i>) × <i>E</i>(<i>B</i> to <i>D</i>)</span>, where we would expect, from our everyday idea of probabilities, that it would be a sum.<sup id="cite_ref-feynbook_2-12" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 112–113">: 112–113 </span></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Propagators">Propagators</h3></div>

<p>Finally, one has to compute <i>P</i>(<i>A</i> to <i>B</i>) and <i>E</i>(<i>C</i> to <i>D</i>) corresponding to the probability amplitudes for the photon and the electron respectively. These are essentially the solutions of the <a href="Dirac_equation" title="Dirac equation">Dirac equation</a>, which describe the behavior of the electron's probability amplitude and the <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a>, which describes the behavior of the photon's probability amplitude. These are called <a href="Propagator" title="Propagator">Feynman propagators</a>. The translation to a notation commonly used in the standard literature is as follows:

</p>

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<annotation encoding="application/x-tex">{\displaystyle P(A{\text{ to }}B)\to D_{F}(x_{B}-x_{A}),\quad E(C{\text{ to }}D)\to S_{F}(x_{D}-x_{C}),}</annotation>

</semantics>

</math></span><img src="./d9c3a93b4ef958f51f9a4c5c691996c3f7d82495.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.938ex; height:2.843ex;" alt="{\displaystyle P(A{\text{ to }}B)\to D_{F}(x_{B}-x_{A}),\quad E(C{\text{ to }}D)\to S_{F}(x_{D}-x_{C}),}" loading="lazy"></span></dd></dl>

<p>where a shorthand symbol such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{A}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{A}}</annotation>

</semantics>

</math></span><img src="./131920dc49fade5cd528c48af190e33e3c7e0a6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.795ex; height:2.009ex;" alt="{\displaystyle x_{A}}" loading="lazy"></span> stands for the four real numbers that give the time and position in three dimensions of the point labeled <i>A</i>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Mass_renormalization">Mass renormalization</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Self-energy" title="Self-energy">Self-energy</a></div>

<p>A problem arose historically which held up progress for twenty years: although we start with the assumption of three basic "simple" actions, the rules of the game say that if we want to calculate the probability amplitude for an electron to get from <i>A</i> to <i>B</i>, we must take into account <i>all</i> the possible ways: all possible Feynman diagrams with those endpoints. Thus there will be a way in which the electron travels to <i>C</i>, emits a photon there and then absorbs it again at <i>D</i> before moving on to <i>B</i>. Or it could do this kind of thing twice, or more. In short, we have a <a href="Fractal" title="Fractal">fractal</a>-like situation in which if we look closely at a line, it breaks up into a collection of "simple" lines, each of which, if looked at closely, are in turn composed of "simple" lines, and so on <i>ad infinitum</i>. This is a challenging situation to handle. If adding that detail only altered things slightly, then it would not have been too bad, but disaster struck when it was found that the simple correction mentioned above led to <i>infinite</i> probability amplitudes. In time this problem was "fixed" by the technique of <a href="Renormalization" title="Renormalization">renormalization</a>. However, Feynman himself remained unhappy about it, calling it a "dippy process",<sup id="cite_ref-feynbook_2-13" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 128">: 128 </span></sup> and Dirac also criticized this procedure, saying "in mathematics one does not get rid of infinities when it does not please you".<sup id="cite_ref-:1_24-1" class="reference"><a href="#cite_note-:1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Conclusions">Conclusions</h3></div>

<p>Within the above framework physicists were then able to calculate to a high degree of accuracy some of the properties of electrons, such as the <a href="Anomalous_magnetic_dipole_moment" title="Anomalous magnetic dipole moment">anomalous magnetic dipole moment</a>. However, as Feynman points out, it fails to explain why particles such as the electron have the masses they do. "There is no theory that adequately explains these numbers. We use the numbers in all our theories, but we don't understand them – what they are, or where they come from. I believe that from a fundamental point of view, this is a very interesting and serious problem."<sup id="cite_ref-feynbook_2-14" class="reference"><a href="#cite_note-feynbook-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 152">: 152 </span></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation">Mathematical formulation</h2></div>

<div class="mw-heading mw-heading3"><h3 id="QED_action">QED action</h3></div>

<p>Mathematically, QED is an <a href="Abelian_group" title="Abelian group">abelian</a> <a href="Gauge_theory" title="Gauge theory">gauge theory</a> with the symmetry group <a href="U(1)" class="mw-redirect" title="U(1)">U(1)</a>, defined on <a href="Minkowski_space" title="Minkowski space">Minkowski space</a> (flat spacetime). The <a href="Gauge_field" class="mw-redirect" title="Gauge field">gauge field</a>, which mediates the interaction between the charged <a href="Spin_(physics)" title="Spin (physics)">spin-1/2</a> <a href="Field_(physics)" title="Field (physics)">fields</a>, is the <a href="Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>.

The QED <a href="Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> for a spin-1/2 field interacting with the electromagnetic field in natural units gives rise to the action<sup id="cite_ref-Peskin_27-0" class="reference"><a href="#cite_note-Peskin-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 78">: 78 </span></sup>

</p>

<div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>QED Action</b>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(i\gamma ^{\mu }D_{\mu }-m)\,\psi \right]}</annotation>

</semantics>

</math></span><img src="./5c55691a9d87188b6e030994eeb7c7b49c783f11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.993ex; height:6.176ex;" alt="{\displaystyle S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(i\gamma ^{\mu }D_{\mu }-m)\,\psi \right]}" loading="lazy"></span>

</p>

</div>

<p>where

</p>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \gamma ^{\mu }}</annotation>

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</math></span><img src="./cb9ad28fd0224d333e10919fe473cf40e52ac6c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.503ex; height:2.843ex;" alt="{\displaystyle \gamma ^{\mu }}" loading="lazy"></span> are <a href="Dirac_matrices" class="mw-redirect" title="Dirac matrices">Dirac matrices</a>.</li>

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math></span><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"></span> a <a href="Bispinor" title="Bispinor">bispinor</a> <a href="Field_(physics)" title="Field (physics)">field</a> of <a href="Spin-1/2" title="Spin-1/2">spin-1/2</a> particles (e.g. <a href="Electron" title="Electron">electron</a>–<a href="Positron" title="Positron">positron</a> field).</li>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}\equiv \psi ^{\dagger }\gamma ^{0}}</annotation>

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</math></span><img src="./cec578aa82a22f6d9230e2c2ca1dd951ae667827.svg" 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 }}\equiv \psi ^{\dagger }\gamma ^{0}}" loading="lazy"></span>, called "psi-bar", is sometimes referred to as the <a href="Dirac_adjoint" title="Dirac adjoint">Dirac adjoint</a>.</li>

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<annotation encoding="application/x-tex">{\displaystyle D_{\mu }\equiv \partial _{\mu }+ieA_{\mu }+ieB_{\mu }}</annotation>

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</math></span><img src="./0e6ef9f45ce6ee3270019b5b58e4ed71bb8fef1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.111ex; height:2.843ex;" alt="{\displaystyle D_{\mu }\equiv \partial _{\mu }+ieA_{\mu }+ieB_{\mu }}" loading="lazy"></span> is the <a href="Gauge_covariant_derivative" title="Gauge covariant derivative">gauge covariant derivative</a>.

<ul><li><i>e</i> is the <a href="Fine-structure_constant" title="Fine-structure constant">coupling constant</a>, equal to the <a href="Electric_charge" title="Electric charge">electric charge</a> of the bispinor field.</li>

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<annotation encoding="application/x-tex">{\displaystyle A_{\mu }}</annotation>

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</math></span><img src="./9277f5286335ab99c040c9c9151ab752d3bedc49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.967ex; height:2.843ex;" alt="{\displaystyle A_{\mu }}" loading="lazy"></span> is the <a href="Lorentz_covariance" title="Lorentz covariance">covariant</a> <a href="Four-potential" class="mw-redirect" title="Four-potential">four-potential</a> of the electromagnetic field generated by the electron itself. It is also known as a gauge field or a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{U}}(1)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation>

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</math></span><img src="./799f83909c3169d94893468b6aa2e83b40fee010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}" loading="lazy"></span> connection.</li>

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<annotation encoding="application/x-tex">{\displaystyle B_{\mu }}</annotation>

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</math></span><img src="./0fe5f2581724819fecf73ea084522e1278fcde4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.987ex; height:2.843ex;" alt="{\displaystyle B_{\mu }}" loading="lazy"></span> is the external field imposed by external source.</li></ul></li>

<li><i>m</i> is the mass of the electron or positron.</li>

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<annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}</annotation>

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</math></span><img src="./77fbf0c39a13f9706357f83b041774749598ded1.svg" 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 }}" loading="lazy"></span> is the <a href="Electromagnetic_field_tensor" class="mw-redirect" title="Electromagnetic field tensor">electromagnetic field tensor</a>. This is also known as the curvature of the gauge field.</li></ul>

<p>Expanding the covariant derivative reveals a second useful form of the Lagrangian (external field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\mu }}">

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<annotation encoding="application/x-tex">{\displaystyle B_{\mu }}</annotation>

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</math></span><img src="./0fe5f2581724819fecf73ea084522e1278fcde4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.987ex; height:2.843ex;" alt="{\displaystyle B_{\mu }}" loading="lazy"></span> set to zero for simplicity)

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi -ej^{\mu }A_{\mu }}</annotation>

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</math></span><img src="./f580a951cf5936100e563c86d269a15124255f0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.482ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi -ej^{\mu }A_{\mu }}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{\mu }}">

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<annotation encoding="application/x-tex">{\displaystyle j^{\mu }}</annotation>

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</math></span><img src="./55d73bb66707f932f561fce1669b06587038fee8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.208ex; height:2.676ex;" alt="{\displaystyle j^{\mu }}" loading="lazy"></span> is the conserved <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{U}}(1)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation>

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</math></span><img src="./799f83909c3169d94893468b6aa2e83b40fee010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}" loading="lazy"></span> current arising from Noether's theorem. It is written

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<annotation encoding="application/x-tex">{\displaystyle j^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .}</annotation>

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</math></span><img src="./53fe2642d329ac8cca25cb173cee2b184b5712c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:11.566ex; height:3.009ex;" alt="{\displaystyle j^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Equations_of_motion">Equations of motion</h3></div>

<p>Expanding the covariant derivative in the Lagrangian gives

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -e{\bar {\psi }}\gamma ^{\mu }A_{\mu }\psi -m{\bar {\psi }}\psi }</annotation>

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</math></span><img src="./0009a0e14387dd51c4ad178a0c33020b66caec0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.212ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -e{\bar {\psi }}\gamma ^{\mu }A_{\mu }\psi -m{\bar {\psi }}\psi }" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle =-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -m{\bar {\psi }}\psi -ej^{\mu }A_{\mu }.}</annotation>

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</math></span><img src="./b04f2b4ef50e17cf87cb62657b3d464f46a4e942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.18ex; height:5.176ex;" alt="{\displaystyle =-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -m{\bar {\psi }}\psi -ej^{\mu }A_{\mu }.}" loading="lazy"></span></dd></dl>

<p>For simplicity, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\mu }}">

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<annotation encoding="application/x-tex">{\displaystyle B_{\mu }}</annotation>

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</math></span><img src="./0fe5f2581724819fecf73ea084522e1278fcde4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.987ex; height:2.843ex;" alt="{\displaystyle B_{\mu }}" loading="lazy"></span> has been set to zero, with no loss of generality. Alternatively, we can absorb <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\mu }}">

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<annotation encoding="application/x-tex">{\displaystyle B_{\mu }}</annotation>

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</math></span><img src="./0fe5f2581724819fecf73ea084522e1278fcde4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.987ex; height:2.843ex;" alt="{\displaystyle B_{\mu }}" loading="lazy"></span> into a new gauge field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'_{\mu }=A_{\mu }+B_{\mu }}">

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<annotation encoding="application/x-tex">{\displaystyle A'_{\mu }=A_{\mu }+B_{\mu }}</annotation>

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</math></span><img src="./7d22eab5a38da686c63ba8a27a5bbeefc5a3d6c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.859ex; height:2.843ex;" alt="{\displaystyle A'_{\mu }=A_{\mu }+B_{\mu }}" loading="lazy"></span> and relabel the new field as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }.}">

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<annotation encoding="application/x-tex">{\displaystyle A_{\mu }.}</annotation>

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</math></span><img src="./d6a4c7dc4d11f6a817307b45afe51df45f227b1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.613ex; height:2.843ex;" alt="{\displaystyle A_{\mu }.}" loading="lazy"></span>

</p><p>From this Lagrangian, the equations of motion for the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math></span><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }}">

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<annotation encoding="application/x-tex">{\displaystyle A_{\mu }}</annotation>

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</math></span><img src="./9277f5286335ab99c040c9c9151ab752d3bedc49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.967ex; height:2.843ex;" alt="{\displaystyle A_{\mu }}" loading="lazy"></span> fields can be obtained.

</p>

<div class="mw-heading mw-heading4"><h4 id="Equation_of_motion_for_ψ">Equation of motion for ψ</h4></div>

<p>These arise most straightforwardly by considering the Euler-Lagrange equation for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\psi }}}">

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</math></span><img src="./890adebe2730294079a81b7bf08b2fe0f2c59909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.596ex; height:2.843ex;" alt="{\displaystyle {\bar {\psi }}}" loading="lazy"></span>. Since the Lagrangian contains no <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }{\bar {\psi }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }})}}=0}</annotation>

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</math></span><img src="./fb2f786ddcb050b68acd3527bf7bfe254ab56969.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.278ex; height:6.509ex;" alt="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }})}}=0}" loading="lazy"></span></dd></dl>

<p>so the equation of motion can be written

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<annotation encoding="application/x-tex">{\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m)\psi =e\gamma ^{\mu }A_{\mu }\psi .}</annotation>

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</math></span><img src="./27af2c0fb754876dac146c3d95d8e94ec21f6be2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.779ex; height:3.009ex;" alt="{\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m)\psi =e\gamma ^{\mu }A_{\mu }\psi .}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading4"><h4 id="Equation_of_motion_for_Aμ">Equation of motion for A<sub>μ</sub></h4></div>

<ul><li>Using the Euler–Lagrange equation for the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }}">

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</style><table role="presentation" class="numblk" id="math_3" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\nu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }A_{\mu })}}\right)-{\frac {\partial {\mathcal {L}}}{\partial A_{\mu }}}=0,}">

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<p>the derivatives this time are

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<annotation encoding="application/x-tex">{\displaystyle \partial _{\nu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }A_{\mu })}}\right)=\partial _{\nu }\left(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }\right),}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial A_{\mu }}}=-e{\bar {\psi }}\gamma ^{\mu }\psi .}</annotation>

</semantics>

</math></span></span>

</p><p>Substituting back into (<b><a href="#math_3">3</a></b>) leads to

</p>

<msub>

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</msub>

<msup>

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</msup>

<mover>

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<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }F^{\mu \nu }=e{\bar {\psi }}\gamma ^{\nu }\psi }</annotation>

</semantics>

</math></span><img src="./4ed24a4ceea613bcfd53208bc3f89b7037c1e48c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.041ex; height:3.176ex;" alt="{\displaystyle \partial _{\mu }F^{\mu \nu }=e{\bar {\psi }}\gamma ^{\nu }\psi }" loading="lazy"></span></dd></dl>

<p>which can be written in terms of the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{U}}(1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation>

</semantics>

</math></span><img src="./799f83909c3169d94893468b6aa2e83b40fee010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}" loading="lazy"></span> current <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{\mu }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j^{\mu }}</annotation>

</semantics>

</math></span><img src="./55d73bb66707f932f561fce1669b06587038fee8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.208ex; height:2.676ex;" alt="{\displaystyle j^{\mu }}" loading="lazy"></span> as

</p>

<msub>

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<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }F^{\mu \nu }=ej^{\nu }.}</annotation>

</semantics>

</math></span><img src="./8513c5f5b07efa6ba3064994463a40686bd081d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.258ex; height:3.009ex;" alt="{\displaystyle \partial _{\mu }F^{\mu \nu }=ej^{\nu }.}" loading="lazy"></span>

</p>

</div>

<p>Now, if we impose the <a href="Lorenz_gauge_condition" title="Lorenz gauge condition">Lorenz gauge condition</a>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }A^{\mu }=0,}</annotation>

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the equations reduce to

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Box A^{\mu }=ej^{\mu },}</annotation>

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</math></span></span>

which is a <a href="Wave_equation" title="Wave equation">wave equation</a> for the four-potential, the QED version of the classical <a href="Maxwell_equations" class="mw-redirect" title="Maxwell equations">Maxwell equations</a> in the <a href="Lorenz_gauge" class="mw-redirect" title="Lorenz gauge">Lorenz gauge</a>. (The square represents the <a href="Wave_operator" class="mw-redirect" title="Wave operator">wave operator</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box =\partial _{\mu }\partial ^{\mu }}">

<msub>

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</msub>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Box =\partial _{\mu }\partial ^{\mu }}</annotation>

</semantics>

</math></span><img src="./4b76feaf22d39d6bdf0aed58fdd164c2e8e288c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.931ex; height:3.009ex;" alt="{\displaystyle \Box =\partial _{\mu }\partial ^{\mu }}" loading="lazy"></span>.)

</p>

<div class="mw-heading mw-heading3"><h3 id="Interaction_picture">Interaction picture</h3></div>

<p>This theory can be straightforwardly quantized by treating bosonic and fermionic sectors as free. This permits us to build a set of asymptotic states that can be used to start computation of the probability amplitudes for different processes. In order to do so, we have to compute an <a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">evolution operator</a>, which for a given initial state <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |i\rangle }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |i\rangle }</annotation>

</semantics>

</math></span><img src="./5c9bee24e938877d1c7bc0099f6bd886c3f10a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.354ex; height:2.843ex;" alt="{\displaystyle |i\rangle }" loading="lazy"></span> will give a final state <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f|}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle f|}</annotation>

</semantics>

</math></span><img src="./6e99dc122035c8052d1ad9b48a7a305f8b2f3351.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.83ex; height:2.843ex;" alt="{\displaystyle \langle f|}" loading="lazy"></span> in such a way to have<sup id="cite_ref-Peskin_27-1" class="reference"><a href="#cite_note-Peskin-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 5">: 5 </span></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{fi}=\langle f|U|i\rangle .}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{fi}=\langle f|U|i\rangle .}</annotation>

</semantics>

</math></span></span>

</p><p>This technique is also known as the <a href="S-matrix" title="S-matrix">S-matrix</a>. The evolution operator is obtained in the <a href="Interaction_picture" title="Interaction picture">interaction picture</a>, where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:<sup id="cite_ref-Peskin_27-2" class="reference"><a href="#cite_note-Peskin-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 123">: 123 </span></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=e\int d^{3}x\,{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },}">

<msup>

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</msup>

<mover>

</mover>

</mrow>

<msup>

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</msup>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=e\int d^{3}x\,{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },}</annotation>

</semantics>

</math></span></span>

</p><p>Which can also be written in terms of an integral over the interaction Hamiltonian density <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{I}=e{\overline {\psi }}\gamma ^{\mu }\psi A_{\mu }}">

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{I}=e{\overline {\psi }}\gamma ^{\mu }\psi A_{\mu }}</annotation>

</semantics>

</math></span><img src="./181d18a91acebc235c6a7e4f8176209711d46ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.818ex; height:3.676ex;" alt="{\displaystyle {\mathcal {H}}_{I}=e{\overline {\psi }}\gamma ^{\mu }\psi A_{\mu }}" loading="lazy"></span>. Thus, one has<sup id="cite_ref-Peskin_27-3" class="reference"><a href="#cite_note-Peskin-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 86">: 86 </span></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=T\exp \left[-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'\,V(t')\right],}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U=T\exp \left[-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'\,V(t')\right],}</annotation>

</semantics>

</math></span></span>

</p><p>where <i>T</i> is the <a href="Path-ordering" title="Path-ordering">time-ordering</a> operator. This evolution operator only has meaning as a series, and what we get here is a <a href="Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation series</a> with the <a href="Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a> as the development parameter. This series expansion of the probability amplitude <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{fi}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{fi}}</annotation>

</semantics>

</math></span><img src="./ca647d0dfe280bbd24147883a0e11b881eb2a614.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.958ex; height:2.843ex;" alt="{\displaystyle M_{fi}}" loading="lazy"></span> is called the <a href="Dyson_series" title="Dyson series">Dyson series</a>, and is given by:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{fi}=\langle f|U|i\rangle =\left\langle f\left|\sum _{n=0}^{\infty }{\frac {(-i)^{n}}{n!}}\int d^{4}x_{1}\cdots \int d^{4}x_{n}T{\bigg \{}{\mathcal {H}}(x_{1})\cdots {\mathcal {H}}(x_{n}){\bigg \}}\right|i\right\rangle }">

<msub>

</mrow>

</msub>

</mrow>

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<mrow>

<mrow>

<mrow>

<mrow>

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</mrow>

</munderover>

<mfrac>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle M_{fi}=\langle f|U|i\rangle =\left\langle f\left|\sum _{n=0}^{\infty }{\frac {(-i)^{n}}{n!}}\int d^{4}x_{1}\cdots \int d^{4}x_{n}T{\bigg \{}{\mathcal {H}}(x_{1})\cdots {\mathcal {H}}(x_{n}){\bigg \}}\right|i\right\rangle }</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Feynman_diagrams">Feynman diagrams</h3></div>

<p>Despite the conceptual clarity of the Feynman approach to QED, almost no early textbooks follow him in their presentation. When performing calculations, it is much easier to work with the <a href="Fourier_transform" title="Fourier transform">Fourier transforms</a> of the <a href="Propagator" title="Propagator">propagators</a>. Experimental tests of quantum electrodynamics are typically scattering experiments. In scattering theory, particles' <a href="Momentum" title="Momentum">momenta</a> rather than their positions are considered, and it is convenient to think of particles as being created or annihilated when they interact. Feynman diagrams then <i>look</i> the same, but the lines have different interpretations. The electron line represents an electron with a given energy and momentum, with a similar interpretation of the photon line. A vertex diagram represents the annihilation of one electron and the creation of another together with the absorption or creation of a photon, each having specified energies and momenta.

</p><p>Using <a href="Wick's_theorem" title="Wick's theorem">Wick's theorem</a> on the terms of the Dyson series, all the terms of the <a href="S-matrix" title="S-matrix">S-matrix</a> for quantum electrodynamics can be computed through the technique of <a href="Feynman_diagrams" class="mw-redirect" title="Feynman diagrams">Feynman diagrams</a>. In this case, rules for drawing are the following<sup id="cite_ref-Peskin_27-4" class="reference"><a href="#cite_note-Peskin-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 801–802">: 801–802 </span></sup>

</p>

<p>To these rules we must add a further one for closed loops that implies an integration on momenta <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int d^{4}p/(2\pi )^{4}}">

<msup>

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</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \int d^{4}p/(2\pi )^{4}}</annotation>

</semantics>

</math></span><img src="./f3df4d81e11654eb58e228ef021d064b29cf4eb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.767ex; height:3.176ex;" alt="{\textstyle \int d^{4}p/(2\pi )^{4}}" loading="lazy"></span>, since these internal ("virtual") particles are not constrained to any specific energy–momentum, even that usually required by special relativity (see <a href="Propagator#Propagators_in_Feynman_diagrams" title="Propagator">Propagator</a> for details). The signature of the metric <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\mu \nu }}">

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \eta _{\mu \nu }}</annotation>

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</math></span><img src="./9a8fb25ea6ee5d591c2d819519401a871ee04bc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.25ex; height:2.343ex;" alt="{\displaystyle \eta _{\mu \nu }}" loading="lazy"></span> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {diag}}(+---)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\rm {diag}}(+---)}</annotation>

</semantics>

</math></span><img src="./76531041dd606d1ed4706bdb8b883bfae77c4205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.338ex; height:2.843ex;" alt="{\displaystyle {\rm {diag}}(+---)}" loading="lazy"></span>.

</p><p>From them, computations of <a href="Probability_amplitude" title="Probability amplitude">probability amplitudes</a> are straightforwardly given. An example is <a href="Compton_scattering" title="Compton scattering">Compton scattering</a>, with an <a href="Electron" title="Electron">electron</a> and a <a href="Photon" title="Photon">photon</a> undergoing <a href="Elastic_scattering" title="Elastic scattering">elastic scattering</a>. Feynman diagrams are in this case<sup id="cite_ref-Peskin_27-5" class="reference"><a href="#cite_note-Peskin-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 158–159">: 158–159 </span></sup>

</p>

<p>and so we are able to get the corresponding amplitude at the first order of a <a href="Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation series</a> for the <a href="S-matrix" title="S-matrix">S-matrix</a>:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{fi}=(ie)^{2}{\overline {u}}({\vec {p}}',s')\epsilon \!\!\!/\,'({\vec {k}}',\lambda ')^{*}{\frac {p\!\!\!/+k\!\!\!/+m_{e}}{(p+k)^{2}-m_{e}^{2}}}\epsilon \!\!\!/({\vec {k}},\lambda )u({\vec {p}},s)+(ie)^{2}{\overline {u}}({\vec {p}}',s')\epsilon \!\!\!/({\vec {k}},\lambda ){\frac {p\!\!\!/-k\!\!\!/'+m_{e}}{(p-k')^{2}-m_{e}^{2}}}\epsilon \!\!\!/\,'({\vec {k}}',\lambda ')^{*}u({\vec {p}},s),}">

<msub>

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<mover>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle M_{fi}=(ie)^{2}{\overline {u}}({\vec {p}}',s')\epsilon \!\!\!/\,'({\vec {k}}',\lambda ')^{*}{\frac {p\!\!\!/+k\!\!\!/+m_{e}}{(p+k)^{2}-m_{e}^{2}}}\epsilon \!\!\!/({\vec {k}},\lambda )u({\vec {p}},s)+(ie)^{2}{\overline {u}}({\vec {p}}',s')\epsilon \!\!\!/({\vec {k}},\lambda ){\frac {p\!\!\!/-k\!\!\!/'+m_{e}}{(p-k')^{2}-m_{e}^{2}}}\epsilon \!\!\!/\,'({\vec {k}}',\lambda ')^{*}u({\vec {p}},s),}</annotation>

</semantics>

</math></span></span>

</p><p>from which we can compute the <a href="Cross_section_(physics)" title="Cross section (physics)">cross section</a> for this scattering.

</p>

<div class="mw-heading mw-heading3"><h3 id="Nonperturbative_phenomena">Nonperturbative phenomena</h3></div>

<p>The predictive success of quantum electrodynamics largely rests on the use of perturbation theory, expressed in Feynman diagrams. However, quantum electrodynamics also leads to predictions beyond perturbation theory. In the presence of very strong electric fields, it predicts that electrons and positrons will be spontaneously produced, so causing the decay of the field. This process, called the <a href="Schwinger_effect" title="Schwinger effect">Schwinger effect</a>,<sup id="cite_ref-Schwinger_28-0" class="reference"><a href="#cite_note-Schwinger-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> cannot be understood in terms of any finite number of Feynman diagrams and hence is described as <a href="Non-perturbative" title="Non-perturbative">nonperturbative</a>. Mathematically, it can be derived by a semiclassical approximation to the <a href="Path_integral_formulation" title="Path integral formulation">path integral</a> of quantum electrodynamics.

</p>

<div class="mw-heading mw-heading2"><h2 id="Renormalizability">Renormalizability</h2></div>

<p>Higher-order terms can be straightforwardly computed for the evolution operator, but these terms display diagrams containing the following simpler ones<sup id="cite_ref-Peskin_27-6" class="reference"><a href="#cite_note-Peskin-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: ch 10">: ch 10 </span></sup>

</p>

<div class="gallerytext"> One-loop contribution to the <a href="Vacuum_polarization" title="Vacuum polarization">vacuum polarization</a> function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math></span><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"></span></div>

</li>

<div class="gallerytext"> One-loop contribution to the electron <a href="Self-energy" title="Self-energy">self-energy</a> function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math></span><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"></span></div>

</li>

<div class="gallerytext"> One-loop contribution to the <a href="Vertex_function" title="Vertex function">vertex function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span></div>

</li>

</ul>

<p>that, being closed loops, imply the presence of diverging <a href="Integral" title="Integral">integrals</a> having no mathematical meaning. To overcome this difficulty, a technique called <a href="Renormalization" title="Renormalization">renormalization</a> has been devised, producing finite results in very close agreement with experiments. A criterion for the theory being meaningful after renormalization is that the number of diverging diagrams is finite. In this case, the theory is said to be "renormalizable". The reason for this is that to get observables renormalized, one needs a finite number of constants to maintain the predictive value of the theory untouched. This is exactly the case of quantum electrodynamics displaying just three diverging diagrams. This procedure gives observables in very close agreement with experiment as seen e.g. for electron <a href="Gyromagnetic_ratio" title="Gyromagnetic ratio">gyromagnetic ratio</a>.

</p><p>Renormalizability has become an essential criterion for a <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a> to be considered as a viable one. All the theories describing <a href="Fundamental_interaction" title="Fundamental interaction">fundamental interactions</a>, except <a href="Gravitation" class="mw-redirect" title="Gravitation">gravitation</a>, whose quantum counterpart is only conjectural and presently under very active research, are renormalizable theories.

</p>

<div class="mw-heading mw-heading2"><h2 id="Nonconvergence_of_series">Nonconvergence of series</h2></div>

<p>An argument by <a href="Freeman_Dyson" title="Freeman Dyson">Freeman Dyson</a> shows that the <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a> of the perturbation series in QED is zero.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> The basic argument goes as follows: if the <a href="Fine-structure_constant" title="Fine-structure constant">coupling constant</a> were negative, this would be equivalent to the <a href="Coulomb_force_constant" class="mw-redirect" title="Coulomb force constant">Coulomb force constant</a> being negative. This would "reverse" the electromagnetic interaction so that <i>like</i> charges would <i>attract</i> and <i>unlike</i> charges would <i>repel</i>. This would render the vacuum unstable against decay into a cluster of electrons on one side of the universe and a cluster of positrons on the other side of the universe. Because the theory is "sick" for any negative value of the coupling constant, the series does not converge but is at best an <a href="Asymptotic_series" class="mw-redirect" title="Asymptotic series">asymptotic series</a>.

</p><p>From a modern perspective, we say that QED is not well defined as a quantum field theory to arbitrarily high energy.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> The coupling constant runs to infinity at finite energy, signalling a <a href="Landau_pole" title="Landau pole">Landau pole</a>. The problem is essentially that QED appears to suffer from <a href="Quantum_triviality" title="Quantum triviality">quantum triviality</a> issues. This is one of the motivations for embedding QED within a <a href="Grand_Unified_Theory" title="Grand Unified Theory">Grand Unified Theory</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Electrodynamics_in_curved_spacetime">Electrodynamics in curved spacetime</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Maxwell's_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell's equations in curved spacetime</a> and <a href="Dirac_equation_in_curved_spacetime" title="Dirac equation in curved spacetime">Dirac equation in curved spacetime</a></div>

<p>This theory can be extended, at least as a classical field theory, to curved spacetime. This arises similarly to the flat spacetime case, from coupling a free electromagnetic theory to a free fermion theory and including an interaction which promotes the partial derivative in the fermion theory to a gauge-covariant derivative.

</p>

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<ul><li><a href="Abraham%E2%80%93Lorentz_force" title="Abraham–Lorentz force">Abraham–Lorentz force</a></li>

<li><a href="Anomalous_magnetic_moment" class="mw-redirect" title="Anomalous magnetic moment">Anomalous magnetic moment</a></li>

<li><a href="Bhabha_scattering" title="Bhabha scattering">Bhabha scattering</a></li>

<li><a href="Cavity_quantum_electrodynamics" title="Cavity quantum electrodynamics">Cavity quantum electrodynamics</a></li>

<li><a href="Circuit_quantum_electrodynamics" title="Circuit quantum electrodynamics">Circuit quantum electrodynamics</a></li>

<li><a href="Compton_scattering" title="Compton scattering">Compton scattering</a></li>

<li><a href="Euler%E2%80%93Heisenberg_Lagrangian" title="Euler–Heisenberg Lagrangian">Euler–Heisenberg Lagrangian</a></li>

<li><a href="Gupta%E2%80%93Bleuler_formalism" title="Gupta–Bleuler formalism">Gupta–Bleuler formalism</a></li>

<li><a href="Lamb_shift" title="Lamb shift">Lamb shift</a></li>

<li><a href="Landau_pole" title="Landau pole">Landau pole</a></li>

<li><a href="Moeller_scattering" class="mw-redirect" title="Moeller scattering">Moeller scattering</a></li>

<li><a href="Non-relativistic_quantum_electrodynamics" title="Non-relativistic quantum electrodynamics">Non-relativistic quantum electrodynamics</a></li>

<li><a href="Photon_polarization" title="Photon polarization">Photon polarization</a></li>

<li><a href="Positronium" title="Positronium">Positronium</a></li>

<li><a href="Precision_tests_of_QED" title="Precision tests of QED">Precision tests of QED</a></li>

<li><a href="QED_vacuum" title="QED vacuum">QED vacuum</a></li>

<li><i><a href="QED%3A_The_Strange_Theory_of_Light_and_Matter" title="QED: The Strange Theory of Light and Matter">QED: The Strange Theory of Light and Matter</a></i></li>

<li><a href="Quantization_of_the_electromagnetic_field" title="Quantization of the electromagnetic field">Quantization of the electromagnetic field</a></li>

<li><a href="Scalar_electrodynamics" title="Scalar electrodynamics">Scalar electrodynamics</a></li>

<li><a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></li>

<li><a href="Schwinger_model" title="Schwinger model">Schwinger model</a></li>

<li><a href="Schwinger%E2%80%93Dyson_equation" title="Schwinger–Dyson equation">Schwinger–Dyson equation</a></li>

<li><a href="Vacuum_polarization" title="Vacuum polarization">Vacuum polarization</a></li>

<li><a href="Vertex_function" title="Vertex function">Vertex function</a></li>

<li><a href="Wheeler%E2%80%93Feynman_absorber_theory" title="Wheeler–Feynman absorber theory">Wheeler–Feynman absorber theory</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFR._P._Feynman1949" class="citation journal cs1"><a href="Richard_Feynman" title="Richard Feynman">R. P. Feynman</a> (1949). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.76.769">"Space–Time Approach to Quantum Electrodynamics"</a>. <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>76</b> (6): <span class="nowrap">769–</span>89. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1949PhRv...76..769F">1949PhRv...76..769F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.76.769">10.1103/PhysRev.76.769</a></span>.</cite></span>

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<li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFFeynman1950" class="citation journal cs1">Feynman, R. P. (1950). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200914231627/https://authors.library.caltech.edu/3528/">"Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction"</a>. <i>Physical Review</i>. <b>80</b> (3): <span class="nowrap">440–</span>457. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1950PhRv...80..440F">1950PhRv...80..440F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.80.440">10.1103/PhysRev.80.440</a>. Archived from <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://authors.library.caltech.edu/3528/">the original</a></span> on 2020-09-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-09-23</span></span>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFVenkataraman1994" class="citation book cs1"><a href="Ganeshan_Venkataraman" title="Ganeshan Venkataraman">Venkataraman, Ganeshan</a> (1994). <i>Quantum Revolution II — QED: The Jewel of Physics</i>. Universities Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-8173710032</bdi>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation journal cs1"><span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.nature.com/articles/d41586-023-02620-7">"Testing the limits of the standard model of particle physics with a heavy, highly charged ion"</a></span>. <i>Nature</i>. 2023-10-05. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fd41586-023-02620-7">10.1038/d41586-023-02620-7</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/37794145">37794145</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:263670732">263670732</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-23</span></span>.</cite></span>

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<cite id="CITEREFP._A._M._Dirac1927" class="citation journal cs1"><a href="Paul_Dirac" title="Paul Dirac">P. A. M. Dirac</a> (1927). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1927.0039">"The Quantum Theory of the Emission and Absorption of Radiation"</a>. <i><a href="Proceedings_of_the_Royal_Society_of_London_A" class="mw-redirect" title="Proceedings of the Royal Society of London A">Proceedings of the Royal Society of London A</a></i>. <b>114</b> (767): <span class="nowrap">243–</span>65. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1927RSPSA.114..243D">1927RSPSA.114..243D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1927.0039">10.1098/rspa.1927.0039</a></span>.</cite></span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFKuhlmann2020" class="citation web cs1">Kuhlmann, Meinard (Aug 10, 2020) [Jun 22, 2006]. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/quantum-field-theory/qft-history.html">"Quantum Field Theory > The History of QFT"</a>. <i>Stanford Encyclopedia of Philosophy</i>. <a rel="nofollow" class="external text" href="https://archive.today/20240616034116/https://plato.stanford.edu/entries/quantum-field-theory/qft-history.html">Archived</a> from the original on 16 Jun 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-22</span></span>.</cite></span>

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<cite id="CITEREFE._Fermi1932" class="citation journal cs1"><a href="Enrico_Fermi" title="Enrico Fermi">E. Fermi</a> (1932). "Quantum Theory of Radiation". <i><a href="Reviews_of_Modern_Physics" title="Reviews of Modern Physics">Reviews of Modern Physics</a></i>. <b>4</b> (1): <span class="nowrap">87–</span>132. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1932RvMP....4...87F">1932RvMP....4...87F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.4.87">10.1103/RevModPhys.4.87</a>.</cite></span>

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<li id="cite_note-bloch-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-bloch_9-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBlochNordsieck1937" class="citation journal cs1"><a href="Felix_Bloch" title="Felix Bloch">Bloch, F.</a>; <a href="Arnold_Nordsieck" title="Arnold Nordsieck">Nordsieck, A.</a> (1937). "Note on the Radiation Field of the Electron". <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>52</b> (2): <span class="nowrap">54–</span>59. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1937PhRv...52...54B">1937PhRv...52...54B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.52.54">10.1103/PhysRev.52.54</a>.</cite></span>

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<li id="cite_note-weisskopf-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-weisskopf_10-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFV._F._Weisskopf1939" class="citation journal cs1"><a href="Victor_Weisskopf" title="Victor Weisskopf">V. F. Weisskopf</a> (1939). "On the Self-Energy and the Electromagnetic Field of the Electron". <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>56</b> (1): <span class="nowrap">72–</span>85. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1939PhRv...56...72W">1939PhRv...56...72W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.56.72">10.1103/PhysRev.56.72</a>.</cite></span>

</li>

<li id="cite_note-oppenheimer-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-oppenheimer_11-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFR._Oppenheimer1930" class="citation journal cs1"><a href="J._Robert_Oppenheimer" title="J. Robert Oppenheimer">R. Oppenheimer</a> (1930). "Note on the Theory of the Interaction of Field and Matter". <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>35</b> (5): <span class="nowrap">461–</span>77. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1930PhRv...35..461O">1930PhRv...35..461O</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.35.461">10.1103/PhysRev.35.461</a>.</cite></span>

</li>

<cite id="CITEREFLambRetherford1947" class="citation journal cs1"><a href="Willis_Lamb" title="Willis Lamb">Lamb, Willis</a>; <a href="Robert_Retherford" title="Robert Retherford">Retherford, Robert</a> (1947). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.72.241">"Fine Structure of the Hydrogen Atom by a Microwave Method"</a>. <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>72</b> (3): <span class="nowrap">241–</span>43. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1947PhRv...72..241L">1947PhRv...72..241L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.72.241">10.1103/PhysRev.72.241</a></span>.</cite></span>

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<cite id="CITEREFFoleyKusch1948" class="citation journal cs1"><a href="Henry_M._Foley" title="Henry M. Foley">Foley, H.M.</a>; <a href="Polykarp_Kusch" title="Polykarp Kusch">Kusch, P.</a> (1948). "On the Intrinsic Moment of the Electron". <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>73</b> (3): 412. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1948PhRv...73..412F">1948PhRv...73..412F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.73.412">10.1103/PhysRev.73.412</a>.</cite></span>

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<cite id="CITEREFH._Bethe1947" class="citation journal cs1"><a href="Hans_Bethe" title="Hans Bethe">H. Bethe</a> (1947). "The Electromagnetic Shift of Energy Levels". <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>72</b> (4): <span class="nowrap">339–</span>41. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1947PhRv...72..339B">1947PhRv...72..339B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.72.339">10.1103/PhysRev.72.339</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120434909">120434909</a>.</cite></span>

</li>

<cite id="CITEREFSchweber1994" class="citation book cs1"><a href="Silvan_Schweber" class="mw-redirect" title="Silvan Schweber">Schweber, Silvan</a> (1994). <a rel="nofollow" class="external text" href="https://archive.org/details/qedmenwhomadeitd0000schw/page/230">"Chapter 5"</a>. <i>QED and the Men Who Did it: Dyson, Feynman, Schwinger, and Tomonaga</i>. Princeton University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/qedmenwhomadeitd0000schw/page/230">230</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-03327-3</bdi>.</cite></span>

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<cite id="CITEREFS._Tomonaga1946" class="citation journal cs1"><a href="Sin-Itiro_Tomonaga" class="mw-redirect" title="Sin-Itiro Tomonaga">S. Tomonaga</a> (1946). <a rel="nofollow" class="external text" href="https://doi.org/10.1143%2FPTP.1.27">"On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields"</a>. <i><a href="Progress_of_Theoretical_Physics" class="mw-redirect" title="Progress of Theoretical Physics">Progress of Theoretical Physics</a></i>. <b>1</b> (2): <span class="nowrap">27–</span>42. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1946PThPh...1...27T">1946PThPh...1...27T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1143%2FPTP.1.27">10.1143/PTP.1.27</a></span>.</cite></span>

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<cite id="CITEREFJ._Schwinger1948" class="citation journal cs1"><a href="Julian_Schwinger" title="Julian Schwinger">J. Schwinger</a> (1948). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.73.416">"On Quantum-Electrodynamics and the Magnetic Moment of the Electron"</a>. <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>73</b> (4): <span class="nowrap">416–</span>17. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1948PhRv...73..416S">1948PhRv...73..416S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.73.416">10.1103/PhysRev.73.416</a></span>.</cite></span>

</li>

<cite id="CITEREFJ._Schwinger1948" class="citation journal cs1"><a href="Julian_Schwinger" title="Julian Schwinger">J. Schwinger</a> (1948). "Quantum Electrodynamics. I. A Covariant Formulation". <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>74</b> (10): <span class="nowrap">1439–</span>61. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1948PhRv...74.1439S">1948PhRv...74.1439S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.74.1439">10.1103/PhysRev.74.1439</a>.</cite></span>

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<li id="cite_note-feynman2-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-feynman2_19-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFR._P._Feynman1949" class="citation journal cs1"><a href="Richard_Feynman" title="Richard Feynman">R. P. Feynman</a> (1949). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220809030941/https://authors.library.caltech.edu/3520/">"The Theory of Positrons"</a>. <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>76</b> (6): <span class="nowrap">749–</span>59. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1949PhRv...76..749F">1949PhRv...76..749F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.76.749">10.1103/PhysRev.76.749</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120117564">120117564</a>. Archived from <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://authors.library.caltech.edu/3520/">the original</a></span> on 2022-08-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2021-11-19</span></span>.</cite></span>

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<cite id="CITEREFR._P._Feynman1950" class="citation journal cs1"><a href="Richard_Feynman" title="Richard Feynman">R. P. Feynman</a> (1950). <a rel="nofollow" class="external text" href="https://authors.library.caltech.edu/3528/1/FEYpr50.pdf">"Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction"</a> <span class="cs1-format">(PDF)</span>. <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>80</b> (3): <span class="nowrap">440–</span>57. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1950PhRv...80..440F">1950PhRv...80..440F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.80.440">10.1103/PhysRev.80.440</a>.</cite></span>

</li>

<cite id="CITEREFF._Dyson1949" class="citation journal cs1"><a href="Freeman_Dyson" title="Freeman Dyson">F. Dyson</a> (1949). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.75.486">"The Radiation Theories of Tomonaga, Schwinger, and Feynman"</a>. <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>75</b> (3): <span class="nowrap">486–</span>502. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1949PhRv...75..486D">1949PhRv...75..486D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.75.486">10.1103/PhysRev.75.486</a></span>.</cite></span>

</li>

<cite id="CITEREFF._Dyson1949" class="citation journal cs1"><a href="Freeman_Dyson" title="Freeman Dyson">F. Dyson</a> (1949). "The S Matrix in Quantum Electrodynamics". <i><a href="Physical_Review" title="Physical Review">Physical Review</a></i>. <b>75</b> (11): <span class="nowrap">1736–</span>55. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1949PhRv...75.1736D">1949PhRv...75.1736D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.75.1736">10.1103/PhysRev.75.1736</a>.</cite></span>

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<li id="cite_note-nobel65-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-nobel65_23-0">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://nobelprize.org/nobel_prizes/physics/laureates/1965/index.html">"The Nobel Prize in Physics 1965"</a>. Nobel Foundation<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-10-09</span></span>.</cite></span>

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<li id="cite_note-:1-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_24-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=Ci86Aps7CMo"><i>The story of the positron - Paul Dirac (1975)</i></a><span class="reference-accessdate">, retrieved <span class="nowrap">2023-07-19</span></span></cite></span>

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<cite id="CITEREFGuralnikHagenKibble1964" class="citation journal cs1">Guralnik, G. S.; Hagen, C. R.; Kibble, T. W. B. (1964). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.13.585">"Global Conservation Laws and Massless Particles"</a>. <i><a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>13</b> (20): <span class="nowrap">585–</span>87. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964PhRvL..13..585G">1964PhRvL..13..585G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.13.585">10.1103/PhysRevLett.13.585</a></span>.</cite></span>

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<cite id="CITEREFGuralnik2009" class="citation journal cs1">Guralnik, G. S. (2009). "The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles". <i><a href="International_Journal_of_Modern_Physics_A" class="mw-redirect" title="International Journal of Modern Physics A">International Journal of Modern Physics A</a></i>. <b>24</b> (14): <span class="nowrap">2601–</span>27. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0907.3466">0907.3466</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009IJMPA..24.2601G">2009IJMPA..24.2601G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0217751X09045431">10.1142/S0217751X09045431</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16298371">16298371</a>.</cite></span>

</li>

<li id="cite_note-Peskin-27"><span class="mw-cite-backlink">^ <a href="#cite_ref-Peskin_27-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Peskin_27-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Peskin_27-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Peskin_27-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Peskin_27-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Peskin_27-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Peskin_27-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFPeskinSchroeder1995" class="citation book cs1">Peskin, Michael; Schroeder, Daniel (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoqu0000pesk"><i>An introduction to quantum field theory</i></a></span> (Reprint ed.). Westview Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0201503975</bdi>.</cite></span>

</li>

<li id="cite_note-Schwinger-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-Schwinger_28-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFSchwinger1951" class="citation journal cs1">Schwinger, Julian (1951-06-01). "On Gauge Invariance and Vacuum Polarization". <i>Physical Review</i>. <b>82</b> (5). American Physical Society (APS): <span class="nowrap">664–</span>679. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1951PhRv...82..664S">1951PhRv...82..664S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrev.82.664">10.1103/physrev.82.664</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-899X">0031-899X</a>.</cite></span>

</li>

<li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><cite id="CITEREFKinoshita1997" class="citation web cs1">Kinoshita, Toichiro (June 5, 1997). <a rel="nofollow" class="external text" href="http://www.lassp.cornell.edu/sethna/Cracks/QED.html">"Quantum Electrodynamics has Zero Radius of Convergence Summarized from Toichiro Kinoshita"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">May 6,</span> 2017</span>.</cite></span>

</li>

<li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><cite id="CITEREFEspriu_and_Tarrach1996" class="citation journal cs1">Espriu and Tarrach (Apr 30, 1996). "Ambiguities in QED: Renormalons versus Triviality". <i>Physics Letters B</i>. <b>383</b> (4): <span class="nowrap">482–</span>486. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ph/9604431">hep-ph/9604431</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996PhLB..383..482E">1996PhLB..383..482E</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2896%2900779-4">10.1016/0370-2693(96)00779-4</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119095192">119095192</a>.</cite></span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Books">Books</h3></div>

<ul><li><cite id="CITEREFBerestetskiiLifshitzPitaevskii1982" class="citation book cs1">Berestetskii, V. B.; <a href="Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, E. M.</a>; <a href="Lev_Pitaevskii" title="Lev Pitaevskii">Pitaevskii, L. P.</a> (1982). <a href="Course_of_Theoretical_Physics" title="Course of Theoretical Physics"><i>Course of Theoretical Physics, Volume 4: Quantum Electrodynamics</i></a> (2 ed.). Elsevier. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7506-3371-0</bdi>.</cite></li>

<li><cite id="CITEREFDe_Broglie1925" class="citation book cs1"><a href="Louis_de_Broglie" title="Louis de Broglie">De Broglie, L.</a> (1925). <i>Recherches sur la theorie des quanta [Research on quantum theory]</i>. France: Wiley-Interscience.</cite></li>

<li><cite id="CITEREFFeynman1998" class="citation book cs1"><a href="Richard_Feynman" title="Richard Feynman">Feynman, R. P.</a> (1998). <i>Quantum Electrodynamics</i> (New ed.). Westview Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-36075-2</bdi>.</cite></li>

<li><cite id="CITEREFGreinerBromleyMüller2000" class="citation book cs1">Greiner, W.; Bromley, D. A.; Müller, B. (2000). <i>Gauge Theory of Weak Interactions</i>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-67672-0</bdi>.</cite></li>

<li><cite id="CITEREFJauchRohrlich1980" class="citation book cs1">Jauch, J. M.; Rohrlich, F. (1980). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/theoryofphotonse0000jauc"><i>The Theory of Photons and Electrons</i></a></span>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-07295-1</bdi>.</cite></li>

<li><cite id="CITEREFKane1993" class="citation book cs1">Kane, G. L. (1993). <i>Modern Elementary Particle Physics</i>. Westview Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-62460-1</bdi>.</cite></li>

<li><cite id="CITEREFMiller1995" class="citation book cs1">Miller, A. I. (1995). <i>Early Quantum Electrodynamics: A Sourcebook</i>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-56891-3</bdi>.</cite></li>

<li><cite id="CITEREFMilonni1994" class="citation book cs1"><a href="Peter_W._Milonni" title="Peter W. Milonni">Milonni, P. W.</a> (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uPHJCgAAQBAJ"><i>The Quantum Vacuum: An Introduction to Quantum Electrodynamics</i></a>. Boston: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0124980805</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/93029780">93029780</a>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/422797902">422797902</a>.</cite></li>

<li><cite id="CITEREFSchweber1994" class="citation book cs1">Schweber, S. S. (1994). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/qedmenwhomadeitd0000schw"><i>QED and the Men Who Made It</i></a></span>. Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-03327-3</bdi>.</cite></li>

<li><cite id="CITEREFSchwinger1958" class="citation book cs1"><a href="Julian_Schwinger" title="Julian Schwinger">Schwinger, J.</a> (1958). <i>Selected Papers on Quantum Electrodynamics</i>. Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-60444-2</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></li>

<li><cite id="CITEREFTannoudji-CohenDupont-RocGrynberg1997" class="citation book cs1"><a href="Claude_Cohen-Tannoudji" title="Claude Cohen-Tannoudji">Tannoudji-Cohen, C.</a>; Dupont-Roc, Jacques; Grynberg, Gilbert (1997). <i>Photons and Atoms: Introduction to Quantum Electrodynamics</i>. Wiley-Interscience. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-18433-1</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Journals">Journals</h3></div>

<ul><li><cite id="CITEREFDudleyKwan1996" class="citation journal cs1">Dudley, J.M.; Kwan, A.M. (1996). "Richard Feynman's popular lectures on quantum electrodynamics: The 1979 Robb Lectures at Auckland University". <i>American Journal of Physics</i>. <b>64</b> (6): <span class="nowrap">694–</span>98. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996AmJPh..64..694D">1996AmJPh..64..694D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.18234">10.1119/1.18234</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.nobelprize.org/prizes/physics/1965/feynman/lecture/">Feynman's Nobel Prize lecture describing the evolution of QED and his role in it</a></li>

<li><a rel="nofollow" class="external text" href="http://www.vega.org.uk/video/subseries/8">Feynman's New Zealand lectures on QED for non-physicists</a></li>

<li><a rel="nofollow" class="external text" href="http://qed.wikina.org/">The Strange Theory of Light | Animation of Feynman pictures light by QED</a> – Animations demonstrating QED</li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_electrodynamics196" 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"><div id="Quantum_electrodynamics196" style="font-size:114%;margin:0 4em"></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formalism</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="Euler%E2%80%93Heisenberg_Lagrangian" title="Euler–Heisenberg Lagrangian">Euler–Heisenberg Lagrangian</a></li>

<li><a href="Feynman_diagram" title="Feynman diagram">Feynman diagram</a></li>

<li><a href="Gupta%E2%80%93Bleuler_formalism" title="Gupta–Bleuler formalism">Gupta–Bleuler formalism</a></li>

<li><a href="Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Particles</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="Dual_photon" title="Dual photon">Dual photon</a></li>

<li><a href="Electron" title="Electron">Electron</a></li>

<li><a href="Faddeev%E2%80%93Popov_ghost" title="Faddeev–Popov ghost">Faddeev–Popov ghost</a></li>

<li><a href="Photon" title="Photon">Photon</a></li>

<li><a href="Positron" title="Positron">Positron</a></li>

<li><a href="Positronium" title="Positronium">Positronium</a></li>

<li><a href="Virtual_particle" title="Virtual particle">Virtual particles</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">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="Anomalous_magnetic_dipole_moment" title="Anomalous magnetic dipole moment">Anomalous magnetic dipole moment</a></li>

<li><a href="Furry's_theorem" title="Furry's theorem">Furry's theorem</a></li>

<li><a href="Klein%E2%80%93Nishina_formula" title="Klein–Nishina formula">Klein–Nishina formula</a></li>

<li><a href="Landau_pole" title="Landau pole">Landau pole</a></li>

<li><a href="QED_vacuum" title="QED vacuum">QED vacuum</a></li>

<li><a href="Self-energy" title="Self-energy">Self-energy</a></li>

<li><a href="Schwinger_limit" title="Schwinger limit">Schwinger limit</a></li>

<li><a href="Uehling_potential" title="Uehling potential">Uehling potential</a></li>

<li><a href="Vacuum_polarization" title="Vacuum polarization">Vacuum polarization</a></li>

<li><a href="Vertex_function" title="Vertex function">Vertex function</a></li>

<li><a href="Ward%E2%80%93Takahashi_identity" title="Ward–Takahashi identity">Ward–Takahashi identity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Processes</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="Bhabha_scattering" title="Bhabha scattering">Bhabha scattering</a></li>

<li><a href="Breit%E2%80%93Wheeler_process" title="Breit–Wheeler process">Breit–Wheeler process</a></li>

<li><a href="Bremsstrahlung" title="Bremsstrahlung">Bremsstrahlung</a></li>

<li><a href="Compton_scattering" title="Compton scattering">Compton scattering</a></li>

<li><a href="Delbr%C3%BCck_scattering" title="Delbrück scattering">Delbrück scattering</a></li>

<li><a href="Lamb_shift" title="Lamb shift">Lamb shift</a></li>

<li><a href="M%C3%B8ller_scattering" title="Møller scattering">Møller scattering</a></li>

<li><a href="Schwinger_effect" title="Schwinger effect">Schwinger effect</a></li>

<li><a href="Two-photon_physics" title="Two-photon physics">Photon-photon scattering</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i>See also:</i> <span class="noviewer" typeof="mw:File"><span title="Template"></span></span> Template:Quantum mechanics topics</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Quantum_field_theories202" 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"><div id="Quantum_field_theories202" style="font-size:114%;margin:0 4em"><a href="Quantum_field_theory" title="Quantum field theory">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="Algebraic_quantum_field_theory" title="Algebraic quantum field theory">Algebraic QFT</a></li>

<li><a href="Axiomatic_quantum_field_theory" title="Axiomatic quantum field theory">Axiomatic QFT</a></li>

<li><a href="Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></li>

<li><a href="Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li>

<li><a href="Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative QFT</a></li>

<li><a href="Gauge_theory" title="Gauge theory">Gauge theory</a></li>

<li><a href="Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">QFT in curved spacetime</a></li>

<li><a href="String_theory" title="String theory">String theory</a></li>

<li><a href="Supergravity" title="Supergravity">Supergravity</a></li>

<li><a href="Thermal_quantum_field_theory" title="Thermal quantum field theory">Thermal QFT</a></li>

<li><a href="Topological_quantum_field_theory" title="Topological quantum field theory">Topological QFT</a></li>

<li><a href="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="Born%E2%80%93Infeld_model" title="Born–Infeld model">Born–Infeld</a></li>

<li><a href="Euler%E2%80%93Heisenberg_Lagrangian" title="Euler–Heisenberg Lagrangian">Euler–Heisenberg</a></li>

<li><a href="Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau</a></li>

<li><a href="Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma</a></li>

<li><a href="Proca_action" title="Proca action">Proca</a></li>

<li><a href="Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li>

<li><a href="Quartic_interaction" title="Quartic interaction">Quartic interaction</a></li>

<li><a href="Scalar_electrodynamics" title="Scalar electrodynamics">Scalar electrodynamics</a></li>

<li><a href="Scalar_chromodynamics" title="Scalar chromodynamics">Scalar chromodynamics</a></li>

<li><a href="Soler_model" title="Soler model">Soler</a></li>

<li><a href="Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills</a></li>

<li><a href="Yang%E2%80%93Mills%E2%80%93Higgs_equations" title="Yang–Mills–Higgs equations">Yang–Mills–Higgs</a></li>

<li><a href="Yukawa_interaction" class="mw-redirect" 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="Two-dimensional_Yang%E2%80%93Mills_theory" title="Two-dimensional Yang–Mills theory">2D Yang–Mills</a></li>

<li><a href="Bullough%E2%80%93Dodd_model" title="Bullough–Dodd model">Bullough–Dodd</a></li>

<li><a href="Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu</a></li>

<li><a href="Schwinger_model" title="Schwinger model">Schwinger</a></li>

<li><a href="Sine-Gordon_equation" title="Sine-Gordon equation">Sine-Gordon</a></li>

<li><a href="Thirring_model" title="Thirring model">Thirring</a></li>

<li><a href="Thirring%E2%80%93Wess_model" title="Thirring–Wess model">Thirring–Wess</a></li>

</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="Massless_free_scalar_bosons_in_two_dimensions" title="Massless free scalar bosons in two dimensions">2D free massless scalar</a></li>

<li><a href="Liouville_field_theory" title="Liouville field theory">Liouville</a></li>

<li><a href="Minimal_model_(physics)" title="Minimal model (physics)">Minimal</a></li>

<li><a href="Polyakov_action" title="Polyakov action">Polyakov</a></li>

<li><a href="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">

<li><a href="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="Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten</a></li>

<li><a href="Super_QCD" title="Super QCD">Super QCD</a></li>

<li><a href="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">

<li><a href="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">

<li><a href="Higher-dimensional_supergravity" title="Higher-dimensional supergravity">Higher dimensional</a></li>

</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">

<li><a href="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="Chiral_model" title="Chiral model">Chiral</a></li>

<li><a href="Fermi's_interaction" title="Fermi's interaction">Fermi</a></li>

<li><a href="Nambu%E2%80%93Jona-Lasinio_model" title="Nambu–Jona-Lasinio model">Nambu–Jona-Lasinio</a></li>

<li><a href="Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li>

<li><a href="Standard_Model" title="Standard Model">Standard Model</a></li>

<li><a href="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="Casimir_effect" title="Casimir effect">Casimir effect</a></li>

<li><a href="Cosmic_string" title="Cosmic string">Cosmic string</a></li>

<li><a href="History_of_quantum_field_theory" title="History of quantum field theory">History</a></li>

<li><a href="Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li>

<li><a href="Loop_quantum_cosmology" title="Loop quantum cosmology">Loop quantum cosmology</a></li>

<li><a href="On_shell_and_off_shell" title="On shell and off shell">On shell and off shell</a></li>

<li><a href="Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li>

<li><a href="Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li>

<li><a href="Quantum_foam" title="Quantum foam">Quantum foam</a></li>

<li><a href="Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuations</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_hadrodynamics" title="Quantum hadrodynamics">Quantum hadrodynamics</a></li>

<li><a href="Quantum_hydrodynamics" title="Quantum hydrodynamics">Quantum hydrodynamics</a></li>

<li><a href="Quantum_information" title="Quantum information">Quantum information</a></li>

<li><a href="Quantum_information_science" title="Quantum information science">Quantum information science</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>

<li><a href="Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i>See also:</i> <span class="noviewer" typeof="mw:File"><span title="Template"></span></span> Template:Quantum mechanics topics</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Quantum_mechanics328" 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"><div id="Quantum_mechanics328" style="font-size:114%;margin:0 4em"><a href="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="Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li>

<li><a href="History_of_quantum_mechanics" title="History of quantum mechanics">History</a>

<ul><li><a href="Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li>

<li><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li>

<li><a href="Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li>

<li><a href="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">

<li><a href="Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li>

<li><a href="Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li>

<li><a href="Density_matrix" title="Density matrix">Density matrix</a></li>

<li><a href="Energy_level" title="Energy level">Energy level</a>

<ul><li><a href="Ground_state" title="Ground state">Ground state</a></li>

<li><a href="Excited_state" title="Excited state">Excited state</a></li>

<li><a href="Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li>

<li><a href="Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li>

<li><a href="Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li>

<li><a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li>

<li><a href="Wave_interference" title="Wave interference">Interference</a></li>

<li><a href="Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li>

<li><a href="Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li>

<li><a href="Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li>

<li><a href="Quantum_state" title="Quantum state">Quantum state</a></li>

<li><a href="Quantum_superposition" title="Quantum superposition">Superposition</a></li>

<li><a href="Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li>

<li><a href="Scattering#Theory" title="Scattering">Scattering theory</a></li>

<li><a href="Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li>

<li><a href="Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li>

<li><a href="Wave_function" title="Wave function">Wave function</a>

<ul><li><a href="Wave_function_collapse" title="Wave function collapse">Collapse</a></li>

<li><a href="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="Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li>

<li><a href="Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li>

<li><a href="Interaction_picture" title="Interaction picture">Interaction</a></li>

<li><a href="Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li>

<li><a href="Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li>

<li><a href="Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li>

<li><a href="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="Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li>

<li><a href="Dirac_equation" title="Dirac equation">Dirac</a></li>

<li><a href="Majorana_equation" title="Majorana equation">Majorana</a></li>

<li><a href="Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li>

<li><a href="Pauli_equation" title="Pauli equation">Pauli</a></li>

<li><a href="Rydberg_formula" title="Rydberg formula">Rydberg</a></li>

<li><a href="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="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="Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li>

<li><a href="Consciousness_causes_collapse" title="Consciousness causes collapse">Consciousness causes collapse</a></li>

<li><a href="Consistent_histories" title="Consistent histories">Consistent histories</a></li>

<li><a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li>

<li><a href="De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li>

<li><a href="Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li>

<li><a href="Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a>

<ul><li><a href="Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a>

<ul><li><a href="Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li>

<li><a href="Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li>

<li><a href="Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li>

<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>

<li><a href="Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li>

<li><a href="Transactional_interpretation" title="Transactional interpretation">Transactional</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">

<li><a href="Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li>

<li><a href="Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li>

<li><a href="Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li>

<li><a href="Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li>

<li><a href="Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li>

<li><a href="Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li>

<li><a href="Popper's_experiment" title="Popper's experiment">Popper</a></li>

<li><a href="Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li>

<li><a href="Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li>

<li><a href="Wheeler's_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="Nanotechnology" title="Nanotechnology">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="Quantum_biology" title="Quantum biology">Quantum biology</a></li>

<li><a href="Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li>

<li><a href="Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li>

<li><a href="Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li>

<li><a href="Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li>

<li><a href="Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li>

<li><a href="Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li>

<li><a href="Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li>

<li><a href="Quantum_mind" title="Quantum mind">Quantum mind</a></li>

<li><a href="Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li>

<li><a href="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="Quantum_engineering" title="Quantum engineering">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="Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li>

<li><a href="Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li>

<li><a href="Quantum_bus" title="Quantum bus">Quantum bus</a></li>

<li><a href="Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a>

<ul><li><a href="Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li>

<li><a href="Quantum_channel" title="Quantum channel">Quantum channel</a></li>

<li><a href="Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li>

<li><a href="Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li>

<li><a href="Quantum_computing" title="Quantum computing">Quantum computing</a>

<ul><li><a href="Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li>

<li><a href="Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li>

<li><a href="Quantum_optics#Quantum_electronics" title="Quantum optics">Quantum electronics</a></li>

<li><a href="Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li>

<li><a href="Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li>

<li><a href="Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li>

<li><a href="Quantum_information" title="Quantum information">Quantum information</a></li>

<li><a href="Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li>

<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>

<li><a href="Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li>

<li><a href="Quantum_machine" title="Quantum machine">Quantum machine</a></li>

<li><a href="Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li>

<li><a href="Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li>

<li><a href="Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li>

<li><a href="Quantum_network" title="Quantum network">Quantum network</a></li>

<li><a href="Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li>

<li><a href="Quantum_optics" title="Quantum optics">Quantum optics</a></li>

<li><a href="Quantum_programming" title="Quantum programming">Quantum programming</a></li>

<li><a href="Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li>

<li><a href="Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li>

<li><a href="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="Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li>

<li><a href="Casimir_effect" title="Casimir effect">Casimir effect</a></li>

<li><a href="Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li>

<li><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theory</a>

<ul><li><a href="History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li>

<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li>

<li><a href="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="Schr%C3%B6dinger's_cat" title="Schrödinger's cat">Schrödinger's cat</a>

<ul><li><a href="Schr%C3%B6dinger's_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li>

<li><a href="Wigner's_friend" title="Wigner's friend">Wigner's friend</a></li>

<li><a href="Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li>

<li><a href="Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Standard_Model377" 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"><div id="Standard_Model377" style="font-size:114%;margin:0 4em"><a href="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="Particle_physics" title="Particle physics">Particle physics</a>

<ul><li><a href="Fermion" title="Fermion">Fermions</a></li>

<li><a href="Gauge_boson" title="Gauge boson">Gauge boson</a></li>

<li><a href="Higgs_boson" title="Higgs boson">Higgs boson</a></li></ul></li>

<li><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li>

<li><a href="Gauge_theory" title="Gauge theory">Gauge theory</a></li>

<li><a href="Strong_interaction" title="Strong interaction">Strong interaction</a>

<ul><li><a href="Color_charge" title="Color charge">Color charge</a></li>

<li><a href="Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li>

<li><a href="Quark_model" title="Quark model">Quark model</a></li></ul></li>

<li><a href="Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a>

<ul><li><a href="Weak_interaction" title="Weak interaction">Weak interaction</a></li>

<li><a href="Fermi's_interaction" title="Fermi's interaction">Fermi's interaction</a></li>

<li><a href="Weak_hypercharge" title="Weak hypercharge">Weak hypercharge</a></li>

<li><a href="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><span typeof="mw:File"><span></span></span></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="Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">CKM matrix</a></li>

<li><a href="Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li>

<li><a href="Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li>

<li><a href="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="Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Beyond the<br>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="Hierarchy_problem" title="Hierarchy problem">Hierarchy problem</a></li>

<li><a href="Dark_matter" title="Dark matter">Dark matter</a></li>

<li><a href="Cosmological_constant" title="Cosmological constant">Cosmological constant</a>

<ul><li><a href="Cosmological_constant_problem" title="Cosmological constant problem">problem</a></li></ul></li>

<li><a href="CP_violation" title="CP violation">Strong CP problem</a></li>

<li><a href="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="Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li>

<li><a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li>

<li><a href="Grand_Unified_Theory" title="Grand Unified Theory">Grand Unified Theory</a></li>

<li><a href="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="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">

<li><a href="Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li>

<li><a href="Split_supersymmetry" title="Split supersymmetry">Split supersymmetry</a></li>

<li><a href="Supergravity" title="Supergravity">Supergravity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="String_theory" title="String theory">String theory</a></li>

<li><a href="Superstring_theory" title="Superstring theory">Superstring theory</a></li>

<li><a href="Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li>

<li><a href="Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li>

<li><a href="Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</a></li>

<li><a href="Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a></li>

<li><a href="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="Laboratori_Nazionali_del_Gran_Sasso" title="Laboratori Nazionali del Gran Sasso">Gran Sasso</a></li>

<li><a href="Super-Kamiokande" title="Super-Kamiokande">Super-K</a></li>

<li><a href="Tevatron" title="Tevatron">Tevatron</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> <b>Category</b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Standard_Model_(physics)" class="extiw external" title="commons:Category:Standard Model (physics)">Commons</a></b></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Major_branches_of_physics48" 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"><div id="Major_branches_of_physics48" style="font-size:114%;margin:0 4em">Major <a href="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">

<li><a href="Applied_physics" title="Applied physics">Applied</a>

<ul><li><a href="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="Experimental_physics" title="Experimental physics">Experimental</a></li>

<li><a href="Theoretical_physics" title="Theoretical physics">Theoretical</a>

<ul><li><a href="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="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="Classical_mechanics" title="Classical mechanics">Classical mechanics</a>

<ul><li><a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newtonian</a></li>

<li><a href="Analytical_mechanics" title="Analytical mechanics">Analytical</a></li>

<li><a href="Celestial_mechanics" title="Celestial mechanics">Celestial</a></li>

<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum</a></li></ul></li>

<li><a href="Acoustics" title="Acoustics">Acoustics</a></li>

<li><a href="Classical_electromagnetism" title="Classical electromagnetism">Classical electromagnetism</a></li>

<li><a href="Classical_optics" class="mw-redirect" title="Classical optics">Classical optics</a>

<li><a href="Thermodynamics" title="Thermodynamics">Thermodynamics</a>

<ul><li><a href="Statistical_mechanics" title="Statistical mechanics">Statistical</a></li>

<li><a href="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="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="Relativistic_mechanics" title="Relativistic mechanics">Relativistic mechanics</a>

<ul><li><a href="Special_relativity" title="Special relativity">Special</a></li>

<li><a href="General_relativity" title="General relativity">General</a></li></ul></li>

<li><a href="Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li>

<li><a href="Particle_physics" title="Particle physics">Particle physics</a></li>

<li><a href="Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li>

<li><a href="Atomic%2C_molecular%2C_and_optical_physics" title="Atomic, molecular, and optical physics">Atomic, molecular, and optical physics</a>

<ul><li><a href="Atomic_physics" title="Atomic physics">Atomic</a></li>

<li><a href="Molecular_physics" title="Molecular physics">Molecular</a></li>

<li><a href="Optics#Modern_optics" title="Optics">Modern optics</a></li></ul></li>

<li><a href="Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a>

<ul><li><a href="Solid-state_physics" title="Solid-state physics">Solid-state physics</a></li>

<li><a href="Crystallography" title="Crystallography">Crystallography</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Interdisciplinary</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="Astrophysics" title="Astrophysics">Astrophysics</a></li>

<li><a href="Atmospheric_physics" title="Atmospheric physics">Atmospheric physics</a></li>

<li><a href="Biophysics" title="Biophysics">Biophysics</a></li>

<li><a href="Chemical_physics" title="Chemical physics">Chemical physics</a></li>

<li><a href="Geophysics" title="Geophysics">Geophysics</a></li>

<li><a href="Materials_science" title="Materials science">Materials science</a></li>

<li><a href="Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li>

<li><a href="Medical_physics" title="Medical physics">Medical physics</a></li>

<li><a href="Physical_oceanography" title="Physical oceanography">Ocean physics</a></li>

<li><a href="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="History_of_physics" title="History of physics">History of physics</a></li>

<li><a href="Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a></li>

<li><a href="Philosophy_of_physics" title="Philosophy of physics">Philosophy of physics</a></li>

<li><a href="Physics_education" title="Physics education">Physics education</a>

<ul><li><a href="Physics_education_research" title="Physics education research">research</a></li></ul></li>

<li><a href="Timeline_of_fundamental_physics_discoveries" title="Timeline of fundamental physics discoveries">Timeline of physics discoveries</a></li></ul>

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