SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Second quantization</title>

</head>

Second quantization

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1126788409">

/* start https://en.wikipedia.org/ */

.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1246091330">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}

/* end https://en.wikipedia.org/ */

</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="Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li>

<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>

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

</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></td></tr></tbody></table>

Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze <a href="Quantum_mechanics" title="Quantum mechanics">quantum</a> <a href="N-body_problem" title="N-body problem">many-body</a> systems. In <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, it is known as <a href="Canonical_quantization" title="Canonical quantization">canonical quantization</a>, in which the fields (typically as the <a href="Wave_function" title="Wave function">wave functions</a> of matter) are thought of as <a href="Field_operator" class="mw-redirect" title="Field operator">field operators</a>, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in <a href="First_quantization" title="First quantization">first quantization</a>. The key ideas of this method were introduced in 1927 by <a href="Paul_Dirac" title="Paul Dirac">Paul Dirac</a>,<a href="#cite_note-Dirac1927-1">[1]</a> and were later developed, most notably, by <a href="Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a><a href="#cite_note-Jordan1928-2">[2]</a> and <a href="Vladimir_Fock" title="Vladimir Fock">Vladimir Fock</a>.<a href="#cite_note-Fock1932-3">[3]</a><a href="#cite_note-Reed1975-4">[4]</a>

In this approach, the quantum many-body states are represented in the <a href="Fock_state" title="Fock state">Fock state</a> basis, which are constructed by filling up each single-particle state with a certain number of identical particles.<a href="#cite_note-Becchi2010-5">[5]</a> The second quantization formalism introduces the <a href="Creation_and_annihilation_operators" title="Creation and annihilation operators">creation and annihilation operators</a> to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory.

<div class="mw-heading mw-heading2"><h2 id="Quantum_many-body_states">Quantum many-body states</h2></div>

The starting point of the second quantization formalism is the notion of <a href="Identical_particles" class="mw-redirect" title="Identical particles">indistinguishability</a> of particles in quantum mechanics. Unlike in classical mechanics, where each particle is labeled by a distinct position vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}}</annotation>

</semantics>

</math><img src="./ed603561819ebd007acd75a0931d3ba401ad677a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}" loading="lazy"> and different configurations of the set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}}</annotation>

</semantics>

</math><img src="./ed603561819ebd007acd75a0931d3ba401ad677a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}" loading="lazy">s correspond to different many-body states, in quantum mechanics, the particles are identical, such that exchanging two particles, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}\leftrightarrow \mathbf {r} _{j}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}\leftrightarrow \mathbf {r} _{j}}</annotation>

</semantics>

</math><img src="./795102ded904efe8484c1b3a579802f0719fae2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.528ex; height:2.509ex;" alt="{\displaystyle \mathbf {r} _{i}\leftrightarrow \mathbf {r} _{j}}" loading="lazy">, does not lead to a different many-body quantum state. This implies that the quantum many-body wave function must be invariant (up to a phase factor) under the exchange of two particles. According to the <a href="Particle_statistics" title="Particle statistics">statistics</a> of the particles, the many-body wave function can either be symmetric or antisymmetric under the particle exchange:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{\rm {B}}(\cdots ,\mathbf {r} _{i},\cdots ,\mathbf {r} _{j},\cdots )=+\Psi _{\rm {B}}(\cdots ,\mathbf {r} _{j},\cdots ,\mathbf {r} _{i},\cdots )}</annotation>

</semantics>

</math><img src="./75971b091b8ef151260d2840e8e21171367ac493.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.92ex; height:3.009ex;" alt="{\displaystyle \Psi _{\rm {B}}(\cdots ,\mathbf {r} _{i},\cdots ,\mathbf {r} _{j},\cdots )=+\Psi _{\rm {B}}(\cdots ,\mathbf {r} _{j},\cdots ,\mathbf {r} _{i},\cdots )}" loading="lazy"> if the particles are <a href="Bosons" class="mw-redirect" title="Bosons">bosons</a>,</dd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{\rm {F}}(\cdots ,\mathbf {r} _{i},\cdots ,\mathbf {r} _{j},\cdots )=-\Psi _{\rm {F}}(\cdots ,\mathbf {r} _{j},\cdots ,\mathbf {r} _{i},\cdots )}</annotation>

</semantics>

</math><img src="./061d5b8de39029c0b7926868f892afda86c32fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.739ex; height:3.009ex;" alt="{\displaystyle \Psi _{\rm {F}}(\cdots ,\mathbf {r} _{i},\cdots ,\mathbf {r} _{j},\cdots )=-\Psi _{\rm {F}}(\cdots ,\mathbf {r} _{j},\cdots ,\mathbf {r} _{i},\cdots )}" loading="lazy"> if the particles are <a href="Fermions" class="mw-redirect" title="Fermions">fermions</a>.</dd></dl>

This exchange symmetry property imposes a constraint on the many-body wave function. Each time a particle is added or removed from the many-body system, the wave function must be properly symmetrized or anti-symmetrized to satisfy the symmetry constraint. In the first quantization formalism, this constraint is guaranteed by representing the wave function as linear combination of <a href="Permanent_(mathematics)" title="Permanent (mathematics)">permanents</a> (for bosons) or <a href="Determinant" title="Determinant">determinants</a> (for fermions) of single-particle states. In the second quantization formalism, the issue of symmetrization is automatically taken care of by the creation and annihilation operators, such that its notation can be much simpler.

<div class="mw-heading mw-heading3"><h3 id="First-quantized_many-body_wave_function">First-quantized many-body wave function</h3></div>

Consider a complete set of single-particle wave functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }(\mathbf {r} )}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{\alpha }(\mathbf {r} )}</annotation>

</semantics>

</math><img src="./2952b9cd365f7e5640b4a3ec96fd5ffb82622bcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.709ex; height:2.843ex;" alt="{\displaystyle \psi _{\alpha }(\mathbf {r} )}" loading="lazy"> labeled by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> (which may be a combined index of a number of quantum numbers). The following wave function

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi [\mathbf {r} _{i}]=\prod _{i=1}^{N}\psi _{\alpha _{i}}(\mathbf {r} _{i})\equiv \psi _{\alpha _{1}}\otimes \psi _{\alpha _{2}}\otimes \cdots \otimes \psi _{\alpha _{N}}}</annotation>

</semantics>

</math><img src="./be77aa66ecac0c2ac4e1cb5020448005c8e96cd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.339ex; height:7.343ex;" alt="{\displaystyle \Psi [\mathbf {r} _{i}]=\prod _{i=1}^{N}\psi _{\alpha _{i}}(\mathbf {r} _{i})\equiv \psi _{\alpha _{1}}\otimes \psi _{\alpha _{2}}\otimes \cdots \otimes \psi _{\alpha _{N}}}" loading="lazy"></dd></dl>

represents an N-particle state with the i-th particle occupying the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\alpha _{i}}\rangle }">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./585b68e91d25063122baabf0dbc78261a489e85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.839ex; height:2.843ex;" alt="{\displaystyle |{\alpha _{i}}\rangle }" loading="lazy">. In the shorthanded notation, the position argument of the wave function may be omitted, and it is assumed that the i-th single-particle wave function describes the state of the i-th particle. The wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5471531a3fe80741a839bc98d49fae862a6439a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" loading="lazy"> has not been symmetrized or anti-symmetrized, thus in general not qualified as a many-body wave function for identical particles. However, it can be brought to the symmetrized (anti-symmetrized) form by operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> for symmetrizer, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"> for <a href="Antisymmetrizer" title="Antisymmetrizer">antisymmetrizer</a>.

For bosons, the many-body wave function must be symmetrized,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</munderover>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{\rm {B}}[\mathbf {r} _{i}]={\mathcal {N}}{\mathcal {S}}\Psi [\mathbf {r} _{i}]={\mathcal {N}}\sum _{\pi \in S_{N}}\prod _{i=1}^{N}\psi _{\alpha _{\pi (i)}}(\mathbf {r} _{i})={\mathcal {N}}\sum _{\pi \in S_{N}}\psi _{\alpha _{\pi (1)}}\otimes \psi _{\alpha _{\pi (2)}}\otimes \cdots \otimes \psi _{\alpha _{\pi (N)}};}</annotation>

</semantics>

</math><img src="./1598d3dd90f20cfe9ec30b4309e51c47af1febe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:80.352ex; height:7.843ex;" alt="{\displaystyle \Psi _{\rm {B}}[\mathbf {r} _{i}]={\mathcal {N}}{\mathcal {S}}\Psi [\mathbf {r} _{i}]={\mathcal {N}}\sum _{\pi \in S_{N}}\prod _{i=1}^{N}\psi _{\alpha _{\pi (i)}}(\mathbf {r} _{i})={\mathcal {N}}\sum _{\pi \in S_{N}}\psi _{\alpha _{\pi (1)}}\otimes \psi _{\alpha _{\pi (2)}}\otimes \cdots \otimes \psi _{\alpha _{\pi (N)}};}" loading="lazy"></dd></dl>

while for fermions, the many-body wave function must be anti-symmetrized,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{\rm {F}}[\mathbf {r} _{i}]={\mathcal {N}}{\mathcal {A}}\Psi [\mathbf {r} _{i}]={\mathcal {N}}\sum _{\pi \in S_{N}}(-1)^{\pi }\prod _{i=1}^{N}\psi _{\alpha _{\pi (i)}}(\mathbf {r} _{i})={\mathcal {N}}\sum _{\pi \in S_{N}}(-1)^{\pi }\psi _{\alpha _{\pi (1)}}\otimes \psi _{\alpha _{\pi (2)}}\otimes \cdots \otimes \psi _{\alpha _{\pi (N)}}.}</annotation>

</semantics>

</math><img src="./b48cfa97cc169725c9d4f0a5b385605f6aeea99d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:92.193ex; height:7.843ex;" alt="{\displaystyle \Psi _{\rm {F}}[\mathbf {r} _{i}]={\mathcal {N}}{\mathcal {A}}\Psi [\mathbf {r} _{i}]={\mathcal {N}}\sum _{\pi \in S_{N}}(-1)^{\pi }\prod _{i=1}^{N}\psi _{\alpha _{\pi (i)}}(\mathbf {r} _{i})={\mathcal {N}}\sum _{\pi \in S_{N}}(-1)^{\pi }\psi _{\alpha _{\pi (1)}}\otimes \psi _{\alpha _{\pi (2)}}\otimes \cdots \otimes \psi _{\alpha _{\pi (N)}}.}" loading="lazy"></dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> is an element in the N-body permutation group (or <a href="Symmetric_group" title="Symmetric group">symmetric group</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{N}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aca805f5a6c6548a825d119230ab751152321ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.116ex; height:2.509ex;" alt="{\displaystyle S_{N}}" loading="lazy">, which performs a <a href="Permutation" title="Permutation">permutation</a> among the state labels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3b1fb627423abe4988b7ed88d4920bf1ec074790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{\pi }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e9b2d5eb0f88fdcb41a08e97343f4016b0e9c742.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.954ex; height:2.843ex;" alt="{\displaystyle (-1)^{\pi }}" loading="lazy"> denotes the corresponding <a href="Parity_of_a_permutation" title="Parity of a permutation">permutation sign</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b7551c7bed2cd2ee83e10536d157c94a5f8f72fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}" loading="lazy"> is the normalization operator that normalizes the wave function. (It is the operator that applies a suitable numerical normalization factor to the symmetrized tensors of degree n; see the next section for its value.)

If one arranges the single-particle wave functions in a matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, such that the row-i column-j matrix element is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{ij}=\psi _{\alpha _{j}}(\mathbf {r} _{i})\equiv \langle \mathbf {r} _{i}|\alpha _{j}\rangle }">

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{ij}=\psi _{\alpha _{j}}(\mathbf {r} _{i})\equiv \langle \mathbf {r} _{i}|\alpha _{j}\rangle }</annotation>

</semantics>

</math><img src="./b69323cab949fb31a02becb96cddb5e63d35d8b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.239ex; height:3.176ex;" alt="{\displaystyle U_{ij}=\psi _{\alpha _{j}}(\mathbf {r} _{i})\equiv \langle \mathbf {r} _{i}|\alpha _{j}\rangle }" loading="lazy">, then the boson many-body wave function can be simply written as a <a href="Permanent_(mathematics)" title="Permanent (mathematics)">permanent</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\rm {B}}={\mathcal {N}}\operatorname {perm} U}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{\rm {B}}={\mathcal {N}}\operatorname {perm} U}</annotation>

</semantics>

</math><img src="./e59714180ec09420eebb99dbe194a4ac9b53d44d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.307ex; height:2.843ex;" alt="{\displaystyle \Psi _{\rm {B}}={\mathcal {N}}\operatorname {perm} U}" loading="lazy">, and the fermion many-body wave function as a <a href="Determinant" title="Determinant">determinant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\rm {F}}={\mathcal {N}}\det U}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{\rm {F}}={\mathcal {N}}\det U}</annotation>

</semantics>

</math><img src="./3e7327dea2b9d8828ad42827fdca840ea7954403.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.273ex; height:2.843ex;" alt="{\displaystyle \Psi _{\rm {F}}={\mathcal {N}}\det U}" loading="lazy"> (also known as the <a href="Slater_determinant" title="Slater determinant">Slater determinant</a>).<a href="#cite_note-Koch2013-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Second-quantized_Fock_states">Second-quantized Fock states</h3></div>

First quantized wave functions involve complicated symmetrization procedures to describe physically realizable many-body states because the language of first quantization is redundant for indistinguishable particles. In the first quantization language, the many-body state is described by answering a series of questions like "Which particle is in which state?". However these are not physical questions, because the particles are identical, and it is impossible to tell which particle is which in the first place. The seemingly different states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}\otimes \psi _{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{1}\otimes \psi _{2}}</annotation>

</semantics>

</math><img src="./94f065bbbfbe9df30870d5aa8a25cda52682482d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.975ex; height:2.509ex;" alt="{\displaystyle \psi _{1}\otimes \psi _{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{2}\otimes \psi _{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{2}\otimes \psi _{1}}</annotation>

</semantics>

</math><img src="./f7506586825cc6ba50ba6fd9aebe852254dfe7a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.975ex; height:2.509ex;" alt="{\displaystyle \psi _{2}\otimes \psi _{1}}" loading="lazy"> are actually redundant names of the same quantum many-body state. So the symmetrization (or anti-symmetrization) must be introduced to eliminate this redundancy in the first quantization description.

In the second quantization language, instead of asking "each particle on which state", one asks "How many particles are there in each state?". Because this description does not refer to the labeling of particles, it contains no redundant information, and hence leads to a precise and simpler description of the quantum many-body state. In this approach, the many-body state is represented in the occupation number basis, and the basis state is labeled by the set of occupation numbers, denoted

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |[n_{\alpha }]\rangle \equiv |n_{1},n_{2},\cdots ,n_{\alpha },\cdots \rangle ,}</annotation>

</semantics>

</math><img src="./28720c90fee4e56aff090ed162f18c9647f52791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.367ex; height:2.843ex;" alt="{\displaystyle |[n_{\alpha }]\rangle \equiv |n_{1},n_{2},\cdots ,n_{\alpha },\cdots \rangle ,}" loading="lazy"></dd></dl>

meaning that there are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./28ba9deb2ca590784ac01c1778371105e371cc88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.679ex; height:2.009ex;" alt="{\displaystyle n_{\alpha }}" loading="lazy"> particles in the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f42032e642ee1c9d27adb318d34c7cc85f7a95d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle |\alpha \rangle }" loading="lazy"> (or as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./eae086daf01dbbd5684898cccb1b1b7baf234dde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }}" loading="lazy">). The occupation numbers sum to the total number of particles, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{\alpha }n_{\alpha }=N}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \sum _{\alpha }n_{\alpha }=N}</annotation>

</semantics>

</math><img src="./754c8558ec99355adde4b56ca932bf31069af2df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.966ex; height:3.009ex;" alt="{\textstyle \sum _{\alpha }n_{\alpha }=N}" loading="lazy">. For <a href="Fermions" class="mw-redirect" title="Fermions">fermions</a>, the occupation number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./28ba9deb2ca590784ac01c1778371105e371cc88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.679ex; height:2.009ex;" alt="{\displaystyle n_{\alpha }}" loading="lazy"> can only be 0 or 1, due to the <a href="Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>; while for <a href="Bosons" class="mw-redirect" title="Bosons">bosons</a> it can be any non-negative integer

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext>fermions,</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext>bosons.</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{\alpha }={\begin{cases}0,1&{\text{fermions,}}\\0,1,2,3,...&{\text{bosons.}}\end{cases}}}</annotation>

</semantics>

</math><img src="./929d8c1008954c37d4005276bdb8a3b3b4c56f7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.351ex; height:6.176ex;" alt="{\displaystyle n_{\alpha }={\begin{cases}0,1&{\text{fermions,}}\\0,1,2,3,...&{\text{bosons.}}\end{cases}}}" loading="lazy"></dd></dl>

The occupation number states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |[n_{\alpha }]\rangle }">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |[n_{\alpha }]\rangle }</annotation>

</semantics>

</math><img src="./684cb65b00674220dd7a6fca42cd0ed4de25865b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.524ex; height:2.843ex;" alt="{\displaystyle |[n_{\alpha }]\rangle }" loading="lazy"> are also known as Fock states. All the Fock states form a complete basis of the many-body Hilbert space, or <a href="Fock_space" title="Fock space">Fock space</a>. Any generic quantum many-body state can be expressed as a linear combination of Fock states.

Note that besides providing a more efficient language, Fock space allows for a variable number of particles. As a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a>, it is isomorphic to the sum of the n-particle bosonic or fermionic tensor spaces described in the previous section, including a one-dimensional zero-particle space C.

The Fock state with all occupation numbers equal to zero is called the <a href="Vacuum_state" class="mw-redirect" title="Vacuum state">vacuum state</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle \equiv |\cdots ,0_{\alpha },\cdots \rangle }">

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |0\rangle \equiv |\cdots ,0_{\alpha },\cdots \rangle }</annotation>

</semantics>

</math><img src="./322d9d0d2e2b03af5304e0609eba5a074d67a3c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.099ex; height:2.843ex;" alt="{\displaystyle |0\rangle \equiv |\cdots ,0_{\alpha },\cdots \rangle }" loading="lazy">. The Fock state with only one non-zero occupation number is a single-mode Fock state, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n_{\alpha }\rangle \equiv |\cdots ,0,n_{\alpha },0,\cdots \rangle }">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |n_{\alpha }\rangle \equiv |\cdots ,0,n_{\alpha },0,\cdots \rangle }</annotation>

</semantics>

</math><img src="./6f7b08ef6f54107276598d46edf63896a461754e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.241ex; height:2.843ex;" alt="{\displaystyle |n_{\alpha }\rangle \equiv |\cdots ,0,n_{\alpha },0,\cdots \rangle }" loading="lazy">. In terms of the first quantized wave function, the vacuum state is the unit tensor product and can be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle =1}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |0\rangle =1}</annotation>

</semantics>

</math><img src="./18176e9b6d96aaff6c6cb08ce5b9e62848e75f77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.975ex; height:2.843ex;" alt="{\displaystyle |0\rangle =1}" loading="lazy">. The single-particle state is reduced to its wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1_{\alpha }\rangle =\psi _{\alpha }}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |1_{\alpha }\rangle =\psi _{\alpha }}</annotation>

</semantics>

</math><img src="./13a69d18aef3cab59bc7e26718e73f4f9631a8c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.894ex; height:2.843ex;" alt="{\displaystyle |1_{\alpha }\rangle =\psi _{\alpha }}" loading="lazy">. Other single-mode many-body (boson) states are just the tensor product of the wave function of that mode, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |2_{\alpha }\rangle =\psi _{\alpha }\otimes \psi _{\alpha }}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |2_{\alpha }\rangle =\psi _{\alpha }\otimes \psi _{\alpha }}</annotation>

</semantics>

</math><img src="./9b91790d76a3533c9998e9cce53eb7c6ed4cd8db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.532ex; height:2.843ex;" alt="{\displaystyle |2_{\alpha }\rangle =\psi _{\alpha }\otimes \psi _{\alpha }}" loading="lazy"> and

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |n_{\alpha }\rangle =\psi _{\alpha }^{\otimes n}}</annotation>

</semantics>

</math><img src="./04ab0fe9aec96a5f750aaee0be6c85b07c5ac6a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.339ex; height:3.009ex;" alt="{\displaystyle |n_{\alpha }\rangle =\psi _{\alpha }^{\otimes n}}" loading="lazy">. For multi-mode Fock states (meaning more than one single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f42032e642ee1c9d27adb318d34c7cc85f7a95d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle |\alpha \rangle }" loading="lazy"> is involved), the corresponding first-quantized wave function will require proper symmetrization according to the particle statistics, e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}+\psi _{2}\psi _{1})/{\sqrt {2}}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}+\psi _{2}\psi _{1})/{\sqrt {2}}}</annotation>

</semantics>

</math><img src="./3e4b1af21bcd594ead3ef90003eeffcde248d41a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.297ex; height:3.176ex;" alt="{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}+\psi _{2}\psi _{1})/{\sqrt {2}}}" loading="lazy"> for a boson state, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}-\psi _{2}\psi _{1})/{\sqrt {2}}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}-\psi _{2}\psi _{1})/{\sqrt {2}}}</annotation>

</semantics>

</math><img src="./6cf923d715aabb5701b2046d8cc306a0ffd14ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.297ex; height:3.176ex;" alt="{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}-\psi _{2}\psi _{1})/{\sqrt {2}}}" loading="lazy"> for a fermion state (the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"> between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{1}}</annotation>

</semantics>

</math><img src="./8cfdde1da54e02a016fe2a230c58b25dfcc014d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{2}}</annotation>

</semantics>

</math><img src="./c5083a526766f85c4f39ab695791b0b739f06897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{2}}" loading="lazy"> is omitted for simplicity). In general, the normalization is found to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {\frac {1}{N!{\prod _{\alpha }{n_{\alpha }!}}}}}}">

<msqrt>

<mfrac>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\sqrt {\frac {1}{N!{\prod _{\alpha }{n_{\alpha }!}}}}}}</annotation>

</semantics>

</math><img src="./f30b0063614ad7cfea9ccb38b7f35b0ee2b7ff8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.495ex; height:4.843ex;" alt="{\textstyle {\sqrt {\frac {1}{N!{\prod _{\alpha }{n_{\alpha }!}}}}}}" loading="lazy">, where N is the total number of particles. For fermion, this expression reduces to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {N!}}}}">

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {N!}}}}</annotation>

</semantics>

</math><img src="./2407d841dce5edeb776a55257b99b21fb373e2be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.122ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {N!}}}}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./28ba9deb2ca590784ac01c1778371105e371cc88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.679ex; height:2.009ex;" alt="{\displaystyle n_{\alpha }}" loading="lazy"> can only be either zero or one. So the first-quantized wave function corresponding to the Fock state reads

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |[n_{\alpha }]\rangle _{\rm {B}}=\left({\frac {1}{N!\prod _{\alpha }n_{\alpha }!}}\right)^{1/2}{\mathcal {S}}\bigotimes \limits _{\alpha }\psi _{\alpha }^{\otimes n_{\alpha }}}</annotation>

</semantics>

</math><img src="./819f392dbb0ebea8383cf40ef2ab9ca86e0faa1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.068ex; height:7.176ex;" alt="{\displaystyle |[n_{\alpha }]\rangle _{\rm {B}}=\left({\frac {1}{N!\prod _{\alpha }n_{\alpha }!}}\right)^{1/2}{\mathcal {S}}\bigotimes \limits _{\alpha }\psi _{\alpha }^{\otimes n_{\alpha }}}" loading="lazy"></dd></dl>

for bosons and

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |[n_{\alpha }]\rangle _{\rm {F}}={\frac {1}{\sqrt {N!}}}{\mathcal {A}}\bigotimes \limits _{\alpha }\psi _{\alpha }^{\otimes n_{\alpha }}}</annotation>

</semantics>

</math><img src="./8ba48c4ff2d4244f8b5549dc89e83f03bd893d11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.627ex; height:6.343ex;" alt="{\displaystyle |[n_{\alpha }]\rangle _{\rm {F}}={\frac {1}{\sqrt {N!}}}{\mathcal {A}}\bigotimes \limits _{\alpha }\psi _{\alpha }^{\otimes n_{\alpha }}}" loading="lazy"></dd></dl>

for fermions. Note that for fermions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\alpha }=0,1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{\alpha }=0,1}</annotation>

</semantics>

</math><img src="./661a7415c222881b49a4acab7f1a553629e8ca0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.136ex; height:2.509ex;" alt="{\displaystyle n_{\alpha }=0,1}" loading="lazy"> only, so the tensor product above is effectively just a product over all occupied single-particle states.

<div class="mw-heading mw-heading2"><h2 id="Creation_and_annihilation_operators">Creation and annihilation operators</h2></div>

The <a href="Creation_and_annihilation_operators" title="Creation and annihilation operators">creation and annihilation operators</a> are introduced to add or remove a particle from the many-body system. These operators lie at the core of the second quantization formalism, bridging the gap between the first- and the second-quantized states. Applying the creation (annihilation) operator to a first-quantized many-body wave function will insert (delete) a single-particle state from the wave function in a symmetrized way depending on the particle statistics. On the other hand, all the second-quantized Fock states can be constructed by applying the creation operators to the vacuum state repeatedly.

The creation and annihilation operators (for bosons) are originally constructed in the context of the <a href="Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> as the raising and lowering operators, which are then generalized to the field operators in the quantum field theory.<a href="#cite_note-Mahan2000-7">[7]</a> They are fundamental to the quantum many-body theory, in the sense that every many-body operator (including the Hamiltonian of the many-body system and all the physical observables) can be expressed in terms of them.

<div class="mw-heading mw-heading3"><h3 id="Insertion_and_deletion_operation">Insertion and deletion operation</h3></div>

The creation and annihilation of a particle is implemented by the insertion and deletion of the single-particle state from the first quantized wave function in an either symmetric or anti-symmetric manner. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./eae086daf01dbbd5684898cccb1b1b7baf234dde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }}" loading="lazy"> be a single-particle state, let 1 be the tensor identity (it is the generator of the zero-particle space C and satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }\equiv 1\otimes \psi _{\alpha }\equiv \psi _{\alpha }\otimes 1}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{\alpha }\equiv 1\otimes \psi _{\alpha }\equiv \psi _{\alpha }\otimes 1}</annotation>

</semantics>

</math><img src="./47d64fcfc7af17f1cecb0a127546e9c7aca5a0a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.595ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }\equiv 1\otimes \psi _{\alpha }\equiv \psi _{\alpha }\otimes 1}" loading="lazy"> in the <a href="Tensor_algebra" title="Tensor algebra">tensor algebra</a> over the fundamental Hilbert space), and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi =\psi _{\alpha _{1}}\otimes \psi _{\alpha _{2}}\otimes \cdots }">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi =\psi _{\alpha _{1}}\otimes \psi _{\alpha _{2}}\otimes \cdots }</annotation>

</semantics>

</math><img src="./50af36f1f34134af96dd6280949eb615e61423c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.568ex; height:2.843ex;" alt="{\displaystyle \Psi =\psi _{\alpha _{1}}\otimes \psi _{\alpha _{2}}\otimes \cdots }" loading="lazy"> be a generic tensor product state. The insertion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes _{\pm }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \otimes _{\pm }}</annotation>

</semantics>

</math><img src="./934bbdb820a982d74e1b3239cdbbd469636bd73a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.319ex; height:2.343ex;" alt="{\displaystyle \otimes _{\pm }}" loading="lazy"> and the deletion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oslash _{\pm }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \oslash _{\pm }}</annotation>

</semantics>

</math><img src="./b278e4019dfe2d733ba000aecc3f7fd31e10c3a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.319ex; height:2.343ex;" alt="{\displaystyle \oslash _{\pm }}" loading="lazy"> operators are linear operators defined by the following recursive equations

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{\alpha }\otimes _{\pm }1=\psi _{\alpha },\quad \psi _{\alpha }\otimes _{\pm }(\psi _{\beta }\otimes \Psi )=\psi _{\alpha }\otimes \psi _{\beta }\otimes \Psi \pm \psi _{\beta }\otimes (\psi _{\alpha }\otimes _{\pm }\Psi );}</annotation>

</semantics>

</math><img src="./26a2035b36e5e1c0c647143fbe0dbe7b7104029a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:69.71ex; height:3.009ex;" alt="{\displaystyle \psi _{\alpha }\otimes _{\pm }1=\psi _{\alpha },\quad \psi _{\alpha }\otimes _{\pm }(\psi _{\beta }\otimes \Psi )=\psi _{\alpha }\otimes \psi _{\beta }\otimes \Psi \pm \psi _{\beta }\otimes (\psi _{\alpha }\otimes _{\pm }\Psi );}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{\alpha }\oslash _{\pm }1=0,\quad \psi _{\alpha }\oslash _{\pm }(\psi _{\beta }\otimes \Psi )=\delta _{\alpha \beta }\Psi \pm \psi _{\beta }\otimes (\psi _{\alpha }\oslash _{\pm }\Psi ).}</annotation>

</semantics>

</math><img src="./23c285094d208719c75d462fc624a206503fdd66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:60.168ex; height:3.009ex;" alt="{\displaystyle \psi _{\alpha }\oslash _{\pm }1=0,\quad \psi _{\alpha }\oslash _{\pm }(\psi _{\beta }\otimes \Psi )=\delta _{\alpha \beta }\Psi \pm \psi _{\beta }\otimes (\psi _{\alpha }\oslash _{\pm }\Psi ).}" loading="lazy"></dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\alpha \beta }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{\alpha \beta }}</annotation>

</semantics>

</math><img src="./12656c1c39d2ddb970601b3f4458023baa011084.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.258ex; height:3.009ex;" alt="{\displaystyle \delta _{\alpha \beta }}" loading="lazy"> is the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a> symbol, which gives 1 if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\beta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ef6894a6c2f414b03c984a1c7f0639063b0020ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.918ex; height:2.509ex;" alt="{\displaystyle \alpha =\beta }" loading="lazy">, and 0 otherwise. The subscript <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./869e366caf596564de4de06cb0ba124056d4064b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \pm }" loading="lazy"> of the insertion or deletion operators indicates whether symmetrization (for bosons) or anti-symmetrization (for fermions) is implemented.

<div class="mw-heading mw-heading3"><h3 id="Boson_creation_and_annihilation_operators">Boson creation and annihilation operators</h3></div>

The boson creation (resp. annihilation) operator is usually denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\alpha }^{\dagger }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }^{\dagger }}</annotation>

</semantics>

</math><img src="./b3aa31d48819ad11601d402dbabcf5a607738cb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.282ex; height:3.176ex;" alt="{\displaystyle b_{\alpha }^{\dagger }}" loading="lazy"> (resp. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fbfba2ff3a4a7b22f2847c829622c28fc5d146d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.282ex; height:2.509ex;" alt="{\displaystyle b_{\alpha }}" loading="lazy">). The creation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\alpha }^{\dagger }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }^{\dagger }}</annotation>

</semantics>

</math><img src="./b3aa31d48819ad11601d402dbabcf5a607738cb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.282ex; height:3.176ex;" alt="{\displaystyle b_{\alpha }^{\dagger }}" loading="lazy"> adds a boson to the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f42032e642ee1c9d27adb318d34c7cc85f7a95d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle |\alpha \rangle }" loading="lazy">, and the annihilation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fbfba2ff3a4a7b22f2847c829622c28fc5d146d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.282ex; height:2.509ex;" alt="{\displaystyle b_{\alpha }}" loading="lazy"> removes a boson from the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f42032e642ee1c9d27adb318d34c7cc85f7a95d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle |\alpha \rangle }" loading="lazy">. The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\alpha }\neq b_{\alpha }^{\dagger }}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }\neq b_{\alpha }^{\dagger }}</annotation>

</semantics>

</math><img src="./750cbf5241c80560ce93b947b23a8eff5be83808.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:3.343ex;" alt="{\displaystyle b_{\alpha }\neq b_{\alpha }^{\dagger }}" loading="lazy">).

<div class="mw-heading mw-heading4"><h4 id="Definition">Definition</h4></div>

The boson creation (annihilation) operator is a linear operator, whose action on a N-particle first-quantized wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5471531a3fe80741a839bc98d49fae862a6439a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" loading="lazy"> is defined as

</mrow>

</mrow>

</msubsup>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }^{\dagger }\Psi ={\frac {1}{\sqrt {N+1}}}\psi _{\alpha }\otimes _{+}\Psi ,}</annotation>

</semantics>

</math><img src="./6a2b9e5768efaa7368058004d3bd2083dd175037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.63ex; height:6.176ex;" alt="{\displaystyle b_{\alpha }^{\dagger }\Psi ={\frac {1}{\sqrt {N+1}}}\psi _{\alpha }\otimes _{+}\Psi ,}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }\Psi ={\frac {1}{\sqrt {N}}}\psi _{\alpha }\oslash _{+}\Psi ,}</annotation>

</semantics>

</math><img src="./2a0aea08cabec5db50431c3d8926d5e6f6ba8934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.627ex; height:6.176ex;" alt="{\displaystyle b_{\alpha }\Psi ={\frac {1}{\sqrt {N}}}\psi _{\alpha }\oslash _{+}\Psi ,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }\otimes _{+}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a535a878d496141e68076c776925b1185d531a94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.116ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }\otimes _{+}}" loading="lazy"> inserts the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./eae086daf01dbbd5684898cccb1b1b7baf234dde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N+1}</annotation>

</semantics>

</math><img src="./fdf2b9cbfe9051fd4e7b50c8028866d497eac35b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N+1}" loading="lazy"> possible insertion positions symmetrically, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }\oslash _{+}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./be4e037e05708d178ce59bec224f74038ed5f2f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.116ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }\oslash _{+}}" loading="lazy"> deletes the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./eae086daf01dbbd5684898cccb1b1b7baf234dde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> possible deletion positions symmetrically.

<div class="mw-heading mw-heading5"><h5 id="Examples">Examples</h5></div>

Hereinafter the tensor symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"> between single-particle states is omitted for simplicity. Take the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}+\psi _{2}\psi _{1})/{\sqrt {2}}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}+\psi _{2}\psi _{1})/{\sqrt {2}}}</annotation>

</semantics>

</math><img src="./3e4b1af21bcd594ead3ef90003eeffcde248d41a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.297ex; height:3.176ex;" alt="{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}+\psi _{2}\psi _{1})/{\sqrt {2}}}" loading="lazy">, create one more boson on the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{1}}</annotation>

</semantics>

</math><img src="./8cfdde1da54e02a016fe2a230c58b25dfcc014d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{1}}" loading="lazy">,

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msqrt>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}b_{1}^{\dagger }|1_{1},1_{2}\rangle =&{\frac {1}{\sqrt {2}}}(b_{1}^{\dagger }\psi _{1}\psi _{2}+b_{1}^{\dagger }\psi _{2}\psi _{1})\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {3}}}\psi _{1}\otimes _{+}\psi _{1}\psi _{2}+{\frac {1}{\sqrt {3}}}\psi _{1}\otimes _{+}\psi _{2}\psi _{1}\right)\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{1}\psi _{2}+\psi _{1}\psi _{1}\psi _{2}+\psi _{1}\psi _{2}\psi _{1})+{\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{2}\psi _{1}+\psi _{2}\psi _{1}\psi _{1}+\psi _{2}\psi _{1}\psi _{1})\right)\\=&{\frac {\sqrt {2}}{\sqrt {3}}}(\psi _{1}\psi _{1}\psi _{2}+\psi _{1}\psi _{2}\psi _{1}+\psi _{2}\psi _{1}\psi _{1})\\=&{\sqrt {2}}|2_{1},1_{2}\rangle .\end{array}}}</annotation>

</semantics>

</math><img src="./c03a5c6bb5472965804c5a17e7fd8da16417740e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.338ex; width:90.875ex; height:23.843ex;" alt="{\displaystyle {\begin{array}{rl}b_{1}^{\dagger }|1_{1},1_{2}\rangle =&{\frac {1}{\sqrt {2}}}(b_{1}^{\dagger }\psi _{1}\psi _{2}+b_{1}^{\dagger }\psi _{2}\psi _{1})\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {3}}}\psi _{1}\otimes _{+}\psi _{1}\psi _{2}+{\frac {1}{\sqrt {3}}}\psi _{1}\otimes _{+}\psi _{2}\psi _{1}\right)\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{1}\psi _{2}+\psi _{1}\psi _{1}\psi _{2}+\psi _{1}\psi _{2}\psi _{1})+{\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{2}\psi _{1}+\psi _{2}\psi _{1}\psi _{1}+\psi _{2}\psi _{1}\psi _{1})\right)\\=&{\frac {\sqrt {2}}{\sqrt {3}}}(\psi _{1}\psi _{1}\psi _{2}+\psi _{1}\psi _{2}\psi _{1}+\psi _{2}\psi _{1}\psi _{1})\\=&{\sqrt {2}}|2_{1},1_{2}\rangle .\end{array}}}" loading="lazy"></dd></dl>

Then annihilate one boson from the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{1}}</annotation>

</semantics>

</math><img src="./8cfdde1da54e02a016fe2a230c58b25dfcc014d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{1}}" loading="lazy">,

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msqrt>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}b_{1}|2_{1},1_{2}\rangle =&{\frac {1}{\sqrt {3}}}(b_{1}\psi _{1}\psi _{1}\psi _{2}+b_{1}\psi _{1}\psi _{2}\psi _{1}+b_{1}\psi _{2}\psi _{1}\psi _{1})\\=&{\frac {1}{\sqrt {3}}}\left({\frac {1}{\sqrt {3}}}\psi _{1}\oslash _{+}\psi _{1}\psi _{1}\psi _{2}+{\frac {1}{\sqrt {3}}}\psi _{1}\oslash _{+}\psi _{1}\psi _{2}\psi _{1}+{\frac {1}{\sqrt {3}}}\psi _{1}\oslash _{+}\psi _{2}\psi _{1}\psi _{1}\right)\\=&{\frac {1}{\sqrt {3}}}\left({\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{2}+\psi _{1}\psi _{2}+0)+{\frac {1}{\sqrt {3}}}(\psi _{2}\psi _{1}+0+\psi _{1}\psi _{2})+{\frac {1}{\sqrt {3}}}(0+\psi _{2}\psi _{1}+\psi _{2}\psi _{1})\right)\\=&\psi _{1}\psi _{2}+\psi _{2}\psi _{1}\\=&{\sqrt {2}}|1_{1},1_{2}\rangle .\end{array}}}</annotation>

</semantics>

</math><img src="./1ae9060ea65787c3ab9b9320bd1256580f3b0d17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.171ex; width:92.316ex; height:21.509ex;" alt="{\displaystyle {\begin{array}{rl}b_{1}|2_{1},1_{2}\rangle =&{\frac {1}{\sqrt {3}}}(b_{1}\psi _{1}\psi _{1}\psi _{2}+b_{1}\psi _{1}\psi _{2}\psi _{1}+b_{1}\psi _{2}\psi _{1}\psi _{1})\\=&{\frac {1}{\sqrt {3}}}\left({\frac {1}{\sqrt {3}}}\psi _{1}\oslash _{+}\psi _{1}\psi _{1}\psi _{2}+{\frac {1}{\sqrt {3}}}\psi _{1}\oslash _{+}\psi _{1}\psi _{2}\psi _{1}+{\frac {1}{\sqrt {3}}}\psi _{1}\oslash _{+}\psi _{2}\psi _{1}\psi _{1}\right)\\=&{\frac {1}{\sqrt {3}}}\left({\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{2}+\psi _{1}\psi _{2}+0)+{\frac {1}{\sqrt {3}}}(\psi _{2}\psi _{1}+0+\psi _{1}\psi _{2})+{\frac {1}{\sqrt {3}}}(0+\psi _{2}\psi _{1}+\psi _{2}\psi _{1})\right)\\=&\psi _{1}\psi _{2}+\psi _{2}\psi _{1}\\=&{\sqrt {2}}|1_{1},1_{2}\rangle .\end{array}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Action_on_Fock_states">Action on Fock states</h4></div>

Starting from the single-mode vacuum state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0_{\alpha }\rangle =1}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |0_{\alpha }\rangle =1}</annotation>

</semantics>

</math><img src="./2f517186ed9e650e33d09028106fbfd07793b6a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.259ex; height:2.843ex;" alt="{\displaystyle |0_{\alpha }\rangle =1}" loading="lazy">, applying the creation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\alpha }^{\dagger }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }^{\dagger }}</annotation>

</semantics>

</math><img src="./b3aa31d48819ad11601d402dbabcf5a607738cb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.282ex; height:3.176ex;" alt="{\displaystyle b_{\alpha }^{\dagger }}" loading="lazy"> repeatedly, one finds

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }^{\dagger }|0_{\alpha }\rangle =\psi _{\alpha }\otimes _{+}1=\psi _{\alpha }=|1_{\alpha }\rangle ,}</annotation>

</semantics>

</math><img src="./c9c2bef2631a32950e61b5e8846b84cac3519379.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.328ex; height:3.343ex;" alt="{\displaystyle b_{\alpha }^{\dagger }|0_{\alpha }\rangle =\psi _{\alpha }\otimes _{+}1=\psi _{\alpha }=|1_{\alpha }\rangle ,}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }^{\dagger }|n_{\alpha }\rangle ={\frac {1}{\sqrt {n_{\alpha }+1}}}\psi _{\alpha }\otimes _{+}\psi _{\alpha }^{\otimes n_{\alpha }}={\sqrt {n_{\alpha }+1}}\psi _{\alpha }^{\otimes (n_{\alpha }+1)}={\sqrt {n_{\alpha }+1}}|n_{\alpha }+1\rangle .}</annotation>

</semantics>

</math><img src="./10db1d7c822aea8d2e92272ea498a4bd50f70258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:72.737ex; height:6.176ex;" alt="{\displaystyle b_{\alpha }^{\dagger }|n_{\alpha }\rangle ={\frac {1}{\sqrt {n_{\alpha }+1}}}\psi _{\alpha }\otimes _{+}\psi _{\alpha }^{\otimes n_{\alpha }}={\sqrt {n_{\alpha }+1}}\psi _{\alpha }^{\otimes (n_{\alpha }+1)}={\sqrt {n_{\alpha }+1}}|n_{\alpha }+1\rangle .}" loading="lazy"></dd></dl>

The creation operator raises the boson occupation number by 1. Therefore, all the occupation number states can be constructed by the boson creation operator from the vacuum state

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |n_{\alpha }\rangle ={\frac {1}{\sqrt {n_{\alpha }!}}}(b_{\alpha }^{\dagger })^{n_{\alpha }}|0_{\alpha }\rangle .}</annotation>

</semantics>

</math><img src="./91cee27a5a6c63a7fd9f5423d749bad1ae6d8e01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.787ex; height:6.509ex;" alt="{\displaystyle |n_{\alpha }\rangle ={\frac {1}{\sqrt {n_{\alpha }!}}}(b_{\alpha }^{\dagger })^{n_{\alpha }}|0_{\alpha }\rangle .}" loading="lazy"></dd></dl>

On the other hand, the annihilation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fbfba2ff3a4a7b22f2847c829622c28fc5d146d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.282ex; height:2.509ex;" alt="{\displaystyle b_{\alpha }}" loading="lazy"> lowers the boson occupation number by 1

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }|n_{\alpha }\rangle ={\frac {1}{\sqrt {n_{\alpha }}}}\psi _{\alpha }\oslash _{+}\psi _{\alpha }^{\otimes n_{\alpha }}={\sqrt {n_{\alpha }}}\psi _{\alpha }^{\otimes (n_{\alpha }-1)}={\sqrt {n_{\alpha }}}|n_{\alpha }-1\rangle .}</annotation>

</semantics>

</math><img src="./bc1661c18f3e03b54e4c2256b149e4ca1480caab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:59.953ex; height:6.176ex;" alt="{\displaystyle b_{\alpha }|n_{\alpha }\rangle ={\frac {1}{\sqrt {n_{\alpha }}}}\psi _{\alpha }\oslash _{+}\psi _{\alpha }^{\otimes n_{\alpha }}={\sqrt {n_{\alpha }}}\psi _{\alpha }^{\otimes (n_{\alpha }-1)}={\sqrt {n_{\alpha }}}|n_{\alpha }-1\rangle .}" loading="lazy"></dd></dl>

It will also quench the vacuum state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\alpha }|0_{\alpha }\rangle =0}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }|0_{\alpha }\rangle =0}</annotation>

</semantics>

</math><img src="./ef36b45f1a2f5ee79ae90efb795abb4559c3b368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.541ex; height:2.843ex;" alt="{\displaystyle b_{\alpha }|0_{\alpha }\rangle =0}" loading="lazy"> as there has been no boson left in the vacuum state to be annihilated. Using the above formulae, it can be shown that

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }^{\dagger }b_{\alpha }|n_{\alpha }\rangle =n_{\alpha }|n_{\alpha }\rangle ,}</annotation>

</semantics>

</math><img src="./ca66671c6354c6f9e390bf65ee5fde04944e3ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.448ex; height:3.343ex;" alt="{\displaystyle b_{\alpha }^{\dagger }b_{\alpha }|n_{\alpha }\rangle =n_{\alpha }|n_{\alpha }\rangle ,}" loading="lazy"></dd></dl>

meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}_{\alpha }=b_{\alpha }^{\dagger }b_{\alpha }}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}_{\alpha }=b_{\alpha }^{\dagger }b_{\alpha }}</annotation>

</semantics>

</math><img src="./4f5ca12a24bba24f3d977010400ee2e0febf2ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.341ex; height:3.176ex;" alt="{\displaystyle {\hat {n}}_{\alpha }=b_{\alpha }^{\dagger }b_{\alpha }}" loading="lazy"> defines the boson number operator.

The above result can be generalized to any Fock state of bosons.

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }^{\dagger }|\cdots ,n_{\beta },n_{\alpha },n_{\gamma },\cdots \rangle ={\sqrt {n_{\alpha }+1}}|\cdots ,n_{\beta },n_{\alpha }+1,n_{\gamma },\cdots \rangle .}</annotation>

</semantics>

</math><img src="./6937d15f4cad1d93ae3be24678003e212ed78054.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:58.386ex; height:3.676ex;" alt="{\displaystyle b_{\alpha }^{\dagger }|\cdots ,n_{\beta },n_{\alpha },n_{\gamma },\cdots \rangle ={\sqrt {n_{\alpha }+1}}|\cdots ,n_{\beta },n_{\alpha }+1,n_{\gamma },\cdots \rangle .}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{\alpha }|\cdots ,n_{\beta },n_{\alpha },n_{\gamma },\cdots \rangle ={\sqrt {n_{\alpha }}}|\cdots ,n_{\beta },n_{\alpha }-1,n_{\gamma },\cdots \rangle .}</annotation>

</semantics>

</math><img src="./6aea491fb6b7c1175f36635583729d3314426788.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:53.995ex; height:3.176ex;" alt="{\displaystyle b_{\alpha }|\cdots ,n_{\beta },n_{\alpha },n_{\gamma },\cdots \rangle ={\sqrt {n_{\alpha }}}|\cdots ,n_{\beta },n_{\alpha }-1,n_{\gamma },\cdots \rangle .}" loading="lazy"></dd></dl>

These two equations can be considered as the defining properties of boson creation and annihilation operators in the second-quantization formalism. The complicated symmetrization of the underlying first-quantized wave function is automatically taken care of by the creation and annihilation operators (when acting on the first-quantized wave function), so that the complexity is not revealed on the second-quantized level, and the second-quantization formulae are simple and neat.

<div class="mw-heading mw-heading4"><h4 id="Operator_identities">Operator identities</h4></div>

The following operator identities follow from the action of the boson creation and annihilation operators on the Fock state,

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [b_{\alpha }^{\dagger },b_{\beta }^{\dagger }]=[b_{\alpha },b_{\beta }]=0,\quad [b_{\alpha },b_{\beta }^{\dagger }]=\delta _{\alpha \beta }.}</annotation>

</semantics>

</math><img src="./a5b2c3d6f35ccd6c64d48b10db034a89b3f90a08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:38.063ex; height:3.843ex;" alt="{\displaystyle [b_{\alpha }^{\dagger },b_{\beta }^{\dagger }]=[b_{\alpha },b_{\beta }]=0,\quad [b_{\alpha },b_{\beta }^{\dagger }]=\delta _{\alpha \beta }.}" loading="lazy"></dd></dl>

These commutation relations can be considered as the algebraic definition of the boson creation and annihilation operators. The fact that the boson many-body wave function is symmetric under particle exchange is also manifested by the commutation of the boson operators.

The raising and lowering operators of the <a href="Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> also satisfy the same set of commutation relations, implying that the bosons can be interpreted as the energy quanta (phonons) of an oscillator. The position and momentum operators of a Harmonic oscillator (or a collection of Harmonic oscillating modes) are given by Hermitian combinations of phonon creation and annihilation operators,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{\alpha }=(b_{\alpha }+b_{\alpha }^{\dagger })/{\sqrt {2}},\quad p_{\alpha }=(b_{\alpha }-b_{\alpha }^{\dagger })/({\sqrt {2}}\mathrm {i} ),}</annotation>

</semantics>

</math><img src="./73b3ac1f0c71edb4ff71040f9c2d203c8f4a5914.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.672ex; height:3.343ex;" alt="{\displaystyle x_{\alpha }=(b_{\alpha }+b_{\alpha }^{\dagger })/{\sqrt {2}},\quad p_{\alpha }=(b_{\alpha }-b_{\alpha }^{\dagger })/({\sqrt {2}}\mathrm {i} ),}" loading="lazy"></dd></dl>

which reproduce the canonical commutation relation between position and momentum operators (with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \hbar =1}</annotation>

</semantics>

</math><img src="./6e5f56ce258c75510831a8a14f3e2970ef0a1467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.567ex; height:2.176ex;" alt="{\displaystyle \hbar =1}" loading="lazy">)

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x_{\alpha },p_{\beta }]=\mathrm {i} \delta _{\alpha \beta },\quad [x_{\alpha },x_{\beta }]=[p_{\alpha },p_{\beta }]=0.}</annotation>

</semantics>

</math><img src="./58581c5d5b15030d6075e7fce226e9bd9e239d3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.222ex; height:3.009ex;" alt="{\displaystyle [x_{\alpha },p_{\beta }]=\mathrm {i} \delta _{\alpha \beta },\quad [x_{\alpha },x_{\beta }]=[p_{\alpha },p_{\beta }]=0.}" loading="lazy"></dd></dl>

This idea is generalized in the <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, which considers each mode of the matter field as an oscillator subject to quantum fluctuations, and the bosons are treated as the excitations (or energy quanta) of the field.

<div class="mw-heading mw-heading3"><h3 id="Fermion_creation_and_annihilation_operators">Fermion creation and annihilation operators</h3></div>

The fermion creation (annihilation) operator is usually denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\alpha }^{\dagger }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }^{\dagger }}</annotation>

</semantics>

</math><img src="./ca31fed66582965ee59d912172fd3b9d260792b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:3.176ex;" alt="{\displaystyle c_{\alpha }^{\dagger }}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6dcb7aadbcfa0ea1be48b6d5e135843344acbc3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:2.009ex;" alt="{\displaystyle c_{\alpha }}" loading="lazy">). The creation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\alpha }^{\dagger }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }^{\dagger }}</annotation>

</semantics>

</math><img src="./ca31fed66582965ee59d912172fd3b9d260792b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:3.176ex;" alt="{\displaystyle c_{\alpha }^{\dagger }}" loading="lazy"> adds a fermion to the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f42032e642ee1c9d27adb318d34c7cc85f7a95d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle |\alpha \rangle }" loading="lazy">, and the annihilation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6dcb7aadbcfa0ea1be48b6d5e135843344acbc3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:2.009ex;" alt="{\displaystyle c_{\alpha }}" loading="lazy"> removes a fermion from the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f42032e642ee1c9d27adb318d34c7cc85f7a95d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle |\alpha \rangle }" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Definition_2">Definition</h4></div>

The fermion creation (annihilation) operator is a linear operator, whose action on a N-particle first-quantized wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5471531a3fe80741a839bc98d49fae862a6439a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" loading="lazy"> is defined as

</mrow>

</mrow>

</msubsup>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }^{\dagger }\Psi ={\frac {1}{\sqrt {N+1}}}\psi _{\alpha }\otimes _{-}\Psi ,}</annotation>

</semantics>

</math><img src="./9e5475c84d2b35c8cb1e9a17a73cabc405ed1f5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.64ex; height:6.176ex;" alt="{\displaystyle c_{\alpha }^{\dagger }\Psi ={\frac {1}{\sqrt {N+1}}}\psi _{\alpha }\otimes _{-}\Psi ,}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }\Psi ={\frac {1}{\sqrt {N}}}\psi _{\alpha }\oslash _{-}\Psi ,}</annotation>

</semantics>

</math><img src="./07084597179747c7feda70c8e14004a4fe443054.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.637ex; height:6.176ex;" alt="{\displaystyle c_{\alpha }\Psi ={\frac {1}{\sqrt {N}}}\psi _{\alpha }\oslash _{-}\Psi ,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }\otimes _{-}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./de3884d684914a255afcf0be256efb04733a2ec2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.116ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }\otimes _{-}}" loading="lazy"> inserts the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./eae086daf01dbbd5684898cccb1b1b7baf234dde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N+1}</annotation>

</semantics>

</math><img src="./fdf2b9cbfe9051fd4e7b50c8028866d497eac35b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N+1}" loading="lazy"> possible insertion positions anti-symmetrically, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }\oslash _{-}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./768203ef9c808f50c88c02a357dd019ee0e5cb78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.116ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }\oslash _{-}}" loading="lazy"> deletes the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./eae086daf01dbbd5684898cccb1b1b7baf234dde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> possible deletion positions anti-symmetrically.

It is particularly instructive to view the results of creation and annihilation operators on states of two (or more) fermions, because they demonstrate the effects of exchange. A few illustrative operations are given in the example below. The complete algebra for creation and annihilation operators on a two-fermion state can be found in Quantum Photonics.<a href="#cite_note-Pearsall2020-8">[8]</a>

<div class="mw-heading mw-heading5"><h5 id="Examples_2">Examples</h5></div>

Hereinafter the tensor symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"> between single-particle states is omitted for simplicity. Take the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}-\psi _{2}\psi _{1})/{\sqrt {2}}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}-\psi _{2}\psi _{1})/{\sqrt {2}}}</annotation>

</semantics>

</math><img src="./6cf923d715aabb5701b2046d8cc306a0ffd14ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.297ex; height:3.176ex;" alt="{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}-\psi _{2}\psi _{1})/{\sqrt {2}}}" loading="lazy">, attempt to create one more fermion on the occupied <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{1}}</annotation>

</semantics>

</math><img src="./8cfdde1da54e02a016fe2a230c58b25dfcc014d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{1}}" loading="lazy"> state will quench the whole many-body wave function,

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}c_{1}^{\dagger }|1_{1},1_{2}\rangle =&{\frac {1}{\sqrt {2}}}(c_{1}^{\dagger }\psi _{1}\psi _{2}-c_{1}^{\dagger }\psi _{2}\psi _{1})\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {3}}}\psi _{1}\otimes _{-}\psi _{1}\psi _{2}-{\frac {1}{\sqrt {3}}}\psi _{1}\otimes _{-}\psi _{2}\psi _{1}\right)\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{1}\psi _{2}-\psi _{1}\psi _{1}\psi _{2}+\psi _{1}\psi _{2}\psi _{1})-{\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{2}\psi _{1}-\psi _{2}\psi _{1}\psi _{1}+\psi _{2}\psi _{1}\psi _{1})\right)\\=&0.\end{array}}}</annotation>

</semantics>

</math><img src="./4688b1db2a73068cde3016e0d2940525beb986c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.505ex; width:90.885ex; height:18.176ex;" alt="{\displaystyle {\begin{array}{rl}c_{1}^{\dagger }|1_{1},1_{2}\rangle =&{\frac {1}{\sqrt {2}}}(c_{1}^{\dagger }\psi _{1}\psi _{2}-c_{1}^{\dagger }\psi _{2}\psi _{1})\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {3}}}\psi _{1}\otimes _{-}\psi _{1}\psi _{2}-{\frac {1}{\sqrt {3}}}\psi _{1}\otimes _{-}\psi _{2}\psi _{1}\right)\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{1}\psi _{2}-\psi _{1}\psi _{1}\psi _{2}+\psi _{1}\psi _{2}\psi _{1})-{\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{2}\psi _{1}-\psi _{2}\psi _{1}\psi _{1}+\psi _{2}\psi _{1}\psi _{1})\right)\\=&0.\end{array}}}" loading="lazy"></dd></dl>

Annihilate a fermion on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{2}}</annotation>

</semantics>

</math><img src="./c5083a526766f85c4f39ab695791b0b739f06897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{2}}" loading="lazy"> state,

take the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}-\psi _{2}\psi _{1})/{\sqrt {2}}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}-\psi _{2}\psi _{1})/{\sqrt {2}}}</annotation>

</semantics>

</math><img src="./6cf923d715aabb5701b2046d8cc306a0ffd14ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.297ex; height:3.176ex;" alt="{\displaystyle |1_{1},1_{2}\rangle =(\psi _{1}\psi _{2}-\psi _{2}\psi _{1})/{\sqrt {2}}}" loading="lazy">,

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}c_{2}|1_{1},1_{2}\rangle =&{\frac {1}{\sqrt {2}}}(c_{2}\psi _{1}\psi _{2}-c_{2}\psi _{2}\psi _{1})\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {2}}}\psi _{2}\oslash _{-}\psi _{1}\psi _{2}-{\frac {1}{\sqrt {2}}}\psi _{2}\oslash _{-}\psi _{2}\psi _{1}\right)\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {2}}}(0-\psi _{1})-{\frac {1}{\sqrt {2}}}(\psi _{1}-0)\right)\\=&-\psi _{1}\\=&-|1_{1},0_{2}\rangle .\end{array}}}</annotation>

</semantics>

</math><img src="./88d25236dbca7f23f32cfd6b5a698e8ae094a615.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:53.798ex; height:21.176ex;" alt="{\displaystyle {\begin{array}{rl}c_{2}|1_{1},1_{2}\rangle =&{\frac {1}{\sqrt {2}}}(c_{2}\psi _{1}\psi _{2}-c_{2}\psi _{2}\psi _{1})\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {2}}}\psi _{2}\oslash _{-}\psi _{1}\psi _{2}-{\frac {1}{\sqrt {2}}}\psi _{2}\oslash _{-}\psi _{2}\psi _{1}\right)\\=&{\frac {1}{\sqrt {2}}}\left({\frac {1}{\sqrt {2}}}(0-\psi _{1})-{\frac {1}{\sqrt {2}}}(\psi _{1}-0)\right)\\=&-\psi _{1}\\=&-|1_{1},0_{2}\rangle .\end{array}}}" loading="lazy"></dd></dl>

The minus sign (known as the fermion sign) appears due to the anti-symmetric property of the fermion wave function.

<div class="mw-heading mw-heading4"><h4 id="Action_on_Fock_states_2">Action on Fock states</h4></div>

Starting from the single-mode vacuum state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0_{\alpha }\rangle =1}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |0_{\alpha }\rangle =1}</annotation>

</semantics>

</math><img src="./2f517186ed9e650e33d09028106fbfd07793b6a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.259ex; height:2.843ex;" alt="{\displaystyle |0_{\alpha }\rangle =1}" loading="lazy">, applying the fermion creation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\alpha }^{\dagger }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }^{\dagger }}</annotation>

</semantics>

</math><img src="./ca31fed66582965ee59d912172fd3b9d260792b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:3.176ex;" alt="{\displaystyle c_{\alpha }^{\dagger }}" loading="lazy">,

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }^{\dagger }|0_{\alpha }\rangle =\psi _{\alpha }\otimes _{-}1=\psi _{\alpha }=|1_{\alpha }\rangle ,}</annotation>

</semantics>

</math><img src="./e6cf7fa9273d7d81ecf1c84f034e330641edba85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.338ex; height:3.343ex;" alt="{\displaystyle c_{\alpha }^{\dagger }|0_{\alpha }\rangle =\psi _{\alpha }\otimes _{-}1=\psi _{\alpha }=|1_{\alpha }\rangle ,}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }^{\dagger }|1_{\alpha }\rangle ={\frac {1}{\sqrt {2}}}\psi _{\alpha }\otimes _{-}\psi _{\alpha }=0.}</annotation>

</semantics>

</math><img src="./38d6176ba5d59796a0dc7f165af9e598e2ee6a90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.176ex; height:6.176ex;" alt="{\displaystyle c_{\alpha }^{\dagger }|1_{\alpha }\rangle ={\frac {1}{\sqrt {2}}}\psi _{\alpha }\otimes _{-}\psi _{\alpha }=0.}" loading="lazy"></dd></dl>

If the single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f42032e642ee1c9d27adb318d34c7cc85f7a95d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle |\alpha \rangle }" loading="lazy"> is empty, the creation operator will fill the state with a fermion. However, if the state is already occupied by a fermion, further application of the creation operator will quench the state, demonstrating the <a href="Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> that two identical fermions can not occupy the same state simultaneously. Nevertheless, the fermion can be removed from the occupied state by the fermion annihilation operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6dcb7aadbcfa0ea1be48b6d5e135843344acbc3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:2.009ex;" alt="{\displaystyle c_{\alpha }}" loading="lazy">,

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }|1_{\alpha }\rangle =\psi _{\alpha }\oslash _{-}\psi _{\alpha }=1=|0_{\alpha }\rangle ,}</annotation>

</semantics>

</math><img src="./d3cac8e967d8b103696c4cdcb77e16e0bf0d0c51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.338ex; height:2.843ex;" alt="{\displaystyle c_{\alpha }|1_{\alpha }\rangle =\psi _{\alpha }\oslash _{-}\psi _{\alpha }=1=|0_{\alpha }\rangle ,}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }|0_{\alpha }\rangle =0.}</annotation>

</semantics>

</math><img src="./c52625e805de5febad84c641f675681084621fbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.197ex; height:2.843ex;" alt="{\displaystyle c_{\alpha }|0_{\alpha }\rangle =0.}" loading="lazy"></dd></dl>

The vacuum state is quenched by the action of the annihilation operator.

Similar to the boson case, the fermion Fock state can be constructed from the vacuum state using the fermion creation operator

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |n_{\alpha }\rangle =(c_{\alpha }^{\dagger })^{n_{\alpha }}|0_{\alpha }\rangle .}</annotation>

</semantics>

</math><img src="./171a7fcd9a924565730087cff85d5fc5fe7cff0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.311ex; height:3.343ex;" alt="{\displaystyle |n_{\alpha }\rangle =(c_{\alpha }^{\dagger })^{n_{\alpha }}|0_{\alpha }\rangle .}" loading="lazy"></dd></dl>

It is easy to check (by enumeration) that

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }^{\dagger }c_{\alpha }|n_{\alpha }\rangle =n_{\alpha }|n_{\alpha }\rangle ,}</annotation>

</semantics>

</math><img src="./d0783be1bb3f1ce6954c20effb364b6b6e364a8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.467ex; height:3.343ex;" alt="{\displaystyle c_{\alpha }^{\dagger }c_{\alpha }|n_{\alpha }\rangle =n_{\alpha }|n_{\alpha }\rangle ,}" loading="lazy"></dd></dl>

meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}_{\alpha }=c_{\alpha }^{\dagger }c_{\alpha }}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}_{\alpha }=c_{\alpha }^{\dagger }c_{\alpha }}</annotation>

</semantics>

</math><img src="./aeb1959a05a14a98995702933198f81688de7c3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.359ex; height:3.176ex;" alt="{\displaystyle {\hat {n}}_{\alpha }=c_{\alpha }^{\dagger }c_{\alpha }}" loading="lazy"> defines the fermion number operator.

The above result can be generalized to any Fock state of fermions.

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }^{\dagger }|\cdots ,n_{\beta },n_{\alpha },n_{\gamma },\cdots \rangle =(-1)^{\sum _{\beta <\alpha }n_{\beta }}{\sqrt {1-n_{\alpha }}}|\cdots ,n_{\beta },1+n_{\alpha },n_{\gamma },\cdots \rangle .}</annotation>

</semantics>

</math><img src="./d4e9ca84b37d6d33d89935613628b17b0cc8b450.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:70.265ex; height:3.676ex;" alt="{\displaystyle c_{\alpha }^{\dagger }|\cdots ,n_{\beta },n_{\alpha },n_{\gamma },\cdots \rangle =(-1)^{\sum _{\beta <\alpha }n_{\beta }}{\sqrt {1-n_{\alpha }}}|\cdots ,n_{\beta },1+n_{\alpha },n_{\gamma },\cdots \rangle .}" loading="lazy"><a href="#cite_note-9">[9]</a></dd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }|\cdots ,n_{\beta },n_{\alpha },n_{\gamma },\cdots \rangle =(-1)^{\sum _{\beta <\alpha }n_{\beta }}{\sqrt {n_{\alpha }}}|\cdots ,n_{\beta },1-n_{\alpha },n_{\gamma },\cdots \rangle .}</annotation>

</semantics>

</math><img src="./89ae2ebf305348e9ebbf52486cbb09b7bb4e142a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:65.875ex; height:3.676ex;" alt="{\displaystyle c_{\alpha }|\cdots ,n_{\beta },n_{\alpha },n_{\gamma },\cdots \rangle =(-1)^{\sum _{\beta <\alpha }n_{\beta }}{\sqrt {n_{\alpha }}}|\cdots ,n_{\beta },1-n_{\alpha },n_{\gamma },\cdots \rangle .}" loading="lazy"></dd></dl>

Recall that the occupation number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./28ba9deb2ca590784ac01c1778371105e371cc88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.679ex; height:2.009ex;" alt="{\displaystyle n_{\alpha }}" loading="lazy"> can only take 0 or 1 for fermions. These two equations can be considered as the defining properties of fermion creation and annihilation operators in the second quantization formalism. Note that the fermion sign structure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{\sum _{\beta <\alpha }n_{\beta }}}">

<msup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-1)^{\sum _{\beta <\alpha }n_{\beta }}}</annotation>

</semantics>

</math><img src="./4431a584dc95011778d7983c1de3e048e8efa39b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.87ex; height:3.343ex;" alt="{\displaystyle (-1)^{\sum _{\beta <\alpha }n_{\beta }}}" loading="lazy">, also known as the <a href="Jordan%E2%80%93Wigner_transformation" title="Jordan–Wigner transformation">Jordan-Wigner string</a>, requires there to exist a predefined ordering of the single-particle states (the <a href="Spin_structure" title="Spin structure">spin structure</a>) and involves a counting of the fermion occupation numbers of all the preceding states; therefore the fermion creation and annihilation operators are considered non-local in some sense. This observation leads to the idea that fermions are emergent particles in the long-range entangled local <a href="Qubit" title="Qubit">qubit</a> system.<a href="#cite_note-10">[10]</a>

<div class="mw-heading mw-heading4"><h4 id="Operator_identities_2">Operator identities</h4></div>

The following operator identities follow from the action of the fermion creation and annihilation operators on the Fock state,

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{c_{\alpha }^{\dagger },c_{\beta }^{\dagger }\}=\{c_{\alpha },c_{\beta }\}=0,\quad \{c_{\alpha },c_{\beta }^{\dagger }\}=\delta _{\alpha \beta }.}</annotation>

</semantics>

</math><img src="./af4f50a8d2b62073e7c819f8e1fe5f264da2ab21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:41.212ex; height:3.843ex;" alt="{\displaystyle \{c_{\alpha }^{\dagger },c_{\beta }^{\dagger }\}=\{c_{\alpha },c_{\beta }\}=0,\quad \{c_{\alpha },c_{\beta }^{\dagger }\}=\delta _{\alpha \beta }.}" loading="lazy"></dd></dl>

These anti-commutation relations can be considered as the algebraic definition of the fermion creation and annihilation operators. The fact that the fermion many-body wave function is anti-symmetric under particle exchange is also manifested by the anti-commutation of the fermion operators.

The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\alpha }\neq c_{\alpha }^{\dagger }}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\alpha }\neq c_{\alpha }^{\dagger }}</annotation>

</semantics>

</math><img src="./102023294be22cff49653a6a384f26e2ca05de5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.68ex; height:3.343ex;" alt="{\displaystyle c_{\alpha }\neq c_{\alpha }^{\dagger }}" loading="lazy">). The Hermitian combination of the fermion creation and annihilation operators

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{\alpha ,{\text{Re}}}=(c_{\alpha }+c_{\alpha }^{\dagger })/{\sqrt {2}},\quad \chi _{\alpha ,{\text{Im}}}=(c_{\alpha }-c_{\alpha }^{\dagger })/({\sqrt {2}}\mathrm {i} ),}</annotation>

</semantics>

</math><img src="./f09d11d657fdcebd8cb31365335c7d7566f50018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.937ex; height:3.509ex;" alt="{\displaystyle \chi _{\alpha ,{\text{Re}}}=(c_{\alpha }+c_{\alpha }^{\dagger })/{\sqrt {2}},\quad \chi _{\alpha ,{\text{Im}}}=(c_{\alpha }-c_{\alpha }^{\dagger })/({\sqrt {2}}\mathrm {i} ),}" loading="lazy"></dd></dl>

are called <a href="Majorana_fermion" title="Majorana fermion">Majorana fermion</a> operators. They can be viewed as the fermionic analog of position and momentum operators of a "fermionic" Harmonic oscillator. They satisfy the anticommutation relation

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\chi _{i},\chi _{j}\}=\delta _{ij},}</annotation>

</semantics>

</math><img src="./31fb27cacf1216252f93c585ca17fd87b7d61d68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.233ex; height:3.009ex;" alt="{\displaystyle \{\chi _{i},\chi _{j}\}=\delta _{ij},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f4cbf8bbc622154cda8208d6e339495fe16a1f9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle i,j}" loading="lazy"> labels any Majorana fermion operators on equal footing (regardless their origin from Re or Im combination of complex fermion operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6dcb7aadbcfa0ea1be48b6d5e135843344acbc3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:2.009ex;" alt="{\displaystyle c_{\alpha }}" loading="lazy">). The anticommutation relation indicates that Majorana fermion operators generates a <a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a>, which can be systematically represented as Pauli operators in the many-body Hilbert space.

<div class="mw-heading mw-heading2"><h2 id="Quantum_field_operators">Quantum field operators</h2></div>

Defining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\nu }^{\dagger }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{\nu }^{\dagger }}</annotation>

</semantics>

</math><img src="./0c734f2c06f98dfef798942d05f04acf1292d0b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.333ex; height:3.176ex;" alt="{\displaystyle a_{\nu }^{\dagger }}" loading="lazy"> as a general annihilation (creation) operator for a single-particle state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> that could be either fermionic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{\nu }^{\dagger })}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (c_{\nu }^{\dagger })}</annotation>

</semantics>

</math><img src="./26c0c88a2a63aa2ba345f9ac4acb1e27218a4d04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.92ex; height:3.343ex;" alt="{\displaystyle (c_{\nu }^{\dagger })}" loading="lazy"> or bosonic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{\nu }^{\dagger })}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (b_{\nu }^{\dagger })}</annotation>

</semantics>

</math><img src="./4a257485e8e2c3d8a8b9e00e44677d9e26452713.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.91ex; height:3.343ex;" alt="{\displaystyle (b_{\nu }^{\dagger })}" loading="lazy">, the <a href="Position_and_momentum_space" class="mw-redirect" title="Position and momentum space">real space representation</a> of the operators defines the <a href="Quantum" title="Quantum">quantum</a> field <a href="Operator_(physics)" title="Operator (physics)">operators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\mathbf {r} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} )}</annotation>

</semantics>

</math><img src="./e478d6f260286a13b6516ffecb4787fb0b2aaae5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.72ex; height:2.843ex;" alt="{\displaystyle \Psi (\mathbf {r} )}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi ^{\dagger }(\mathbf {r} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi ^{\dagger }(\mathbf {r} )}</annotation>

</semantics>

</math><img src="./4dd9f5ead305297426c991a45eb86338418f46b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.682ex; height:3.176ex;" alt="{\displaystyle \Psi ^{\dagger }(\mathbf {r} )}" loading="lazy"> by

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} )=\sum _{\nu }\psi _{\nu }\left(\mathbf {r} \right)a_{\nu }}</annotation>

</semantics>

</math><img src="./f4f008dbf5817545bca49bb6f7fa47e16203bd7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.196ex; height:5.509ex;" alt="{\displaystyle \Psi (\mathbf {r} )=\sum _{\nu }\psi _{\nu }\left(\mathbf {r} \right)a_{\nu }}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi ^{\dagger }(\mathbf {r} )=\sum _{\nu }\psi _{\nu }^{*}\left(\mathbf {r} \right)a_{\nu }^{\dagger }}</annotation>

</semantics>

</math><img src="./55285f64c56d5ff436b91525afe6db710d92dc1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.158ex; height:5.509ex;" alt="{\displaystyle \Psi ^{\dagger }(\mathbf {r} )=\sum _{\nu }\psi _{\nu }^{*}\left(\mathbf {r} \right)a_{\nu }^{\dagger }}" loading="lazy"></dd></dl>

These are second quantization operators, with coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\nu }\left(\mathbf {r} \right)}">

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{\nu }\left(\mathbf {r} \right)}</annotation>

</semantics>

</math><img src="./120cd5ab34898cf69a2109e991de3bb411cd0639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.915ex; height:2.843ex;" alt="{\displaystyle \psi _{\nu }\left(\mathbf {r} \right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\nu }^{*}\left(\mathbf {r} \right)}">

</mrow>

</mrow>

</msubsup>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{\nu }^{*}\left(\mathbf {r} \right)}</annotation>

</semantics>

</math><img src="./7749212bdc05910415d253f79a8f24502662ca0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.915ex; height:2.843ex;" alt="{\displaystyle \psi _{\nu }^{*}\left(\mathbf {r} \right)}" loading="lazy"> that are ordinary <a href="First_quantization" title="First quantization">first-quantization</a> <a href="Wavefunctions" class="mw-redirect" title="Wavefunctions">wavefunctions</a>. Thus, for example, any expectation values will be ordinary first-quantization wavefunctions. Loosely speaking, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi ^{\dagger }(\mathbf {r} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi ^{\dagger }(\mathbf {r} )}</annotation>

</semantics>

</math><img src="./4dd9f5ead305297426c991a45eb86338418f46b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.682ex; height:3.176ex;" alt="{\displaystyle \Psi ^{\dagger }(\mathbf {r} )}" loading="lazy"> is the sum of all possible ways to add a particle to the system at position r through any of the basis states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\nu }\left(\mathbf {r} \right)}">

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{\nu }\left(\mathbf {r} \right)}</annotation>

</semantics>

</math><img src="./120cd5ab34898cf69a2109e991de3bb411cd0639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.915ex; height:2.843ex;" alt="{\displaystyle \psi _{\nu }\left(\mathbf {r} \right)}" loading="lazy">, not necessarily plane waves, as below.

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\mathbf {r} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} )}</annotation>

</semantics>

</math><img src="./e478d6f260286a13b6516ffecb4787fb0b2aaae5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.72ex; height:2.843ex;" alt="{\displaystyle \Psi (\mathbf {r} )}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi ^{\dagger }(\mathbf {r} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi ^{\dagger }(\mathbf {r} )}</annotation>

</semantics>

</math><img src="./4dd9f5ead305297426c991a45eb86338418f46b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.682ex; height:3.176ex;" alt="{\displaystyle \Psi ^{\dagger }(\mathbf {r} )}" loading="lazy"> are second quantization operators defined in every point in space they are called <a href="Quantum_field" class="mw-redirect" title="Quantum field">quantum field</a> operators. They obey the following fundamental commutator and anti-commutator relations,

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[\Psi (\mathbf {r} _{1}),\Psi ^{\dagger }(\mathbf {r} _{2})\right]=\delta (\mathbf {r} _{1}-\mathbf {r} _{2})}</annotation>

</semantics>

</math><img src="./adfab9b2bae3d90b9a2aaacecf82f9882e3c4eca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.592ex; height:3.343ex;" alt="{\displaystyle \left[\Psi (\mathbf {r} _{1}),\Psi ^{\dagger }(\mathbf {r} _{2})\right]=\delta (\mathbf {r} _{1}-\mathbf {r} _{2})}" loading="lazy"> boson fields,</dd>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\Psi (\mathbf {r} _{1}),\Psi ^{\dagger }(\mathbf {r} _{2})\}=\delta (\mathbf {r} _{1}-\mathbf {r} _{2})}</annotation>

</semantics>

</math><img src="./dfc012cdc1a87642b4e6e67dae64ce85476c997d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.978ex; height:3.176ex;" alt="{\displaystyle \{\Psi (\mathbf {r} _{1}),\Psi ^{\dagger }(\mathbf {r} _{2})\}=\delta (\mathbf {r} _{1}-\mathbf {r} _{2})}" loading="lazy"> fermion fields.</dd></dl>

For homogeneous systems it is often desirable to transform between real space and the momentum representations, hence, the quantum fields operators in <a href="Fourier_transform" title="Fourier transform">Fourier basis</a> yields:

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} )={1 \over {\sqrt {V}}}\sum _{\mathbf {k} }e^{i\mathbf {k\cdot r} }a_{\mathbf {k} }}</annotation>

</semantics>

</math><img src="./f6d1424faf283a53b7dd180ace9c8e8e40521633.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.084ex; height:6.343ex;" alt="{\displaystyle \Psi (\mathbf {r} )={1 \over {\sqrt {V}}}\sum _{\mathbf {k} }e^{i\mathbf {k\cdot r} }a_{\mathbf {k} }}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi ^{\dagger }(\mathbf {r} )={1 \over {\sqrt {V}}}\sum _{\mathbf {k} }e^{-i\mathbf {k\cdot r} }a_{\mathbf {k} }^{\dagger }}</annotation>

</semantics>

</math><img src="./c082dcc7018ec0de340941253e12e583a2655f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.325ex; height:6.343ex;" alt="{\displaystyle \Psi ^{\dagger }(\mathbf {r} )={1 \over {\sqrt {V}}}\sum _{\mathbf {k} }e^{-i\mathbf {k\cdot r} }a_{\mathbf {k} }^{\dagger }}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Comment_on_nomenclature">Comment on nomenclature</h2></div>

The term "second quantization", introduced by Jordan,<a href="#cite_note-Todorov2012-11">[11]</a> is a misnomer that has persisted for historical reasons. At the origin of quantum field theory, it was inappositely thought that the <a href="Dirac_equation" title="Dirac equation">Dirac equation</a> described a relativistic wavefunction (hence the obsolete "Dirac sea" interpretation), rather than a classical spinor field which, when quantized (like the scalar field), yielded a fermionic quantum field (vs. a bosonic quantum field).

One is not quantizing "again", as the term "second" might suggest; the field that is being quantized is not a <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger wave function</a> that was produced as the result of quantizing a particle, but is a classical field (such as the electromagnetic field or <a href="Dirac_spinor" title="Dirac spinor">Dirac spinor</a> field), essentially an assembly of coupled oscillators, that was not previously quantized. One is merely quantizing each oscillator in this assembly, shifting from a <a href="Semiclassical_physics" title="Semiclassical physics">semiclassical</a> treatment of the system to a fully quantum-mechanical one.

<ul><li><a href="Canonical_quantization" title="Canonical quantization">Canonical quantization</a></li>

<li><a href="First_quantization" title="First quantization">First quantization</a></li>

<li><a href="Geometric_quantization" title="Geometric quantization">Geometric quantization</a></li>

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

<li><a href="Schr%C3%B6dinger_functional" title="Schrödinger functional">Schrödinger functional</a></li>

<li><a href="Scalar_field_theory" title="Scalar field theory">Scalar field theory</a></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .infobox .side-box{font-size:100%}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1237033735">

/* start https://en.wikipedia.org/ */

@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}

/* end https://en.wikipedia.org/ */

</style><div class="side-box side-box-right sistersitebox">

<div class="side-box-text plainlist"><a href="https://en.wikiversity.org/wiki/Quantum_mechanics/Quantum_field_theory_on_a_violin_string" class="extiw external" title="v:Quantum mechanics/Quantum field theory on a violin string">Second quantization on Wikiversity</a></div></div>

</div>

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

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist reflist-columns references-column-width reflist-columns-3">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFDirac1927" class="citation journal cs1">Dirac, Paul Adrien Maurice (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>. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 114 (767): 243–265. <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>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1927.0039">10.1098/rspa.1927.0039</a>.</cite>

</li>

<li id="cite_note-Jordan1928-2"><a href="#cite_ref-Jordan1928_2-0">^</a> <cite id="CITEREFJordanWigner1928" class="citation journal cs1 cs1-prop-foreign-lang-source">Jordan, Pascual; Wigner, Eugene (1928). "Über das Paulische Äquivalenzverbot". Zeitschrift für Physik (in German). 47 (9): 631–651. <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/1928ZPhy...47..631J">1928ZPhy...47..631J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01331938">10.1007/bf01331938</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:126400679">126400679</a>.</cite>

</li>

<li id="cite_note-Fock1932-3"><a href="#cite_ref-Fock1932_3-0">^</a> <cite id="CITEREFFock1932" class="citation journal cs1 cs1-prop-foreign-lang-source">Fock, Vladimir Aleksandrovich (1932). "Konfigurationsraum und zweite Quantelung". Zeitschrift für Physik (in German). 75 (9–10): 622–647. <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/1932ZPhy...75..622F">1932ZPhy...75..622F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01344458">10.1007/bf01344458</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186238995">186238995</a>.</cite>

</li>

<li id="cite_note-Reed1975-4"><a href="#cite_ref-Reed1975_4-0">^</a> <cite id="CITEREFReedSimon1975" class="citation book cs1"><a href="Michael_C._Reed" title="Michael C. Reed">Reed, Michael</a>; <a href="Barry_Simon" title="Barry Simon">Simon, Barry</a> (1975). Methods of Modern Mathematical Physics. Volume II: Fourier Analysis, Self-Adjointness. San Diego: Academic Press. p. 328. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780080925370</bdi>.</cite>

</li>

<li id="cite_note-Becchi2010-5"><a href="#cite_ref-Becchi2010_5-0">^</a> <cite id="CITEREFBecchi2010" class="citation journal cs1">Becchi, Carlo Maria (2010). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.7902">"Second quantization"</a>. Scholarpedia. 5 (6): 7902. <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/2010SchpJ...5.7902B">2010SchpJ...5.7902B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.7902">10.4249/scholarpedia.7902</a>.</cite>

</li>

<li id="cite_note-Koch2013-6"><a href="#cite_ref-Koch2013_6-0">^</a> <cite id="CITEREFKoch,_Erik2013" class="citation book cs1">Koch, Erik (2013). "Many-electron states". In Pavarini, Eva; Koch, Erik; Schollwöck, Ulrich (eds.). <a rel="nofollow" class="external text" href="http://hdl.handle.net/2128/5389">Emergent Phenomena in Correlated Matter</a>. Modeling and Simulation. Vol. 3. Jülich: Verlag des Forschungszentrum Jülich. pp. 2.1 – 2.26. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2128%2F5389">2128/5389</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-89336-884-6</bdi>.</cite>

</li>

<li id="cite_note-Mahan2000-7"><a href="#cite_ref-Mahan2000_7-0">^</a> <cite id="CITEREFMahan,_Gerald_D.2000" class="citation book cs1">Mahan, Gerald D. (2000). Many-Particle Physics. Physics of Solids and Liquids (3rd ed.). New York: Springer. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-5714-9">10.1007/978-1-4757-5714-9</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4757-5714-9</bdi>.</cite>

</li>

<li id="cite_note-Pearsall2020-8"><a href="#cite_ref-Pearsall2020_8-0">^</a> <cite id="CITEREFPearsall,_Thomas_P.2020" class="citation book cs1">Pearsall, Thomas P. (2020). Quantum Photonics. Graduate Texts in Physics (2nd ed.). Cham, Switzerland: Springer. pp. 301–302. <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/2020quph.book.....P">2020quph.book.....P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-47325-9">10.1007/978-3-030-47325-9</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-030-47325-9</bdi>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> Book "Nuclear Models" of Greiner and Maruhn p53 equation 3.47 : <a rel="nofollow" class="external free" href="http://xn--webducation-dbb.com/wp-content/uploads/2019/02/Walter-Greiner-Joachim-A.-Maruhn-D.A.-Bromley-Nuclear-Models-Springer-Verlag-1996.pdf">http://xn--webducation-dbb.com/wp-content/uploads/2019/02/Walter-Greiner-Joachim-A.-Maruhn-D.A.-Bromley-Nuclear-Models-Springer-Verlag-1996.pdf</a>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFLevinWen2003" class="citation journal cs1">Levin, M.; Wen, X. G. (2003). "Fermions, strings, and gauge fields in lattice spin models". Physical Review B. 67 (24): 245316. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0302460">cond-mat/0302460</a>. <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/2003PhRvB..67x5316L">2003PhRvB..67x5316L</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%2FPhysRevB.67.245316">10.1103/PhysRevB.67.245316</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:29180411">29180411</a>.</cite>

</li>

<li id="cite_note-Todorov2012-11"><a href="#cite_ref-Todorov2012_11-0">^</a> <cite id="CITEREFTodorov,_Ivan2012" class="citation journal cs1">Todorov, Ivan (2012). "Quantization is a mystery". Bulgarian Journal of Physics. 39 (2): 107–149. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1206.3116">1206.3116</a>.</cite>

</li>

</ol></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-08" href="https://en.wikipedia.org/wiki/?title=Second_quantization&oldid=1299528151">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>