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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> <h4 id="quantum_field_theory">Quantum field theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></strong></p> <h2 id="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a></p> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#for_sigmamodels'>For sigma-models</a></li> </ul> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="general">General</h3> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, the <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> in (<a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre</a>-)<a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> whose <a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a> is described by the <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> may in general depend on certain fixed <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">structure</a> which also has an interpretation of a <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a>, but one that is fixed and not regarded as having dynamics in the given <a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">model (in theoretical physics)</a>. These are often called <strong>background fields</strong>.</p> <h3 id="for_sigmamodels">For sigma-models</h3> <p>In particuar a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> describes the propagation of a <a class="existingWikiWord" href="/nlab/show/brane">brane</a> on some <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Typically the dynamics is controled by a <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> under which the brane is <a class="existingWikiWord" href="/nlab/show/charge">charge</a>d. This is the <strong>background gauge field</strong> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model. The term “background” alludes to the fact that this background gauge field is fixed and only serves to influence the dynamics of the brane coupled to it. More generally, one could also study the quantum dynamics of the possible background fields themselves. This would be given by the corresponding <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <h2 id="definition">Definition</h2> <p>A formalization of the notion is discussed at <em><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a></em> in <em><a href="field%20%28physics%29#DefinitionPhysicalField">Definition – Physical fields</a></em>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>The electrically charged <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a> is described by a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> with background gauge field the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/string">string</a> of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> may be described by a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> with background gauge field the <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> may be understood as the <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> of a <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> charged under a background field that is a <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a></p> </li> <li> <p><strong>background field</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 27, 2016 at 09:27:14. See the <a href="/nlab/history/background+field" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/background+field" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/background+field/5" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/background+field" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/background+field" accesskey="S" class="navlink" id="history" rel="nofollow">History (5 revisions)</a> <a href="/nlab/show/background+field/cite" style="color: black">Cite</a> <a href="/nlab/print/background+field" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/background+field" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>