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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="aqft">AQFT</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>) <a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a> <h2 id="concepts">Concepts</h2> <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+history">field history</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> </li> </ul> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>, <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+net">field net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> <a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>) </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> </li> </ul> </div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <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> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a> <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> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a 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particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <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> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <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> </li> <li> examples <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> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <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> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <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> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <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> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#3category_of_conformal_nets'>3-Category of conformal nets</a></li> <li><a href='#RelationToVertexOperatorAlgebras'>Relation to vertex operator algebras</a></li> <li><a href='#relation_to_spectral_triples'>Relation to spectral triples</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#free_fermion'>Free fermion</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#fundamentals'>Fundamentals</a></li> <li><a href='#ReferencesClassification'>Classification</a></li> <li><a href='#representation_theory'>Representation theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> A conformal net is a <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> that describes (<a class="existingWikiWord" href="/nlab/show/2d+CFT">2-dimensional</a>) <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> in the framework of <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a>. <h2 id="definition">Definition</h2> (…) <h2 id="properties">Properties</h2> <h3 id="3category_of_conformal_nets">3-Category of conformal nets</h3> (…) <ul> <li><a class="existingWikiWord" href="/nlab/show/3-category+of+fermionic+conformal+nets">3-category of fermionic conformal nets</a></li> </ul> <h3 id="RelationToVertexOperatorAlgebras">Relation to vertex operator algebras</h3> Subject to some conditions a <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a> may be turned into a conformal net and vice versa (<a href="#CarpiKawahigahshiLongoWeiner15">Carpi-Kawahigahshi-Longo-Weiner 15</a>, <a href="#Carpi16">Carpi 16</a>) also (<a href="#Gui18">Gui 18</a>). <h3 id="relation_to_spectral_triples">Relation to spectral triples</h3> Superconformal nets encoding <a class="existingWikiWord" href="/nlab/show/2d+SCFTs">2d SCFTs</a> may be related to <a class="existingWikiWord" href="/nlab/show/local+nets">local nets</a> of <a class="existingWikiWord" href="/nlab/show/spectral+triples">spectral triples</a> (<a href="#CHKL09">CHKL 09</a>) (an incarnation of “<a class="existingWikiWord" href="/nlab/show/2-spectral+triples">2-spectral triples</a>”, see there for more). <h2 id="examples">Examples</h2> <h3 id="free_fermion">Free fermion</h3> (<a href="#DouglasHenriquesGeometric">DH, section 4</a>) <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Fer</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Cl</mi><mo stretchy="false">(</mo><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>⊗</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Fer_n(I) \coloneqq Cl(L^2(I, \mathbb{R}^n \otimes S)) </annotation></semantics></math></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/conformal+structure">conformal structure</a>, <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+bootstrap">conformal bootstrap</a> </li> </ul> <h2 id="references">References</h2> <h3 id="fundamentals">Fundamentals</h3> Standard <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> references on conformal nets include <ul> <li><a class="existingWikiWord" href="/nlab/show/Roberto+Longo">Roberto Longo</a>, Conformal Subnets and Intermediate Subfactors (<a href="http://arxiv.org/abs/math/0102196">arXiv:0102196</a>)</li> </ul> and lots of others… A somewhat novel take on conformal nets is presented and studied in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Arthur+Bartels">Arthur Bartels</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Douglas">Chris Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Henriques">Andre Henriques</a>, Conformal nets and local field theory (<a href="http://arxiv.org/abs/0912.5307">arXiv:0912.5307</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Arthur+Bartels">Arthur Bartels</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Douglas">Chris Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Henriques">Andre Henriques</a>, Conformal nets I: coordinate-free nets (<a href="http://arxiv.org/abs/1302.2604">arXiv:1302.2604</a>) </li> </ul> where conformal nets are arranged into a <a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a> with <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal structure</a> that is a <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> of <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a>s and <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>s between these. Among other things, this work connects the <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> notion of conformal nets with the <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> notion of <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> representations. More on this in <ul id="DouglasHenriquesGeometric"> <li> <a class="existingWikiWord" href="/nlab/show/Chris+Douglas">Chris Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Henriques">André Henriques</a>, Topological modular forms and conformal nets , in <a class="existingWikiWord" href="/nlab/show/Branislav+Jur%C4%8Do">Branislav Jurčo</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.), <a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chris+Douglas">Chris Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Henriques">André Henriques</a>, Geometric string structures (<a href="http://www.staff.science.uu.nl/~henri105/PDF/TringWP.pdf">pdf</a>) </li> </ul> Relation to <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebras">vertex operator algebras</a> is discussed in <ul> <li id="CarpiKawahigahshiLongoWeiner15"> <a class="existingWikiWord" href="/nlab/show/Sebastiano+Carpi">Sebastiano Carpi</a>, <a class="existingWikiWord" href="/nlab/show/Yasuyuki+Kawahigashi">Yasuyuki Kawahigashi</a>, <a class="existingWikiWord" href="/nlab/show/Roberto+Longo">Roberto Longo</a>, Mihály Weiner, From vertex operator algebras to conformal nets and back, Memoirs of the American Mathematical Society, Volume 254, Number 1213 2018; (<a href="https://doi.org/10.1090/memo/1213">doi:10.1090/memo/1213</a>, <a href="http://arxiv.org/abs/1503.01260">arXiv:1503.01260</a>) </li> <li id="Carpi16"> <a class="existingWikiWord" href="/nlab/show/Sebastiano+Carpi">Sebastiano Carpi</a>, Operator algebras and vertex operator algebras, Contribution to the Proceedings of the 14th Marcel Grossmann Meeting - MG14 (Rome, 2015) (<a href="http://arxiv.org/abs/1603.06742">arXiv:1603.06742</a>) </li> </ul> and for <a class="existingWikiWord" href="/nlab/show/super+vertex+operator+algebras">super vertex operator algebras</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Sebastiano+Carpi">Sebastiano Carpi</a>, <a class="existingWikiWord" href="/nlab/show/Tiziano+Gaudio">Tiziano Gaudio</a>, <a class="existingWikiWord" href="/nlab/show/Robin+Hillier">Robin Hillier</a>, From vertex operator superalgebras to graded-local conformal nets and back [<a href="https://arxiv.org/abs/2304.14263">arXiv:2304.14263</a>]</li> </ul> See also <ul> <li>James E. Tener, Representation theory in chiral conformal field theory: from fields to observables (<a href="https://arxiv.org/abs/1810.08168">arXiv:1810.08168</a>)</li> </ul> Relation of the corresponding <a class="existingWikiWord" href="/nlab/show/ribbon+categories">ribbon categories</a>: <ul> <li id="Gui18">Bin Gui, Categorical extensions of conformal nets (<a href="https://arxiv.org/abs/1812.04470">arXiv:1812.04470</a>)</li> </ul> Relation to <a class="existingWikiWord" href="/nlab/show/2-spectral+triples">2-spectral triples</a> is discussed in terms of <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebras">vertex operator algebras</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yan+Soibelman">Yan Soibelman</a>, Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry , in <a class="existingWikiWord" href="/nlab/show/Branislav+Jur%C4%8Do">Branislav Jurčo</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.), <a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></li> </ul> and in terms of actual <a class="existingWikiWord" href="/nlab/show/local+nets">local nets</a> in <ul> <li id="CHKL09"><a class="existingWikiWord" href="/nlab/show/Sebastiano+Carpi">Sebastiano Carpi</a>, Robin Hillier, <a class="existingWikiWord" href="/nlab/show/Yasuyuki+Kawahigashi">Yasuyuki Kawahigashi</a>, <a class="existingWikiWord" href="/nlab/show/Roberto+Longo">Roberto Longo</a>, Spectral triples and the super-Virasoro algebra, Commun.Math.Phys.295:71-97 (2010) (<a href="http://arxiv.org/abs/0811.4128">arXiv:0811.4128</a>)</li> </ul> <h3 id="ReferencesClassification">Classification</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Yasuyuki+Kawahigashi">Yasuyuki Kawahigashi</a>, Classification of operator algebraic conformal field theories in dimensions one and two (<a href="http://arxiv.org/abs/math-ph/0308029">arXiv:math-ph/0308029</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Yasuyuki+Kawahigashi">Yasuyuki Kawahigashi</a>, <a class="existingWikiWord" href="/nlab/show/Roberto+Longo">Roberto Longo</a>, Classification of Two-dimensional Local Conformal Nets with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo><</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">c \lt 1</annotation></semantics></math> and 2-cohomology Vanishing for Tensor Categories (<a href="http://arxiv.org/abs/math-ph/0304022">arXiv</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Yasuyuki+Kawahigashi">Yasuyuki Kawahigashi</a>, <a class="existingWikiWord" href="/nlab/show/Roberto+Longo">Roberto Longo</a>, Classification of Local Conformal Nets. Case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo><</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">c \lt 1</annotation></semantics></math> (<a href="http://arxiv.org/abs/math-ph/0201015">arXiv</a>) </li> </ul> <h3 id="representation_theory">Representation theory</h3> Articles that discuss the <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> of conformal nets include <ul> <li><a class="existingWikiWord" href="/nlab/show/Sebastiano+Carpi">Sebastiano Carpi</a>, <a class="existingWikiWord" href="/nlab/show/Roberto+Conti">Roberto Conti</a>, Robin Hillier, Mihaly Weiner, Representations of Conformal Nets, Universal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-Algebras and K-Theory (<a href="http://arxiv.org/abs/1202.2543">arXiv:1202.2543</a>)</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on April 28, 2023 at 09:42:08. 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