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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="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 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> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <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> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </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/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <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> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of 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> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#InAlgebraicQuantumFieldTheory'>In algebraic quantum field theory</a></li> <li><a href='#in_smatrix_theories_and_string_theory'>In S-matrix theories and string theory</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#in_local_quantum_field_theory'>In local quantum field theory</a></li> <li><a href='#in_algebraic_quantum_field_theory_2'>In algebraic quantum field theory</a></li> <li><a href='#in_smatrix_theories_and_string_theory_2'>In S-matrix theories and string theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> A basic characteristic of <a class="existingWikiWord" href="/nlab/show/physics">physics</a> in the context of <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> and <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> is that causal influences on a <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian manifold</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> propagate in <a class="existingWikiWord" href="/nlab/show/timelike">timelike</a> or <a class="existingWikiWord" href="/nlab/show/lightlike">lightlike</a> directions but not <a class="existingWikiWord" href="/nlab/show/spacelike">spacelike</a>. The fact that any two spacelike-separated regions of spacetime thus behave like <a href="http://ncatlab.org/nlab/show/quantum+mechanics#Subsystems">independent subsystems</a> is called causal locality or, with a slightly stronger technical definition, Einstein causality. (One sometimes sees a further criterion to causality, that the causal influences in timelike and lightlike directions only propagate into the <a class="existingWikiWord" href="/nlab/show/future">future</a>, but this is not so simply dealt with; it probably only makes sense as a statement about coarse-grained <a class="existingWikiWord" href="/nlab/show/entropy">entropy</a> in <a class="existingWikiWord" href="/nlab/show/statistical+physics">statistical physics</a>.) From (<a href="#Grigoriev">Grigor’ev 197x</a>): <blockquote> Microcausality condition a requirement that the causality condition (which states that cause must precede effect) be satisfied down to an arbitrarily small distance and time interval. The microcausality condition usually refers to distances ≲ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>16</mn></mrow></msup></mrow><annotation encoding="application/x-tex">10^{-16}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/meter">cm</a> and to times ≲ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>24</mn></mrow></msup></mrow><annotation encoding="application/x-tex">10^{-24}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/second">sec</a>. It is shown in the theory of <a class="existingWikiWord" href="/nlab/show/special+relativity">relativity</a> that the assumption of the existence of physical signals that propagate with a velocity greater than the velocity of <a class="existingWikiWord" href="/nlab/show/light">light</a> leads to violation of the causality requirement. Thus, the microcausality condition prohibits the propagation of signals at a velocity greater than the velocity of <a class="existingWikiWord" href="/nlab/show/light">light</a> “in the small”. In <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum theory</a>, where <a class="existingWikiWord" href="/nlab/show/quantum+operator">operators</a> correspond to physical quantities, the microcausality condition requires the interchangeability of any operators that pertain to two points of <a class="existingWikiWord" href="/nlab/show/space-time">space-time</a> if these points cannot be linked by a light signal. This interchangeability means that the physical quantities to which these operators correspond can be precisely determined independently and simultaneously. The microcausality condition is important in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>, especially in the dispersion and <a class="existingWikiWord" href="/nlab/show/AQFT">axiomatic approaches</a>; these approaches are not based on specific <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model</a> concepts of interaction and therefore can be used for direct verification of the microcausality condition. In the most highly developed branch of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> — <a class="existingWikiWord" href="/nlab/show/quantum+electrodynamics">quantum electrodynamics</a> — the microcausality condition has been experimentally verified for distances ≲ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>15</mn></mrow></msup></mrow><annotation encoding="application/x-tex">10^{-15}</annotation></semantics></math> cm (and, correspondingly, for times ≲ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>25</mn></mrow></msup></mrow><annotation encoding="application/x-tex">10^{-25}</annotation></semantics></math> sec). The violation of the microcausality condition would make it necessary to radically alter the method of describing physical processes and to reject the dynamic description used in modern theories, in which the state of a physical system at a given moment of time (the effect) is determined by the states of the system at preceding times (the cause). </blockquote> Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>15</mn></mrow></msup></mrow><annotation encoding="application/x-tex">10^{-15}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/cm">cm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>17</mn></mrow></msup><mi>m</mi><mo>=</mo><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> = 10^{-17}m = 10^{-2}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/fm">fm</a> and that the (charge) <a class="existingWikiWord" href="/nlab/show/radius">radius</a> of the <a class="existingWikiWord" href="/nlab/show/proton">proton</a> is about 0.8 <a class="existingWikiWord" href="/nlab/show/fm">fm</a>. So the bound cited by (<a href="#Grigoriev">Grigor’ev 197x</a>) in the above quote is about 1/100 the diameter of a proton. It seems that <a href="#Grigoriev">Grigor’ev 197x</a> just cited the length scale resolution of particle accelerators at that time. More recently, the <a class="existingWikiWord" href="/nlab/show/LHC">LHC</a> (see there) probes scales <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>20</mn></mrow></msup><mi>m</mi></mrow><annotation encoding="application/x-tex">\simeq 10^{-20}m</annotation></semantics></math>. <h2 id="InAlgebraicQuantumFieldTheory">In algebraic quantum field theory</h2> In <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> causal locality is formalized as follows. This is a key statement in the <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> on <a class="existingWikiWord" href="/nlab/show/causally+local+nets+of+quantum+observables">causally local nets of quantum observables</a>: <div class="num_defn" id="CausalLocalityOfAlgebrasOfObservables"> <h6 id="definition">Definition</h6> (causal locality of algebras of observables) A <a class="existingWikiWord" href="/nlab/show/co-presheaf">co-presheaf</a> of <a class="existingWikiWord" href="/nlab/show/algebras+of+quantum+observables">algebras of quantum observables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> on some <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is causally local if the algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}(\mathcal{O})</annotation></semantics></math> localized in <a class="existingWikiWord" href="/nlab/show/spacelike">spacelike</a> separated <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> regions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math> commute with each other (inside any of the algebras of observables localized in the causal closure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math> of the union of the two spacetime regions). <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>𝒪</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mtext>spacelike separated from</mtext><mspace width="thickmathspace"></mspace><mi>𝒪</mi><mn>2</mn><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo stretchy="false">[</mo><mi>𝒜</mi><mo stretchy="false">(</mo><msub><mi>𝒪</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>𝒜</mi><mo stretchy="false">(</mo><msub><mi>𝒪</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>𝒜</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \mathcal{O}_1 \;\text{spacelike separated from} \; \mathcal{O}2 \right) \;\Rightarrow\; \left( [\mathcal{A}(\mathcal{O}_1), \mathcal{A}(\mathcal{O}_2)] \;=\; 0 \;\;\;\; \in \mathcal{A}( \mathcal{O} ) \right) \,. </annotation></semantics></math></div></div> Under <a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>, this causal locality becomes “statistical locality” (see at <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a>). In <a class="existingWikiWord" href="/nlab/show/perturbative+algebraic+quantum+field+theory">perturbative algebraic quantum field theory</a> this condition follows from the <a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a> of the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> (see there the section <a href="S-matrix#CausalLocality">Causal locality and Quantum obsrvables</a>). There are variants that one may consider: <div class="num_defn" id="StrongLocality"> <h6 id="definition_2">Definition</h6> (strong causal locality of algebras of observables) A <a class="existingWikiWord" href="/nlab/show/local+net+of+quantum+observables">local net of quantum observables</a> is strongly causally local if it is causally local in that algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>=</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_1 = A(O_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_2 = A(O_2)</annotation></semantics></math> associated with spacelike separated regions commute with each other, and in addition for all commutative subalgebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>⊂</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1 \subset A_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>2</mn></msub><mo>⊂</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">C_2 \subset A_2</annotation></semantics></math> the algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>⊂</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_1 \vee C_2 \subset A(O_1 \vee O_2)</annotation></semantics></math> satisfies <ol> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>C</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∩</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">(C_1 \vee C_2) \cap A_1 = C_1</annotation></semantics></math> </li> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>C</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∩</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">(C_1 \vee C_2) \cap A_2 = C_2</annotation></semantics></math>. </li> </ol> </div> This is (<a href="#Nuiten11">Nuiten 11, def. 14</a>). There have been various proposals to understand these conditions from other principles: <ol> <li> In (<a href="#Schreiber09">Schreiber 09</a>) the condition <a class="maruku-ref" href="#CausalLocalityOfAlgebrasOfObservables"></a> is related to <a class="existingWikiWord" href="/nlab/show/n-functor">n-functoriality</a> of a corresponding (<a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a>) <a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a>. </li> <li> In (<a href="#Nuiten11">Nuiten 11, theorem 4.2</a>) the condition <a class="maruku-ref" href="#StrongLocality"></a> us shown to be implied by the associated pre-sheaf of <a class="existingWikiWord" href="/nlab/show/Bohr+toposes">Bohr toposes</a> satisfying spatial <a class="existingWikiWord" href="/nlab/show/descent">descent</a> by <a class="existingWikiWord" href="/nlab/show/local+geometric+morphisms">local geometric morphisms</a>. </li> <li> In (<a href="#BrunettiFredenhagenImaniRejzner12">Brunetti-Fredenhagen-Imani-Rejzner 12</a>) condition def. <a class="maruku-ref" href="#CausalLocalityOfAlgebrasOfObservables"></a> is shown to be equivalent to the <a class="existingWikiWord" href="/nlab/show/co-presheaf">co-presheaf</a> of observables being a <a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a> is a suitable way. </li> </ol> <h2 id="in_smatrix_theories_and_string_theory">In S-matrix theories and string theory</h2> In <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> theories <a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a> is meant to also be incarnated in terms of <a class="existingWikiWord" href="/nlab/show/analytic+function">analyticity</a> properties of the <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a> (for this reason one often speaks of “the analytic S-matrix”). One S-matrix theory is <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theory</a>. Discussion of causality in string theory includes <a href="#Martinec95">Martinec 95</a> and (<a href="#ErlerGross04">Erler-Gross 04</a>). The latter write in their introduction: <blockquote> Perhaps then it comes as a surprise that critical string theory produces an analytic <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> consistent with macroscopic causality. In absence of any other known theoretical mechanism which might explain this, despite appearances one is lead to believe that string interactions must be, in some sense, local. </blockquote> and <blockquote> We find that string theory avoids problems with nonlocality in a surprising way. In particular, we find that the Witten vertex is “local enough” to allow for a nonsingular description of the theory which is completely local along a single null direction. </blockquote> and <blockquote> unlike lightcone string field theory, it is clear that cubic string field theory at least has a local limit where all spacetime coordinates are taken to the midpoint. We investigate this limit with a careful choice of regulator and show that at any stage the theory is nonsingular but arbitrarily close to being local and manifestly causal. We believe that the existence of this limit, though singular, must account for the macroscopic causality of the string S-matrix. Thus, string theory is local enough to avoid the inconsistencies of a theory which is acausal and nonlocal in time, but is nonlocal enough to make string theory different from quantum field theory </blockquote> Then they comment on Martinec’s account above, and other’s, by saying: <blockquote> To motivate our particular perspective, it seems appropriate to discuss earlier attempts to understand the role of locality, causality and time in string theory, and explain why we feel these approaches do not adequately address the problems just raised. </blockquote> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+order">causal order</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+structure">causal structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+closure">causal closure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+prequantum+field+theory">local prequantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+quantum+field+theory">local quantum field theory</a> </li> <li> in <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>: <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a>, <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> </li> <li> in <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>: <a class="existingWikiWord" href="/nlab/show/extended+field+theory">extended field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory+FAQ">string theory FAQ</a>: <a href="string+theory+FAQ#IsStringTheoryCausal">Is string theory causal, given that it is not local on the string scale?</a> </li> </ul> <h2 id="references">References</h2> <h3 id="in_local_quantum_field_theory">In local quantum field theory</h3> <ul> <li id="Grigoriev"> V. I. Grigor’ev, Microcausality condition, The Great Soviet Encyclopedia, 3rd Edition (1970-1979) (<a href="http://encyclopedia2.thefreedictionary.com/Microcausality+Condition">web</a>) </li> <li> Jessey Wright, Quantum field theory: Motivating the Axiom of Microcausality, PhD thesis 2012 (<a href="http://uwspace.uwaterloo.ca/bitstream/10012/6998/1/Wright_Jessey.pdf">pdf</a>) </li> <li> Anthony Duncan, The Conceptual Framework of Quantum Field Theory – Dynamics IV: Aspects of locality: clustering, microcausality, and analyticity, Oxford Scholarship Online (<a href="http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199573264.001.0001/acprof-9780199573264-chapter-6">web</a>) </li> <li id="Bell75"> <a class="existingWikiWord" href="/nlab/show/John+Bell">John Bell</a>, The theory of local beables (1975) (<a href="http://cds.cern.ch/record/980036/files/197508125.pdf">pdf</a>) </li> <li> <a href="http://ptp.oxfordjournals.org/content/52/1/335.full.pdf">pdf</a> </li> </ul> <h3 id="in_algebraic_quantum_field_theory_2">In algebraic quantum field theory</h3> For references on the traditional discussion in <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> see at <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> and at <a class="existingWikiWord" href="/nlab/show/causally+local+net+of+observables">causally local net of observables</a>. Proposals to understand the causal locality axiom from other principles include <ul> <li id="Schreiber09"> <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, AQFT from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-functorial QFT , Comm. Math. Phys., Volume 291, Issue 2, pp.357-401 2009 (<a href="http://ncatlab.org/schreiber/files/AQFTfromFQFT.pdf">pdf</a>) </li> <li id="Nuiten11"> <a class="existingWikiWord" href="/nlab/show/Joost+Nuiten">Joost Nuiten</a>, <a class="existingWikiWord" href="/schreiber/show/bachelor+thesis+Nuiten">Bohrification of local nets</a>, Proceedings of <a href="http://qpl.science.ru.nl/">QPL 2011</a>, <a href="http://rvg.web.cse.unsw.edu.au/eptcs/content.cgi?QPL2011">EPTCS 95, 2012</a>, pp. 211-218 (<a href="http://arxiv.org/abs/1109.1397">arXiv:1109.1397</a>) </li> <li id="BrunettiFredenhagenImaniRejzner12"> <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Paniz Imani, <a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, The Locality Axiom in Quantum Field Theory and Tensor Products of <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^*</annotation></semantics></math>-algebras (<a href="https://arxiv.org/abs/1206.5484">arXiv:1206.5484</a>) </li> </ul> <h3 id="in_smatrix_theories_and_string_theory_2">In S-matrix theories and string theory</h3> Discussion of causality in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> includes the following: A brief look at the causality property of the string 2-point function is in <ul> <li id="Martinec95"><a class="existingWikiWord" href="/nlab/show/Emil+Martinec">Emil Martinec</a>, Strings and Causality (<a href="http://cds.cern.ch/record/569030/files/9311129.pdf">pdf</a>) in L. Baulieu, V. Dotsenko, V. Kazakov, P. Windey (eds.) Quantum Field Theory and String Theory , NATO ASI Series B: Physics Vol. 328 (1995)</li> </ul> An in-depth discussion of causality of the <a class="existingWikiWord" href="/nlab/show/string+scattering+amplitude">string scattering</a> <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> via <a class="existingWikiWord" href="/nlab/show/open+string+field+theory">open string field theory</a> is in <ul> <li id="ErlerGross04"><a class="existingWikiWord" href="/nlab/show/Theodore+Erler">Theodore Erler</a>, <a class="existingWikiWord" href="/nlab/show/David+Gross">David Gross</a>, Locality, Causality, and an Initial Value Formulation for Open String Field Theory (<a href="https://arxiv.org/abs/hep-th/0406199">arXiv:hep-th/0406199</a>)</li> </ul> This rediscovered some facts that had earlier been noticed in <ul> <li>M. Maeno, Canonical quantization of Witten’s string field theory using midpoint light-cone time, Phys. Rev. D43 no. 12 (1991).</li> </ul> See also <ul> <li>Gianluca Calcagni, Leonardo Modesto, Nonlocality in string theory, J. Phys. A: Math. Theor. 47 (2014) 355402 (<a href="https://arxiv.org/abs/1310.4957">arXiv:1310.4957</a>)</li> </ul> Discussion in theories with <a class="existingWikiWord" href="/nlab/show/higher+curvature+corrections">higher curvature corrections</a> of the gravitational background and in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> includes <ul> <li> Xian O. Camanho, Jose D. Edelstein, <a class="existingWikiWord" href="/nlab/show/Juan+Maldacena">Juan Maldacena</a>, Alexander Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling (<a href="http://arxiv.org/abs/1407.5597">arXiv:</a>) </li> <li> Giuseppe D’Appollonio, <a class="existingWikiWord" href="/nlab/show/Paolo+Di+Vecchia">Paolo Di Vecchia</a>, <a class="existingWikiWord" href="/nlab/show/Rodolfo+Russo">Rodolfo Russo</a>, <a class="existingWikiWord" href="/nlab/show/Gabriele+Veneziano">Gabriele Veneziano</a>, Regge behavior saves String Theory from causality violations (<a href="http://arxiv.org/abs/1502.01254">arXiv:1502.01254</a>) </li> </ul> </body></html> </div> <div class="revisedby"> Last revised on March 31, 2023 at 09:10:40. 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