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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="algbraic_quantum_field_theory">Algbraic Quantum Field Theory</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 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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <ul> <li><a href='#Terminology'>Terminology</a></li> <li><a href='#Motivation'>Motivation</a></li> </ul> <li><a href='#Formulation'>Formulation</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#structural_theorems'>Structural theorems</a></li> <li><a href='#relation_to_wightman_axioms'>Relation to Wightman axioms</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> The Haag–Kastler axioms (<a href="#HaagKastler64">Haag-Kastler 64</a>) (sometimes also called Araki–Haag–Kastler axioms) try to capture in a mathematically precise way the notion of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> (QFT), by axiomatizing how its algebras of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> should depend on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> regions, namely as <a class="existingWikiWord" href="/nlab/show/local+nets+of+observables">local nets of observables</a>. The main point of these axioms is to say that <ol> <li> to every <a class="existingWikiWord" href="/nlab/show/causally+closed+subset">causally closed subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{O} \subset X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> there is associated a <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a> <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>; </li> <li> for every inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mn>1</mn></msub><mo>↪</mo><msub><mi>𝒪</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_1 \hookrightarrow \mathcal{O}_2</annotation></semantics></math> of such spacetime regions there is a corresponding inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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></mrow><annotation encoding="application/x-tex">\mathcal{A}(\mathcal{O}_1) \hookrightarrow \mathcal{A}(\mathcal{O}_2)</annotation></semantics></math> </li> <li> such that this respects <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of inclusions (hence such that <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> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> of <a class="existingWikiWord" href="/nlab/show/causally+closed+subsets">causally closed subsets</a> to <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a>) </li> <li> (<a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a>) and such that whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>𝒪</mi> <mn>2</mn></msub><mo>⊂</mo><mi>𝒪</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}_1, \mathcal{O}_2 \subset \mathcal{O} \subset X</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/spacelike">spacelike</a> separated, then the elements of the corresponding algebras of observables (graded-)commute with each other: <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><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><mo>=</mo><mi>t</mi><mn>0</mn><mphantom><mi>AAAA</mi></mphantom><mo>∈</mo><mi>𝒜</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\mathcal{A}(\mathcal{O}_1), \mathcal{A}(\mathcal{O}_2)] = t 0 \phantom{AAAA} \in \mathcal{A}(\mathcal{O}) \,. </annotation></semantics></math></div></li> </ol> Moreover one wants these assignments to behave well with spacetime symmetry. The formulation of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> via these axioms has come to be known as <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> or <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>, for short. There are various further axioms in the list such as the <a class="existingWikiWord" href="/nlab/show/time+slice+axiom">time slice axiom</a>. The precise details of the list of axioms is in flux as the theory develops. The Haag-Kastler axioms in their original form aim for the description of <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative quantum field theory</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>. They could be used to give rigorous <a class="existingWikiWord" href="/nlab/show/proof">proof</a> of structural statements of <a class="existingWikiWord" href="/nlab/show/QFT">QFT</a> such as the <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> or the <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a>, but examples of interacting field theories in dimensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\geq 4</annotation></semantics></math> are missing. There are however variants: <ul> <li>In <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> one drops the requirement of <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> and instead considers just <a class="existingWikiWord" href="/nlab/show/formal+power+series+algebras">formal power series algebras</a> (in <a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math>) in order to describe <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> and connect to the traditional construction of pQFTs via the <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a>.</li> </ul> The observation that <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> yields <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> that satisfy the Haag-Kastler axioms, except that <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> are replaced by <a class="existingWikiWord" href="/nlab/show/formal+power+series+algebras">formal power series algebras</a>, is due to (<a href="#IlinSlavnov78">Il’in-Slavnov 78</a>, <a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00</a>, <a href="#BrunettiDuetschFredenhagen09">Brunetti-Dütsch-Fredenhagen 09</a>). See at <a href="S-matrix#CausalLocality">S-Matrix – Causal Locality and Quantum Observables</a>. <ul> <li> In <a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+AQFT">locally covariant perturbative AQFT</a> one generalizes from <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> to more general <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetimes">globally hyperbolic spacetimes</a> in order to describe <a class="existingWikiWord" href="/nlab/show/QFT+on+curved+spacetimes">QFT on curved spacetimes</a>. </li> <li> In <a class="existingWikiWord" href="/nlab/show/homotopical+AQFT">homotopical AQFT</a> one consider <a class="existingWikiWord" href="/nlab/show/homotopical+algebras">homotopical algebras</a> and commutativity only up to <a class="existingWikiWord" href="/nlab/show/coherence+law">coherent</a> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> in order to discuss <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> with non-trivial topological (<a class="existingWikiWord" href="/nlab/show/instanton">instanton</a>) sectors. </li> </ul> <h3 id="Terminology">Terminology</h3> The approch to quantum field theory based on these axioms is often called <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> : either for axiomatic quantum field theory (since it was among the first attempts to put the edifice of QFT on solid <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a>atic grounds) or algebraic quantum field theory (since it amplifies the algebras of local <a class="existingWikiWord" href="/nlab/show/observable">observable</a>s over the spaces of <a class="existingWikiWord" href="/nlab/show/state">state</a>s). Neither of these terms is very descriptive. First there is another, dual, axiomatization which does axiomatize the propagation of <a class="existingWikiWord" href="/nlab/show/states">states</a> – see <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> – which, second, is also “algebraic” in some sense, even though algebras of observables to not appear directly. Another common term for these axioms is local quantum field theory (see the title of the standard textbook (<a href="#Haag">Haag</a>)) since, as becomes clear <a href="#Formulation">below</a>, they are focused on encoding the locality properties of QFT in terms of the algebras of observables. However, also the core aspect of <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended</a> <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> is all about the notion of locality of QFT. Therefore neither of the traditional terms for QFT as axiomatized by Haag-Kastler is truly descriptive in that it genuinely distinguishes from the other, the Atiyah-Segal axiomatization by <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>. What does distinguish the two approaches may be characterized in traditional terminology of quantum theory as follows (<a href="#Schreiber">Schreiber</a>, <a href="#SatiSchreiber">SatiSchreiber</a>): <ul> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> axiomatizes the <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a> of QFT, which encodes the propagation of <a class="existingWikiWord" href="/nlab/show/state">state</a>s through spacetimes; </li> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> axiomatizes the <a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a> of QFT, which encodes the way that <a class="existingWikiWord" href="/nlab/show/observable">observable</a>s depend on spacetime. </li> </ul> <h3 id="Motivation">Motivation</h3> A central difference between the Haag-Kastler axioms and traditionally more widespread formulations of QFT (usually far from being formalized in any way) is the emphasis of the <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a> of a QFT (Heisenberg picture) and the de-emphasis of the (<a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert</a>) <a class="existingWikiWord" href="/nlab/show/spaces+of+states">spaces of states</a> (<a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a>). This emphasis receives motivation from the the fact that many technical problems of QFT simply disappear when one is not trying to form its spaces of states, while at the same time no real information about the theory is lost. Examples of technical problems that formulation in terms of spaces of states bring with them are the following: <ul> <li> In <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> as opposed to <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, the <a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a> fails, making the <a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a> of the <a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a> on the spaces of states non-unique, hence requiring an explicit choice of representation. There is no generally good theory available for how to make this choice. </li> <li> More seriously, <a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a> says that at a crucial step in <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> where one wants to pass from the representation “free fields” to that of “interacting fields”, the two representations are necessarily inequivalent, contrary to what is (silently or explicitly) assumed in much traditional QFT literature (see <a href="#EarmanFraser">EarmanFraser</a>). </li> <li> In the <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> or <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> the formulation in terms of states brings with it infrared problems that are simply absent when formulating renormalization just in terms of observables (<a href="#DuetschFredenhagen">DuetschFredenhagen</a>). </li> </ul> <h2 id="Formulation">Formulation</h2> We formulate the ideas of the core axioms of Haag-Kastler, and their intended physical meaning. For more details see <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a>. <ol> <li> spacetime locality Since the fields in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> (such as the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>) exhibit and are characterized by their local excitations (for instance the value of the electric/magnetic <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> at any point) having effects only locally (the field excitations at two points a finite distance apart do not directly influence each other) the fields over any region of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> form a <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a> of the fields of any larger region and in particular of the total system. If “<a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>” is formalized as “<a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>” (of <a class="existingWikiWord" href="/nlab/show/observables">observables</a>) then “<a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a>” translates to “sub-<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>-algebra”. Therefore the above sentence translates into: quantum fields form a <a class="existingWikiWord" href="/nlab/show/copresheaf">copresheaf</a> of <a class="existingWikiWord" href="/nlab/show/C-star+algebras">C-star algebras</a> on <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> whose co-restriction morphisms are <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>s. In <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> such is called an <a class="existingWikiWord" href="/nlab/show/local+net">isotonic net of algebras</a> . There are different approaches to define what kind of spacetime regions the algebras of observables are assigned to, hence different approaches as to what exactly the <a class="existingWikiWord" href="/nlab/show/site">site</a> is on which the co-presheaf is defined. A common approach is to take all <a class="existingWikiWord" href="/nlab/show/bounded+space">bounded</a> <a class="existingWikiWord" href="/nlab/show/open+subspace">open</a> subsets of <a class="existingWikiWord" href="/nlab/show/Minkowski+space">Minkowski</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>. For more general setups see <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetimes">AQFT on curved spacetimes</a>. </li> <li> causal locality If two regions of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> are <a class="existingWikiWord" href="/nlab/show/spacelike">spacelike</a> separated, then there can be no influence between them whatsoever. Not only do the field excitations in one of the two regions not directly influence those in the other region (as per item 1), but they do not even influence indirectly : no <a class="existingWikiWord" href="/nlab/show/waves">waves</a> of excitations (for instance <a class="existingWikiWord" href="/nlab/show/electromagnetic+waves">electromagnetic waves</a>: <a class="existingWikiWord" href="/nlab/show/light">light</a>) can run from one region to a spacelike separated region. Therefore the two <a class="existingWikiWord" href="/nlab/show/subsystems">subsystems</a> constituted by these two regions accordording to the first point are even <a class="existingWikiWord" href="/nlab/show/independent+subsystems">independent subsystems</a> . This is called <a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a>. The formalization of “two <a class="existingWikiWord" href="/nlab/show/independent+subsystems">independent subsystems</a>” in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> is: two subalgebras that commute with each other inside the larger <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>. (And usually one adds: and such that the algebra they generate in the larger algebra is <a class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a> to their <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>.) Therefore this translates into the axiom: quantum fields on a spacetime form an isotonic copresheaf of algebras such that the algebras assigned to any two spacelike separated regions commute with each other inside the algebra assigned to any larger region containing these two regions. </li> <li> spacetime covariance The geometric symmetry operations map the algebra of a region onto the algebra of the transformed region. (this is not an extra axiom if one defines the <a class="existingWikiWord" href="/nlab/show/site">site</a> of spacetime regions general enough…) In Minkowski spacetime the geometric symmetry group is usually taken to be the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+group">Poincaré group</a>, but note that some authors consider subgroups of the full Poincaré group, like the subgroup of translations (Borchers: “Translation group and particle representations in quantum field theory”). </li> <li> positivity of energy An axiom is needed to ensure that only nonnegative energies occur – one possibility is the “spectrum condition”, which says that the spectrum (to be more precise: the support of the <a class="existingWikiWord" href="/nlab/show/spectral+measure">spectral measure</a>) of the operator associated with a translation is contained in the closed forward light cone, for all translations. </li> </ol> <h2 id="properties">Properties</h2> <h3 id="structural_theorems">Structural theorems</h3> It is possible to prove both a <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> and a <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> in the Haag-Kastler approach. The mathematically precise, model independent statements and their proofs are considered to be a major breakthrough of the theory. <h3 id="relation_to_wightman_axioms">Relation to Wightman axioms</h3> Unlike the <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a>, the Haag–Kastler axioms do not need the notion of “<a class="existingWikiWord" href="/nlab/show/physical+field">field</a>”: the fields in the Wightman axioms are – from the Haag–Kastler point of view – only necessary to describe how the algebras of observables are constructed; any way to consistently construct the net of algebras would suffice. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> <ul> <li> Haag-Kastler axioms </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a>, <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+AQFT">locally covariant perturbative AQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> </li> </ul> </li> </ul> <h2 id="references">References</h2> The original article that introduced these axioms is <ul> <li id="Haag64"><a class="existingWikiWord" href="/nlab/show/Rudolf+Haag">Rudolf Haag</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kastler">Daniel Kastler</a>, An algebraic approach to quantum field theory, Journal of Mathematical Physics, Bd.5, 1964, S.848-861 (<a href="https://doi.org/10.1063/1.1704187">doi:10.1063/1.1704187</a>, <a href="https://inspirehep.net/literature/9124">spire:9124</a>)</li> </ul> following <ul> <li id="Haag59"><a class="existingWikiWord" href="/nlab/show/Rudolf+Haag">Rudolf Haag</a>, Discussion des “axiomes” et des propriétés asymptotiques d’une théorie des champs locales avec particules composées, Les Problèmes Mathématiques de la Théorie Quantique des Champs, Colloque Internationaux du CNRS LXXV (Lille 1957), CNRS Paris (1959), 151.</li> </ul> translated to English as: <ul> <li><a class="existingWikiWord" href="/nlab/show/Rudolf+Haag">Rudolf Haag</a>, Discussion of the ‘axioms’ and the asymptotic properties of a local field theory with composite particles, EPJ H 35, 243–253 (2010) (<a href="https://doi.org/10.1140/epjh/e2010-10041-3">doi:10.1140/epjh/e2010-10041-3</a>)</li> </ul> Textbook accounts: <ul> <li id="Haag"> <a class="existingWikiWord" href="/nlab/show/Rudolf+Haag">Rudolf Haag</a>, <a class="existingWikiWord" href="/nlab/show/Local+Quantum+Physics+--+Fields%2C+Particles%2C+Algebras">Local Quantum Physics – Fields, Particles, Algebras</a> Springer (1992) 2nd., rev. and enlarged ed. Springer (1996) (<a href="https://www.springer.com/gp/book/9783540610496">ISBN 978-3-642-61458-3</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Huzihiro+Araki">Huzihiro Araki</a>: <a class="existingWikiWord" href="/nlab/show/Mathematical+Theory+of+Quantum+Fields">Mathematical Theory of Quantum Fields</a> Oxford University Press 1999 <a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0998.81501&format=complete">ZMATH entry</a>. </li> </ul> and seminar notes: <ul> <li><a class="existingWikiWord" href="/nlab/show/Garth+Warner">Garth Warner</a>: Quantum Field Theory Seminar (School of Haag-Kastler et al.), seminar notes, University of Washington [<a href="https://sites.math.washington.edu//~warner/QFT2Seminar_Warner.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Warner-HaagKastlerQFT.pdf" title="pdf">pdf</a>]</li> </ul> Precursors include: <ul> <li>Bogolyubov, Logunov, Oksak, Todorov: General principles of quantum field theory. (<a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0732.46040&format=complete">ZMATH entry</a>)</li> </ul> An online reference page is here: <ul> <li>Stephen J. Summers: <a href="http://www.math.ufl.edu/~sjs/aqft.html">AQFT online</a></li> </ul> See also the references at <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> and at <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a>. One of the founding fathers of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> wrote: <ul> <li id="Dyson72"> <a class="existingWikiWord" href="/nlab/show/Freeman+Dyson">Freeman Dyson</a>Missed opportunities, Bulletin of the AMS, Volume 78, Number 5, September 1972 (<a href="https://www.math.uh.edu/~tomforde/Articles/Missed-Opportunities-Dyson.pdf">pdf</a>) <blockquote> These axioms, taken together with the axioms defining a C-algebra are a distillation into abstract mathematical language of all the general truths that we have learned about the physics of microscopic systems during the last 50 years. They describe a mathematical structure of great elegance whose properties correspond in many respects to the facts of experimental physics. In some sense, the axioms represent the most serious attempt that has yet been made to define precisely what physicists mean by the words “observability, causality, locality, relativistic invariance,” which they are constantly using or abusing in their everyday speech. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math>…<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> I therefore propose as an outstanding opportunity still open to the pure mathematicians, to create a mathematical structure preserving the main features of the Haag-Kastler axioms but possessing E-invariance instead of P-invariance. </blockquote> (on the latter see <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetimes">AQFT on curved spacetimes</a>) </li> </ul> The observation that <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> yields <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> that satisfy the Haag-Kastler axioms, except that <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> are replaced by <a class="existingWikiWord" href="/nlab/show/formal+power+series+algebras">formal power series algebras</a> is due to <ul> <li id="IlinSlavnov78"> V. A. Il’in and D. S. Slavnov, Observable algebras in the S-matrix approach, Theor. Math. Phys. 36 (1978) 32. (<a href="http://inspirehep.net/record/135575">spire</a>, <a href="http://dx.doi.org/10.1007/BF01035870">doi</a>) </li> <li id="BrunettiFredenhagen00"> <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208 : 623-661, 2000 (<a href="https://arxiv.org/abs/math-ph/9903028">math-ph/9903028</a>) </li> <li id="BrunettiDuetschFredenhagen09"> <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (<a href="http://arxiv.org/abs/0901.2038">arXiv:0901.2038</a>) </li> </ul> An introduction into <a class="existingWikiWord" href="/nlab/show/Tomita-Takesaki+modular+theory">Tomita-Takesaki modular theory</a>: <ul> <li>Stephen J. Summers: “Tomita-Takesaki Modular Theory” (<a href="http://xxx.uni-augsburg.de/abs/math-ph/0511034">arXiv</a>)</li> </ul> and on its role in AQFT: <ul> <li><a class="existingWikiWord" href="/nlab/show/Hans-J%C3%BCrgen+Borchers">Hans-Jürgen Borchers</a>, On Revolutionizing of Quantum Field Theory with Tomita’s Modular Theory, ESI Preprint 773 (1999) [<a href="https://www.mat.univie.ac.at/~esiprpr/esi773.pdf">pdf</a>]</li> </ul> A discussion of how the Haag-Kastler axioms (those concerning locality) follow from an extended <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> with Lorentzian structure is in <ul> <li id="Schreiber"><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 (<a href="http://ncatlab.org/schreiber/files/AQFTfromFQFT.pdf">pdf</a>)</li> </ul> A discussion putting the state of the art of the <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>-axiomatization in context with that of the <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>-axiomatization is in <ul> <li id="SatiSchreiber"><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+String+Theory">Mathematical Foundations of Quantum Field and String Theory</a></li> </ul> A discussion of <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> and <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> in terms of the Haag-Kastler axioms is in <ul> <li id="DuetschFredenhagen"><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, A local (perturbative) construction of observables in gauge theores: the example of qed_ , Commun. Math. Phys. 203 (1999), no.1, 71-105, (<a href="http://xxx.uni-augsburg.de/ps/hep-th/9807078">arXiv:hep-th/9807078</a>).</li> </ul> <a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a> and its meaning and implication is discussed thoroughly in <ul> <li id="EarmanFraser">John Earman, Doreer Fraser, Haag’s theorem and its implications for the foundations of quantum field theory (<a href="http://philsci-archive.pitt.edu/2673/1/earmanfraserfinalrevd.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on July 29, 2024 at 11:40:49. 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