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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="algebraic_quantum_field_theory">Algebraic 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="quantum_systems">Quantum systems</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, <a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">dependent</a> <a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a> <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">†-compact categories</a> <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a> (<a class="existingWikiWord" href="/nlab/show/parameterized+quantum+systems">parameterized</a>, <a class="existingWikiWord" href="/nlab/show/open+quantum+system">open</a>) <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">quantum state collapse</a> <a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a> <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a> <a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a> <a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+many-body+physics">quantum many-body physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-</a>)<a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a> <a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a> (<a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological</a>) <a class="existingWikiWord" href="/nlab/show/phases+of+matter">phases of matter</a> </li> </ul> <div> <a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a> </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/quasi-state">quasi-state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> <a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a> <a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a> <a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a> <a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a> <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</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> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a> </li> <li> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/theorems">theorems</a> <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/Gleason%27s+theorem">Gleason's theorem</a> </li> <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> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> </ul> </li> </ul> </div> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+technology">quantum technology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+sensing">quantum sensing</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+communication">quantum communication</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+cryptography">quantum cryptography</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+machine+learning">quantum machine learning</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/qdit">qdit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+gate">quantum gate</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+quantum+computation">adiabatic quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/measurement-based+quantum+computation">measurement-based quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+programming+language">quantum programming language</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/HaPPY+code">HaPPY code</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Majorana+dimer+code">Majorana dimer code</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a> </li> <li> quantum algorithms: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Grover%27s+algorithm">Grover's algorithm</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Shor%27s+algorithm">Shor's algorithm</a> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#axioms'>Axioms</a></li> <li><a href='#theorems'>Theorems</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#FreeScalarField'>Free scalar field / Klein Gordon field</a></li> <li><a href='#free_fermion__dirac_field'>Free fermion / Dirac field</a></li> <li><a href='#electromagnetic_field'>Electromagnetic field</a></li> <li><a href='#proca_field'>Proca field</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#axioms_2'>Axioms</a></li> <li><a href='#lecture_notes_and_textbooks'>Lecture notes and Textbooks</a></li> <li><a href='#reviews'>Reviews</a></li> <li><a href='#ExamplesReferences'>Examples</a></li> <li><a href='#LocalGaugeTheory'>Local gauge theory</a></li> <li><a href='#ReferencesPerturbationTheory'>Perturbation theory and renormalization</a></li> <li><a href='#further_developments'>Further developments</a></li> <li><a href='#relation_to_holographic_entanglement_entropy'>Relation to holographic entanglement entropy</a></li> <li><a href='#ReferencesRelationBetweenAQFTAndFQFT'>Relation between algebraic and functorial field theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> Algebraic Quantum Field Theory or Axiomatic Quantum Field Theory or AQFT for short is a formalization of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> (and specifically full, hence <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative quantum field theory</a>) that axiomatizes the assignment of <a class="existingWikiWord" href="/nlab/show/algebras+of+observables">algebras of observables</a> to patches of parameter space (<a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>) that one expects a quantum field theory to provide. As such, the approach of AQFT is roughly dual to that of <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>, where instead spaces of states are assigned to boundaries of <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s and propagation maps between state spaces to cobordisms themselves. One may roughly think of AQFT as being a formalization of what in basic <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> textbooks is called the <a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a> of quantum mechanics. On the other hand <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> axiomatizes the <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a> . The axioms of traditional AQFT encode the properties of a <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> of observables and are called the <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a>. They are one of the oldest systems of axioms that seriously attempt to put <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> on a solid conceptual footing. From the <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a> we may think of a <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> as a co-flabby <a class="existingWikiWord" href="/nlab/show/presheaf">copresheaf</a> of <a class="existingWikiWord" href="/nlab/show/algebra">algebras</a> on spacetime which satisfies a certain locality axiom with respect to the <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">Lorentzian structure</a> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>: <ul> <li>locality: algebras assigned to spacelike separated regions commute with each other when embedded into any joint superalgebra.</li> </ul> This is traditionally formulated (implicitly) as a structure in ordinary <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. More recently, with the proof of the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> and the corresponding <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a>-formulation of <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> also <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher categorical</a> versions of systems of local algebras of observables are being put forward and studied. Three structures are curently being studied, that are all conceptually very similar and similar to the Haag-Kastler axioms: <ul> <li> <a class="existingWikiWord" href="/nlab/show/factorization+algebra">factorization algebra</a>s </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+chiral+homology">topological chiral homology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/blob+homology">blob homology</a>. </li> </ul> Initially, all three of these encoded what in physics are called Euclidean quantum field theories, whereas only the notion of <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> incorporated the fact that the underlying spacetime of a quantum field theory is a <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+space">smooth Lorentzian space</a>. Recent developments in the formalism of <a class="existingWikiWord" href="/nlab/show/factorization+algebra">factorization algebra</a>s have extended their theory to globally hyperbolic <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifolds">Lorentzian manifolds</a>. In the context of the Haag-Kastler axioms there is a precise theorem, the <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a>, relating the Euclidean to the Lorentzian formulation: this is the operation known as <a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>. Sheaves are used explicitly in: <ul> <li> Roberts, John E.: <a href="http://books.google.com/books?id=IFjzuLjE43kC&lpg=PA297&ots=5Ld1B3I45m&dq=Operator%20algebras%20and%20applications%2C%20Part%202&pg=PA523#v=onepage&q=Operator%20algebras%20and%20applications,%20Part%202&f=false">New light on the mathematical structure of algebraic field theory.</a> Operator algebras and applications, Part 2 (Kingston, Ont., 1980), pp. 523–550, Proc. Sympos. Pure Math., 38, Amer. Math. Soc., Providence, R.I., 1982. </li> <li> Roberts, John E.: <a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-85/issue-1/Localization-in-algebraic-field-theory/cmp/1103921341.full">Localization in algebraic field theory</a>. Comm. Math. Phys. 85 (1982), no. 1, 87–98. </li> </ul> — much information to be filled in — <h2 id="axioms">Axioms</h2> <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> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> </li> </ul> <h2 id="properties">Properties</h2> Generically the algebra of a relativistic AQFT turns out to be a (<a class="existingWikiWord" href="/nlab/show/generalized+the">the</a>) hyperfinite type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>III</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">III_1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra+factor">von Neumann algebra factor</a>. See (<a href="#Yngvason">Yngvason</a>) <h2 id="Examples">Examples</h2> Examples of AQFT <a class="existingWikiWord" href="/nlab/show/local+nets+of+observables">local nets of observables</a> that encode interacting quantum field theories are not easy to construct. The construction of free field theories is well understood, see the references <a href="#ExamplesReferences">below</a>. In <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> also interacting theories can be constructed, see the references <a href="#ReferencesPerturbationTheory">here</a>. <h3 id="FreeScalarField">Free scalar field / Klein Gordon field</h3> A survey of the AQFT description of the free<a href="/nlab/new/free+field+theor">?</a> <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> is in (<a href="#Montoya">Motoya, slides 11-17</a>). Discussion in more general context of <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetimes">AQFT on curved spacetimes</a> in (<a href="#BrunettiFredenhagen">Brunetti-Fredenhagen, section 5.2</a>) <h3 id="free_fermion__dirac_field">Free fermion / Dirac field</h3> The <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a> and its deformations is discussed for instance in (<a href="#DLM">DLM, section 3.2</a>), (<a href="#Dimock82">Dimock 83</a>). <h3 id="electromagnetic_field">Electromagnetic field</h3> The quantized <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> is discussed for instance in (<a href="#Dimock92">Dimock 92</a>). <h3 id="proca_field">Proca field</h3> (<a href="#Furliani">Furliani</a>) <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> <ul> <li> AQFT <ul> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a>, <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a>, <a class="existingWikiWord" href="/nlab/show/field+net">field net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/time+slice+axiom">time slice axiom</a>, <a class="existingWikiWord" href="/nlab/show/split+property">split property</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+lattice+systems">quantum lattice systems</a>, <a class="existingWikiWord" href="/nlab/show/string-localized+quantum+field">string-localized quantum field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+covariant+AQFT">locally covariant AQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical algebraic quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Ruelle+scattering+theory">Haag-Ruelle scattering theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/constructive+quantum+field+theory">constructive quantum field theory</a> </li> </ul> <div> <a class="existingWikiWord" href="/nlab/show/duality+between+algebra+and+geometry">duality between</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/category">category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dual+category">dual category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a></mtext></mover><msubsup><mi>Alg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a></mtext></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup><mo>,</mo><mi>comm</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">comm. C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncomm. topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCTopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">NCTopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>general <a class="existingWikiWord" href="/nlab/show/C-star-algebra">C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>Schemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}Schemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op} </annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A} \phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncomm. algebraic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCSchemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">NCSchemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mrow><mi>fin</mi><mo>,</mo><mi>red</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/finitely+generated+algebra">fin. gen.</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothManifolds</mi></mrow><annotation encoding="application/x-tex">SmoothManifolds</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mi>comm</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>SuperSpaces</mi> <mi>Cart</mi></msub></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mphantom><mtext>Pursell's theorem</mtext></mphantom></mover></mtd> <mtd><msubsup><mi>Alg</mi> <mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mphantom><mi>AAAA</mi></mphantom></mrow> <mi>op</mi></msubsup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>q</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \overset{\phantom{\text{Pursell's theorem}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/formal+moduli+problem">formal</a> <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>(<a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super Lie theory</a>)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mi>Super</mi><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>fin</mi></msub></mtd></mtr> <mtr><mtd><mi>𝔤</mi></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><mtext><a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a></mtext><mphantom><mi>A</mi></mphantom></mrow></mover></mtd> <mtd><msup><mi>sdgcAlg</mi> <mi>op</mi></msup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">FDAs</a>”)</td></tr> </tbody></table> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>: <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/space+of+states">space of states</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/n-plectic+manifold">n-plectic manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/En-algebras">En-algebras</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Beilinson-Drinfeld+algebra">BD</a>-<a class="existingWikiWord" href="/nlab/show/BV+quantization">BV quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+algebra+of+observables">factorization algebra of observables</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/extended+quantum+field+theory">extended quantum field theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+homology">factorization homology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism representation</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="axioms_2">Axioms</h3> The original article that introduced the <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> 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, 5 (1964) 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> The generalization of the <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/site">site</a> from open in <a class="existingWikiWord" href="/nlab/show/Minkowski+space">Minkowski space</a> to more general and <a class="existingWikiWord" href="/nlab/show/curvature">curved</a> spacetimes (see <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetimes">AQFT on curved spacetimes</a>) is due to <ul> <li id="BrunettiFredenhagen"> <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Quantum field theory on curved spacetimes <a href="http://arxiv.org/abs/0901.2063">arXiv:0901.2063</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Rainer+Verch">Rainer Verch</a>, The generally covariant locality principle – A new paradigm for local quantum physics Commun. Math. Phys. 237:31-68 (2003) (<a href="http://arxiv.org/abs/math-ph/0112041">arXiv:math-ph/0112041</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Quantum Field Theory on Curved Backgrounds , Proceedings of the Kompaktkurs “Quantenfeldtheorie auf gekruemmten Raumzeiten” held at Universitaet Potsdam, Germany, in 8.-12.10.2007, organized by C. Baer and K. Fredenhagen </li> </ul> See also <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetimes">AQFT on curved spacetimes</a> . <h3 id="lecture_notes_and_textbooks">Lecture notes and Textbooks</h3> Introductory lecture 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> <li id="Fredenhagen03"> <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Algebraische Quantenfeldtheorie, lecture notes, 2003 (<a class="existingWikiWord" href="/nlab/files/FredenhagenAQFT2003.pdf" title="pdf">pdf</a>) </li> <li id="FewsterRejzner19"> <a class="existingWikiWord" href="/nlab/show/Christopher+Fewster">Christopher Fewster</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, Algebraic Quantum Field Theory - an introduction (<a href="https://arxiv.org/abs/1904.04051">arXiv:1904.04051</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Jonathan+Sorce">Jonathan Sorce</a>: Bootstrap 2024: Lectures on “The algebraic approach: when, how, and why?” [<a href="https://arxiv.org/abs/2408.07994">arXiv:2408.07994</a>] <blockquote> (aimed at non-mathematically inclined field theorists) </blockquote> </li> </ul> and for just <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> in the algebraic perspective: <ul> <li id="Gleason09"> <a class="existingWikiWord" href="/nlab/show/Jonathan+Gleason">Jonathan Gleason</a>, The <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>-algebraic formalism of quantum mechanics (2009) [<a class="existingWikiWord" href="/nlab/files/Gleason09.pdf" title="pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GleasonAlgebraic.pdf" title="pdf">pdf</a>] </li> <li id="Gleason11"> <a class="existingWikiWord" href="/nlab/show/Jonathan+Gleason">Jonathan Gleason</a>, From Classical to Quantum: The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">F^\ast</annotation></semantics></math>-Algebraic Approach, contribution to <a href="http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2011.html">VIGRE REU 2011</a>, Chicago (2011) [<a href="https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GleasonFAlgebraic.pdf" title="pdf">pdf</a>] </li> </ul> Textbook accounts: <ul> <li id="BratteliRobinson79"> <a class="existingWikiWord" href="/nlab/show/Ola+Bratteli">Ola Bratteli</a>, <a class="existingWikiWord" href="/nlab/show/Derek+W.+Robinson">Derek W. Robinson</a>, Operator Algebras and Quantum Statistical Mechanics – vol 1: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^\ast</annotation></semantics></math>-Algebras. Symmetry Groups. Decomposition of States., Springer (1979, 1987, 2002) [<a href="https://doi.org/10.1007/978-3-662-02520-8">doi:10.1007/978-3-662-02520-8</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Raymond+F.+Streater">Raymond F. Streater</a>, <a class="existingWikiWord" href="/nlab/show/Arthur+S.+Wightman">Arthur S. Wightman</a>, PCT, Spin and Statistics, and All That, Princeton University Press (1989, 2000) [<a href="https://press.princeton.edu/books/paperback/9780691070629/pct-spin-and-statistics-and-all-that">ISBN:9780691070629</a>, <a href="https://www.jstor.org/stable/j.ctt1cx3vcq">jstor:j.ctt1cx3vcq</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Nikolay+Bogolyubov">Nikolay Bogolyubov</a>, A. A. Logunov, A. I. Oksak, I. T. Todorov, General principles of quantum field theory, Mathematical Physics and Applied Mathematics 10, Kluwer (1990) [<a href="https://doi.org/10.1007/978-94-009-0491-0">doi:10.1007/978-94-009-0491-0</a>] </li> <li> <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>, Texts and Monographs in Physics, Springer (1992) </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> (1999) </li> <li> <a class="existingWikiWord" href="/nlab/show/Hans+Halvorson">Hans Halvorson</a> (with an appendix by <a class="existingWikiWord" href="/nlab/show/Michael+M%C3%BCger">Michael Müger</a>): Algebraic Quantum Field Theory, in Philosophy of Physics, Handbook of the Philosophy of Science (2007) 731-864 [<a href="https://doi.org/10.1016/B978-044451560-5/50011-7">doi:10.1016/B978-044451560-5/50011-7</a>, <a href="http://arxiv.org/abs/math-ph/0602036">arXiv:math-ph/0602036</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Franco+Strocchi">Franco Strocchi</a>, An Introduction to Non-Perturbative Foundations of Quantum Field Theory, Oxford University Press (2013) [<a href="https://doi.org/10.1093/acprof:oso/9780199671571.001.0001">doi:10.1093/acprof:oso/9780199671571.001.0001</a>] </li> </ul> An account written by mathematicians for mathematicians: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hellmut+Baumg%C3%A4rtel">Hellmut Baumgärtel</a>, Manfred Wollenberg, Causal nets of operator algebras. Berlin: Akademie Verlag 1992 (<a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0749.46038&format=complete">ZMATH entry</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Hellmut+Baumg%C3%A4rtel">Hellmut Baumgärtel</a>, Operator algebraic Methods in Quantum Field Theory. A series of lectures. Akademie Verlag 1995 (<a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0839.46063&format=complete">ZMATH entry</a>) </li> </ul> <h3 id="reviews">Reviews</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Franco+Strocchi">Franco Strocchi</a>, Relativistic Quantum Mechanics and Field Theory, Found. Phys. 34 (2004) 501-527 [<a href="http://arxiv.org/abs/hep-th/0401143">arXiv:hep-th/0401143</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Sergio+Doplicher">Sergio Doplicher</a>, The principle of locality: Effectiveness, fate, and challenges, J. Math. Phys. 51, 015218 (2010), <a href="http://dx.doi.org/10.1063/1.3276100">doi</a> </li> <li id="Montoya"> Edison Montoya, Algebraic quantum field theory (2009) (<a href="http://www.matmor.unam.mx/~robert/sem/20091021_Montoya.pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Detlev+Buchholz">Detlev Buchholz</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Algebraic quantum field theory: objectives, methods, and results, in <a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a>, Elsevier (2024) [<a href="https://arxiv.org/abs/2305.12923">arXiv:2305.12923</a>] </li> </ul> More on the role of <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra+factors">von Neumann algebra factors</a> in AQFT <ul> <li id="Yngvason">J. Yngvason, The role of type III factors in quantum field theory [<a href="http://arxiv.org/abs/math-ph/0411058">arXiv:math-ph/0411058</a>]</li> </ul> <h3 id="ExamplesReferences">Examples</h3> Construction of examples is considered for instance in <ul id="Dimock82"> <li><a class="existingWikiWord" href="/nlab/show/Jonathan+Dimock">Jonathan Dimock</a>, Dirac quantum fields on a manifold, Trans. Amer. Math. Soc. 269 (1982), 133-147. (<a href="http://www.ams.org/journals/tran/1982-269-01/S0002-9947-1982-0637032-8/home.html">web</a>)</li> </ul> <ul id="Dimock92"> <li><a class="existingWikiWord" href="/nlab/show/Jonathan+Dimock">Jonathan Dimock</a>, Quantized electromagnetic field on a manifdold, Reviews in mathematical physics, Volume 4, Issue 2 (1992) (<a href="http://www.worldscinet.com/rmp/04/0402/S0129055X92000078.html">web</a>)</li> </ul> <ul id="Furliani"> <li>Edward Furliani, Quantization of massive vector fields in curved space–time, J. Math. Phys. 40, 2611 (1999) (<a href="http://jmp.aip.org/resource/1/jmapaq/v40/i6/p2611_s1?isAuthorized=no">web</a>)</li> </ul> General discussion of AQFT quantization of <a class="existingWikiWord" href="/nlab/show/free+quantum+fields">free quantum fields</a> is in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Christian+B%C3%A4r">Christian Bär</a>, N. Ginoux, <a class="existingWikiWord" href="/nlab/show/Frank+Pf%C3%A4ffle">Frank Pfäffle</a>, Wave Equations on Lorentzian Manifolds and Quantization, (EMS, 2007) (<a href="http://arxiv.org/abs/0806.1036">arXiv:0806.1036</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Christian+B%C3%A4r">Christian Bär</a>, N. Ginoux, Classical and quantum fields on lorentzian manifolds (2011) (<a href="http://arxiv.org/abs/1104.1158">arXiv:1104.1158</a>) </li> </ul> Examples of <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/interacting+quantum+field+theory">interacting</a> <a class="existingWikiWord" href="/nlab/show/scalar+field+theory">scalar field theory</a> in any <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> (in particular in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">d \geq 4</annotation></semantics></math>) are claimed in <ul> <li id="BuchholtzFredenhagen20"><a class="existingWikiWord" href="/nlab/show/Detlev+Buchholz">Detlev Buchholz</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebraic approach to interacting quantum field theories, Commun. Math. Phys. 377 (2020) 947–969 [<a href="https://arxiv.org/abs/1902.06062">arXiv:1902.06062</a>, <a href="https://doi.org/10.1007/s00220-020-03700-9">doi:10.1007/s00220-020-03700-9</a>]</li> </ul> <h3 id="LocalGaugeTheory">Local gauge theory</h3> Discussion of aspects of <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> includes <ul> <li> Fabio Ciolli, <a class="existingWikiWord" href="/nlab/show/Giuseppe+Ruzzi">Giuseppe Ruzzi</a>, Ezio Vasselli, Causal posets, loops and the construction of nets of local algebras for QFT (<a href="http://arxiv.org/abs/1109.4824">arXiv:1109.4824</a>) </li> <li> Fabio Ciolli, <a class="existingWikiWord" href="/nlab/show/Giuseppe+Ruzzi">Giuseppe Ruzzi</a>, Ezio Vasselli, QED Representation for the Net of Causal Loops (<a href="http://arxiv.org/abs/1305.7059">arXiv:1305.7059</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Giuseppe+Ruzzi">Giuseppe Ruzzi</a>, Nets of local algebras and gauge theories, 2014 (<a href="http://www.aqft14.eu/wp-content/uploads/2014/05/Ruzzi.pdf">pdf slides</a>) </li> </ul> Construction and axiomatization of gauge field AQFT via <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/homotopical+algebra">homotopical algebra</a> (see also at <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a>) is being developed in <ul> <li id="BDS"> <a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Claudio+Dappiaggi">Claudio Dappiaggi</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, Quantized Abelian principal connections on Lorentzian manifolds, Communications in Mathematical Physics 2013 (<a href="http://arxiv.org/abs/1303.2515">arXiv:1303.2515</a>) </li> <li id="BeniniSchenkelSzabo15"> <a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, Homotopy colimits and global observables in Abelian gauge theory (<a href="http://arxiv.org/abs/1503.08839">arXiv:1503.08839</a>) </li> <li id="BeniniSchenkel16"> <a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, Quantum field theories on categories fibered in groupoids (<a href="https://arxiv.org/abs/1610.06071">arXiv:1610.06071</a>) </li> </ul> The issue of the tension between local gauge invariance and locality and the need to pass to <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>/<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> is made explicit in <ul> <li id="Schenkel14"><a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, On the problem of gauge theories in locally covariant QFT_, talk at <a href="http://www.science.unitn.it/~moretti/convegno/convegno.html">Operator and Geometric Analysis on Quantum Theory</a> Trento, 2014 (<a class="existingWikiWord" href="/nlab/files/SchenkelTrento2014.pdf" title="pdf">pdf</a>) (with further emphasis on this point in the companion talk <a href="field+bundle#Schreiber14">Schreiber 14</a>) </li> </ul> Further development of this <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical algebraic quantum field theory</a> includes <ul> <li id="BeniniSchenkel16"><a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, Quantum field theories on categories fibered in groupoids (<a href="https://arxiv.org/abs/1610.06071">arXiv:1610.06071</a>)</li> </ul> <h3 id="ReferencesPerturbationTheory">Perturbation theory and renormalization</h3> <a class="existingWikiWord" href="/nlab/show/perturbation+theory">Perturbation theory</a> and <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> in the context of AQFT and is discussed in the following articles. The observation that in <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> the <a class="existingWikiWord" href="/nlab/show/renormalization">Stückelberg-Bogoliubov-Epstein-Glaser</a> local <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrices</a> yield a <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> was first made in <ul> <li>V. Il’in, D. Slavnov, Observable algebras in the S-matrix approach Theor. Math. Phys. 36 , 32 (1978)</li> </ul> which was however mostly ignored and forgotten. It is taken up again in <ul> <li><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="http://arxiv.org/abs/math-ph/9903028">arXiv</a>)</li> </ul> (a quick survey is in section 8, details are in section 2). Further developments along these lines are in <ul> <li><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 deformation quantization, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) (<a href="http://xxx.uni-augsburg.de/abs/hep-th/0101079">arXiv:hep-th/0101079</a>)</li> </ul> (relation to <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a>) <ul> <li> <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="http://arxiv.org/abs/math-ph/9903028">arXiv</a>) </li> <li> <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> (relation to <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a>) <ul> <li><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/abs/hep-th/9807078">arXiv:hep-th/9807078</a>).</li> </ul> (relation to <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> and <a class="existingWikiWord" href="/nlab/show/QED">QED</a>) Lecture notes are in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, Perturbative algebraic quantum field theory, In Mathematical Aspects of Quantum Field Theories, Springer 2016 (<a href="https://arxiv.org/abs/1208.1428">arXiv:1208.1428</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, Perturbative Construction of Models of Algebraic Quantum Field Theory (<a href="https://arxiv.org/abs/1503.07814">arXiv:1503.07814</a>) </li> </ul> and a textbook acount is in <ul> <li><a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (<a href="https://link.springer.com/book/10.1007%2F978-3-319-25901-7">pdf</a>)</li> </ul> <h3 id="further_developments">Further developments</h3> <ul> <li id="DLM"> <a class="existingWikiWord" href="/nlab/show/Claudio+Dappiaggi">Claudio Dappiaggi</a>, <a class="existingWikiWord" href="/nlab/show/Gandalf+Lechner">Gandalf Lechner</a>, E. Morfa-Morales, Deformations of quantum field theories on spacetimes with Killing vector fields, Commun.Math.Phys.305:99-130, (2011), (<a href="http://arxiv.org/abs/1006.3548">arXiv:1006.3548</a>) </li> <li> Angelos Anastopoulos, <a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>. Gluing algebraic quantum field theories on manifolds (2024). (<a href="https://arxiv.org/abs/2404.09638">arXiv:2404.09638</a>). </li> </ul> Relation to <a class="existingWikiWord" href="/nlab/show/factorization+algebras">factorization algebras</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Marco+Perin">Marco Perin</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds, Communications in Mathematical Physics volume 377, pages 971–997 (2020) (<a href="https://arxiv.org/abs/1903.03396">arXiv:1903.03396v2</a>, <a href="https://doi.org/10.1007/s00220-019-03561-x">doi:10.1007/s00220-019-03561-x</a>)</li> </ul> Relation to <a class="existingWikiWord" href="/nlab/show/smooth+stacks">smooth stacks</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Marco+Perin">Marco Perin</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, Smooth 1-dimensional algebraic quantum field theories (<a href="https://arxiv.org/abs/2010.13808">arXiv:2010.13808</a>)</li> </ul> <h3 id="relation_to_holographic_entanglement_entropy">Relation to holographic entanglement entropy</h3> Discussion of <a class="existingWikiWord" href="/nlab/show/local+nets+of+observables">local nets of observables</a> in <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> as the natural language for grasping <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Notes on Some Entanglement Properties of Quantum Field Theory, Rev. Mod. Phys. 90, 45003 (2018) (<a href="https://arxiv.org/abs/1803.04993">arXiv:1803.04993</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Thomas+Faulkner">Thomas Faulkner</a>, The holographic map as a conditional expectation (<a href="https://arxiv.org/abs/2008.04810">arXiv:2008.04810</a>) </li> </ul> <div> <h3 id="ReferencesRelationBetweenAQFTAndFQFT">Relation between algebraic and functorial field theory</h3> On the relation between <a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a> (axiomatizing the <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a> of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>) and <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (axiomatizing the <a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a>): <ul> <li id="Schreiber09"> <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/files/AQFTfromFQFT.pdf" title="AQFT from n-Functorial QFT">AQFT from n-Functorial QFT</a>, Comm. Math. Phys. 291 2 (2009) 357-401 &lbrack;<a href="https://arxiv.org/abs/0806.1079">arXiv:0806.1079</a>, <a href="https://doi.org/10.1007/s00220-009-0840-2">doi:10.1007/s00220-009-0840-2</a>&rbrack; </li> <li id="Johnson-Freyd21"> <a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, Heisenberg-picture quantum field theory, in Representation Theory, Mathematical Physics, and Integrable Systems, Progress in Mathematics 340 (2021) &lbrack;<a href="https://arxiv.org/abs/1508.05908">arXiv:1508.05908</a>, <a href="https://doi.org/10.1007/978-3-030-78148-4_13">doi:10.1007/978-3-030-78148-4_13</a>&rbrack; </li> <li id="BunkMacManusSchenkel23"> <a class="existingWikiWord" href="/nlab/show/Severin+Bunk">Severin Bunk</a>, <a class="existingWikiWord" href="/nlab/show/James+MacManus">James MacManus</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, Lorentzian bordisms in algebraic quantum field theory &lbrack;<a href="https://arxiv.org/abs/2308.01026">arXiv:2308.01026</a>&rbrack; </li> </ul> </div></body></html> </div> <div class="revisedby"> Last revised on January 29, 2025 at 16:29:06. See the <a href="/nlab/history/AQFT" style="color: #005c19">history</a> of this page for a list of all contributions to it. </div> <div class="navigation navfoot"> <a href="/nlab/edit/AQFT" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/2837/#Item_23">Discuss</a><a href="/nlab/revision/AQFT/92" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a><a href="/nlab/show/diff/AQFT" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/AQFT" accesskey="S" class="navlink" id="history" rel="nofollow">History (92 revisions)</a> <a href="/nlab/show/AQFT/cite" style="color: black">Cite</a> <a href="/nlab/print/AQFT" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/AQFT" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>