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this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#details'>Details</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesNonConvergenceOfThePerturbationSeries'>Non-convergence of the perturbation series</a></li> <li><a href='#linfinity_algebra_structure'>L-infinity algebra structure</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <blockquote> “God does not do perturbation theory, perturbation theory is what we do because we don’t know any better.” [<a class="existingWikiWord" href="/nlab/show/Duff+interview+at+M-Theory-Mathematics+2020">Duff 2020</a>] </blockquote> What is called perturbative quantum field theory (pQFT) is <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> where the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> (between <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>/<a class="existingWikiWord" href="/nlab/show/particles">particles</a>) is treated as a tiny <a class="existingWikiWord" href="/nlab/show/perturbation">perturbation</a> of the “<a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a>” where no <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> is assumed to take place (“<a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a>”). This is meant to be an approximation to the actual <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative quantum field theory</a>. However, the latter remains elusive except for toy examples of low spacetime dimension, vanishing <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> and/or <a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological invariance</a> and most of the “quantum field theory” in the literature is tacitly understood to be perturbative. Hence pQFT studies the <a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhood">infinitesimal neighbourhood</a> (also called the <a class="existingWikiWord" href="/nlab/show/formal+neighbourhood">formal neighbourhood</a>) of <a class="existingWikiWord" href="/nlab/show/free+quantum+field+theories">free</a> <a class="existingWikiWord" href="/nlab/show/classical+field+theories">classical field theories</a> in the parameter space of all quantum field theories, the rest of the space being the realm of <a class="existingWikiWord" href="/nlab/show/non-perturbative+QFT">non-perturbative QFT</a>: Mathematically this means that the resulting <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> in pQFT (typically: <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a> or “<a class="existingWikiWord" href="/nlab/show/S-matrices">S-matrices</a>” encoded by <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a>) are <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> in the <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> which measures the strength of the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> (as well as in <a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a>, which measures the general strength of <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum</a>) effects. This distinguishes perturbative quantum field theory from <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative quantum field theory</a>, where the algebras of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> are supposed to be not formal power series algebras, but <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a>. <center> <img src="https://ncatlab.org/nlab/files/NonPerturbativeParameterSpace-230613.jpg" width="360" /> </center> While just describing a tiny (infinitesimal) part of QFT, perturbative QFT is way better understood than full <a class="existingWikiWord" href="/nlab/show/non-perturbative+QFT">non-perturbative QFT</a> (which remains ill-understood to the extent that it has been named a <a href="mass+gap#ReferencesMassGapProblem">Millennium Problem</a>), and has the reputation of still being dramatically successful in practice – though this applies to some field theories (such as <a class="existingWikiWord" href="/nlab/show/QED">QED</a>) much more than to others (such as <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a>), see <a href="#ReferencesForInverseOfCouplingArgument">below</a>. The key object of perturbative QFT is the perturbative <a class="existingWikiWord" href="/nlab/show/scattering+matrix">scattering matrix</a> which expresses, as a <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> in the ratio of the <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a> over <a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a>, the <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a> of <a class="existingWikiWord" href="/nlab/show/scattering">scattering</a> processes, namely of processes where <a class="existingWikiWord" href="/nlab/show/free+field+theory">free fields</a> in a certain <a class="existingWikiWord" href="/nlab/show/quantum+state">state</a> come in from the far past, interact and hence scatter off each other, and then go off in some other <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> into the far future. The <a class="existingWikiWord" href="/nlab/show/scattering+cross+sections">scattering cross sections</a> thus defined are the quantities which may be directly measured in scattering <a class="existingWikiWord" href="/nlab/show/experiments">experiments</a>, such as the <a class="existingWikiWord" href="/nlab/show/LHC">LHC</a> accelerator. The perturbative <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> turns out to have an expression as a sum over separate <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a> for elementary processes labeled by <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a>, each of which depicts one specific way for fields (<a class="existingWikiWord" href="/nlab/show/particles">particles</a>) to interact with each other. That the full S-matrix is the sum over all amplitudes for all these possible scattering processes, the <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a>, is an incarnation of the informal heuristic of the <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> and the <a class="existingWikiWord" href="/nlab/show/superposition+principle">superposition principle</a> in <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>, which says that the <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a> for a specific outcome is the sum over the probability amplitudes of all the possible processes that can contribute to this outcome. For all interesting <a class="existingWikiWord" href="/nlab/show/interacting+field+theories">interacting field theories</a>, such as <a class="existingWikiWord" href="/nlab/show/quantum+electrodynamics">quantum electrodynamics</a> and <a class="existingWikiWord" href="/nlab/show/quantum+chromodynamics">quantum chromodynamics</a>, this <a class="existingWikiWord" href="/nlab/show/scattering+matrix">scattering matrix</a> <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> necessarily has vanishing <a class="existingWikiWord" href="/nlab/show/radius+of+convergence">radius of convergence</a> (<a href="#Dyson52">Dyson 52</a>). If it is assumed that the <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal</a> <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> is the <a class="existingWikiWord" href="/nlab/show/Taylor+series">Taylor series</a> of an actual <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> given by the actual <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative quantum field theory</a> that is being approximated, then this means that it is at least an <a class="existingWikiWord" href="/nlab/show/asymptotic+series">asymptotic series</a> (by <a href="asymptotic+series#TaylorSeriesOfSmoothFunctionIsAsymptoticSeries">this example</a>) whose first couple of terms could sum to a good approximation of the actual value to be computed. Indeed, the sum of the first few <a class="existingWikiWord" href="/nlab/show/loop+orders">loop orders</a> in the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> for <a class="existingWikiWord" href="/nlab/show/QED">QED</a> and <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a> in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> turns out to be in agreement with <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a> to good precision. (There are however known <a class="existingWikiWord" href="/nlab/show/non-perturbative+effects">non-perturbative effects</a> which are not captured in perturbation theory, such as <a class="existingWikiWord" href="/nlab/show/confinement">confinement</a> in <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a>, supposedly related to <a class="existingWikiWord" href="/nlab/show/instantons+in+QCD">instantons in QCD</a>. In <a class="existingWikiWord" href="/nlab/show/resurgence+theory">resurgence theory</a> one tries to identify these from the <a class="existingWikiWord" href="/nlab/show/asymptotic+series">asymptotic</a> nature of the <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a>.) A key step in the construction of perturbative quantum field theory is the <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> of the point interactions. This comes about because given <ol> <li> a <a class="existingWikiWord" href="/nlab/show/local+Lagrangian+density">local Lagrangian density</a> defining the nature of the <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> and their <a class="existingWikiWord" href="/nlab/show/interactions">interactions</a>, </li> <li> a <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> (generally: <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a>) that defines the <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> to be perturbed about </li> </ol> it turns out that the construction of the perturbative <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> (the <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a>) still involves at each order a finite-dimensional space of choices to be made. Physically, these are the specification of further high energy interactions not seen in the original <a class="existingWikiWord" href="/nlab/show/local+Lagrangian+density">local Lagrangian density</a>; mathematically, this is the choice of extending the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> of the interaction, which is an <a class="existingWikiWord" href="/nlab/show/operator-valued+distribution">operator-valued distribution</a>, to the locus of coinciding interaction points, in the sense of <a class="existingWikiWord" href="/nlab/show/extensions+of+distributions">extensions of distributions</a>. Historically, perturbative quantum field theory as originally conceived informally by <a class="existingWikiWord" href="/nlab/show/Schwinger-Tomonaga-Feynman-Dyson">Schwinger-Tomonaga-Feynman-Dyson</a> in the 1940s, had been notorious for the mysterious conceptual nature of its mathematical principles (“divergences”). The mathematically rigorous formulation of <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> (“removal of <a class="existingWikiWord" href="/nlab/show/UV-divergences">UV-divergences</a>”) in perturbative quantum field theory on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> was established by <a href="#EpsteinGlaser73">Epstein-Glaser 73</a>, based on <a href="#BogoliubovShirkov59">Bogoliubov-Shirkov 59</a> and <a href="#Stueckelberg51">Stückelberg 51</a>), now known as <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>; laid out in the seminal Erice summer school proceeding (<a href="#VeloWightman76">Velo-Wightman 76</a>). The correct definition of the <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatic limit</a> (“removal of IR divergencies”) was understood in <a href="#IlinSlavnov78">Il’in-Slavnov 78</a> and eventually developed by <a href="#DuetschFredenhagen01">Dütsch-Fredenhagen 01</a>, <a href="#BrunettiDuetschFredenhagen09">Brunetti-DütschFredenhagen 09</a>, this is now called <a class="existingWikiWord" href="/nlab/show/perturbative+algebraic+quantum+field+theory">perturbative algebraic quantum field theory</a>. The rigorous derivation of the previously informal <a class="existingWikiWord" href="/nlab/show/Feynman+rules">Feynman rules</a> and their <a class="existingWikiWord" href="/nlab/show/dimensional+regularization">dimensional regularization</a> for computation of <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a> was achieved in <a href="#Keller10">Keller 10 (IV.12)</a>, <a href="#DuetschFredenhagenKellerRejzner14">Dütsch-Fredenhagen-Keller-Rejzner 14</a>. Quantization of <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>) in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> was then discussed (for trivial <a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a> and restricted to <a class="existingWikiWord" href="/nlab/show/gauge+invariant+observables">gauge invariant observables</a>) in the spirit of <a class="existingWikiWord" href="/nlab/show/BRST-complex">BRST-complex</a>/<a class="existingWikiWord" href="/nlab/show/BV-formalism">BV-formalism</a> in (<a href="#FredenhagenRejzner11b">Fredenhagen-Rejzner 11b</a>). The generalization of all these constructions from <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> to perturbative quantum fields on more general <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> (i.e. for more general <a class="existingWikiWord" href="/nlab/show/gravity">gravitational</a> <a class="existingWikiWord" href="/nlab/show/background+fields">background fields</a> such as appearing in <a class="existingWikiWord" href="/nlab/show/cosmology">cosmology</a> or <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> physics) was made possible due to the identification of the proper generalization of <a class="existingWikiWord" href="/nlab/show/vacuum+states">vacuum states</a> and their <a class="existingWikiWord" href="/nlab/show/Feynman+propagators">Feynman propagators</a> to <a class="existingWikiWord" href="/nlab/show/Hadamard+states">Hadamard states</a> on <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetimes">globally hyperbolic spacetimes</a> in <a href="#Radzikowski96">Radzikowski 96</a>. The resulting rigorous perturbative <a class="existingWikiWord" href="/nlab/show/QFT+on+curved+spacetimes">QFT on curved spacetimes</a> was developed in a long series of articles by <a class="existingWikiWord" href="/nlab/show/Stefan+Hollands">Hollands</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Wald">Wald</a>, <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Fredenhagen</a> and others, now called <a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+AQFT">locally covariant perturbative AQFT</a>. While this establishes a rigorous construction of perturbative quantum field theory on general gravitational backgrounds, the construction principles had remained somewhat ad-hoc: The <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> for the perturbative <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> (equivalently for the <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> or <a class="existingWikiWord" href="/nlab/show/retarded+products">retarded products</a> of field operators) were well motivated by comparison with the <a class="existingWikiWord" href="/nlab/show/Dyson+series">Dyson series</a> in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, by the heuristics of the <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> and not the least by their excellent confirmation by <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, but had not been derived from first principles of <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a>. Then in <a href="#DuetschFredenhagen01">Dütsch Fredenhagen 01</a> it was observed that the <a class="existingWikiWord" href="/nlab/show/Wick+algebras">Wick algebras</a> of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> in <a class="existingWikiWord" href="/nlab/show/free+quantum+field+theory">free quantum field theory</a> are equivalently the <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a> of the canonical <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> (the <a class="existingWikiWord" href="/nlab/show/Peierls+bracket">Peierls bracket</a> or <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a>) on the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> of the free field theory (or rather of a choice of <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> for it) and <a href="#Collini16">Collini 16</a> showed that under suitable conditions the perturbative <a class="existingWikiWord" href="/nlab/show/interacting+observable+algebra">interacting observable algebra</a> is the <a class="existingWikiWord" href="/nlab/show/Fedosov+deformation+quantization">Fedosov deformation quantization</a> of <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> of the interacting theory. A general argument to this extent was given in <a href="#HawkinsRejzner16">Hawkins-Rejzner 16</a>. This suggests that the construction of the full <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative quantum field theory</a> ought to be given by a <a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantization</a> of the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>. But presently no example of such for non-trivial interaction in <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\geq 4</annotation></semantics></math> is known. In particular the <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenologically</a> interesting case of a complete construction of interacting field theories on 4-dimensional spacetimes is presently unknown. For the case of <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> this open problem to go beyond perturbative quantum field theory is one of the “Millennium Problems” (see at <a class="existingWikiWord" href="/nlab/show/quantization+of+Yang-Mills+theory">quantization of Yang-Mills theory</a>). For the case of <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a> this is possibly the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>10</mn> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">10^4</annotation></semantics></math>-year problem that the field is facing. But observe that as a perturbative (<a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective</a>“) quantum field theory, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a> does fit into the framework of perturbative QFT, is mathematically well-defined and makes predictions, see the references <a href="quantum%20gravity#ReferencesAsAnEffectiveFieldTheory">there</a>. <h2 id="details">Details</h2> A comprehensive introduction is at <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+perturbative+quantum+field+theory">geometry of physics – perturbative quantum field theory</a>. <h2 id="properties">Properties</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization+theory">main theorem of perturbative renormalization theory</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a>, <a class="existingWikiWord" href="/nlab/show/radiative+correction">radiative correction</a> </li> <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> </ul> <div> <table><thead><tr><th>product in <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_27ee5fe71b396df84a9f4b19030bbbae0b7c742f_1"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,</annotation></semantics></math> induces</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> (<a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> (<a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/interacting+quantum+observables">interacting quantum observables</a></td></tr> </tbody></table> </div> <ul> <li><a class="existingWikiWord" href="/nlab/show/non-perturbative+field+theory">non-perturbative field theory</a></li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> The original informal conception of perturbative QFT is due to <a class="existingWikiWord" href="/nlab/show/Schwinger-Tomonaga-Feynman-Dyson">Schwinger-Tomonaga-Feynman-Dyson</a>: <ul> <li id="Dyson49"><a class="existingWikiWord" href="/nlab/show/Freeman+Dyson">Freeman Dyson</a>, The raditation theories of Tomonaga, Schwinger and Feynman, Phys. Rev. 75, 486, 1949 (<a href="http://web.ihep.su/dbserv/compas/src/dyson49b/eng.pdf">pdf</a>)</li> </ul> Non-rigorous but widely used textbooks: <ul> <li> <a class="existingWikiWord" href="/nlab/show/James+D.+Bjorken">James D. Bjorken</a>, <a class="existingWikiWord" href="/nlab/show/Sidney+D.+Drell">Sidney D. Drell</a>: Relativistic Quantum Mechanics, McGrawHill (1964) [<a href="https://archive.org/details/relativisticquan0000bjor/page/n1/mode/2up">ark:/13960/t5fc2v05h</a>, <a href="https://emineter.wordpress.com/wp-content/uploads/2018/10/james-d-bjorken-sidney-d-drell-relativistic-quantum-mechanics-1964.pdf">pdf</a>, <a href="http://www.mmmut.ac.in/News_content/14331tpnews_11122020.pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/James+D.+Bjorken">James D. Bjorken</a>, <a class="existingWikiWord" href="/nlab/show/Sidney+D.+Drell">Sidney D. Drell</a>: Relativistic Quantum Fields, McGraw-Hill (1965) [<a href="https://archive.org/details/relativisticquan0000bjor_c5q0">ark:/13960/t9t22sx2r</a>, <a href="https://ivlabs.github.io/resources/physics/books/Relativistic%20Quantum%20Fields%20by%20James%20D.%20Bjorken,%20Sidney%20D.%20Drell.pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Michael+Peskin">Michael Peskin</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Schroeder">Daniel Schroeder</a>: An Introduction to Quantum Field Theory, CRC Press (1995) [<a href="https://doi.org/10.1201/9780429503559">doi:10.1201/9780429503559</a>, <a href="https://inspirehep.net/literature/407703">spire:407703</a>, <a href="https://www.routledge.com/An-Introduction-To-Quantum-Field-Theory/Peskin-Schroeder/p/book/9780201503975">ISBN 9780201503975</a>] </li> <li id="Weinberg95"> <a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, The Quantum Theory of Fields Vol 1: Foundations, Cambridge University Press (1995) [<a href="https://doi.org/10.1017/CBO9781139644167">doi:10.1017/CBO9781139644167</a>] </li> <li id="Weinberg96"> <a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, The Quantum Theory of Fields Vol 2: Applications, Cambridge University Press (1996) [<a href="https://doi.org/10.1017/CBO9781139644174">doi:10.1017/CBO9781139644174</a>] </li> </ul> The rigorous formulation of renormalized perturbative quantum field theory in terms of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> was first accomplished in <ul> <li id="EpsteinGlaser73"><a class="existingWikiWord" href="/nlab/show/Henri+Epstein">Henri Epstein</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Glaser">Vladimir Glaser</a>, <a class="existingWikiWord" href="/nlab/show/The+Role+of+locality+in+perturbation+theory">The Role of locality in perturbation theory</a>, Annales Poincaré Phys. Theor. A 19 (1973) 211 (<a href="http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0">Numdam</a>)</li> </ul> with precursors in <ul> <li id="Stueckelberg49"> <a class="existingWikiWord" href="/nlab/show/Ernst+St%C3%BCckelberg">Ernst Stückelberg</a>, D. Rivier, Helv. Phys. Acta, 22 (1949) 215. </li> <li id="Stueckelberg51"> <a class="existingWikiWord" href="/nlab/show/Ernst+St%C3%BCckelberg">Ernst Stückelberg</a>, J. Green, Helv. Phys. Acta, 24 (1951) 153. </li> <li> <a class="existingWikiWord" href="/nlab/show/Ernst+St%C3%BCckelberg">Ernst Stückelberg</a>, A. Peterman, , La normalisation des constants dans la theorie des quanta, Helv. Phys. Acta 26, 499 (1953); </li> <li id="BogoliubovShirkov59"> <a class="existingWikiWord" href="/nlab/show/Nikolay+Bogoliubov">Nikolay Bogoliubov</a>, <a class="existingWikiWord" href="/nlab/show/Dmitry+Shirkov">Dmitry Shirkov</a>, Introduction to the Theory of Quantized Fields, New York (1959) </li> </ul> Traditional review includes: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Matthew+Schwartz">Matthew Schwartz</a>, Quantum Field Theory and the Standard Model, Cambridge University Press (2014) [<a href="https://www.cambridge.org/de/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/quantum-field-theory-and-standard-model?format=HB&isbn=9781107034730">ISBN:9781107034730</a>, <a href="https://doi.org/10.1017/9781139540940">doi:10.1017/9781139540940</a>, front matter:<a href="https://assets.cambridge.org/97811070/34730/frontmatter/9781107034730_frontmatter.pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Michel+Talagrand">Michel Talagrand</a>, What is a Quantum Field Theory? – A first Introduction for Mathematicians, Cambridge University Press (2022) [<a href="https://doi.org/10.1017/9781108225144">doi:10.1017/9781108225144</a>] </li> </ul> A seminal compilation of the resulting rigorous understanding of <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> is <ul> <li id="VeloWightman76">G. Velo and <a class="existingWikiWord" href="/nlab/show/Arthur+Wightman">Arthur Wightman</a> (eds.): Renormalization Theory, Proceedings of the 1975 Erice summer school, NATO ASI Series C 23, D. Reidel, Dordrecht, 1976</li> </ul> Concrete computations in rigorous <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> have been spelled out for <a class="existingWikiWord" href="/nlab/show/quantum+electrodynamics">quantum electrodynamics</a> in <ul> <li id="Scharf95"><a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Scharf">Günter Scharf</a>, <a class="existingWikiWord" href="/nlab/show/Finite+Quantum+Electrodynamics+--+The+Causal+Approach">Finite Quantum Electrodynamics – The Causal Approach</a>, Berlin: Springer-Verlag, 1995, 2nd edition</li> </ul> and for <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+chromodynamics">quantum chromodynamics</a> and <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+gravity">perturbative quantum gravity</a> in <ul> <li id="Scharf01"><a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Scharf">Günter Scharf</a>, <a class="existingWikiWord" href="/nlab/show/Quantum+Gauge+Theories+--+A+True+Ghost+Story">Quantum Gauge Theories – A True Ghost Story</a>, Wiley 2001</li> </ul> The treatment of the IR-divergencies by organizing the perturbative <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> into a <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> was first suggested in <ul> <li id="IlinSlavnov78">V. A. Il’in and D. S. Slavnov, Observable algebras in the S-matrix approach, Theor. Math. Phys. 36 (1978) 32 (<a href="http://inspirehep.net/record/135575">spire</a>, <a href="http://dx.doi.org/10.1007/BF01035870">doi</a>)</li> </ul> and then developed to <a class="existingWikiWord" href="/nlab/show/perturbative+algebraic+quantum+field+theory">perturbative algebraic quantum field theory</a> in <ul> <li id="DuetschFredenhagen01"> <a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion, Commun. Math. Phys. 219 (2001) 5-30 (<a href="https://arxiv.org/abs/hep-th/0001129">arXiv:hep-th/0001129</a>) </li> <li id="BrunettiDuetschFredenhagen09"> <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (<a href="http://arxiv.org/abs/0901.2038">arXiv:0901.2038</a>) </li> </ul> Quantization of <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>) in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> is discussed (for trivial <a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a> and restricted to <a class="existingWikiWord" href="/nlab/show/gauge+invariant+observables">gauge invariant observables</a>) in the spirit of <a class="existingWikiWord" href="/nlab/show/BRST-complex">BRST-complex</a>/<a class="existingWikiWord" href="/nlab/show/BV-formalism">BV-formalism</a> in <ul> <li id="FredenhagenRejzner11a"> <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, Batalin-Vilkovisky formalism in the functional approach to classical field theory, Commun. Math. Phys. 314(1), 93–127 (2012) (<a href="https://arxiv.org/abs/1101.5112">arXiv:1101.5112</a>) </li> <li id="FredenhagenRejzner11b"> <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun. Math. Phys. 317(3), 697–725 (2012) (<a href="https://arxiv.org/abs/1110.5232">arXiv:1110.5232</a>) </li> </ul> and surveyed in: <ul> <li id="Rejzner16"> <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, section 7 of <a class="existingWikiWord" href="/nlab/show/Perturbative+Algebraic+Quantum+Field+Theory">Perturbative Algebraic Quantum Field Theory</a>, Springer 2016 </li> <li> <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+perturbative+quantum+field+theory">geometry of physics – perturbative quantum field theory</a>, 2017 </li> <li id="Duetsch18"> <a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/From+classical+field+theory+to+perturbative+quantum+field+theory">From classical field theory to perturbative quantum field theory</a>, 2018 </li> </ul> The generalization of all these constructions to quantum fields on general <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetimes">globally hyperbolic spacetimes</a> (perturbative <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetimes">AQFT on curved spacetimes</a>) was made possible by the results on <a class="existingWikiWord" href="/nlab/show/Hadamard+states">Hadamard states</a> and <a class="existingWikiWord" href="/nlab/show/Feynman+propagators">Feynman propagators</a> in <ul> <li id="Radzikowski96"><a class="existingWikiWord" href="/nlab/show/Marek+Radzikowski">Marek Radzikowski</a>, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996), 529–553 (<a href="http://projecteuclid.org/euclid.cmp/1104287114">Euclid</a>)</li> </ul> and then developed in a long series of articles by <a class="existingWikiWord" href="/nlab/show/Stefan+Hollands">Stefan Hollands</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Wald">Robert Wald</a>, <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a> and others. For this see the references at <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetimes">AQFT on curved spacetimes</a>. The observation that perturbative quantum field theory is equivalently the <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a> of the defining <a class="existingWikiWord" href="/nlab/show/local+Lagrangian+density">local Lagrangian density</a> is for <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a> due to <ul> <li id="DuetschFredenhagen01"> <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> <li id="HirschfeldHenselder02"> A. C. Hirshfeld, P. Henselder, Star Products and Perturbative Quantum Field Theory, Annals Phys. 298 (2002) 382-393 (<a href="https://arxiv.org/abs/hep-th/0208194">arXiv:hep-th/0208194</a>) </li> </ul> and for interacting field theories (<a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a>) due <ul> <li id="Collini16"> <a class="existingWikiWord" href="/nlab/show/Giovanni+Collini">Giovanni Collini</a>, Fedosov Quantization and Perturbative Quantum Field Theory (<a href="https://arxiv.org/abs/1603.09626">arXiv:1603.09626</a>) </li> <li id="HawkinsRejzner16"> <a class="existingWikiWord" href="/nlab/show/Eli+Hawkins">Eli Hawkins</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, The Star Product in Interacting Quantum Field Theory (<a href="https://arxiv.org/abs/1612.09157">arXiv:1612.09157</a>) </li> </ul> For more see the references at <a class="existingWikiWord" href="/nlab/show/perturbative+algebraic+quantum+field+theory">perturbative algebraic quantum field theory</a>. The relation of the construction via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> to the <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> in terms of <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> was understood in <ul> <li id="GarciaBondiaLazzarini00"> <a class="existingWikiWord" href="/nlab/show/Jose+Gracia-Bondia">Jose Gracia-Bondia</a>, S. Lazzarini, Connes-Kreimer-Epstein-Glaser Renormalization (<a href="https://arxiv.org/abs/hep-th/0006106">arXiv:hep-th/0006106</a>) </li> <li id="Keller10"> <a class="existingWikiWord" href="/nlab/show/Kai+Keller">Kai Keller</a>, chapter IV of Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization, PhD thesis (<a href="https://arxiv.org/abs/1006.2148">arXxiv:1006.2148</a>) </li> <li id="DuetschFredenhagenKellerRejzner14"> <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 class="existingWikiWord" href="/nlab/show/Kai+Keller">Kai Keller</a>, <a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization, J. Math. Phy. 55(12), 122303 (2014) (<a href="https://arxiv.org/abs/1311.5424">arXiv:1311.5424</a>) </li> </ul> (…) <h3 id="ReferencesNonConvergenceOfThePerturbationSeries">Non-convergence of the perturbation series</h3> The argument that the perturbation series of realistic pQFTs necessarily <a class="existingWikiWord" href="/nlab/show/divergent+series">diverges</a>, in fact has vanishing <a class="existingWikiWord" href="/nlab/show/radius+of+convergence">radius of convergence</a> (is at best an <a class="existingWikiWord" href="/nlab/show/asymptotic+series">asymptotic series</a>) goes back to <ul> <li id="Dyson52"><a class="existingWikiWord" href="/nlab/show/Freeman+Dyson">Freeman Dyson</a>, Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. 85, 631, 1952 (<a href="http://inspirehep.net/record/29799?ln=en">spire</a>)</li> </ul> and is made more precise in <ul> <li id="Lipatov77"><a class="existingWikiWord" href="/nlab/show/Lev+Lipatov">Lev Lipatov</a>, Divergence of the Perturbation Theory Series and the Quasiclassical Theory, Sov.Phys.JETP 45 (1977) 216–223 (<a href="http://jetp.ac.ru/cgi-bin/dn/e_045_02_0216.pdf">pdf</a>)</li> </ul> recalled for instance in <ul> <li id="Suslov05"> <a class="existingWikiWord" href="/nlab/show/Igor+Suslov">Igor Suslov</a>, section 1 of Divergent perturbation series, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 (<a href="https://arxiv.org/abs/hep-ph/0510142">arXiv:hep-ph/0510142</a>) </li> <li> Justin Bond, last section of Perturbative QFT is Asymptotic; is Divergent; is Problematic in Principle (<a href="https://mcgreevy.physics.ucsd.edu/s13/final-papers/2013S-215C-Bond-Justin.pdf">pdf</a>) </li> <li id="FloryHellingSluka12"> <a class="existingWikiWord" href="/nlab/show/Mario+Flory">Mario Flory</a>, <a class="existingWikiWord" href="/nlab/show/Robert+C.+Helling">Robert C. Helling</a>, Constantin Sluka, How I Learned to Stop Worrying and Love QFT [<a href="https://arxiv.org/abs/1201.2714">arXiv:1201.2714</a>] </li> <li id="Strocchi13"> <a class="existingWikiWord" href="/nlab/show/Franco+Strocchi">Franco Strocchi</a>, §2.2 of: 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> <li id="HollandsWald14"> <a class="existingWikiWord" href="/nlab/show/Stefan+Hollands">Stefan Hollands</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Wald">Robert Wald</a>, Section 4.1 of: Quantum fields in curved spacetime, Physics Reports 574 (2015) 1-35 [<a href="https://arxiv.org/abs/1401.2026">arXiv:1401.2026</a>, <a href="https://doi.org/10.1016/j.physrep.2015.02.001">doi:10.1016/j.physrep.2015.02.001</a>] </li> <li> Marco Serone, from 2:46 on in A look at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ϕ</mi> <mn>2</mn> <mn>4</mn></msubsup></mrow><annotation encoding="application/x-tex">\phi^4_2</annotation></semantics></math> using perturbation theory (<a href="https://www.youtube.com/watch?v=J4nxvY1rOhI">recording</a>) </li> </ul> The argument that the perturbation series should be trustworthy for number of terms smaller than the inverse of the <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a> is recalled in <a href="#FloryHellingSluka12">Flory, Helling & Sluka 2012, p. 8 & eq. (34) & Sec. 2.5</a>. Exposition also in: <ul> <li><a class="existingWikiWord" href="/nlab/show/Jakob+Schwichtenberg">Jakob Schwichtenberg</a>, <a href="http://jakobschwichtenberg.com/divergence-perturbation-series-qft">divergence-perturbation-series-qft</a></li> </ul> For the example of <a class="existingWikiWord" href="/nlab/show/phi%5E4+theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msup><mi>ϕ</mi> <mn>4</mn></msup> </mrow> <annotation encoding="application/x-tex">\phi^4</annotation> </semantics> </math>-theory</a> this non-convergence of the perturbation series is discussed in <ul> <li id="Helling"> <a class="existingWikiWord" href="/nlab/show/Robert+C.+Helling">Robert C. Helling</a>, p. 4 of Solving classical field equations (<a href="http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/HellingClassicalQFT.pdf" title="pdf">pdf</a>) </li> <li id="BakulevShirkov10"> Alexander P. Bakulev, <a class="existingWikiWord" href="/nlab/show/Dmitry+Shirkov">Dmitry Shirkov</a>, section 1.1 of Inevitability and Importance of Non-Perturbative Elements in Quantum Field Theory, Proceedings of the 6th Mathematical Physics Meeting, Sept. 14–23, 2010, Belgrade, Serbia (ISBN 978-86-82441-30-4), pp. 27–54 (<a href="https://arxiv.org/abs/1102.2380">arXiv:1102.2380</a>) </li> </ul> See also <ul> <li>Carl M. Bender, Carlo Heissenberg, Convergent and Divergent Series in Physics (<a href="https://arxiv.org/abs/1703.05164">arXiv:1703.05164</a>)</li> </ul> And see at <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> – <a href="perturbation+theory#ReferencesDivergenceConvergence">On divergence/convergence</a> Discussion of further issues, even when <a class="existingWikiWord" href="/nlab/show/resummation">resummation</a> is thought to apply, arising for <a class="existingWikiWord" href="/nlab/show/n-point+functions">n-point functions</a> at large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> (large number of external particles in a scattering process): failure of unitarity (for <a class="existingWikiWord" href="/nlab/show/phi%5En+interaction"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msup><mi>ϕ</mi> <mi>n</mi></msup> </mrow> <annotation encoding="application/x-tex">\phi^n</annotation> </semantics> </math>-theory</a>): <ul> <li>Sebastian Schenk, The Breakdown of Resummed Perturbation Theory at High Energies (<a href="https://arxiv.org/abs/2109.00549">arXiv:2109.00549</a>)</li> </ul> failure of <a class="existingWikiWord" href="/nlab/show/locality">locality</a> (for <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+gravity">perturbative quantum gravity</a> and <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theory</a>): <ul> <li> <a class="existingWikiWord" href="/nlab/show/Sudip+Ghosh">Sudip Ghosh</a>, <a class="existingWikiWord" href="/nlab/show/Suvrat+Raju">Suvrat Raju</a>, Loss of locality in gravitational correlators with a large number of insertions, Phys. Rev. D 96, 066033 (2017) (<a href="https://arxiv.org/abs/1706.07424">arXiv:1706.07424</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Sudip+Ghosh">Sudip Ghosh</a>, <a class="existingWikiWord" href="/nlab/show/Suvrat+Raju">Suvrat Raju</a>, The Breakdown of String Perturbation Theory for Many External Particles, Phys. Rev. Lett. 118, 131602 (2017) (<a href="https://arxiv.org/abs/1611.08003">arXiv:1611.08003</a>) </li> </ul> <h3 id="linfinity_algebra_structure">L-infinity algebra structure</h3> Further identification of <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra">L-infinity algebra</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> in the <a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a>/<a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> of <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian</a> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>: <ul> <li id="Froeb18"> <a class="existingWikiWord" href="/nlab/show/Markus+Fr%C3%B6b">Markus Fröb</a>, Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories (<a href="https://arxiv.org/abs/1803.10235">arXiv:1803.10235</a>) </li> <li id="Arvanitakis19"> <a class="existingWikiWord" href="/nlab/show/Alex+Arvanitakis">Alex Arvanitakis</a>, The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra of the S-matrix (<a href="https://arxiv.org/abs/1903.05643">arXiv:1903.05643</a>) </li> </ul> </body></html> </div> <div class="revisedby"> Last revised on August 16, 2024 at 06:28:57. 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