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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> effective quantum field theory </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/2960/#Item_17" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </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> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></strong> (<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>)</p> <p><a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a></p> <h2 id="concepts">Concepts</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a></strong>:</p> <ul> <li> <p><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></p> </li> <li> <p><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></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+history">field history</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/observables">observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+net">field net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> </ul> </li> <li> <p><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></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a></p> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a></p> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> <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> <p><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a></p> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>)</p> </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a></p> </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> <ul> <li><a href='#InCausalPerturbationTheory'>In causal perturbation theory</a></li> <ul> <li><a href='#UVRegularization'>(“Re”-)Normalization via UV-Regularization</a></li> <li><a href='#RelativeEffectiveAction'>Effective quantum field theory</a></li> <li><a href='#renormalization_via_wilsonian_rg_flow'>(“Re”-)Normalization via Wilsonian RG flow</a></li> </ul> <li><a href='#TraditionalInformalDiscussion'>Traditional informal discussion</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#lightbylight_scattering'>Light-by-light scattering</a></li> <li><a href='#rayleigh_scattering'>Rayleigh scattering</a></li> <li><a href='#FermiTheoryOfWeakInteractions'>Fermi theory of weak interactions</a></li> <li><a href='#for_nuclear_physics'>For nuclear physics</a></li> <li><a href='#in_lattice_qcd'>In lattice QCD</a></li> <li><a href='#neutrino_masses_'>Neutrino masses (?)</a></li> <li><a href='#StringTheoryAndGravityCoupledToGaugeTheory'>String theory and gravity coupled to gauge theory</a></li> </ul> <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='#ReferencesInCausalPerturbationTheory'>In causal perturbation theory</a></li> <li><a href='#for_string_theory'>For string theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>While fundamental <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is at some level well described by <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>, a typical <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> used to define such a QFT can reasonably be expected to define only degrees of freedom and interactions that are relevant up to some given <a class="existingWikiWord" href="/nlab/show/energy">energy</a> scale. In this perspective one speaks of the theory as being the <em>effective quantum field theory</em> of some – possibly known but possibly unspecified – more fundamental theory.</p> <p>An example (historically the first to be successfully considered) is the <a class="existingWikiWord" href="/nlab/show/Fermi+theory+of+beta+decay">Fermi theory of beta decay</a> of hadrons: this contains interactions of four <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a>s at a time, for instance a process in which a <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> decays into a collection consisting of a <a class="existingWikiWord" href="/nlab/show/proton">proton</a>, an <a class="existingWikiWord" href="/nlab/show/electron">electron</a> and a <a class="existingWikiWord" href="/nlab/show/neutrino">neutrino</a>. Later it was discovered that, more fundamentally, this is not a single reaction but is composed out of several other interactions that involve exchanges of <a class="existingWikiWord" href="/nlab/show/W-boson">W-boson</a>s between these four particles. Nevertheless, Fermi’s original <em>effective</em> theory made very precise predictions at energy scales less than 10 <a class="existingWikiWord" href="/nlab/show/MeV">MeV</a>. The reason is that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-boson has mass several orders of magnitude higher than that (about 80 <a class="existingWikiWord" href="/nlab/show/GeV">GeV</a>) and was thus <em>effectively</em> invisible at these low energies.</p> <p>The low energy expansion of any unitary, relativistic, <a class="existingWikiWord" href="/nlab/show/crossing+symmetry">crossing symmetric</a> <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> can be described by an effective quantum field theory.</p> <p>In the perspective of effective field theory notably <a class="existingWikiWord" href="/nlab/show/renormalizable+interaction">non-renormalizable</a> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> Lagrangians can still make perfect sense as effective theories and give rise to well defined predictions: they can be effective approximations to <a class="existingWikiWord" href="/nlab/show/renormalizable+interaction">renormalizable</a> more fundamental theories. This is sometimes called a <a class="existingWikiWord" href="/nlab/show/UV+completion">UV completion</a> of the given effective theory.</p> <p>For instance <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a> – which is notoriously <span class="newWikiWord">non-renormalizable<a href="/nlab/new/non-renormalizable+interaction">?</a></span> – makes perfect sense as an effective field theory (see for instance the introduction in (<a href="#DonoghueIntroduction">Donoghue</a>). It is in principle possible that there is some more fundamental theory with plenty of excitations at high energies that is however degreewise finite in <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a>, whose <em>effective</em> description at low energy is given by the <a class="existingWikiWord" href="/nlab/show/renormalizable+interaction">non-renormalizable</a> <a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a>. (For instance, <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> is meant to be such a theory.)</p> <h2 id="details">Details</h2> <p>The concept of effective <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> has a precise formulation in the rigoruous context of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a>:</p> <ul> <li><em><a href="#InCausalPerturbationTheory">Effective pQFT in Causal perturbation theory</a></em></li> </ul> <p>Effective quantum field theory has traditioanlly been discussed informally, referring to <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> intuition:</p> <ul> <li><em><a href="#TraditionalInformalDiscussion">Traditional informal arguments</a>-</em></li> </ul> <h3 id="InCausalPerturbationTheory">In causal perturbation theory</h3> <p>We discuss the rigorous formulation of effective <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> in terms of <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 to (<a href="#BrunettiDuetschFredenhagen09">Brunetti-Dütsch-Fredenhagen 09, section 5.2</a>, <a href="#Duetsch10">Dütsch 10</a>), reviewed in <a href="#Duetsch18">Dütsch 18, section 3.8</a>).</p> <h4 id="UVRegularization">(“Re”-)Normalization via UV-Regularization</h4> <div class="num_defn" id="CutoffsUVForPerturbativeQFT"> <h6 id="definition">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/UV+cutoffs">UV cutoffs</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo>,</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> over <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> (according to <a href="S-matrix#VacuumFree">this def.</a>), where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H</annotation></semantics></math> is the corresponding <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> inducing the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub><mo>∈</mo><mi>Γ</mi><msub><mo>′</mo> <mrow><mi>Σ</mi><mo>×</mo><mi>Σ</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>⊠</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta_F \in \Gamma'_{\Sigma \times \Sigma}(E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}}) </annotation></semantics></math></div> <p>by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H</annotation></semantics></math>.</p> <p>Then a choice of <em><a class="existingWikiWord" href="/nlab/show/UV+cutoffs">UV cutoffs</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a></em> around this vacuum is a collection of <a class="existingWikiWord" href="/nlab/show/non-singular+distributions">non-singular distributions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\Lambda}</annotation></semantics></math> parameterized by <a class="existingWikiWord" href="/nlab/show/positive+real+numbers">positive real numbers</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>×</mo><mi>Σ</mi><mo>,</mo><mi>cp</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>⊠</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>Λ</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (0, \infty) &\overset{}{\longrightarrow}& \Gamma_{\Sigma \times \Sigma,cp}(E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}}) \\ \Lambda &\mapsto& \Delta_{F,\Lambda} } </annotation></semantics></math></div> <p>such that:</p> <ol> <li> <p>each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\Lambda}</annotation></semantics></math> satisfies the following basic properties</p> <ol> <li> <p>(translation invariance)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta_{F,\Lambda}(x,y) = \Delta_{F,\Lambda}(x-y) </annotation></semantics></math></div></li> <li> <p>(symmetry)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow> <mrow><mi>b</mi><mi>a</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta^{b a}_{F,\Lambda}(y, x) \;=\; \Delta^{a b}_{F,\Lambda}(x, y) </annotation></semantics></math></div> <p>i.e.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow> <mrow><mi>b</mi><mi>a</mi></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta_{F,\Lambda}^{b a}(-x) \;=\; \Delta_{F,\Lambda}^{a b}(x) </annotation></semantics></math></div></li> </ol> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\Lambda}</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Lambda \to 0</annotation></semantics></math> exists and is zero</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda} \;=\; 0 \,. </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\Lambda}</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">\Lambda \to \infty</annotation></semantics></math> exists and is the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>Δ</mi> <mi>F</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda} \;=\; \Delta_F \,. </annotation></semantics></math></div></li> </ol> </div> <p>(<a href="#Duetsch10">Dütsch 10, section 4</a>)</p> <p>example: relativistic momentum cutoff with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>-regularization (<a href="#KellerKopperSchophaus97">Keller-Kopper-Schophaus 97, section 6.1</a>, <a href="#Duetsch18">Dütsch 18, example 3.126</a>)</p> <div class="num_defn" id="SMatrixEffective"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/effective+S-matrix+scheme">effective S-matrix scheme</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo>,</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (according to <a href="S-matrix#VacuumFree">this def.</a>) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}</annotation></semantics></math> be a choice of <a class="existingWikiWord" href="/nlab/show/UV+cutoffs">UV cutoffs</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> around this vacuum (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>).</p> <p>We say that the <em><a class="existingWikiWord" href="/nlab/show/effective+S-matrix+scheme">effective S-matrix scheme</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\Lambda</annotation></semantics></math> at cutoff scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda \in [0,\infty)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub></mrow></mover></mtd> <mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mi>O</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ] &\overset{\mathcal{S}_{\Lambda}}{\longrightarrow}& PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ] \\ O &\mapsto& \mathcal{S}_\Lambda(O) } </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/exponential+series">exponential series</a></p> <div class="maruku-equation" id="eq:EffectiveSMatrixScheme"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><msub><mi>exp</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mrow><mo>(</mo><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac><mi>O</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac><mi>O</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow></mfrac><mi>O</mi><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mi>O</mi><mo>+</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mn>3</mn></msup></mrow></mfrac><mi>O</mi><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mi>O</mi><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mn>0</mn><mo>+</mo><mi>⋯</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathcal{S}_\Lambda(O) & \coloneqq \exp_{F,\Lambda}\left( \frac{1}{i \hbar} O \right) \\ & = 1 + \frac{1}{i \hbar} O + \frac{1}{2} \frac{1}{(i \hbar)^2} O \star_{F,\Lambda} O + \frac{1}{3!} \frac{1}{(i \hbar)^3} O \star_{F,\Lambda} O \star_{F,\Lambda} 0 + \cdots \end{aligned} \,. </annotation></semantics></math></div> <p>with respect to the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\star_{F,\Lambda}</annotation></semantics></math> induced by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\Lambda}</annotation></semantics></math> (<a href="star+product#PropagatorStarProduct">this def.</a>).</p> <p>This is evidently defined on all <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> as shown, and restricts to an endomorphism on <a class="existingWikiWord" href="/nlab/show/microcausal+polynomial+observables">microcausal polynomial observables</a> as shown, since the contraction coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\Lambda}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/non-singular+distributions">non-singular distributions</a>, by definition of <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a>.</p> </div> <p>(<a href="#Duetsch10">Dütsch 10, (4.2)</a>)</p> <div class="num_prop" id="UVRegularization"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> via <a class="existingWikiWord" href="/nlab/show/UV+regularization">UV regularization</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo>,</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (according to <a href="S-matrix#VacuumFree">this def.</a>) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo>∈</mo><mi>LocObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle</annotation></semantics></math> a polynomial <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a>, regarded as an <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>.</p> <p>Let moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo stretchy="false">}</mo> <mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\{\Delta_{F,\Lambda}\}_{\Lambda \in [0,\infty)}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>); with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\Lambda</annotation></semantics></math> the induced <a class="existingWikiWord" href="/nlab/show/effective+S-matrix+schemes">effective S-matrix schemes</a> <a class="maruku-eqref" href="#eq:EffectiveSMatrixScheme">(1)</a>.</p> <p>Then</p> <ol> <li> <p>there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[0,\infty)</annotation></semantics></math>-parameterized <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>𝒵</mi> <mi>Λ</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>Λ</mi><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\{\mathcal{Z}_\Lambda\}_{\Lambda \in \mathbb{R}_{\geq 0}}</annotation></semantics></math> (<a href="Stückelberg-Petermann+renormalization+group#InteractionVertexRedefinition">this def.</a>) such that the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> of <a class="existingWikiWord" href="/nlab/show/effective+S-matrix+schemes">effective S-matrix schemes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_{\Lambda}</annotation></semantics></math> <a class="maruku-eqref" href="#eq:EffectiveSMatrixScheme">(1)</a> applied to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒵</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{Z}_\Lambda</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vertex+redefinition">redefined interactions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><mrow><mo>(</mo><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mi>Λ</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathcal{S}_\infty \;\coloneqq\; \underset{\Lambda \to \infty}{\lim} \left( \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda \right) </annotation></semantics></math></div> <p>exists and is a genuine <a class="existingWikiWord" href="/nlab/show/S-matrix+scheme">S-matrix scheme</a> around the given vacuum (<a href="S-matrix#LagrangianFieldTheoryPerturbativeScattering">this def.</a>);</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/S-matrix+scheme">S-matrix scheme</a> around the given vacuum arises this way.</p> </li> </ol> <p>These <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒵</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{Z}_\Lambda</annotation></semantics></math> are called <em><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></em> (remark <a class="maruku-ref" href="#TermCounter"></a> below) and the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda</annotation></semantics></math> is called a <em><a class="existingWikiWord" href="/nlab/show/UV+regularization">UV regularization</a></em> of the <a class="existingWikiWord" href="/nlab/show/effective+S-matrices">effective S-matrices</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\Lambda</annotation></semantics></math>.</p> <p>Hence <a class="existingWikiWord" href="/nlab/show/UV-regularization">UV-regularization</a> via <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a> is a method of <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> of <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> (<a href="S-matrix#ExtensionOfTimeOrderedProoductsRenormalization">this def.</a>).</p> </div> <p>This was claimed in (<a href="#BrunettiDuetschFredenhagen09">Brunetti-Dütsch-Fredenhagen 09, (75)</a>), a proof was indicated in (<a href="#DuetschFredenhagenKellerRejzner14">Dütsch-Fredenhagen-Keller-Rejzner 14, theorem A.1</a>).</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mi>k</mi></msub></mrow></msub><msub><mo stretchy="false">}</mo> <mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{p_{\rho_{k}}\}_{k \in \mathbb{N}}</annotation></semantics></math> be a sequence of projection maps as in <a class="maruku-eqref" href="#eq:ForExtensionOfDistributionsProjectionMaps">(?)</a> defining an <a class="existingWikiWord" href="/nlab/show/Epstein-Glaser+renormalization">Epstein-Glaser ("re"-)normalization</a> (prop. <a class="maruku-ref" href="#RenormalizationIsInductivelyExtensionToDiagonal"></a>) of <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>T</mi> <mi>k</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{T_k\}_{k \in \mathbb{N}}</annotation></semantics></math> as <a class="existingWikiWord" href="/nlab/show/extensions+of+distributions">extensions of distributions</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">T_k</annotation></semantics></math>, regarded as distributions via remark <a class="maruku-ref" href="#TimeOrderedProductOfFixedInteraction"></a>, by the choice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>q</mi> <mi>k</mi> <mi>α</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">q_k^\alpha = 0</annotation></semantics></math> in <a class="maruku-eqref" href="#eq:ExtensionOfDitstributionsPointFixedAndChoice">(?)</a>.</p> <p>We will construct that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒵</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{Z}_\Lambda</annotation></semantics></math> in terms of these projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">p_\rho</annotation></semantics></math>.</p> <p>First consider some convenient shorthand:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mn>1</mn><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow></munder><mfrac><mn>1</mn><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><msub><mi>Z</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{Z}_{\leq n} \coloneqq \underset{1 \in \{1, \cdots, n\}}{\sum} \frac{1}{n!} Z_n</annotation></semantics></math>. Moreover, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">(T_\Lambda \circ \mathcal{Z}_{\leq n})_k</annotation></semantics></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ary coefficient in the expansion of the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\Lambda \circ \mathcal{Z}_{\leq n}</annotation></semantics></math>, as in equation <a class="maruku-eqref" href="#eq:MainTheoremPerturbativeRenormalizationInductionStep">(?)</a> in the proof of the <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> (theorem <a class="maruku-ref" href="#PerturbativeRenormalizationMainTheorem"></a>).</p> <p>In this notation we need to find <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒵</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{Z}_\Lambda</annotation></semantics></math> such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> we have</p> <div class="maruku-equation" id="eq:CountertermsInductionAssumption"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><msub><mrow><mo>(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>)</mo></mrow> <mi>n</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>T</mi> <mi>n</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{\Lambda \to \infty}{\lim} \left( T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda} \right)_n \;=\; T_n \,. </annotation></semantics></math></div> <p>We proceed by <a class="existingWikiWord" href="/nlab/show/induction">induction</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>.</p> <p>Since by definition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub><mo>=</mo><msub><mi>const</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_0 = const_1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">T_1 = id</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mn>0</mn></msub><mo>=</mo><msub><mi>const</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">Z_0 = const_0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mn>1</mn></msub><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">Z_1 = id</annotation></semantics></math> the statement is trivially true for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math>.</p> <p>So assume now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Z</mi> <mi>k</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{Z_{k}\}_{k \leq n}</annotation></semantics></math> has been found such that <a class="maruku-eqref" href="#eq:CountertermsInductionAssumption">(2)</a> holds.</p> <p>Observe that with the chosen renormalizing projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub></mrow><annotation encoding="application/x-tex">p_{\rho_{n+1}}</annotation></semantics></math> the time-ordered product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">T_{n+1}</annotation></semantics></math> may be expressed as follows:</p> <div class="maruku-equation" id="eq:RenormalizedSMatrixAsLimitOfEffectiveSMatricesEvaluatedOnProjection"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>T</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></mrow><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>,</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><msub><mi>χ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><msub><mo>⋆</mo> <mi>F</mi></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo>⊗</mo><mi>⋯</mi><mo>⊗</mo><mi>O</mi><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></mrow><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>,</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><msub><mi>χ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo>⊗</mo><mi>⋯</mi><mo>⊗</mo><mi>O</mi><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} T_{n+1}(O, \cdots, O) & = \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} T_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F} \left( {\, \atop \,} T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_k}(O \otimes \cdots \otimes O) \right\rangle \\ & = \left\langle \underset{\Lambda \to \infty}{\lim} \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} T_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_k}(O \otimes \cdots \otimes O) \right\rangle \end{aligned} \,. </annotation></semantics></math></div> <p>Here in the first step we inserted the causal decomposition <a class="maruku-eqref" href="#eq:TimeOrderedProductsAwayFromDiagonalByInduction">(?)</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">T_{n+1}</annotation></semantics></math> in terms of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>T</mi> <mi>k</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{T_k\}_{k \leq n}</annotation></semantics></math> away from the diagonal, as in the proof of prop. <a class="maruku-ref" href="#RenormalizationIsInductivelyExtensionToDiagonal"></a>, which is admissible because the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub></mrow><annotation encoding="application/x-tex">p_{\rho_{n+1}}</annotation></semantics></math> vanishes on the diagonal. In the second step we replaced the star-product of the Feynman propagator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_F</annotation></semantics></math> with the limit over the star-products of the regularized propagators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\Lambda}</annotation></semantics></math>, which converges by the nature of the <a class="existingWikiWord" href="/nlab/show/H%C3%B6rmander+topology">Hörmander topology</a> (which is assumed by def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>).</p> <p>Hence it is sufficient to find <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Z_{n+1,\Lambda}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">K_{n+1,\Lambda}</annotation></semantics></math> such that</p> <div class="maruku-equation" id="eq:CountertermsAndCorrectionTerm"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mrow><mo>⟨</mo><msub><mrow><mo>(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mi>Λ</mi></msub><mo>)</mo></mrow> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></mrow><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>,</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><msub><mi>χ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mi>k</mi></msub></mrow></msub><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>)</mo></mrow><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\Lambda} \right)_{n+1} \,,\, (-, \cdots, -) \right\rangle & = \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} T_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_{k}}\left( -, \cdots, - \right) \right\rangle \\ & \phantom{=} + K_{n+1,\Lambda}(-, \cdots, -) \end{aligned} </annotation></semantics></math></div> <p>subject to these two conditions:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒵</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{Z}_{n+1,\Lambda}</annotation></semantics></math> is local;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\underset{\Lambda \to \infty}{\lim} K_{n+1,\Lambda} = 0</annotation></semantics></math>.</p> </li> </ol> <p>Now by expanding out the left hand side of <a class="maruku-eqref" href="#eq:CountertermsAndCorrectionTerm">(4)</a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mi>Λ</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>Z</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> (T_\Lambda \circ \mathcal{Z}_\Lambda)_{n+1} \;=\; Z_{n+1,\Lambda} \;+\; (T_\Lambda \circ Z_{\leq n, \Lambda})_{n+1} </annotation></semantics></math></div> <p>(which uses the condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">T_1 = id</annotation></semantics></math>) we find the unique solution of <a class="maruku-eqref" href="#eq:CountertermsAndCorrectionTerm">(4)</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Z_{n+1,\Lambda}</annotation></semantics></math>, in terms of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Z</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{Z_{\leq n,\Lambda}\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">K_{n+1,\Lambda}</annotation></semantics></math> (the latter still to be chosen) to be:</p> <div class="maruku-equation" id="eq:CountertermOrderByOrderInTermsOfCorrectionTerm"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mrow><mo>⟨</mo><msub><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></mrow><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>,</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><msub><mi>χ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mrow><mo>⟨</mo><msub><mrow><mo>(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>)</mo></mrow> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mrow><mo>⟨</mo><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left\langle Z_{n+1,\Lambda} , (-,\cdots, -) \right\rangle & = \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} T_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_{n+1}}(-, \cdots, -) \right\rangle \\ & \phantom{=} - \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\leq n,\Lambda} \right)_{n+1} \,,\, (-, \cdots, -) \right\rangle \\ & \phantom{=} + \left\langle K_{n+1, \Lambda}, (-, \cdots, -) \right\rangle \end{aligned} \,. </annotation></semantics></math></div> <p>We claim that the following choice works:</p> <div class="maruku-equation" id="eq:LocalityCorrection"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mrow><mo>⟨</mo><msub><mrow><mo>(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>)</mo></mrow> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mrow><mo>⟨</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></mrow><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>,</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><msub><mi>χ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} K_{n+1, \Lambda}(-, \cdots, -) & \coloneqq \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda} \right)_{n+1} \,,\, p_{\rho_{n+1}}(-, \cdots, -) \right\rangle \\ & \phantom{=} - \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} T_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_{n+1}}(-, \cdots, -) \right\rangle \end{aligned} \,. </annotation></semantics></math></div> <p>To prove this, we need to show that 1) the resulting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Z_{n+1,\Lambda}</annotation></semantics></math> is local and 2) the limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">K_{n+1,\Lambda}</annotation></semantics></math> vanishes as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">\Lambda \to \infty</annotation></semantics></math>.</p> <p>First regarding the locality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Z_{n+1,\Lambda}</annotation></semantics></math>: By inserting <a class="maruku-eqref" href="#eq:LocalityCorrection">(6)</a> into <a class="maruku-eqref" href="#eq:CountertermOrderByOrderInTermsOfCorrectionTerm">(5)</a> we obtain</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mrow><mo>⟨</mo><msub><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><msub><mrow><mo>(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>p</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow><mo>−</mo><mrow><mo>⟨</mo><msub><mrow><mo>(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><msub><mrow><mo>(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>−</mo><mi>id</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left\langle Z_{n+1,\Lambda} \,,\, (-,\cdots,-) \right\rangle & = \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\leq n} \right)_{n+1} \,,\, p(-, \cdots, -) \right\rangle - \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\leq n} \right)_{n+1} \,,\, (-, \cdots, -) \right\rangle \\ & = \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\leq n} \right)_{n+1} \,,\, ( p_{\rho_{n+1}} - id)(-, \cdots, -) \right\rangle \end{aligned} </annotation></semantics></math></div> <p>By definition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>−</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">p_{\rho_{n+1}} - id</annotation></semantics></math> is the identity on test functions (adiabatic switchings) that vanish at the diagonal. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Z_{n+1,\Lambda}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">supported</a> on the diagonal, and is hence local.</p> <p>Second we need to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\underset{\Lambda \to \infty}{\lim} K_{n+1,\Lambda} = 0</annotation></semantics></math>:</p> <p>By applying the analogous causal decomposition <a class="maruku-eqref" href="#eq:TimeOrderedProductsAwayFromDiagonalByInduction">(?)</a> to the regularized products, we find</p> <div class="maruku-equation" id="eq:InductionStepForCounterterms"><span class="maruku-eq-number">(7)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mrow><mo>⟨</mo><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></mrow><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>,</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><msub><mi>χ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \left\langle (T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda})_{n+1} \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \\ & = \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \,. \end{aligned} </annotation></semantics></math></div> <p>Using this we compute as follows:</p> <div class="maruku-equation" id="eq:CorrectionTermForCountertermsVanishesAsCutoffIsRemoved"><span class="maruku-eq-number">(8)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mrow><mo>⟨</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></mrow><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>,</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><msub><mi>χ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></mrow><mrow><mrow><mi>I</mi><mo>,</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><msub><mi>χ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><munder><munder><mrow><mo>(</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⏟</mo></munder><mrow><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo></mrow></munder><mrow><mo>(</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>)</mo></mrow><munder><munder><mrow><mo>(</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo>≤</mo><mi>n</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⏟</mo></munder><mrow><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow></munder><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></mrow><mrow><mrow><mi>I</mi><mo>,</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><msub><mi>χ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mi>T</mi> <mrow><mo stretchy="false">|</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>p</mi> <mrow><msub><mi>ρ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \left\langle \underset{\Lambda \to \infty}{\lim} (T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda})_{n+1} \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \\ & = \left\langle \underset{\Lambda \to \infty}{\lim} \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \\ & = \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { I, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \chi_i(\mathbf{X})\, \underset{ T_{{\vert \mathbf{I}\vert}}(\mathbf{I}) }{ \underbrace{ \left( \underset{\Lambda \to \infty}{\lim} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I}) \right) }} \left( \underset{\Lambda \to \infty}{\lim} \star_{F,\Lambda} \right) \underset{ T_{{\vert \overline{\mathbf{I}}\vert}}(\overline{\mathbf{I}}) }{ \underbrace{ \left( \underset{\Lambda \to \infty}{\lim} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) \right) }} \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \\ & = \left\langle \underset{\Lambda \to \infty}{\lim} \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { I, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \chi_i(\mathbf{X})\, T_{ { \vert \mathbf{I} \vert } }( \mathbf{I} ) \star_{F,\Lambda} T_{ {\vert \overline{\mathbf{I}} \vert} }( \overline{\mathbf{I}} ) \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \end{aligned} \,. </annotation></semantics></math></div> <p>Here in the first step we inserted <a class="maruku-eqref" href="#eq:InductionStepForCounterterms">(7)</a>; in the second step we used that in the <a class="existingWikiWord" href="/nlab/show/H%C3%B6rmander+topology">Hörmander topology</a> the <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> preserves limits in each variable and in the third step we used the induction assumption <a class="maruku-eqref" href="#eq:CountertermsInductionAssumption">(2)</a> and the definition of <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>).</p> <p>Inserting this for the first summand in <a class="maruku-eqref" href="#eq:LocalityCorrection">(6)</a> shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>Λ</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\underset{\Lambda \to \infty}{\lim} K_{n+1, \Lambda} = 0</annotation></semantics></math>.</p> <p>In conclusion this shows that a consistent choice of <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒵</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{Z}_\Lambda</annotation></semantics></math> exists to produce <em>some</em> S-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><mo stretchy="false">(</mo><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mi>Λ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{S} = \underset{\Lambda \to \infty }{\lim} (\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda)</annotation></semantics></math>.</p> <p id="CountertermsForArbitrarySMatrixFromAnyGivenOnes"> It just remains to see that for <em>every</em> other S-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝒮</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{\mathcal{S}}</annotation></semantics></math> there exist counterterms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>𝒵</mi><mo>˜</mo></mover> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">\widetilde{\mathcal{Z}}_\lambda</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝒮</mi><mo>˜</mo></mover><mo>=</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><mo stretchy="false">(</mo><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mover><mi>𝒵</mi><mo>˜</mo></mover> <mi>Λ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde{\mathcal{S}} = \underset{\Lambda \to \infty }{\lim} (\mathcal{S}_\Lambda \circ \widetilde{\mathcal{Z}}_\Lambda)</annotation></semantics></math>.</p> <p>But by the <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> (theorem <a class="maruku-ref" href="#PerturbativeRenormalizationMainTheorem"></a>) we know that there exists a <a class="existingWikiWord" href="/nlab/show/vertex+redefinition">vertex redefinition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒵</mi></mrow><annotation encoding="application/x-tex">\mathcal{Z}</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mover><mi>𝒮</mi><mo>˜</mo></mover></mtd> <mtd><mo>=</mo><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><mrow><mo>(</mo><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mi>Λ</mi></msub><mo>)</mo></mrow><mo>∘</mo><mi>𝒵</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><mo stretchy="false">(</mo><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo>∘</mo><mo stretchy="false">(</mo><munder><munder><mrow><msub><mi>𝒵</mi> <mi>Λ</mi></msub><mo>∘</mo><mi>𝒵</mi></mrow><mo>⏟</mo></munder><mrow><msub><mover><mi>𝒵</mi><mo>˜</mo></mover> <mi>Λ</mi></msub></mrow></munder><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \widetilde{\mathcal{S}} & = \mathcal{S} \circ \mathcal{Z} \\ & = \underset{\Lambda \to \infty}{\lim} \left( \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda \right) \circ \mathcal{Z} \\ & = \underset{\Lambda \to \infty}{\lim} ( \mathcal{S}_\Lambda \circ ( \underset{ \widetilde{\mathcal{Z}}_\Lambda }{ \underbrace{ \mathcal{Z}_\Lambda \circ \mathcal{Z} } } ) ) \end{aligned} </annotation></semantics></math></div> <p>and hence with counterterms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒵</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{Z}_\Lambda</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi></mrow><annotation encoding="application/x-tex">\mathcal{S}</annotation></semantics></math> given, then counterterms for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝒮</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{\mathcal{S}}</annotation></semantics></math> are given by the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>𝒵</mi><mo>˜</mo></mover> <mi>Λ</mi></msub><mo>≔</mo><msub><mi>𝒵</mi> <mi>Λ</mi></msub><mo>∘</mo><mi>𝒵</mi></mrow><annotation encoding="application/x-tex">\widetilde{\mathcal{Z}}_\Lambda \coloneqq \mathcal{Z}_\Lambda \circ \mathcal{Z}</annotation></semantics></math>.</p> </div> <div class="num_remark" id="TermCounter"> <h6 id="remark">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo>,</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (according to <a href="S-matrix#VacuumFree">this def.</a>) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}</annotation></semantics></math> be a choice of <a class="existingWikiWord" href="/nlab/show/UV+cutoffs">UV cutoffs</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> around this vacuum (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>).</p> <p>Consider</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>LocObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a>, regarded as an <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>.</p> <p>Then prop. <a class="maruku-ref" href="#UVRegularization"></a> says that there exist <a class="existingWikiWord" href="/nlab/show/vertex+redefinitions">vertex redefinitions</a> of this <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒵</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>LocObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> \mathcal{Z}_\Lambda(g S_{int} + j A) \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle </annotation></semantics></math></div> <p>parameterized by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda \in [0,\infty)</annotation></semantics></math>, such that the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mrow><mo>(</mo><msub><mi>𝒵</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathcal{S}_\infty(g S_{int} + j A) \;\coloneqq\; \underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda\left( \mathcal{Z}_\Lambda( g S_{int} + j A )\right) </annotation></semantics></math></div> <p>exists and is an <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> with the given <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">g S_{int} + j A</annotation></semantics></math>.</p> <p>In this case the difference</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>S</mi> <mrow><mi>counter</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mtd> <mtd><mo>≔</mo><mrow><mo>(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>−</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒵</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>Loc</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><msup><mi>g</mi> <mn>2</mn></msup><mo>,</mo><msup><mi>j</mi> <mn>2</mn></msup><mo>,</mo><mi>g</mi><mi>j</mi><mo stretchy="false">⟩</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} S_{counter, \Lambda} & \coloneqq \left( g S_{int} + j A \right) \;-\; \mathcal{Z}_{\Lambda}(g S_{int} + j A) \;\;\;\;\;\in\; Loc(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g^2, j^2, g j\rangle \end{aligned} </annotation></semantics></math></div> <p>(which by the axiom “perturbation” in <a href="Stückelberg-Petermann+renormalization+group#InteractionVertexRedefinition">this def.</a> is at least of second order in the <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a>/<a class="existingWikiWord" href="/nlab/show/source+field">source field</a>, as shown) is called a choice of <em><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></em> at cutoff scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math>. These are new interactions which are added to the given interaction at cutoff scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒵</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><msub><mi>S</mi> <mrow><mi>counter</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{Z}_{\Lambda}(g S_{int} + j A) \;=\; g S_{int} + j A \;+\; S_{counter,\Lambda} \,. </annotation></semantics></math></div> <p>In this language prop. <a class="maruku-ref" href="#UVRegularization"></a> says that for every free field vacuum and every choice of local interaction, there is a choice of counterterms to the interaction that defines a corresponding <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalized</a> <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a>, and every <a class="existingWikiWord" href="/nlab/show/renormalization">(re"-)normalized</a> <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> arises from some choice of counterterms.</p> </div> <h4 id="RelativeEffectiveAction">Effective quantum field theory</h4> <div class="num_prop" id="EffectiveSmatrixSchemeInvertible"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/effective+S-matrix+schemes">effective S-matrix schemes</a> are <a class="existingWikiWord" href="/nlab/show/inverse">invertible functions</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo>,</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (according to <a href="S-matrix#VacuumFree">this def.</a>) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}</annotation></semantics></math> be a choice of <a class="existingWikiWord" href="/nlab/show/UV+cutoffs">UV cutoffs</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> around this vacuum (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>).</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo><mo>↪</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle \hookrightarrow PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ] </annotation></semantics></math></div> <p>for the subspace of the space of <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">\hbar, g, j</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> on those which are at least of first order in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>,</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">g,j</annotation></semantics></math>, i.e. those that vanish for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g, j = 0</annotation></semantics></math> (as in <a href="S-matrix#FormalParameters">this def.</a>).</p> <p>Write moreover</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>+</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo><mo>↪</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> 1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle \hookrightarrow PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ] </annotation></semantics></math></div> <p>for the subspace of polynomial observables which are the sum of 1 (the multiplicative unit) with an observable at least linear n <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>,</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">g,j</annotation></semantics></math>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/effective+S-matrix+schemes">effective S-matrix schemes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\Lambda</annotation></semantics></math> (def. <a class="maruku-ref" href="#SMatrixEffective"></a>) <a class="existingWikiWord" href="/nlab/show/restriction">restrict</a> to <a class="existingWikiWord" href="/nlab/show/linear+isomorphisms">linear isomorphisms</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub></mrow></munderover><mn>1</mn><mo>+</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle \underoverset{\simeq}{\mathcal{S}_\Lambda}{\longrightarrow} 1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle \,. </annotation></semantics></math></div></div> <p>(<a href="#Duetsch10">Dütsch 10, (4.7)</a>)</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Since each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\Lambda}</annotation></semantics></math> is symmetric (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>) if follows by general properties of <a class="existingWikiWord" href="/nlab/show/star+products">star products</a> (<a href="star+product#SymmetricContribution">this prop.</a>) just as for the genuine <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> (<a href="Wick+algebra#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise">this prop.</a>) that eeach the “effective time-ordered product” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\star_{F,\Lambda}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the pointwise product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-)\cdot (-)</annotation></semantics></math> (<a href="A+first+idea+of+quantum+field+theory#Observable">this def.</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mi>A</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒯</mi> <mi>Λ</mi></msub><mrow><mo>(</mo><msubsup><mi>𝒯</mi> <mi>Λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><msubsup><mi>𝒯</mi> <mi>Λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> A_1 \star_{F,\Lambda} A_2 \;=\; \mathcal{T}_\Lambda \left( \mathcal{T}_\Lambda^{-1}(A_1) \cdot \mathcal{T}_\Lambda^{-1}(A_2) \right) </annotation></semantics></math></div> <p>for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒯</mi> <mi>Λ</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>exp</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mi>ℏ</mi><munder><mo>∫</mo><mi>Σ</mi></munder><msubsup><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mfrac><mrow><msup><mi>δ</mi> <mn>2</mn></msup></mrow><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathcal{T}_\Lambda \;\coloneqq\; \exp \left( \tfrac{1}{2}\hbar \underset{\Sigma}{\int} \Delta_{F,\Lambda}^{a b}(x,y) \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)} \right) </annotation></semantics></math></div> <p>(as in <a href="Wick+algebra#eq:OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism">this equation</a>).</p> <p>In particular this means that the <a class="existingWikiWord" href="/nlab/show/effective+S-matrix">effective S-matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\Lambda</annotation></semantics></math> arises from the <a class="existingWikiWord" href="/nlab/show/exponential+series">exponential series</a> for the pointwise product by <a class="existingWikiWord" href="/nlab/show/conjugation">conjugation</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒯</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{T}_\Lambda</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒯</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>exp</mi> <mo>⋅</mo></msub><mrow><mo>(</mo><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>∘</mo><msubsup><mi>𝒯</mi> <mi>Λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex"> \mathcal{S}_\Lambda \;=\; \mathcal{T}_\Lambda \circ \exp_\cdot\left( \frac{1}{i \hbar}(-) \right) \circ \mathcal{T}_\Lambda^{-1} </annotation></semantics></math></div> <p>(just as for the genuine S-matrix on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> in <a href="S-matrix#OnRegularObservablesPerturbativeSMatrix">this def.</a>).</p> <p>Now the exponential of the pointwise product on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>+</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">1 + PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle</annotation></semantics></math> has as <a class="existingWikiWord" href="/nlab/show/inverse+function">inverse function</a> the <a class="existingWikiWord" href="/nlab/show/natural+logarithm">natural logarithm</a> <a class="existingWikiWord" href="/nlab/show/power+series">power series</a>, and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> evidently preserves powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>,</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">g,j</annotation></semantics></math> this <a class="existingWikiWord" href="/nlab/show/conjugation">conjugates</a> to an inverse at each UV cutoff scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math>:</p> <div class="maruku-equation" id="eq:InverseOfEffectiveSMatrixByLogarithm"><span class="maruku-eq-number">(9)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒮</mi> <mi>Λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒯</mi> <mi>Λ</mi></msub><mo>∘</mo><mi>ln</mi><mrow><mo>(</mo><mi>i</mi><mi>ℏ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>∘</mo><msubsup><mi>𝒯</mi> <mi>Λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}_\Lambda^{-1} \;=\; \mathcal{T}_\Lambda \circ \ln\left( i \hbar (-) \right) \circ \mathcal{T}_\Lambda^{-1} \,. </annotation></semantics></math></div></div> <div class="num_defn" id="EffectiveActionRelative"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo>,</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (according to <a href="S-matrix#VacuumFree">this def.</a>) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}</annotation></semantics></math> be a choice of <a class="existingWikiWord" href="/nlab/show/UV+cutoffs">UV cutoffs</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> around this vacuum (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>).</p> <p>Consider</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>LocObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BrST</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BrST}})[ [ \hbar, g, j] ]\langle g, j\rangle </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a> regarded as an <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>.</p> <p>Then for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>Λ</mi> <mi>vac</mi></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Lambda,\, \Lambda_{vac} \;\in\; (0, \infty) </annotation></semantics></math></div> <p>two <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a>-scale parameters, we say the <em><a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi><mo>,</mo><msub><mi>Λ</mi> <mn>0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">S_{eff, \Lambda, \Lambda_0}</annotation></semantics></math> is the image of this interaction under the <a class="existingWikiWord" href="/nlab/show/composition">composite</a> of the <a class="existingWikiWord" href="/nlab/show/effective+S-matrix+scheme">effective S-matrix scheme</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mrow><msub><mi>Λ</mi> <mn>0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_{\Lambda_0}</annotation></semantics></math> at scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Λ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Lambda_0</annotation></semantics></math> <a class="maruku-eqref" href="#eq:EffectiveSMatrixScheme">(1)</a> and the <a class="existingWikiWord" href="/nlab/show/inverse+function">inverse function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒮</mi> <mi>Λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{S}_\Lambda^{-1}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/effective+S-matrix+scheme">effective S-matrix scheme</a> at scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> (via prop. <a class="maruku-ref" href="#EffectiveSmatrixSchemeInvertible"></a>):</p> <div class="maruku-equation" id="eq:RelativeEffectiveActionComposite"><span class="maruku-eq-number">(10)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi><mo>,</mo><msub><mi>Λ</mi> <mn>0</mn></msub></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msubsup><mi>𝒮</mi> <mi>Λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>𝒮</mi> <mrow><msub><mi>Λ</mi> <mn>0</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo><mphantom><mi>AAA</mi></mphantom><mi>Λ</mi><mo>,</mo><msub><mi>Λ</mi> <mn>0</mn></msub><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S_{eff,\Lambda, \Lambda_0} \;\coloneqq\; \mathcal{S}_{\Lambda}^{-1} \circ \mathcal{S}_{\Lambda_0}(g S_{int} + j A) \phantom{AAA} \Lambda, \Lambda_0 \in [0,\infty) \,. </annotation></semantics></math></div> <p>For chosen <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a> (remark <a class="maruku-ref" href="#TermCounter"></a>) hence for chosen <a class="existingWikiWord" href="/nlab/show/UV+regularization">UV regularization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\infty</annotation></semantics></math> (prop. <a class="maruku-ref" href="#UVRegularization"></a>) this makes sense also for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Λ</mi> <mn>0</mn></msub><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">\Lambda_0 = \infty</annotation></semantics></math> and we write:</p> <div class="maruku-equation" id="eq:RelativeEffectiveActionRelativeToInfinity"><span class="maruku-eq-number">(11)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi><mo>,</mo><mn>∞</mn></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msubsup><mi>𝒮</mi> <mi>Λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>𝒮</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo><mphantom><mi>AAA</mi></mphantom><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S_{eff,\Lambda} \;\coloneqq\; S_{eff,\Lambda, \infty} \;\coloneqq\; \mathcal{S}_{\Lambda}^{-1} \circ \mathcal{S}_{\infty}(g S_{int} + j A) \phantom{AAA} \Lambda \in [0,\infty) </annotation></semantics></math></div></div> <p>(<a href="#Duetsch10">Dütsch 10, (5.4)</a>)</p> <div class="num_remark" id="pQFTEffective"> <h6 id="remark_2">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo>,</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (according to <a href="S-matrix#VacuumFree">this def.</a>), let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}</annotation></semantics></math> be a choice of <a class="existingWikiWord" href="/nlab/show/UV+cutoffs">UV cutoffs</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> around this vacuum (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>), and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub><mo>=</mo><munder><mi>lim</mi><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow></munder><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo>∘</mo><msub><mi>𝒵</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\infty = \underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda</annotation></semantics></math> be a corresponding <a class="existingWikiWord" href="/nlab/show/UV+regularization">UV regularization</a> (prop. <a class="maruku-ref" href="#UVRegularization"></a>).</p> <p>Consider a <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>LocObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BrST</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BrST}})[ [ \hbar, g, j] ]\langle g, j\rangle </annotation></semantics></math></div> <p>regarded as an <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>.</p> <p>Then def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a> and def. <a class="maruku-ref" href="#EffectiveActionRelative"></a> say that for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda \in (0,\infty)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/effective+S-matrix">effective S-matrix</a> <a class="maruku-eqref" href="#eq:EffectiveSMatrixScheme">(1)</a> of the <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> <a class="maruku-eqref" href="#eq:RelativeEffectiveActionComposite">(10)</a> equals the genuine <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\infty</annotation></semantics></math> of the genuine <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">g S_{int} + j A</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒮</mi> <mn>∞</mn></msub><mrow><mo>(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}_\Lambda( S_{eff,\Lambda} ) \;=\; \mathcal{S}_\infty\left( g S_{int} + j A \right) \,. </annotation></semantics></math></div> <p>In other words the <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">S_{eff,\Lambda}</annotation></semantics></math> encodes what the actual <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub><mrow><mo>(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathcal{S}_\infty\left( g S_{int} + j A \right)</annotation></semantics></math> <em>effectively</em> looks like at <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math>.</p> <p>Therefore one says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">S_{eff,\Lambda}</annotation></semantics></math> defines <em><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></em> at <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math>.</p> <p>Notice that in general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">S_{eff,\Lambda}</annotation></semantics></math> is <em>not a <a class="existingWikiWord" href="/nlab/show/local+observable">local</a> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></em> anymore: By prop. <a class="maruku-ref" href="#EffectiveSmatrixSchemeInvertible"></a> the <a class="existingWikiWord" href="/nlab/show/image">image</a> of the <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒮</mi> <mi>Λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{S}^{-1}_\Lambda</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/effective+S-matrix">effective S-matrix</a> is <a class="existingWikiWord" href="/nlab/show/microcausal+polynomial+observables">microcausal polynomial observables</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>+</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle</annotation></semantics></math> and there is no guarantee that this lands in the subspace of <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a>.</p> <p>Therefore <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theories">effective quantum field theories</a> at finite <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a>-scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda \in [0,\infty)</annotation></semantics></math> are in general <em>not</em> <a class="existingWikiWord" href="/nlab/show/local+field+theories">local field theories</a>, even if their <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>→</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">\Lambda \to \infty</annotation></semantics></math> is, via prop. <a class="maruku-ref" href="#UVRegularization"></a>.</p> </div> <div class="num_prop" id="EffectiveActionAsRelativeEffectiveAction"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a> is <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Lambda = 0</annotation></semantics></math>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo>,</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (according to <a href="S-matrix#VacuumFree">this def.</a>) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}</annotation></semantics></math> be a choice of <a class="existingWikiWord" href="/nlab/show/UV+cutoffs">UV cutoffs</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> around this vacuum (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>).</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> (def. <a class="maruku-ref" href="#EffectiveActionRelative"></a>) at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Lambda = 0</annotation></semantics></math> is the actual <a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a> (<a href="S-matrix#InPerturbationTheoryActionEffective">this def.</a>) being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">i \hbar</annotation></semantics></math> times the <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> of <a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(g S_{int} + j A)</annotation></semantics></math> for <a class="existingWikiWord" href="/nlab/show/connected+graph">connected</a> <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mn>0</mn></mrow></msub></mtd> <mtd><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mn>0</mn><mo>,</mo><mn>∞</mn></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>S</mi> <mi>eff</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>Γ</mi><mo>∈</mo><msub><mi>Γ</mi> <mi>conn</mi></msub></mrow></munder><mi>Γ</mi><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} S_{eff,0} & \coloneqq\; S_{eff,0,\infty} \\ & = S_{eff} \;\coloneqq\; \underset{\Gamma \in \Gamma_{conn}}{\sum} \Gamma(g S_{int} + j A) \end{aligned} \,. </annotation></semantics></math></div></div> <p>(<a href="#Duetsch18">Dütsch 18, (3.473)</a>)</p> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Observe that the <a class="existingWikiWord" href="/nlab/show/effective+S-matrix+scheme">effective S-matrix scheme</a> at scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Lambda = 0</annotation></semantics></math> <a class="maruku-eqref" href="#eq:EffectiveSMatrixScheme">(1)</a> is the <a class="existingWikiWord" href="/nlab/show/exponential+series">exponential series</a> with respect to the pointwise product (<a href="A+first+idea+of+quantum+field+theory#Observable">this def.</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>exp</mi> <mo>⋅</mo></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}_0(O) = \exp_\cdot( O ) \,. </annotation></semantics></math></div> <p>Therefore the statement to be proven says equivalently that the <a class="existingWikiWord" href="/nlab/show/exponential+series">exponential series</a> of the <a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a> with respect to the pointwise product is the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>exp</mi> <mo>⋅</mo></msub><mrow><mo>(</mo><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac><msub><mi>S</mi> <mi>eff</mi></msub><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒮</mi> <mn>∞</mn></msub><mrow><mo>(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp_\cdot\left( \frac{1}{i \hbar} S_{eff} \right) \;=\; \mathcal{S}_\infty\left( g S_{int} + j A \right) \,. </annotation></semantics></math></div> <p>That this is the case is the statement of <a href="S-matrix#LogarithmEffectiveAction">this prop.</a>.</p> </div> <h4 id="renormalization_via_wilsonian_rg_flow">(“Re”-)Normalization via Wilsonian RG flow</h4> <p>The definition of the <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>≔</mo><msub><mi>𝒮</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi><mo>,</mo><mn>∞</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_{eff,\Lambda} \coloneqq \mathcal{S}_{eff,\Lambda, \infty}</annotation></semantics></math> in def. <a class="maruku-ref" href="#EffectiveActionRelative"></a> invokes a choice of <a class="existingWikiWord" href="/nlab/show/UV+regularization">UV regularization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\infty</annotation></semantics></math> (prop. <a class="maruku-ref" href="#UVRegularization"></a>). While (by that proposition and the <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> this is guaranteed to exist, in practice one is after methods for constructing this without specifying it a priori.</p> <p>But the collection <a class="existingWikiWord" href="/nlab/show/relative+effective+actions">relative effective actions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi><mo>,</mo><msub><mi>Λ</mi> <mn>0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_{eff,\Lambda, \Lambda_0}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Λ</mi> <mn>0</mn></msub><mo><</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">\Lambda_0 \lt \infty</annotation></semantics></math> “flows” with the cutoff-parameters <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> and in particular also with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Λ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Lambda_0</annotation></semantics></math> (remark <a class="maruku-ref" href="#GroupoidOfEFTs"></a> below) which suggests that examination of this flow yields information about full theory at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\infty</annotation></semantics></math>.</p> <p>This is made precise by <em><a class="existingWikiWord" href="/nlab/show/Polchinski%27s+flow+equation">Polchinski's flow equation</a></em> (prop. <a class="maruku-ref" href="#FlowEquationPolchinski"></a> below), which is the <a class="existingWikiWord" href="/nlab/show/infinitesimal">infinitesimal</a> version of the “<a class="existingWikiWord" href="/nlab/show/Wilsonian+RG+flow">Wilsonian RG flow</a>” (remark <a class="maruku-ref" href="#GroupoidOfEFTs"></a>). As a <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a> it is <em>independent</em> of the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_{\infty}</annotation></semantics></math> and hence may be used to solve for the <a class="existingWikiWord" href="/nlab/show/Wilsonian+RG+flow">Wilsonian RG flow</a> without knowing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\infty</annotation></semantics></math> in advance.</p> <p>The freedom in choosing the initial values of this differential equation corresponds to the <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization freedom</a> in choosing the <a class="existingWikiWord" href="/nlab/show/UV+regularization">UV regularization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\infty</annotation></semantics></math>. In this sense “<a class="existingWikiWord" href="/nlab/show/Wilsonian+RG+flow">Wilsonian RG flow</a>” is a method of <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> of <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> (<a href="S-matrix#ExtensionOfTimeOrderedProoductsRenormalization">this def.</a>).</p> <div class="num_remark" id="GroupoidOfEFTs"> <h6 id="remark_3">Remark</h6> <p><strong>(Wilsonian <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> of <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theories">effective quantum field theories</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo>,</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (according to <a href="S-matrix#VacuumFree">this def.</a>) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}</annotation></semantics></math> be a choice of <a class="existingWikiWord" href="/nlab/show/UV+cutoffs">UV cutoffs</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> around this vacuum (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>).</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/relative+effective+actions">relative effective actions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi><mo>,</mo><msub><mi>Λ</mi> <mn>0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_{eff,\Lambda, \Lambda_0}</annotation></semantics></math> (def. <a class="maruku-ref" href="#EffectiveActionRelative"></a>) satisfy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi><mo>′</mo><mo>,</mo><msub><mi>Λ</mi> <mn>0</mn></msub></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msubsup><mi>𝒮</mi> <mrow><mi>Λ</mi><mo>′</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo>)</mo></mrow><mrow><mo>(</mo><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi><mo>,</mo><msub><mi>Λ</mi> <mn>0</mn></msub></mrow></msub><mo>)</mo></mrow><mphantom><mi>AAA</mi></mphantom><mtext>for</mtext><mspace width="thinmathspace"></mspace><mi>Λ</mi><mo>,</mo><mi>Λ</mi><mo>′</mo><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>Λ</mi> <mn>0</mn></msub><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>⊔</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S_{eff, \Lambda', \Lambda_0} \;=\; \left( \mathcal{S}_{\Lambda'}^{-1} \circ \mathcal{S}_\Lambda \right) \left( S_{eff, \Lambda, \Lambda_0} \right) \phantom{AAA} \text{for} \, \Lambda,\Lambda' \in [0,\infty) \,,\, \Lambda_0 \in [0,\infty) \sqcup \{\infty\} \,. </annotation></semantics></math></div> <p>This is similar to a <a class="existingWikiWord" href="/nlab/show/group">group</a> of UV-cutoff scale-transformations. But since the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operations are only sensible when the UV-cutoff labels match, as shown, it is really a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <a class="existingWikiWord" href="/nlab/show/groupoid+action">action</a>.</p> <p>This is often called the <em>Wilsonian RG</em>, following (<a href="#Wilson71">Wilson 71</a>).</p> </div> <p>We now consider the <a class="existingWikiWord" href="/nlab/show/infinitesimal">infinitesimal</a> version of this “flow”:</p> <div class="num_prop" id="FlowEquationPolchinski"> <h6 id="proposition_4">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Polchinski%27s+flow+equation">Polchinski's flow equation</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo>,</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (according to <a href="S-matrix#VacuumFree">this def.</a>), let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>Λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}</annotation></semantics></math> be a choice of <a class="existingWikiWord" href="/nlab/show/UV+cutoffs">UV cutoffs</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> around this vacuum (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a>), such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>↦</mo><msub><mi>Λ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Lambda \mapsto \Lambda_{F,\Lambda}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable</a>.</p> <p>Then for <em>every</em> choice of <a class="existingWikiWord" href="/nlab/show/UV+regularization">UV regularization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_\infty</annotation></semantics></math> (prop. <a class="maruku-ref" href="#UVRegularization"></a>) the corresponding <a class="existingWikiWord" href="/nlab/show/relative+effective+actions">relative effective actions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">S_{eff,\Lambda}</annotation></semantics></math> (def. <a class="maruku-ref" href="#EffectiveActionRelative"></a>) satisfy the following <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi></mrow></mfrac><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi><mo>′</mo></mrow></mfrac><mrow><mo>(</mo><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi><mo>′</mo></mrow></msub><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>)</mo></mrow><msub><mo stretchy="false">|</mo> <mrow><mi>Λ</mi><mo>′</mo><mo>=</mo><mi>Λ</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \frac{d}{d \Lambda} S_{eff,\Lambda} \;=\; - \frac{1}{2} \frac{1}{i \hbar} \frac{d}{d \Lambda'} \left( S_{eff,\Lambda} \star_{F,\Lambda'} S_{eff,\Lambda} \right)\vert_{\Lambda' = \Lambda} \,, </annotation></semantics></math></div> <p>where on the right we have the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\Lambda'}</annotation></semantics></math> (<a href="star+product#PropagatorStarProduct">this def.</a>).</p> </div> <p>This goes back to (<a href="#Polchinski84">Polchinski 84, (27)</a>). The rigorous formulation and proof is due to (<a href="#BrunettiDuetschFredenhagen09">Brunetti-Dütsch-Fredenhagen 09, prop. 5.2</a>, <a href="#Duetsch10">Dütsch 10, theorem 2</a>).</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>First observe that for any <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo>∈</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">O \in PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi></mrow></mfrac><mo stretchy="false">(</mo><munder><munder><mrow><mi>O</mi><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mi>⋯</mi><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mi>O</mi></mrow><mo>⏟</mo></munder><mrow><mi>k</mi><mo>+</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mtext>factors</mtext></mrow></munder><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi></mrow></mfrac><mrow><mo>(</mo><mi>prod</mi><mo>∘</mo><mi>exp</mi><mrow><mo>(</mo><mi>ℏ</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>k</mi></mrow></munder><mrow><mo>⟨</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>,</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msub><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>i</mi></msub></mrow></mfrac><mfrac><mi>δ</mi><mrow><mi>δ</mi><msub><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>j</mi></msub></mrow></mfrac><mo>⟩</mo></mrow><mo>)</mo></mrow><mo stretchy="false">(</mo><munder><munder><mrow><mi>O</mi><mo>⊗</mo><mi>⋯</mi><mo>⊗</mo><mi>O</mi></mrow><mo>⏟</mo></munder><mrow><mi>k</mi><mo>+</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mtext>factors</mtext></mrow></munder><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><munder><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mn>1</mn><mrow><mi>k</mi><mo>!</mo></mrow></mfrac></mrow></munder><mrow><mo>(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi></mrow></mfrac><mi>O</mi><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mi>O</mi><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><munder><munder><mrow><mi>O</mi><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mi>⋯</mi><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mi>O</mi></mrow><mo>⏟</mo></munder><mrow><mi>k</mi><mspace width="thinmathspace"></mspace><mtext>factors</mtext></mrow></munder></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \frac{1}{(k+2)!} \frac{d}{d \Lambda} ( \underset{ k+2 \, \text{factors} }{ \underbrace{ O \star_{F,\Lambda} \cdots \star_{F,\Lambda} O } } ) \\ & = \frac{1}{(k+2)!} \frac{d}{d \Lambda} \left( prod \circ \exp\left( \hbar \underset{1 \leq i \lt j \leq k}{\sum} \left\langle \Delta_{F,\Lambda} , \frac{\delta}{\delta \mathbf{\Phi}_i} \frac{\delta}{\delta \mathbf{\Phi}_j} \right\rangle \right) ( \underset{ k + 2 \, \text{factors} }{ \underbrace{ O \otimes \cdots \otimes O } } ) \right) \\ & = \underset{ = \frac{1}{2} \frac{1}{k!} }{ \underbrace{ \frac{1}{(k+2)!} \left( k + 2 \atop 2 \right) }} \left( \frac{d}{d \Lambda} O \star_{F,\Lambda} O \right) \star_{F,\Lambda} \underset{ k \, \text{factors} }{ \underbrace{ O \star_{F,\Lambda} \cdots \star_{F,\Lambda} O } } \end{aligned} </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mi>δ</mi><mrow><mi>δ</mi><msub><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>i</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\delta}{\delta \mathbf{\Phi}_i}</annotation></semantics></math> denotes the functional derivative of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th tensor factor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math>, and the binomial coefficient counts the number of ways that an unordered pair of distinct labels of tensor factors may be chosen from a total of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">k+2</annotation></semantics></math> tensor factors, where we use that the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\star_{F,\Lambda}</annotation></semantics></math> is commutative (by symmetry of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\Lambda}</annotation></semantics></math>) and associative (by <a href="star+product#AssociativeAndUnitalStarProduct">this prop.</a>).</p> <p>With this and the defining equality <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>𝒮</mi><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}_\Lambda(S_{eff,\Lambda}) = \mathcal{S}(g S_{int} + j A)</annotation></semantics></math> <a class="maruku-eqref" href="#eq:RelativeEffectiveActionRelativeToInfinity">(11)</a> we compute as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mn>0</mn></mtd> <mtd><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi></mrow></mfrac><mi>𝒮</mi><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi></mrow></mfrac><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi></mrow></mfrac><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mrow><mo>(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi></mrow></mfrac><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo>)</mo></mrow><mrow><mo>(</mo><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi></mrow></mfrac><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mi>Λ</mi><mo>′</mo></mrow></mfrac><mrow><mo>(</mo><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi><mo>′</mo></mrow></msub><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>)</mo></mrow><msub><mo stretchy="false">|</mo> <mrow><mi>Λ</mi><mo>′</mo><mo>=</mo><mi>Λ</mi></mrow></msub><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mrow><mo>(</mo><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} 0 & = \frac{d}{d \Lambda} \mathcal{S}(g S_{int} + j A) \\ & = \frac{d}{d \Lambda} \mathcal{S}_\Lambda(S_{eff,\Lambda}) \\ & = \left( \frac{1}{i \hbar} \frac{d}{d \Lambda} S_{eff,\Lambda} \right) \star_{F,\Lambda} \mathcal{S}_\Lambda(S_{eff,\Lambda}) + \left( \frac{d}{d \Lambda} \mathcal{S}_{\Lambda} \right) \left( S_{eff, \Lambda} \right) \\ & = \left( \frac{1}{i \hbar} \frac{d}{d \Lambda} S_{eff,\Lambda} \right) \star_{F,\Lambda} \mathcal{S}_\Lambda(S_{eff,\Lambda}) \;+\; \frac{1}{2} \frac{d}{d \Lambda'} \left( \frac{1}{i \hbar} S_{eff,\Lambda} \star_{F,\Lambda'} \frac{1}{i \hbar} S_{eff, \Lambda} \right) \vert_{\Lambda' = \Lambda} \star_{F,\Lambda} \mathcal{S}_\Lambda \left( S_{eff, \Lambda} \right) \end{aligned} </annotation></semantics></math></div> <p>Acting on this equation with the multiplicative inverse <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mrow><mi>eff</mi><mo>,</mo><mi>Λ</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-) \star_{F,\Lambda} \mathcal{S}_\Lambda( - S_{eff,\Lambda} )</annotation></semantics></math> (using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\star_{F,\Lambda}</annotation></semantics></math> is a commutative product, so that exponentials behave as usual) this yields the claimed equation.</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="TraditionalInformalDiscussion">Traditional informal discussion</h3> <p>Traditional informal discussion of effective field theory proceeds from the following claim</p> <p><em>For a given set of asymptotic states, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> with the most general <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> containing all terms allowed by the assumed symmetries will yield the most general <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> elements consistent with analyticity, <span class="newWikiWord">perturbative unitarity<a href="/nlab/new/perturbative+unitarity">?</a></span>, <a class="existingWikiWord" href="/nlab/show/cluster+decomposition">cluster decomposition</a> and the assumed symmetries.</em></p> <p>This is due to (<a href="#Weinberg79">Weinberg 1979</a>) and (<a href="#Leutwyler94">Leutwyler94</a>); reviewed in <a href="#Pich">Pich, p. 6</a>.</p> <p>Based on this, one argues to obtains an effective approximation to a given more fundamental theory (which may or may not be actually known) by</p> <ol> <li> <p>choosing the (sub)set of fields to be considered;</p> </li> <li> <p>writing down a <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>eff</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><msub><mi>c</mi> <mi>i</mi></msub><msub><mi>O</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> L_{eff} = \sum_i c_i O_i </annotation></semantics></math></div> <p>that contains <em>all</em> the possible polynomial interaction terms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">O_i</annotation></semantics></math> of these fields scaled by their expected/known energy scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>O</mi> <mi>i</mi></msub><mo stretchy="false">]</mo><mo>=</mo><msub><mi>d</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">[O_i] = d_i</annotation></semantics></math>, up to a maximal energy scale</p> <p>(this will in general contain lots of direct interaction that in the fundamental theory are really compound interactions)</p> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>i</mi></msub><mo>∝</mo><mfrac><mn>1</mn><mrow><msup><mi>Λ</mi> <mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>−</mo><mi>dim</mi><mi>X</mi></mrow></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">c_i \propto \frac{1}{\Lambda^{d_i - dim X}}</annotation></semantics></math>;</p> </li> <li> <p>finally one fixes all the coupling constants of all these interactions by</p> <ul> <li> <p>either deriving them from a known fundamental theory by <em>integrating out</em> higher energy effects in that theory;</p> </li> <li> <p>or, otherwise, measuring them in the laboratory. The point being that due to the energy cutoff, this is guaranteed to be a finite number of parameters. After these have been determined, all remaining quantities given by the Lagrangian are then predictions of the effective theory.</p> </li> </ul> </li> </ol> <h2 id="examples">Examples</h2> <h3 id="lightbylight_scattering">Light-by-light scattering</h3> <p>(<a href="#Pich">Pich, section 2.1</a>)</p> <h3 id="rayleigh_scattering">Rayleigh scattering</h3> <p>(<a href="#Pich">Pich, section 2.2</a>)</p> <h3 id="FermiTheoryOfWeakInteractions">Fermi theory of weak interactions</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Fermi+theory+of+beta+decay">Fermi theory of beta decay</a></li> </ul> <p>(<a href="#Pich">Pich, section 2.3</a>)</p> <h3 id="for_nuclear_physics">For nuclear physics</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/effective+field+theories">effective field theories</a> of <a class="existingWikiWord" href="/nlab/show/nuclear+physics">nuclear physics</a>, hence for <a class="existingWikiWord" href="/nlab/show/confinement">confined</a>-<a class="existingWikiWord" href="/nlab/show/phase+of+matter">phase</a> <a class="existingWikiWord" href="/nlab/show/quantum+chromodynamics">quantum chromodynamics</a></strong>:</p> <ul> <li> <p>with effective <a class="existingWikiWord" href="/nlab/show/light+meson">ligh</a> <a class="existingWikiWord" href="/nlab/show/meson">meson</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+meson+dominance">vector meson dominance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hidden+local+symmetry">hidden local symmetry</a></p> </li> </ul> </li> <li> <p>with emergent (<a class="existingWikiWord" href="/nlab/show/soliton">solitonic</a>) <a class="existingWikiWord" href="/nlab/show/baryon">baryon</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quark+bag+model">quark bag model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Witten-Sakai-Sugimoto+model">Witten-Sakai-Sugimoto model</a></p> </li> </ul> </li> <li> <p>with explicit <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective</a> <a class="existingWikiWord" href="/nlab/show/baryon">baryon</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/baryon+chiral+perturbation+theory">baryon chiral perturbation theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Walecka+model">Walecka model</a></li> </ul> </li> </ul> </li> <li> <p>with explicit <a class="existingWikiWord" href="/nlab/show/quark">quark</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quark-meson+coupling+model">quark-meson coupling model</a></li> </ul> </li> </ul> </li> </ul> </div> <h3 id="in_lattice_qcd">In lattice QCD</h3> <p>In <a class="existingWikiWord" href="/nlab/show/lattice+QCD">lattice QCD</a>: see <em><a class="existingWikiWord" href="/nlab/show/Symanzik+effective+field+theory">Symanzik effective field theory</a></em></p> <h3 id="neutrino_masses_">Neutrino masses (?)</h3> <p>On <a class="existingWikiWord" href="/nlab/show/neutrino">neutrino</a> <a class="existingWikiWord" href="/nlab/show/masses">masses</a> and the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> as an effective field theory:</p> <blockquote> <p>I also noted at the same time that interactions between a pair of lepton doublets and a pair of scalar doublets can generate a neutrino mass, which is suppressed only by a factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">M^{-1}</annotation></semantics></math>, and that therefore with a reasonable estimate of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> could produce observable neutrino oscillations. The subsequent confirmation of neutrino oscillations lends support to the view of the Standard Model as an effective field theory, with M somewhere in the neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>10</mn> <mn>16</mn></msup><mi>GeV</mi></mrow><annotation encoding="application/x-tex">10^{16} GeV</annotation></semantics></math>. (<a href="#Weinberg09">Weinberg 09, p. 15</a>)</p> </blockquote> <h3 id="StringTheoryAndGravityCoupledToGaugeTheory">String theory and gravity coupled to gauge theory</h3> <p>The <a class="existingWikiWord" href="/nlab/show/string+scattering+amplitudes">string scattering amplitudes</a> for <a class="existingWikiWord" href="/nlab/show/superstrings">superstrings</a> are finite (fully proven so for low loop order and with various plausibility arguments for higher loop order, see at <em><a class="existingWikiWord" href="/nlab/show/string+scattering+amplitudes">string scattering amplitudes</a></em> for more), hence define a UV-complete <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>. The corresponding low energy effective field theories are theories of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> coupled to <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>. (<a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a>, <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a>).</p> <p>See also at <em><a href="string+theory+FAQ#WhatIsStringTheory">string theory FAQ – What is string theory?</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasiparticle">quasiparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+order+curvature+corrections">higher order curvature corrections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/threshold+correction">threshold correction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/swampland">swampland</a></p> </li> <li> <p><a href="string+theory+FAQ#RelationshipBetweenQuantumFieldTheoryAndStringTheory">string theory FAQ – What is the relationship between quantum field theory and string theory?</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The modern picture of effective low-energy QFT goes back to</p> <ul> <li> <p>L. P. Kadanoff, <em>Scaling laws for Ising models near <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">T_c</annotation></semantics></math></em> , Physica 2 (1966);</p> </li> <li id="Wilson71"> <p><a class="existingWikiWord" href="/nlab/show/Kenneth+Wilson">Kenneth Wilson</a>, <em>Renormalization group and critical phenomena 1. Renormalization group and the Kadanoff scaling picture</em> , , Physical review B 4(9) (1971) (<a href="https://doi.org/10.1103/PhysRevB.4.3174">doi:10.1103/PhysRevB.4.3174</a>)</p> </li> <li id="Wilson71b"> <p><a class="existingWikiWord" href="/nlab/show/Kenneth+Wilson">Kenneth Wilson</a>, <em>Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior</em>, Phys. Rev. B4 , 3184 (1971) (<a href="https://doi.org/10.1103/PhysRevB.4.3184">doi:10.1103/PhysRevB.4.3184</a>)</p> </li> <li id="Weinberg79"> <p><a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, <em>Phenomenological Lagrangians</em>, Physica A: Statistical Mechanics and its Applications Volume 96, Issues 1–2, April 1979, Pages 327-340 (<a href="https://doi.org/10.1016/0378-4371(79)90223-1">doi:10.1016/0378-4371(79)90223-1</a>)</p> </li> <li id="Polchinski84"> <p><a class="existingWikiWord" href="/nlab/show/Joseph+Polchinski">Joseph Polchinski</a>, <em>Renormalization and effective Lagrangians</em> , Nuclear Phys. B B231, 1984 (<a href="http://max2.physics.sunysb.edu/~rastelli/2016/Polchinski.pdf">pdf</a>)</p> </li> <li id="Leutwyler94"> <p>H. Leutwyler, Ann. Phys., NY 235 (1994) 165.</p> </li> </ul> <p>Early history:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, <em>On the Development of Effective Field Theory</em> (<a href="https://arxiv.org/abs/2101.04241">arXiv:2101.04241</a>)</li> </ul> <p>Review:</p> <ul> <li id="Weinberg09"> <p><a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, <em>Effective Field Theory, Past and Future</em> (<a href="http://arxiv.org/abs/0908.1964">arXiv:0908.1964</a>)</p> </li> <li id="Pich"> <p>A. Pich, <em>Effective Field Theory</em> (<a href="http://arxiv.org/abs/hep-ph/9806303">arXiv:hep-ph/9806303</a>)</p> </li> <li id="Abdesselam13"> <p><a class="existingWikiWord" href="/nlab/show/Abdelmalek+Abdesselam">Abdelmalek Abdesselam</a>, <em>QFT, RG, and all that, for mathematicians, in eleven pages</em> (<a href="https://arxiv.org/abs/1311.4897">arXiv:1311.4897</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>Lecture 5 of <a class="existingWikiWord" href="/nlab/show/Five+lectures+on+supersymmetry">Five lectures on supersymmetry</a></em></p> </li> <li> <p>Andrey Grozin, <em>Effective field theories</em> (<a href="https://arxiv.org/abs/2001.00434">arXiv:2001.00434</a>)</p> </li> <li> <p>Riccardo Penco, <em>An Introduction to Effective Field Theories</em> (<a href="https://arxiv.org/abs/2006.16285">arXiv:2006.16285</a>)</p> </li> <li> <p>C. P. Burgess, <em>Introduction to effective field theory</em>, Cambridge University Press 2020 (<a href="https://arxiv.org/abs/hep-th/0701053">arXiv:hep-th/0701053</a>, <a href="https://www.cambridge.org/core/books/introduction-to-effective-field-theory/A9CDB35F4AA7921E3A9CFD573EBA8B64">ISBN:9781139048040</a>)</p> </li> </ul> <p>A classical textbook adopting the EFT perspective is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, <em>The Quantum Theory of Fields</em> (Cambridge University <p>Press,Cambridge,1995).</p> </li> </ul> <p>whose author describes his goal as:</p> <blockquote> <p>This is intended to be a book on quantum field theory for the era of effective field theory.</p> </blockquote> <p>Another book which takes the effective-field-theory approach to QFT is</p> <ul> <li>Anthony Zee, <em>Quantum Field Theory in a Nutshell</em> (Princeton University Press, second edition, 2010).</li> </ul> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/nuclear+physics">nuclear physics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <em>Lectures on Effective Field Theories for Nuclei, Nuclear Matter and Dense Matter</em>, 2002 (<a href="https://arxiv.org/abs/nucl-th/0202078">arXiv:nucl-th/0202078</a>, <a href="https://cds.cern.ch/record/539674">cds:539674</a>)</li> </ul> <p>Discussion with an eye towards <a class="existingWikiWord" href="/nlab/show/condensed+matter+physics">condensed matter physics</a> is in</p> <ul> <li id="Shankar99"><a class="existingWikiWord" href="/nlab/show/Ramamurti+Shankar">Ramamurti Shankar</a>, <em>Effective Field Theory in Condensed Matter Physics</em> in <em>Conceptual Foundations of Quantum Field Theory</em>, 1999 (<a href="http://arxiv.org/abs/cond-mat/9703210">arXiv:cond-mat/9703210</a>)</li> </ul> <p>and with an eye towards <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> and the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</p> <ul> <li id="PetrovBlechman16"><a class="existingWikiWord" href="/nlab/show/Alexey+Petrov">Alexey Petrov</a>, Andrew E Blechman, <em>Effective Field Theories</em>, World Scientific 2016 (<a href="https://doi.org/10.1142/8619">doi:10.1142/8619</a>)</li> </ul> <p>The point that perturbatively <a class="existingWikiWord" href="/nlab/show/renormalization">non-renormalizable</a> theories may be regarded as effective field theories at each energy scale was highligted in</p> <ul> <li id="GomisWeinberg95">J. Gomis and <a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, <em>Are nonrenormalizable gauge theories renormalizable?</em>, Nucl. Phys. B 469 (1996) 473 (<a href="https://arxiv.org/abs/hep-th/9510087">arXiv:hep-th/9510087</a>)</li> </ul> <p>Notably the theory of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> based on the standard <a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a> may be regarded as just an effective QFT, which makes some of its notorious problems be non-problems:</p> <ul> <li id="DonoghueIntroduction"> <p><a class="existingWikiWord" href="/nlab/show/John+Donoghue">John Donoghue</a>, <em>Introduction to the Effective Field Theory Description of Gravity</em> (<a href="http://arxiv.org/abs/gr-qc/9512024">arXiv:gr-qc/9512024</a>)</p> </li> <li id="AtanceCortes96"> <p>Mario Atance, Jose Luis Cortes, <em>Effective Field Theory of pure Gravity and the Renormalization Group</em> (<a href="http://arxiv.org/abs/hep-th/9604076">arXiv:hep-th/9604076</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+F.+Donoghue">John F. Donoghue</a>, <em>Quantum General Relativity and Effective Field Theory</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a></em>, Springer (2023) [<a href="https://arxiv.org/abs/2211.09902">arXiv:2211.09902</a>]</p> </li> </ul> <p>and in the context of <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> in <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>:</p> <ul> <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/Katarzyna+Rejzner">Katarzyna Rejzner</a>, <em>Quantum gravity from the point of view of locally covariant quantum field theory</em> (<a href="http://arxiv.org/abs/1306.1058">arXiv:1306.1058</a>)</li> </ul> <p>Comments on this point are also in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, blog posts</p> <ul> <li> <p><em><a href="http://golem.ph.utexas.edu/~distler/blog/archives/000639.html">Motivation</a></em></p> </li> <li> <p><em><a href="http://golem.ph.utexas.edu/~distler/blog/archives/001255.html">Effective field theory and gravity</a></em></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em><a class="existingWikiWord" href="/nlab/show/Renormalization+and+Effective+Field+Theory">Renormalization and Effective Field Theory</a></em></p> </li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alastair+Hamilton">Alastair Hamilton</a>, <em>On two constructions of an effective field theory</em> (<a href="http://arxiv.org/abs/1502.05790">arXiv:1502.05790</a>)</li> </ul> <h3 id="ReferencesInCausalPerturbationTheory">In causal perturbation theory</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative</a> effective QFT in the rigorous context of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> and its relation to the <a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a> is due to</p> <ul> <li id="BrunettiDuetschFredenhagen09"> <p><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>, section 5.2 of <em>Perturbative Algebraic Quantum Field Theory and the Renormalization Groups</em>, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (<a href="https://arxiv.org/abs/0901.2038">arXiv:0901.2038</a>)</p> </li> <li id="Duetsch10"> <p><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <em>Connection between the renormalization groups of Stückelberg-Petermann and Wilson</em>, Confluentes Mathematici, Vol. 4, No. 1 (2012) 12400014 (<a href="https://arxiv.org/abs/1012.5604">arXiv:1012.5604</a>)</p> </li> <li id="DuetschFredenhagenKellerRejzner14"> <p><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>, appendix A of <em>Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization</em>, J. Math. Phy. 55(12), 122303 (2014) (<a href="https://arxiv.org/abs/1311.5424">arXiv:1311.5424</a>)</p> </li> </ul> <p>reviewed in</p> <ul> <li id="Duetsch18"><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, section 3.8 of <em><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></em>, 2018</li> </ul> <p>See also</p> <ul> <li id="KellerKopperSchophaus97">Georg Keller, Christoph Kopper, Clemens Schophaus, <em>Perturbative Renormalization with Flow Equations in Minkowski Space</em>, Helv.Phys.Acta 70 (1997) 247-274 (<a href="https://arxiv.org/abs/hep-th/9605137">arXiv.hep-th/9605137</a>)</li> </ul> <h3 id="for_string_theory">For string theory</h3> <p>Discussion of the effective field theories induced by <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> includes the following:</p> <p>Via <a class="existingWikiWord" href="/nlab/show/string+scattering+amplitudes">string scattering amplitudes</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ralph+Blumenhagen">Ralph Blumenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Dieter+L%C3%BCst">Dieter Lüst</a>, <a class="existingWikiWord" href="/nlab/show/Stefan+Theisen">Stefan Theisen</a>, <em>String Scattering Amplitudes and Low Energy Effective Field Theory</em>, chapter 16 in <em>Basic Concepts of String Theory</em> Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013 (<a href="https://link.springer.com/content/pdf/bfm%3A978-3-642-29497-6%2F1.pdf">TOC pdf</a>, <a href="http://www.springer.com/gp/book/9783642294969">publisher page</a>)</li> </ul> <p>Via <a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a>:</p> <ul> <li> <p>R. Brustein, S.P.De Alwis, <em>Renormalization group equation and non-perturbative effects in string-field theory</em>, Nuclear Physics B Volume 352, Issue 2, 25 March 1991, Pages 451-468 (<a href="https://doi.org/10.1016/0550-3213(91)90451-3">doi:10.1016/0550-3213(91)90451-3</a>)</p> </li> <li> <p>Brustein and K. Roland, “Space-time versus world sheet renormalization group equation in string theory,” Nucl. Phys. B372, 201 (1992) (<a href="https://doi.org/10.1016/0550-3213(92)90317-5">doi:10.1016/0550-3213(92)90317-5</a>)</p> </li> <li id="Sen16"> <p><a class="existingWikiWord" href="/nlab/show/Ashoke+Sen">Ashoke Sen</a>, <em>Wilsonian Effective Action of Superstring Theory</em>, J. High Energ. Phys. (2017) 2017: 108 (<a href="https://arxiv.org/abs/1609.00459">arXiv:1609.00459</a>)</p> </li> </ul> <p>Discussion of possible criteria for which effective field theory do <em>not</em> arise as effective field theories of a string theory:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eran+Palti">Eran Palti</a>, <em>The Swampland: Introduction and Review</em>, lecture notes (<a href="https://arxiv.org/abs/1903.06239">arXiv:1903.06239</a>)</li> </ul> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on June 16, 2023 at 06:14:57. 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