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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> renormalization </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/4710/#Item_10" 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='#EpsteinGlaserRenormalization'>Epstein-Glaser normalization</a></li> <li><a href='#SPRenormalizationGroup'>Stückelberg-Petermann renormalization group</a></li> <li><a href='#UVRegularizationViaZ'>UV-Regularization via Conterterms</a></li> <li><a href='#EffectiveQFTFlowWislonian'>Wilson-Polchinski effective QFT flow</a></li> <li><a href='#RGFlowGeneral'>Renormalization group flow</a></li> <li><a href='#ScalingTransformatinRGFlow'>Gell-Mann & Low RG flow</a></li> <li><a href='#dimensional_regularization'>Dimensional regularization</a></li> <li><a href='#conneskreimer_renormalization'>Connes-Kreimer renormalization</a></li> </ul> <li><a href='#BPHZRenormalization'>BPHZ and Hopf-algebraic renormalization</a></li> <li><a href='#lattice_renormalization'>Lattice renormalization</a></li> <li><a href='#OfTheoriesInBVForm'>Of theories in BV-CS form</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#vacuum_energy_and_cosmological_constant'>Vacuum energy and Cosmological constant</a></li> <li><a href='#chernsimons_level'>Chern-Simons level</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#General'>Review</a></li> <li><a href='#in_causal_perturbation_theory_2'>In causal perturbation theory</a></li> <li><a href='#bphz_renormalization'>BPHZ Renormalization</a></li> <li><a href='#ReferencesInBVFormalism'>In BV formalism</a></li> <li><a href='#ReferencesOnCompactifiedConfigurationSpaces'>On compactified configuration spaces</a></li> <li><a href='#operadic_description'>Operadic description</a></li> <li><a href='#relations_to_motives_polylogarithms_positivity'>Relations to motives, polylogarithms, positivity</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>The construction of a <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> from a given <a class="existingWikiWord" href="/nlab/show/local+Lagrangian+density">local Lagrangian density</a> (rigorously via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>) involves ambiguities associated with the detailed nature of the quantum processes at point interactions. What is called <em>renormalization</em> is making a choice of fixing these ambiguities to produce a <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> (<a href="#Wightman76">Wightman et al. 76</a>).</p> <p>In <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> the renormalization ambiguities are understood as the freedom of extending the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> of <a class="existingWikiWord" href="/nlab/show/operator-valued+distributions">operator-valued distributions</a> to the interaction points, in the sense of <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a>. The notorious “infinities that plague quantum field theory” arise only if this extension is not handled correctly and has to be fixed. Hence what historically is called “renormalization” could from a mathematical point of view just be called “normalization” (a point made vividly for instance in <a class="existingWikiWord" href="/nlab/show/Finite+Quantum+Electrodynamics+--+The+Causal+Approach">Scharf 95</a>, <a class="existingWikiWord" href="/nlab/show/Quantum+Gauge+Theories+--+A+True+Ghost+Story">Scharf 01</a>).</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization+theory">main theorem of perturbative renormalization theory</a></em> (theorem <a class="maruku-ref" href="#PerturbativeRenormalizationMainTheorem"></a> below) states that any two choices of such (re-)normalizations are uniquely related by a re-definition of the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> which introduces further point interactions of higher order (“counter terms”),</p> <p>The <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> of the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> may naturally be organized via <a class="existingWikiWord" href="/nlab/show/graphs">graphs</a>, the <em><a class="existingWikiWord" href="/nlab/show/Feynman+graphs">Feynman graphs</a></em> (<a href="#GarciaBondiaLazzarini00">Garcia-Bondia & Lazzarini 00</a>, <a href="#Keller10">Keller 10, chapter IV</a>), and hence the renormalized perturbative <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> defining the <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> is expressed as a <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> in renormalized <a class="existingWikiWord" href="/nlab/show/Feynman+graphs">Feynman graphs</a>, the <em><a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a></em> (<a href="#Keller10">Keller 10 (IV.12)</a>).</p> <p>Historically the <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> was motivated from intuition about the would-be <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>, and this is still a popular point of view, despite its lack of rigorous formulation. One may understand the axiomatics on the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> as defining the result of the path integral without actually doing an integration over field configurations.</p> <p>But while <a class="existingWikiWord" href="/nlab/show/path+integral+quantization">path integral quantization</a> for <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> remains elusive, it has been shown that the (re-)normalized <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> thus constructed via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> is, at least under favorable circumstances, equivalently the (<a class="existingWikiWord" href="/nlab/show/Fedosov+deformation+quantization">Fedosov</a>) <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a> of the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> induced by the given interacting <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> (<a href="#Collini16">Collini 16</a>). This identifies the (re-)normalization freedom with the usual freedom in choosing <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a>.</p> <p>This also suggest that the construction of the full <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative quantum field theory</a> ought to be given by a <a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantization</a> of the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>. But presently no example of such for non-trivial interaction in <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\geq 4</annotation></semantics></math> is known. In particular the <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenologically</a> interesting case of a complete construction of interacting field theories on 4-dimensional spacetimes is presently unknown. For the case of <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> this open problem is one of the “Millennium Problems” (see at <em><a class="existingWikiWord" href="/nlab/show/quantization+of+Yang-Mills+theory">quantization of Yang-Mills theory</a></em>).</p> <p>The following is taken from <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+A+first+idea+of+quantum+field+theory">geometry of physics – A first idea of quantum field theory</a></em>. See there for more backround.</p> <h2 id="details">Details</h2> <p>There are different formulations of renormalization:</p> <ol> <li> <p><em><a href="#InCausalPerturbationTheory">In causal perturbation theory</a></em></p> </li> <li> <p><em><a href="#BPHZRenormalization">BPHZ and Hopf-Algebraic renormalization</a></em></p> </li> <li> <p><em><a href="#OfTheoriesInBVForm">Of theories in BV-CS form</a></em></p> </li> </ol> <h3 id="InCausalPerturbationTheory">In causal perturbation theory</h3> <p>We discuss (re-)normalization of <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> in the rigorous formulation 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> <p>In this rigorous discussion no “infinite divergent quantities” (as in the original informal discussion due to <a class="existingWikiWord" href="/nlab/show/Schwinger-Tomonaga-Feynman-Dyson">Schwinger-Tomonaga-Feynman-Dyson</a>) that need to be “re-normalized” to finite well-defined quantities are ever considered, instead finite well-defined quantities are considered right away, and the available space of choices is determined. Therefore making such choices is really a <em>normalization</em> of the <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>/<a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> (as prominently highlighted in <a href="causal+perturbation+theoryscatt#Scharf95">Scharf 95, see title, introduction, and section 4.3</a>). Actual re-normalization is the the change of such normalizations:</p> <ul> <li> <p>Normalization</p> <ul> <li> <p><em><a href="#EpsteinGlaserRenormalization">Epstein-Glaser renormalization</a></em>.</p> </li> <li> <p><em><a href="#UVRegularizationViaZ">UV-Regularization via Counterterms</a></em>.</p> </li> <li> <p><em><a href="#EffectiveQFTFlowWislonian">Wilson-Polchinski effective QFT flow</a></em>.</p> </li> </ul> </li> <li> <p>Re-Normalization</p> <ul> <li> <p><em><a href="#SPRenormalizationGroup">Stückelberg-Petermann re-normalization</a></em></p> </li> <li> <p><em><a href="#RGFlowGeneral">Renormalization group flow</a></em></p> </li> <li> <p><em><a href="#ScalingTransformatinRGFlow">Gell-Mann & Low RG Flow</a></em></p> </li> </ul> </li> </ul> <p>The construction of <a class="existingWikiWord" href="/nlab/show/perturbative+QFTs">perturbative QFTs</a> may be explicitly described by an <a class="existingWikiWord" href="/nlab/show/induction">inductive</a> <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> of <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>/<a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> to coinciding interaction points. This type of construction is called</p> <ul> <li><em><a href="#EpsteinGlaserRenormalization">Epstein-Glaser renormalization</a></em>.</li> </ul> <p>This inductive construction has the advantage that it gives accurate control over the space of available choices of renormalizations (theorem <a class="maruku-ref" href="#ExistenceRenormalization"></a> below) but it leaves the nature of the “new interactions” that are to be chosen at coinciding interaction points somwewhat implicit.</p> <p>Alternatively, one may <a class="existingWikiWord" href="/nlab/show/vertex+redefinition">re-define the interactions</a> explicitly (by adding “<a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a>”), depending on a chosen <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a>-scale, and construct the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> as the “cutoff is removed”. This is called (“re”-)normalization by</p> <ul> <li><em><a href="#UVRegularizationViaZ">UV-Regularization via Counterterms</a></em>.</li> </ul> <p>This still leaves open the question how to choose the <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a>. For this it serves to understand the <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> induced by the choice of <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> at any given cutoff scale. The <a class="existingWikiWord" href="/nlab/show/infinitesimal">infinitesimal</a> change of these <a class="existingWikiWord" href="/nlab/show/relative+effective+actions">relative effective actions</a> follows a universal <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski%27s+flow+equation">Polchinski's flow equation</a> (prop. <a class="maruku-ref" href="#FlowEquationPolchinski"></a> below). This makes the problem of (“re”-)normalization be that of solving this <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a> subject to chosen initial data. This is the perspective on (“re”-)normalization called</p> <ul> <li><em><a href="#EffectiveQFTFlowWislonian">Wilson-Polchinski effective QFT flow</a></em>.</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> states that different <a class="existingWikiWord" href="/nlab/show/S-matrix+schemes">S-matrix schemes</a> are precisely related by <a class="existingWikiWord" href="/nlab/show/vertex+redefinitions">vertex redefinitions</a>. This yields the</p> <ul> <li><em><a href="#SPRenormalizationGroup">Stückelberg-Petermann renormalization group</a></em>.</li> </ul> <p>If a sub-collection of <a class="existingWikiWord" href="/nlab/show/renormalization+schemes">renormalization schemes</a> is parameterized by some <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi></mrow><annotation encoding="application/x-tex">RG</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem</a> implies <a class="existingWikiWord" href="/nlab/show/vertex+redefinitions">vertex redefinitions</a> depending on pairs of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi></mrow><annotation encoding="application/x-tex">RG</annotation></semantics></math> this is known as</p> <ul> <li><em><a href="#RGFlowGeneral">Renormalization group flow</a></em></li> </ul> <p>Specifically <a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> yields such a collection of <a class="existingWikiWord" href="/nlab/show/renormalization+schemes">renormalization schemes</a>; the corresponding <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a> is known as</p> <ul> <li><em><a href="#ScalingTransformatinRGFlow">Gell-Mann & Low RG flow</a></em>.</li> </ul> <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> <h4 id="EpsteinGlaserRenormalization">Epstein-Glaser normalization</h4> <p>The construction of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theories">perturbative quantum field theories</a> around a given <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 field</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> is equivalently, by <a href="S-matrix#InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization">this prop.</a>, the construction of <a class="existingWikiWord" href="/nlab/show/S-matrices">S-matrices</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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}(g S_{int} + j A)</annotation></semantics></math> in the sense of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> (<a href="S-matrix#LagrangianFieldTheoryPerturbativeScattering">this def.</a>) for the given <a class="existingWikiWord" href="/nlab/show/local+observable">local</a> <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>. By prop. <a class="maruku-ref" href="#RenormalizationIsInductivelyExtensionToDiagonal"></a>, the construction of these <a class="existingWikiWord" href="/nlab/show/S-matrices">S-matrices</a> is <a class="existingWikiWord" href="/nlab/show/induction">inductively</a> in <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> a choice of <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> (remark <a class="maruku-ref" href="#TimeOrderedProductOfFixedInteraction"></a> and def. <a class="maruku-ref" href="#ExtensionOfDistributions"></a> below) of the corresponding <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 <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> of the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> to the locus of coinciding interaction points (the <a class="existingWikiWord" href="/nlab/show/fat+diagonal">fat diagonal</a>). An inductive construction of the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> this way is called <em><a class="existingWikiWord" href="/nlab/show/Epstein-Glaser+renormalization">Epstein-Glaser-("re"-)normalization</a></em> (<a href="S-matrix#ExtensionOfTimeOrderedProoductsRenormalization">this def.</a>).</p> <p>By paying attention to the <a class="existingWikiWord" href="/nlab/show/scaling+degree+of+distributions">scaling degree</a> (def. <a class="maruku-ref" href="#ScalingDegree"></a> below) one may precisely characterize the space of choices in the <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> (prop. <a class="maruku-ref" href="#SpaceOfPointExtensions"></a> below): For a given <a class="existingWikiWord" href="/nlab/show/local+observable">local</a> <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> it is inductively in <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> a <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite-dimensional</a> <a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>. This conclusion is theorem <a class="maruku-ref" href="#ExistenceRenormalization"></a> below.</p> <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> <div class="num_prop" id="RenormalizationIsInductivelyExtensionToDiagonal"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> is <a class="existingWikiWord" href="/nlab/show/induction">inductive</a> <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension</a> of <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> to <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</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>.</p> <p>Assume that 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>, <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 \leq n}</annotation></semantics></math> of arity <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 \leq n</annotation></semantics></math> have been constructed in the sense of <a href="S-matrix#TimeOrderedProduct">this def.</a>. Then 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> of arity <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> is uniquely fixed on the <a class="existingWikiWord" href="/nlab/show/complement">complement</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∖</mo><mi>diag</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>∈</mo><mi>Σ</mi><msubsup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><munder><mo>∃</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>≠</mo><msub><mi>x</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \Sigma^{n+1} \setminus diag(n) \;=\; \left\{ (x_i \in \Sigma)_{i = 1}^n \;\vert\; \underset{i,j}{\exists} (x_i \neq x_j) \right\} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/image">image</a> of the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mover><mo>⟶</mo><mi>diag</mi></mover><msup><mi>Σ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Sigma \overset{diag}{\longrightarrow} \Sigma^{n}</annotation></semantics></math> (where we regarded <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> as a <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Sigma^{n+1}</annotation></semantics></math> according to <a href="S-matrix#NotationForTimeOrderedProductsAsGeneralizedFunctions">this remark</a>).</p> </div> <p>This statement appears in (<a href="#PopineauStora82">Popineau-Stora 82</a>), with (unpublished) details in (<a href="#Stora93">Stora 93</a>), following personal communication by <a class="existingWikiWord" href="/nlab/show/Henri+Epstein">Henri Epstein</a> (according to <a href="#Duetsch18">Dütsch 18, footnote 57</a>). Following this, statement and detailed proof appeared in (<a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00</a>).</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>We will construct an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∖</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma^{n+1} \setminus \Sigma</annotation></semantics></math> by subsets <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><msup><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}_I \subset \Sigma^{n+1}</annotation></semantics></math> which are <a class="existingWikiWord" href="/nlab/show/disjoint+unions">disjoint unions</a> of <a class="existingWikiWord" href="/nlab/show/inhabited+set">non-empty</a> sets that are in <a class="existingWikiWord" href="/nlab/show/causal+order">causal order</a>, so that by <a class="existingWikiWord" href="/nlab/show/causal+factorization">causal factorization</a> the time-ordered products <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> on these subsets are uniquely given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>T</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T_{k}(-) \star_H T_{n-k}(-)</annotation></semantics></math>. Then we show that these unique products on these special subsets do coincide on <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a>. This yields the claim by a <a class="existingWikiWord" href="/nlab/show/partition+of+unity">partition of unity</a>.</p> <p>We now say this in detail:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><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><annotation encoding="application/x-tex">I \subset \{1, \cdots, n+1\}</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>I</mi><mo>¯</mo></mover><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><mo>∖</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\overline{I} \coloneqq \{1, \cdots, n+1\} \setminus I</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>,</mo><mover><mi>I</mi><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">I, \overline{I} \neq \emptyset</annotation></semantics></math>, define the subset</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>I</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><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></msub><mo>∈</mo><msup><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><msub><mi>x</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mi>j</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><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><mo>∖</mo><mi>I</mi></mrow></msub><mo>}</mo></mrow><mspace width="thickmathspace"></mspace><mo>⊂</mo><mspace width="thickmathspace"></mspace><msup><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C}_I \;\coloneqq\; \left\{ (x_i)_{i \in \{1, \cdots, n+1\}} \in \Sigma^{n+1} \;\vert\; \{x_i\}_{i \in I} {\vee\!\!\!\wedge} \{x_j\}_{j \in \{1, \cdots, n+1\} \setminus I} \right\} \;\subset\; \Sigma^{n+1} \,. </annotation></semantics></math></div> <p>Since the <a class="existingWikiWord" href="/nlab/show/causal+order">causal order</a>-relation involves the <a class="existingWikiWord" href="/nlab/show/closed+future+cones">closed future cones</a>/<a class="existingWikiWord" href="/nlab/show/closed+past+cones">closed past cones</a>, respectively, it is clear that these are <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a>. Moreover it is immediate that they form an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> of the <a class="existingWikiWord" href="/nlab/show/complement">complement</a> of the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>∪</mo><mfrac linethickness="0"><mrow><mi>I</mi><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><mi>I</mi><mo>,</mo><mover><mi>I</mi><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><msub><mi>𝒞</mi> <mi>I</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∖</mo><mi>diag</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{ { I \subset \{1, \cdots, n+1\} \atop { I, \overline{I} \neq \emptyset } } }{\cup} \mathcal{C}_I \;=\; \Sigma^{n+1} \setminus diag(\Sigma) \,. </annotation></semantics></math></div> <p>(Because any two distinct points in the <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetime">globally hyperbolic 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> may be causally separated by a <a class="existingWikiWord" href="/nlab/show/Cauchy+surface">Cauchy surface</a>, and any such may be deformed a little such as not to intersect any of a given finite set of points. )</p> <p>Hence the condition of <a class="existingWikiWord" href="/nlab/show/causal+factorization">causal factorization</a> on <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> implies that <a class="existingWikiWord" href="/nlab/show/restriction+of+distributions">restricted</a> to any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{I}</annotation></semantics></math> these have to be given (in the condensed <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a>-notation from <a href="S-matrix#NotationForTimeOrderedProductsAsGeneralizedFunctions">this remark</a>) on any unordered tuple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>=</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">}</mo><mo>∈</mo><msub><mi>𝒞</mi> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{X} = \{x_1, \cdots, x_{n+1}\} \in \mathcal{C}_I</annotation></semantics></math> with corresponding induced tuples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>≔</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{I} \coloneqq \{x_i\}_{i \in I}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo>≔</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mover><mi>I</mi><mo>¯</mo></mover></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{\mathbf{I}} \coloneqq \{x_i\}_{i \in \overline{I}}</annotation></semantics></math> by</p> <div class="maruku-equation" id="eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>T</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><mi>T</mi><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom><mtext>for</mtext><mphantom><mi>A</mi></mphantom><mi>𝒳</mi><mo>∈</mo><msub><mi>𝒞</mi> <mi>I</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T_{n+1}( \mathbf{X} ) \;=\; T(\mathbf{I}) T(\overline{\mathbf{I}}) \phantom{AA} \text{for} \phantom{A} \mathcal{X} \in \mathcal{C}_I \,. </annotation></semantics></math></div> <p>This shows that <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> is unique on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∖</mo><mi>diag</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma^{n+1} \setminus diag(\Sigma)</annotation></semantics></math> if it exists at all, hence if these local identifications glue to a global definition 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>. To see that this is the case, we have to consider any two such subsets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>I</mi> <mn>2</mn></msub><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><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AA</mi></mphantom><msub><mi>I</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>I</mi> <mn>2</mn></msub><mo>,</mo><mover><mrow><msub><mi>I</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>,</mo><mover><mrow><msub><mi>I</mi> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> I_1, I_2 \subset \{1, \cdots, n+1\} \,, \phantom{AA} I_1, I_2, \overline{I_1}, \overline{I_2} \neq \emptyset \,. </annotation></semantics></math></div> <p>By definition this implies that for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>∈</mo><msub><mi>𝒞</mi> <mrow><msub><mi>I</mi> <mn>1</mn></msub></mrow></msub><mo>∩</mo><msub><mi>𝒞</mi> <mrow><msub><mi>I</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2} </annotation></semantics></math></div> <p>a tuple of spacetime points which decomposes into causal order with respect to both these subsets, the corresponding mixed intersections of tuples are spacelike separated:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub><mo>∩</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mspace width="thickmathspace"></mspace><mrow><mo>></mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo><</mo></mrow><mspace width="thickmathspace"></mspace><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>∩</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{I}_1 \cap \overline{\mathbf{I}_2} \; {\gt\!\!\!\!\lt} \; \overline{\mathbf{I}_1} \cap \mathbf{I}_2 \,. </annotation></semantics></math></div> <p>By the assumption that 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 \neq n}</annotation></semantics></math> satisfy causal factorization, this implies that the corresponding time-ordered products commute:</p> <div class="maruku-equation" id="eq:TimeOrderedProductsOfMixedIntersectionsCommute"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub><mo>∩</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>T</mi><mo stretchy="false">(</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>∩</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>T</mi><mo stretchy="false">(</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>∩</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>T</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub><mo>∩</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2}) \, T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2) \;=\; T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2) \, T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2}) \,. </annotation></semantics></math></div> <p>Using this we find that the identifications 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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><msub><mi>I</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{I_1}</annotation></semantics></math> and on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><msub><mi>I</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{I_2}</annotation></semantics></math>, accrding to <a class="maruku-eqref" href="#eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal">(1)</a>, agree on the intersection: in that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>∈</mo><msub><mi>𝒞</mi> <mrow><msub><mi>I</mi> <mn>1</mn></msub></mrow></msub><mo>∩</mo><msub><mi>𝒞</mi> <mrow><msub><mi>I</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}</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><mi>T</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>T</mi><mo stretchy="false">(</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>T</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub><mo>∩</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>T</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub><mo>∩</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>T</mi><mo stretchy="false">(</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>∩</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>T</mi><mo stretchy="false">(</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>∩</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>T</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub><mo>∩</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">)</mo><munder><mrow><mi>T</mi><mo stretchy="false">(</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>∩</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>T</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub><mo>∩</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mi>T</mi><mo stretchy="false">(</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>∩</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>T</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>T</mi><mo stretchy="false">(</mo><mover><mrow><msub><mstyle mathvariant="bold"><mi>I</mi></mstyle> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} T( \mathbf{I}_1 ) T( \overline{\mathbf{I}_1} ) & = T( \mathbf{I}_1 \cap \mathbf{I}_2 ) T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} ) \, T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 ) T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} ) \\ & = T( \mathbf{I}_1 \cap \mathbf{I}_2 ) \underbrace{ T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 ) T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} ) } T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} ) \\ & = T( \mathbf{I}_2 ) T( \overline{\mathbf{I}_2} ) \end{aligned} </annotation></semantics></math></div> <p>Here in the first step we expanded out the two factors using <a class="maruku-eqref" href="#eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal">(1)</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">I_2</annotation></semantics></math>, then under the brace we used <a class="maruku-eqref" href="#eq:TimeOrderedProductsOfMixedIntersectionsCommute">(2)</a> and in the last step we used again <a class="maruku-eqref" href="#eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal">(1)</a>, but now for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">I_1</annotation></semantics></math>.</p> <p>To conclude, let</p> <div class="maruku-equation" id="eq:PartitionCausalOfUnityForComplementOfDiagonal"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>(</mo><msub><mi>χ</mi> <mi>I</mi></msub><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>χ</mi> <mi>I</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>𝒞</mi> <mi>i</mi></msub><mo>)</mo></mrow> <mfrac linethickness="0"><mrow><mrow><mi>I</mi><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><mi>I</mi><mo>¯</mo></mover><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></msub></mrow><annotation encoding="application/x-tex"> \left( \chi_I \in C^\infty_{cp}(\Sigma^{n+1}), \, supp(\chi_I) \subset \mathcal{C}_i \right)_{ { I \subset \{1, \cdots, n+1\} } \atop { I, \overline{I} \neq \emptyset } } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/partition+of+unity">partition of unity</a> subordinate to the <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> formed by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_I</annotation></semantics></math>:</p> <p>Then the above implies that setting for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>∈</mo><msup><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∖</mo><mi>diag</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{X} \in \Sigma^{n+1} \setminus diag(\Sigma)</annotation></semantics></math></p> <div class="maruku-equation" id="eq:TimeOrderedProductsAwayFromDiagonalByInduction"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>I</mi><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><mi>I</mi><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><mi>T</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo stretchy="false">)</mo><mi>T</mi><mo stretchy="false">(</mo><mover><mstyle mathvariant="bold"><mi>I</mi></mstyle><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> T_{n+1}(\mathbf{X}) \;\coloneqq\; \underset{ { I \in \{1, \cdots, n+1\} } \atop { I, \overline{I} \neq \emptyset } }{\sum} \chi_i(\mathbf{X}) T( \mathbf{I} ) T( \overline{\mathbf{I}} ) </annotation></semantics></math></div> <p>is well defined and satisfies causal factorization.</p> </div> <div class="num_remark" id="TimeOrderedProductOfFixedInteraction"> <h6 id="remark">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> of fixed <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> as <a class="existingWikiWord" href="/nlab/show/distributions">distributions</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 assume that the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> is a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a> (<a href="A+first+idea+of+quantum+field+theory#TrivialVectorBundleAsAFieldBundle">this example</a>)</p> <p>and let</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>LoObs</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\; LoObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle </annotation></semantics></math></div> <p>be a polynomial <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a>, to be regarded as a <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>. This means that there is a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><mi>int</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mstyle mathvariant="bold"><mi>α</mi></mstyle> <mrow><mi>i</mi><mo>′</mo></mrow></msub><mo>∈</mo><msubsup><mi>Ω</mi> <mi>Σ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo><mo>}</mo></mrow> <mrow><mi>i</mi><mo>,</mo><mi>i</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \left\{ \mathbf{L}_{int,i}, \mathbf{\alpha}_{i'} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}}) \right\}_{i,i'} </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/Lagrangian+densities">Lagrangian densities</a> which are monomials in the field and jet coordinates, and a corresponding finite set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><msub><mi>g</mi> <mrow><mi>sw</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>j</mi> <mrow><mi>sw</mi><mo>,</mo><mi>i</mi><mo>′</mo></mrow></msub><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo stretchy="false">⟨</mo><mi>j</mi><mo stretchy="false">⟩</mo><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ g_{sw,i} \in C^\infty_{cp}(\Sigma)\langle g \rangle \,,\, j_{sw,i'} \in C^\infty_{cp}(\Sigma)\langle j \rangle \right\} </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/adiabatic+switchings">adiabatic switchings</a>, such that</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><msub><mi>τ</mi> <mi>Σ</mi></msub><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>i</mi></munder><msub><mi>g</mi> <mrow><mi>sw</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><mi>int</mi><mo>,</mo><mi>i</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>′</mo></mrow></munder><msub><mi>j</mi> <mrow><mi>sw</mi><mo>,</mo><mi>i</mi><mo>′</mo></mrow></msub><msub><mstyle mathvariant="bold"><mi>α</mi></mstyle> <mrow><mi>i</mi><mo>′</mo></mrow></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> g S_{int} + j A \;=\; \tau_{\Sigma} \left( \underset{i}{\sum} g_{sw,i} \mathbf{L}_{int,i} \;+\; \underset{i'}{\sum} j_{sw,i'} \mathbf{\alpha}_{i'} \right) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/transgression+of+variational+differential+forms">transgression of variational differential forms</a> (<a href="A+first+idea+of+quantum+field+theory#TransgressionOfVariationalDifferentialFormsToConfigrationSpaces">this def.</a>) of the sum of the products of these <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatic switching</a> with these <a class="existingWikiWord" href="/nlab/show/Lagrangian+densities">Lagrangian densities</a>.</p> <p>In order to discuss the <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><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}(g S_{int} + j A)</annotation></semantics></math> and hence the <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> of the special form <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><mo>(</mo><munder><munder><mrow><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi></mrow><mo>⏟</mo></munder><mrow><mi>k</mi><mspace width="thinmathspace"></mspace><mtext>factors</mtext></mrow></munder><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">T_k\left( \underset{k \, \text{factors}}{\underbrace{g S_{int} + j A, \cdots, g S_{int} + j A }} \right)</annotation></semantics></math> it is sufficient to restrict attention to the <a class="existingWikiWord" href="/nlab/show/restriction">restriction</a> of each <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> to the subspace of <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> induced by the finite set of <a class="existingWikiWord" href="/nlab/show/Lagrangian+densities">Lagrangian densities</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><mi>int</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mstyle mathvariant="bold"><mi>α</mi></mstyle> <mrow><mi>i</mi><mo>′</mo></mrow></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>,</mo><mi>i</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\{\mathbf{L}_{int,i}, \mathbf{\alpha}_{i'}\}_{i,i'}</annotation></semantics></math>.</p> <p>This restriction is a <a class="existingWikiWord" href="/nlab/show/continuous+linear+functional">continuous linear functional</a> on the corresponding space of <a class="existingWikiWord" href="/nlab/show/bump+functions">bump functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>g</mi> <mrow><mi>sw</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mi>j</mi> <mrow><mi>sw</mi><mo>,</mo><mi>i</mi><mo>′</mo></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{g_{sw,i}, j_{sw,i'}\}</annotation></semantics></math>, hence a <a class="existingWikiWord" href="/nlab/show/distribution">dstributional</a> <a class="existingWikiWord" href="/nlab/show/section">section</a> of a corresponding <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>.</p> <p>In terms of this, prop. <a class="maruku-ref" href="#RenormalizationIsInductivelyExtensionToDiagonal"></a> says that the choice 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><msub><mi>T</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">T_k</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/induction">inductively</a> in <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> a choice of <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> to the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>=</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Sigma = \mathbb{R}^{p,1}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> and we impose the <a class="existingWikiWord" href="/nlab/show/renormalization+condition">renormalization condition</a> “translation invariance” (<a href="S-matrix#RenormalizationConditions">this def.</a>) then each <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> is a distribution on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>ℝ</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\Sigma^{k-1} = \mathbb{R}^{(p+1)(k-1)}</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> is from the complement of the origina <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><msup><mi>ℝ</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">0 \in \mathbb{R}^{(p+1)(k-1)}</annotation></semantics></math>.</p> </div> <p>Therefore we now discuss <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> (def. <a class="maruku-ref" href="#ExtensionOfDistributions"></a> below) on <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> from the complement of the origin to the origin. Since the space of choices of such extensions turns out to depend on the <em><a class="existingWikiWord" href="/nlab/show/scaling+degree+of+distributions">scaling degree of distributions</a></em>, we first discuss that (def. <a class="maruku-ref" href="#ScalingDegree"></a> below).</p> <div class="num_defn" id="RescaledDistribution"> <h6 id="definition">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/scaling+degree+of+distributions">rescaled distribution</a>)</strong></p> <p>Let <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>. For <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><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\lambda \in (0,\infty) \subset \mathbb{R}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/real+number">real number</a> write</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><msup><mi>ℝ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>s</mi> <mi>λ</mi></msub></mrow></mover></mtd> <mtd><msup><mi>ℝ</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>λ</mi><mi>x</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{R}^n &\overset{s_\lambda}{\longrightarrow}& \mathbb{R}^n \\ x &\mapsto& \lambda x } </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> given by multiplication with <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>, using the canonical <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a>-structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>.</p> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u \in \mathcal{D}'(\mathbb{R}^n)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> on the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> the <em>rescaled distribution</em> is the <a class="existingWikiWord" href="/nlab/show/pullback+of+a+distribution">pullback</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">m_\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>u</mi> <mi>λ</mi></msub><mo>≔</mo><msubsup><mi>s</mi> <mi>λ</mi> <mo>*</mo></msubsup><mi>u</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> u_\lambda \coloneqq s_\lambda^\ast u \;\in\; \mathcal{D}'(\mathbb{R}^n) \,. </annotation></semantics></math></div> <p>Explicitly, this is given by</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>𝒟</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">⟨</mo><msub><mi>u</mi> <mi>λ</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow></mover></mtd> <mtd><mi>ℝ</mi></mtd></mtr> <mtr><mtd><mi>b</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msup><mi>λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup><mo stretchy="false">⟨</mo><mi>u</mi><mo>,</mo><mi>b</mi><mo stretchy="false">(</mo><msup><mi>λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{D}(\mathbb{R}^n) &\overset{ \langle u_\lambda, - \rangle}{\longrightarrow}& \mathbb{R} \\ b &\mapsto& \lambda^{-n} \langle u , b(\lambda^{-1}\cdot (-))\rangle } \,. </annotation></semantics></math></div> <p>Similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X \subset \mathbb{R}^n</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> which is invariant under <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">s_\lambda</annotation></semantics></math>, the rescaling of a distribution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u \in \mathcal{D}'(X)</annotation></semantics></math> is is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mi>λ</mi></msub><mo>≔</mo><msubsup><mi>s</mi> <mi>λ</mi> <mo>*</mo></msubsup><mi>u</mi></mrow><annotation encoding="application/x-tex">u_\lambda \coloneqq s_\lambda^\ast u</annotation></semantics></math>.</p> </div> <div class="num_defn" id="ScalingDegree"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/scaling+degree+of+a+distribution">scaling degree of a distribution</a>)</strong></p> <p>Let <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 let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X \subset \mathbb{R}^n</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> which is invariant under <a class="existingWikiWord" href="/nlab/show/rescaling">rescaling</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">s_\lambda</annotation></semantics></math> (def. <a class="maruku-ref" href="#RescaledDistribution"></a>) for all <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>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u \in \mathcal{D}'(X)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> on this subset. Then</p> <ol> <li> <p>The <em><a class="existingWikiWord" href="/nlab/show/scaling+degree+of+a+distribution">scaling degree</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/infimum">infimum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sd</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>inf</mi><mrow><mo>{</mo><mi>ω</mi><mo>∈</mo><mi>ℝ</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mrow><mi>λ</mi><mo>→</mo><mn>0</mn></mrow></munder><msup><mi>λ</mi> <mi>ω</mi></msup><msub><mi>u</mi> <mi>λ</mi></msub><mo>=</mo><mn>0</mn><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> sd(u) \;\coloneqq\; inf \left\{ \omega \in \mathbb{R} \;\vert\; \underset{\lambda \to 0}{\lim} \lambda^\omega u_\lambda = 0 \right\} </annotation></semantics></math></div> <p>of the set of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> of the rescaled distribution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>λ</mi> <mi>ω</mi></msup><msub><mi>u</mi> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">\lambda^\omega u_\lambda</annotation></semantics></math> (def. <a class="maruku-ref" href="#RescaledDistribution"></a>) vanishes. If there is no such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> one sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sd</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>≔</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">sd(u) \coloneqq \infty</annotation></semantics></math>.</p> </li> <li> <p>The <em><a class="existingWikiWord" href="/nlab/show/degree+of+divergence+of+a+distribution">degree of divergence</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is the difference of the scaling degree by the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of the underlying space:</p> </li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>sd</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>−</mo><mi>n</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> deg(u) \coloneqq sd(u) - n \,. </annotation></semantics></math></div></div> <div class="num_example" id="NonSingularDistributionsScalingDegree"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/scaling+degree+of+distributions">scaling degree</a> of <a class="existingWikiWord" href="/nlab/show/non-singular+distributions">non-singular distributions</a>)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>=</mo><msub><mi>u</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">u = u_f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/non-singular+distribution">non-singular distribution</a> given by <a class="existingWikiWord" href="/nlab/show/bump+function">bump function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C^\infty(X) \subset \mathcal{D}'(X)</annotation></semantics></math>, then its <a class="existingWikiWord" href="/nlab/show/scaling+degree+of+a+distribution">scaling degree</a> (def. <a class="maruku-ref" href="#ScalingDegree"></a>) is non-<a class="existingWikiWord" href="/nlab/show/positive+number">positive</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sd</mi><mo stretchy="false">(</mo><msub><mi>u</mi> <mi>f</mi></msub><mo stretchy="false">)</mo><mo>≤</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> sd(u_f) \leq 0 \,. </annotation></semantics></math></div> <p>Specifically if the first non-vanishing <a class="existingWikiWord" href="/nlab/show/partial+derivative">partial derivative</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>α</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial_\alpha f(0)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> at 0 occurs at order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">{\vert \alpha\vert} \in \mathbb{N}</annotation></semantics></math>, then the scaling degree of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">u_f</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">-{\vert \alpha\vert}</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By definition we have for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b \in C^\infty_{cp}(\mathbb{R}^n)</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/bump+function">bump function</a> 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><mrow><mo>⟨</mo><msup><mi>λ</mi> <mi>ω</mi></msup><mo stretchy="false">(</mo><msub><mi>u</mi> <mi>f</mi></msub><msub><mo stretchy="false">)</mo> <mi>λ</mi></msub><mo>,</mo><mi>n</mi><mo>⟩</mo></mrow></mtd> <mtd><mo>=</mo><msup><mi>λ</mi> <mrow><mi>ω</mi><mo>−</mo><mi>n</mi></mrow></msup><munder><mo>∫</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><msup><mi>λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>x</mi><mo stretchy="false">)</mo><msup><mi>d</mi> <mi>n</mi></msup><mi>x</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msup><mi>λ</mi> <mi>ω</mi></msup><munder><mo>∫</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>d</mi> <mi>n</mi></msup><mi>x</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left\langle \lambda^{\omega} (u_f)_\lambda, n \right\rangle & = \lambda^{\omega-n} \underset{\mathbb{R}^n}{\int} f(x) g(\lambda^{-1} x) d^n x \\ & = \lambda^{\omega} \underset{\mathbb{R}^n}{\int} f(\lambda x) g(x) d^n x \end{aligned} \,, </annotation></semantics></math></div> <p>where in last line we applied <a class="existingWikiWord" href="/nlab/show/change+of+integration+variables">change of integration variables</a>.</p> <p>The limit of this expression is clearly zero for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\omega \gt 0</annotation></semantics></math>, which shows the first claim.</p> <p>If moreover the first non-vanishing <a class="existingWikiWord" href="/nlab/show/partial+derivative">partial derivative</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> occurs at order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">{\vert \alpha \vert} = k</annotation></semantics></math>, then <a class="existingWikiWord" href="/nlab/show/Hadamard%27s+lemma">Hadamard's lemma</a> says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is 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>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>i</mi></munder><msub><mi>α</mi> <mi>i</mi></msub><mo>!</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>α</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>i</mi></msup><msup><mo stretchy="false">)</mo> <mrow><msub><mi>α</mi> <mi>i</mi></msub></mrow></msup><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>β</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>β</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>+</mo><mn>1</mn></mrow></mrow></mfrac></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>i</mi></msup><msup><mo stretchy="false">)</mo> <mrow><msub><mi>β</mi> <mi>i</mi></msub></mrow></msup><msub><mi>h</mi> <mi>β</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(x) \;=\; \left( \underset{i}{\prod} \alpha_i ! \right)^{-1} (\partial_\alpha f(0)) \underset{i}{\prod} (x^i)^{\alpha_i} + \underset{ {\beta \in \mathbb{N}^n} \atop { {\vert \beta\vert} = {\vert \alpha \vert} + 1 } }{\sum} \underset{i}{\prod} (x^i)^{\beta_i} h_{\beta}(x) </annotation></semantics></math></div> <p>where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">h_{\beta}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>. Hence in this case</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><msup><mi>λ</mi> <mi>ω</mi></msup><mo stretchy="false">(</mo><msub><mi>u</mi> <mi>f</mi></msub><msub><mo stretchy="false">)</mo> <mi>λ</mi></msub><mo>,</mo><mi>n</mi><mo>⟩</mo></mrow></mtd> <mtd><mo>=</mo><msup><mi>λ</mi> <mrow><mi>ω</mi><mo>+</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow></msup><munder><mo>∫</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msup><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>i</mi></munder><msub><mi>α</mi> <mi>i</mi></msub><mo>!</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>α</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>i</mi></msup><msup><mo stretchy="false">)</mo> <mrow><msub><mi>α</mi> <mi>i</mi></msub></mrow></msup><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>d</mi> <mi>n</mi></msup><mi>x</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><msup><mi>λ</mi> <mrow><mi>ω</mi><mo>+</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>+</mo><mn>1</mn></mrow></msup><munder><mo>∫</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>i</mi></msup><msup><mo stretchy="false">)</mo> <mrow><msub><mi>β</mi> <mi>i</mi></msub></mrow></msup><msub><mi>h</mi> <mi>β</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>d</mi> <mi>n</mi></msup><mi>x</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left\langle \lambda^{\omega} (u_f)_\lambda, n \right\rangle & = \lambda^{\omega + {\vert \alpha\vert }} \underset{\mathbb{R}^n}{\int} \left( \underset{i}{\prod} \alpha_i ! \right)^{-1} (\partial_\alpha f(0)) \underset{i}{\prod} (x^i)^{\alpha_i} b(x) d^n x \\ & \phantom{=} + \lambda^{\omega + {\vert \alpha\vert} + 1} \underset{\mathbb{R}^n}{\int} \underset{i}{\prod} (x^i)^{\beta_i} h_{\beta}(x) b(x) d^n x \end{aligned} \,. </annotation></semantics></math></div> <p>This makes manifest that the expression goes to zero with <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> precisely for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>></mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">\omega \gt - {\vert \alpha \vert}</annotation></semantics></math>, which means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sd</mi><mo stretchy="false">(</mo><msub><mi>u</mi> <mi>f</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex"> sd(u_f) = -{\vert \alpha \vert} </annotation></semantics></math></div> <p>in this case.</p> </div> <div class="num_example" id="DerivativesOfDeltaDistributionScalingDegree"> <h6 id="example_2">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/scaling+degree+of+a+distribution">scaling degree</a> of <a class="existingWikiWord" href="/nlab/show/derivative+of+a+distribution">derivatives</a> of <a class="existingWikiWord" href="/nlab/show/delta-distributions">delta-distributions</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\alpha \in \mathbb{N}^n</annotation></semantics></math> be a multi-index and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>α</mi></msub><mi>δ</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial_\alpha \delta \in \mathcal{D}'(X)</annotation></semantics></math> the corresponding <a class="existingWikiWord" href="/nlab/show/partial+derivative">partial</a> <a class="existingWikiWord" href="/nlab/show/derivative+of+distributions">derivatives</a> of the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mn>0</mn></msub><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\delta_0 \in \mathcal{D}'(\mathbb{R}^n)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">supported</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>. Then the <a class="existingWikiWord" href="/nlab/show/degree+of+divergence+of+a+distribution">degree of divergence</a> (def. <a class="maruku-ref" href="#ScalingDegree"></a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\partial_\alpha \delta_0</annotation></semantics></math> is the total order the derivatives</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex"> deg\left( {\, \atop \,} \partial_\alpha\delta_0{\, \atop \,} \right) \;=\; {\vert \alpha \vert} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>i</mi></munder><msub><mi>α</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">{\vert \alpha\vert} \coloneqq \underset{i}{\sum} \alpha_i</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By definition we have for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b \in C^\infty_{cp}(\mathbb{R}^n)</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/bump+function">bump function</a> 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><mrow><mo>⟨</mo><msup><mi>λ</mi> <mi>ω</mi></msup><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mi>λ</mi></msub><mo>,</mo><mi>b</mi><mo>⟩</mo></mrow></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></msup><msup><mi>λ</mi> <mrow><mi>ω</mi><mo>−</mo><mi>n</mi></mrow></msup><msub><mrow><mo>(</mo><mfrac><mrow><msup><mo>∂</mo> <mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></msup></mrow><mrow><msup><mo>∂</mo> <mrow><msub><mi>α</mi> <mn>1</mn></msub></mrow></msup><msup><mi>x</mi> <mn>1</mn></msup><mi>⋯</mi><msup><mo>∂</mo> <mrow><msub><mi>α</mi> <mi>n</mi></msub></mrow></msup><msup><mi>x</mi> <mi>n</mi></msup></mrow></mfrac><mi>b</mi><mo stretchy="false">(</mo><msup><mi>λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>x</mi><mo stretchy="false">)</mo><mo>)</mo></mrow> <mrow><mo stretchy="false">|</mo><mi>x</mi><mo>=</mo><mn>0</mn></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></msup><msup><mi>λ</mi> <mrow><mi>ω</mi><mo>−</mo><mi>n</mi><mo>−</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow></msup><mfrac><mrow><msup><mo>∂</mo> <mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></msup></mrow><mrow><msup><mo>∂</mo> <mrow><msub><mi>α</mi> <mn>1</mn></msub></mrow></msup><msup><mi>x</mi> <mn>1</mn></msup><mi>⋯</mi><msup><mo>∂</mo> <mrow><msub><mi>α</mi> <mi>n</mi></msub></mrow></msup><msup><mi>x</mi> <mi>n</mi></msup></mrow></mfrac><mi>b</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left\langle \lambda^\omega (\partial_\alpha \delta_0)_\lambda, b \right\rangle & = (-1)^{{\vert \alpha \vert}} \lambda^{\omega-n} \left( \frac{ \partial^{{\vert \alpha \vert}} }{ \partial^{\alpha_1} x^1 \cdots \partial^{\alpha_n}x^n } b(\lambda^{-1}x) \right)_{\vert x = 0} \\ & = (-1)^{{\vert \alpha \vert}} \lambda^{\omega - n - {\vert \alpha\vert}} \frac{ \partial^{{\vert \alpha \vert}} }{ \partial^{\alpha_1} x^1 \cdots \partial^{\alpha_n}x^n } b(0) \end{aligned} \,, </annotation></semantics></math></div> <p>where in the last step we used the <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a> of <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>. It is clear that this goes to zero with <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> as long as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>></mo><mi>n</mi><mo>+</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">\omega \gt n + {\vert \alpha\vert}</annotation></semantics></math>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sd</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi><mo>+</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">sd(\partial_{\alpha} \delta_0) = n + {\vert \alpha \vert}</annotation></semantics></math>.</p> </div> <div class="num_example" id="FeynmanPropagatorOnMinkowskiScalingDegree"> <h6 id="example_3">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/scaling+degree+of+a+distribution">scaling degree</a> of <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>Let</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 stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>x</mi> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>+</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex"> \Delta_F(x) \;=\; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> for the massive <a class="existingWikiWord" href="/nlab/show/free+field">free</a> <a class="existingWikiWord" href="/nlab/show/real+scalar+field">real scalar field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mi>p</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = p+1</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> (<a href="Feynman+propagator#FeynmanPropagatorAsACauchyPrincipalvalue">this prop.</a>). Its <a class="existingWikiWord" href="/nlab/show/scaling+degree+of+a+distribution">scaling degree</a> is</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><mi>sd</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>n</mi><mo>−</mo><mn>2</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>p</mi><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} sd(\Delta_{F}) & = n - 2 \\ & = p -1 \end{aligned} \,. </annotation></semantics></math></div></div> <p>(<a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00, example 3 on p. 22</a>)</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Regarding <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> as a <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a> via the given <a class="existingWikiWord" href="/nlab/show/Fourier+transform+of+distributions">Fourier-transform</a> expression, we find by <a class="existingWikiWord" href="/nlab/show/change+of+integration+variables">change of integration variables</a> in the Fourier integral that in the scaling limit the Feynman propagator becomes that for vannishing <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, which scales homogeneously:</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><munder><mi>lim</mi><mrow><mi>λ</mi><mo>→</mo><mn>0</mn></mrow></munder><mrow><mo>(</mo><msup><mi>λ</mi> <mi>ω</mi></msup><mspace width="thickmathspace"></mspace><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mrow><mi>λ</mi><mo>→</mo><mn>0</mn></mrow></munder><mrow><mo>(</mo><msup><mo lspace="0em" rspace="thinmathspace">lamba</mo> <mi>ω</mi></msup><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mi>λ</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>+</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mrow><mi>λ</mi><mo>→</mo><mn>0</mn></mrow></munder><mrow><mo>(</mo><msup><mi>λ</mi> <mrow><mi>ω</mi><mo>−</mo><mi>n</mi></mrow></msup><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mi>λ</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><msup><mi>λ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>+</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mrow><mi>λ</mi><mo>→</mo><mn>0</mn></mrow></munder><mrow><mo>(</mo><msup><mi>λ</mi> <mrow><mi>ω</mi><mo>−</mo><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msup><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mi>λ</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>+</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \underset{\lambda \to 0}{\lim} \left( \lambda^\omega \; \Delta_F(\lambda x) \right) & = \underset{\lambda \to 0}{\lim} \left( \lamba^{\omega} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \lambda x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k \right) \\ & = \underset{\lambda \to 0}{\lim} \left( \lambda^{\omega-n} \; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \lambda x^\mu} }{ - (\lambda^{-2}) k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k \right) \\ & = \underset{\lambda \to 0}{\lim} \left( \lambda^{\omega-n + 2 } \; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \lambda x^\mu} }{ - k_\mu k^\mu + i \epsilon } \, d k_0 \, d^p \vec k \right) \,. \end{aligned} </annotation></semantics></math></div></div> <div class="num_prop" id="ScalingDegreeOfDistributionsBasicProperties"> <h6 id="proposition_2">Proposition</h6> <p><strong>(basic properties of <a class="existingWikiWord" href="/nlab/show/scaling+degree+of+distributions">scaling degree of distributions</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X \subset \mathbb{R}^n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u \in \mathcal{D}'(X)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> as in def. <a class="maruku-ref" href="#RescaledDistribution"></a>, such that its <a class="existingWikiWord" href="/nlab/show/scaling+degree+of+a+distribution">scaling degree</a> is finite: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sd</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo><</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">sd(u) \lt \infty</annotation></semantics></math> (def. <a class="maruku-ref" href="#ScalingDegree"></a>). Then</p> <ol> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\alpha \in \mathbb{N}^n</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/partial+derivative">partial</a> <a class="existingWikiWord" href="/nlab/show/derivative+of+distributions">derivative of distributions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">\partial_\alpha</annotation></semantics></math> increases scaling degree at most by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert \alpha\vert }</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>α</mi></msub><mi>u</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≤</mo><mspace width="thickmathspace"></mspace><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>+</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex"> deg(\partial_\alpha u) \;\leq\; deg(u) + {\vert \alpha\vert} </annotation></semantics></math></div></li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\alpha \in \mathbb{N}^n</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> with the smooth coordinate functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">x^\alpha</annotation></semantics></math> decreases scaling degree at least by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert \alpha\vert }</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>α</mi></msup><mi>u</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≤</mo><mspace width="thickmathspace"></mspace><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>−</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex"> deg(x^\alpha u) \;\leq\; deg(u) - {\vert \alpha\vert} </annotation></semantics></math></div></li> <li> <p>Under <a class="existingWikiWord" href="/nlab/show/tensor+product+of+distributions">tensor product of distributions</a> their scaling degrees add:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sd</mi><mo stretchy="false">(</mo><mi>u</mi><mo>⊗</mo><mi>v</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>sd</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>+</mo><mi>sd</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> sd(u \otimes v) \leq sd(u) + sd(v) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in \mathcal{D}'(Y)</annotation></semantics></math> another distribution on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Y \subset \mathbb{R}^{n'}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>f</mi><mi>u</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">deg(f u) \leq deg(u) - k</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C^\infty(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f^{(\alpha)}(0) = 0</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">{\vert \alpha\vert} \leq k-1</annotation></semantics></math>;</p> </li> </ol> </div> <p>(<a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00, lemma 5.1</a>, <a href="#Duetsch18">Dütsch 18, exercise 3.34</a>)</p> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>The first three statements follow with manipulations as in example <a class="maruku-ref" href="#NonSingularDistributionsScalingDegree"></a> and example <a class="maruku-ref" href="#DerivativesOfDeltaDistributionScalingDegree"></a>.</p> <p>For the fourth…</p> </div> <div class="num_prop" id="ScalingDegreeOfProductDistribution"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/scaling+degree+of+distributions">scaling degree</a> of <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product distribution</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u,v \in \mathcal{D}'(\mathbb{R}^n)</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/distributions">distributions</a> such that</p> <ol> <li> <p>both have finite <a class="existingWikiWord" href="/nlab/show/degree+of+divergence+of+a+distribution">degree of divergence</a> (def. <a class="maruku-ref" href="#ScalingDegree"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>,</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo><</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex"> deg(u), deg(v) \lt \infty </annotation></semantics></math></div></li> <li> <p>their <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> is well-defined</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> u v \in \mathcal{D}'(\mathbb{R}^n) </annotation></semantics></math></div> <p>(in that their <a class="existingWikiWord" href="/nlab/show/wave+front+sets">wave front sets</a> satisfy <a class="existingWikiWord" href="/nlab/show/H%C3%B6rmander%27s+criterion">Hörmander's criterion</a>)</p> </li> </ol> <p>then the product distribution has <a class="existingWikiWord" href="/nlab/show/degree+of+divergence+of+a+distribution">degree of divergence</a> bounded by the sum of the separate degrees:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mi>v</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≤</mo><mspace width="thickmathspace"></mspace><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>+</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> deg(u v) \;\leq\; deg(u) + deg(v) \,. </annotation></semantics></math></div></div> <p>With the concept of <a class="existingWikiWord" href="/nlab/show/scaling+degree+of+distributions">scaling degree of distributions</a> in hand, we may now discuss <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a>:</p> <div class="num_defn" id="ExtensionOfDistributions"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⊂</mo><mi>ι</mi></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">X \overset{\iota}{\subset} \hat X</annotation></semantics></math> be an inclusion of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of some <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>. This induces the operation of <a class="existingWikiWord" href="/nlab/show/restriction+of+distributions">restriction of distributions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>ι</mi> <mo>*</mo></msup></mrow></mover><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{D}'(\hat X) \overset{\iota^\ast}{\longrightarrow} \mathcal{D}'(X) \,. </annotation></semantics></math></div> <p>Given a <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u \in \mathcal{D}'(X)</annotation></semantics></math>, then an <em><a class="existingWikiWord" href="/nlab/show/extension">extension</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat X</annotation></semantics></math> is a distribution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat u \in \mathcal{D}'(\hat X)</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><msup><mi>ι</mi> <mo>*</mo></msup><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>u</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \iota^\ast \hat u \;=\; u \,. </annotation></semantics></math></div></div> <div class="num_prop" id="ExtensionUniqueNonPositiveDegreeOfDivergence"> <h6 id="proposition_4">Proposition</h6> <p><strong>(unique <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> with negative <a class="existingWikiWord" href="/nlab/show/degree+of+divergence+of+a+distribution">degree of divergence</a>)</strong></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>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> on the <a class="existingWikiWord" href="/nlab/show/complement">complement</a> of the origin, with <a class="existingWikiWord" href="/nlab/show/negative+number">negative</a> <a class="existingWikiWord" href="/nlab/show/degree+of+divergence+of+a+distribution">degree of divergence</a> at the origin</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> deg(u) \lt 0 \,. </annotation></semantics></math></div> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> has a <em>unique</em> <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat u \in \mathcal{D}'(\mathbb{R}^n)</annotation></semantics></math> to the origin with the same degree of divergence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> deg(\hat u) = deg(u) \,. </annotation></semantics></math></div></div> <p>(<a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00, theorem 5.2</a>, <a href="#Duetsch18">Dütsch 18, theorem 3.35 a)</a>)</p> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>Regarding uniqueness:</p> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>u</mi><mo stretchy="false">^</mo></mover> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><annotation encoding="application/x-tex">{\hat u}^\prime</annotation></semantics></math> are two extensions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>deg</mi><mo stretchy="false">(</mo><msup><mover><mi>u</mi><mo stretchy="false">^</mo></mover> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">deg(\hat u) = deg({\hat u}^\prime)</annotation></semantics></math>. Both being extensions of a distribution defined on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \setminus \{0\}</annotation></semantics></math>, this difference has <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">support</a> at the origin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\{0\} \subset \mathbb{R}^n</annotation></semantics></math>. By <a href="point-supported+distribution#PointSupportedDistributionsAreSumsOfDerivativesOfDeltaDistibutions">this prop.</a> this implies that it is a linear combination of <a class="existingWikiWord" href="/nlab/show/derivative+of+a+distribution">derivatives</a> of the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">supported</a> at the origin:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>u</mi><mo stretchy="false">^</mo></mover> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></munder><msup><mi>c</mi> <mi>α</mi></msup><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> {\hat u}^\prime - \hat u \;=\; \underset{ {\alpha \in \mathbb{N}^n} }{\sum} c^\alpha \partial_\alpha \delta_0 </annotation></semantics></math></div> <p>for constants <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>c</mi> <mi>α</mi></msup><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">c^\alpha \in \mathbb{C}</annotation></semantics></math>. But by <a href="scaling+degree+of+a+distribution#DerivativesOfDeltaDistributionScalingDegree">this example</a> the <a class="existingWikiWord" href="/nlab/show/degree+of+divergence+of+a+distribution">degree of divergence</a> of these <a class="existingWikiWord" href="/nlab/show/point-supported+distributions">point-supported distributions</a> is non-negative</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≥</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> deg( \partial_\alpha \delta_0) = {\vert \alpha\vert} \geq 0 \,. </annotation></semantics></math></div> <p>This implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>c</mi> <mi>α</mi></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c^\alpha = 0</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>, hence that the two extensions coincide.</p> <p>Regarding existence:</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> b \in C^\infty_{cp}(\mathbb{R}^n) </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/bump+function">bump function</a> which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\leq 1</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/constant+function">constant</a> on 1 over a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> of the origin. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>≔</mo><mn>1</mn><mo>−</mo><mi>b</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \chi \coloneqq 1 - b \;\in\; C^\infty(\mathbb{R}^n) </annotation></semantics></math></div><center> <img src="https://ncatlab.org/nlab/files/PointExtensionOfDistributions.png" /> </center> <blockquote> <p>graphics grabbed from <a href="#Duetsch18">Dütsch 18, p. 108</a></p> </blockquote> <p>and for <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> a <a class="existingWikiWord" href="/nlab/show/positive+real+number">positive real number</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>λ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \chi_\lambda(x) \coloneqq \chi(\lambda x) \,. </annotation></semantics></math></div> <p>Since the <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>λ</mi></msub><mi>u</mi></mrow><annotation encoding="application/x-tex">\chi_\lambda u</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">support of a distribution</a> on a <a class="existingWikiWord" href="/nlab/show/complement">complement</a> of a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> of the origin, we may extend it by zero to a distribution on all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>, which we will denote by the same symbols:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>λ</mi></msub><mi>u</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \chi_\lambda u \in \mathcal{D}'(\mathbb{R}^n) \,. </annotation></semantics></math></div> <p>By construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">xhi</mo> <mi>λ</mi></msub><mi>u</mi></mrow><annotation encoding="application/x-tex">\xhi_\lambda u</annotation></semantics></math> coincides with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> away from a neighbourhood of the origin, which moreover becomes arbitrarily small as <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> increases. This means that if the following <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> exists</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover><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>χ</mi> <mi>λ</mi></msub><mi>u</mi></mrow><annotation encoding="application/x-tex"> \hat u \;\coloneqq\; \underset{\lambda \to \infty}{\lim} \chi_\lambda u </annotation></semantics></math></div> <p>then it is an extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>.</p> <p>To see that the limit exists, it is sufficient to observe that we have a <a class="existingWikiWord" href="/nlab/show/Cauchy+sequence">Cauchy sequence</a>, hence that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b\in C^\infty_{cp}(\mathbb{R}^n)</annotation></semantics></math> the difference</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>χ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>u</mi><mo>−</mo><msub><mi>χ</mi> <mi>n</mi></msub><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>u</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>χ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>χ</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\chi_{n+1} u - \chi_n u)(b) \;=\; u(b)( \chi_{n+1} + \chi_n ) </annotation></semantics></math></div> <p>becomes arbitrarily small.</p> <p>It remains to see that the unique extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat u</annotation></semantics></math> thus established has the same scaling degree as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>. This is shown in (<a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00, p. 24</a>).</p> </div> <div class="num_prop" id="SpaceOfPointExtensions"> <h6 id="proposition_5">Proposition</h6> <p><strong>(space of <a class="existingWikiWord" href="/nlab/show/point-extensions+of+distributions">point-extensions of distributions</a>)</strong></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>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> of <a class="existingWikiWord" href="/nlab/show/scaling+degree+of+a+distribution">degree of divergence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo><</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">deg(u) \lt \infty</annotation></semantics></math>.</p> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> does admit at least one <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension</a> (def. <a class="maruku-ref" href="#ExtensionOfDistributions"></a>) to a distribution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat u \in \mathcal{D}'(\mathbb{R}^n)</annotation></semantics></math>, and every choice of extension has the same <a class="existingWikiWord" href="/nlab/show/degree+of+divergence+of+a+distribution">degree of divergence</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> deg(\hat u) = deg(u) \,. </annotation></semantics></math></div> <p>Moreover, any two such extensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>u</mi><mo stretchy="false">^</mo></mover> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><annotation encoding="application/x-tex">{\hat u}^\prime</annotation></semantics></math> differ by a linear combination of <a class="existingWikiWord" href="/nlab/show/partial+derivatives">partial</a> <a class="existingWikiWord" href="/nlab/show/derivatives+of+distributions">derivatives of distributions</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\leq deg(u)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\delta_0</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">supported</a> at the origin:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>u</mi><mo stretchy="false">^</mo></mover> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></munder><msup><mi>q</mi> <mi>α</mi></msup><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> {\hat u}^\prime - \hat u \;=\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq deg(u) } }{\sum} q^\alpha \partial_\alpha \delta_0 \,, </annotation></semantics></math></div> <p>for a finite number of constants <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>q</mi> <mi>α</mi></msup><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">q^\alpha \in \mathbb{C}</annotation></semantics></math>.</p> </div> <p>This is essentially (<a href="#Hoermander90">Hörmander 90, thm. 3.2.4</a>). We follow (<a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00, theorem 5.3</a>), which was inspired by (<a href="#EpsteinGlaser73">Epstein-Glaser 73, section 5</a>). Review of this approach is in (<a href="#Duetsch18">Dütsch 18, theorem 3.35 (b)</a>), see also remark <a class="maruku-ref" href="#WExtensions"></a> below.</p> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C^\infty(\mathbb{R}^n)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\rho \in \mathbb{N}</annotation></semantics></math>, we say that <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> vanishes to order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></em> at the origin if all <a class="existingWikiWord" href="/nlab/show/partial+derivatives">partial derivatives</a> with multi-index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\alpha \in \mathbb{N}^n</annotation></semantics></math> of total order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">{\vert \alpha\vert} \leq \rho</annotation></semantics></math> vanish at the origin:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>α</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mphantom><mi>AAA</mi></mphantom><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial_\alpha f (0) = 0 \phantom{AAA} {\vert \alpha\vert} \leq \rho \,. </annotation></semantics></math></div> <p>By <a class="existingWikiWord" href="/nlab/show/Hadamard%27s+lemma">Hadamard's lemma</a>, such a function may be written in the form</p> <div class="maruku-equation" id="eq:ForVanishingOrderRhoHadamardExpansion"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mi>ρ</mi><mo>+</mo><mn>1</mn></mrow></mrow></mfrac></munder><msup><mi>x</mi> <mi>α</mi></msup><msub><mi>r</mi> <mi>α</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(x) \;=\; \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha \vert} = \rho + 1 } }{\sum} x^\alpha r_\alpha(x) </annotation></semantics></math></div> <p>for <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>α</mi></msub><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_\alpha \in C^\infty_{cp}(\mathbb{R}^n)</annotation></semantics></math>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒟</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>↪</mo><mi>𝒟</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≔</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{D}_\rho(\mathbb{R}^n) \hookrightarrow \mathcal{D}(\mathbb{R}^n) \coloneqq C^\infty_{cp}(\mathbb{R}^n) </annotation></semantics></math></div> <p>for the subspace of that of all <a class="existingWikiWord" href="/nlab/show/bump+functions">bump functions</a> on those that vanish to order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> at the origin.</p> <p>By definition this is equivalently the joint <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of the <a class="existingWikiWord" href="/nlab/show/partial+derivative">partial</a> <a class="existingWikiWord" href="/nlab/show/derivatives+of+distributions">derivatives of distributions</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert \alpha\vert}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\delta_0</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">supported</a> at the origin:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>𝒟</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom><mo>⇔</mo><mphantom><mi>AA</mi></mphantom><munder><mo>∀</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow></mrow></mfrac></munder><mrow><mo>⟨</mo><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mo>,</mo><mi>b</mi><mo>⟩</mo></mrow><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> b \in \mathcal{D}_\rho(\mathbb{R}^n) \phantom{AA} \Leftrightarrow \phantom{AA} \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq \rho } } {\forall} \left\langle \partial_\alpha \delta_0, b \right\rangle = 0 \,. </annotation></semantics></math></div> <p>Therefore every <a class="existingWikiWord" href="/nlab/show/continuous+linear+map">continuous linear</a> <a class="existingWikiWord" href="/nlab/show/projection">projection</a></p> <div class="maruku-equation" id="eq:ForExtensionOfDistributionsProjectionMaps"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>ρ</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒟</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>𝒟</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> p_\rho \;\colon\; \mathcal{D}(\mathbb{R}^n) \longrightarrow \mathcal{D}_\rho(\mathbb{R}^n) </annotation></semantics></math></div> <p>may be obtained from a choice of <em>dual basis</em> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\partial_\alpha \delta_0\}</annotation></semantics></math>, hence a choice of smooth functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msup><mi>w</mi> <mi>β</mi></msup><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>}</mo></mrow> <mfrac linethickness="0"><mrow><mrow><mi>β</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>β</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow></mrow></mfrac></msub></mrow><annotation encoding="application/x-tex"> \left\{ w^\beta \in C^\infty_{cp}(\mathbb{R}^n) \right\}_{ { \beta \in \mathbb{N}^n } \atop { {\vert \beta\vert} \leq \rho } } </annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>⟨</mo><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>w</mi> <mi>β</mi></msup><mo>⟩</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msubsup><mi>δ</mi> <mi>α</mi> <mi>β</mi></msubsup><mphantom><mi>AAA</mi></mphantom><mo>⇔</mo><mphantom><mi>AAA</mi></mphantom><msub><mo>∂</mo> <mi>α</mi></msub><msup><mi>w</mi> <mi>β</mi></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msubsup><mi>δ</mi> <mi>α</mi> <mi>β</mi></msubsup><mphantom><mi>AAAA</mi></mphantom><mtext>for</mtext><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \left\langle \partial_\alpha \delta_0 \,,\, w^\beta \right\rangle \;=\; \delta_\alpha^\beta \phantom{AAA} \Leftrightarrow \phantom{AAA} \partial_\alpha w^\beta(0) \;=\; \delta_\alpha^\beta \phantom{AAAA} \text{for}\, {\vert \alpha\vert} \leq \rho \,, </annotation></semantics></math></div> <p>by setting</p> <div class="maruku-equation" id="eq:SpaceOfSmoothFunctionsOfGivenVaishingOrderProjector"><span class="maruku-eq-number">(7)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>ρ</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>id</mi><mspace width="thickmathspace"></mspace><mo>−</mo><mspace width="thickmathspace"></mspace><mrow><mo>⟨</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow></mrow></mfrac></munder><msup><mi>w</mi> <mi>α</mi></msup><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> p_\rho \;\coloneqq\; id \;-\; \left\langle \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq \rho } }{\sum} w^\alpha \partial_\alpha \delta_0 \,,\, (-) \right\rangle \,, </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>ρ</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>b</mi><mo>↦</mo><mi>b</mi><mo>−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow></mrow></mfrac></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></msup><msup><mi>w</mi> <mi>α</mi></msup><msub><mo>∂</mo> <mi>α</mi></msub><mi>b</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> p_\rho \;\colon\; b \mapsto b - \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq \rho } }{\sum} (-1)^{{\vert \alpha\vert}} w^\alpha \partial_\alpha b(0) \,. </annotation></semantics></math></div> <p>Together with <a class="existingWikiWord" href="/nlab/show/Hadamard%27s+lemma">Hadamard's lemma</a> in the form <a class="maruku-eqref" href="#eq:ForVanishingOrderRhoHadamardExpansion">(5)</a> this means that every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>𝒟</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b \in \mathcal{D}(\mathbb{R}^n)</annotation></semantics></math> is decomposed as</p> <div class="maruku-equation" id="eq:ForExtensionOfDistributionsTestFunctionDecomposition"><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><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>p</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>id</mi><mo>−</mo><msub><mi>p</mi> <mi>ρ</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mi>ρ</mi><mo>+</mo><mn>1</mn></mrow></mrow></mfrac></munder><msup><mi>x</mi> <mi>α</mi></msup><msub><mi>r</mi> <mi>α</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow></mrow></mfrac></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow></msup><msup><mi>w</mi> <mi>α</mi></msup><msub><mo>∂</mo> <mi>α</mi></msub><mi>b</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} b(x) & = p_\rho(b)(x) \;+\; (id - p_\rho)(b)(x) \\ & = \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha \vert} = \rho + 1 } }{\sum} x^\alpha r_\alpha(x) \;+\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha \vert} \leq \rho } }{\sum} (-1)^{{\vert \alpha \vert}} w^\alpha \partial_\alpha b(0) \end{aligned} </annotation></semantics></math></div> <p>Now let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>deg</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho \;\coloneqq\; deg(u) \,. </annotation></semantics></math></div> <p>Observe that (by <a href="scaling+degree+of+a+distribution#ScalingDegreeOfDistributionsBasicProperties">this prop.</a>) the <a class="existingWikiWord" href="/nlab/show/degree+of+divergence+of+a+distribution">degree of divergence</a> of the <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>α</mi></msup><mi>u</mi></mrow><annotation encoding="application/x-tex">x^\alpha u</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mi>ρ</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">{\vert \alpha\vert} = \rho + 1</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/negative+number">negative</a></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><mi>deg</mi><mrow><mo>(</mo><msup><mi>x</mi> <mi>α</mi></msup><mi>u</mi><mo>)</mo></mrow></mtd> <mtd><mo>=</mo><mi>ρ</mi><mo>−</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} deg\left( x^\alpha u \right) & = \rho - {\vert \alpha \vert} \leq -1 \end{aligned} </annotation></semantics></math></div> <p>Therefore prop. <a class="maruku-ref" href="#ExtensionUniqueNonPositiveDegreeOfDivergence"></a> says that each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>α</mi></msup><mi>u</mi></mrow><annotation encoding="application/x-tex">x^\alpha u</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mi>ρ</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">{\vert \alpha\vert} = \rho + 1</annotation></semantics></math> has a unique extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><msup><mi>x</mi> <mi>α</mi></msup><mi>u</mi></mrow><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{ x^\alpha u}</annotation></semantics></math> to the origin. Accordingly the composition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∘</mo><msub><mi>p</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">u \circ p_\rho</annotation></semantics></math> has a unique extension, by <a class="maruku-eqref" href="#eq:ForExtensionOfDistributionsTestFunctionDecomposition">(8)</a>:</p> <div class="maruku-equation" id="eq:ExtensionOfDitstributionsPointFixedAndChoice"><span class="maruku-eq-number">(9)</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><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>b</mi><mo>⟩</mo></mrow></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo>,</mo><msub><mi>p</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⟩</mo></mrow><mo>+</mo><mrow><mo>⟨</mo><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mo stretchy="false">(</mo><mi>id</mi><mo>−</mo><msub><mi>p</mi> <mi>ρ</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mi>ρ</mi><mo>+</mo><mn>1</mn></mrow></mrow></mfrac></munder><munder><munder><mrow><mo>⟨</mo><mover><mrow><msup><mi>x</mi> <mi>α</mi></msup><mi>u</mi></mrow><mo>^</mo></mover><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>r</mi> <mi>α</mi></msub><mo>⟩</mo></mrow><mo>⏟</mo></munder><mtext>unique</mtext></munder><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow></mrow></mfrac></munder><munder><munder><mrow><mo stretchy="false">⟨</mo><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>w</mi> <mi>α</mi></msup><mo stretchy="false">⟩</mo></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mrow><msup><mi>q</mi> <mi>α</mi></msup></mrow></mrow><mrow><mtext>choice</mtext></mrow></mfrac></munder><mrow><mo>⟨</mo><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>b</mi><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left\langle \hat u \,,\, b \right\rangle & = \left\langle \hat u , p_\rho(b) \right\rangle + \left\langle \hat u , (id - p_\rho)(b) \right\rangle \\ & = \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha \vert} = \rho + 1 } }{\sum} \underset{ \text{unique} }{ \underbrace{ \left\langle \widehat{x^\alpha u} \,,\, r_\alpha \right\rangle } } \;+\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq \rho } }{\sum} \underset{ { q^\alpha } \atop { \text{choice} } }{ \underbrace{ \langle \hat u \,,\, w^\alpha \rangle } } \left\langle \partial_\alpha \delta_0 \,,\, b \right\rangle \end{aligned} </annotation></semantics></math></div> <p>That says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat u</annotation></semantics></math> is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><munder><mover><mrow><mi>u</mi><mo>∘</mo><msub><mi>p</mi> <mi>ρ</mi></msub></mrow><mo>^</mo></mover><mo>⏟</mo></munder><mtext>unique</mtext></munder><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow></mrow></mfrac></munder><msup><mi>c</mi> <mi>α</mi></msup><mspace width="thinmathspace"></mspace><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> \hat u \;=\; \underset{ \text{unique} }{ \underbrace{ \widehat{ u \circ p_\rho } } } + \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq \rho } }{\sum} c^\alpha \, \partial_\alpha \delta_0 </annotation></semantics></math></div> <p>for a finite number of constants <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>c</mi> <mi>α</mi></msup><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">c^\alpha \in \mathbb{C}</annotation></semantics></math>.</p> <p>Notice that for any extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat u</annotation></semantics></math> the exact value of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>c</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">c^\alpha</annotation></semantics></math> here depends on the arbitrary choice of dual basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>w</mi> <mi>α</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{w^\alpha\}</annotation></semantics></math> used for this construction. But the uniqueness of the first summand means that for any two choices of extensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>u</mi><mo stretchy="false">^</mo></mover> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><annotation encoding="application/x-tex">{\hat u}^\prime</annotation></semantics></math>, their difference is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>u</mi><mo stretchy="false">^</mo></mover> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow></mrow></mfrac></munder><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>α</mi></msup><mo>−</mo><msup><mi>c</mi> <mi>α</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> {\hat u}^\prime - \hat u \;=\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq \rho } }{\sum} ( (c')^\alpha - c^\alpha ) \, \partial_\alpha \delta_0 \,, </annotation></semantics></math></div> <p>where the constants <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>q</mi> <mi>α</mi></msup><mo>≔</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>α</mi></msup><mo>−</mo><msup><mi>c</mi> <mi>α</mi></msup><mo stretchy="false">)</mo><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">q^\alpha \coloneqq ( (c')^\alpha - c^\alpha ) \in \mathbb{C}</annotation></semantics></math> are independent of any choices.</p> <p>It remains to see that all these <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat u</annotation></semantics></math> in fact have the same degree of divergence as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>.</p> <p>By <a href="scaling+degree+of+a+distribution#DerivativesOfDeltaDistributionScalingDegree">this example</a> the degree of divergence of the point-supported distributions on the right is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>α</mi></msub><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">deg(\partial_\alpha \delta_0) = {\vert \alpha\vert} \leq \rho</annotation></semantics></math>.</p> <p>Therefore to conclude it is now sufficient to show that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mrow><mo>(</mo><mover><mrow><mi>u</mi><mo>∘</mo><msub><mi>p</mi> <mi>ρ</mi></msub></mrow><mo>^</mo></mover><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>ρ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> deg\left( \widehat{ u \circ p_\rho } \right) \;=\; \rho \,. </annotation></semantics></math></div> <p>This is shown in (<a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00, p. 25</a>).</p> </div> <div class="num_remark" id="WExtensions"> <h6 id="remark_2">Remark</h6> <p><strong>(“W-extensions”)</strong></p> <p>Since in <a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00, (38)</a> the projectors <a class="maruku-eqref" href="#eq:SpaceOfSmoothFunctionsOfGivenVaishingOrderProjector">(7)</a> are denoted “<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>”, the construction of <a class="existingWikiWord" href="/nlab/show/extensions+of+distributions">extensions of distributions</a> via the proof of prop. <a class="maruku-ref" href="#SpaceOfPointExtensions"></a> has come to be called “W-extensions” (e.g <a href="#Duetsch18">Dütsch 18</a>).</p> </div> <p>In conclusion we obtain the central theorem of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>:</p> <div class="num_theorem" id="ExistenceRenormalization"> <h6 id="theorem">Theorem</h6> <p><strong>(existence and choices of <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> of <a class="existingWikiWord" href="/nlab/show/S-matrices">S-matrices</a>/<a class="existingWikiWord" href="/nlab/show/perturbative+QFTs">perturbative QFTs</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>, such that the underlying <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> and the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math> is translation-invariant.</p> <p>Then:</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/S-matrix+scheme">S-matrix scheme</a> <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> (<a href="S-matrix#LagrangianFieldTheoryPerturbativeScattering">this def.</a>) around this vacuum exists;</p> </li> <li> <p>for <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 <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>, the space of possible choices of <a class="existingWikiWord" href="/nlab/show/S-matrices">S-matrices</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><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><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><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"> \mathcal{S}(g S_{int} + j A) \;\in\; PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ] </annotation></semantics></math></div> <p>hence of the corresponding <a class="existingWikiWord" href="/nlab/show/perturbative+QFTs">perturbative QFTs</a>, by <a href="S-matrix#InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization">this prop.</a>, is, <a class="existingWikiWord" href="/nlab/show/induction">inductively</a> in <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>, a <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional</a> <a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, parameterizing the <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension</a> of the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> <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> to the locus of coinciding interaction points (the <a class="existingWikiWord" href="/nlab/show/fat+diagonal">fat diagonal</a>).</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>By prop. <a class="maruku-ref" href="#FeynmanPropagatorOnMinkowskiScalingDegree"></a> the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> is finite <a class="existingWikiWord" href="/nlab/show/scaling+degree+of+a+distribution">scaling degree of a distribution</a>, so that by prop. <a class="maruku-ref" href="#ScalingDegreeOfProductDistribution"></a> the binary <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> away from the diagonal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><msup><mi>Σ</mi> <mn>2</mn></msup><mo>∖</mo><mi>diag</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T_2(-,-)\vert_{\Sigma^2 \setminus diag(\Sigma)} = (-) \star_{F} (-)</annotation></semantics></math> has finite scaling degree.</p> <p>By prop. <a class="maruku-ref" href="#ScalingDegreeOfProductDistribution"></a> this implies that in the inductive description of the time-ordered products by prop. <a class="maruku-ref" href="#RenormalizationIsInductivelyExtensionToDiagonal"></a>, each induction step is the <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> of finite <a class="existingWikiWord" href="/nlab/show/scaling+degree+of+a+distribution">scaling degree of a distribution</a> to the point. By prop. <a class="maruku-ref" href="#SpaceOfPointExtensions"></a> this always exists.</p> <p>This proves the first statement.</p> <p>Now if a polynomial local interaction is fixed, then via remark <a class="maruku-ref" href="#TimeOrderedProductOfFixedInteraction"></a> each induction step involved extending a finite number of distributions, each of finite scaling degree. By prop. <a class="maruku-ref" href="#SpaceOfPointExtensions"></a> the corresponding space of choices is in each step a finite-dimensional affine space.</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> <h4 id="SPRenormalizationGroup">Stückelberg-Petermann renormalization group</h4> <p>A genuine re-normalization is the passage from one <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re"-)normalization scheme</a> <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> to another such scheme <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}'</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/induction">inductive</a> <a class="existingWikiWord" href="/nlab/show/Epstein-Glaser+renormalization">Epstein-Glaser ("re"-normalization)</a> construction (prop. <a class="maruku-ref" href="#RenormalizationIsInductivelyExtensionToDiagonal"></a>) shows that the difference between any <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}'</annotation></semantics></math> is inductively in <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> a choice of extra term in the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> of <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> factors, equivalently in the <a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> for <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> with <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> <a class="existingWikiWord" href="/nlab/show/vertices">vertices</a>, that contributes when all <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> of these vertices coincide in <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> (prop. <a class="maruku-ref" href="#SpaceOfPointExtensions"></a>).</p> <p>A natural question is whether these additional interactions that appear when several interaction vertices coincide may be absorbed into a re-definition of the original interaction <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>. Such an <em><a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></em> (def. <a class="maruku-ref" href="#InteractionVertexRedefinition"></a> below)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒵</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</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><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><mtext>higher order corrections</mtext></mrow><annotation encoding="application/x-tex"> \mathcal{Z} \;\colon\; g S_{int} + j A \;\mapsto\; g S_{int} + j A \;+\; \text{higher order corrections} </annotation></semantics></math></div> <p>should perturbatively send <a class="existingWikiWord" href="/nlab/show/local+observables">local</a> interactions to local interactions with higher order corrections.</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a></em> (theorem <a class="maruku-ref" href="#PerturbativeRenormalizationMainTheorem"></a> below) says that indeed under mild conditions every re-normalization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>↦</mo><mi>𝒮</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{S} \mapsto \mathcal{S}'</annotation></semantics></math> is induced by such an <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a> in that there exists a <em>unique</em> such redefinition <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> so that for every local interaction <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> we have that <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a> for the interaction <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> computed with the <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re"-)normalization scheme</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}'</annotation></semantics></math> equal those computed with <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> but applied to the <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">re-defined interaction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒵</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{Z}(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><mi>𝒮</mi><mo>′</mo><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>𝒮</mi><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mi>𝒵</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><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></mrow><annotation encoding="application/x-tex"> \mathcal{S}' \left( {\, \atop \,} g S_{int} + j A {\, \atop \,} \right) \;=\; \mathcal{S}\left( {\, \atop \,} \mathcal{Z}(g S_{int} + j A) {\, \atop \,} \right) \,. </annotation></semantics></math></div> <p>This means that the <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinitions">interaction vertex redefinitions</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> form a <a class="existingWikiWord" href="/nlab/show/group">group</a> under <a class="existingWikiWord" href="/nlab/show/composition">composition</a> which <a class="existingWikiWord" href="/nlab/show/action">acts</a> <a class="existingWikiWord" href="/nlab/show/transitive+action">transitively</a> and <a class="existingWikiWord" href="/nlab/show/free+action">freely</a>, hence <a class="existingWikiWord" href="/nlab/show/regular+action">regularly</a>, on the set of <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> <a class="existingWikiWord" href="/nlab/show/renormalization+schemes">("re"-)normalization schemes</a>; this is called the <em><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></em> (theorem <a class="maruku-ref" href="#PerturbativeRenormalizationMainTheorem"></a> below).</p> <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> <div class="num_defn" id="InteractionVertexRedefinition"> <h6 id="definition_4">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/perturbative+interaction+vertex+redefinition">perturbative interaction vertex redefinition</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/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (<a href="S-matrix#VacuumFree">this def.</a>).</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/perturbative+interaction+vertex+redefinition">perturbative interaction vertex redefinition</a></em> (or just <em><a class="existingWikiWord" href="/nlab/show/vertex+redefinition">vertex redefinition</a></em>, for short) is an <a class="existingWikiWord" href="/nlab/show/endofunction">endofunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒵</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</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><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"> \mathcal{Z} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle </annotation></semantics></math></div> <p>on <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> with formal parameters adjoined (<a href="S-matrix#FormalParameters">this def.</a>) such that there exists a sequence <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 \in \mathbb{N}}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/continuous+linear+functionals">continuous linear functionals</a>, symmetric in their arguments, of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><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><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow> <mrow><msubsup><mo>⊗</mo> <mrow><mi>ℂ</mi><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> <mi>k</mi></msubsup></mrow></msup><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"> \left( {\, \atop \,} LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle {\, \atop \,} \right)^{\otimes^k_{\mathbb{C}[ [ \hbar, g, j] ]}} \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle </annotation></semantics></math></div> <p>such that for all <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> the following conditions hold:</p> <ol> <li> <p>(perturbation)</p> <ol> <li> <p><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 stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mrow><mi>int</mi><mo>+</mo><mi>j</mi><mi>A</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Z_0(g S_{int + j A}) = 0</annotation></semantics></math></p> </li> <li> <p><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 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><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">Z_1(g S_{int} + j A) = g S_{int} + j A</annotation></semantics></math></p> </li> <li> <p>and</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><mi>𝒵</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> <mtd><mo>=</mo><mi>Z</mi><msub><mi>exp</mi> <mo>⊗</mo></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></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mfrac><mn>1</mn><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><msub><mi>Z</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><munder><munder><mrow><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi></mrow><mo>⏟</mo></munder><mrow><mi>k</mi><mspace width="thinmathspace"></mspace><mtext>args</mtext></mrow></munder><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathcal{Z}(g S_{int} + j A) & = Z \exp_\otimes( g S_{int} + j A ) \\ & \coloneqq \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} Z_k( \underset{ k \, \text{args} }{ \underbrace{ g S_{int} + j A , \cdots, g S_{int} + j A } } ) \end{aligned} </annotation></semantics></math></div></li> </ol> </li> <li> <p>(field independence) The <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒵</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{Z}(g S_{int} + j A)</annotation></semantics></math> depends on the <a class="existingWikiWord" href="/nlab/show/field+histories">field histories</a> only through its argument <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>, hence by the <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a>:</p> <div class="maruku-equation" id="eq:FieldIndependenceVertexRedefinition"><span class="maruku-eq-number">(10)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>δ</mi><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></mrow></mfrac><mi>𝒵</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><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>𝒵</mi><msub><mo>′</mo> <mrow><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>j</mi><mi>A</mi></mrow></msub><mrow><mo>(</mo><mfrac><mi>δ</mi><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></mrow></mfrac><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"> \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \mathcal{Z}(g S_{int} + j A) \;=\; \mathcal{Z}'_{g S_{int} + j A} \left( \frac{\delta}{\delta \mathbf{\Phi}^a(x)} (g S_{int} + j A) \right) </annotation></semantics></math></div></li> </ol> </div> <p>The following proposition should be compared to the axiom of <em><a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a></em> of the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> scheme (<a href="S-matrix#eq:CausalAdditivity">this equation</a>):</p> <div class="num_prop" id="InteractionVertexRedefinitionAdditivity"> <h6 id="proposition_6">Proposition</h6> <p><strong>(local additivity of <a class="existingWikiWord" href="/nlab/show/vertex+redefinitions">vertex redefinitions</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/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (<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>𝒵</mi></mrow><annotation encoding="application/x-tex">\mathcal{Z}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/vertex+redefinition">vertex redefinition</a> (def. <a class="maruku-ref" href="#InteractionVertexRedefinition"></a>).</p> <p>Then for all <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>O</mi> <mn>2</mn></msub><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">O_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j\rangle</annotation></semantics></math> with spacetime support denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">supp(O_i) \subset \Sigma</annotation></semantics></math> (<a href="A+first+idea+of+quantum+field+theory#SpacetimeSupport">this def.</a>) we have</p> <ol> <li> <p>(local additivity)</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><mrow><mo>(</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∩</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇒</mo><mphantom><mi>AA</mi></mphantom><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \left( supp(O_1) \cap supp(O_2) = \emptyset \right) \\ & \Rightarrow \phantom{AA} \mathcal{Z}( O_0 + O_1 + O_2) = \mathcal{Z}( O_0 + O_1 ) - \mathcal{Z}(O_0) + \mathcal{Z}(O_0 + O_2) \end{aligned} \,. </annotation></semantics></math></div></li> <li> <p>(preservation of spacetime support)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>0</mn></msub><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="thickmathspace"></mspace><mo>⊂</mo><mspace width="thickmathspace"></mspace><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> supp \left( {\, \atop \,} \mathcal{Z}(O_0 + O_1) - \mathcal{Z}(O_0) {\, \atop \,} \right) \;\subset\; supp(O_1) </annotation></semantics></math></div> <p>hence in particular</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mo>=</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> supp \left( {\, \atop \,} \mathcal{Z}(O_1) {\, \atop \,} \right) = supp(O_1) </annotation></semantics></math></div></li> </ol> </div> <p>(<a href="#Duetsch18">Dütsch 18, exercise 3.98</a>)</p> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>Under the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>LocObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo><mo>↪</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> LocObs(E_{\text{BV-BRST}}) \hookrightarrow PolyObs(E_{\text{BV-BRST}}) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> into <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> we may think of each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Z_k</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a>, as for <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> in <a href="S-matrix#NotationForTimeOrderedProductsAsGeneralizedFunctions">this remark</a>.</p> <p>Hence if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>j</mi></msub><mo>=</mo><munder><mo>∫</mo><mi>Σ</mi></munder><msubsup><mi>j</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> O_j = \underset{\Sigma}{\int} j^\infty_\Sigma( \mathbf{L}_j ) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/transgression+of+variational+differential+forms">transgression</a> of a <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{L}</annotation></semantics></math> we get</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>k</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><msub><mi>j</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>j</mi> <mi>k</mi></msub><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow></munder><munder><mo>∫</mo><mrow><msup><mi>Σ</mi> <mi>k</mi></msup></mrow></munder><mi>Z</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><msub><mi>j</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Z_k( (O_1 + O_2 + O_3) , \cdots , (O_1 + O_2 + O_3) ) = \underset{ j_1, \cdots, j_k \in \{0,1,2\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \,. </annotation></semantics></math></div> <p>Now by definition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z_k(\cdots)</annotation></semantics></math> is in the subspace of <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a>, i.e. those <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> whose <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> <a class="existingWikiWord" href="/nlab/show/distributions">distributions</a> are <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">supported</a> on the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a>, which means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>δ</mi><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></mrow></mfrac><mfrac><mi>δ</mi><mrow><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><msub><mi>Z</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>⋯</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mphantom><mi>AA</mi></mphantom><mtext>for</mtext><mphantom><mi>AA</mi></mphantom><mi>x</mi><mo>≠</mo><mi>y</mi></mrow><annotation encoding="application/x-tex"> \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \frac{\delta}{\delta \mathbf{\Phi}^b(y)} Z_{k}(\cdots) = 0 \phantom{AA} \text{for} \phantom{AA} x \neq y </annotation></semantics></math></div> <p>Together with the axiom “field independence” <a class="maruku-eqref" href="#eq:FieldIndependenceVertexRedefinition">(10)</a> this means that the support of these generalized functions in the <a class="existingWikiWord" href="/nlab/show/integrand">integrand</a> here must be on the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>=</mo><mi>⋯</mi><mo>=</mo><msub><mi>x</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_1 = \cdots = x_k</annotation></semantics></math>.</p> <p>By the assumption that the spacetime supports of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">O_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">O_2</annotation></semantics></math> are disjoint, this means that only the summands with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>j</mi> <mi>k</mi></msub><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">j_1, \cdots, j_k \in \{0,1\}</annotation></semantics></math> and those with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>j</mi> <mi>k</mi></msub><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">j_1, \cdots, j_k \in \{0,2\}</annotation></semantics></math> contribute to the above sum. Removing the overcounting of those summands where all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>j</mi> <mi>k</mi></msub><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">j_1, \cdots, j_k \in \{0\}</annotation></semantics></math> we get</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><msub><mi>Z</mi> <mi>k</mi></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>O</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><msub><mi>j</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>j</mi> <mi>k</mi></msub><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></munder><munder><mo>∫</mo><mrow><msup><mi>Σ</mi> <mi>k</mi></msup></mrow></munder><mi>Z</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><msub><mi>j</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><msub><mi>j</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>j</mi> <mi>k</mi></msub><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow></munder><munder><mo>∫</mo><mrow><msup><mi>Σ</mi> <mi>k</mi></msup></mrow></munder><mi>Z</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><msub><mi>j</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><msub><mi>j</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>j</mi> <mi>k</mi></msub><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow></munder><munder><mo>∫</mo><mrow><msup><mi>Σ</mi> <mi>k</mi></msup></mrow></munder><mi>Z</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><msub><mi>j</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Z</mi> <mi>k</mi></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>O</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mo>−</mo><msub><mi>Z</mi> <mi>k</mi></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>O</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>O</mi> <mn>0</mn></msub><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mo>+</mo><msub><mi>Z</mi> <mi>k</mi></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>O</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} & Z_k\left( {\, \atop \,} (O_1 + O_2 + O_3) , \cdots , (O_1 + O_2 + O_3) {\, \atop \,} \right) \\ & = \underset{ j_1, \cdots, j_k \in \{0,1\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \\ & \phantom{=} - \underset{ j_1, \cdots, j_k \in \{0\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \\ & \phantom{=} - \underset{ j_1, \cdots, j_k \in \{0,2\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \\ & = Z_k\left( {\, \atop \,} (O_0 + O_1), \cdots, (O_0 + O_1) {\, \atop \,}\right) - Z_k\left( {\, \atop \,} O_0, \cdots, O_0 {\, \atop \,} \right) + Z_k\left( {\, \atop \,} (O_0 + O_2), \cdots, (O_0 + O_2) {\, \atop \,} \right) \end{aligned} \,. </annotation></semantics></math></div> <p>This directly implies the claim.</p> </div> <p>As a corollary we obtain:</p> <div class="num_prop" id="CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition"> <h6 id="proposition_7">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> scheme with <a class="existingWikiWord" href="/nlab/show/vertex+redefinition">vertex redefinition</a> is again <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> scheme)</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/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (<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>𝒵</mi></mrow><annotation encoding="application/x-tex">\mathcal{Z}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/vertex+redefinition">vertex redefinition</a> (def. <a class="maruku-ref" href="#InteractionVertexRedefinition"></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>𝒮</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</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><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 stretchy="false">)</mo><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 stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle \longrightarrow PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g,j ] ] </annotation></semantics></math></div> <p>and <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> scheme (<a href="S-matrix#LagrangianFieldTheoryPerturbativeScattering">this def.</a>), the <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</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><mover><mo>⟶</mo><mi>𝒵</mi></mover><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><mover><mo>⟶</mo><mi>𝒮</mi></mover><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 stretchy="false">)</mo><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 stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S} \circ \mathcal{Z} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle \overset{\mathcal{Z}}{\longrightarrow} LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle \overset{\mathcal{S}}{\longrightarrow} PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g,j ] ] </annotation></semantics></math></div> <p>is again an <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> scheme.</p> <p>Moreover, if <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> satisfies the <a class="existingWikiWord" href="/nlab/show/renormalization+condition">renormalization condition</a> “field independence” (<a href="S-matrix#BasicConditionsRenormalization">this prop.</a>), then so does <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi></mrow><annotation encoding="application/x-tex">\mathcal{S} \circ \mathcal{Z}</annotation></semantics></math>.</p> </div> <p>(e.g <a href="#Duetsch18">Dütsch 18, theorem 3.99 (b)</a>)</p> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>It is clear that <a class="existingWikiWord" href="/nlab/show/causal+order">causal order</a> of the spacetime supports implies that they are in particular <a class="existingWikiWord" href="/nlab/show/disjoint+subset">disjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><mphantom><mi>AA</mi></mphantom><mo>⇒</mo><mphantom><mi>AA</mi></mphantom><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∩</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mo stretchy="false">)</mo></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>∅</mi><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( {\, \atop \,} supp(O_1) {\vee\!\!\!\wedge} supp(O_2) {\, \atop \,} \right) \phantom{AA} \Rightarrow \phantom{AA} \left( {\, \atop \,} supp(O_1) \cap supp(O_) \;=\; \emptyset {\, \atop \,} \right) </annotation></semantics></math></div> <p>Therefore the local additivity of <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> (prop. <a class="maruku-ref" href="#InteractionVertexRedefinitionAdditivity"></a>) and the <a class="existingWikiWord" href="/nlab/show/causal+factorization">causal factorization</a> of the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> (<a href="S-matrix#DysonCausalFactorization">this remark</a>) imply the causal factorization of the composite:</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><mi>𝒮</mi><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow></mtd> <mtd><mo>=</mo><mi>𝒮</mi><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>𝒮</mi><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><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><mi>𝒮</mi><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><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></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathcal{S} \left( {\, \atop \,} \mathcal{Z}(O_1 + O_2) {\, \atop \,} \right) & = \mathcal{S} \left( {\, \atop \,} \mathcal{Z}(O_1) + \mathcal{Z}(O_2) {\, \atop \,} \right) \\ & = \mathcal{S} \left( {\, \atop \,} \mathcal{Z}(O_1) {\, \atop \,} \right) \, \mathcal{S} \left( {\, \atop \,} \mathcal{Z}(O_2) {\, \atop \,} \right) \,. \end{aligned} </annotation></semantics></math></div> <p>But by <a href="S-matrix#CausalFactorizationAlreadyImpliesSMatrix">this prop.</a> this implies in turn <a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a> and hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi></mrow><annotation encoding="application/x-tex">\mathcal{S} \circ \mathcal{Z}</annotation></semantics></math> is itself an S-matrix scheme.</p> <p>Finally that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi></mrow><annotation encoding="application/x-tex">\mathcal{S} \circ \mathcal{Z}</annotation></semantics></math> satisfies “field indepndence” if <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> does is immediate by the <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a>, given that <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> satisfies this condition by definition.</p> </div> <div class="num_prop" id="AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition"> <h6 id="proposition_8">Proposition</h6> <p><strong>(any two <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> <a class="existingWikiWord" href="/nlab/show/renormalization+schemes">renormalization schemes</a> differ by unique <a class="existingWikiWord" href="/nlab/show/vertex+redefinition">vertex redefinition</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/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (<a href="S-matrix#VacuumFree">this def.</a>).</p> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>,</mo><mi>𝒮</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}, \mathcal{S}'</annotation></semantics></math> any two <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> schemes (<a href="S-matrix#LagrangianFieldTheoryPerturbativeScattering">this def.</a>) which both satisfy the <a class="existingWikiWord" href="/nlab/show/renormalization+condition">renormalization condition</a> “field independence”, the there exists a unique <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> (def. <a class="maruku-ref" href="#InteractionVertexRedefinition"></a>) relating them by <a class="existingWikiWord" href="/nlab/show/composition">composition</a>, i. e. such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}' \;=\; \mathcal{S} \circ \mathcal{Z} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_11">Proof</h6> <p>By applying both sides of the equation to linear combinations of local observables of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mn>1</mn></msub><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><mi>⋯</mi><mo>+</mo><msub><mi>κ</mi> <mi>k</mi></msub><msub><mi>O</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\kappa_1 O_1 + \cdots + \kappa_k O_k</annotation></semantics></math> and then taking <a class="existingWikiWord" href="/nlab/show/derivatives">derivatives</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mi>j</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\kappa_j = 0</annotation></semantics></math> (as in <a href="S-matrix#TimeOrderedProductsFromSMatrixScheme">this example</a>) we get that the equation in question implies</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><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mfrac><mrow><msup><mo>∂</mo> <mi>k</mi></msup></mrow><mrow><mo>∂</mo><msub><mi>κ</mi> <mn>1</mn></msub><mi>⋯</mi><mo>∂</mo><msub><mi>κ</mi> <mi>k</mi></msub></mrow></mfrac><mi>𝒮</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>κ</mi> <mn>1</mn></msub><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><mi>⋯</mi><mo>+</mo><msub><mi>κ</mi> <mi>k</mi></msub><msub><mi>O</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><msub><mi>κ</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>κ</mi> <mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mfrac><mrow><msup><mo>∂</mo> <mi>k</mi></msup></mrow><mrow><mo>∂</mo><msub><mi>κ</mi> <mn>1</mn></msub><mi>⋯</mi><mo>∂</mo><msub><mi>κ</mi> <mi>k</mi></msub></mrow></mfrac><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi><mo stretchy="false">(</mo><msub><mi>κ</mi> <mn>1</mn></msub><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><mi>⋯</mi><mo>+</mo><msub><mi>κ</mi> <mi>k</mi></msub><msub><mi>O</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><msub><mi>κ</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>κ</mi> <mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> (i \hbar)^k \frac{ \partial^k }{ \partial \kappa_1 \cdots \partial \kappa_k } \mathcal{S}'( \kappa_1 O_1 + \cdots + \kappa_k O_k ) \vert_{\kappa_1, \cdots, \kappa_k = 0} \;=\; (i \hbar)^k \frac{ \partial^k }{ \partial \kappa_1 \cdots \partial \kappa_k } \mathcal{S} \circ \mathcal{Z}( \kappa_1 O_1 + \cdots + \kappa_k O_k ) \vert_{\kappa_1, \cdots, \kappa_k = 0} </annotation></semantics></math></div> <p>which in components means 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><mi>T</mi><msub><mo>′</mo> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>O</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>k</mi></mrow></munder><mfrac><mn>1</mn><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mrow><mi>k</mi><mo>−</mo><mi>n</mi></mrow></msup><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mrow><msub><mi>I</mi> <mn>1</mn></msub><mo>⊔</mo><mi>⋯</mi><mo>⊔</mo><msub><mi>I</mi> <mi>n</mi></msub></mrow></mrow><mrow><mrow><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>k</mi><mo stretchy="false">}</mo><mo>,</mo></mrow></mrow></mfrac></mrow><mrow><mrow><msub><mi>I</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>I</mi> <mi>n</mi></msub><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><msub><mi>T</mi> <mi>n</mi></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>Z</mi> <mrow><mo stretchy="false">|</mo><msub><mi>I</mi> <mn>1</mn></msub><mo stretchy="false">|</mo></mrow></msub><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>O</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>I</mi> <mn>1</mn></msub></mrow></msub><mo>)</mo></mrow><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>Z</mi> <mrow><mo stretchy="false">|</mo><msub><mi>I</mi> <mi>n</mi></msub><mo stretchy="false">|</mo></mrow></msub><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>O</mi> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>∈</mo><msub><mi>I</mi> <mi>n</mi></msub></mrow></msub><mo>)</mo></mrow><mo>,</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><msub><mi>Z</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>O</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} T'_k( O_1, \cdots, O_k ) & = \underset{ 2 \leq n \leq k }{\sum} \frac{1}{n!} (i \hbar)^{k-n} \underset{ { { I_1 \sqcup \cdots \sqcup I_n } \atop { = \{1, \cdots, k\}, } } \atop { I_1, \cdots, I_n \neq \emptyset } }{\sum} T_n \left( {\, \atop \,} Z_{{\vert I_1\vert}}\left( (O_{i_1})_{i_1 \in I_1} \right), \cdots, Z_{{\vert I_n\vert}}\left( (O_{i_n})_{i_n \in I_n} \right), {\, \atop \,} \right) \\ & \phantom{=} + Z_k( O_1,\cdots, O_k ) \end{aligned} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>T</mi><msub><mo>′</mo> <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> are the <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}'</annotation></semantics></math> (by <a href="S-matrix#TimeOrderedProductsFromSMatrixScheme">this example</a>) and <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 \mathcal{N}}</annotation></semantics></math> those correspondong to <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>.</p> <p>Here the sum on the right runs over all ways that in the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi></mrow><annotation encoding="application/x-tex">\mathcal{S} \circ \mathcal{Z}</annotation></semantics></math> a <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 operation arises as the composite of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-ary time-ordered product applied to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msub><mi>I</mi> <mi>i</mi></msub><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert I_i\vert}</annotation></semantics></math>-ary components of <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>, for <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> running from 1 to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>; except for the case <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 = n</annotation></semantics></math>, which is displayed separately in the second line</p> <p>This shows that if <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> exists, then it is unique, because its coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Z_k</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/induction">inductively</a> in <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> given by the expressions</p> <div class="maruku-equation" id="eq:MainTheoremPerturbativeRenormalizationInductionStep"><span class="maruku-eq-number">(11)</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><msub><mi>Z</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>O</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>T</mi><msub><mo>′</mo> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>O</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>k</mi></mrow></munder><mfrac><mn>1</mn><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mrow><mi>k</mi><mo>−</mo><mi>n</mi></mrow></msup><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mrow><msub><mi>I</mi> <mn>1</mn></msub><mo>⊔</mo><mi>⋯</mi><mo>⊔</mo><msub><mi>I</mi> <mi>n</mi></msub></mrow></mrow><mrow><mrow><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>k</mi><mo stretchy="false">}</mo><mo>,</mo></mrow></mrow></mfrac></mrow><mrow><mrow><msub><mi>I</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>I</mi> <mi>n</mi></msub><mo>≠</mo><mi>∅</mi></mrow></mrow></mfrac></munder><msub><mi>T</mi> <mi>n</mi></msub><mrow><mo>(</mo><msub><mi>Z</mi> <mrow><mo stretchy="false">|</mo><msub><mi>I</mi> <mn>1</mn></msub><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>O</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>I</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>Z</mi> <mrow><mo stretchy="false">|</mo><msub><mi>I</mi> <mi>n</mi></msub><mo stretchy="false">|</mo></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>O</mi> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>∈</mo><msub><mi>I</mi> <mi>n</mi></msub></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mo>)</mo></mrow></mrow><mo>⏟</mo></munder><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo><</mo><mi>k</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>k</mi></msub></mrow></munder></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & Z_k( O_1,\cdots, O_k ) \\ & = T'_k( O_1, \cdots, O_k ) \;-\; \underset{ (T \circ \mathcal{Z}_{\lt k})_k }{ \underbrace{ \underset{ 2 \leq n \leq k }{\sum} \frac{1}{n!} (i \hbar)^{k-n} \underset{ { { I_1 \sqcup \cdots \sqcup I_n } \atop { = \{1, \cdots, k\}, } } \atop { I_1, \cdots, I_n \neq \emptyset } }{\sum} T_n \left( Z_{{\vert I_1\vert}}( (O_{i_1})_{i_1 \in I_1} ), \cdots, Z_{{\vert I_n\vert}}( (O_{i_n})_{i_n \in I_n} ), \right) } } \end{aligned} </annotation></semantics></math></div> <p>(The symbol under the brace is introduced as a convenient shorthand for the term above the brace.)</p> <p>Hence it remains to see that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Z_k</annotation></semantics></math> defined this way satisfy the conditions in def. <a class="maruku-ref" href="#InteractionVertexRedefinition"></a>.</p> <p>The condition “perturbation” is immediate from the corresponding condition on <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}'</annotation></semantics></math>.</p> <p>Similarly the condition “field independence” follows immediately from the assumoption that <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}'</annotation></semantics></math> satisfy this condition.</p> <p>It only remains to see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Z_k</annotation></semantics></math> indeed takes values in <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a>. Given that the <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> a priori take values in the larrger space of <a class="existingWikiWord" href="/nlab/show/microcausal+polynomial+observables">microcausal polynomial observables</a> this means to show that the spacetime support of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Z_k</annotation></semantics></math> is on the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a>.</p> <p>But observe that, as indicated in the above formula, the term over the brace may be understood as the coefficient at order <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> of the <a class="existingWikiWord" href="/nlab/show/exponential+series">exponential series</a>-expansion of the <a class="existingWikiWord" href="/nlab/show/composition">composite</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo><</mo><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{S} \circ \mathcal{Z}_{\lt k}</annotation></semantics></math>, where</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> <mrow><mo><</mo><mi>k</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>n</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><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}_{\lt k} \;\coloneqq\; \underset{ n \in \{1, \cdots, k-1\} }{\sum} \frac{1}{n!} Z_n </annotation></semantics></math></div> <p>is the truncation of the <a class="existingWikiWord" href="/nlab/show/vertex+redefinition">vertex redefinition</a> to degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo><</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\lt k</annotation></semantics></math>. This truncation is clearly itself still a vertex redefinition (according to def. <a class="maruku-ref" href="#InteractionVertexRedefinition"></a>) so that the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo><</mo><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{S} \circ \mathcal{Z}_{\lt k}</annotation></semantics></math> is still an <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> scheme (by prop. <a class="maruku-ref" href="#CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition"></a>) so that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo><</mo><mi>k</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">(T \circ \mathcal{Z}_{\lt k})_k</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> (by <a href="S-matrix#TimeOrderedProductsFromSMatrixScheme">this example</a>).</p> <p>So as we solve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo><mo>=</mo><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi></mrow><annotation encoding="application/x-tex">\mathcal{S}' = \mathcal{S} \circ \mathcal{Z}</annotation></semantics></math> inductively in degree <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>, then for the induction step in degree <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> the expressions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><msub><mo>′</mo> <mrow><mo><</mo><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">T'_{\lt k}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>∘</mo><mi>𝒵</mi><msub><mo stretchy="false">)</mo> <mrow><mo><</mo><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(T \circ \mathcal{Z})_{\lt k}</annotation></semantics></math> agree and are both time-ordered products. By <a href="S-matrix#RenormalizationIsInductivelyExtensionToDiagonal">this prop.</a> this implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><msub><mo>′</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">T'_{k}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>∘</mo><msub><mi>𝒵</mi> <mrow><mo><</mo><mi>k</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">(T \circ \mathcal{Z}_{\lt k})_{k}</annotation></semantics></math> agree away from the diagonal. This means that their difference <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Z_k</annotation></semantics></math> is supported on the diagonal, and hence is indeed local.</p> </div> <p>In conclusion this establishes the following pivotal statement of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>:</p> <div class="num_theorem" id="PerturbativeRenormalizationMainTheorem"> <h6 id="theorem_2">Theorem</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> – <a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a> of <a class="existingWikiWord" href="/nlab/show/vertex+redefinitions">vertex redefinitions</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/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (<a href="S-matrix#VacuumFree">this def.</a>).</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/vertex+redefinitions">vertex redefinitions</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> (def. <a class="maruku-ref" href="#InteractionVertexRedefinition"></a>) form a <a class="existingWikiWord" href="/nlab/show/group">group</a> under <a class="existingWikiWord" href="/nlab/show/composition">composition</a>;</p> </li> <li> <p>the set of <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> <a class="existingWikiWord" href="/nlab/show/renormalization+schemes">("re"-)normalization schemes</a> (<a href="S-matrix#LagrangianFieldTheoryPerturbativeScattering">this def.</a>, <a href="S-matrix#calSFunctionIsRenormalizationScheme">this remark</a>) satisfying the <a class="existingWikiWord" href="/nlab/show/renormalization+condition">renormalization condition</a> “field independence” (<a href="S-matrix#BasicConditionsRenormalization">this prop.</a>) is a <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a> over this group, hence equipped with a <a class="existingWikiWord" href="/nlab/show/regular+action">regular action</a> in that</p> <ol> <li> <p>the set of <a class="existingWikiWord" href="/nlab/show/S-matrix+schemes">S-matrix schemes</a> is <a class="existingWikiWord" href="/nlab/show/inhabited+set">non-empty</a>;</p> </li> <li> <p>any two <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re"-)normalization schemes</a> <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}'</annotation></semantics></math> are related by a <em>unique</em> <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> via <a class="existingWikiWord" href="/nlab/show/composition">composition</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}' \;=\; \mathcal{S} \circ \mathcal{Z} \,. </annotation></semantics></math></div></li> </ol> </li> </ol> <p>This group is called the <em><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></em>.</p> <p>Typically one imposes a set of <a class="existingWikiWord" href="/nlab/show/renormalization+conditions">renormalization conditions</a> (<a href="S-matrix#RenormalizationConditions">this def.</a>) and considers the corresponding <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of <a class="existingWikiWord" href="/nlab/show/vertex+redefinitions">vertex redefinitions</a> preserving these conditions.</p> </div> <div class="proof"> <h6 id="proof_12">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/group">group</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> and <a class="existingWikiWord" href="/nlab/show/regular+action">regular action</a> is given by prop. <a class="maruku-ref" href="#CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition"></a> and prop. <a class="maruku-ref" href="#AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition"></a>. The existence of S-matrices follows is the statement of <a class="existingWikiWord" href="/nlab/show/Epstein-Glaser+renormalization">Epstein-Glaser ("re"-)normalization</a> in theorem <a class="maruku-ref" href="#ExistenceRenormalization"></a>.</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> <h4 id="UVRegularizationViaZ">UV-Regularization via Conterterms</h4> <p>While <a class="existingWikiWord" href="/nlab/show/Epstein-Glaser+renormalization">Epstein-Glaser renormalization</a> (prop. <a class="maruku-ref" href="#RenormalizationIsInductivelyExtensionToDiagonal"></a>) gives a transparent picture on the space of choices in <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> (theorem <a class="maruku-ref" href="#ExistenceRenormalization"></a>) the physical nature of the higher interactions that it introduces at coincident interaction points (via the <a class="existingWikiWord" href="/nlab/show/extensions+of+distributions">extensions of distributions</a> in prop. <a class="maruku-ref" href="#SpaceOfPointExtensions"></a>) remains more implicit. But 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>), which re-expresses the <em>difference</em> between any two such choices as an <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a>, suggests that already the choice of <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> itself should have an incarnation in terms of <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinitions">interaction vertex redefinitions</a>.</p> <p>This may be realized via a construction of <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> in terms of <em><a class="existingWikiWord" href="/nlab/show/UV-regularization">UV-regularization</a></em> (prop. <a class="maruku-ref" href="#UVRegularization"></a> below): For any choice of “<a class="existingWikiWord" href="/nlab/show/UV-cutoff">UV-cutoff</a>”, given by an approximation of the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> <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> by <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> (def. <a class="maruku-ref" href="#CutoffsUVForPerturbativeQFT"></a> below) there is a unique “<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> induced at each cutoff scale (def. <a class="maruku-ref" href="#SMatrixEffective"></a> below). While the “UV-limit” <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>𝒮</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex">\underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda</annotation></semantics></math> does not in general exist, it may be “regularized” by applying suitable <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinitions">interaction vertex redefinitions</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>; if the higher-order corrections that these introduce serve to “<a class="existingWikiWord" href="/nlab/show/counterterms">counter</a>” (remark <a class="maruku-ref" href="#TermCounter"></a> below) the coresponding UV-divergences.</p> <p>This perspective of <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization via</a> via <em><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></em> is often regarded as the primary one. Its elegant proof in prop. <a class="maruku-ref" href="#UVRegularization"></a> below, however relies on the <a class="existingWikiWord" href="/nlab/show/Epstein-Glaser+renormalization">Epstein-Glaser renormalization</a> via inductive <a class="existingWikiWord" href="/nlab/show/extensions+of+distributions">extensions of distributions</a> and uses the same kind of argument as 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>) that establishes the <a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a>.</p> <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> <div class="num_defn" id="CutoffsUVForPerturbativeQFT"> <h6 id="definition_5">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 <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> interpolate between zero and the Feynman propagator, in that, in the <a class="existingWikiWord" href="/nlab/show/H%C3%B6rmander+topology">Hörmander topology</a>:</p> <ol> <li> <p>the <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>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> 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> </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="UV+regularization#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_6">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">(12)</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_9">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">(12)</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">(12)</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_13">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">(6)</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">(9)</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">(11)</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">(13)</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">(13)</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">(14)</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">(4)</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">(15)</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">(15)</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">(15)</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">(16)</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">(17)</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">(17)</a> into <a class="maruku-eqref" href="#eq:CountertermOrderByOrderInTermsOfCorrectionTerm">(16)</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">(4)</a> to the regularized products, we find</p> <div class="maruku-equation" id="eq:InductionStepForCounterterms"><span class="maruku-eq-number">(18)</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">(19)</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">(18)</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">(13)</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">(17)</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_3">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> <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> <h4 id="EffectiveQFTFlowWislonian">Wilson-Polchinski effective QFT flow</h4> <p>We have seen <a href="#UVRegularizationViaZ">above</a> that a choice of <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> induces <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> 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> (def. <a class="maruku-ref" href="#SMatrixEffective"></a>). To these one may associated non-local <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> below) which are such that their effective <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</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> coincide with the true scattering amplitudes of a genuine <a class="existingWikiWord" href="/nlab/show/local+observable">local</a> interaction as the cutoff is removed. This is the Wilsonian picture of <em><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></em> at a given cutoff scale (remark <a class="maruku-ref" href="#pQFTEffective"></a> below). Crucially the “flow” of the <a class="existingWikiWord" href="/nlab/show/relative+effective+actions">relative effective actions</a> with the cutoff scale satisfies a <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a> that in itself is independent of the full UV-theory; this is <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). Solving this equation for given choice of initial value data is hence another way of choosing <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> constants.</p> <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> <div class="num_prop" id="EffectiveSmatrixSchemeInvertible"> <h6 id="proposition_10">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><mphantom><mi>AA</mi></mphantom><msub><mi>𝒮</mi> <mi>Λ</mi></msub><mphantom><mi>AA</mi></mphantom></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}{\phantom{AA}\mathcal{S}_\Lambda \phantom{AA} }{\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_14">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">(20)</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_7">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">(12)</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">(21)</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">(22)</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_4">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">(12)</a> of the <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> <a class="maruku-eqref" href="#eq:RelativeEffectiveActionComposite">(21)</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_11">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>) in the sense of the 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> (<a href="S-matrix#FeynmanPerturbationSeriesAwayFromCoincidingPoints">this def.</a>) 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><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></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> <p>More generally this holds true 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><mo>⊔</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Lambda \in [0, \infty) \sqcup \{\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 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><mo>,</mo><mi>Λ</mi></mrow></msub></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>Γ</mi><mo>∈</mo><msub><mi>Γ</mi> <mi>conn</mi></msub></mrow></munder><msub><mi>Γ</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="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} S_{eff,0,\Lambda} & = \underset{\Gamma \in \Gamma_{conn}}{\sum} \Gamma_\Lambda(g S_{int} + j A) \,, \end{aligned} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</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></mrow><annotation encoding="application/x-tex">\Gamma_\Lambda( g S_{int} + j A)</annotation></semantics></math> denotes the evident version of the <a class="existingWikiWord" href="/nlab/show/Feynman+amplitude">Feynman amplitude</a> (<a href="S-matrix#FeynmanPerturbationSeriesAwayFromCoincidingPoints">this def.</a>) with <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> replaced by effective time ordered product 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> as in (def. <a class="maruku-ref" href="#SMatrixEffective"></a>).</p> </div> <p>(<a href="#Duetsch18">Dütsch 18, (3.473)</a>)</p> <div class="proof"> <h6 id="proof_15">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">(12)</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> <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> (theorem <a class="maruku-ref" href="#PerturbativeRenormalizationMainTheorem"></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 “Wilsonian RG flow” (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 Wilsonian RG flow 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 “Wilsonian RG flow” 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_5">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>.</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_12">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>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Lambda \mapsto \Delta_{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_16">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">(22)</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> <h4 id="RGFlowGeneral">Renormalization group flow</h4> <p>In <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> the construction of the <a class="existingWikiWord" href="/nlab/show/scattering+matrix">scattering matrix</a> <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>, hence of the <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a> for a 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></mrow><annotation encoding="application/x-tex">g S_{int}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbing</a> around a given <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, involves choices of <em>normalization</em> of <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>/<a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> (traditionally called <em><a class="existingWikiWord" href="/nlab/show/renormalization">"re"-normalizations</a></em>) encoding new <a class="existingWikiWord" href="/nlab/show/interactions">interactions</a> that appear where several of the original interaction vertices defined by <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></mrow><annotation encoding="application/x-tex">g S_{int}</annotation></semantics></math> coincide.</p> <p>Whenever a <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi></mrow><annotation encoding="application/x-tex">RG</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/action">acts</a> on the space of <a class="existingWikiWord" href="/nlab/show/observables">observables</a> of the theory such that <a class="existingWikiWord" href="/nlab/show/conjugation">conjugation</a> by this action takes <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re"-)normalization schemes</a> into each other, then these choices of <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> are parameterized by – or “flow with” – the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi></mrow><annotation encoding="application/x-tex">RG</annotation></semantics></math>. This is called <em>renormalization group flow</em> (prop. <a class="maruku-ref" href="#FlowRenormalizationGroup"></a> below); often called <em>RG flow</em>, for short.</p> <p>The archetypical example here is the <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi></mrow><annotation encoding="application/x-tex">RG</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> (def. <a class="maruku-ref" href="#ScalingTransformations"></a> below), which induces a <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a> (prop. <a class="maruku-ref" href="#RGFlowScalingTransformations"></a> below) due to the particular nature of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> resp. <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> (example <a class="maruku-ref" href="#ScalarFieldMassDimensionOnMinkowskiSpacetime"></a> below). In this case the choice of <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> hence “flows with scale”.</p> <p>Now the <em><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> states that (if only the basic <a class="existingWikiWord" href="/nlab/show/renormalization+condition">renormalization condition</a> called “field independence” is satisfied) any two choices of <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re"-)normalization schemes</a> <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}'</annotation></semantics></math> are related by a unique <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><mi>𝒵</mi></mrow><annotation encoding="application/x-tex">\mathcal{Z}</annotation></semantics></math>, as</em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>′</mo><mo>=</mo><mi>𝒮</mi><mo>∘</mo><mi>𝒵</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}' = \mathcal{S} \circ \mathcal{Z} \,. </annotation></semantics></math></div> <p>Applied to a parameterization/flow of renormalization choices by a group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi></mrow><annotation encoding="application/x-tex">RG</annotation></semantics></math> this hence induces an <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi></mrow><annotation encoding="application/x-tex">RG</annotation></semantics></math>. One may think of the shape of the interaction vertices as fixed and only their (<a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a>) <a class="existingWikiWord" href="/nlab/show/coupling+constants">coupling constants</a> as changing under such an <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a>, and hence then one has <a class="existingWikiWord" href="/nlab/show/coupling+constants">coupling constants</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">g_j</annotation></semantics></math> that are parameterized by elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi></mrow><annotation encoding="application/x-tex">RG</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><msub><mi>g</mi> <mi>j</mi></msub><mo stretchy="false">}</mo><mo>↦</mo><mo stretchy="false">{</mo><msub><mi>g</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \mathcal{Z}_{\rho_{vac}}^\rho \;\colon\; \{g_j\} \mapsto \{g_j(\rho)\} </annotation></semantics></math></div> <p>This dependendence is called <em>running of the coupling constants</em> under the renormalization group flow (def. <a class="maruku-ref" href="#CouplingRunning"></a> below).</p> <p>One example of <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a> is that induced by <a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a> (prop. <a class="maruku-ref" href="#RGFlowScalingTransformations"></a> below). This is the original and main example of the concept (<a href="#GellMannLow54">Gell-Mann & Low 54</a>)</p> <p>In this case the <a class="existingWikiWord" href="/nlab/show/running+of+the+coupling+constants">running of the coupling constants</a> may be understood as expressing how “more” <a class="existingWikiWord" href="/nlab/show/interactions">interactions</a> (at higher energy/shorter <a class="existingWikiWord" href="/nlab/show/wavelength">wavelength</a>) become visible (say to <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>) as the scale resolution is increased. In this case the dependence of the coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_j(\rho)</annotation></semantics></math> on the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> happens to be <a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable</a>; its <a class="existingWikiWord" href="/nlab/show/logarithm">logarithmic</a> <a class="existingWikiWord" href="/nlab/show/derivative">derivative</a> (denoted “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math>” in <a href="#GellMannLow54">Gell-Mann & Low 54</a>) is known as the <em><a class="existingWikiWord" href="/nlab/show/beta+function">beta function</a></em> (<a href="#Callan70">Callan 70</a>, <a href="#Symanzik70">Symanzik 70</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>ρ</mi><mfrac><mrow><mo>∂</mo><msub><mi>g</mi> <mi>j</mi></msub></mrow><mrow><mo>∂</mo><mi>ρ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \beta(g) \coloneqq \rho \frac{\partial g_j}{\partial \rho} \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/running+of+the+coupling+constants">running of the coupling constants</a> is not quite a <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of the <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>, but it is a “twisted” representation, namely a <a class="existingWikiWord" href="/nlab/show/group+cocycle">group 1-cocycle</a> (prop. <a class="maruku-ref" href="#CocycleRunningCoupling"></a> below). For the case of <a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a> this may be called the <em><a class="existingWikiWord" href="/nlab/show/Gell-Mann-Low+renormalization+cocycle">Gell-Mann-Low renormalization cocycle</a></em> (<a href="#BrunettiDuetschFredenhagen09">Brunetti-Dütsch-Fredenhagen 09</a>).</p> <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> <div class="num_prop" id="FlowRenormalizationGroup"> <h6 id="proposition_13">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>vac</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><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"> vac \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H ) </annotation></semantics></math></div> <p>be 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>) around which we consider <a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting</a> <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a>.</p> <p>Consider a <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi></mrow><annotation encoding="application/x-tex">RG</annotation></semantics></math> equipped with an <a class="existingWikiWord" href="/nlab/show/action">action</a> on the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> of <a class="existingWikiWord" href="/nlab/show/off-shell">off-shell</a> <a class="existingWikiWord" href="/nlab/show/microcausal+polynomial+observables">microcausal polynomial observables</a> with formal parameters adjoined (as in <a href="S-matrix#FormalParameters">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>rg</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>RG</mi><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 stretchy="false">)</mo><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 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 stretchy="false">)</mo><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><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> rg_{(-)} \;\colon\; RG \times PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ \hbar, g, j ] ] \,, </annotation></semantics></math></div> <p>hence for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">\rho \in RG</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/continuous+linear+map">continuous linear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>rg</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">rg_\rho</annotation></semantics></math> which has an <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>rg</mi> <mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">rg_\rho^{-1} \in RG</annotation></semantics></math> and is a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a>-product (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> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\star_H</annotation></semantics></math> induced by the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> of the given vauum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>vac</mi></mrow><annotation encoding="application/x-tex">vac</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>rg</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>rg</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>rg</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> rg_\rho( A_1 \star_H A_2 ) \;=\; rg_\rho(A_1) \star_H rg_\rho(A_2) </annotation></semantics></math></div> <p>such that the following conditions hold:</p> <ol> <li> <p>the action preserves the subspace of <a class="existingWikiWord" href="/nlab/show/off-shell">off-shell</a> polynomial <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a>, hence it <a class="existingWikiWord" href="/nlab/show/restriction">restricts</a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>rg</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>RG</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><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"> rg_{(-)} \;\colon\; RG \times LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle </annotation></semantics></math></div></li> <li> <p>the action respects the <a class="existingWikiWord" href="/nlab/show/causal+order">causal order</a> of the spacetime support (<a href="A+first+idea+of+quantum+field+theory#SpacetimeSupport">this def.</a>) of local observables, in that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>O</mi> <mn>2</mn></msub><mo>∈</mo><mi>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></mrow><annotation encoding="application/x-tex">O_1, O_2 \in LocObs(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><mo>(</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mspace width="thinmathspace"></mspace><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mphantom><mi>A</mi></mphantom><mo>⇒</mo><mphantom><mi>A</mi></mphantom><mrow><mo>(</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>rg</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mspace width="thinmathspace"></mspace><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>rg</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( supp(O_1) \,{\vee\!\!\!\wedge}\, supp(O_2) \right) \phantom{A} \Rightarrow \phantom{A} \left( supp(rg_\rho(O_1)) \,{\vee\!\!\!\wedge}\, supp(rg_\rho(O_2)) \right) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">\rho \in RG</annotation></semantics></math>.</p> </li> </ol> <p>Then:</p> <p>The operation of <a class="existingWikiWord" href="/nlab/show/conjugation">conjugation</a> by this action on <a class="existingWikiWord" href="/nlab/show/observables">observables</a> induces an <a class="existingWikiWord" href="/nlab/show/action">action</a> on the <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> <a class="existingWikiWord" href="/nlab/show/renormalization+schemes">renormalization schemes</a> (<a href="S-matrix#LagrangianFieldTheoryPerturbativeScattering">this def.</a>, <a href="S-matrix#calSFunctionIsRenormalizationScheme">this remark</a>), in that for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</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>⟶</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 stretchy="false">)</mo><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 stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ] </annotation></semantics></math></div> <p>a perturbative <a class="existingWikiWord" href="/nlab/show/S-matrix+scheme">S-matrix scheme</a> around the given <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>vac</mi></mrow><annotation encoding="application/x-tex">vac</annotation></semantics></math>, also the <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝒮</mi> <mi>ρ</mi></msup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>rg</mi> <mi>ρ</mi></msub><mo>∘</mo><mi>𝒮</mi><mo>∘</mo><msubsup><mi>rg</mi> <mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</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>⟶</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 stretchy="false">)</mo><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 stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}^\rho \;\coloneqq\; rg_\rho \circ \mathcal{S} \circ rg_{\rho}^{-1} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ] </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> scheme, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">\rho \in RG</annotation></semantics></math>.</p> <p>More generally, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>vac</mi> <mi>ρ</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo>′</mo> <mi>ρ</mi></msub><mo>,</mo><msub><mi>Δ</mi> <mrow><mi>H</mi><mo>,</mo><mi>ρ</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> vac_\rho \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}'_\rho, \Delta_{H,\rho} ) </annotation></semantics></math></div> <p>be a collection of <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/vacua">vacua</a> parameterized by elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">\rho \in RG</annotation></semantics></math>, all with the same underlying <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a>; and consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>rg</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">rg_\rho</annotation></semantics></math> as above, except that it is not an <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> of any <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a>, but an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> between the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a>-structures on various vacua, in that</p> <div class="maruku-equation" id="eq:IntertwiningWickProductsActionRG"><span class="maruku-eq-number">(23)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>rg</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mrow><mi>H</mi><mo>,</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>rg</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mrow><mi>H</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><msub><mi>rg</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> rg_{\rho}( A_1 \star_{H, \rho^{-1} \rho_{vac}} A_2 ) \;=\; rg_{\rho}(A_1) \star_{H, \rho_{vac}} rg_{\rho}(A_2) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">\rho, \rho_{vac} \in RG</annotation></semantics></math></p> <p>Then if</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>𝒮</mi> <mi>ρ</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>ρ</mi><mo>∈</mo><mi>RG</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \{ \mathcal{S}_{\rho} \}_{\rho \in RG} </annotation></semantics></math></div> <p>is a collection of <a class="existingWikiWord" href="/nlab/show/S-matrix+schemes">S-matrix schemes</a>, one around each of the <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/vacua">vacua</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>vac</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">vac_\rho</annotation></semantics></math>, it follows that for all pairs of group elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>vac</mi></msub><mo>,</mo><mi>ρ</mi><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">\rho_{vac}, \rho \in RG</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <div class="maruku-equation" id="eq:RGConjugateSmatrix"><span class="maruku-eq-number">(24)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>rg</mi> <mi>ρ</mi></msub><mo>∘</mo><msub><mi>𝒮</mi> <mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mo>∘</mo><msubsup><mi>rg</mi> <mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex"> \mathcal{S}_{\rho_{vac}}^\rho \;\coloneqq\; rg_\rho \circ \mathcal{S}_{\rho^{-1}\rho_{vac}} \circ rg_\rho^{-1} </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/S-matrix+scheme">S-matrix scheme</a> around the vacuum labeled by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow><annotation encoding="application/x-tex">\rho_{vac}</annotation></semantics></math>.</p> <p>Since therefore each element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">\rho \in RG</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi></mrow><annotation encoding="application/x-tex">RG</annotation></semantics></math> picks a different choice of <a class="existingWikiWord" href="/nlab/show/renormalization">normalization</a> of the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> scheme around a given vacuum at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow><annotation encoding="application/x-tex">\rho_{vac}</annotation></semantics></math>, we call the assignment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>↦</mo><msubsup><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup></mrow><annotation encoding="application/x-tex">\rho \mapsto \mathcal{S}_{\rho_{vac}}^{\rho}</annotation></semantics></math> a <em><a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">re-normalization group flow</a></em>.</p> </div> <p>(<a href="#BrunettiDuetschFredenhagen09">Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1</a>, <a href="#Duetsch18">Dütsch 18, section 3.5.3</a>)</p> <div class="proof"> <h6 id="proof_17">Proof</h6> <p>It is clear from the definition that each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{S}^{\rho}_{\rho_{vac}}</annotation></semantics></math> satisfies the axiom “perturbation” (in <a href="S-matrix#LagrangianFieldTheoryPerturbativeScattering">this def.</a>).</p> <p>In order to verify the axiom “<a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a>”, observe, for convenience, that by <a href="S-matrix#CausalFactorizationAlreadyImpliesSMatrix">this prop.</a> it is sufficient to check <a class="existingWikiWord" href="/nlab/show/causal+factorization">causal factorization</a>.</p> <p>So consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>O</mi> <mn>2</mn></msub><mo>∈</mo><mi>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">O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle</annotation></semantics></math> two local observables whose spacetime support is in <a class="existingWikiWord" href="/nlab/show/causal+order">causal order</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mspace width="thickmathspace"></mspace><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> supp(O_1) \;{\vee\!\!\!\wedge}\; supp(O_2) \,. </annotation></semantics></math></div> <p>We need to show that the</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mrow><mi>H</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><msubsup><mi>𝒮</mi> <mrow><msub><mi>vac</mi> <mi>e</mi></msub></mrow> <mi>ρ</mi></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}_{\rho_{vac}}^{\rho}(O_1 + O_2) = \mathcal{S}_{\rho_{vac}}^\rho(O_1) \star_{H,\rho_{vac}} \mathcal{S}_{vac_e}^\rho(O_2) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">\rho, \rho_{vac} \in RG</annotation></semantics></math>.</p> <p>Using the defining properties of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>rg</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">rg_{(-)}</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/causal+factorization">causal factorization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{S}_{\rho^{-1}\rho_{vac}}</annotation></semantics></math> we directly 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><msubsup><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>rg</mi> <mi>ρ</mi></msub><mo>∘</mo><msub><mi>𝒮</mi> <mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mo>∘</mo><msubsup><mi>rg</mi> <mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>rg</mi> <mi>ρ</mi></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>𝒮</mi> <mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mrow><mo>(</mo><msubsup><mi>rg</mi> <mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>rg</mi> <mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>rg</mi> <mi>ρ</mi></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mrow><mo>(</mo><msub><mi>𝒮</mi> <mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mrow><mo>(</mo><msubsup><mi>rg</mi> <mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>H</mi><mo>,</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mrow><mo>(</mo><msub><mi>𝒮</mi> <mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mrow><mo>(</mo><msubsup><mi>rg</mi> <mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>rg</mi> <mi>ρ</mi></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>𝒮</mi> <mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mrow><mo>(</mo><msub><mi>rg</mi> <mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow><msub><mo>⋆</mo> <mrow><mi>H</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><msub><mi>rg</mi> <mi>ρ</mi></msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><msub><mi>𝒮</mi> <mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mrow><mo>(</mo><msubsup><mi>rg</mi> <mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mo>⋆</mo> <mrow><mi>H</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mspace width="thinmathspace"></mspace><msubsup><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mo stretchy="false">(</mo><msub><mi>O</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathcal{S}_{\rho_{vac}}^\rho(O_1 + O_2) & = rg_\rho \circ \mathcal{S}_{\rho^{-1} \rho_{vac}} \circ rg_\rho^{-1}( O_1 + O_2 ) \\ & = rg_\rho \left( {\, \atop \,} \mathcal{S}_{\rho^{-1}\rho_{vac}} \left( rg_\rho^{-1}(O_1) + rg_\rho^{-1}(O_2) \right) {\, \atop \,} \right) \\ & = rg_\rho \left( {\, \atop \,} \left( \mathcal{S}_{\rho^{-1}\rho_{vac}}\left(rg_\rho^{-1}(O_1)\right) \right) \star_{H, \rho^{-1} \rho_{vac}} \left( \mathcal{S}_{ \rho^{-1} \rho_{vac} }\left(rg_\rho^{-1}(O_2)\right) \right) {\, \atop \,} \right) \\ & = rg_\rho \left( {\, \atop \,} \mathcal{S}_{\rho^{-1} \rho_{vac}}\left(rg_{\rho^{-1}}(O_1)\right) {\, \atop \,} \right) \star_{H, \rho_{vac}} rg_\rho \left( {\, \atop \,} \mathcal{S}_{\rho^{-1} \rho_{vac}}\left( rg_\rho^{-1}(O_2)\right) {\, \atop \,} \right) \\ & = \mathcal{S}^\rho_{\rho_{vac}}( O_1 ) \, \star_{H, \rho_{vac}} \, \mathcal{S}_{\rho_{vac}}^\rho(O_2) \,. \end{aligned} </annotation></semantics></math></div></div> <div class="num_defn" id="CouplingRunning"> <h6 id="definition_8">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a>)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>vac</mi><mo>≔</mo><msub><mi>vac</mi> <mi>e</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><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"> vac \coloneqq vac_e \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H ) </annotation></semantics></math></div> <p>be 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>) around which we consider <a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting</a> <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a>, let <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> be an <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> scheme around this vacuum and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>rg</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">rg_{(-)}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a> according to prop. <a class="maruku-ref" href="#FlowRenormalizationGroup"></a>, such that each re-normalized <a class="existingWikiWord" href="/nlab/show/S-matrix+scheme">S-matrix scheme</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒮</mi> <mi>vac</mi> <mi>ρ</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{S}_{vac}^\rho</annotation></semantics></math> satisfies the <a class="existingWikiWord" href="/nlab/show/renormalization+condition">renormalization condition</a> “field independence”.</p> <p>Then by the <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> (<a href="Stückelberg-Petermann+renormalization+group#AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition">this prop.</a>) there is for every <a class="existingWikiWord" href="/nlab/show/pair">pair</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>2</mn></msub><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">\rho_1, \rho_2 \in RG</annotation></semantics></math> a unique <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</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>⟶</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></mrow><annotation encoding="application/x-tex"> \mathcal{Z}_{\rho_{vac}}^{\rho} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ] \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ] </annotation></semantics></math></div> <p>which relates the corresponding two <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> schemes via</p> <div class="maruku-equation" id="eq:SMatrixScemesRelatedByRunningFunction"><span class="maruku-eq-number">(25)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mo>∘</mo><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}_{\rho_{vac}}^{\rho} \;=\; \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^\rho \,. </annotation></semantics></math></div> <p>If one thinks of an <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> vertex, hence a <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</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>, as specified by the (<a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a>) <a class="existingWikiWord" href="/nlab/show/coupling+constants">coupling constants</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>j</mi></msub><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">g_j \in C^\infty_{cp}(\Sigma)\langle g \rangle</annotation></semantics></math> multiplying the corresponding <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+densities">Lagrangian densities</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><mi>int</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>∈</mo><msubsup><mi>Ω</mi> <mi>Σ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{L}_{int,j} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})</annotation></semantics></math> as</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><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>j</mi></munder><msub><mi>τ</mi> <mi>Σ</mi></msub><mrow><mo>(</mo><msub><mi>g</mi> <mi>j</mi></msub><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><mi>int</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> g S_{int} \;=\; \underset{j}{\sum} \tau_\Sigma \left( g_j \mathbf{L}_{int,j} \right) </annotation></semantics></math></div> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">\tau_\Sigma</annotation></semantics></math> denotes <a class="existingWikiWord" href="/nlab/show/transgression+of+variational+differential+forms">transgression of variational differential forms</a>) then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{Z}_{\rho_1}^{\rho_2}</annotation></semantics></math> exhibits a dependency of the (<a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a>) <a class="existingWikiWord" href="/nlab/show/coupling+constants">coupling constants</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">g_j</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a> parameterized by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>. The corresponding functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mo stretchy="false">(</mo><mi>g</mi><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>g</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{Z}_{\rho_{vac}}^{\rho}(g S_{int}) \;\colon\; (g_j) \mapsto (g_j(\rho)) </annotation></semantics></math></div> <p>are then called <em><a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a></em>.</p> </div> <p>(<a href="#BrunettiDuetschFredenhagen09">Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1</a>, <a href="#Duetsch18">Dütsch 18, section 3.5.3</a>)</p> <div class="num_prop" id="CocycleRunningCoupling"> <h6 id="proposition_14">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a> are <a class="existingWikiWord" href="/nlab/show/group+cocycle">group cocycle</a> over <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>)</strong></p> <p>Consider <a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mi>ρ</mi></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>g</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{Z}_{\rho_{vac}}^{\rho} \;\colon\; (g_j) \mapsto (g_j(\rho)) </annotation></semantics></math></div> <p>as in def. <a class="maruku-ref" href="#CouplingRunning"></a>. Then for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>vac</mi></msub><mo>,</mo><msub><mi>ρ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>2</mn></msub><mo>∈</mo><mi>RG</mi></mrow><annotation encoding="application/x-tex">\rho_{vac}, \rho_1, \rho_2 \in RG</annotation></semantics></math> the following equality is satisfied by the “running functions” <a class="maruku-eqref" href="#eq:SMatrixScemesRelatedByRunningFunction">(25)</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></msubsup><mo>∘</mo><mrow><mo>(</mo><msub><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></msub><mo>∘</mo><msubsup><mi>𝒵</mi> <mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></msubsup><mo>∘</mo><msubsup><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2} \;=\; \mathcal{Z}_{\rho_{vac}}^{\rho_1} \circ \left( \sigma_{\rho_1} \circ \mathcal{Z}_{\rho^{-1} \rho_{vac}}^{\rho_2} \circ \sigma_{\rho_1}^{-1} \right) \,. </annotation></semantics></math></div></div> <p>(<a href="#BrunettiDuetschFredenhagen09">Brunetti-Dütsch-Fredenhagen 09 (69)</a>, <a href="#Duetsch18">Dütsch 18, (3.325)</a>)</p> <div class="proof"> <h6 id="proof_18">Proof</h6> <p>Directly using the definitions, 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><msub><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mo>∘</mo><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></msubsup></mtd> <mtd><mo>=</mo><msubsup><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></msub><mo>∘</mo><munder><munder><mrow><msub><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></msub><mo>∘</mo><msub><mi>𝒮</mi> <mrow><msubsup><mi>ρ</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msubsup><mi>ρ</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mo>∘</mo><msubsup><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><msubsup><mi>𝒮</mi> <mrow><msubsup><mi>ρ</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></msubsup><mo>=</mo><msub><mi>𝒮</mi> <mrow><msubsup><mi>ρ</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mo>∘</mo><msubsup><mi>𝒵</mi> <mrow><msubsup><mi>ρ</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></msubsup></mrow></munder><mo>∘</mo><msubsup><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><munder><mrow><msub><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></msub><mo>∘</mo><msub><mi>𝒮</mi> <mrow><msubsup><mi>ρ</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mo>∘</mo><mover><mover><mrow><msubsup><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></msub></mrow><mo>⏞</mo></mover><mrow><mo>=</mo><mi>id</mi></mrow></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><msub><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mo>∘</mo><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></msubsup><mo>∘</mo><msub><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></msub></mrow></munder><mo>∘</mo><msubsup><mi>𝒵</mi> <mrow><msubsup><mi>ρ</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></msubsup><mo>∘</mo><msubsup><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mo>∘</mo><msubsup><mi>𝒵</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></msubsup><mo>∘</mo><munder><mrow><msub><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></msub><mo>∘</mo><msubsup><mi>𝒵</mi> <mrow><msubsup><mi>ρ</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow> <mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></msubsup><mo>∘</mo><msubsup><mi>σ</mi> <mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><mo>⏟</mo></munder></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2} & = \mathcal{S}_{\rho_{vac}}^{\rho_1 \rho_2 } \\ & = \sigma_{\rho_1} \circ \underset{ = \mathcal{S}_{\rho_1^{-1} \rho_{vac}}^{\rho_2} = \mathcal{S}_{\rho_1^{-1} \rho_{vac}} \circ \mathcal{Z}_{\rho_1^{-1} \rho_vac}^{\rho_2} }{ \underbrace{ \sigma_{\rho_2} \circ \mathcal{S}_{\rho_2^{-1}\rho_1^{-1}\rho_{vac}} \circ \sigma_{\rho_2}^{-1} }} \circ \sigma_{\rho_1}^{-1} \\ & = \underset{ = \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^{\rho_1} \circ \sigma_{\rho_1} }{ \underbrace{ \sigma_{\rho_1} \circ \mathcal{S}_{\rho_1^{-1} \rho_{vac}} \circ \overset{ = id }{ \overbrace{ \sigma_{\rho_1}^{-1} \circ \sigma_{\rho_1} } } }} \circ \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2} \circ \sigma_{\rho_1}^{-1} \\ & = \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^{\rho_1} \circ \underbrace{ \sigma_{\rho_1} \circ \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2} \circ \sigma_{\rho_1}^{-1} } \end{aligned} </annotation></semantics></math></div> <p>This demonstrates the equation between vertex redefinitions to be shown after <a class="existingWikiWord" href="/nlab/show/composition">composition</a> with an S-matrix scheme. But by the uniqueness-clause in 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>) the composition operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒮</mi> <mrow><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}_{\rho_{vac}} \circ (-)</annotation></semantics></math> as a function from <a class="existingWikiWord" href="/nlab/show/vertex+redefinitions">vertex redefinitions</a> to S-matrix schemes is <a class="existingWikiWord" href="/nlab/show/injective+function">injective</a>. This implies the equation itself.</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> <h4 id="ScalingTransformatinRGFlow">Gell-Mann & Low RG flow</h4> <p>We discuss (prop. <a class="maruku-ref" href="#RGFlowScalingTransformations"></a> below) that, if the field species involved have well-defined <a class="existingWikiWord" href="/nlab/show/mass+dimension">mass dimension</a> (example <a class="maruku-ref" href="#ScalarFieldMassDimensionOnMinkowskiSpacetime"></a> below) then <a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> (example <a class="maruku-ref" href="#ScalingTransformations"></a> below) induce a <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a> (def. <a class="maruku-ref" href="#FlowRenormalizationGroup"></a>). This is the original and main example of <a class="existingWikiWord" href="/nlab/show/renormalization+group+flows">renormalization group flows</a> (<a href="#GellMannLow54">Gell-Mann& Low 54</a>).</p> <div class="num_example" id="ScalingTransformations"> <h6 id="example_4">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a> and <a class="existingWikiWord" href="/nlab/show/mass+dimension">mass dimension</a>)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>⟶</mo><mi>fb</mi></mover><mi>Σ</mi></mrow><annotation encoding="application/x-tex"> E \overset{fb}{\longrightarrow} \Sigma </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> which is a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</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><mo>=</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><msub><mo>≃</mo> <mi>ℝ</mi></msub><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Sigma = \mathbb{R}^{p,1} \simeq_{\mathbb{R}} \mathbb{R}^{p+1}</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\rho \in (0,\infty) \subset \mathbb{R}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/positive+real+number">positive real number</a>, write</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>Σ</mi></mtd> <mtd><mover><mo>⟶</mo><mi>ρ</mi></mover></mtd> <mtd><mi>Σ</mi></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>ρ</mi><mi>x</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Sigma &\overset{\rho}{\longrightarrow}& \Sigma \\ x &\mapsto& \rho x } </annotation></semantics></math></div> <p>for the operation of multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> using the <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p+1}</annotation></semantics></math> underlying <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>.</p> <p>By <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback</a> this acts on <a class="existingWikiWord" href="/nlab/show/field+histories">field histories</a> (<a class="existingWikiWord" href="/nlab/show/sections">sections</a> of the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a>) via</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><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>ρ</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>Φ</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>Φ</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Gamma_\Sigma(E) &\overset{\rho^\ast}{\longrightarrow}& \Gamma_\Sigma(E) \\ \Phi &\mapsto& \Phi(\rho(-)) } \,. </annotation></semantics></math></div> <p>Let then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>↦</mo><msub><mi>vac</mi> <mi>ρ</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo>′</mo> <mi>ρ</mi></msub><mo>,</mo><msub><mi>Δ</mi> <mrow><mi>H</mi><mo>,</mo><mi>ρ</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \rho \mapsto vac_\rho \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}'_{\rho}, \Delta_{H,\rho} ) </annotation></semantics></math></div> <p>be a 1-parameter collection of <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/vacua">vacua</a> on that field bundle, according to <a href="S-matrix#VacuumFree">this def.</a>, and consider a decomposition into a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi></mrow><annotation encoding="application/x-tex">Spec</annotation></semantics></math> of field species (<a href="S-matrix#VerticesAndFieldSpecies">this def.</a>) such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sp</mi><mo>∈</mo><mi>Spec</mi></mrow><annotation encoding="application/x-tex">sp \in Spec</annotation></semantics></math> the collection of <a class="existingWikiWord" href="/nlab/show/Feynman+propagators">Feynman propagators</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>ρ</mi><mo>,</mo><mi>sp</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,\rho,sp}</annotation></semantics></math> for that species <em>scales homogeneously</em> in that there exists</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>sp</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> dim(sp) \in \mathbb{R} </annotation></semantics></math></div> <p>such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> we have (using <a class="existingWikiWord" href="/nlab/show/generalized+functions">generalized functions</a>-notation)</p> <div class="maruku-equation" id="eq:FeynmanPropagatorScalingBehaviour"><span class="maruku-eq-number">(26)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ρ</mi> <mrow><mn>2</mn><mi>dim</mi><mo stretchy="false">(</mo><mi>sp</mi><mo stretchy="false">)</mo></mrow></msup><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mn>1</mn><mo stretchy="false">/</mo><mi>ρ</mi><mo>,</mo><mi>sp</mi></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>sp</mi><mo>,</mo><mi>ρ</mi><mo>=</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho^{ 2 dim(sp) } \Delta_{F, 1/\rho, sp}( \rho x ) \;=\; \Delta_{F,sp, \rho = 1}(x) \,. </annotation></semantics></math></div> <p>Typically <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> rescales a <a class="existingWikiWord" href="/nlab/show/mass">mass</a> parameter, in which case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>sp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">dim(sp)</annotation></semantics></math> is also called the <em><a class="existingWikiWord" href="/nlab/show/mass+dimension">mass dimension</a></em> of the field species <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sp</mi></mrow><annotation encoding="application/x-tex">sp</annotation></semantics></math>.</p> <p>Let finally</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><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>σ</mi> <mi>ρ</mi></msub></mrow></mover></mtd> <mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msubsup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>sp</mi> <mi>a</mi></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>sp</mi><mo stretchy="false">)</mo></mrow></msup><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ PolyObs(E) & \overset{ \sigma_\rho }{\longrightarrow} & PolyObs(E) \\ \mathbf{\Phi}_{sp}^a(x) &\mapsto& \rho^{- dim(sp)} \mathbf{\Phi}^a( \rho^{-1} x ) } </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/function">function</a> on <a class="existingWikiWord" href="/nlab/show/off-shell">off-shell</a> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> given on <a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Phi</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Phi}^a(x)</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\rho^{-1}</annotation></semantics></math> followed by multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> taken to the negative power of the <a class="existingWikiWord" href="/nlab/show/mass+dimension">mass dimension</a>, and extended from there to all <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> as an <a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a>.</p> <p>This constitutes an <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/group">group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>RG</mi><mo>≔</mo><mrow><mo>(</mo><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>,</mo><mo>⋅</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> RG \coloneqq \left( \mathbb{R}_+, \cdot \right) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/positive+real+numbers">positive real numbers</a> (under <a class="existingWikiWord" href="/nlab/show/multiplication">multiplication</a>) on <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a>, called the group of <em><a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a></em> for the given choice of field species and <a class="existingWikiWord" href="/nlab/show/mass">mass</a> parameters.</p> </div> <p>(<a href="#Duetsch18">Dütsch 18, def. 3.19</a>)</p> <div class="num_example" id="ScalarFieldMassDimensionOnMinkowskiSpacetime"> <h6 id="example_5">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/mass+dimension">mass dimension</a> of <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>)</strong></p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</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>m</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{F,m}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/real+scalar+field">real scalar field</a> on <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><mo>=</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Sigma = \mathbb{R}^{p,1}</annotation></semantics></math> for <a class="existingWikiWord" href="/nlab/show/mass">mass</a> parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">m \in (0,\infty)</annotation></semantics></math>; a <a class="existingWikiWord" href="/nlab/show/Green+function">Green function</a> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a>.</p> <p>Let the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RG \coloneqq (\mathbb{R}_+, \cdots)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><msub><mi>ℝ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\rho \in \mathbb{R}_+</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> (def. <a class="maruku-ref" href="#ScalingTransformations"></a>) act on the mass parameter by inverse multiplication</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><mi>ρ</mi><mo>,</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>m</mi></mrow></msub><mo stretchy="false">)</mo><mo>↦</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\rho , \Delta_{F,m}) \mapsto \Delta_{F,\rho^{-1}m}(\rho (-)) \,. </annotation></semantics></math></div> <p>Then we have</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><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn></mrow></msup><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta_{F,\rho^{-1}m}(\rho (-)) \;=\; \rho^{-(p+1) + 2} \Delta_{F,1}(x) </annotation></semantics></math></div> <p>and hence the corresponding <a class="existingWikiWord" href="/nlab/show/mass+dimension">mass dimension</a> (def. <a class="maruku-ref" href="#ScalingTransformations"></a>) of the <a class="existingWikiWord" href="/nlab/show/real+scalar+field">real scalar field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p,1}</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mtext>scalar field</mtext><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mn>2</mn><mo>−</mo><mn>1</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> dim(\text{scalar field}) = (p+1)/2 - 1 \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_19">Proof</h6> <p>By (<a href="Feynman+propagator#FeynmanPropagatorAsACauchyPrincipalvalue">this prop.</a>) the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> in question is given by the <a class="existingWikiWord" href="/nlab/show/Cauchy+principal+value">Cauchy principal value</a>-formula (in <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a>-notation)</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>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>x</mi> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>±</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_{F,m}(x) & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \,. \end{aligned} </annotation></semantics></math></div> <p>By applying <a class="existingWikiWord" href="/nlab/show/change+of+integration+variables">change of integration variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>↦</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi></mrow><annotation encoding="application/x-tex">k \mapsto \rho^{-1} k</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/Fourier+transform+of+distributions">Fourier transform</a> this becomes</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>Δ</mi> <mrow><mi>F</mi><mo>,</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mi>ρ</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>±</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>x</mi> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msup><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>±</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn></mrow></msup><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>x</mi> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>±</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn></mrow></msup><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_{F,\rho^{-1}m}(\rho x) & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \rho x^\mu} }{ - k_\mu k^\mu - \left( \rho^{-1} \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \\ & = \rho^{-(p+1)} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - \rho^{-2} k_\mu k^\mu - \rho^{-2} \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \\ & = \rho^{-(p+1)+2} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \\ & = \rho^{-(p+1) + 2} \Delta_{F,m}(x) \end{aligned} </annotation></semantics></math></div></div> <div class="num_prop" id="RGFlowScalingTransformations"> <h6 id="proposition_15">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a> are <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>vac</mi><mo>≔</mo><msub><mi>vac</mi> <mi>m</mi></msub><mo>≔</mo><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> <mrow><mi>H</mi><mo>,</mo><mi>m</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> vac \coloneqq vac_m \coloneqq (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_{H,m}) </annotation></semantics></math></div> <p>be 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/vacua">vacua</a> on that field bundle, according to <a href="S-matrix#VacuumFree">this def.</a> equipped with a decomposition into a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi></mrow><annotation encoding="application/x-tex">Spec</annotation></semantics></math> of field species (<a href="S-matrix#VerticesAndFieldSpecies">this def.</a>) such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sp</mi><mo>∈</mo><mi>Spec</mi></mrow><annotation encoding="application/x-tex">sp \in Spec</annotation></semantics></math> the collection of <a class="existingWikiWord" href="/nlab/show/Feynman+propagators">Feynman propagators</a> the corresponding field species has a well-defined <a class="existingWikiWord" href="/nlab/show/mass+dimension">mass dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>sp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">dim(sp)</annotation></semantics></math> (def. <a class="maruku-ref" href="#ScalingTransformations"></a>)</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RG</mi><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RG \coloneqq (\mathbb{R}_+, \cdot)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a> (def. <a class="maruku-ref" href="#ScalingTransformations"></a>) is a <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a> in the sense of <a href="renormalization+group+flow#FlowRenormalizationGroup">this prop.</a>.</p> </div> <p>(<a href="#Duetsch18">Dütsch 18, exercise 3.20</a>)</p> <div class="proof"> <h6 id="proof_20">Proof</h6> <p>It is clear that rescaling preserves <a class="existingWikiWord" href="/nlab/show/causal+order">causal order</a> and the <a class="existingWikiWord" href="/nlab/show/renormalization+condition">renormalization condition</a> of “field indepencen”.</p> <p>The condition we need to check is that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><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">A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/microcausal+polynomial+observables">microcausal polynomial observables</a> we have for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub><mo>∈</mo><msub><mi>ℝ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\rho, \rho_{vac} \in \mathbb{R}_+</annotation></semantics></math> that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>ρ</mi></msub><mrow><mo>(</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mrow><mi>H</mi><mo>,</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub><mi>c</mi></mrow></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>σ</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mrow><mi>H</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub></mrow></msub><msub><mi>σ</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sigma_\rho \left( A_1 \star_{H, \rho^{-1} \rho_{vac} c} A_2 \right) \;=\; \sigma_\rho(A_1) \star_{H,\rho_{vac}} \sigma_\rho(A_2) \,. </annotation></semantics></math></div> <p>By the assumption of decomposition into free field species <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sp</mi><mo>∈</mo><mi>Spec</mi></mrow><annotation encoding="application/x-tex">sp \in Spec</annotation></semantics></math>, it is sufficient to check this for each species <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>H</mi><mo>,</mo><mi>sp</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{H,sp}</annotation></semantics></math>. Moreover, by the nature of the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> on <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a>, which is given by iterated contractions with the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a>, it is sufficient to check this for one such contraction.</p> <p>Observe that the scaling behaviour of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>H</mi><mo>,</mo><mi>m</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{H,m}</annotation></semantics></math> is the same as the behaviour <a class="maruku-eqref" href="#eq:FeynmanPropagatorScalingBehaviour">(26)</a> of the correspponding <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a>. With this we directly 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><msub><mi>σ</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub><mi>m</mi></mrow></msub><msub><mi>σ</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn><mi>dim</mi></mrow></msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub><mi>m</mi></mrow></msub><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn><mi>dim</mi></mrow></msup><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><msub><mi>ρ</mi> <mi>vac</mi></msub><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>rg</mi> <mi>ρ</mi></msub><mrow><mo>(</mo><msub><mi>Δ</mi> <mrow><mi>F</mi><mo>,</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>rg</mi> <mi>ρ</mi></msub><mrow><mo>(</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mrow><mi>F</mi><mo>,</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>ρ</mi> <mi>vac</mi></msub><mi>m</mi></mrow></msub><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</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} \sigma_\rho (\mathbf{\Phi}(x)) \star_{F, \rho_{vac} m} \sigma_\rho (\mathbf{\Phi}(y) & = \rho^{-2 dim } \mathbf{\Phi}(\rho^{-1} x) \star_{F, \rho_{vac} m} \mathbf{\Phi}(\rho^{-1} y) \\ & = \rho^{-2 dim } \Delta_{F, \rho_{vac} m}(\rho^{-1}(x-y)) \\ & = \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1} \\ & = rg_{\rho}\left( \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1} \right) \\ & = rg_{\rho} \left( \mathbf{\Phi}(x) \star_{F, \rho^{-1} \rho_{vac} m} \mathbf{\Phi}(y) \right) \end{aligned} \,. </annotation></semantics></math></div></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> <h4 id="dimensional_regularization">Dimensional regularization</h4> <p>Discussion of renormalization via <a class="existingWikiWord" href="/nlab/show/dimensional+regularization">dimensional regularization</a> in the rigorous context of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> is due to <a href="#DuetschFredenhagenKellerRejzner14">Dütsch-Fredenhagen-Keller-Rejzner 14, section 4</a>.</p> <h4 id="conneskreimer_renormalization">Connes-Kreimer renormalization</h4> <p>Discussion of the Connes-Kreimer Hopf algebraic renormalization in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> is in <a href="#DuetschFredenhagenKellerRejzner14">Dütsch-Fredenhagen-Keller-Rejzner 14, section 5</a>.</p> <p>(…)</p> <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="BPHZRenormalization">BPHZ and Hopf-algebraic renormalization</h3> <p><strong>The phenomenon</strong></p> <p>In the study of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> one is concerned with functions – called <em>amplitudes</em> – that take a collection of graphs – called <a class="existingWikiWord" href="/nlab/show/Feynman+graph">Feynman graph</a>s – to <a class="existingWikiWord" href="/nlab/show/Laurent+polynomial">Laurent polynomial</a>s in a complex variable <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">z</annotation></semantics></math> – called the <strong>(dimensional) <a class="existingWikiWord" href="/nlab/show/regularization+%28physics%29">regularization</a> parameter</strong> –</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Amplitude</mi><mo>:</mo><mi>CertainGraphs</mi><mo>→</mo><mi>LaurentPolynomials</mi></mrow><annotation encoding="application/x-tex"> Amplitude : CertainGraphs \to LaurentPolynomials </annotation></semantics></math></div> <p>and wishes to extract a “meaningful” finite component when evaluated at vanishing regularization parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">z = 0</annotation></semantics></math>.</p> <p>A prescription – called <strong>renormalization scheme</strong> – for adding to a given amplitude in a certain recursive fashion further terms – called <strong>counterterms</strong> – such that the resulting modified amplitude – called the <strong>renormalized amplitude</strong> – is finite at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">z=0</annotation></semantics></math> was once given by physicists and is called the <strong>BPHZ-procedure</strong> .</p> <p>This procedure justifies itself mainly through the remarkable fact that the numbers obtained from it match certain numbers measured in particle accelerators to fantastic accuracy.</p> <p><strong>Its combinatorial Hopf-algebraic interpretation</strong></p> <p>The combinatorial Hopf algebraic approach to perturbative quantum field theory, see for instance</p> <ul> <li>Hector Figueroa, <a class="existingWikiWord" href="/nlab/show/Jose+Gracia-Bondia">Jose Gracia-Bondia</a>, <em>Combinatorial Hopf algebras in quantum field theory I</em> (<a href="http://arxiv.org/abs/hep-th/0408145">arXiv</a>),</li> </ul> <p>starts with the observation that the BPHZ-procedure can be understood</p> <ul> <li> <p>by noticing that there is secretly a natural <a class="existingWikiWord" href="/nlab/show/group">group</a> structure on the collection of amplitudes;</p> </li> <li> <p>which is induced from the fact that there is secretly a natural <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> structure on the vector space whose basis consists of graphs;</p> </li> <li> <p>and with respect to which the BPHZ-procedure is simply the <a class="existingWikiWord" href="/nlab/show/Birkhoff+decomposition">Birkhoff decomposition</a> of group valued functions on the circle into a divergent and a finite part.</p> </li> </ul> <p>The Hopf algebra structure on the vector space whose basis consists of graphs can be understood most conceptually in terms of <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebras</a>.</p> <p><strong>The Connes-Kreimer theorem</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/Birkhoff+decomposition">Birkhoff decomposition</a> of a loop <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\phi : S^1 \to G</annotation></semantics></math> in a complex group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a continuation of the loop to</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/holomorphic+function">holomorphic function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\phi_+</annotation></semantics></math> on the standard disk inside the circle;</p> </li> <li> <p>a holomorphic function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">\phi_-</annotation></semantics></math> on the complement of this disk in the projective <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a></p> </li> <li> <p>such that on the unit circle the original loop is reproduced as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>=</mo><msub><mi>ϕ</mi> <mo>+</mo></msub><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mo>−</mo></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \phi = \phi_+ \cdot (\phi_-)^{-1} \,, </annotation></semantics></math></div> <p>with the product and the inverse on the right taken in the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>Notice that by the assumption of holomorphicity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi_+(0)</annotation></semantics></math> is a well defined element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> </li> </ul> <div class="num_theorem"> <h6 id="theorem_3">Theorem</h6> <p><strong>(Connes-Kreimer)</strong></p> <ol> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is the group of <a class="existingWikiWord" href="/nlab/show/character">character</a>s on any graded connected commutative <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>H</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G = Hom(H,\mathbb{C}) </annotation></semantics></math></div> <p>then the Birkhoff decomposition always exists and is given by the formula</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mo>−</mo></msub><mo>:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∈</mo><mi>H</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>Counit</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>−</mo><mi>PolePartOf</mi><mo stretchy="false">(</mo><mi>Product</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mo>−</mo></msub><mo>⊗</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mn>1</mn><mo>⊗</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>Counit</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>Coproduct</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi_- : (X \in H) \mapsto Counit(X) - PolePartOf( Product(\phi_- \otimes \phi) \circ (1 \otimes (1 - Counit)) \circ Coproduct (X) ) \,. </annotation></semantics></math></div></li> <li> <p>There is naturally the structure of a <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><mi>Graphs</mi></mrow><annotation encoding="application/x-tex">H = Graphs</annotation></semantics></math>, on the graphs considered in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. As an algebra this is the free commutative algebra on the “1-particle irreducible graphs”. Hence QFT amplitudes can be regarded as characters on this Hopf algebra.</p> </li> <li> <p>The BPHZ renormalization-procedure for amplitudes is nothing but the first item applied to the special case of the second item.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_21">Proof</h6> <p>The proof is given in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a>, <em>Renormalization in quantum field theory I</em> (<a href="http://arxiv.org/abs/hep-th/9912092">arXiv</a>)</li> </ul> </div> <p><strong>The Hopf-algebra perspective on QFT</strong></p> <p>This result first of all makes <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> an organizational principle for (re-)expressing familiar operations in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>.</p> <p>Computing the renormalization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\phi_+</annotation></semantics></math> of an amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> amounts to using the above formula to compute the counterterm <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">\phi_-</annotation></semantics></math> and then evaluating the right hand side of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><munder><mrow><msub><mi>ϕ</mi> <mo>+</mo></msub></mrow><mo>⏟</mo></munder> <mrow><mi>renormalized</mi><mi>amplitude</mi></mrow></msub><mo>=</mo><msub><munder><mi>ϕ</mi><mo>⏟</mo></munder> <mi>amplitude</mi></msub><msub><munder><mo>⋅</mo><mo>⏟</mo></munder> <mrow><mi>convolution</mi><mi>product</mi></mrow></msub><msub><munder><mrow><msub><mi>ϕ</mi> <mo>−</mo></msub></mrow><mo>⏟</mo></munder> <mi>counterterm</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \underbrace{\phi_+}_{renormalized amplitude} = \underbrace{\phi}_{amplitude} \underbrace{\cdot}_{convolution product} \underbrace{\phi_-}_{counterterm} \,, </annotation></semantics></math></div> <p>where the product is the group product on characters, hence the <a class="existingWikiWord" href="/nlab/show/convolution+product">convolution product</a> of characters.</p> <p>Every elegant reformulation has in it the potential of going beyond mere reformulation by allowing to see structures invisible in a less natural formulation. For instance <a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a> <a href="http://golem.ph.utexas.edu/category/2007/03/recent_developments_in_quantum.html#c010903">claims</a> that the Hopf algebra language allows him to see patterns in perturbative quantum gravity previously missed.</p> <p><strong>Gauge theory and BV-BRST with Hopf algebra</strong></p> <p>Walter von Suijlekom is thinking about the Hopf-algebraic formulation of <a class="existingWikiWord" href="/nlab/show/BV+theory">BRST-BV methods</a> in nonabelian <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <p>In his nicely readable</p> <ul> <li><span class="newWikiWord">Walter von Suijlekom<a href="/nlab/new/Walter+von+Suijlekom">?</a></span>, <em>Renormalization of gauge fields using Hopf algebra</em>, (<a href="http://arxiv.org/abs/0801.3170">arXiv</a>)</li> </ul> <p>he reviews the central idea: the <a class="existingWikiWord" href="/nlab/show/BRST">BRST</a> formulation of <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> manifests itself at the level of the resulting <em>bare</em> i.e. unnormalized amplitudes in certain relations satisfied by these, the <strong>Slavnov-Taylor identities</strong> .</p> <p>Renormalization of gauge theories is consistent only if these relations are still respected by renormalized amplitudes, too. We can reformulate this in terms of Hopf algebra now:</p> <p>the relations between amplitudes to be preserved under renormalization must define a <a class="existingWikiWord" href="/nlab/show/Hopf+ideal">Hopf ideal</a> in the Hopf algebra of graphs.</p> <p>Walter von Suijlekom proves this to be the case for Slavnov-Taylor in his <a href="http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.3170v1.pdf#page=12">theorem 9 on p. 12</a></p> <p>As a payoff, he obtains a very transparent way to prove the generalization of <strong>Dyson’s formula</strong> to nonabelian gauge theory, which expresses renormalized Green’s functions in terms of unrenormalized Green’s functions “at bare coupling”. This is his <a href="http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.3170v1.pdf#page=12">corollary 12 on p. 13</a>.</p> <p>In the context of <a class="existingWikiWord" href="/nlab/show/BV+theory">BRST-BV quantization</a> these statements are subsumed, he says, by the structure encoded in the Hopf ideal which corresponds to imposing the BV-master equation. See also (<a href="#Suijlekom">Suijlekom</a>).</p> <h3 id="lattice_renormalization">Lattice renormalization</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/lattice+renormalization">lattice renormalization</a></li> </ul> <h3 id="OfTheoriesInBVForm">Of theories in BV-CS form</h3> <p>In (<a href="#Costello">Costello 07</a>) a renormalization procedure is discussed that applies to <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theories</a> that are given by <a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a> which can be given in the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>ϕ</mi><mo>,</mo><mi>Q</mi><mi>ϕ</mi><mo stretchy="false">⟩</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S(\phi) = \langle \phi , Q \phi \rangle + I(\phi) </annotation></semantics></math></div> <p>where</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> are sections of a graded <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> on which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/differential">differential</a>, <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>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,-\rangle</annotation></semantics></math> a compatible <a class="existingWikiWord" href="/nlab/show/antibracket">antibracket</a> pairing such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mo stretchy="false">⟨</mo><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,Q, \langle \rangle)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a> (as discussed there) in <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a>;</p> </li> <li> <p><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> is an <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> that is at least cubic.</p> </li> </ol> <p>These are action functionals that are well adapted to <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> and for which there is a <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> to a <a class="existingWikiWord" href="/nlab/show/factorization+algebra+of+observables">factorization algebra of observables</a>.</p> <p>Most of the fundamental <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theories</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> are of this form, notably <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>. In particular also all theories of <a class="existingWikiWord" href="/schreiber/show/infinity-Chern-Simons+theory">infinity-Chern-Simons theory</a>-type coming from binary <a class="existingWikiWord" href="/nlab/show/invariant+polynomials">invariant polynomials</a> are perturbatively of this form, notably ordinary 3d <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>.</p> <p>For a discussion of just the simple special case of 3d CS see (<a href="#Costello11">Costello 11, chapter 5.4 and 5.14</a>).</p> <p>For comparison of the following with other renormalization schemes, see at (<a href="#Costello">Costello 07, section 1.7</a>).</p> <ol> <li> <p><a href="#TheSetup">The setup</a></p> </li> <li> <p><a href="#OperatorKernelsAndPropagators">Operator (heat) kernels and propagators</a></p> </li> <li> <p><a href="#TheRenormalizationGroupOperator">The renormalization group operator</a></p> </li> <li> <p><a href="#ThePathIntegral">The path integral</a></p> </li> <li> <p><a href="#RenormalizedAction">Renormalized action</a></p> </li> <li> <p><a href="#Renormalization">Renormalization</a></p> </li> </ol> <p id="TheSetup"><strong>The setup</strong></p> <div class="num_defn"> <h6 id="definition_9">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mo stretchy="false">⟨</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mo>.</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \langle, -,-\rangle. Q)</annotation></semantics></math>:</p> <p><strong><a class="existingWikiWord" href="/nlab/show/kinematics">kinematics</a></strong>:</p> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>”/“<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>”);</p> </li> <li> <p>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">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+object">graded</a> <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex</a> <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> (the “<a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a>” containing also in general <a class="existingWikiWord" href="/nlab/show/antifields">antifields</a> and <a class="existingWikiWord" href="/nlab/show/ghosts">ghosts</a>);</p> </li> <li> <p>equipped with a <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> (the “<a class="existingWikiWord" href="/nlab/show/antibracket">antibracket</a> <a class="existingWikiWord" href="/nlab/show/density">density</a>”)</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mi>loc</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>E</mi><mo>×</mo><mi>E</mi><mo>→</mo><msub><mi>Dens</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex"> \langle -,-\rangle_{loc} \;\colon\; E \times E \to Dens_X </annotation></semantics></math></div> <p>from the fiberwise <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> with itself to the compex <a class="existingWikiWord" href="/nlab/show/density+bundle">density bundle</a> which is fiberwise</p> <ul> <li> <p>non-degenerate</p> </li> <li> <p>anti-symmetric</p> </li> <li> <p>of degree -1</p> </li> </ul> </li> </ul> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℰ</mi> <mi>c</mi></msub><mo>≔</mo><msub><mi>Γ</mi> <mi>cp</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{E}_c \coloneqq \Gamma_{cp}(E)</annotation></semantics></math> for the space of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> of <a class="existingWikiWord" href="/nlab/show/compact+support">compact support</a>. Write</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>ℰ</mi> <mi>c</mi></msub><mo>⊗</mo><msub><mi>ℰ</mi> <mi>c</mi></msub><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \langle -,-\rangle \;\colon\; \mathcal{E}_c \otimes \mathcal{E}_c \to \mathbb{C} </annotation></semantics></math></div> <p>for the induced pairing on sections</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><mi>ϕ</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">⟩</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></msub><mo stretchy="false">⟨</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">⟩</mo> <mi>loc</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle \phi, \psi\rangle = \int_{x \in X} \langle \phi(x), \psi(x)\rangle_{loc} \,. </annotation></semantics></math></div> <p>The paring being non-degenerate means that we have an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>E</mi> <mo>*</mo></msup><mo>⊗</mo><msub><mi>Dens</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">E \stackrel{\simeq}{\to} E^* \otimes Dens_X</annotation></semantics></math> and we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>!</mo></msup><mo>≔</mo><msup><mi>E</mi> <mo>*</mo></msup><mo>⊗</mo><msub><mi>Dens</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^! \coloneqq E^* \otimes Dens_X \,. </annotation></semantics></math></div> <p><strong><a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a></strong></p> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a> on sections of the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℰ</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> Q \;\colon\; \mathcal{E} \to \mathcal{C} </annotation></semantics></math></div> <p>of degree 1 such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℰ</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{E}, Q)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/elliptic+complex">elliptic complex</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint</a> with respect to <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>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,-\rangle</annotation></semantics></math> in that for all <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>,</mo><mi>ψ</mi><mo>∈</mo><msub><mi>ℰ</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\phi,\psi \in \mathcal{E}_c</annotation></semantics></math> of homogeneous degree we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>ϕ</mi><mo>,</mo><mi>Q</mi><mi>ψ</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>ϕ</mi><mo stretchy="false">|</mo></mrow></msup><mo stretchy="false">⟨</mo><mi>Q</mi><mi>ϕ</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle \phi , Q \psi\rangle = (-1)^{{\vert \phi\vert}} \langle Q \phi, \psi\rangle</annotation></semantics></math>.</p> </li> </ol> </li> </ul> <p>From this data we obtain:</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo lspace="verythinmathspace">:</mo><msub><mi>ℰ</mi> <mi>c</mi></msub><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">S \colon \mathcal{E}_c \to \mathbb{C}</annotation></semantics></math> of this corresponding free field theory is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ϕ</mi><mo>↦</mo><msub><mo>∫</mo> <mi>X</mi></msub><mo stretchy="false">⟨</mo><mi>ϕ</mi><mo>,</mo><mi>Q</mi><mi>ϕ</mi><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S \;\colon\; \phi \mapsto \int_X \langle \phi, Q \phi\rangle \,. </annotation></semantics></math></div></li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/classical+BV-complex">classical BV-complex</a> is the <a class="existingWikiWord" href="/nlab/show/symmetric+algebra">symmetric algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sym</mi><msup><mi>ℰ</mi> <mo>!</mo></msup></mrow><annotation encoding="application/x-tex">Sym \mathcal{E}^!</annotation></semantics></math> of sections of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>!</mo></msup></mrow><annotation encoding="application/x-tex">E^!</annotation></semantics></math> equipped with the induced action of the differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> and the pairing</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><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">}</mo><mo>≔</mo><msub><mo>∫</mo> <mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></msub><mo stretchy="false">⟨</mo><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>β</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{\alpha,\beta\} \coloneqq \int_{x \in X} \langle \alpha(x), \beta(x)\rangle \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_defn"> <h6 id="definition_10">Definition</h6> <p>An <strong><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></strong> term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∈</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex">I \in \cdots</annotation></semantics></math></p> <p>(…)</p> </div> <div class="num_defn" id="LaplaceOperatorFromGaugeFixing"> <h6 id="definition_11">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/gauge+fixing+operator">gauge fixing operator</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Q</mi> <mi>GF</mi></msup></mrow><annotation encoding="application/x-tex">Q^{GF}</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><mi>H</mi><mo>≔</mo><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><msup><mi>Q</mi> <mi>GF</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> H \coloneqq [Q, Q^{GF}] </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/generalized+Laplace+operator">generalized Laplace operator</a>.</p> </div> <p id="OperatorKernelsAndPropagators"><strong>Operator (heat) kernels and propagators</strong></p> <div class="num_defn" id="ConvolutionOfOperatorKernel"> <h6 id="definition_12">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>K</mi><mo>′</mo><mo>⊗</mo><mi>K</mi><mo>″</mo><mo>∈</mo><mi>ℰ</mi><mo>⊗</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">K = \sum K' \otimes K'' \in \mathcal{E} \otimes \mathcal{E}</annotation></semantics></math> the corresponding <strong>convolution operator</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⋆</mo><mo lspace="verythinmathspace">:</mo><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">K \star \colon \mathcal{E} \to \mathcal{E}</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⋆</mo><mi>e</mi><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>e</mi><mo stretchy="false">|</mo></mrow></msup><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>K</mi><mo>′</mo><mo>⊗</mo><mo stretchy="false">⟨</mo><mi>K</mi><mo>″</mo><mo>,</mo><mi>e</mi><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K \star e = (-1)^{\vert e\vert} \sum K' \otimes \langle K'', e\rangle \,. </annotation></semantics></math></div></div> <div class="num_defn" id="HeatKernel"> <h6 id="definition_13">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo lspace="verythinmathspace">:</mo><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">H \colon \mathcal{E} \to \mathcal{E}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a>, a <strong><a class="existingWikiWord" href="/nlab/show/heat+kernel">heat kernel</a></strong> for it is a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo lspace="verythinmathspace">:</mo><msub><mi>ℝ</mi> <mrow><mo>></mo><mn>0</mn></mrow></msub><mo>→</mo><mi>ℰ</mi><mo>⊗</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">K_{(-)} \colon \mathbb{R}_{\gt 0} \to \mathcal{E}\otimes \mathcal{E}</annotation></semantics></math> such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo>></mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">t \in \mathbb{R}_{\gt 0}</annotation></semantics></math> the convolution with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">K_t</annotation></semantics></math>, def. <a class="maruku-ref" href="#ConvolutionOfOperatorKernel"></a>, reproduces the exponential of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo lspace="0em" rspace="thinmathspace">cdotH</mo></mrow><annotation encoding="application/x-tex">t\cdotH</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>K</mi> <mi>t</mi></msub><mo>⋆</mo><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>t</mi><mi>H</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><mi>t</mi><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo>></mo><mn>0</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K_t \star = \exp(-t H) \;\;\;\;\; \forall t \in \mathbb{R}_{\gt 0} \,. </annotation></semantics></math></div></div> <div class="num_prop" id="UniquenessOfHeatKernel"> <h6 id="proposition_16">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/generalized+Laplace+operator">generalized Laplace operator</a> such as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><msup><mi>Q</mi> <mi>GF</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[Q,Q^{GF}]</annotation></semantics></math> of def. <a class="maruku-ref" href="#LaplaceOperatorFromGaugeFixing"></a> there is a unique <a class="existingWikiWord" href="/nlab/show/heat+kernel">heat kernel</a> which is moreover a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> of <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">t</annotation></semantics></math>.</p> </div> <div class="num_defn" id="DifferentialOperatorByKernel"> <h6 id="definition_14">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>=</mo><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>ϕ</mi><mo>′</mo><mo>⊗</mo><mi>ϕ</mi><mo>″</mo><mo>∈</mo><msup><mi>Sym</mi> <mn>2</mn></msup><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\phi = \sum \phi' \otimes \phi'' \in Sym^2 \mathcal{E}</annotation></semantics></math> write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>ϕ</mi></msub><mo>≔</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><msub><mo>∂</mo> <mrow><mi>ϕ</mi><mo>″</mo></mrow></msub><msub><mo>∂</mo> <mrow><mi>ϕ</mi><mo>′</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial_{\phi} \coloneqq \frac{1}{2} \sum \partial_{\phi''} \partial_{\phi'} \,. </annotation></semantics></math></div></div> <p>This is (<a href="#Costello">Costello 07, p. 32</a>).</p> <div class="num_prop"> <h6 id="proposition_17">Proposition</h6> <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><mo>∂</mo> <mi>ϕ</mi></msub><mo>,</mo><mi>Q</mi><mo stretchy="false">]</mo><mo>=</mo><msub><mo>∂</mo> <mrow><mi>Q</mi><mi>ϕ</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\partial_\phi, Q] = \partial_{Q \phi} \,. </annotation></semantics></math></div></div> <p id="TheRenormalizationGroupOperator"><strong>The renormalization group operator</strong></p> <div class="num_defn"> <h6 id="definition_15">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Q</mi> <mi>GF</mi></msup></mrow><annotation encoding="application/x-tex">Q^{GF}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/gauge+fixing+operator">gauge fixing operator</a> of def. <a class="maruku-ref" href="#LaplaceOperatorFromGaugeFixing"></a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">K_t</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/heat+kernel">heat kernel</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/generalized+Laplace+operator">generalized Laplace operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><msup><mi>Q</mi> <mi>GF</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">H = [Q, Q^{GF}]</annotation></semantics></math> by prop. <a class="maruku-ref" href="#UniquenessOfHeatKernel"></a>, write for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>,</mo><mi>T</mi><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo>></mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\epsilon, T \in \mathbb{R}_{\gt 0}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>≔</mo><msubsup><mo>∫</mo> <mi>ϵ</mi> <mi>T</mi></msubsup><mo stretchy="false">(</mo><msup><mi>Q</mi> <mi>GF</mi></msup><mo>⊗</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>K</mi> <mi>t</mi></msub><mspace width="thickmathspace"></mspace><mi>d</mi><mi>t</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>ℰ</mi><mo>⊗</mo><mi>ℰ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> P(\epsilon, T) \coloneqq \int_{\epsilon}^T (Q^{GF} \otimes 1) K_t \; d t \;\;\;\;\; \in \mathcal{E} \otimes \mathcal{E}\,. </annotation></semantics></math></div></div> <p>(<a href="#Costello">Costello 07, p. 33</a>)</p> <div class="num_defn" id="RenormalizationOperator"> <h6 id="definition_16">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>ℏ</mi><mi>log</mi><mrow><mo>(</mo><mi>exp</mi><mrow><mo>(</mo><mi>ℏ</mi><msub><mo>∂</mo> <mrow><mi>P</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow><mi>exp</mi><mrow><mo>(</mo><mi>I</mi><mo stretchy="false">/</mo><mi>ℏ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>ℰ</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Gamma(P(\epsilon,T), S) \coloneqq \hbar log \left( \exp\left(\hbar \partial_{P(\epsilon,T)}\right) \exp\left(I/\hbar\right) \right) \;\; \in \mathcal{O}(\mathcal{E}, \mathbb{C}[ [ \hbar ] ]) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>⋯</mi></msub></mrow><annotation encoding="application/x-tex">\partial_{\cdots}</annotation></semantics></math> is given by def. <a class="maruku-ref" href="#DifferentialOperatorByKernel"></a>.</p> </div> <p>(<a href="#Costello">Costello 07, def. 6.6.1</a>)</p> <p id="ThePathIntegral"><strong>The path integral</strong></p> <div class="num_prop" id="PathIntegralByLimitInDimension0"> <h6 id="proposition_18">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X = *</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/point">point</a>, then the <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> over the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> exists as an ordinary <a class="existingWikiWord" href="/nlab/show/integral">integral</a> and is equal to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi><mi>log</mi><msub><mo>∫</mo> <mrow><mi>x</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>Im</mi><msup><mi>Q</mi> <mi>GF</mi></msup><msub><mo stretchy="false">)</mo> <mi>ℝ</mi></msub></mrow></msub><mi>exp</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>Q</mi><mi>x</mi><mo stretchy="false">⟩</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo stretchy="false">)</mo><mi>d</mi><mi>μ</mi><mo>)</mo></mrow><mo>=</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \hbar log \int_{x \in (Im Q^{GF})_{\mathbb{R}}} \exp\left( \tfrac{1}{2} \langle x, Q x\rangle + I(x + a) d \mu \right) = \Gamma(P(0,\infty), I)(a) </annotation></semantics></math></div></div> <p>This is (<a href="#Costello">Costello 07, lemma 6.6.2</a>).</p> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of positive <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>, the <a class="existingWikiWord" href="/nlab/show/limit">limit</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>ϵ</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>Γ</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underset{\epsilon \to 0}{\lim} \Gamma(P(\epsilon,\infty), I)(a) </annotation></semantics></math></div> <p>does not in general exist. Renormalization is the process of adding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math>-corrections to the action – the <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a> – such as to make it exist after all. In this case we may regard the limit, by prop. <a class="maruku-ref" href="#PathIntegralByLimitInDimension0"></a>, as the <em>definition</em> of the <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>.</p> </div> <p id="RenormalizedAction"><strong>Renormalized action</strong></p> <div class="num_defn" id="RenormalizedActionAndCounterterms"> <h6 id="definition_17">Definition</h6> <p>Given the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo lspace="verythinmathspace">:</mo><mi>ϕ</mi><mo>↦</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">⟨</mo><mi>ϕ</mi><mo>,</mo><mi>Q</mi><mi>phi</mi><mo stretchy="false">⟩</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \colon \phi \mapsto \frac{1}{2}\langle \phi, Q phi\rangle + I(\phi)</annotation></semantics></math>, a <strong>renormalization</strong> is a <a class="existingWikiWord" href="/nlab/show/power+series">power series</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mi>R</mi></msup><mo stretchy="false">(</mo><mi>ℏ</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mi>ℏ</mi><mo stretchy="false">)</mo><mo>−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>></mo><mn>0</mn></mrow></mrow><mrow><mrow><mi>k</mi><mo>≥</mo><mi>k</mi></mrow></mrow></mfrac></munder><msup><mi>ℏ</mi> <mi>i</mi></msup><msubsup><mi>I</mi> <mrow><mi>i</mi><mo>,</mo><mi>k</mi></mrow> <mi>CT</mi></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> I^R(\hbar, \epsilon) = I(\hbar) - \underset{{i \gt 0} \atop { k \geq k}}{\sum} \hbar^i I^{CT}_{i,k}(\epsilon) </annotation></semantics></math></div> <p>such that the <a class="existingWikiWord" href="/nlab/show/limit">limit</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>ϵ</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>Γ</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>I</mi> <mi>R</mi></msup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \underset{\epsilon \to 0}{\lim} \Gamma(P(\epsilon,T), I^R(\epsilon)) \,, </annotation></semantics></math></div> <p>by def. <a class="maruku-ref" href="#RenormalizationOperator"></a>, exists.</p> </div> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>I</mi> <mrow><mi>i</mi><mo>,</mo><mi>k</mi></mrow> <mi>CT</mi></msubsup></mrow><annotation encoding="application/x-tex">I^{CT}_{i,k}</annotation></semantics></math> are called the <em><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></em>.</p> <p id="Renormalization"><strong>Renormalization</strong></p> <div class="num_defn" id="RenormalizationScheme"> <h6 id="definition_18">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">renormalization scheme</a> is a decomposition of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> on <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>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math>, as a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, into a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>≃</mo><msub><mi>𝒜</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo>⊕</mo><msub><mi>𝒜</mi> <mrow><mo>></mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathcal{A} \simeq \mathcal{A}_{\geq 0} \oplus \mathcal{A}_{\gt 0} </annotation></semantics></math></div> <p>such that the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mi>𝒜</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f \in \mathcal{A}_{\geq 0}</annotation></semantics></math> are non-singular in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{\epsilon \to 0}{\lim} f(\epsilon)</annotation></semantics></math> exists.</p> </div> <p>Hence this is a choice of picking the <a class="existingWikiWord" href="/nlab/show/singularities">singularities</a> in functions that are not necessarily defined at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\epsilon = 0</annotation></semantics></math>.</p> <div class="num_theorem"> <h6 id="theorem_4">Theorem</h6> <p>Given any choice of <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">renormalization scheme</a>, def. <a class="maruku-ref" href="#RenormalizationScheme"></a>, there exists a unique 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><mo stretchy="false">{</mo><msubsup><mi>I</mi> <mrow><mi>k</mi><mo>,</mo><mi>i</mi></mrow> <mi>CT</mi></msubsup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{I^{CT}_{k,i}\}</annotation></semantics></math>, def. <a class="maruku-ref" href="#RenormalizedActionAndCounterterms"></a> such that</p> <ul> <li> <p>each counterterm is in the chosen <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒜</mi> <mrow><mo>></mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{A}_{\gt 0}</annotation></semantics></math> as a function of <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>;</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k \gt 0</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/germ">germ</a> of a counterterm at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> depends only on the <a class="existingWikiWord" href="/nlab/show/germ">germ</a> of the given field theory data <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>Q</mi><mo>,</mo><msup><mi>Q</mi> <mi>GF</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, Q, Q^{GF})</annotation></semantics></math> at that point.</p> </li> </ul> </div> <p>This is (<a href="#Costello">Costello 07, theorem B, p. 38</a>).</p> <div class="num_defn"> <h6 id="definition_19">Definition</h6> <p>Given a renormalization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mi>R</mi></msup></mrow><annotation encoding="application/x-tex">I^{R}</annotation></semantics></math>, write for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo lspace="0em" rspace="thinmathspace">ht</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">T \in \mathbb{R}_{\ht 0}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Γ</mi> <mi>R</mi></msup><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo>→</mo><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>Γ</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo>−</mo><msup><mi>I</mi> <mi>CT</mi></msup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma^R(P(0,T), I) \coloneqq \underset{\epsilon \to 0}{\to} \Gamma(P(\epsilon, R), I - I^{CT}(\epsilon)) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>We think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Γ</mi> <mi>R</mi></msup><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma^R(P(0,T), I)</annotation></semantics></math> as the renormalized <a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a> of the original action at <a class="existingWikiWord" href="/nlab/show/scale">scale</a> <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">T</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition_19">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Γ</mi> <mi>R</mi></msup><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>T</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thickmathspace"></mspace><msup><mi>Γ</mi> <mi>R</mi></msup><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Gamma^R(P(0,T'), I) = \Gamma(P(T,T'), \; \Gamma^R(P(0,T), I )) </annotation></semantics></math></div> <p>holds.</p> </div> <p>(<a href="#Costellon">Costello 07, lemma 9.0.6</a>).</p> <h2 id="examples">Examples</h2> <h3 id="vacuum_energy_and_cosmological_constant">Vacuum energy and Cosmological constant</h3> <p>The renormalization freedom in <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> (<a class="existingWikiWord" href="/nlab/show/perturbative+quantum+gravity">perturbative quantum gravity</a>) induces freedom in the choice of <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a> of the <a class="existingWikiWord" href="/nlab/show/stress-energy+tensor">stress-energy tensor</a> and hence in the <a class="existingWikiWord" href="/nlab/show/cosmological+constant">cosmological constant</a>.</p> <p>For details see <a href="cosmological+constant#InPerturbativeQuantumGravity">there</a>.</p> <h3 id="chernsimons_level">Chern-Simons level</h3> <p>See at <em><a href="Chern-Simons+level#LevelRenormalization">Chern-Simons level renormalization</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a>, <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">renormalization scheme</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field-strength+renormalization">field-strength renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalon">renormalon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regularization+%28physics%29">regularization (physics)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV+quantization">BV quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a>, <a class="existingWikiWord" href="/nlab/show/Gribov+ambiguity">Gribov ambiguity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmic+Galois+group">cosmic Galois group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+renormalization">differential renormalization</a></p> </li> </ul> <h2 id="References">References</h2> <p>After the original informal suggestions by <a class="existingWikiWord" href="/nlab/show/Schwinger-Tomonaga-Feynman-Dyson">Schwinger-Tomonaga-Feynman-Dyson</a></p> <ul> <li id="Dyson49"><a class="existingWikiWord" href="/nlab/show/Freeman+Dyson">Freeman Dyson</a>, <em>The raditation theories of Tomonaga, Schwinger and Feynman</em>, Phys. Rev. 75, 486, 1949 (<a href="http://web.ihep.su/dbserv/compas/src/dyson49b/eng.pdf">pdf</a>)</li> </ul> <p>the mathematics of renormalization was finally understood and summarized in the 1975 Erice Majorana School:</p> <ul> <li id="Wightman76">G. Velo and <a class="existingWikiWord" href="/nlab/show/Arthur+Wightman">Arthur Wightman</a> (eds.) <em>Renormalization Theory</em> Proceedings of the 1975 Erice summer school, NATO ASI Series C 23, D. Reidel, Dordrecht, 1976</li> </ul> <p>which included <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/BPHZ+renormalization">BPHZ renormalization</a>, proof of the <span class="newWikiWord">forest formula<a href="/nlab/new/forest+formula">?</a></span> and the <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a> method for <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>.</p> <p>Little advancement happened until the identification of <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> structure in the <span class="newWikiWord">forest formula<a href="/nlab/new/forest+formula">?</a></span> due to</p> <ul> <li id="Kreimer97"><a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a>, <em>On the Hopf algebra structure of perturbative quantum field theories</em>, Adv. Theor. Math. Phys. 2 , 303 (1998) (<a href="https://arxiv.org/abs/q-alg/9707029">q-alg/9707029</a>)</li> </ul> <p>This finally triggered the formulation of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> in terms of <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> in</p> <ul> <li id="GarciaBondiaLazzarini00"> <p><a class="existingWikiWord" href="/nlab/show/Jose+Gracia-Bondia">Jose Gracia-Bondia</a>, S. Lazzarini, <em>Connes-Kreimer-Epstein-Glaser Renormalization</em> (<a href="https://arxiv.org/abs/hep-th/0006106">arXiv:hep-th/0006106</a>)</p> </li> <li id="Keller10"> <p><a class="existingWikiWord" href="/nlab/show/Kai+Keller">Kai Keller</a>, chapter IV of <em>Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization</em>, PhD thesis (<a href="https://arxiv.org/abs/1006.2148">arXxiv:1006.2148</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>, <em>Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization</em>, J. Math. Phy.</p> <p>55(12), 122303 (2014) (<a href="https://arxiv.org/abs/1311.5424">arXiv:1311.5424</a>)</p> </li> </ul> <p>Thus the original “dark art” of the construction of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> via renormalization finds a complete rigorous formulation.</p> <p>In</p> <ul> <li id="Collini16"><a class="existingWikiWord" href="/nlab/show/Giovanni+Collini">Giovanni Collini</a>, <em>Fedosov Quantization and Perturbative Quantum Field Theory</em> (<a href="https://arxiv.org/abs/1603.09626">arXiv:1603.09626</a>)</li> </ul> <p>it is shown that (at least under favorable circumstances) the construction of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> is equivalently <a class="existingWikiWord" href="/nlab/show/Fedosov+deformation+quantization">Fedosov deformation quantization</a> of the given interacting <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a>, thus identifying the renormalization freedom with the Freedom in choosing a <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a>.</p> <h3 id="General">Review</h3> <p>A textbook account of <a class="existingWikiWord" href="/nlab/show/renormalization">("re"-)normalization</a> in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> is in</p> <ul> <li id="Duetsch18"><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, chapters 3 and 4 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>Informal introductions:</p> <ul> <li> <p>G. Peter Lepage, <em>What is Renormalization?</em>, talk 1989 (<a href="http://arxiv.org/abs/hep-ph/0506330">arXiv:hep-ph/0506330</a>)</p> </li> <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> <p><a class="existingWikiWord" href="/nlab/show/R.+E.+Borcherds">R. E. Borcherds</a>, <em>Renormalization and quantum field theory</em>, (<a href="http://arxiv.org/abs/1008.0129">arxiv/1008.0129</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Arnold+Neumaier">Arnold Neumaier</a>, <em>Renormalizatin without infinities – an elementary tutorial</em> (<a href="http://www.mat.univie.ac.at/~neum/ms/ren.pdf">pdf</a>)</p> </li> </ul> <p>Further review:</p> <ul> <li>V. Mastropietro, <em>Renormalization: general theory</em>, in <em><a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a></em>, Elsevier (2024) [<a href="https://arxiv.org/abs/2312.11400">arXiv:2312.11400</a>]</li> </ul> <h3 id="in_causal_perturbation_theory_2">In causal perturbation theory</h3> <p>The idea of Epstein-Glaser renormalization is due to</p> <ul> <li id="EpsteinGlaser73"><a class="existingWikiWord" href="/nlab/show/Henri+Epstein">Henri Epstein</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Glaser">Vladimir Glaser</a>, <em><a class="existingWikiWord" href="/nlab/show/The+Role+of+locality+in+perturbation+theory">The Role of locality in perturbation theory</a></em>, Annales Poincaré Phys. Theor. A 19 (1973) 211.</li> </ul> <p>following precursors in</p> <ul> <li id="StueckelbergPetermann53"> <p><a class="existingWikiWord" href="/nlab/show/Ernst+St%C3%BCckelberg">Ernst Stückelberg</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Petermann">André Petermann</a>, , <em>La normalisation des constants dans la theorie des quanta</em>, Helv. Phys. Acta 26, 499 (1953); and earlier references therein</p> <p>(see also <em><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></em>)</p> </li> <li id="BogoliubovShirkov76"> <p><a class="existingWikiWord" href="/nlab/show/Nikolay+Bogoliubov">Nikolay Bogoliubov</a>, <a class="existingWikiWord" href="/nlab/show/Dmitry+Shirkov">Dmitry Shirkov</a>,, <em>Introduction to the Theory of Quantized Fiels</em>, New York: John Wiley and Sons, 1976, 3rd edition</p> </li> </ul> <p>This was formulated in terms of splittings of distributions. The equivalent formulation in terms of <a class="existingWikiWord" href="/nlab/show/extensions+of+distributions">extensions of distributions</a> is due to</p> <ul> <li id="PopineauStora82"> <p>G. Popineau and <a class="existingWikiWord" href="/nlab/show/Raymond+Stora">Raymond Stora</a>, <em>A pedagogical remark on the main theorem of perturbative renormalization theory</em>, Nucl. Phys. B 912 (2016), 70–78, preprint: LAPP–TH, Lyon (1982).</p> </li> <li id="Stora93"> <p><a class="existingWikiWord" href="/nlab/show/Raymond+Stora">Raymond Stora</a>, <em>Differential algebras in Lagrangean field theory</em>, Lectures at ETH, Zürich, 1993, unpublished</p> </li> <li id="BrunettiFredenhagen00"> <p><a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <em>Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds</em>, Commun. Math. Phys. 208:623-661 (2000) (<a href="http://arxiv.org/abs/math-ph/9903028">arXiv:math-ph/9903028</a>)</p> </li> </ul> <p>Exposition includes</p> <ul> <li id="Brouder10"><a class="existingWikiWord" href="/nlab/show/Christian+Brouder">Christian Brouder</a>, <em>Multiplication of distributions</em>, 2010 (<a class="existingWikiWord" href="/nlab/files/BrouderProductOfDistributions.pdf" title="pdf">pdf</a>)</li> </ul> <p>The resulting <a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a> is due to</p> <ul> <li id="StueckelbergPetermann53"><a class="existingWikiWord" href="/nlab/show/Ernst+St%C3%BCckelberg">Ernst Stückelberg</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Petermann">André Petermann</a>, <em>La normalisation des constantes dans la theorie des quanta</em>, Helv. Phys. Acta 26 (1953), 499–520</li> </ul> <p>The relation of Epstein-Glaser/Stückelberg-Petermann to the <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a> of</p> <ul> <li id="GellMannLow54"><a class="existingWikiWord" href="/nlab/show/Murray+Gell-Mann">Murray Gell-Mann</a> and F. E. Low, <em>Quantum Electrodynamics at Small Distances</em>, Phys. Rev. 95 (5) (1954), 1300–1312 (<a href="http://www.fafnir.phyast.pitt.edu/py3765/GellManLow.pdf">pdf</a>)</li> </ul> <p>and the <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> of</p> <ul> <li id="Wilson71"> <p><a class="existingWikiWord" href="/nlab/show/Kenneth+Wilson">Kenneth Wilson</a>, <em>Renormalization group and critical phenomena</em> , I., Physical review B 4(9) (1971).</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> </ul> <p>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>Review is 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>For more see at <em><a href="perturbation+theory#ReferencesInAQFT">perturbation theory – In AQFT</a></em>.</p> <p>Applications to the renormalization of <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> on curved <a class="existingWikiWord" href="/nlab/show/background+field">background</a> <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> is accomplished in</p> <ul> <li id="Hollands07"><a class="existingWikiWord" href="/nlab/show/Stefan+Hollands">Stefan Hollands</a>, <em>Renormalized Quantum Yang-Mills Fields in Curved Spacetime</em>, Rev.Math.Phys.20:1033-1172,2008 (<a href="https://arxiv.org/abs/0705.3340">arXiv:0705.3340</a>)</li> </ul> <h3 id="bphz_renormalization">BPHZ Renormalization</h3> <p>BPHZ renormalization was introduced in particular in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Klaus+Hepp">Klaus Hepp</a>, <em>Théorie de la Renormalisation</em> Lect. Notes in Phys. Springer (1969)</li> </ul> <p>Review:</p> <ul> <li> <p>Klaus Sibold, <em>Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme</em>, Scholarpedia, 5(5):7306. <a href="http://dx.doi.org/10.4249/scholarpedia.7306">doi:10.4249/scholarpedia.7306</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Matilde+Marcolli">Matilde Marcolli</a> <em><a class="existingWikiWord" href="/nlab/show/Noncommutative+Geometry%2C+Quantum+Fields+and+Motives">Noncommutative Geometry, Quantum Fields and Motives</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michel+Talagrand">Michel Talagrand</a>, §16 in: <em>What is a Quantum Field Theory? – A first Introduction for Mathematicians</em>, Cambridge University Press (2022) [<a href="https://doi.org/10.1017/9781108225144">doi:10.1017/9781108225144</a>]</p> </li> </ul> <p>The original articles on the Hopf-algebraic formulation:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a>, <em>Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem</em>, Comm. Math. Phys. <strong>210</strong> (2000), no. 1, 249–273, <a href="http://arxiv.org/abs/hep-th/9912092">hep-th/9912092</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=1748177">MR2002f:81070</a>, <a href="http://dx.doi.org/10.1007/s002200050779">doi</a>, <em>II. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math>-function, diffeomorphisms and the renormalization group</em>, Comm. Math. Phys. <strong>216</strong> (2001), no. 1, 215–241; <a href="http://arxiv.org/abs/hep-th/0003188">hep-th/0003188</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=1748177">MR2002f:81071</a>, <a href="http://dx.doi.org/10.1007/PL00005547">doi</a></li> </ul> <p>An introduction and review to the Hopf-algebraic description of renormalization is in</p> <ul> <li>Christian Brouder, <em>Quantum field theory meets Hopf algebra</em> (<a href="http://de.arxiv.org/abs/hep-th/0611153">arxiv:hep-th/0611153</a>)</li> </ul> <p>A textbook treatment is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a>, <em>Knots and Fenyman diagrams</em> , Cambridge Lecture Notes in Physics. 13. Cambridge: Cambridge University Press.</li> <li>Joseph C. Varilly, <em>The interface of noncommutative geometry and physics</em>, <a href="http://arxiv.org/abs/hep-th/0206007">hep-th/0206007</a></li> </ul> <p>Some heavywheight automated computations using this formalism are discussed in</p> <ul> <li>D. J. Broadhurst, <a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a>, <em>Renormalization automated by Hopf algebra</em> (<a href="http://arxiv.org/abs/hep-th/9810087">arXiv:hep-th/9810087</a>)</li> </ul> <p>See also</p> <ul id="Suijlekom"> <li>W. van Suijlekom, <em>Representing Feynman graphs on BV-algebras</em> , (<a href="http://arxiv.org/abs/0807.0999">arXiv</a>)</li> </ul> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a>, Karen Yeats, <em>Diffeomorphisms of quantum fields</em>, <a href="https://arxiv.org/abs/1610.01837">arxiv/1610.01837</a></li> </ul> <h3 id="ReferencesInBVFormalism">In BV formalism</h3> <p>Discussion of renormalization in the context of <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> is for <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic field theory</a> in the rigorous framework 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</p> <ul> <li id="FredenhagenRejzner11a"> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, <em>Batalin-Vilkovisky formalism in the functional approach to classical field theory</em>, Commun. Math. Phys. 314(1), 93–127 (2012) (<a href="https://arxiv.org/abs/1101.5112">arXiv:1101.5112</a>)</p> </li> <li id="FredenhagenRejzner11b"> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, <em>Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory</em>, Commun. Math. Phys. 317(3), 697–725 (2012) (<a href="https://arxiv.org/abs/1110.5232">arXiv:1110.5232</a>)</p> </li> <li id="Rejzner11"> <p><a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, <em>Batalin-Vilkovisky formalism in locally covariant field theory</em> (<a href="https://arxiv.org/abs/1111.5130">arXiv:1111.5130</a>)</p> </li> <li id="Rejzner13"> <p><a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, <em>Remarks on local symmetry invariance in perturbative algebraic quantum field theory</em> (<a href="https://arxiv.org/abs/1301.7037">arXiv:1301.7037</a>)</p> </li> </ul> <p>and surveyed in</p> <ul> <li id="Rejzner16"><a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, section 7 of <em>Perturbative algebraic quantum field theory</em> Springer 2016 (<a href="https://link.springer.com/book/10.1007%2F978-3-319-25901-7">web</a>)</li> </ul> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean field theory</a> is in</p> <ul> <li id="Costello11"><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> AMS (2011) (<a href="http://www.ams.org/publications/authors/books/postpub/surv-170">publisher webpage</a>)</li> </ul> <p>building on</p> <ul> <li id="Costello"><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em>Renormalisation and the Batalin-Vilkovisky formalism</em> (<a href="http://arxiv.org/abs/0706.1533">arXiv:0706.1533</a>)</li> </ul> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/factorization+algebra+of+observables">factorization algebra of observables</a></em>.</p> <p>A vaguely related approach earlier appeared in</p> <ul> <li id="Tamarkin"><a class="existingWikiWord" href="/nlab/show/Dmitry+Tamarkin">Dmitry Tamarkin</a>, <em>A formalism for the renormalization procedure</em> (<a href="http://arxiv.org/abs/math/0312219">arXiv:math/0312219</a>)</li> </ul> <h3 id="ReferencesOnCompactifiedConfigurationSpaces">On compactified configuration spaces</h3> <p>For <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean field theory</a>, an alternative to regarding <a class="existingWikiWord" href="/nlab/show/propagators">propagators</a>/<a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>/<a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> as <a class="existingWikiWord" href="/nlab/show/distributions+of+several+variables">distributions of several variables</a> with <a class="existingWikiWord" href="/nlab/show/singularities">singularities</a> at (in particular) coincident points, one may <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback</a> these distributions to <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on <a class="existingWikiWord" href="/nlab/show/Fulton-MacPherson+compactifications">Fulton-MacPherson compactifications</a> of <a class="existingWikiWord" href="/nlab/show/configuration+spaces">configuration spaces</a> of points and study renormalization in that perspective.</p> <p>This approach was originally considered specifically for <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> in</p> <ul> <li id="AxelrodSinger93"><a class="existingWikiWord" href="/nlab/show/Scott+Axelrod">Scott Axelrod</a>, <a class="existingWikiWord" href="/nlab/show/Isadore+Singer">Isadore Singer</a>, <em>Chern–Simons Perturbation Theory II</em>, J. Diff. Geom. 39 (1994) 173-213 (<a href="http://arxiv.org/abs/hep-th/9304087">arXiv:hep-th/9304087</a>)</li> </ul> <p>which was re-amplified in</p> <ul> <li id="BottCattaneo97"> <p><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, Remark 3.6 in <em>Integral invariants of 3-manifolds</em>, J. Diff. Geom., 48 (1998) 91-133 (<a href="https://arxiv.org/abs/dg-ga/9710001">arXiv:dg-ga/9710001</a>)</p> </li> <li id="CattaneoMnev10"> <p><a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Mnev">Pavel Mnev</a>, Remark 11 in <em>Remarks on Chern-Simons invariants</em>, Commun.Math.Phys.293:803-836,2010 (<a href="https://arxiv.org/abs/0811.2045">arXiv:0811.2045</a>)</p> </li> </ul> <p>A systematic development of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> from this perspective is discussed in</p> <ul> <li id="BergbauerBrunettiKreimer09"> <p><a class="existingWikiWord" href="/nlab/show/Christoph+Bergbauer">Christoph Bergbauer</a>, <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a>, <em>Renormalization and resolution of singularities</em> (<a href="https://arxiv.org/abs/0908.0633">arXiv:0908.0633</a>)</p> </li> <li id="Bergbauer09"> <p><a class="existingWikiWord" href="/nlab/show/Christoph+Bergbauer">Christoph Bergbauer</a>, <em>Renormalization and resolution of singularities</em>, talks as IHES and Boston, 2009 (<a href="http://www.emg.uni-mainz.de/Dateien/bergbauer.pdf">pdf</a>)</p> </li> <li id="Berghoff14a"> <p><a class="existingWikiWord" href="/nlab/show/Marko+Berghoff">Marko Berghoff</a>, <em>Wonderful renormalization</em>, 2014 (<a href="http://www2.mathematik.hu-berlin.de/%7Ekreimer/wp-content/uploads/Berghoff-Marko.pdf">pdf</a>, <a href="https://doi.org/10.18452/17160">doi:10.18452/17160</a>)</p> </li> <li id="Berghoff14b"> <p><a class="existingWikiWord" href="/nlab/show/Marko+Berghoff">Marko Berghoff</a>, <em>Wonderful compactifications in quantum field theory</em>, Communications in Number Theory and Physics Volume 9 (2015) Number 3 (<a href="https://arxiv.org/abs/1411.5583">arXiv:1411.5583</a>)</p> </li> </ul> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes+on+compactified+configuration+spaces+of+points">Feynman amplitudes on compactified configuration spaces of points</a></em>.</p> <h3 id="operadic_description">Operadic description</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, Nikolay M. Nikolov, <em>Operadic construction of the renormalization group</em> (<a href="http://arxiv.org/abs/1202.1206">arXiv:1202.1206</a>)</li> </ul> <h3 id="relations_to_motives_polylogarithms_positivity">Relations to motives, polylogarithms, positivity</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Goncharov">Alexander Goncharov</a>, Marcus Spradlin, C. Vergu, Anastasia Volovich, <em>Classical polylogarithms for amplitudes and Wilson loops</em>, Phys.Rev.Lett.105:151605,2010 <a href="http://arxiv.org/abs/1006.5703">arxiv/1006.5703</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nima+Arkani-Hamed">Nima Arkani-Hamed</a>, Jacob L. Bourjaily, Freddy Cachazo, <a class="existingWikiWord" href="/nlab/show/Alexander+Goncharov">Alexander Goncharov</a>, Alexander Postnikov, Jaroslav Trnka, <em>Scattering amplitudes and the positive Grassmannian</em>, <a href="http://arxiv.org/abs/1212.5605">arxiv/1212.5605</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spencer+Bloch">Spencer Bloch</a>, <a class="existingWikiWord" href="/nlab/show/H%C3%A9l%C3%A8ne+Esnault">Hélène Esnault</a>, <a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a>, <em>On motives associated to graph polynomials</em>, Commun.Math.Phys. <strong>267</strong> (2006) 181-225 <a href="http://arxiv.org/abs/math/0510011">math.AG/0510011</a> <a href="http:&&dx.doi.org/10.1007/s00220-006-0040-2">doi</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 22, 2025 at 16:06:03. 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