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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> renormalization group flow </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/10213/#Item_1" 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='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#ScalingTransformationsRGFlow'>Scaling transformations</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <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>Notice that this is related to, but conceptually different from, <em><a class="existingWikiWord" href="/nlab/show/Polchinski%27s+flow+equation">Polchinski's flow equation</a></em> in the context of <a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>.</p> <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>For more see at</p> <ul> <li><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> the section <em><a href="geometry+of+physics+--+A+first+idea+of+quantum+field+theory#Renormalization">Renormalization</a></em></li> </ul> <h2 id="definition">Definition</h2> <div class="num_prop" id="FlowRenormalizationGroup"> <h6 id="proposition">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">(1)</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">(2)</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">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_2">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">(3)</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> <h2 id="properties">Properties</h2> <div class="num_prop" id="CocycleRunningCoupling"> <h6 id="proposition_2">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">(3)</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_2">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> 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> <h2 id="examples">Examples</h2> <h3 id="ScalingTransformationsRGFlow">Scaling transformations</h3> <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">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">(4)</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_2">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_3">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_3">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_4">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">(4)</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> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+coupling+unification">gauge coupling unification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/threshold+correction">threshold correction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a></p> </li> </ul> <h2 id="references">References</h2> <p>The original informal discussion for RG-flow along <a class="existingWikiWord" href="/nlab/show/scaling+transformations">scaling transformations</a> is due to</p> <ul> <li id="GellMannLow54"> <p><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>)</p> </li> <li id="Callan70"> <p><a class="existingWikiWord" href="/nlab/show/Curtis+Callan">Curtis Callan</a>, <em>Broken Scale Invariance in Scalar Field Theory</em>, Phys. Rev. D 2, 1541, 1970 (<a href="https://doi.org/10.1103/PhysRevD.2.1541">doi:10.1103/PhysRevD.2.1541</a>)</p> </li> <li id="Symanzik70"> <p><a class="existingWikiWord" href="/nlab/show/Kurt+Symanzik">Kurt Symanzik</a>, <em>Small distance behaviour in field theory and power counting</em>, Communications in Mathematical Physics. 18 (3): 227–246 (<a href="https://doi.org/10.1007/BF01649434">doi:10.1007/BF01649434</a>)</p> </li> </ul> <p>Formulation in the rigorous context of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/pAQFT">pAQFT</a>, via the <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a>, is due to</p> <ul> <li id="BrunettiDuetschFredenhagen09"><a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <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>)</li> </ul> <p>reviewed in</p> <ul> <li id="Duetsch18"><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, section 3.5.3 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>In the context of <a class="existingWikiWord" href="/nlab/show/factorization+algebras">factorization algebras</a>, this is given by the book <a class="existingWikiWord" href="/nlab/show/Renormalization+and+Effective+Field+Theory">Renormalization and Effective Field Theory</a></p> </body></html> </div> <div class="revisedby"> <p> Last revised on August 5, 2019 at 18:08:23. 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