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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='#in_causal_perturbation_theory'>In causal perturbation theory</a></li> <ul> <li><a href='#background'>Background</a></li> <li><a href='#interacting_quantum_bvdifferential'>Interacting quantum BV-differential</a></li> <li><a href='#RenormalizationAndMasterWardIdentity'>Renormalization and Master ward identity</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> formulated in terms of <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a>, the <em>classical master equation</em> expresses the nilpotency of the <a class="existingWikiWord" href="/nlab/show/BV-differential">BV-differential</a> before <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a>, with the latter regarded as a <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a> with respect to the <em><a class="existingWikiWord" href="/nlab/show/antibracket">antibracket</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{-,-\}</annotation></semantics></math>, for “Hamiltonian” the BV-BRST-extended <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>+</mo><msub><mi>S</mi> <mi>BRST</mi></msub></mrow><annotation encoding="application/x-tex">S + S_{BRST}</annotation></semantics></math>”:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow><mo>(</mo><msub><mi>s</mi> <mi>BV</mi></msub><mo>)</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mphantom><mi>AA</mi></mphantom><mo>⇔</mo><mphantom><mi>AA</mi></mphantom><msup><mrow><mo>(</mo><mo stretchy="false">{</mo><mi>S</mi><mo>+</mo><msub><mi>S</mi> <mi>BRST</mi></msub><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>)</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mphantom><mi>AA</mi></mphantom><mo>⇔</mo><mphantom><mi>AA</mi></mphantom><mo stretchy="false">{</mo><mi>S</mi><mo>+</mo><msub><mi>S</mi> <mi>BRST</mi></msub><mo>,</mo><mi>S</mi><mo>+</mo><msub><mi>S</mi> <mi>BRST</mi></msub><mo stretchy="false">}</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( s_{BV} \right)^2 = 0 \phantom{AA} \Leftrightarrow \phantom{AA} \left( \{S + S_{BRST},(-)\}\right)^2 = 0 \phantom{AA} \Leftrightarrow \phantom{AA} \{S + S_{BRST}, S + S_{BRST}\} = 0 \,. </annotation></semantics></math></div> <p>The <em>quantum master equation</em> (prop. <a class="maruku-ref" href="#QuantumMasterEquation"></a> below) is the version of this equation after <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a>, in which case the the <a class="existingWikiWord" href="/nlab/show/BV-differential">BV-differential</a> picks up a quantum correction of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a>) by the <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>BV</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_{BV}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow><mo>(</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">{</mo><mi>S</mi><mo>+</mo><msub><mi>S</mi> <mi>BRST</mi></msub><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mspace width="thinmathspace"></mspace><mo>)</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left(\, \{S + S_{BRST},(-)\} + i \hbar \Delta_{BV} \, \right)^2 = 0 \,. </annotation></semantics></math></div> <h2 id="in_causal_perturbation_theory">In causal perturbation theory</h2> <p>We discuss the quantum master equation in the rigorous formulation of <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> (<a href="#FredenhagenRejzner11b">Fredenhagen-Rejzner 11b</a>, <a href="#Rejzner11">Rejzner 11</a>).</p> <p>First we consider all structure just on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a>, hence excluding non-linear <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> such as the usual point-<a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/extension">extension</a> of all structures from regular to <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> is the <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> step, discussed <a href="#RenormalizationAndMasterWardIdentity">furthter below</a>.</p> <h3 id="background">Background</h3> <p>Throughout, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BST</mtext></msub><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{\text{BV-BST}},\mathbf{L}')</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> with global <a class="existingWikiWord" href="/nlab/show/BV-differential">BV-differential</a> (<a href="A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal">this def.</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</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>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \{-S', -\} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] </annotation></semantics></math></div> <p>Hence <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">S'</annotation></semantics></math> denotes the gauge fixed free action functional.</p> <p>Moreover, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math> be a compatible choice of <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> with associated <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_F</annotation></semantics></math>.</p> <div class="num_lemma" id="DerivationBVDifferentialForWickAlgebra"> <h6 id="lemma">Lemma</h6> <p><strong>(global <a class="existingWikiWord" href="/nlab/show/BV-differential">BV-differential</a> is <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> on <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a>)</strong></p> <p>The global <a class="existingWikiWord" href="/nlab/show/BV-differential">BV-differential</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</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>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \{-S',-\} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> also with respect to the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</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>}</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>}</mo></mrow><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\{ -S', A_1 \star_H A_2 \right\} \;=\; \left\{ -S', A_1 \right\} \star_H A_2 \;+\; A_1 \star_H \left\{ -S', A_2 \right\} \,. </annotation></semantics></math></div></div> <p>(<a href="#FredenhagenRejzner11b">Fredenhagen-Rejzner 11b, below (37)</a>, <a href="#Rejzner11">Rejzner 11, below (5.28)</a>) For <strong>proof</strong> see <a href="A+first+idea+of+quantum+field+theory#OnMicrocausalObservablesGlobalBVDifferential">this prop</a></p> <div class="num_defn" id="OnRegularObservablesPerturbativeSMatrix"> <h6 id="definition">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/perturbative+S-matrix">perturbative S-matrix</a> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a>)</strong></p> <p>The <em><a class="existingWikiWord" href="/nlab/show/perturbative+S-matrix">perturbative S-matrix</a></em> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> is the <a class="existingWikiWord" href="/nlab/show/exponential">exponential</a> with respect to the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</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>reg</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>ℏ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar)) </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>exp</mi> <mrow><msub><mo>⋆</mo> <mi>F</mi></msub></mrow></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>≔</mo><mn>1</mn><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mo lspace="0em" rspace="thinmathspace">i</mo><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>S</mi> <mi>int</mi></msub><mo>+</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}(S_{int}) = \exp_{\star_F} \left( \tfrac{1}{i \hbar} S_{int}) \right) \coloneqq 1 + \tfrac{1}{\i \hbar} S_{int} + \tfrac{1}{2} \tfrac{1}{(i \hbar)^2} S_{int} \star_F S_{int} + \cdots \,. </annotation></semantics></math></div> <p>We think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">S_{int}</annotation></semantics></math> here as an <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> non-point-<a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>.</p> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{S}(S_{int})^{-1}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> with respect to the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick product</a> (which exists by <a href="S-matrix#PerturbativeSMatrixInverse">this remark</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mo>⋆</mo> <mi>H</mi></msub><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}(S_{int})^{-1} \star_H \mathcal{S}(S_{int}) = 1 \,. </annotation></semantics></math></div> <p>Notice that this is in general different form the inverse with respect to the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⋆</mo> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\star_F</annotation></semantics></math>, which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}(-S_{int})</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}(-S_{int}) \star_F \mathcal{S}(S_{int}) = 1 \,. </annotation></semantics></math></div></div> <div class="num_defn" id="MollerOperatorOnRegularPolynomialObservables"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a>)</strong></p> <p>Given an <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> non-point-<a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> in the form of a <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observable">regular polynomial observable</a> of degree 0</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>int</mi></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mfrac linethickness="0"><mrow><mi>reg</mi></mrow><mrow><mrow><mi>deg</mi><mo>=</mo><mn>0</mn></mrow></mrow></mfrac></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> S_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ] </annotation></semantics></math></div> <p>then the corresponding <em><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></em> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</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>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathcal{R}^{-1} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] </annotation></semantics></math></div> <p>is given by the <a class="existingWikiWord" href="/nlab/show/derivative">derivative</a> of <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mo>⋆</mo> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{R}^{-1} \;\coloneqq\; \mathcal{S}(S_{int})^{-1} \star_H (\mathcal{S}(S_{int}) \star_F (-)) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathcal{S}(S_{int}) = \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int} \right)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/perturbative+S-matrix">perturbative S-matrix</a> from def. <a class="maruku-ref" href="#OnRegularObservablesPerturbativeSMatrix"></a>.</p> <p>This indeed lands in <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> in <a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math> (by <a href="Bogoliubov's+formula#PowersInPlancksConstant">this remark</a>), instead of in more general <a class="existingWikiWord" href="/nlab/show/Laurent+series">Laurent series</a> as the <a class="existingWikiWord" href="/nlab/show/perturbative+S-matrix">perturbative S-matrix</a> does (def. <a class="maruku-ref" href="#OnRegularObservablesPerturbativeSMatrix"></a>).</p> <p>Hence the inverse map is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>𝒮</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>⋆</mo><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"> \mathcal{R} \;=\; \mathcal{S}(-S_{int}) \star_F ( \mathcal{S}(S_{int}) \star(-) ) \,. </annotation></semantics></math></div></div> <p>(<a href="Bogoliubov's+formula#BogoliubovShirkov59">Bogoliubov-Shirkov 59</a>; the above terminology follows <a href="Møller+operator#HawkinsRejzner16">Hawkins-Rejzner 16, below def. 5.1</a>)</p> <p>Notice that compared to Fredenhagen-Rejzner et. al. we have changed notation conventions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi><mo>↔</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{R} \leftrightarrow \mathcal{R}^{-1}</annotation></semantics></math> in order to bring out the analogy to (the conventions for the) <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo><mi>𝒯</mi><mo stretchy="false">(</mo><msup><mi>𝒯</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>𝒯</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_1 \star_F A_2 = \mathcal{T}(\mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2))</annotation></semantics></math> on regular polynomial observables.</p> <p>notice the implicit dependencies</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a> of <br /> <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a></th><th>meaning</th><th>depends on choice of</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\phantom{AA}\mathcal{T}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordering</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> and <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mi>𝒮</mi></mrow><annotation encoding="application/x-tex">\phantom{AA}\mathcal{S}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> and <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\phantom{AA}\mathcal{R}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> and <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> and <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></td></tr> </tbody></table> <div class="num_defn" id="FieldAlgebraObservablesInteracting"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a>)</strong></p> <p>Given an <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> non-point-<a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> in the form of a <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observable">regular polynomial observable</a> in degree 0</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>int</mi></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mfrac linethickness="0"><mrow><mi>reg</mi></mrow><mrow><mrow><mi>deg</mi><mo>=</mo><mn>0</mn></mrow></mrow></mfrac></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> S_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ] \,, </annotation></semantics></math></div> <p>then the <em><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a></em> <a class="existingWikiWord" href="/nlab/show/structure">structure</a> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</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>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mover><mo>⟶</mo><mrow><msub><mo>⋆</mo> <mi>int</mi></msub></mrow></mover><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \otimes PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \overset{ \star_{int} }{\longrightarrow} PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/conjugation">conjugation</a> of the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> by the <a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a> (def. <a class="maruku-ref" href="#MollerOperatorOnRegularPolynomialObservables"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>int</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>ℛ</mi><mrow><mo>(</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>H</mi></msub><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> A_1 \star_{int} A_2 \;\coloneqq\; \mathcal{R} \left( \mathcal{R}^{-1}(A_1) \star_H \mathcal{R}^{-1}(A_2) \right) </annotation></semantics></math></div></div> <p>(e.g. <a href="#FredenhagenRejzner11b">Fredenhagen-Rejzner 11b, (19)</a>)</p> <h3 id="interacting_quantum_bvdifferential">Interacting quantum BV-differential</h3> <p>Recall how the global BV-differential</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>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</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>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \{S',-\} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] </annotation></semantics></math></div> <p>on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> (<a href="A+first+idea+of+quantum+field+theory#BVDifferentialGlobal">this def.</a>) is conjugated into the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> via the time ordering operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi><mo>∘</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo>∘</mo><msup><mi>𝒯</mi> <mo lspace="verythinmathspace" rspace="0em">−</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-}</annotation></semantics></math> (<a href="BV-operator#GaugeFixedActionFunctionalTimeOrderedAntibracket">this prop.</a>).</p> <p>In the same way we may use the <a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operators">quantum Møller operators</a> to conjugate the BV-differential into the regular part of the <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>:</p> <div class="num_defn" id="BVDifferentialInteractingQuantum"> <h6 id="definition_4">Definition</h6> <p><strong>(interacting quantum <a class="existingWikiWord" href="/nlab/show/BV-differential">BV-differential</a>)</strong></p> <p>Given an <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> non-point-<a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> in the form of a <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observable">regular polynomial observable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">S_{int}</annotation></semantics></math>, then the <em>interacting quantum BV-differential</em> on the <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a> (def. <a class="maruku-ref" href="#FieldAlgebraObservablesInteracting"></a>) on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> is the <a class="existingWikiWord" href="/nlab/show/conjugation">conjugation</a> of the plain <a class="existingWikiWord" href="/nlab/show/BV-differential">BV-differential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{-S',-\}</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a> induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">S_{int}</annotation></semantics></math> (def. <a class="maruku-ref" href="#MollerOperatorOnRegularPolynomialObservables"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi><mo>∘</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>∘</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mi>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</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>reg</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \,. </annotation></semantics></math></div></div> <p>(<a href="#Rejzner11">Rejzner 11, (5.38)</a>)</p> <div class="num_prop" id="QuantumMasterEquation"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a> and <a class="existingWikiWord" href="/nlab/show/quantum+master+Ward+identity">quantum master Ward identity</a> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a>)</strong></p> <p>Consider an <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> non-point-<a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> in the form of a <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observable">regular polynomial observable</a> in degree 0</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>int</mi></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mtext>BV-BRST</mtext></msub><msub><mo stretchy="false">)</mo> <mfrac linethickness="0"><mrow><mi>reg</mi></mrow><mrow><mrow><mi>deg</mi><mo>=</mo><mn>0</mn></mrow></mrow></mfrac></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> S_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ] \,, </annotation></semantics></math></div> <p>Then the following are equivalent:</p> <ol> <li> <p>The <em><a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a></em> (QME)</p> <div class="maruku-equation" id="eq:OnRegularObservablesQuantumMasterEquation"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}} + i \hbar \Delta_{BV}( S' + S_{int} ) \;=\; 0 \,. </annotation></semantics></math></div></li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/perturbative+S-matrix">perturbative S-matrix</a> (def. <a class="maruku-ref" href="#OnRegularObservablesPerturbativeSMatrix"></a>) is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BV</mi></mrow><annotation encoding="application/x-tex">BV</annotation></semantics></math>-closed</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{-S', \mathcal{S}(S_{int})\} = 0 \,. </annotation></semantics></math></div></li> <li> <p>The quantum <em><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></em> (MWI) on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> <em>in terms of <a class="existingWikiWord" href="/nlab/show/retarded+products">retarded products</a></em>:</p> <div class="maruku-equation" id="eq:OnRegularObservablesQuantumMasterWardIdentity"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi><mo>∘</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>∘</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>}</mo></mrow> <mi>𝒯</mi></msub><mo>−</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub></mrow><annotation encoding="application/x-tex"> \mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \;=\; \left\{ -(S' + S_{int}) \,,\, (-) \right\}_{\mathcal{T}} - i \hbar \Delta_{BV} </annotation></semantics></math></div> <p>(<a href="Ward+identity#Duetsch18">Dütsch 18, (4.2)</a>)</p> <p>expressing the interacting quantum <a class="existingWikiWord" href="/nlab/show/BV-differential">BV-differential</a> (def. <a class="maruku-ref" href="#BVDifferentialInteractingQuantum"></a>) as the sum of the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered</a> <a class="existingWikiWord" href="/nlab/show/antibracket">antibracket</a> (<a href="BV-operator#AntibracketTimeOrdered">this def.</a>) with the <em>total</em> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">S' + S_{int}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">i \hbar</annotation></semantics></math> times the <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> (<a href="BV-operator#ForGaugeFixedFreeLagrangianFieldTheoryBVOperator">BV-operator</a>).</p> </li> <li> <p>The quantum <em><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></em> (MWI) on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> <em>in terms of <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a></em>:</p> <div class="maruku-equation" id="eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mrow><mo>(</mo><msub><mrow><mo>{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>}</mo></mrow> <mi>𝒯</mi></msub><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathcal{S}(-S_{int}) \star_F \{-S', \mathcal{S}(S_{int}) \star_F (-)\} \;=\; - \left( \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right) </annotation></semantics></math></div> <p>(<a href="Ward+identity#Duetsch18">Dütsch 18, (4.8)</a>)</p> </li> </ol> </div> <p>(<a href="#Rejzner11">Rejzner 11, (5.35) - (5.38)</a>, following <a href="Ward+identity#Hollands07">Hollands 07, (342)-(345)</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>To see that the first two conditions are equivalent, we compute as follows</p> <div class="maruku-equation" id="eq:QuantumMasterOnRegularObservablesBVDifferentialOfSMatrixInTerms"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>}</mo></mrow></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><mo>}</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><munder><mrow><msub><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><mo>}</mo></mrow> <mi>𝒯</mi></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mrow><mstyle displaystyle="false"><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>S</mi><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub></mrow></mrow><mrow><mrow><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow></mrow></mrow></mfrac></munder><mo>−</mo><mi>i</mi><mi>ℏ</mi><munder><munder><mrow><msub><mi>Δ</mi> <mi>BV</mi></msub><mrow><mo>(</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow></mfrac></mstyle><msub><mrow><mo>{</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>}</mo></mrow> <mi>𝒯</mi></msub><mo>)</mo></mrow></mrow><mrow><mrow><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow></mrow></mrow></mfrac></munder></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><munder><munder><mrow><mo>(</mo><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo>,</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">}</mo><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">{</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">}</mo><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⏟</mo></munder><mtext>QME</mtext></munder><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left\{ -S', \mathcal{S}(S_{int}) \right\} & = \left\{ -S' , \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \right\} \\ & = \underset{ { \tfrac{-1}{i \hbar} \{S',S\}_{\mathcal{T}} } \atop { \star_F \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) } }{ \underbrace{ \left\{ -S' , \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \right\}_{\mathcal{T}} } } - i \hbar \underset{ { \left( \tfrac{1}{i \hbar} \Delta_{BV}(S_{int}) + \tfrac{1}{2 (i \hbar)^2} \left\{ S_{int}, S_{int} \right\}_{\mathcal{T}} \right) } \atop { \star_{F} \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) } }{ \underbrace{ \Delta_{BV} \left( \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \right) } } \\ & = \tfrac{-1}{i \hbar} \underset{ \text{QME} }{ \underbrace{ \left( \{S',S_{int}\} + \tfrac{1}{2}\{S_{int}, S_{int}\} + i \hbar \Delta_{BV}(S_{int}) \right) } } \star_F \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \end{aligned} </annotation></semantics></math></div> <p>Here in the first step we used the definition of the <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> (<a href="ForGaugeFixedFreeLagrangianFieldTheoryBVOperator">this def.</a>) to rewrite the plain antibracket in terms of the time-ordered antibracket (<a href="BV-operator#AntibracketTimeOrdered">this def.</a>), then under the second brace we used that the time-ordered antibracket is the failure of the BV-operator to be a derivation (<a href="BV-operator#AntibracketBVOperatorRelation">this prop</a>) and under the first brace the consequence of this statement for application to exponentials (<a href="BV-operator#TimeOrderedExponentialBVOperator">this example</a>). Finally we collected terms, and to “complete the square” we added the terms on the left of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><munder><munder><mrow><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>S</mi><mo>′</mo><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mo>−</mo><mi>i</mi><mi>ℏ</mi><munder><munder><mrow><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \frac{1}{2} \underset{= 0}{\underbrace{\{S', S'\}_{\mathcal{T}}}} - i \hbar \underset{ = 0}{\underbrace{ \Delta_{BV}(S')}} = 0 </annotation></semantics></math></div> <p>which vanish because, by definition of <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> (<a href="A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity">this def.</a>), the free gauge-fixed action functional <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">S'</annotation></semantics></math> is independent of <a class="existingWikiWord" href="/nlab/show/antifields">antifields</a>.</p> <p>But since the operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int} \right)</annotation></semantics></math>, this implies the claim.</p> <p>Next we show that the <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a> implies the <a class="existingWikiWord" href="/nlab/show/quantum+master+Ward+identities">quantum master Ward identities</a>.</p> <p>We use that the BV-differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{-S',-\}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> product <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> (lemma <a class="maruku-ref" href="#DerivationBVDifferentialForWickAlgebra"></a>).</p> <p>First of all this implies that with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\{-S', \mathcal{S}(S_{int})\} = 0</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">}</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\{-S', \mathcal{S}(S_{int})^{-1}\} = 0</annotation></semantics></math>.</p> <p>Thus 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><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo>∘</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mo>−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mo>⋆</mo> <mi>H</mi></msub><mrow><mo>(</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mi>a</mi><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><mphantom><mo lspace="verythinmathspace" rspace="0em">+</mo></mphantom><munder><munder><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>}</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><msub><mo>⋆</mo> <mi>H</mi></msub><mrow><mo>(</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mo>⋆</mo> <mi>H</mi></msub><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi><mo>}</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mo>⋆</mo> <mi>H</mi></msub><mrow><mo>(</mo><munder><munder><mrow><mi>𝒮</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mi>𝒮</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>1</mn></mrow></munder><msub><mo>⋆</mo> <mi>F</mi></msub><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi><mo>}</mo></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mo>⋆</mo> <mi>H</mi></msub><mrow><mo>(</mo><mi>𝒮</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><munder><munder><mrow><mi>𝒮</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi><mo>}</mo></mrow></mrow><mo>⏟</mo></munder><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></munder><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><munder><munder><mrow><mphantom><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac></mphantom><mi>𝒮</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi><mo>}</mo></mrow></mrow><mo>⏟</mo></munder><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></munder><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \{-S', -\} \circ \mathcal{R}^{-1}(A) & = \{-S', \mathcal{R}^{-1}(A)\} \\ & = \left\{ { \, \atop \, } -S', \mathcal{S}(S_{int})^{-1} \star_H \left( \mathcal{S}(S_{int}) \star_F a \right) {\, \atop \,} \right\} \\ & = \phantom{+} \underset{ = 0 }{ \underbrace{ \left\{ -S', \mathcal{S}(S_{int})^{-1} \right\} } } \star_H \left( \mathcal{S}(S_{int}) \star_F A \right) \\ & \phantom{=} + \mathcal{S}(S_{int})^{-1} \star_H \left\{ -S', \mathcal{S}(S_{int}) \star_F A \right\} \\ & = \mathcal{S}(S_{int})^{-1} \star_H \left( \underset{ = 1 }{ \underbrace{ \mathcal{S}(+ S_{int}) \star_F \mathcal{S}(- S_{int}) } } \star_F \left\{ -S', \mathcal{S}(S_{int}) \star_F A \right\} \right) \\ & = \mathcal{S}(S_{int})^{-1} \star_H \left( \mathcal{S}(+ S_{int}) \star_F \underset{ (\ast) }{ \underbrace{ \mathcal{S}(- S_{int}) \star_F \left\{ -S', \mathcal{S}(S_{int}) \star_F A \right\} } } \right) \\ & = \mathcal{R}^{-1} \left( \underset{ (\ast) }{ \underbrace{ \phantom{\, \atop \,} \mathcal{S}(-S_{int}) \star_F \left\{ -S', \mathcal{S}(S_{int}) \star_F A \right\} } } \right) \end{aligned} </annotation></semantics></math></div> <p>By applying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> to both sides of this equation, this means first of all that the interacting quantum BV-differential is equivalently given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi><mo>∘</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>∘</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>𝒮</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1} \;=\; \mathcal{S}(-S_{int}) \star_F \{-S', \mathcal{S}(S_{int}) \star_F (-)\} \,, </annotation></semantics></math></div> <p>hence that if either version <a class="maruku-eqref" href="#eq:OnRegularObservablesQuantumMasterWardIdentity">(2)</a> or <a class="maruku-eqref" href="#eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered">(3)</a> of the <a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a> holds, it implies the other.</p> <p>Now expanding out the definition of <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> (def. <a class="maruku-ref" href="#OnRegularObservablesPerturbativeSMatrix"></a>) and expressing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{-S',-\}</annotation></semantics></math> via the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered</a> <a class="existingWikiWord" href="/nlab/show/antibracket">antibracket</a> (<a href="BV-operator#AntibracketTimeOrdered">this def.</a>) and the <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>BV</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_{BV}</annotation></semantics></math> (<a href="BV-operator#ForGaugeFixedFreeLagrangianFieldTheoryBVOperator">this prop.</a>) as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>−</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub></mrow><annotation encoding="application/x-tex"> \{-S',-\} \;=\; \{-S',-\}_{\mathcal{T}} - i \hbar \Delta_{BV} </annotation></semantics></math></div> <p>(on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a>), we continue computing as follows:</p> <div class="maruku-equation" id="eq:QMESecondStep"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mi>ℛ</mi><mo>∘</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>∘</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><msub><mo>⋆</mo> <mi>F</mi></msub><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi><mo>}</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><msub><mo>⋆</mo> <mi>F</mi></msub><mrow><mo>(</mo><msub><mrow><mo>{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi><mo>}</mo></mrow> <mi>𝒯</mi></msub><mo>−</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mrow><mo>(</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi><mo>)</mo></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo lspace="verythinmathspace" rspace="0em">+</mo></mphantom><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><msub><mi>S</mi> <mi>int</mi></msub><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi><mo>+</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>A</mi><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mi>i</mi><mi>ℏ</mi><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><msub><mo>⋆</mo> <mi>F</mi></msub><mrow><mo>(</mo><munder><munder><mrow><msub><mi>Δ</mi> <mi>BV</mi></msub><mrow><mo>(</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow></mfrac></mstyle><mrow><mo>{</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow><mrow><mrow><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow></mrow></mrow></mfrac></munder><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi><mspace width="thinmathspace"></mspace><mo>+</mo><mspace width="thinmathspace"></mspace><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>+</mo><mspace width="thinmathspace"></mspace><munder><munder><mrow><msub><mrow><mo>{</mo><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo>}</mo></mrow> <mi>𝒯</mi></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mrow><msub><mi>exp</mi> <mi>𝒯</mi></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow></mrow></mrow><mrow><mrow><msub><mo>⋆</mo> <mi>F</mi></msub><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><mo stretchy="false">{</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">}</mo></mrow></mrow></mfrac></munder><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mrow><mo>(</mo><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><munder><munder><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⏟</mo></munder><mtext>QME</mtext></munder><msub><mo>⋆</mo> <mi>F</mi></msub><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mrow><mo>(</mo><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}( A ) \\ & = \exp_{\mathcal{T}} \left( \tfrac{-1}{i \hbar} S_{int} \right) \star_F \left\{ -S', \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \star_F A \right\} \\ & = \exp_{\mathcal{T}} \left( \tfrac{-1}{i \hbar} S_{int} \right) \star_F \left( \left\{ -S', \exp_{\mathcal{T}} \left( \tfrac{ 1 }{i \hbar} S_{int} \right) \star_F A \right\}_{\mathcal{T}} - i \hbar \Delta_{BV} \left( \exp_{\mathcal{T}} \left( \tfrac{ 1 }{i \hbar} S_{int} \right) \star_F A \right) \right) \\ & \phantom{+} = \tfrac{1}{i \hbar} \{ -S', S_{int} \}_{\mathcal{T}} \star_F A + \{-S', A\}_{\mathcal{T}} \\ & \phantom{=} - i \hbar \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int}\right) \star_F \left( \underset{ { \left( \tfrac{1}{i \hbar}\Delta_{BV}(S_{int}) + \tfrac{1}{2 (i \hbar)^2} \left\{ S_{int}, S_{int} \right\} \right) } \atop { \star_F \exp_{\mathcal{T}}\left( \tfrac{ 1 }{i \hbar} S_{int} \right) } }{ \underbrace{ \Delta_{BV} \left( \exp_{\mathcal{T}} \left( \tfrac{ 1}{i \hbar} S_{int} \right) \right) } } \star_F A \,+\, \exp_{\mathcal{T}} \left( \tfrac{ 1}{i \hbar} S_{int} \right) \star_F \Delta_{BV}(A) \,+\, \underset{ { \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right) } \atop { \star_F \tfrac{ 1}{i \hbar} \{S_{int}, A\} } }{ \underbrace{ \left\{ \exp_{\mathcal{T}} \left( \tfrac{ 1}{i \hbar} S_{int} \right) \,,\, A \right\}_{\mathcal{T}} } } \right) \\ & = - \left( \{ S' + S_{int}\,,\, A\}_{\mathcal{T}} + i \hbar \Delta_{BV}(A) \right) \\ & \phantom{=} - \tfrac{1}{i \hbar} \underset{ \text{QME} }{ \underbrace{ \left( \tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}} + i \hbar \Delta_{BV}( S' + S_{int} ) \right) }} \star_F A \\ & = - \left( \{ S' + S_{int}\,,\, A\}_{\mathcal{T}} + i \hbar \Delta_{BV}(A) \right) \end{aligned} </annotation></semantics></math></div> <p>Here in the line with the braces we used that the <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> is a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> up to correction by the time-ordered <a class="existingWikiWord" href="/nlab/show/antibracket">antibracket</a> (<a href="BV-operator#AntibracketBVOperatorRelation">this prop.</a>), and under the first brace we used the effect of that property on time-ordered exponentials (<a href="BV-operator#TimeOrderedExponentialBVOperator">this example</a>), while under the second brace we used that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub></mrow><annotation encoding="application/x-tex">\{(-),A\}_{\mathcal{T}}</annotation></semantics></math> is a derivation of the time-ordered product. Finally we have collected terms, added <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>=</mo><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>S</mi><mo>′</mo><mo stretchy="false">}</mo><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0 = \{S',S'\} + i \hbar \Delta_{BV}(S')</annotation></semantics></math> as before, and then used the QME.</p> <p>This shows that the quantum <a class="existingWikiWord" href="/nlab/show/master+Ward+identities">master Ward identities</a> follow from the <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a>. To conclude, it is now sufficient to show that, conversely, the MWI in terms of, say, retarded products implies the QME.</p> <p>To see this, observe that with the BV-differential being nilpotent, also its conjugation by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> is, so that with the above we have:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><msup><mrow><mo>(</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo>)</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><msup><mrow><mo>(</mo><mi>ℛ</mi><mo>∘</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>∘</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><munder><mrow><msup><mrow><mo>(</mo><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mo>)</mo></mrow> <mn>2</mn></msup></mrow><mo>⏟</mo></munder><mrow><mo>{</mo><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mspace width="thinmathspace"></mspace></mrow></mfrac><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>}</mo></mrow></munder><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \left( \{-S',-\}\right)^2 = 0 \\ \Leftrightarrow \; & \left( \mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \right)^2 = 0 \\ \Leftrightarrow \; & \underset{ \left\{ {\, \atop \,} \tfrac{1}{2}\{S' + S_{int}, S' + S_{int}\}_{\mathcal{T}} + i \hbar \Delta_{BV}(S' + S_{int}) \,,\, (-) \right\} }{ \underbrace{ \left( \{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right)^2 } } = 0 \end{aligned} </annotation></semantics></math></div> <p>Here under the brace we computed 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><msup><mrow><mo>(</mo><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>+</mo><mi>i</mi><mi>ℏ</mi><msub><mi>Δ</mi> <mi>BV</mi></msub><mo>)</mo></mrow> <mn>2</mn></msup></mtd> <mtd><mo>=</mo><mphantom><mo lspace="verythinmathspace" rspace="0em">+</mo></mphantom><munder><munder><mrow><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub></mrow><mo>⏟</mo></munder><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>S</mi><mo>,</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>S</mi><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub></mrow></munder></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mi>i</mi><mi>ℏ</mi><munder><munder><mrow><mo>(</mo><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>∘</mo><msub><mi>Δ</mi> <mi>BV</mi></msub><mo>+</mo><msub><mi>Δ</mi> <mi>BV</mi></msub><mo>∘</mo><mo stretchy="false">{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub><mo>)</mo></mrow><mo>⏟</mo></munder><mrow><mo stretchy="false">{</mo><msub><mi>Δ</mi> <mi>BV</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>S</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mi>𝒯</mi></msub></mrow></munder></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><munder><munder><mrow><msub><mi>Δ</mi> <mi>BV</mi></msub><mo>∘</mo><msub><mi>Δ</mi> <mi>BV</mi></msub></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left( \{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right)^2 & = \phantom{+} \underset{ \tfrac{1}{2} \{ \{S' + S, S'+ S\}_{\mathcal{T}}, (-) \}_{\mathcal{T}} }{ \underbrace{ \{S' + S_{int}, \{S' + S_{int}\}_{\mathcal{T}}, (-) \}_{\mathcal{T}} }} \\ & \phantom{=} + i \hbar \underset{ \{ \Delta_{BV}(S'+ S)\,,\, (-) \}_{\mathcal{T}} }{ \underbrace{ \left( \{S' + S_{int}, (-)\}_{\mathcal{T}} \circ \Delta_{BV} + \Delta_{BV} \circ \{S' + S_{int}, (-)\}_{\mathcal{T}} \right) }} \\ & \phantom{=} + (i \hbar)^2 \underset{= 0} { \underbrace{ \Delta_{BV} \circ \Delta_{BV} } } \end{aligned} \,. </annotation></semantics></math></div> <p>where, in turn, the term under the first brace follows by the graded <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a>, the one under the second brace by Henneaux-Teitelboim (15.105c) and the one under the third brace by Henneaux-Teitelboim (15.105b).</p> </div> <div class="num_example" id="MasterWardIdentityClassical"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/classical+master+Ward+identity">classical master Ward identity</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/classical+limit">classical limit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\hbar \to 0</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/quantum+master+Ward+identity">quantum master Ward identity</a> <a class="maruku-eqref" href="#eq:OnRegularObservablesQuantumMasterWardIdentity">(2)</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>∘</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>−</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mo>{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>}</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{-S',(-)\} \circ \mathcal{R}^{-1} \;=\; - \mathcal{R}^{-1} \left( \left\{ S' + S_{int} \,,\, (-) \right\} \right) \,. </annotation></semantics></math></div> <p>Applied to an observable which is linear in the <a class="existingWikiWord" href="/nlab/show/antifields">antifields</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo>∫</mo><mi>Σ</mi></munder><msup><mi>A</mi> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msubsup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi> <mo>‡</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A \;=\; \underset{\Sigma}{\int} A^a(x) \mathbf{\Phi}^\ddagger_a(x) \, dvol_\Sigma(x) </annotation></semantics></math></div> <p>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><mn>0</mn></mtd> <mtd><mo>=</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo>′</mo><mo>,</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>+</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><msub><mrow><mo>{</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo>}</mo></mrow> <mi>𝒯</mi></msub><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo>∫</mo><mi>Σ</mi></munder><mfrac><mrow><mi>δ</mi><mi>S</mi><mo>′</mo></mrow><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>A</mi> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>ℛ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><munder><mo>∫</mo><mi>Σ</mi></munder><msup><mi>A</mi> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>δ</mi><mo stretchy="false">(</mo><mi>S</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} 0 & = \{-S', \mathcal{R}^{-1}(A)\} + \mathcal{R}^{-1} \left( \left\{ S' + S_{int} \,,\, A \right\}_{\mathcal{T}} \right) \\ & = \underset{\Sigma}{\int} \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \mathcal{R}^{-1}(A^a(x)) \, dvol_\Sigma(x) + \mathcal{R}^{-1} \left( \underset{\Sigma}{\int} A^a(x) \frac{\delta (S' + S_{int})}{\delta \mathbf{\Phi}^a(x)} \, dvol_\Sigma(x) \right) \end{aligned} </annotation></semantics></math></div> <p>In this form the <em>classical Master Ward identity</em> was originally identified in (<a href="master+Ward+identity#DuetschFredenhagen02">Dütsch-Fredenhagen 02, (90)</a>, <a href="master+Ward+identity#BrennecketDuetsch07">Brennecke-Dütsch 07, (5.5)</a>, following <a href="master+Ward+identity#DuetschBoas02">Dütsch-Boas 02</a>).</p> </div> <h3 id="RenormalizationAndMasterWardIdentity">Renormalization and Master ward identity</h3> <p>The quantum master equation in the form of prop. <a class="maruku-ref" href="#QuantumMasterEquation"></a> is derived on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a>, in particular hence for non-point-<a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">S_{int}</annotation></semantics></math>. But the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> terms of interest are point-interactions, hence are <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a>. The <a class="existingWikiWord" href="/nlab/show/extension">extension</a> of the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> and hence of the <a class="existingWikiWord" href="/nlab/show/perturbative+S-matrix">perturbative S-matrix</a> from regular to local onservables exsists but involves choices, these are the <em><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></em> choices in the formulation of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>.</p> <p>Since for <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauged fixed</a> <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> this physically relevant <a class="existingWikiWord" href="/nlab/show/observables">observables</a> are not the plain (<a class="existingWikiWord" href="/nlab/show/microcausal+polynomial+observables">mcirocausal</a>) <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a>, but the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/BV-BRST+differential">BV-BRST differential</a> on them, one needs to require for gauge theories that the <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a> still holds after <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a>. This is closely related to the <a class="existingWikiWord" href="/nlab/show/renormalization+condition">renormalization condition</a> called the <em><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></em> (<a href="#Rejzner11">Rejzner 11 (prop. 5.3.1) and following paragraphs</a>). If the quantum master equation cannot be retained in <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> one says that the field theory suffers from a <em><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></li> </ul> <h2 id="references">References</h2> <p>The concept originates with</p> <ul> <li id="BatalinVilkovisky81"><a class="existingWikiWord" href="/nlab/show/Igor+Batalin">Igor Batalin</a>, <a class="existingWikiWord" href="/nlab/show/Grigori+Vilkovisky">Grigori Vilkovisky</a>, <em>Gauge Algebra and Quantization</em>, Phys. Lett. B 102 (1): 27–31, 1981 (<a href="https://doi.org/10.1016/0370-2693(81)90205-7">doi:10.1016/0370-2693(81)90205-7</a>)</li> </ul> <p>Traditional review includes</p> <ul> <li id="HenneauxTeitelboim92"><a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <a class="existingWikiWord" href="/nlab/show/Claudio+Teitelboim">Claudio Teitelboim</a>, section 15.5.3 of <em><a class="existingWikiWord" href="/nlab/show/Quantization+of+Gauge+Systems">Quantization of Gauge Systems</a></em>, Princeton University Press, 1992</li> </ul> <p>Discussion in the rigorous context of <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a> <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a> formulated in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> is in:</p> <ul> <li id="FredenhagenRejzner11a"> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, <em>Batalin-Vilkovisky formalism in the functional approach to classical field theory</em>, Commun. Math. Phys. 314(1), 93–127 (2012) (<a href="https://arxiv.org/abs/1101.5112">arXiv:1101.5112</a>)</p> </li> <li id="FredenhagenRejzner11b"> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, <em>Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory</em>, Commun. Math. Phys. 317(3), 697–725 (2012) (<a href="https://arxiv.org/abs/1110.5232">arXiv:1110.5232</a>)</p> </li> <li id="Rejzner11"> <p><a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, <em>Batalin-Vilkovisky formalism in locally covariant field theory</em> (<a href="https://arxiv.org/abs/1111.5130">arXiv:1111.5130</a>)</p> </li> <li id="Rejzner13"> <p><a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, <em>Remarks on local symmetry invariance in perturbative algebraic quantum field theory</em> (<a href="https://arxiv.org/abs/1301.7037">arXiv:1301.7037</a>)</p> </li> </ul> <p>and surveyed in</p> <ul> <li id="Rejzner16"><a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, section 7 of <em>Perturbative algebraic quantum field theory</em> Springer 2016 (<a href="https://link.springer.com/book/10.1007%2F978-3-319-25901-7">web</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 20, 2021 at 16:12:14. 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