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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Weyl relation </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/8207/#Item_3" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebraic_quantum_field_theory">Algebraic Quantum Field Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a></p> <h2 id="concepts">Concepts</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a></strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+history">field history</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/observables">observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+net">field net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> <p><a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a></p> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a></p> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a></p> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>)</p> </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#in_the_wick_algebra_of_free_quantum_fields'>In the Wick algebra of free quantum fields</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>What are called <em>Weyl relations</em> is the incarnation of <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a> under passing to <a class="existingWikiWord" href="/nlab/show/exponentials">exponentials</a>, constituting the <em><a class="existingWikiWord" href="/nlab/show/Weyl+algebra">Weyl algebra</a></em>.</p> <p>For example if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">a, a^\ast</annotation></semantics></math> are two elements of an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> with <a class="existingWikiWord" href="/nlab/show/commutator">commutator</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><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mo>*</mo></msup><mo stretchy="false">]</mo><mo>=</mo><mi>ℏ</mi></mrow><annotation encoding="application/x-tex"> [a,a^\ast] = \hbar </annotation></semantics></math></div> <p>then the corresponding Weyl relation is, by the <a class="existingWikiWord" href="/nlab/show/Baker-Campbell-Hausdorff+formula">Baker-Campbell-Hausdorff 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>e</mi> <mrow><mi>z</mi><mi>a</mi></mrow></msup><msup><mi>e</mi> <mrow><msup><mi>z</mi> <mo>*</mo></msup><msup><mi>a</mi> <mo>*</mo></msup></mrow></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mi>e</mi> <mrow><msup><mi>z</mi> <mo>*</mo></msup><msup><mi>a</mi> <mo>*</mo></msup></mrow></msup><msup><mi>e</mi> <mrow><mi>z</mi><mi>a</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>ℏ</mi><mi>z</mi><msup><mi>z</mi> <mo>*</mo></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> e^{z a} e^{z^\ast a^\ast} \;=\; e^{z^\ast a^\ast} e^{z a} e^{\hbar z z^\ast} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>,</mo><msup><mi>z</mi> <mo>*</mo></msup><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">z,z^\ast \in \mathbb{C}</annotation></semantics></math>.</p> <h2 id="in_the_wick_algebra_of_free_quantum_fields">In the Wick algebra of free quantum fields</h2> <div class="num_prop" id="MoyalStarProductOnMicrocausal"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Hadamard+distribution">Hadamard</a>-<a class="existingWikiWord" href="/nlab/show/Moyal+star+product">Moyal star product</a> on <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> – <a class="existingWikiWord" href="/nlab/show/abstract+Wick+algebra">abstract Wick algebra</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,\mathbf{L})</annotation></semantics></math> 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 <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+equation">Green hyperbolic</a> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>Φ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">P \Phi = 0</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mi>Δ</mi><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex"> \Delta_H \;=\; \tfrac{i}{2}\Delta + H </annotation></semantics></math></div> <p>be a corresponding <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> (<a class="existingWikiWord" href="/nlab/show/Hadamard+2-point+function">Hadamard 2-point function</a>).</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⋆</mo> <mi>H</mi></msub><mi>A</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>prod</mi><mo>∘</mo><mi>exp</mi><mrow><mo>(</mo><msub><mo>∫</mo> <mrow><msup><mi>X</mi> <mn>2</mn></msup></mrow></msub><mi>ℏ</mi><msubsup><mi>Δ</mi> <mi>H</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mi>Φ</mi> <mi>a</mi></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>⊗</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mi>Φ</mi> <mi>b</mi></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mfrac><msub><mi>dvol</mi> <mi>g</mi></msub><mo>)</mo></mrow><mo stretchy="false">(</mo><msub><mi>P</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>P</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A \star_H A \;\coloneqq\; prod \circ \exp\left( \int_{X^2} \hbar \Delta_H^{a b}(x_1, x_2) \frac{\delta}{\delta \Phi^a(x_1)} \otimes \frac{\delta}{\delta \Phi^b(x_2)} dvol_g \right) (P_1 \otimes P_2) </annotation></semantics></math></div> <p>on <a class="existingWikiWord" href="/nlab/show/off-shell">off-shell</a> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</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><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>ℱ</mi> <mi>mc</mi></msub></mrow><annotation encoding="application/x-tex">A_1, A_2 \in \mathcal{F}_{mc}</annotation></semantics></math> is well defined in that the <a class="existingWikiWord" href="/nlab/show/wave+front+sets">wave front sets</a> involved in the <a class="existingWikiWord" href="/nlab/show/products+of+distributions">products of distributions</a> that appear in expanding out the <a class="existingWikiWord" href="/nlab/show/exponential">exponential</a> satisfy <a class="existingWikiWord" href="/nlab/show/H%C3%B6rmander%27s+criterion">Hörmander's criterion</a>.</p> <p>Hence by the general properties of <a class="existingWikiWord" href="/nlab/show/star+products">star products</a> (<a href="star+product#AssociativeAndUnitalStarProduct">this prop.</a>) this yields a <a class="existingWikiWord" href="/nlab/show/unital+algebra">unital</a> <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> <a class="existingWikiWord" href="/nlab/show/structure">structure</a> on the space of <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/off-shell">off-shell</a> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mo>⋆</mo> <mi>H</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( PolyObs(E)_{mc}[ [\hbar] ] \,,\, \star_H \right) \,. </annotation></semantics></math></div> <p>This is the <em><a class="existingWikiWord" href="/nlab/show/off-shell">off-shell</a> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></em> corresponding to the choice of <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>.</p> <p>Moreover the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is an ideal with respect to this algebra structure, so that it descends to the <a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> to yield the <em><a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mo>⋆</mo> <mi>H</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( PolyObs(E,\mathbf{L})_{mc}[ [ \hbar ] ] \,,\, \star_H \right) \,. </annotation></semantics></math></div> <p>Finally, under <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</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><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(-)^\ast</annotation></semantics></math> these are <a class="existingWikiWord" href="/nlab/show/star+algebras">star algebras</a> in that</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>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>)</mo></mrow> <mo>*</mo></msup><mo>=</mo><msubsup><mi>A</mi> <mn>2</mn> <mo>*</mo></msubsup><msub><mo>⋆</mo> <mi>H</mi></msub><msubsup><mi>A</mi> <mn>1</mn> <mo>*</mo></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( A_1 \star_H A_2 \right)^\ast = A_2^\ast \star_H A_1^\ast \,. </annotation></semantics></math></div></div> <p>For <strong>proof</strong> see at <em><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></em> <a href="Wick+algebra#MoyalStarProductOnMicrocausal">this prop.</a>.</p> <div class="num_remark" id="WickAlgebraIsFormalDeformationQuantization"> <h6 id="remark">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> is <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a> of <a class="existingWikiWord" href="/nlab/show/Poisson-Peierls+bracket">Poisson-Peierls algebra of observables</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,\mathbf{L})</annotation></semantics></math> 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 <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+equation">Green hyperbolic</a> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>Φ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">P \Phi = 0</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> and 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><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mi>Δ</mi><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta_H \;=\; \tfrac{i}{2}\Delta + H</annotation></semantics></math> be a corresponding <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> (<a class="existingWikiWord" href="/nlab/show/Hadamard+2-point+function">Hadamard 2-point function</a>).</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mo>⋆</mo> <mi>H</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\left( PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ] \,,\, \star_H \right)</annotation></semantics></math> from prop. <a class="maruku-ref" href="#MoyalStarProductOnMicrocausal"></a> is a <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a> of the <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a> on the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> given by the <a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> equipped with the <a class="existingWikiWord" href="/nlab/show/Poisson-Peierls+bracket">Poisson-Peierls bracket</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo>⊗</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo>→</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub></mrow><annotation encoding="application/x-tex">\{-,-\} \;\colon\; PolyObs(E,\mathbf{L})_{mc} \otimes PolyObs(E,\mathbf{L})_{mc} \to PolyObs(E,\mathbf{L})_{mc}</annotation></semantics></math> in that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>∈</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub></mrow><annotation encoding="application/x-tex">A_1, A_2 \in PolyObs(E,\mathbf{L})_{mc}</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><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><mo>⋅</mo><msub><mi>A</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mi>mod</mi><mspace width="thickmathspace"></mspace><mi>ℏ</mi></mrow><annotation encoding="application/x-tex"> A_1 \star_H A_2 \;=\; A_1 \cdot A_2 \;mod\; \hbar </annotation></semantics></math></div> <p>and</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>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>A</mi> <mn>2</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>i</mi><mi>ℏ</mi><mo stretchy="false">{</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mi>mod</mi><mspace width="thickmathspace"></mspace><msup><mi>ℏ</mi> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A_1 \star_H A_2 - A_2 \star_H A_1 \;=\; i \hbar \{A_1, A_2\} \;mod\; \hbar^2 \,. </annotation></semantics></math></div></div> <p>(<a href="#Dito90">Dito 90</a>, <a href="Wick+algebra#DutschFredenhagen01">Dütsch-Fredenhagen 01</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>By prop. <a class="maruku-ref" href="#MoyalStarProductOnMicrocausal"></a> this is immediate from the general properties of the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> (<a href="A+first+idea+of+quantum+field+theory+--+Quantization#MoyalStarProductIsFormalDeformationQuantization">this example</a>).</p> <p>Explicitly, consider, without restriction of generality, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>=</mo><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>a</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><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_1 = \int (\alpha_1)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>a</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><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_2 = \int (\alpha_2)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)</annotation></semantics></math> be two linear observables. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub></mtd> <mtd><mo>=</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>+</mo><mi>ℏ</mi><mo>∫</mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msup><mi>Δ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msup><mi>H</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mfrac><mrow><mo>∂</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow><mrow><mo>∂</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mfrac><mfrac><mrow><mo>∂</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><mrow><mo>∂</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thickmathspace"></mspace><mi>mod</mi><mspace width="thickmathspace"></mspace><msup><mi>ℏ</mi> <mn>2</mn></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>+</mo><mi>ℏ</mi><mrow><mo>(</mo><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msup><mi>Δ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msup><mi>H</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mi>mod</mi><mspace width="thickmathspace"></mspace><msup><mi>ℏ</mi> <mn>2</mn></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} A_1 \star_H A_2 & = A_1 A_2 + \hbar \int \left( \tfrac{i}{2} \Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1,x_2) \right) \frac{\partial A_1}{\partial \mathbf{\Phi}^{a_1}(x_1)} \frac{\partial A_2}{\partial \mathbf{\Phi}^{a_2}(x_2)} \;mod\; \hbar^2 \\ & = A_1 A_2 + \hbar \left( \int (\alpha_1)_{a_1}(x_1) \left( \tfrac{i}{2}\Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1, x_2) \right) (\alpha_2)_{a_2}(x_2) \right) \;mod\; \hbar^2 \end{aligned} </annotation></semantics></math></div> <p>Now since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> is skew-symmetric while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is symmetric is follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><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><msub><mi>A</mi> <mn>2</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd><mo>=</mo><mi>i</mi><mi>ℏ</mi><mrow><mo>(</mo><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msup><mi>Δ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mi>mod</mi><mspace width="thickmathspace"></mspace><msup><mi>ℏ</mi> <mn>2</mn></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>i</mi><mi>ℏ</mi><mspace width="thinmathspace"></mspace><mrow><mo>{</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>}</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} A_1 \star_H A_2 - A_2 \star_H A_1 & = i \hbar \left( \int (\alpha_1)_{a_1}(x_1) \Delta^{a_1 a_2}(x_1, x_2) (\alpha_2)_{a_2}(x_2) \right) \;mod\; \hbar^2 \\ & = i \hbar \, \left\{ A_1, A_2\right\} \end{aligned} \,. </annotation></semantics></math></div> <p>The right hand side is the <a class="existingWikiWord" href="/nlab/show/integral+kernel">integral kernel</a>-expression for the <a class="existingWikiWord" href="/nlab/show/Poisson-Peierls+bracket">Poisson-Peierls bracket</a>, as shown in the second line.</p> </div> <div class="num_example"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Hadamard+vacuum+state">Hadamard vacuum state</a> <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a>)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>i</mi></msub><mo>∈</mo><mi>LinObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo>↪</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub></mrow><annotation encoding="application/x-tex"> A_i \in LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">i \in \{1,2\}</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/linear+observable">linear</a> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> represented by <a class="existingWikiWord" href="/nlab/show/distributions">distributions</a> which in <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a>-notation are given by</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> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>a</mi> <mi>i</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mrow><msub><mi>a</mi> <mi>i</mi></msub></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A_i \;=\; \int (\alpha_i)_{a_i}(x_i) \mathbf{\Phi}^{a_i}(x_i) \, dvol_\Sigma(x_i) \,. </annotation></semantics></math></div> <p>Then their Hadamard-Moyal <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> (prop. <a class="maruku-ref" href="#MoyalStarProductOnMicrocausal"></a>) is the <a class="existingWikiWord" href="/nlab/show/sum">sum</a> of their pointwise product with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mi>i</mi><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\tfrac{1}{2} i \hbar</annotation></semantics></math> times the evaluation</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><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo></mtd> <mtd><mo>≔</mo><mo>∫</mo><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mrow><mo>⟨</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟩</mo></mrow><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mi>i</mi><mi>ℏ</mi><mo>∫</mo><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msubsup><mi>Δ</mi> <mi>H</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>2</mn></msub></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \langle A_1 A_2\rangle & \coloneqq \int \int (\alpha_1)_{a_1}(x_1) \, \left\langle \mathbf{\Phi}^{a_1}(x_1) \mathbf{\Phi}^{a_2}(x_2)\right\rangle \, (\alpha_2)_{a_2}(x_2) \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \\ & \coloneqq \tfrac{1}{2} i \hbar \int \int (\alpha_1)_{a_1}(x_1) \Delta_H^{a_1 a_2}(x_1,x_2) (\alpha_2)_{a_2}(x_2) \,dvol_\Sigma(x_1) \,dvol_\Sigma(x_2) \end{aligned} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math>:</p> <div class="maruku-equation" id="eq:StarProductOfTwoLinearObservables"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>+</mo><mo stretchy="false">⟨</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> A_1 \star_H A_2 = A_1 \cdot A_2 + \langle A_1 A_2\rangle </annotation></semantics></math></div> <p>Further <a href="#HadamardVacuumStatesOnWickAlgebras">below</a> we see that this evaluation is the <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a> of a <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state</a> on the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a>.</p> </div> <div class="num_example" id="WeylRelations"> <h6 id="example_2">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,\mathbf{L})</annotation></semantics></math> 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 <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+equation">Green hyperbolic</a> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> and with <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math>.</p> <p>Then for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>LinObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo>↪</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub></mrow><annotation encoding="application/x-tex"> A_1, A_2 \;\in\; LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc} </annotation></semantics></math></div> <p>two <a class="existingWikiWord" href="/nlab/show/linear+observables">linear</a> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a>, the Hadamard-Moyal star product (def. <a class="maruku-ref" href="#MoyalStarProductOnMicrocausal"></a>) of their <a class="existingWikiWord" href="/nlab/show/exponentials">exponentials</a> exhibits the <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>e</mi> <mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow></msup><msub><mo>⋆</mo> <mi>H</mi></msub><msup><mi>e</mi> <mrow><msub><mi>A</mi> <mn>2</mn></msub></mrow></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mi>e</mi> <mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow></msup><mspace width="thickmathspace"></mspace><msup><mi>e</mi> <mrow><mo stretchy="false">⟨</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> e^{A_1} \star_H e^{A_2} \;=\; e^{A_1 + A_2} \; e^{\langle A_1 A_2\rangle} </annotation></semantics></math></div> <p>where on the right we have the <a class="existingWikiWord" href="/nlab/show/exponential">exponential</a> <a class="existingWikiWord" href="/nlab/show/Wightman+2-point+function">Wightman 2-point function</a> <a class="maruku-eqref" href="#eq:StarProductOfTwoLinearObservables">(1)</a>.</p> </div> <p>(e.g. <a href="#Duetsch18">Dütsch 18, exercise 2.3</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Weyl+algebra">Weyl algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+algebra">Heisenberg algebra</a></p> </li> </ul> <h2 id="references">References</h2> <blockquote> <p>For more references see at <em><a class="existingWikiWord" href="/nlab/show/Weyl+algebra">Weyl algebra</a></em>.</p> </blockquote> <p>The notion goes back to</p> <ul> <li id="Weyl27"> <p><a class="existingWikiWord" href="/nlab/show/Hermann+Weyl">Hermann Weyl</a>, (46) in: <em>Quantenmechanik und Gruppentheorie</em>, Zeitschrift für Physik <strong>46</strong> (1927) 1–46 [<a href="https://doi.org/10.1007/BF02055756">doi:10.1007/BF02055756</a>]</p> </li> <li id="vonNeumann31"> <p><a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a>, p. 571 of: <em>Die Eindeutigkeit der Schrödingerschen Operatoren</em>, Mathematische Annalen <strong>104</strong> (1931) 570–578 [<a href="https://doi.org/10.1007/BF01457956">doi:10.1007/BF01457956</a>]</p> <blockquote> <p>(proving the <a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a>)</p> </blockquote> </li> </ul> <p>See also:</p> <ul> <li id="Duetsch18"><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, exercise 2.3 in: <em><a class="existingWikiWord" href="/nlab/show/From+classical+field+theory+to+perturbative+quantum+field+theory">From classical field theory to perturbative quantum field theory</a></em>, 2018</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 7, 2023 at 16:17:04. 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