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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='#Definition'>Definition</a></li> <ul> <li><a href='#AbstractWickAlgebra'>Abstract Wick algebra</a></li> <li><a href='#AbstractTimeOrderedProduct'>Abstract time-ordered product</a></li> <li><a href='#OperatorProductAndNormalOrderedProduct'>Operator product notation</a></li> <li><a href='#HadamardVacuumStatesOnWickAlgebras'>Hadamard vacuum states</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>A <em>Wick algebra</em> is an <a class="existingWikiWord" href="/nlab/show/algebra+of+quantum+observables">algebra of quantum observables</a> of <a class="existingWikiWord" href="/nlab/show/free+fields">free</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum fields</a>.</p> <p>In the <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> of <a class="existingWikiWord" href="/nlab/show/harmonic+oscillators">harmonic oscillators</a> or in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> of <a class="existingWikiWord" href="/nlab/show/free+fields">free fields</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> one encounters <a class="existingWikiWord" href="/nlab/show/linear+operators">linear operators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>a</mi> <mi>k</mi></msub><mo>,</mo><msubsup><mi>a</mi> <mi>k</mi> <mo>*</mo></msubsup><msub><mo stretchy="false">}</mo> <mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{a_k, a^\ast_k\}_{k \in K}</annotation></semantics></math> that satisfy the <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relation">canonical commutation relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>a</mi> <mi>i</mi></msub><mo>,</mo><msubsup><mi>a</mi> <mi>j</mi> <mo>*</mo></msubsup><mo stretchy="false">]</mo><mo>=</mo><mi>diag</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[a_i, a^\ast_j] = diag((c_k))_{i j}</annotation></semantics></math>. Then by a <em>normal ordered polynomial</em> or <em>Wick polynomial</em> (<a href="#Wick50">Wick 50</a>) one means a <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>:</mo><mi>P</mi><mo>:</mo></mrow><annotation encoding="application/x-tex">:P:</annotation></semantics></math>, which is obtained from a polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mi>k</mi></msub><mo>,</mo><msubsup><mi>a</mi> <mi>k</mi> <mo>*</mo></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P((a_k, a^\ast_k))</annotation></semantics></math> in these operators by ordering all “creation operators” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>a</mi> <mi>k</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">a_k^\ast</annotation></semantics></math> to the left of all “annihiliation operators”. For example focusing on a single mode <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> 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><mtr><mtd><mo>:</mo><msup><mi>a</mi> <mo>*</mo></msup><mo>:</mo><mo>=</mo><msup><mi>a</mi> <mo>*</mo></msup></mtd></mtr> <mtr><mtd><mo>:</mo><mi>a</mi><mo>:</mo><mo>=</mo><mi>a</mi></mtd></mtr> <mtr><mtd><mo>:</mo><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi><mo>:</mo><mo>=</mo><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi></mtd></mtr> <mtr><mtd><mo>:</mo><mi>a</mi><msup><mi>a</mi> <mo>*</mo></msup><mo>:</mo><mo>=</mo><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi></mtd></mtr> <mtr><mtd><mo>:</mo><mi>a</mi><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi><mo>:</mo><mo>=</mo><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi><mi>a</mi></mtd></mtr> <mtr><mtd><mi>etc</mi><mo>.</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex"> \array{ :a^\ast: = a^\ast \\ :a: = a \\ :a^\ast a: = a^\ast a \\ :a a^\ast: = a^\ast a \\ :a a^\ast a: = a^\ast a a \\ etc. } \, </annotation></semantics></math></div> <p>The intuitive idea is that these operators span a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</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">\mathcal{H}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> from a <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>vac</mi><mo stretchy="false">⟩</mo><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\vert vac \rangle \in \mathcal{H}</annotation></semantics></math> characterized by the condition</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>k</mi></msub><mo stretchy="false">|</mo><mi>vac</mi><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn><mphantom><mi>AAA</mi></mphantom><mtext>for all</mtext><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>k</mi></mrow><annotation encoding="application/x-tex"> a_k \vert vac \rangle = 0 \phantom{AAA} \text{for all} \,\, k </annotation></semantics></math></div> <p>hence (if we think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">a_k</annotation></semantics></math> as acting by “removing a quantum in mode <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>”) by the condition that it contains no quanta. So the normal ordered Wick polynomials represent the <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> with vanishing <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>. In <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> they model <a class="existingWikiWord" href="/nlab/show/scattering">scattering</a> processes where quanta enter a reaction process (the modes corresponding to the “annihilation” operators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">a_k</annotation></semantics></math>) and other particles come out of the reaction (the modes corresponding to the “creation” operators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>a</mi> <mi>k</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">a^\ast_k</annotation></semantics></math>).</p> <p>The product of two Wick polynomials, computed in the ambient operator algebra and then re-expressed as a Wick polynomial, is given by computing the relevant sequence of <a class="existingWikiWord" href="/nlab/show/commutators">commutators</a> by <a class="existingWikiWord" href="/nlab/show/Wick%27s+lemma">Wick's lemma</a>, for example</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>:</mo><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi><mo>:</mo></mrow><mspace width="thinmathspace"></mspace><mrow><mo>:</mo><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi><mo>:</mo></mrow><mo>=</mo><mo>:</mo><msup><mi>a</mi> <mo>*</mo></msup><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi><mi>a</mi><mo>:</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>ℏ</mi><mspace width="thinmathspace"></mspace><mo>:</mo><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi><mo>:</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> {:a^\ast a:} \, {:a^\ast a:} = :a^\ast a^\ast a a: + \hbar \, :a^\ast a: \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi><mo>=</mo><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mo>*</mo></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\hbar = [a, a^\ast]</annotation></semantics></math> is the value of the <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutator</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> thus obtained is hence called the <em>algebra of normal ordered operators</em> or <em>Wick polynomial algebra</em> or just <em>Wick algebra</em>.</p> <p>This plays a central role in <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, where the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> of <a class="existingWikiWord" href="/nlab/show/free+fields">free fields</a> is traditionally <em>defined</em> as the corresponding Wick algebra.</p> <p>But the Wick algebra in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> may also be understood more systematically from first principles of <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a>. It turns out that it is <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a> of the canonical <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> on the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> of the <a class="existingWikiWord" href="/nlab/show/free+field">free field</a>, which is the <a class="existingWikiWord" href="/nlab/show/Peierls+bracket">Peierls bracket</a> modified to an <a class="existingWikiWord" href="/nlab/show/almost+K%C3%A4hler+structure">almost Kähler structure</a> by the <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a> of a <a class="existingWikiWord" href="/nlab/show/quasi-free+Hadamard+state">quasi-free Hadamard state</a> (<a href="#Dito90">Dito 90</a>, <a href="#DutschFredenhagen01">Dütsch-Fredenhagen 01</a>).</p> <p>In <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a> with <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+equation">Green hyperbolic</a> <a class="existingWikiWord" href="/nlab/show/equations">equations</a> of motion, the analog of the star product tensor (<a href="star+product#eq:InStarProductTensorInvertingHermitianForm">this equation</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msup><mi>ω</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \pi \;=\; \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1} </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> according to <a href="Hadamard+distribution#eq:DeompositionOfHadamardPropagatorOnMinkowkski">this equation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>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><msub><mi>Δ</mi> <mi>S</mi></msub><mo>+</mo><mi>H</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_H \;=\; \tfrac{i}{2}\Delta_S + H \,. </annotation></semantics></math></div> <p>Understood in this form the construction directly generalizes to <a class="existingWikiWord" href="/nlab/show/quantum+field+theory+on+curved+spacetimes">quantum field theory on curved spacetimes</a> (<a href="#BrunettiFredenhagen95">Brunetti-Fredenhagen 95</a>, <a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00</a>, <a href="#HollandsWald01">Hollands-Wald 01</a>).</p> <p>Finally, the shift by the <a class="existingWikiWord" href="/nlab/show/quasi-free+Hadamard+state">quasi-free Hadamard state</a>, which is the very source of the “normal ordering”, was understood as an example of the almost-Kähler version of the quantization recipe of <a class="existingWikiWord" href="/nlab/show/Fedosov+deformation+quantization">Fedosov deformation quantization</a> (<a href="#Collini16">Collini 16</a>). For more on this see at <em><a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+quantum+field+theory">locally covariant perturbative quantum field theory</a></em>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h2 id="Definition">Definition</h2> <p>Traditionally the Wick algebra is regarded as an <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> acting on a <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>. However, it is useful to realize the Wick algebra directly as an <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/microcausal+polynomial+observables">microcausal polynomial observables</a>. This “abstract” Wick algebra (meaning: not <a class="existingWikiWord" href="/nlab/show/representation">represented</a> yet on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>) we discuss in</p> <ul> <li><em><a href="#AbstractWickAlgebra">Abstract Wick algebra</a></em>.</li> </ul> <p>That the abstract Wick algebra indeed has a <a class="existingWikiWord" href="/nlab/show/faithful+representation">faithful representation</a> on <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a> is <em><a class="existingWikiWord" href="/nlab/show/Wick%27s+lemma">Wick's lemma</a></em>.</p> <p>Similarly there is the</p> <ul> <li><em><a href="#AbstractTimeOrderedProduct">Abstract time-ordered product</a></em></li> </ul> <p>The traditional notation for the <a class="existingWikiWord" href="/nlab/show/operator+products">operator products</a> on the <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a> may be carried across the <a class="existingWikiWord" href="/nlab/show/representation">representation</a> map to the abstract Wick algebra:</p> <ul> <li><em><a href="#OperatorProductAndNormalOrderedProduct">Operator product notation</a></em></li> </ul> <p>The abstract Wick algebra carries a canonical <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, whose <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a> is just the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> that the abstract Wick algebra structure is constructed from. This we discuss in</p> <ul> <li><em><a href="#HadamardVacuumStatesOnWickAlgebras">Hadamard vacuum states</a></em>.</li> </ul> <p>The only non-trivial part of the proof of the <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state</a>-property (prop. <a class="maruku-ref" href="#WickAlgebraCanonicalState"></a>) below is positivity. This however is immediate from the <a class="existingWikiWord" href="/nlab/show/representation">representation</a> on the <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, observing that under this identification the state is represented by the <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> of the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> (<a href="#Duetsch18">Dütsch 18, remark 2.20</a>). From the point of view of <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a>, the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>-structure mainly just serves as a technical tool for establishing this positivity property.</p> <h3 id="AbstractWickAlgebra">Abstract Wick algebra</h3> <p>The abstract <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> of a <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a> with <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+equation">Green hyperbolic differential equation</a> is directly analogous to the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a>-algebra induced by a <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional</a> <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+vector+space">Kähler vector space</a> (<a href="star+product#WickAlgebraOfAlmostKaehlerVectorSpace">this def.</a>) under the following identification of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> with the <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+space">Kähler space</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a>:</p> <div class="num_remark" id="WightmanPropagatorAsKaehlerVectorSpaceStructure"> <h6 id="remark">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> as <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+vector+space">Kähler vector space</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</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> be 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> whose <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange</a> <a class="existingWikiWord" href="/nlab/show/equation+of+motion">equation of motion</a> is a <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+equation">Green hyperbolic differential equation</a>. Then the corresponding <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> is analogous to the rank-2 tensor on a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+vector+space">Kähler vector space</a> as follows:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> <br /> of <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+equation">Green hyperbolic</a> <br /> <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></th><th><a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional</a> <br /> <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+vector+space">Kähler vector space</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_\Sigma(E)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2n}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><msub><mi>Σ</mi> <mi>p</mi></msub></mrow></msub><msub><mi>Ω</mi> <mi>BFV</mi></msub></mrow><annotation encoding="application/x-tex">\tau_{\Sigma_p} \Omega_{BFV}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+form">Kähler form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ω</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\omega^{-1}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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><mo>=</mo><mo>∫</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><mfrac><mrow><mi>δ</mi><msub><mi>A</mi> <mn>1</mn></msub></mrow><mrow><mi>δ</mi><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><mi>δ</mi><msub><mi>A</mi> <mn>2</mn></msub></mrow><mrow><mi>δ</mi><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><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,A_2\} = \int \Delta^{a_1 a_2}(x_1,x_2) \frac{\delta A_1}{\delta \mathbf{\Phi}^{a_1}(x_1)} \frac{\delta A_2}{\delta \mathbf{\Phi}^{a_2}(x_2)} dvol_\Sigma(x)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mi>Δ</mi><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta_H = \tfrac{i}{2} \Delta + H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msup><mi>ω</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\pi = \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1}</annotation></semantics></math></td></tr> </tbody></table></div> <p>(<a href="pAQFT#FredenhagenRejzner15">Fredenhagen-Rejzner 15, section 3.6</a>, <a href="pAQFT#Collini16">Collini 16, table 2.1</a>)</p> <div class="num_defn" id="MicrocausalObservable"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal</a> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a>)</strong> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>fb</mi></mover></mrow><annotation encoding="application/x-tex">E \overset{fb}{\to}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> which is a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>. An <a class="existingWikiWord" href="/nlab/show/off-shell">off-shell</a> <em><a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a></em> is a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</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 lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> A \;\colon\; \Gamma_\Sigma(E) \longrightarrow \mathbb{C} </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> <a class="existingWikiWord" href="/nlab/show/space+of+sections">space of sections</a> of the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>fb</mi></mover><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \overset{fb}{\to} \Sigma</annotation></semantics></math> (space of field histories) which may be expressed as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>A</mi><mo stretchy="false">(</mo><mi>Φ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mphantom><mo lspace="verythinmathspace" rspace="0em">+</mo></mphantom><msup><mi>α</mi> <mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><msubsup><mi>α</mi> <mi>a</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>Φ</mi> <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></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mo>∫</mo> <mi>Σ</mi></msub><msubsup><mi>α</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>2</mn></msub></mrow> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></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><msup><mi>Φ</mi> <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><mi>Φ</mi> <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><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><mphantom><mo>=</mo></mphantom><mo>+</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} A(\Phi) & = \phantom{+} \alpha^{(0)} \\ & \phantom{=} + \int_\Sigma \alpha^{(1)}_a(x) \Phi^a(x) \, dvol_\Sigma(x) \\ & \phantom{=} + \int_\Sigma \int_\Sigma \alpha^{(2)}_{a_1 a_2}(x_1, x_2) \Phi^{a_1}(x_1) \Phi^{a_2}(x_2) \,dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \\ & \phantom{=} + \cdots \,, \end{aligned} </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>α</mi> <mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>∈</mo><mi>Γ</mi><msub><mo>′</mo> <mrow><msup><mi>Σ</mi> <mi>k</mi></msup></mrow></msub><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><msup><mo stretchy="false">)</mo> <mrow><msubsup><mo>⊠</mo> <mi>sym</mi> <mi>k</mi></msubsup></mrow></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \alpha^{(k)} \in \Gamma'_{\Sigma^k}\left((E^\ast)^{\boxtimes^k_{sym}} \right) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/compactly+supported+distribution">compactly supported distribution</a> <a class="existingWikiWord" href="/nlab/show/distribution+of+two+variables">of k variables</a> on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-fold graded-symmetric <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product of vector bundles</a> of the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> with itself.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>Obs</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> PolyObs(E) \hookrightarrow Obs(E) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of off-shell polynomial observables onside all off-shell <a class="existingWikiWord" href="/nlab/show/observables">observables</a>.</p> <p>Let moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><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> be 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> whose <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> are <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+equations">Green hyperbolic differential equations</a>. Then an <em><a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> polynomial observable</em> is the <a class="existingWikiWord" href="/nlab/show/restriction">restriction</a> of an off-shell polynomial observable along the inclusion of the <a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> <a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>=</mo><mn>0</mn></mrow></msub><mo>↪</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)</annotation></semantics></math>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo stretchy="false">)</mo><mo>↪</mo><mi>Obs</mi><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"> PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L}) </annotation></semantics></math></div> <p>for the subspace of all on-shell polynomial observables inside all on-shell <a class="existingWikiWord" href="/nlab/show/observables">observables</a>.</p> <p>By <a href="Green+hyperbolic+partial+differential+equation#DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions">this prop.</a> restriction yields an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> between polynomial on-shell observables and polynomial off-shell observables modulo the image of the <a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a> <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>:</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><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo stretchy="false">)</mo><munderover><mo>⟵</mo><mo>≃</mo><mtext>restriction</mtext></munderover><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PolyObs(E,\mathbf{L}) \underoverset{\simeq}{\text{restriction}}{\longleftarrow} PolyObs(E)/im(P) \,. </annotation></semantics></math></div> <p>Finally a polynomial observable is a <em><a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a></em> if each <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>α</mi> <mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\alpha^{(k)}</annotation></semantics></math> as above has <a class="existingWikiWord" href="/nlab/show/wave+front+set">wave front set</a> away from those points where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/wave+vectors">wave vectors</a> are all in the <a class="existingWikiWord" href="/nlab/show/future+cone">future cone</a> or all in the <a class="existingWikiWord" href="/nlab/show/past+cone">past cone</a>. We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><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><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ PolyObs(E)_{mc} &\hookrightarrow& PolyObs(E) \\ PolyObs(E,\mathbf{L})_{mc} \simeq PolyObs(E)_{mc}/im(P) &\hookrightarrow& PolyObs(E,\mathbf{L}) } </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of <a class="existingWikiWord" href="/nlab/show/off-shell">off-shell</a>/<a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> inside all <a class="existingWikiWord" href="/nlab/show/off-shell">off-shell</a>/<a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a>.</p> </div> <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><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <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><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <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 \mathbf{\Phi}^a(x_1)} \otimes \frac{\delta}{\delta \mathbf{\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> (def. <a class="maruku-ref" href="#MicrocausalObservable"></a>) 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>(<a href="#Dito90">Dito 90</a>, <a href="#DuetschFredenhagen00">Dütsch-Fredenhagen 00</a> <a href="#DuetschFredenhagen01">Dütsch-Fredenhagen 01</a>, <a href="#HirschfeldHenselder02">Hirshfeld-Henselder 02</a>, see <a href="#Collini16">Collini 16, p. 25-26</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>By definition of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> (or else by <a href="Hadamard+distribution#WaveFronSetsForKGPropagatorsOnMinkowski">this prop.</a>), the <a class="existingWikiWord" href="/nlab/show/wave+front+set">wave front set</a> of powers of <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> has all cotangent <a class="existingWikiWord" href="/nlab/show/wave+vectors">wave vectors</a> on the first variables in the <a class="existingWikiWord" href="/nlab/show/closed+future+cone">closed future cone</a> at the given base point (which itself is on the <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a>)</p> <center> <img src="https://ncatlab.org/nlab/files/HadamardPropagator.png" width="60" /> </center> <p>and hence all those on the second variables in the <a class="existingWikiWord" href="/nlab/show/closed+past+cone">closed past cone</a>.</p> <p>The first variables are integrated against those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1</annotation></semantics></math> and the second against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_2</annotation></semantics></math>. By definition of <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> (def. <a class="maruku-ref" href="#MicrocausalObservable"></a>), the wave front sets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1</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></mrow><annotation encoding="application/x-tex">A_2</annotation></semantics></math> are disjoint from the subsets where all components are in the <a class="existingWikiWord" href="/nlab/show/closed+future+cone">closed future cone</a> or all components are in the <a class="existingWikiWord" href="/nlab/show/closed+past+cone">closed past cone</a>. Therefore the relevant sum of of the wave front covectors never vanishes and hence <a class="existingWikiWord" href="/nlab/show/H%C3%B6rmander%27s+criterion">Hörmander's criterion</a> for partial <a class="existingWikiWord" href="/nlab/show/products+of+distributions">products of</a> <a class="existingWikiWord" href="/nlab/show/distributions+of+several+variables">distributions of several variables</a> (<a href="product+of+distributions#PartialProductOfDistributionsOfSeveralVariables">this prop.</a>) is met and the star product is well defined.</p> <p>It remains to see that the star product <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>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_1 \star_H A_2</annotation></semantics></math> is itself again a <a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a>. It is clear that it is again a <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a> and that it respects the ideal generated by the equations of motion. That it still satisfies the condition on the <a class="existingWikiWord" href="/nlab/show/wave+front+set">wave front set</a> follows directly from the fact that the wave front set of a <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> is inside the fiberwise sum of elements of the factor wave front sets (<a href="product+of+distributions#WaveFrontSetOfProductOfDistributionsInsideFiberProductOfFactorWaveFrontSets">this prop.</a>, <a href=""></a>).</p> <p>Finally the <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>-structure follows via remark <a class="maruku-ref" href="#WightmanPropagatorAsKaehlerVectorSpaceStructure"></a> as in <a href="star+product#StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra">this prop.</a>.</p> </div> <div class="num_remark" id="WickAlgebraIsFormalDeformationQuantization"> <h6 id="remark_2">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="#DutschFredenhagen01">Dütsch-Fredenhagen 01</a>)</p> <div class="proof"> <h6 id="proof_2">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> <h3 id="AbstractTimeOrderedProduct">Abstract time-ordered product</h3> <div class="num_defn" id="OnRegularPolynomialObservablesTimeOrderedProduct"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a>)</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> be 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> over a <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> and with <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+equation">Green-hyperbolic</a> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange</a> <a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>; write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub><mo>=</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_S = \Delta_+ - \Delta_-</annotation></semantics></math> for the induced <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a>. Let moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mi>S</mi></msub><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta_H = \tfrac{i}{2}\Delta_S + H </annotation></semantics></math> be a compatible <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H</annotation></semantics></math> for the induced <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a>.</p> <p>Then the <em><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a></em> on the space of <a class="existingWikiWord" href="/nlab/show/off-shell">off-shell</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><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><msub><mo stretchy="false">)</mo> <mi>reg</mi></msub></mrow><annotation encoding="application/x-tex">PolyObs(E)_{reg}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> induced by the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> (via <a href="star+product#PropagatorStarProduct">this prop.</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><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><mi>E</mi><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></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><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></mtd></mtr> <mtr><mtd><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></mtd> <mtd><mo>↦</mo></mtd> <mtd><mphantom><mo>≔</mo></mphantom><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] \\ (A_1, A_2) &\mapsto& \phantom{\coloneqq} A_1 \star_F A_2 } </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>exp</mi><mrow><mo>(</mo><munder><mo>∫</mo><mrow><mi>Σ</mi><mo>×</mo><mi>Σ</mi></mrow></munder><msubsup><mi>Δ</mi> <mi>F</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⊗</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub><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>y</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> A_1 \star_F A_2 \; \coloneqq \; ((-)\cdot(-)) \circ \exp\left( \underset{\Sigma \times \Sigma}{\int} \Delta_F^{a b}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \, dvol_\Sigma(x) \, dvol_\Sigma(y) \right) </annotation></semantics></math></div> <p>(Notice that this does not descend to the <a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> observables, since the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> is not a solution to the <em>homogeneous</em> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a>.)</p> </div> <div class="num_prop" id="CausalOrderingTimeOrderedProductOnRegular"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> is indeed causally ordered <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> product)</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> be 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> over a <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> and with <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+equation">Green-hyperbolic</a> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange</a> <a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>; write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub><mo>=</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_S = \Delta_+ - \Delta_-</annotation></semantics></math> for the induced <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a>. Let moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mi>S</mi></msub><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta_H = \tfrac{i}{2}\Delta_S + H </annotation></semantics></math> be a compatible <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H</annotation></semantics></math> for the induced <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> (def. <a class="maruku-ref" href="#OnRegularPolynomialObservablesTimeOrderedProduct"></a>) is indeed a time-ordering 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> in that for all <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> of <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><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><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"> A_1, A_2 \in PolyObs(E)_{reg}[ [\hbar] ] </annotation></semantics></math></div> <p>with <a class="existingWikiWord" href="/nlab/show/disjoint+subset">disjoint</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/support">support</a> 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>F</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><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 stretchy="false">|</mo></mtd> <mtd><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><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 stretchy="false">|</mo></mtd> <mtd><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A_1 \star_F A_2 \;=\; \left\{ \array{ A_1 \star_H A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \star_H A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \,. </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>1</mn></msub><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><msub><mi>S</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S_1 {\vee\!\!\!\wedge} S_2</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/causal+order">causal order</a> relation (“<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">S_1</annotation></semantics></math> does not intersect the <a class="existingWikiWord" href="/nlab/show/past+cone">past cone</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S_2</annotation></semantics></math>”). Beware that for general <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>S</mi> <mn>1</mn></msub><mo>,</mo><mi>S</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S_1, S-2)</annotation></semantics></math> of subsets neither <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>1</mn></msub><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><msub><mi>S</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S_1 {\vee\!\!\!\wedge} S_2</annotation></semantics></math> nor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>2</mn></msub><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><msub><mi>S</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">S_2 {\vee\!\!\!\wedge} S_1</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Recall the following facts:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagators">advanced and retarded propagators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_{\pm}</annotation></semantics></math> by definition are <a class="existingWikiWord" href="/nlab/show/support">supported</a> in the <a class="existingWikiWord" href="/nlab/show/future+cone">future cone</a>/<a class="existingWikiWord" href="/nlab/show/past+cone">past cone</a>, respectively</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">)</mo><mo>⊂</mo><msup><mover><mi>V</mi><mo>¯</mo></mover> <mo>±</mo></msup></mrow><annotation encoding="application/x-tex"> supp(\Delta_{\pm}) \subset \overline{V}^{\pm} </annotation></semantics></math></div></li> <li> <p>they turn into each other under exchange of their arguments (<a href="causal+propagator#CausalPropagatorIsSkewSymmetric">this cor.</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Δ</mi> <mo>∓</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_\pm(y,x) = \Delta_{\mp}(x,y) \,. </annotation></semantics></math></div></li> <li> <p>the real part <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> of the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a>, which by definition is the real part of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> is symmetric (by definition or else by <a href="Hadamard+distribution#SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime">this prop.</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H(x,y) = H(y,x) </annotation></semantics></math></div></li> </ol> <p>Using this we compute as follows:</p> <div class="maruku-equation" id="eq:CausallyOrderedWickProductViaFeynmanPropagator"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><munder><mo>⋆</mo><mrow><msub><mi>Δ</mi> <mi>F</mi></msub></mrow></munder><msub><mi>A</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace></mtd> <mtd><mo>=</mo><msub><mi>A</mi> <mn>1</mn></msub><munder><mo>⋆</mo><mrow><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi></mrow></munder><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><munder><mo>⋆</mo><mrow><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>H</mi></mrow></munder><msub><mi>A</mi> <mn>2</mn></msub></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><munder><mo>⋆</mo><mrow><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mo>−</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>H</mi></mrow></munder><msub><mi>A</mi> <mn>2</mn></msub></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><munder><mo>⋆</mo><mrow><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>H</mi></mrow></munder><msub><mi>A</mi> <mn>2</mn></msub></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>2</mn></msub><munder><mo>⋆</mo><mrow><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>H</mi></mrow></munder><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><munder><mo>⋆</mo><mrow><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi></mrow></munder><msub><mi>A</mi> <mn>2</mn></msub></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>2</mn></msub><munder><mo>⋆</mo><mrow><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi></mrow></munder><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><munder><mo>⋆</mo><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow></munder><msub><mi>A</mi> <mn>2</mn></msub></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>2</mn></msub><munder><mo>⋆</mo><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow></munder><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mrow><mo>∨</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo>∧</mo></mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} A_1 \underset{\Delta_{F}}{\star} A_2 \; & = A_1 \underset{\tfrac{i}{2}(\Delta_+ + \Delta_-) + H}{\star} A_2 \\ & = \left\{ \array{ A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_1 \underset{\tfrac{i}{2}\Delta_- + H}{\star} A_2 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \array{ A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \array{ A_1 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \array{ A_1 \underset{\Delta_H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\Delta_H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \end{aligned} </annotation></semantics></math></div></div> <div class="num_prop" id="IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to pointwise product)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> (def. <a class="maruku-ref" href="#OnRegularPolynomialObservablesTimeOrderedProduct"></a>) is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the pointwise product of <a class="existingWikiWord" href="/nlab/show/observables">observables</a> (<a href="A+first+idea+of+quantum+field+theory#Observable">this def.</a>) via the <a class="existingWikiWord" href="/nlab/show/linear+isomorphism">linear isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><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><mi>E</mi><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{T} \;\colon\; PolyObs(E)_{reg}[ [\hbar] ] \longrightarrow PolyObs(E)_{reg}[ [\hbar] ] </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation" id="eq:OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi><mi>A</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>exp</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mi>ℏ</mi><munder><mo>∫</mo><mi>Σ</mi></munder><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mfrac><mrow><msup><mi>δ</mi> <mn>2</mn></msup></mrow><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow><mi>A</mi></mrow><annotation encoding="application/x-tex"> \mathcal{T}A \;\coloneqq\; \exp\left( \tfrac{1}{2} \hbar \underset{\Sigma}{\int} \Delta_F(x,y)^{a b} \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)} \right) A </annotation></semantics></math></div> <p>in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>T</mi><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><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><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></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} T(A_1 A_2) & \coloneqq A_1 \star_{F} A_2 \\ & = \mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2) ) \end{aligned} </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><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><mi>E</mi><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></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><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></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mo>≃</mo> <mpadded width="0" lspace="-100%width"><mrow><mi>𝒯</mi><mo>⊗</mo><mi>𝒯</mi></mrow></mpadded></msubsup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mo>≃</mo> <mpadded width="0"><mi>𝒯</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><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><mi>E</mi><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></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><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></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{(-)\cdot (-)}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] \\ {}^{\mathllap{\mathcal{T} \otimes \mathcal{T}}}_\simeq\Big\downarrow && \downarrow^{\mathrlap{\mathcal{T}}}_\simeq \\ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{(-) \star_F (-)}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] } </annotation></semantics></math></div></div> <p>(<a href="time-ordered+product#BrunettiDuetschFredenhagen09">Brunetti-Dütsch-Fredenhagen 09, (12)-(13)</a>, <a href="time-ordered+product#FredenhagenRejzner11b">Fredenhagen-Rejzner 11b, (14)</a>)</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Since the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> is symmetric (<a href="A+first+idea+of+quantum+field+theory#SymmetricFeynmanPropagator">this prop.</a>), the statement is a special case of <a href="star+product#SymmetricContribution">this prop.</a>).</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> of <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> on <a class="existingWikiWord" href="/nlab/show/regular+polynomial+observables">regular polynomial observables</a> from prop. <a class="maruku-ref" href="#OnRegularPolynomialObservablesTimeOrderedProduct"></a> extends to a product on <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial</a> <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a>, then taking values in <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><mi>T</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PolyLocObs</mi><mo stretchy="false">(</mo><mi>E</mi><msup><mo stretchy="false">)</mo> <mrow><msub><mo>⊗</mo> <mi>n</mi></msub></mrow></msup><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><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></mrow><annotation encoding="application/x-tex"> T \;\colon\; PolyLocObs(E)^{\otimes_n}[ [\hbar] ] \longrightarrow PolyObs(E)_{mc}[ [\hbar] ] \,. </annotation></semantics></math></div> <p>This extension is not unique. A choice of such an extension, satisfying some evident compatibility conditions, is a choice of <em><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">renormalization scheme</a></em> for the given <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>. Every such choice corresponds to a choice of <a class="existingWikiWord" href="/nlab/show/perturbative+S-matrix">perturbative S-matrix</a> for the theory. This construction is called <em><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a></em>.</p> </div> <h3 id="OperatorProductAndNormalOrderedProduct">Operator product notation</h3> <div class="num_defn" id="NormalOrderedProductNotation"> <h6 id="definition_4">Definition</h6> <p><strong>(notation for <a class="existingWikiWord" href="/nlab/show/operator+product">operator product</a> and <a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>)</strong></p> <p>It is traditional to use the following alternative notation for the product structures on <a class="existingWikiWord" href="/nlab/show/microcausal+polynomial+observables">microcausal polynomial observables</a>:</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a>-product, hence the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⋆</mo> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\star_H</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> (def. <a class="maruku-ref" href="#MoyalStarProductOnMicrocausal"></a>), is rewritten as plain juxtaposition:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtext>"operator product"</mtext><mphantom><mi>AAA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mphantom><mi>AA</mi></mphantom><mo>≔</mo><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>1</mn></msub><mphantom><mi>AAAA</mi></mphantom><mrow><mtable><mtr><mtd><mtext>star product of</mtext></mtd></mtr> <mtr><mtd><mtext>Wightman propagator</mtext></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \text{"operator product"} \phantom{AAA} A_1 A_2 \phantom{AA} \coloneqq \phantom{AA} A_1 \star_H A_1 \phantom{AAAA} \array{ \text{star product of} \\ \text{Wightman propagator} } \,. </annotation></semantics></math></div></li> <li> <p>The pointwise product of observables (<a href="A+first+idea+of+quantum+field+theory#Observable">this def.</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></mrow><annotation encoding="application/x-tex">A_1 \cdot A_2</annotation></semantics></math> is equivalently written as plain juxtaposition enclosed by colons:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mtext>"normal-ordered</mtext></mtd></mtr> <mtr><mtd><mtext>product"</mtext></mtd></mtr></mtable></mrow><mphantom><mi>AAAA</mi></mphantom><mo>:</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>:</mo><mphantom><mi>AA</mi></mphantom><mo>≔</mo><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>A</mi> <mn>2</mn></msub><mphantom><mi>AAAA</mi></mphantom><mphantom><mi>AAa</mi></mphantom><mtext>pointwise product</mtext><mphantom><mi>AAa</mi></mphantom></mrow><annotation encoding="application/x-tex"> \array{ \text{"normal-ordered} \\ \text{product"} } \phantom{AAAA} :A_1 A_2: \phantom{AA}\coloneqq\phantom{AA} A_1 \cdot A_2 \phantom{AAAA} \phantom{AAa}\text{pointwise product}\phantom{AAa} </annotation></semantics></math></div></li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, hence the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> for the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⋆</mo> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\star_F</annotation></semantics></math> (def. <a class="maruku-ref" href="#OnRegularPolynomialObservablesTimeOrderedProduct"></a>) is equivalently written as plain juxtaposition prefixed by a “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>”</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mtext>"time-ordered</mtext></mtd></mtr> <mtr><mtd><mtext>product"</mtext></mtd></mtr></mtable></mrow><mphantom><mi>AAAA</mi></mphantom><mi>T</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom><mo>≔</mo><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mphantom><mi>AAAA</mi></mphantom><mrow><mtable><mtr><mtd><mtext>star product of</mtext></mtd></mtr> <mtr><mtd><mtext>Feynman propagator</mtext></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \text{"time-ordered} \\ \text{product"} } \phantom{AAAA} T(A_1 A_2) \phantom{AA}\coloneqq\phantom{AA} A_1 \star_F A_2 \phantom{AAAA} \array{ \text{star product of} \\ \text{Feynman propagator} } </annotation></semantics></math></div></li> </ol> <p>Under <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> on a <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> by <a class="existingWikiWord" href="/nlab/show/linear+operators">linear operators</a> the first product become the <em><a class="existingWikiWord" href="/nlab/show/operator+product">operator product</a></em>, while the second becomes the operator poduct applied after suitable re-ordering, called “<a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal odering</a>” of the factors.</p> <p>Disregarding the <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>-representation, which is <a class="existingWikiWord" href="/nlab/show/faithful+representation">faithful</a>, we may still refer to these “abstract” products as the “operator product” and the “normal-ordered product”, respectively.</p> </div> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/algebra+of+quantum+observables">algebra of quantum observables</a></th><th><a class="existingWikiWord" href="/nlab/show/physics">physics</a> terminology</th><th><a class="existingWikiWord" href="/nlab/show/mathematics">maths</a> terminology</th></tr></thead><tbody><tr><td style="text-align: left;">1)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative product</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_1"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mo>:</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>:</mo></mrow><annotation encoding="application/x-tex">\phantom{AA} :A_1 A_2:</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_2"><semantics><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phantom{AA} A_1 \cdot A_2</annotation></semantics></math> <br /> pointwise product of functionals</td></tr> <tr><td style="text-align: left;">2)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/non-commutative+algebra">non-commutative product</a> <br /> (<a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation</a> induced by <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_3"><semantics><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phantom{AA} A_1 A_2</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/operator+product">operator product</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_4"><semantics><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phantom{AA} A_1 \star_H A_2</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> for <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></td></tr> <tr><td style="text-align: left;">3)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_5"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mi>T</mi><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">\phantom{AA} T(A_1 A_2)</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_6"><semantics><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phantom{AA} A_1 \star_F A_2</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> for <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative expansion</a> <br /> of 2) via 1)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Wick%27s+lemma">Wick's lemma</a> <br /> <img src="https://ncatlab.org/nlab/files/WickTheorem.png" width="350" /></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Moyal+star+product">Moyal product</a> for <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_7"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math><br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_8"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <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></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>exp</mi><mrow><mo>(</mo><mi>ℏ</mi><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>H</mi></msub><msup><mo stretchy="false">)</mo> <mi>ab</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⊗</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow><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></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & A_1 \star_H A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_H)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative expansion</a> <br /> of 3) via 1)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> <br /> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramGlobal.jpg" width="350" /></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Moyal+star+product">Moyal product</a> for <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_9"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_F</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_10"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><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></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>exp</mi><mrow><mo>(</mo><mi>ℏ</mi><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>F</mi></msub><msup><mo stretchy="false">)</mo> <mi>ab</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⊗</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow><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></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & A_1 \star_F A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_F)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}</annotation></semantics></math></td></tr> </tbody></table> </div> <h3 id="HadamardVacuumStatesOnWickAlgebras">Hadamard vacuum states</h3> <div class="num_prop" id="WickAlgebraCanonicalState"> <h6 id="proposition_4">Proposition</h6> <p><strong>(canonical <a class="existingWikiWord" href="/nlab/show/Hadamard+vacuum+state">vacuum</a> <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">states</a> on abstract <a class="existingWikiWord" href="/nlab/show/Wick+algebra">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> be 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/Euler-Lagrange+equation">Euler-Lagrange</a> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a>; 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></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math> be a compatible <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a>.</p> <p>For</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mn>0</mn></msub><mo>∈</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>=</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> \Phi_0 \in \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} </annotation></semantics></math></div> <p>any <a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> <a class="existingWikiWord" href="/nlab/show/field+history">field history</a> (i.e. solving the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a>), consider the function from the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> to <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> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> which evaluates any <a class="existingWikiWord" href="/nlab/show/microcausal+polynomial+observable">microcausal polynomial observable</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Phi_0</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><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><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mrow><msub><mi>Φ</mi> <mn>0</mn></msub></mrow></msub></mrow></mover></mtd> <mtd><mi>ℂ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>A</mi><mo stretchy="false">(</mo><msub><mi>Φ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ] &\overset{\langle -\rangle_{\Phi_0}}{\longrightarrow}& \mathbb{C}[ [\hbar] ] \\ A &\mapsto& A(\Phi_0) } </annotation></semantics></math></div> <p>Specifically for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Phi_0 = 0</annotation></semantics></math> (which is a solution of the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> by the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,\mathbf{L})</annotation></semantics></math> defines a <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a>) this is the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</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><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>ℂ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>A</mi></mtd> <mtd><mo>=</mo><msup><mi>α</mi> <mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><munder><mo>∫</mo><mi>Σ</mi></munder><msubsup><mi>α</mi> <mi>a</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><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></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mi>⋯</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>A</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mi>α</mi> <mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ] &\overset{\langle -\rangle_0}{\longrightarrow}& \mathbb{C}[ [\hbar] ] \\ \left. \begin{aligned} A & = \alpha^{(0)} \\ & \phantom{=} + \underset{\Sigma}{\int} \alpha^{(1)}_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x) \\ & \phantom{=} + \cdots \end{aligned} \right\} &\mapsto& A(0) = \alpha^{(0)} } </annotation></semantics></math></div> <p>which sends each <a class="existingWikiWord" href="/nlab/show/microcausal+polynomial+observable">microcausal polynomial observable</a> to its value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>Φ</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(\Phi = 0)</annotation></semantics></math> on the zero <a class="existingWikiWord" href="/nlab/show/field+history">field history</a>, hence to the constant contribution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>α</mi> <mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\alpha^{(0)}</annotation></semantics></math> in its <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> expansion.</p> <p>The function <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><msub><mo stretchy="false">⟩</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\langle -\rangle_0</annotation></semantics></math> is</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+function">linear</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><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">\mathbb{C}[ [\hbar] ]</annotation></semantics></math>;</p> </li> <li> <p>real, in that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><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></mrow><annotation encoding="application/x-tex">A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]</annotation></semantics></math></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><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>A</mi><msup><mo stretchy="false">⟩</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> \langle A^\ast \rangle = \langle A \rangle^\ast </annotation></semantics></math></div></li> <li> <p>positive, in that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><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></mrow><annotation encoding="application/x-tex">A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]</annotation></semantics></math> there exist a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>A</mi></msub><mo>∈</mo><mi>ℂ</mi><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">c_A \in \mathbb{C}[ [\hbar] ]</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msup><mi>A</mi> <mo>*</mo></msup><msub><mo>⋆</mo> <mi>H</mi></msub><mi>A</mi><msub><mo stretchy="false">⟩</mo> <mrow><msub><mi>Φ</mi> <mn>0</mn></msub></mrow></msub><mo>=</mo><msubsup><mi>c</mi> <mi>A</mi> <mo>*</mo></msubsup><mo>⋅</mo><msub><mi>c</mi> <mi>A</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \langle A^\ast \star_H A\rangle_{\Phi_0} = c_A^\ast \cdot c_A \,, </annotation></semantics></math></div></li> <li> <p>normalized, in that</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><mn>1</mn><msub><mo stretchy="false">⟩</mo> <mi>H</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> \langle 1\rangle_H = 1 </annotation></semantics></math></div></li> </ol> <p>where <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> denotes componet-wise <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a>.</p> <p>This means that <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><msub><mo stretchy="false">⟩</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\langle -\rangle_{0}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">states</a> on the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick</a> <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>PolyObs</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>mc</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>,</mo><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> (prop. <a class="maruku-ref" href="#MoyalStarProductOnMicrocausal"></a>). One says that</p> <ul> <li><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><msub><mo stretchy="false">⟩</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\langle - \rangle_0</annotation></semantics></math> is a <em><a class="existingWikiWord" href="/nlab/show/Hadamard+vacuum+state">Hadamard vacuum state</a></em>;</li> </ul> <p>and generally</p> <ul> <li><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><msub><mo stretchy="false">⟩</mo> <mrow><msub><mi>Φ</mi> <mn>0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\langle - \rangle_{\Phi_0}</annotation></semantics></math> is called a <em><a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a></em>.</li> </ul> </div> <p>(<a href="#Duetsch18">Dütsch 18, def. 2.12, remark 2.20, def. 5.28, exercise 5.30 and equations (5.178)</a>)</p> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>The properties of linearity, reality and normalization are obvious, what requires proof is positivity. This is proven by exhibiting a <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of the Wick algebra on a <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> (this algebra <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> is <em><a class="existingWikiWord" href="/nlab/show/Wick%27s+lemma">Wick's lemma</a></em>), with formal powers 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> suitably taken care of, and showing that under this representation the function <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><msub><mo stretchy="false">⟩</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\langle -\rangle_0</annotation></semantics></math> is represented, degreewise 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>, by the <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> of the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>.</p> </div> <div class="num_example" id="HadamardMoyalStarProductOfTwoLinearObservables"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/operator+product">operator product</a> of two <a class="existingWikiWord" href="/nlab/show/linear+observables">linear observables</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 their value</p> <div class="maruku-equation" id="eq:EvaluatingLinearObservablesInWightmanPropagator"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><msub><mo stretchy="false">⟩</mo> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><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></mrow><annotation encoding="application/x-tex"> \langle A_1 \star_H A_2 \rangle_0 \;\coloneqq\; 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) </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a>, which is the value of the <a class="existingWikiWord" href="/nlab/show/Hadamard+vacuum+state">Hadamard vacuum state</a> from prop. <a class="maruku-ref" href="#WickAlgebraCanonicalState"></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>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><mo>+</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">⟨</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><msub><mo stretchy="false">⟩</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> A_1 \star_H A_2 \;=\; A_1 \cdot A_2 \;+\; \langle A_1 \star_H A_2 \rangle_0 </annotation></semantics></math></div> <p>In the <a class="existingWikiWord" href="/nlab/show/operator+product">operator product</a>/<a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>-notation of def. <a class="maruku-ref" href="#NormalOrderedProductNotation"></a> this reads</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><mi>A</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>:</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>:</mo><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">⟨</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A_1 A_2 \;=\; :A_1 A_2: \;+\; \langle A_1 A_2\rangle \,. </annotation></semantics></math></div></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><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><msub><mo stretchy="false">⟩</mo> <mn>0</mn></msub></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 \star_H A_2\rangle_0} </annotation></semantics></math></div> <p>where on the right we have the <a class="existingWikiWord" href="/nlab/show/exponential">exponential</a> of the value of the <a class="existingWikiWord" href="/nlab/show/Hadamard+vacuum+state">Hadamard vacuum state</a> (prop. <a class="maruku-ref" href="#WickAlgebraCanonicalState"></a>) as in example <a class="maruku-ref" href="#HadamardMoyalStarProductOfTwoLinearObservables"></a>.</p> </div> <p>(e.g. <a href="#Duetsch18">Dütsch 18, exercise 2.3</a>)</p> <div class="num_example"> <h6 id="example_3">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> is <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a> in the <a class="existingWikiWord" href="/nlab/show/Hadamard+vacuum+state">Hadamard vacuum state</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> be 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/Euler-Lagrange+equation">Euler-Lagrange</a> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a>; 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></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math> be a compatible <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a>.</p> <p>With respect to the induced <a class="existingWikiWord" href="/nlab/show/Hadamard+vacuum+state">Hadamard vacuum state</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><msub><mo stretchy="false">⟩</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\langle - \rangle_0</annotation></semantics></math> from prop. <a class="maruku-ref" href="#WickAlgebraCanonicalState"></a>, 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><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta_H(x,y)</annotation></semantics></math> itself is the <em><a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a></em>, namely the <a class="existingWikiWord" href="/nlab/show/distribution">distributional</a> <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a> of the operator product of two <a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>⟨</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>H</mi></msub><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⟩</mo></mrow> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><mo>⟨</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⟩</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mo>+</mo><munder><munder><mrow><mo>⟨</mo><mi>ℏ</mi><munder><mo>∫</mo><mrow><mi>Σ</mi><mo>×</mo><mi>Σ</mi></mrow></munder><mi>δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>x</mi><mo>′</mo><mo stretchy="false">)</mo><msubsup><mi>Δ</mi> <mi>H</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>y</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mi>ℏ</mi><msubsup><mi>Δ</mi> <mi>H</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mrow><annotation encoding="application/x-tex"> \left\langle \mathbf{\Phi}^a(x) \star_H \mathbf{\Phi}^b(y) \right\rangle_0 \;=\; \underset{ = 0 }{ \underbrace{ \left\langle \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y) \right\rangle }} + \underset{ = \hbar \Delta^{a b}_H(x,y) }{ \underbrace{ \left \langle \hbar \underset{\Sigma \times \Sigma}{\int} \delta(x-x') \Delta^{a b}_H(x,y) \delta(y-y') \right\rangle }} </annotation></semantics></math></div> <p>by example <a class="maruku-ref" href="#HadamardMoyalStarProductOfTwoLinearObservables"></a>.</p> <p>Equivalently in the <a class="existingWikiWord" href="/nlab/show/operator+product">operator product</a>-notation of def. <a class="maruku-ref" href="#NormalOrderedProductNotation"></a> this reads:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>⟨</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⟩</mo></mrow> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>ℏ</mi><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\langle \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y) \right\rangle_0 \;=\; \hbar \Delta_H(x,y) \,. </annotation></semantics></math></div></div> <p>Similarly:</p> <div class="num_example"> <h6 id="example_4">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> is time-ordered <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a> in the <a class="existingWikiWord" href="/nlab/show/Hadamard+vacuum+state">Hadamard vacuum state</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> be 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/Euler-Lagrange+equation">Euler-Lagrange</a> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a>; 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></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math> be a compatible <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> with induced <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> <p>With respect to the induced <a class="existingWikiWord" href="/nlab/show/Hadamard+vacuum+state">Hadamard vacuum state</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><msub><mo stretchy="false">⟩</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\langle - \rangle_0</annotation></semantics></math> from prop. <a class="maruku-ref" href="#WickAlgebraCanonicalState"></a>, the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta_F(x,y)</annotation></semantics></math> itself is the <em>time-ordered <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a></em>, namely the <a class="existingWikiWord" href="/nlab/show/distribution">distributional</a> <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a> of the <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a> (def. <a class="maruku-ref" href="#OnRegularPolynomialObservablesTimeOrderedProduct"></a>) of two <a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>⟨</mo><mi>T</mi><mrow><mo>(</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⋆</mo> <mi>F</mi></msub><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⟩</mo></mrow> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><mo>⟨</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⟩</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mo>+</mo><munder><munder><mrow><mo>⟨</mo><mi>ℏ</mi><munder><mo>∫</mo><mrow><mi>Σ</mi><mo>×</mo><mi>Σ</mi></mrow></munder><mi>δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>x</mi><mo>′</mo><mo stretchy="false">)</mo><msubsup><mi>Δ</mi> <mi>F</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>y</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⟩</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mi>ℏ</mi><msubsup><mi>Δ</mi> <mi>H</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mrow><annotation encoding="application/x-tex"> \left\langle T\left( \mathbf{\Phi}^a(x) \star_F \mathbf{\Phi}^b(y) \right) \right\rangle_0 \;=\; \underset{ = 0 }{ \underbrace{ \left\langle \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y) \right\rangle }} + \underset{ = \hbar \Delta^{a b}_H(x,y) }{ \underbrace{ \left \langle \hbar \underset{\Sigma \times \Sigma}{\int} \delta(x-x') \Delta^{a b}_F(x,y) \delta(y-y') \right\rangle }} </annotation></semantics></math></div> <p>analogous to example <a class="maruku-ref" href="#HadamardMoyalStarProductOfTwoLinearObservables"></a>.</p> <p>Equivalently in the <a class="existingWikiWord" href="/nlab/show/operator+product">operator product</a>-notation of def. <a class="maruku-ref" href="#NormalOrderedProductNotation"></a> this reads:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>⟨</mo><mi>T</mi><mrow><mo>(</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⟩</mo></mrow> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>ℏ</mi><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\langle T\left( \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y) \right) \right\rangle_0 \;=\; \hbar \Delta_F(x,y) \,. </annotation></semantics></math></div></div> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/propagators">propagators</a> (i.e. <a class="existingWikiWord" href="/nlab/show/integral+kernels">integral kernels</a> of <a class="existingWikiWord" href="/nlab/show/Green+functions">Green functions</a>)</strong> <br /> <strong>for the <a class="existingWikiWord" href="/nlab/show/wave+operator">wave operator</a> and <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a></strong> <br /> <strong>on a <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetime">globally hyperbolic spacetime</a> such as <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>:</strong></p> <table><thead><tr><th>name</th><th>symbol</th><th><a class="existingWikiWord" href="/nlab/show/wave+front+set">wave front set</a></th><th>as <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum exp. value</a> <br /> of <a class="existingWikiWord" href="/nlab/show/operator-valued+distribution">field operators</a></th><th>as a <a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">product</a> of <br /> <a class="existingWikiWord" href="/nlab/show/operator-valued+distribution">field operators</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_1"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd><msub><mi>Δ</mi> <mi>S</mi></msub></mtd> <mtd><mo>=</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned}\Delta_S & = \Delta_+ - \Delta_- \end{aligned}</annotation></semantics></math></td><td style="text-align: left;"><img src="https://ncatlab.org/nlab/files/RetGreenFunction.png" width="60" /> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_2"><semantics><mrow><mphantom><mi>A</mi></mphantom><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>−</mo></mrow><annotation encoding="application/x-tex">\phantom{A}\,\,\,-</annotation></semantics></math> <br /> <img src="https://ncatlab.org/nlab/files/AdvancedGreenFunction.png" width="60" /></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_3"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><mi>i</mi><mi>ℏ</mi><mspace width="thinmathspace"></mspace><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} & i \hbar \, \Delta_S(x,y) = \\ & \left\langle \;\left[\mathbf{\Phi}(x),\mathbf{\Phi}(y)\right]\; \right\rangle \end{aligned} </annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Peierls+bracket">Peierls-Poisson bracket</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/advanced+propagator">advanced propagator</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_4"><semantics><mrow><msub><mi>Δ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_+</annotation></semantics></math></td><td style="text-align: left;"><img src="https://ncatlab.org/nlab/files/RetGreenFunction.png" width="60" /></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_5"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><mi>i</mi><mi>ℏ</mi><mspace width="thinmathspace"></mspace><msub><mi>Δ</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>x</mi><mo>≥</mo><mi>y</mi></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>y</mi><mo>≥</mo><mi>x</mi></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} & i \hbar \, \Delta_+(x,y) = \\ & \left\{ \array{ \left\langle \; \left[ \mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& x \geq y \\ 0 &\vert& y \geq x } \right. \end{aligned} </annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/future">future</a> part of <br /> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/retarded+propagator">retarded propagator</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_6"><semantics><mrow><msub><mi>Δ</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_-</annotation></semantics></math></td><td style="text-align: left;"><img src="https://ncatlab.org/nlab/files/AdvancedGreenFunction.png" width="60" /></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_7"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><mi>i</mi><mi>ℏ</mi><mspace width="thinmathspace"></mspace><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>y</mi><mo>≥</mo><mi>x</mi></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>x</mi><mo>≥</mo><mi>y</mi></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} & i \hbar \, \Delta_-(x,y) = \\ & \left\{ \array{ \left\langle \; \left[\mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& y \geq x \\ 0 &\vert& x \geq y } \right. \end{aligned}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/past">past</a> part of <br /> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_8"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mrow><mo>(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo>)</mo></mrow><mo>+</mo><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mi>S</mi></msub><mo>+</mo><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Δ</mi> <mi>F</mi></msub><mo>−</mo><mi>i</mi><msub><mi>Δ</mi> <mo>−</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} \Delta_H &= \tfrac{i}{2}\left( \Delta_+ - \Delta_-\right) + H\\ & = \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_- \end{aligned}</annotation></semantics></math></td><td style="text-align: left;"><img src="https://ncatlab.org/nlab/files/HadamardPropagator.png" width="60" /></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_9"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><mi>ℏ</mi><mspace width="thinmathspace"></mspace><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><munder><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mo>:</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>:</mo><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} & \hbar \, \Delta_H(x,y) \\ & = \left\langle \; \mathbf{\Phi}(x) \mathbf{\Phi}(y) \; \right\rangle \\ & = \underset{ = 0 }{\underbrace{\left\langle \; : \mathbf{\Phi}(x) \mathbf{\Phi}(y) : \; \right\rangle}} \\ & \phantom{=} + \left\langle \; \left[ \mathbf{\Phi}^{(-)}(x), \mathbf{\Phi}^{(+)}(y) \right] \; \right\rangle \end{aligned} </annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/positive+real+number">positive</a> <a class="existingWikiWord" href="/nlab/show/frequency">frequency</a> of <br /> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a>, <br /> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a>-product, <br /> <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_10"><semantics><mrow><mphantom><mo>=</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{=}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_11"><semantics><mrow><mphantom><mo>=</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{=}</annotation></semantics></math> or generally of <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_12"><semantics><mrow><mphantom><mo>=</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{=}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_13"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd><msub><mi>Δ</mi> <mi>F</mi></msub></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mrow><mo>(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo>)</mo></mrow><mo>+</mo><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>i</mi><msub><mi>Δ</mi> <mi>D</mi></msub><mo>+</mo><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo>+</mo><mi>i</mi><msub><mi>Δ</mi> <mo>−</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned}\Delta_F & = \tfrac{i}{2}\left( \Delta_+ + \Delta_- \right) + H \\ & = i \Delta_D + H \\ & = \Delta_H + i \Delta_- \end{aligned}</annotation></semantics></math></td><td style="text-align: left;"><img src="https://ncatlab.org/nlab/files/FeynmanPropagator.png" width="60" /></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_14"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><mi>ℏ</mi><mspace width="thinmathspace"></mspace><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mi>T</mi><mrow><mo>(</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>x</mi><mo>≥</mo><mi>y</mi></mtd></mtr> <mtr><mtd><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>y</mi><mo>≥</mo><mi>x</mi></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} & \hbar \, \Delta_F(x,y) \\ & = \left\langle \; T\left( \; \mathbf{\Phi}(x)\mathbf{\Phi}(y) \;\right) \; \right\rangle \\ & = \left\{ \array{ \left\langle \; \mathbf{\Phi}(x)\mathbf{\Phi}(x) \; \right\rangle &\vert& x \geq y \\ \left\langle \; \mathbf{\Phi}(y) \mathbf{\Phi}(x) \; \right\rangle &\vert& y \geq x } \right.\end{aligned}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a></td></tr> </tbody></table> <p>(see also <a class="existingWikiWord" href="/nlab/show/Mikica+Kocic">Kocic</a>‘s overview: <a class="existingWikiWord" href="/nlab/files/KGPropagatorsOnMinkowskiTable.pdf" title="pdf">pdf</a>)</p></div> <h2 id="related_concepts">Related concepts</h2> <div> <table><thead><tr><th>product in <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative QFT</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_27ee5fe71b396df84a9f4b19030bbbae0b7c742f_1"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,</annotation></semantics></math> induces</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> (<a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> (<a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/interacting+quantum+observables">interacting quantum observables</a></td></tr> </tbody></table> </div><div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></p> </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/quasi-state">quasi-state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> <p><a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a></p> <p><a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</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> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a></p> </li> <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/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a></p> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/observables">observables</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a></p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorems">theorems</a></p> <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/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <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> <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/Nuiten%27s+lemma">Nuiten's lemma</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> </ul> </li> </ul> </div> <h2 id="references">References</h2> <p>The construction goes back to</p> <ul> <li id="Wick50"><a class="existingWikiWord" href="/nlab/show/Gian-Carlo+Wick">Gian-Carlo Wick</a>, <em>The evaluation of the collision matrix</em>, Phys. Rev. 80, 268-272 (1950)</li> </ul> <p>Its realization as the <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a> of the <a class="existingWikiWord" href="/nlab/show/Peierls+bracket">Peierls bracket</a> shifted by a <a class="existingWikiWord" href="/nlab/show/quasi-free+Hadamard+state">quasi-free Hadamard state</a> is due to</p> <ul> <li id="Dito90">Joseph Dito, <em>Star-product approach to quantum field theory: The free scalar field</em>. Letters in Mathematical Physics, 20(2):125–134, 1990 (<a href="https://inspirehep.net/record/303898/">spire</a>)</li> </ul> <p>further amplified in</p> <ul> <li id="DuetschFredenhagen00"> <p><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, section 5.1 of <em>Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion</em>, Commun.Math.Phys. 219 (2001) 5-30 (<a href="https://arxiv.org/abs/hep-th/0001129">arXiv:hep-th/0001129</a>)</p> </li> <li id="DuetschFredenhagen01"> <p><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <em>Perturbative algebraic field theory, and deformation quantization</em>, in <a class="existingWikiWord" href="/nlab/show/Roberto+Longo">Roberto Longo</a> (ed.), <em>Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects</em>, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001 (<a href="https://arxiv.org/abs/hep-th/0101079">arXiv:hep-th/0101079</a>)</p> </li> <li id="HirschfeldHenselder02"> <p>A. C. Hirshfeld, P. Henselder, <em>Star Products and Perturbative Quantum Field Theory</em>, Annals Phys. 298 (2002) 382-393 (<a href="https://arxiv.org/abs/hep-th/0208194">arXiv:hep-th/0208194</a>)</p> </li> <li id="Collini16"> <p><a class="existingWikiWord" href="/nlab/show/Giovanni+Collini">Giovanni Collini</a>, <em>Fedosov Quantization and Perturbative Quantum Field Theory</em> (<a href="https://arxiv.org/abs/1603.09626">arXiv:1603.09626</a>)</p> </li> </ul> <p>and the generalization to <a class="existingWikiWord" href="/nlab/show/quantum+field+theory+on+curved+spacetime">quantum field theory on curved spacetime</a> is discussed in</p> <ul> <li id="BrunettiFredenhagen95"> <p><a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, M. Köhler, <em>The microlocal spectrum condition and Wick polynomials on curved spacetimes</em>, Commun. Math. Phys. 180, 633-652, 1996 (<a href="https://arxiv.org/abs/gr-qc/9510056">arXiv:gr-qc/9510056</a>)</p> </li> <li id="BrunettiFredenhagen00"> <p><a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <em>Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds</em>, Commun. Math. Phys. 208 : 623-661, 2000 (<a href="https://arxiv.org/abs/math-ph/9903028">math-ph/9903028</a>)</p> </li> <li id="HollandsWald01"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Hollands">Stefan Hollands</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Wald">Robert Wald</a>, <em>Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime</em>, Commun. Math. Phys. 223:289-326, 2001 (<a href="https://arxiv.org/abs/gr-qc/0103074">arXiv:gr-qc/0103074</a>)</p> </li> </ul> <p>Review is in</p> <ul> <li id="Duetsch18"><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, section 2.1 of <em><a class="existingWikiWord" href="/nlab/show/From+classical+field+theory+to+perturbative+quantum+field+theory">From classical field theory to perturbative quantum field theory</a></em>, 2018</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 20, 2024 at 19:49:28. See the <a href="/nlab/history/Wick+algebra" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Wick+algebra" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/7954/#Item_4">Discuss</a><span class="backintime"><a href="/nlab/revision/Wick+algebra/61" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Wick+algebra" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Wick+algebra" accesskey="S" class="navlink" id="history" rel="nofollow">History (61 revisions)</a> <a href="/nlab/show/Wick+algebra/cite" style="color: black">Cite</a> <a href="/nlab/print/Wick+algebra" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Wick+algebra" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>