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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> causal additivity </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/8031/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebraic_qunantum_field_theory">Algebraic Qunantum 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='#causal_additivity'>Causal additivity</a></li> <li><a href='#causal_factorization'>Causal factorization</a></li> <li><a href='#causal_locality_of_quantum_observables'>Causal locality of quantum observables</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The formalization of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> states the key properties that the <a class="existingWikiWord" href="/nlab/show/scattering+matrix">scattering matrix</a> of the theory is supposed to have as <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> (instead of imagining that the scattering matrix is defined by a <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>). The key such axiom is the statement that the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> be <em>causally additive</em> (which gives <em><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a></em> its name). This essentially encodes the idea that effects in the quantum theory propagate causally, hence within the <a class="existingWikiWord" href="/nlab/show/future+cone">future cone</a> and <a class="existingWikiWord" href="/nlab/show/past+cone">past cone</a> of the region which causes the effect.</p> <h2 id="causal_additivity">Causal additivity</h2> <p>The <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> (as discussed there) is a <a class="existingWikiWord" href="/nlab/show/functional">functional</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>ℱ</mi> <mi>loc</mi></msub><mo>⟶</mo><mi>𝒲</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mstyle displaystyle="false"><mfrac><mi>g</mi><mi>ℏ</mi></mfrac></mstyle><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> S \;\colon\; \mathcal{F}_{loc} \longrightarrow \mathcal{W}[ [ \tfrac{g}{\hbar} ] ] </annotation></semantics></math></div> <p>sending <a href="#Wick+algebra#CompactlySupportedPolynomialLocalDensities">local observables</a> with <a class="existingWikiWord" href="/nlab/show/compact+support">compact support</a> in <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> (i.e. <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+densities">Lagrangian densities</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">L_{int}</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/source+fields">source fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>) to <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> in the ratio of a <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in 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><mi>𝒲</mi></mrow><annotation encoding="application/x-tex">\mathcal{W}</annotation></semantics></math> of the underlying <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a>.</p> <p>Causal additivity (<a href="#EpsteinGlaser73">Epstein-Glaser 73</a>) is the statement/requirement that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>≥</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><munder><mo>∀</mo><mrow><mi>L</mi><mo>∈</mo><msub><mi>ℱ</mi> <mi>loc</mi></msub></mrow></munder><mrow><mo>(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>L</mi><mo>+</mo><msub><mi>J</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>J</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mi>L</mi><mo>+</mo><msub><mi>J</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>S</mi><mo stretchy="false">(</mo><mi>L</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>S</mi><mo stretchy="false">(</mo><mi>L</mi><mo>+</mo><msub><mi>J</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( supp(J_1) \geq supp(J_2) \right) \;\; \Rightarrow \;\; \left( \underset{L \in \mathcal{F}_{loc}}{\forall} \left( S(L + J_1 + J_2) = S(L + J_1) S(L)^{-1} S(L + J_2) \right) \right) \,. </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(-)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/support">support</a> of the given <a href="#Wick+algebra#CompactlySupportedPolynomialLocalDensities">local observables</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>≥</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(J_1) \geq supp(J_2)</annotation></semantics></math> means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(J_1)</annotation></semantics></math> does not intersect the <a class="existingWikiWord" href="/nlab/show/causal+past">causal past</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(J_2)</annotation></semantics></math>. The product on the right is the product in the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> (hence the <a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a>).</p> <h2 id="causal_factorization">Causal factorization</h2> <p>As a special case, causal additivity immediately implies <strong>causal factorization</strong>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>≥</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>L</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( supp(L_1) \geq supp(L_2) \right) \;\; \Rightarrow \;\; \left( S(L_1 + L_2) = S(L_1) S(L_2) \right) \,. </annotation></semantics></math></div> <p>This in turn directly implies that the perturbative expansion of the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> is given by <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>.</p> <h2 id="causal_locality_of_quantum_observables">Causal locality of quantum observables</h2> <p>But the reason why the full condition is called <em>causal additivity</em> is that it is equivalent to a simpler-looking condition on the <a class="existingWikiWord" href="/nlab/show/generating+function">generating function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>L</mi></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>S</mi><mo stretchy="false">(</mo><mi>L</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>S</mi><mo stretchy="false">(</mo><mi>L</mi><mo>+</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Z_{L}(J) \;\colon\; S(L)^{-1} S(L + J) </annotation></semantics></math></div> <p>that is induced by the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>. In terms of these, causal additivity is equivalently (by <a href="S-matrix#CausalLocalityOfThePerturbativeSMatrix">this lemma</a>) the condition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>≥</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><munder><mo>∀</mo><mrow><mi>L</mi><mo>∈</mo><msub><mi>ℱ</mi> <mi>loc</mi></msub></mrow></munder><mrow><mo>(</mo><msub><mi>Z</mi> <mi>L</mi></msub><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>J</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Z</mi> <mi>L</mi></msub><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>Z</mi> <mi>L</mi></msub><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( supp(J_1) \geq supp(J_2) \right) \;\; \Rightarrow \;\; \left( \underset{L \in \mathcal{F}_{loc}}{\forall} \left( Z_{L}(J_1 + J_2) = Z_L(J_1) Z_L(J_2) \right) \right) \,. </annotation></semantics></math></div> <p>Notice that what these generating functions generate is the perturbative <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">^</mo></mover><mo>≔</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>ϵ</mi></mrow></mfrac><msub><mi>T</mi> <mrow><mstyle displaystyle="false"><mfrac><mi>g</mi><mi>ℏ</mi></mfrac></mstyle><msub><mi>L</mi> <mi>int</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>ϵ</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mo stretchy="false">|</mo> <mrow><mi>ϵ</mi><mo>=</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> \hat A \coloneqq \frac{d}{d \epsilon} T_{\tfrac{g}{\hbar}L_{int}}( \epsilon A ) \vert|_{\epsilon = 0} </annotation></semantics></math></div> <p>and causal additivity implies that as long as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">L_{int}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatically switched</a> only outside the <a class="existingWikiWord" href="/nlab/show/causal+closure">causal closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(A)</annotation></semantics></math>, then the system of quantum observables satisfies <strong><a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a></strong> and hence forms a <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> (<a href="#S-matrix#PerturbativeQuantumObservablesIsLocalnet">this prop.</a>).</p> <p>This is how <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> gives rise to <em><a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></em>.</p> <h2 id="references">References</h2> <p>The concept goes back to</p> <ul> <li id="EpsteinGlaser73"><a class="existingWikiWord" href="/nlab/show/Henri+Epstein">Henri Epstein</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Glaser">Vladimir Glaser</a>, <em><a class="existingWikiWord" href="/nlab/show/The+Role+of+locality+in+perturbation+theory">The Role of locality in perturbation theory</a></em>, Annales Poincaré Phys. Theor. A 19 (1973) 211 (<a href="http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0">Numdam</a>)</li> </ul> <p>Lecture notes include</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+perturbative+quantum+field+theory">geometry of physics – perturbative quantum field theory</a></em> (around <a href="geometry+of+physics+--+perturbative+quantum+field+theory#eq:CausalAdditivity">this equation</a>)</li> </ul> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a></em>, <em><a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></em> and <em><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on May 13, 2019 at 05:58:52. 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