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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="aqft">AQFT</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<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>) <a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a> <h2 id="concepts">Concepts</h2> <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a>: <ul> <li> <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> </li> <li> <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> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+history">field history</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> </li> </ul> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>, <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+net">field net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> </ul> </li> <li> <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> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> <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> <a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>) </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> </li> </ul> </div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a> <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a 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particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" 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href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#abstract'>Abstract</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#statement'>Statement</a></li> <li><a href='#preconditions_needed_for_the_proof'>Preconditions Needed for the Proof</a></li> <li><a href='#consequences'>Consequences</a></li> <li><a href='#related_theorems'>Related theorems</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> The Reeh-Schlieder theorem is a <a class="existingWikiWord" href="/nlab/show/theorem">theorem</a> about <a class="existingWikiWord" href="/nlab/show/local+nets+of+observables">local nets of observables</a> in the <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+approach">Haag-Kastler approach</a> to <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>). It states that for certain nets the set of <a class="existingWikiWord" href="/nlab/show/vectors">vectors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo><mi>Ω</mi><mo>↪</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O}) \Omega \hookrightarrow \mathcal{H}</annotation></semantics></math> <blockquote> (the local <a class="existingWikiWord" href="/nlab/show/linear+operator">operator</a> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}</annotation></semantics></math> for a <a class="existingWikiWord" href="/nlab/show/bounded+set">bounded</a> <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> applied to the <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><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math>, cf. <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+vacuum+representation">Haag-Kastler vacuum representation</a> for definitions and details) </blockquote> is <a class="existingWikiWord" href="/nlab/show/dense+subset">dense</a> in the given <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">state space</a>, 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>. The Reeh-Schlieder theorem can be proven to be valid in the <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+vacuum+representation">Haag-Kastler vacuum representation</a>, but the statement itself is sometimes used as an <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a> and called the Reeh-Schlieder property in this case. The Reeh-Schlieder theorem is of central importance for the mathematical structure theory of the <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+approach">Haag-Kastler approach</a>. Its physical interpretation is counterintuitive and therefore to a certain degree controversial: Intuitively one might expect that for a localized observable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in \mathcal{M}(\mathcal{O})</annotation></semantics></math> the vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Ω</mi></mrow><annotation encoding="application/x-tex">A \Omega</annotation></semantics></math> should be localized in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math>, in that the state should still look like the vacuum in the <a class="existingWikiWord" href="/nlab/show/causal+complement">causal complement</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math>. But the Reeh-Schlieder theorem says that every arbitrary state can be approximated by states of the kind <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Ω</mi></mrow><annotation encoding="application/x-tex">A \Omega</annotation></semantics></math>. This shows that the concept of localized states is nontrivial in <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> and needs to be handled with care. Complementary statements about the asymptotic “vacuum-like appearance” of localized observables exist, too, and are commonly referred to as cluster theorems. In other parts of the physics literature, notably in the context of <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a>, the Reeh-Schlieder property is also called the <a class="existingWikiWord" href="/nlab/show/state-operator+correspondence">state-operator correspondence</a>, cf. <a href="#Schroer">Schroer, footnote 14, page 34</a>. <h2 id="abstract">Abstract</h2> We state the theorem for a vacuum representation on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>. Since the theorem has some controverse consequences for the physical interpretation of the theory, some of which will be mentioned in the corresponding paragraph, we take a quick look at which axioms really enter into the proof of the theorem. The references include work on the generalization of the theorem to more general <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> as well as work concerned with dodging the theorem and its consequences by choosing an alternative set of axioms. <h2 id="definition">Definition</h2> All necessary definitions can be found at <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+vacuum+representation">Haag-Kastler vacuum representation</a>. <h2 id="statement">Statement</h2> Both the statement and the proof we mention here refer to the <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+vacuum+representation">Haag-Kastler vacuum representation</a>, but both the statement and the proof can be generalized resp. translated to different contexts, like e.g. to a <a class="existingWikiWord" href="/nlab/show/Wightman+theory">Wightman theory</a>, to more general spacetimes and to other states than the vacuum vector. <div class='num_theorem'> <h6>Theorem</h6> (Reeh-Schlieder) In the <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+vacuum+representation">Haag-Kastler vacuum representation</a> the vacuum vector <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> is <a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic</a> and <a class="existingWikiWord" href="/nlab/show/separating+vector">separating</a> for every local algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O})</annotation></semantics></math>. </div> The following proof does not strive for maximal generality, for example we specialize to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">d = 4</annotation></semantics></math> dimensions. The generalization to other dimensions can be done by the reader. We will comment on which part of the axioms of the vacuum representation are actually used below. <div class="proof"> <h6 id="proof">Proof</h6> First some preliminary observations: The covariance axiom says that we have a strongly continuous representation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of the Poincare group, the SNAG-Theorem (see <a class="existingWikiWord" href="/nlab/show/spectral+measure">spectral measure</a>) therefore provides us with a spectral measure for the translation subgroup <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">\mathcal{T}</annotation></semantics></math>, that we identify with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mi>k</mi><mo>∈</mo><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow></msub><msup><mi>e</mi> <mrow><mi>i</mi><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">⟩</mo></mrow></msup><mi>𝒫</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="2em"></mspace><mo>∀</mo><mi>x</mi><mo>∈</mo><mi>𝒯</mi><mo>≅</mo><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex"> \mathcal{U}(x) = \int_{k\in \mathbb{R}^4} e^{i \langle x, k\rangle} \mathcal{P}(k) \qquad \forall x \in \mathcal{T} \cong \mathbb{R}^4 </annotation></semantics></math></div> We may restrict the domain of integration to the closure of the forward lightcone <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cloV</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">cloV_+</annotation></semantics></math> by the spectrum condition. The spectral measure can be used to define operators for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi><mo>∈</mo><msup><mi>ℛ</mi> <mn>4</mn></msup><mo>+</mo><mi>i</mi><msub><mi>V</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">z = x + i y \in \mathcal{R}^4 + i V_+</annotation></semantics></math> via <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mo>∫</mo> <mrow><msub><mi>cloV</mi> <mo>+</mo></msub></mrow></msub><msup><mi>e</mi> <mrow><mi>i</mi><mo stretchy="false">⟨</mo><mi>z</mi><mo>,</mo><mi>k</mi><mo stretchy="false">⟩</mo></mrow></msup><mi>𝒫</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> U(z) \;\coloneqq\; \int_{cloV_+} e^{i \langle z, k\rangle} \mathcal{P}(k) </annotation></semantics></math></div> The dominated convergence theorem for spectral integrals (see <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>) tells us that the strong limit of the left side of the following equation exists and is equal to the right side: <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mrow><mi>y</mi><mo stretchy="false">↓</mo><mn>0</mn></mrow></munder><mi>U</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \lim_{y \downarrow 0} U(x + i y) = U (x) </annotation></semantics></math></div> Next we choose a vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">u \in \mathcal{H}</annotation></semantics></math> from our Hilbert space, a bounded operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> on <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> and form the complex-valued function <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mi>u</mi><mo>,</mo><mi>A</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">⟨</mo><mi>u</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mi>A</mi><mi>Ω</mi><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> f_{u, A}(z) \;\coloneqq\; \langle u, U(z) A \Omega \rangle \,, </annotation></semantics></math></div> which is holomorphic on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℛ</mi> <mn>4</mn></msup><mo>+</mo><mi>i</mi><msub><mi>V</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{R}^4 + i V_+</annotation></semantics></math> by definition and continuous on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℛ</mi> <mn>4</mn></msup><mo>+</mo><mi>i</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>+</mo></msub><mo>∪</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{R}^4 + i (V_+ \cup \{0 \}) </annotation></semantics></math> by our last observation. We can define a second such function <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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℛ</mi> <mn>4</mn></msup><mo>−</mo><mi>i</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>+</mo></msub><mo>∪</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{R}^4 - i (V_+ \cup \{0 \}) </annotation></semantics></math> by complex conjugation: <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mi>u</mi><mo>,</mo><mi>A</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mover><mrow><msub><mi>f</mi> <mrow><mi>u</mi><mo>,</mo><mi>A</mi></mrow></msub><mo stretchy="false">(</mo><mover><mi>z</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex"> g_{u, A}(z) = \overline{f_{u, A}(\overline z)} </annotation></semantics></math></div> If there is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">U \subset \mathbb{R}^4</annotation></semantics></math> open such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mi>u</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f_{u, A}</annotation></semantics></math> is real valued on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mi>u</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f_{u, A}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mi>u</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex">g_{u, A}</annotation></semantics></math> coincide on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> and we can invoke a suitable version of the edge-of-the-wedge theorem<a href="/nlab/new/edge-of-the-wedge+theorem">?</a> as stated on <a class="existingWikiWord" href="/nlab/show/analytic+geometry">analytic geometry</a> to conclude that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mi>u</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f_{u, A}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mi>u</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex">g_{u, A}</annotation></semantics></math> are the branches of a unique holomorphic function, that is holomorphic at least on a complex neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. Now the proof that <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> is <a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O})</annotation></semantics></math>: Choose a bounded open <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_0</annotation></semantics></math> and a neighborhood of zero <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊂</mo><msup><mi>ℛ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">V \subset \mathcal{R}^4</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mn>0</mn></msub><mo>+</mo><mi>V</mi><mo>⊂</mo><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}_0 + V \subset \mathcal{O}</annotation></semantics></math>. Suppose that <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> is not cyclic for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O})</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O}) \Omega</annotation></semantics></math> is not dense in <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> and we can choose a vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>ℋ</mi><mo>,</mo><mi>v</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">v \in \mathcal{H}, v \neq 0</annotation></semantics></math> such that <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>v</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mi>A</mi><mi>Ω</mi><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \langle v, U(z) A \Omega \rangle = 0 </annotation></semantics></math></div> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in \mathcal{M}(\mathcal{O})</annotation></semantics></math>. In particular we have for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>0</mn></msub><mo>∈</mo><mi>ℳ</mi><mo stretchy="false">(</mo><msub><mi>𝒪</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_0 \in \mathcal{M}(\mathcal{O}_0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">x \in V</annotation></semantics></math> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mi>v</mi><mo>,</mo><msub><mi>A</mi> <mn>0</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>v</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>A</mi> <mn>0</mn></msub><mi>Ω</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>v</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>A</mi> <mn>0</mn></msub><mi>U</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>Ω</mi><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> f_{v, A_0} (x) = \langle v, U(x) A_0 \Omega \rangle = \langle v, U(x) A_0 U(x)^{-1} \Omega \rangle = 0 </annotation></semantics></math></div> Now we see from our previous considerations that there is a function holomorphic in a complex neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> that restricts to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mi>v</mi><mo>,</mo><msub><mi>A</mi> <mn>0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">f_{v, A_0}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, vanishes on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> and has therefore to vanish everywhere. That means the previous equality holds for arbitrary translations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo>∈</mo><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">U(x), x \in \mathbb{R}^4</annotation></semantics></math>. Recall that weak additivity holds in the vacuum representation. This together with the previous statement implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>v</mi><mo>,</mo><mi>R</mi><mi>Ω</mi><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\langle v, R \Omega \rangle = 0</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">R \in \mathcal{R}</annotation></semantics></math>, the global algebra. But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\mathcal{R} \Omega</annotation></semantics></math> is dense in <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> by assumption (see the axiom about the existence of a vacuum vector), we get that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> must be zero, contradiction: <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> has to be cyclic for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O})</annotation></semantics></math>. Now the proof that <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> is <a class="existingWikiWord" href="/nlab/show/separating+vector">separating</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O})</annotation></semantics></math>: Choose a bounded open set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_2</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo>⊥</mo><msub><mi>𝒪</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{O} \perp \mathcal{O}_2</annotation></semantics></math>, then by locality we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo><mo>⊆</mo><mo stretchy="false">(</mo><mi>ℳ</mi><mo stretchy="false">(</mo><msub><mi>𝒪</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O}) \subseteq (\mathcal{M}(\mathcal{O}_2))'</annotation></semantics></math>. We know already that <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> is cyclic for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><msub><mi>𝒪</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O}_2)</annotation></semantics></math>, therefore it is separating for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O})</annotation></semantics></math>. </div> <h2 id="preconditions_needed_for_the_proof">Preconditions Needed for the Proof</h2> Since the theorem and its consequences are counterintuitive for the physical interpretation of the theory, it is worthwhile to take a look which axioms enter the proof: Is the theorem a true feature of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> or is it an artifact caused by ill chosen assumptions? The proof that the vacuum vector is cyclic given in the previous paragraph makes use of: <ol> <li>Strong representation of the translation subgroup and the spectrum condition.</li> </ol> Note that we do not need the representation of the whole Poincare group, but really only that of the translations. <ol> <li> isotony, </li> <li> weak additivity and </li> <li> of course the existence of the vacuum vector itself and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\mathcal{R} \Omega</annotation></semantics></math> is dense in <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>. </li> </ol> Now weak additivity can be proven to hold using isotony and additiviy. It is sometimes argued that the Reeh-Schlieder theorem - that is the part that the vacuum vector is cyclic for every local algebra - is somehow a contradiction to locality . The remarkable fact in this context is that the locality axiom does not enter the proof that the vacuum vector is cyclic. (It was used for the proof that the vacuum vector is also separating, though). For some further elaboration of this point see the paper by Halvorson cited in the references. <h2 id="consequences">Consequences</h2> The fact that the vacuum vector is separating implies that there cannot be a number operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> associated to any bounded open set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math>, because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> would annihilate the vacuum vector. This implies that the notion of particles in relativistic quantum physics cannot be quite as simple as in classical physics or in nonrelativistic quantum physics. More generally there cannot be a nonzero localized observable that annihilates the vacuum. This implies that the stress-energy tensor, if localized to a bounded region, cannot be a positive operator, since its expectation value in the vacuum is zero, in fact it cannot be bounded from below. This is sometimes mentioned as a pathology occuring in general spacetimes that do not have a global timelike <a class="existingWikiWord" href="/nlab/show/Killing+vector">Killing vector</a> field, but in our context it actually holds in <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> in the vacuum state. The fact that the vacuum vector is cyclic means that any arbitrary state in the vacuum representation can be approximated by measurements in an arbitrary small bounded open region applied to the vacuum vector. This fact is sometimes referred to as the existence of <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuations">vacuum fluctuations</a>. A direct consequence of the Reeh-Schlieder theorem is therefore that to any regions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>𝒪</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_1, \mathcal{O}_2</annotation></semantics></math>, no matter how far apart, there are many projections in the corresponding local algebras that are positively correlated in the vacuum state. <h2 id="related_theorems">Related theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/state-field+correspondence">state-field correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS-construction">GNS-construction</a> </li> </ul> <h2 id="references">References</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Stephen+J.+Summers">Stephen J. Summers</a>: Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State, in Deep Beauty – Understanding the Quantum World through Mathematical Innovation, Cambridge University Press (2011) pp 317-342 [<a href="http://arxiv.org/abs/0802.1854">arXiv:0802.1854</a>, <a href="https://doi.org/10.1017/CBO9780511976971.009">doi:10.1017/CBO9780511976971.009</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, section 2 of: Notes on Some Entanglement Properties of Quantum Field Theory [<a href="https://arxiv.org/abs/1803.04993">arXiv:1803.04993</a>] </li> </ul> See also <ul> <li>Wikipedia on the <a href="http://en.wikipedia.org/wiki/Reeh%E2%80%93Schlieder_theorem">Reeh-Schlieder Theorem</a></li> </ul> The Reeh-Schlieder theorem can be generalized from <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> to more general <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a>, see for example: <ul> <li>Alexander Strohmaier, Rainer Verch, Manfred Wollenberg: Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems (<a href="http://xxx.uni-augsburg.de/abs/math-ph/0202003">arXiv</a>)</li> </ul> On efforts to avoid the Reeh-Schlieder theorem and its counter-intuitive implications: <ul> <li><a class="existingWikiWord" href="/nlab/show/Hans+Halvorson">Hans Halvorson</a>, Reeh-Schlieder Defeats Newton-Wigner: On alternative localization schemes in relativistic quantum field theory (<a href="http://xxx.uni-augsburg.de/abs/quant-ph/0007060">arXiv</a>)</li> </ul> The observation that the Reeh-Schlieder property describes what elsewhere is called the <a class="existingWikiWord" href="/nlab/show/operator-state+correspondence">operator-state correspondence</a> is made explicit in: <ul> <li id="Schroer"><a class="existingWikiWord" href="/nlab/show/Bert+Schroer">Bert Schroer</a>, footnote 14 on page 34 of: Particle Physics and QFT at the Turn of the Century: Old principles with new concepts [<a href="http://cds.cern.ch/record/367824/files/9810080.pdf">pdf</a>,<a href="https://arxiv.org/abs/hep-th/9810080">arXiv:hep-th/9810080</a>]</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on December 29, 2024 at 19:13:07. 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