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this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </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> The following surveys how basic <a class="existingWikiWord" href="/nlab/show/theorems">theorems</a> about the standard foundation of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> imply an accurate <a class="existingWikiWord" href="/nlab/show/geometry">geometric</a> incarnation of the “<a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>” by an <a class="existingWikiWord" href="/nlab/show/order+theory">order-theoretic structure</a> that combines with an <a class="existingWikiWord" href="/nlab/show/algebra">algebraic</a> <a class="existingWikiWord" href="/nlab/show/structure">structure</a> to a <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a>, the “<a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a>”. While the notion of <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> has been motivated by the <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a>, the point here is to highlight that taking into account further theorems about the standard foundations of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, the notion effectively follows automatically and provides an accurate and useful description of the geometry of “quantum phase space” also in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> formulated in the style of <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>. <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <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 class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </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/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder 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/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of 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" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> </li> <li> examples <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" 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> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#QuantumPhaseSpaceAsJordanGeometry'>The quantum phase space as a Jordan-geometry given by a ringed topos</a></li> <li><a href='#RelationToTheNonCommutativePhaseSpace'>Relation to the traditional non-commutative geometry</a></li> <li><a href='#ListOfTheoremsInvoked'>List of theorems invoked</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="QuantumPhaseSpaceAsJordanGeometry">The quantum phase space as a Jordan-geometry given by a ringed topos</h2> Back when <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> was discovered in the first half of the 20th century, it was eventually formalized as <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a> and the traditional modern formulation emerged, where, in <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> perspective, <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> are represented by suitable <a class="existingWikiWord" href="/nlab/show/linear+operators">linear operators</a> on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> and more generally by elements of a <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">of observables</a>, and where <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> are certain (namely positive and normalized) linear functionals on these <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*</a>-<a class="existingWikiWord" href="/nlab/show/algebras+of+observables">algebras of observables</a> (<a class="existingWikiWord" href="/nlab/show/states+on+star-algebras">states on star-algebras</a>). But one may still ask if the <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> in the definition of <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a> accurately capture the intended <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. This or similar questions were discussed back in the middle of the 20th century, when these notions were still in flux. Specifically in the 1930s <a class="existingWikiWord" href="/nlab/show/Pascual+Jordan">Pascual Jordan</a> argued (<a href="#Jordan32">Jordan 32</a>) that the <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> <a class="existingWikiWord" href="/nlab/show/structure">structure</a> on the observables is more <a class="existingWikiWord" href="/nlab/show/structure">structure</a> than supported by the physics of states and observation, that instead only the underlying structure of what is now called the <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> should matter. Much later in 1978 this idea was formally validated by the <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> (<a href="#AlfsenShultz78">Alfsen-Shultz 78</a>). This states that the <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> for given <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> depends indeed only on the underlying <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> structure. This is not too surprising: the definition of a <a class="existingWikiWord" href="/nlab/show/state+on+an+operator+algebra">state on an operator algebra</a> does not even mention the <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> structure but mentions only the <a class="existingWikiWord" href="/nlab/show/positive+operator">positivity</a> structure, which is what the <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> captures. Despite these insights, <a class="existingWikiWord" href="/nlab/show/Jordan+algebras">Jordan algebras</a> found and find only marginal attention in <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>. In a <a href="#ASReview04">review from 2004</a> of the book of Alfsen and Shultz it says that back then Jordan algebras were hoped to shed light on conceptual problems of genuine <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>, but that these hopes never materialized. However, more recent developments change this picture a bit, we come to that below. Much more recently in 2010, the <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> sheds a new light on the role of <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> structure. This theorem states mild conditions under which a <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> structure on <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> is equivalently encoded in the <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a> of the full <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>. These commutative subalgebras themselves are of course of old fame in quantum mechanics, they are the “<a class="existingWikiWord" href="/nlab/show/classical+contexts">classical contexts</a>” given by tuples of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> that all commute with each other and hence which can all be <a class="existingWikiWord" href="/nlab/show/measurement">measured</a> simultaneously without the <a class="existingWikiWord" href="/nlab/show/uncertainty+principle">uncertainty principle</a> interfering. These <a class="existingWikiWord" href="/nlab/show/classical+contexts">classical contexts</a> played a crucial role in the discussion of the foundation of quantum mechanics in the first half of the 20th century: back then people argued that for <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> which do not commute with each other it is unclear what it means physically to form their sum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A + B</annotation></semantics></math> or their product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">A B</annotation></semantics></math>, hence that a <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> should be demanded to be a linear (and positive normed) <a class="existingWikiWord" href="/nlab/show/linear+functional">functional</a> on all <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">commutative subalgebras</a>, but not necessarily on the whole non-commutative algebra. Such a notion of “quantum state on all <a class="existingWikiWord" href="/nlab/show/classical+contexts">classical contexts</a>” was called a <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a>. The issue of whether quasi-states are a more accurate description of quantum <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a> was settled in 1957 by <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> (<a href="#Gleason57">Gleason 57</a>), which says that given a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> greater than 2, then the <a class="existingWikiWord" href="/nlab/show/quasi-states">quasi-states</a> are automatically <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> also on the full non-commutative <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>. Typically this is viewed as making the notion of <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a> obsolete, but since that is a formally weaker notion the opposite attitude makes sense: what is traditionally taken as the definition of <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> is more accurately thought of as being a <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a>, hence something that is intrinsically related not to a non-commutative algebra of observables, but to the “<a class="existingWikiWord" href="/nlab/show/classical+contexts">classical contexts</a>” of its <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a>. Indeed, by combining the <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> with the <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a>, we have (under the pertinent mild assumptions) that two algebras of observables have the same <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> already when they have the same <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a>. Notice that where <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> only involves the commutative subalgebras themselves, this Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem crucially involves their <a class="existingWikiWord" href="/nlab/show/order">order</a> of inclusions, hence the actual <a class="existingWikiWord" href="/nlab/show/poset">poset</a> structure of their inclusions. Therefore there is an intrinsic <a class="existingWikiWord" href="/nlab/show/order+theory">order theoretic</a> aspect in the standard foundations of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>. But it is not just order theory, for it is not the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> structure of inclusions alone that matters in the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem, but that poset structure together with the actual <a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">commutative C*-algebra</a> structure of each <a class="existingWikiWord" href="/nlab/show/classical+context">classical context</a>. There is an elegant way to combine these two aspects: a system of <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative algebras</a> together with the <a class="existingWikiWord" href="/nlab/show/order">order</a> of their inclusions is equivalently a single <a class="existingWikiWord" href="/nlab/show/algebra+object">algebra object</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> over the <a class="existingWikiWord" href="/nlab/show/poset">poset</a>. With some basics of <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> in hand this is a trivial statement, but in view of the above it is worthwhile to make explicit: the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem (<a href="#HardingDoering10">Harding-Döring 10</a>, <a href="#Hamhalter11">Hamhalter 11</a>) says that the collection of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> is accurately formalized by a single <a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">commutative C*-algebra</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to a <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> over a <a class="existingWikiWord" href="/nlab/show/poset">poset</a>. It has been argued that this serves as a formalization of the views on <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> that in the middle of the 20th century <a class="existingWikiWord" href="/nlab/show/Niels+Bohr">Niels Bohr</a> expressed in extensive but informal writing (see <a href="#Scheibe73">Scheibe 73</a>): he said roughly that whatever <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> is, it must be expressible and must be expressed through classical contexts. In honor of this intuition, the above <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a> deserve to be called <a class="existingWikiWord" href="/nlab/show/Bohr+toposes">Bohr toposes</a>, following the term “Bohrification” in (<a href="#HeunenLandsmanSpitters09">Heunen-Landsman-Spitters 09</a>). In fact, a <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> is fairly trivial as far as <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a> go, since, by the above, it is just a reflection – precisely: the “<a class="existingWikiWord" href="/nlab/show/localic+reflection">localic reflection</a>” – of a purely <a class="existingWikiWord" href="/nlab/show/order+theory">order theoretic structure</a>. But the key is that passing to the topos over the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> provides a home for the context-wise <a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">commutative C*-algebra</a> structure which makes the <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> have the additional structure of a <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a>. This is the additional <a class="existingWikiWord" href="/nlab/show/algebra">algebraic</a> datum on top of the purely <a class="existingWikiWord" href="/nlab/show/order+theory">order theoretic</a> datum in the <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> structure of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a>. Ringed toposes have of course a long tradition in <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> (most famously in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>). By <a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a>‘s foundational work, laid out in the <a href="#Hakim72">thesis</a> of <a class="existingWikiWord" href="/nlab/show/Monique+Hakim">Monique Hakim</a>, ringed toposes form a general foundation for structured <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>. More recently this was further strengthened and refined by <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a> by the notion of <a class="existingWikiWord" href="/nlab/show/structured+%28infinity%2C1%29-topos">higher ringed toposes</a>, which we will see appear in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> below. In as far as the <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> are supposed to be the <a class="existingWikiWord" href="/nlab/show/Isbell+duality">dual</a> of the <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>, it is therefore natural to have this “quantum phase space” be realized as a <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a>. Notice that this is a different geometric interpretation of “quantum phase space” than the traditional idea that quantum phase space is an object in <a class="existingWikiWord" href="/nlab/show/non-commutative+geometry">non-commutative geometry</a>! Here by the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem we see that it is not actually accurate to say that quantum phase space is dually given by a non-commutative <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, as in fact it is given dually just by a <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a>. The <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed</a> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> provides a natural <a class="existingWikiWord" href="/nlab/show/geometry">geometric</a> interpretation of this, one might call it “Jordan geometry”. This geometric nature of the Bohr topos becomes more manifest as we consider its <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a>. By <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> this carries not an internal <a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">commutative C*-algebra</a> but its <a class="existingWikiWord" href="/nlab/show/Gelfand+spectrum">Gelfand spectrum</a>: an actual internal <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. This internal topological space has been called the <a class="existingWikiWord" href="/nlab/show/spectral+presheaf">spectral presheaf</a> (<a href="#ButterfieldHamiltonIsham98">Butterfield-Hamilton-Isham 98</a>). While here we find this <a class="existingWikiWord" href="/nlab/show/spectral+presheaf">spectral presheaf</a> as an accurate dual description of the space of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> based on the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem, interest in the spectral presheaf originally came from the observation that it provides a clear geometric formulation of yet another theorem about the foundations of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, namely of the <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> (<a href="#KochenSpecker67">Kochen-Specker 67</a>). This again amplifies the role of the “classical contexts” of commutative subalgebras: one may ask if there is a <a class="existingWikiWord" href="/nlab/show/hidden+variable">hidden variable</a> description of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> that allows to assign actual values to all <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> such that in all <a class="existingWikiWord" href="/nlab/show/classical+contexts">classical contexts</a> this assignment behaves as an actual <a class="existingWikiWord" href="/nlab/show/classical+observable">classical observable</a> in that it provides a (star-)<a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a> from the <a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">commutative C*-algebra</a> of the classical context to the real numbers. The <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> rules out such a <a class="existingWikiWord" href="/nlab/show/hidden+variable+theory">hidden variable theory</a> by stating that when the <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a> is that of <a class="existingWikiWord" href="/nlab/show/bounded+operators">bounded operators</a> on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> greater than 2, then such a “hidden variable” assignment cannot exist. In (<a href="#IshamButterfield98">Butterfield-Isham 98</a>) it was observed, that this statement has a natural geometric interpretation in the <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a>: it simply says that the <a class="existingWikiWord" href="/nlab/show/spectral+presheaf">spectral presheaf</a>, hence the “Jordan-algebraic geometry” incarnation of quantum phase space, has no <a class="existingWikiWord" href="/nlab/show/global+element">global element</a>. This statement in turn is a characterization of how the quantum phase space is “exotic” as far as <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> go. It behaves like a non-trivial space, and yet there is no way to map a point into it as a whole, maps of points into it exist only locally. In summary, the <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> incarnation of the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter characterization of quantum observables not only accurately captures the nature of quantum observables, but also makes other subtle nature of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> becomes more explicitly evident than in other formulations. To see how one can get more out of the <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> incarnation of the quantum phase space, it serves to pass from plain <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> to the more general context of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. Here the original <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> of <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> demand that to a region of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is to be assigned the <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> as 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>. But by the above discussion it is rather only the underlying <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> structure that matters, hence the <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a>. In light of this a <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> in <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> is naturally regarded as a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> of <a class="existingWikiWord" href="/nlab/show/ringed+toposes">ringed toposes</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, assigning the respective <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> of local observables to each local region of spacetime. Such “Bohrification of local nets of observables” were analyzed in (<a href="#Nuiten12">Nuiten 12</a>). There it was found that the natural structure of the transition functions of local nets of Bohr toposes of observables by <a class="existingWikiWord" href="/nlab/show/geometric+morphisms">geometric morphisms</a> automatically capture the <a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a> condition of <a class="existingWikiWord" href="/nlab/show/local+quantum+field+theory">local quantum field theory</a>. This is now called “Nuiten’s lemma” in (<a href="#WoltersHalvorson13">Wolters-Halvorson 13</a>). Moreover, (<a href="#Nuiten12">Nuiten 12</a>) shows that a natural <a class="existingWikiWord" href="/nlab/show/descent">descent</a> condition on spacetime nets of <a class="existingWikiWord" href="/nlab/show/Bohr+toposes">Bohr toposes</a> is equivalent to <a href="local+net#StrongLocality">strong locality</a> of the <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>, something slightly weaker than <a href="local+net#EinsteinLocality">Einstein causality</a>, which implies it. Since, by the above, the <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> is the geometric incarnation of the <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> structure on <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a>, one might see this as a reply to the alleged lack of implications (in <a href="#ASReview04">the AS review, 04</a>) of <a class="existingWikiWord" href="/nlab/show/Pascual+Jordan">Pascual Jordan's</a> ideas from the 1930s to quantum field theory. Be that as it may, notice that generally local systems of <a class="existingWikiWord" href="/nlab/show/ringed+toposes">ringed toposes</a> are to be expected to naturally encode <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> on general grounds: the modern <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>-style formulation of <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a>/<a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a> via <a class="existingWikiWord" href="/nlab/show/factorization+algebras">factorization algebras</a> or similar assigns to subsets of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> the <a class="existingWikiWord" href="/nlab/show/derived+critical+locus">derived critical locus</a> of local <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> extremizing the given <a class="existingWikiWord" href="/nlab/show/local+action+functional">local action functional</a>, hence the <a class="existingWikiWord" href="/nlab/show/derived+geometry">derived space</a> of solutions to the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> of motion: the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> (pre-quantum). In <a class="existingWikiWord" href="/nlab/show/physics">physics</a> this <a class="existingWikiWord" href="/nlab/show/derived+critical+locus">derived critical locus</a> is modeled explicitly as a <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-complex</a>, but when realized in the full technical beauty of <a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a> it is in fact a <a class="existingWikiWord" href="/nlab/show/structured+%28infinity%2C1%29-topos">higher ringed topos</a>, as explained by <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>. In view of this incarnation of <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a> in <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>-perspective as a net of <a class="existingWikiWord" href="/nlab/show/structured+%28infinity%2C1%29-topos">higher ringed topos</a>, it seems rather natural that under <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> it remains a net of ringed toposes, sending ringed toposes incarnating classical <a class="existingWikiWord" href="/nlab/show/covariant+phase+spaces">covariant phase spaces</a> to ringed toposes incarnating their quantized version as quantum phase spaces. <h2 id="RelationToTheNonCommutativePhaseSpace">Relation to the traditional non-commutative geometry</h2> In the above we highlighted that by <a href="#ListOfTheoremsInvoked">the fundamental theorems</a> on the foundations of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> – going back to insights of <a class="existingWikiWord" href="/nlab/show/Pascual+Jordan">Pascual Jordan</a> way back in 1930 and formally affirmed by more recent results by Alfsen-Shultz and Harding-Döring – it follows that, accurately speaking, quantum phase space is not really an object in <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a>, but rather in a kind of non-associative “Jordan geometry” which is naturally captured by the <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed</a> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> over the <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a>. This observation indeed puts doubt on the long and widely held believe that the quantum phase space is an object in <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a>, a belief that in fact motivated much of the development of <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a> in the first place. But that this is not really true was “well known” all along: it is pointed out for instance in the foundational text (<a href="#BatesWeinstein97">Bates-Weinstein 97</a>) on <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>. On page 80 there is highlighted how given a classical <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> represented by a <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \{-,-\})</annotation></semantics></math>, hence of a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> that carries <ol> <li> a commutative algebra of classical observables </li> <li> equipped with a compatible non-commutative <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie 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></mrow><annotation encoding="application/x-tex">\{-,-\}</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>) </li> </ol> the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of this data is to be thought of as applying to these two items separately: <ol> <li> a non-associative but commutative <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> structure of quantum observables is the <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> of the commutative algebra structure of classical observables; </li> <li> a non-commutative <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> structure is the deformation quantization of the Poisson bracket </li> </ol> and that if the quantization of both items is given by a single <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a> structure, then the non-associative commutative Jordan algebra structure is the one induced by the anticommutator <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∘</mo><mi>B</mi><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo>+</mo><mi>B</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A \circ B = \tfrac{1}{2}(A B + B A) </annotation></semantics></math></div> while the non-commutative algebra structure is that given by the commutator <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo>−</mo><mi>B</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [A,B] = \tfrac{1}{2}(A B - B A) </annotation></semantics></math></div> See also at <a href="Jordan+algebra#OriginInQuantumPhysics">Jordan algebra – Origin in quantum physics</a>. This splitting of the notion of quantization into a Lie-algebraic and a Jordan algebraic aspect is formalized in the notion of <a class="existingWikiWord" href="/nlab/show/Jordan-Lie-Banach+algebra">Jordan-Lie-Banach algebra</a>. To understand that this makes good sense notice that this decomposition is that into <a class="existingWikiWord" href="/nlab/show/kinematics">kinematics</a> and <a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a> of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>: <ol> <li> <a class="existingWikiWord" href="/nlab/show/kinematics">kinematics</a>— the construction of the <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> and of the <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> alone indeed does not involve the associative product of the <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>; </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a>— the commutator <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> structure is used to impose quantum <a class="existingWikiWord" href="/nlab/show/Hamiltonian+flows">Hamiltonian flows</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp([A,-])</annotation></semantics></math>, hence (time) propagation along the <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a> generated by the observables <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>. </li> </ol> But then observe in addition that when we pass from <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> to <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> axiomatized as <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>, then in fact time propagation is no longer implemented by <a class="existingWikiWord" href="/nlab/show/inner+automorphisms">inner automorphisms</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp([A,-])</annotation></semantics></math>. Indeed it is impossible for any realistic <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> with infinitely many degrees of freedom (such as a <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a>) to have time evolution given by an <a class="existingWikiWord" href="/nlab/show/inner+automorphism">inner automorphism</a>. That this does work for <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> is really an artifact of the degenerate case of finitely many degrees of freedom considered there. Instead, in <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> the <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>-evolution of the quantum fields is all encoded in the transition functions of the <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a>. By the <a href="#QuantumPhaseSpaceAsJordanGeometry">above</a> discussion, this is however, accurately speaking, not actually to be thought of as a net of <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a>, but rather as a net of <a class="existingWikiWord" href="/nlab/show/Jordan+algebras">Jordan algebras</a>, hence as a net of <a class="existingWikiWord" href="/nlab/show/Bohr+toposes">Bohr toposes</a>. This way the need for the non-commutative algebraic structure disappears and only the non-associative commutative <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> structure remains. In this context it may be noteworthy to recall what is a well-kept secret: despite much work on <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> with the traditional <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> that demand to assign <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> to regions of spacetime, there is to date not a single non-<a class="existingWikiWord" href="/nlab/show/free+field+theory">free field</a> example in <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> greater than 3 of these axioms. (There are plenty of interesting example in dimension 2, though, see at <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> and pointers given there.) On the other hand, more recently the variant of the <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> axioms known as <a class="existingWikiWord" href="/nlab/show/factorization+algebras">factorization algebras</a> has been shown to admit plenty of interesting examples of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. Comparison of the axioms is not straightforward and should be taken with a grain of salt, but it is maybe noteworthy that a <a class="existingWikiWord" href="/nlab/show/factorization+algebra">factorization algebra</a> is indeed a net that assigns to a region of spacetime not an algebra structure. The algebra structure there is instead all encoded into the way that spacetime regions are included into each other. <h2 id="ListOfTheoremsInvoked">List of theorems invoked</h2> For reference, the following lists those theorems about the standard foundations of quantum mechanics that are being referred to <a href="#QuantumPhaseSpaceAsJordanGeometry">above</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <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> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a> </li> </ul> <h2 id="related_entries">Related entries</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/interpretation+of+quantum+mechanics">interpretation of quantum mechanics</a></li> </ul> <div> <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> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a> </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/quasi-state">quasi-state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> <a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a> <a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a> <a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a> <a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a> <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</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> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> <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> <a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a> </li> <li> <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> </li> <li> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> </li> <li> <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> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/theorems">theorems</a> <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/Gleason%27s+theorem">Gleason's theorem</a> </li> <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> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> </ul> </li> </ul> </div> <h2 id="references">References</h2> <ul id="Jordan32"> <li><a class="existingWikiWord" href="/nlab/show/Pascual+Jordan">Pascual Jordan</a>, Über eine Klasse nichtassociativer hyperkomplexer Algebren, Nachr. Ges. Wiss. Göttingen (1932), 569–575. </li> </ul> <ul id="JordanvNeumannWigner34"> <li><a class="existingWikiWord" href="/nlab/show/Pascual+Jordan">Pascual Jordan</a>, <a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a> and <a class="existingWikiWord" href="/nlab/show/Eugene+Wigner">Eugene Wigner</a>, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29–64.</li> </ul> <ul id="Scheibe73"> <li> A.M. Gleason, Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Mechanics, Indiana Univ. Math. J. 6 No. 4 (1957), 885–893 (<a href="http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050">web</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Simon+Kochen">Simon Kochen</a>, <a class="existingWikiWord" href="/nlab/show/Ernst+Specker">Ernst Specker</a>, The problem of hidden variables in quantum mechanics , Journal of Mathematics and Mechanics 17, 59–87 (1967), (<a href="http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1968/17/17004/pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Monique+Hakim">Monique Hakim</a>, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972). </li> <li> Erhard Scheibe, The logical analysis of quantum mechanics . Oxford: Pergamon Press, 1973. </li> </ul> <ul id="ASReview04"> <li> <a class="existingWikiWord" href="/nlab/show/Erik+Alfsen">Erik Alfsen</a>, <a class="existingWikiWord" href="/nlab/show/Frederic+Shultz">Frederic Shultz</a>, A Gelfand Neumark theorem for Jordan algebras, Advances in Math., 28 (1978), 11-56. </li> <li> <a class="existingWikiWord" href="/nlab/show/Erik+Alfsen">Erik Alfsen</a>, H. 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