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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topos_theory">Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="operator_algebra">Operator algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a></p> <h2 id="concepts">Concepts</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a></strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+history">field history</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/observables">observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+net">field net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> <p><a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a></p> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a></p> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a></p> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>)</p> </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a></p> </li> </ul> </div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><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></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><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></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <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> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <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> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <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> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><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></p> </li> <li> <p>examples</p> <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> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <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> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <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> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><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></p> </li> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> <h4 id="noncommutative_geometry">Noncommutative geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></strong></p> <p>(<a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>)</p> <h2 id="topology">Topology</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a></li> <li><a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></li> <li><a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> </ul> <h2 id="smooth_and_riemannian_geometry">Smooth and Riemannian geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/dense+subalgebra">dense subalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+triple">spectral triple</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-spectral+triple">2-spectral triple</a></p> </li> </ul> <h2 id="algebraic_geometry">Algebraic geometry</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a></li> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+scheme">noncommutative scheme</a></li> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+motive">noncommutative motive</a></li> <li><a class="existingWikiWord" href="/nlab/show/quantum+flag+variety">quantum flag variety</a></li> <li><a class="existingWikiWord" href="/nlab/show/quantum+Schubert+cell">quantum Schubert cell</a></li> </ul> <h2 id="homotopy_theory">Homotopy theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+stable+homotopy+theory">noncommutative stable homotopy theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operator+algebras">model structure on operator algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+structure+on+C%2A-algebras">homotopical structure on C*-algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a>, <a class="existingWikiWord" href="/nlab/show/E-theory">E-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <h2 id="relation_to_physics">Relation to physics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/noncommutative+geometry+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <ul> <li><a href='#IdeaInBrief'>Brief</a></li> <li><a href='#more_detailed'>More detailed</a></li> </ul> <li><a href='#outline'>Outline</a></li> <li><a href='#BohrToposOfQMSystem'>Bohr topos of a quantum mechanical system</a></li> <ul> <li><a href='#algebras'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras</a></li> <li><a href='#BohrSite'>The Bohr site</a></li> <li><a href='#BohrTopos'>The Bohr topos</a></li> </ul> <li><a href='#KinematicsOnBohrTopos'>Kinematics in a Bohr topos</a></li> <ul> <li><a href='#ThePhaseSpace'>The phase space</a></li> <li><a href='#TheObservables'>The observables</a></li> <li><a href='#the_states'>The states</a></li> <ul> <li><a href='#observation'>Observation</a></li> <li><a href='#observation_2'>Observation</a></li> </ul> </ul> <li><a href='#SheafOfBohrToposesOfQFT'>(Pre-)Sheaf of Bohr toposes of a quantum field theory</a></li> <li><a href='#contravariant_functors_on_open_subsets'>Contravariant functors on open subsets</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p>A <em>Bohr topos</em> is a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> associated with any <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>, which is such that</p> <ul> <li>the <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> and <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> of the <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>, hence its <a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a>,</li> </ul> <p>are equivalently</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/classical+physics">classical</a> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and classical <a class="existingWikiWord" href="/nlab/show/states">states</a>, hence its classical <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a>,</p> <p><em><a class="existingWikiWord" href="/nlab/show/internalization">internal</a></em> to the Bohr topos, hence in the <em><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></em> of the Bohr topos.</p> </li> </ul> <p>A detailed motivation/derivation of this from classical theorems on the foundations of standard <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> is at <em><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></em>.</p> <p>One might think of a Bohr topos as (part of) a formalization of the “<a class="existingWikiWord" href="/nlab/show/coordination">coordination</a>” of the <a class="existingWikiWord" href="/nlab/show/physical+theory">physical theory</a> of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, providing a formalized prescription of how to map the theory to <a class="existingWikiWord" href="/nlab/show/propositions">propositions</a> about (<a class="existingWikiWord" href="/nlab/show/experiment">experimental</a>) <a class="existingWikiWord" href="/nlab/show/observables">observables</a> of the system. The <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> of Bohr toposes has been argued (e.g. <a href="#HeunenLandsmanSpitters09">Heunen-Landsman-Spitters 09</a>) to be a better formal context for such considerations than the old <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a> going back to <a class="existingWikiWord" href="/nlab/show/von+Neumann">von Neumann</a>.</p> <p>The idea of Bohr toposes goes back to <a href="#ButterfieldHamiltonIsham">Butterfield-Hamilton-Isham</a>, <a href="#IshamDoering07">Isham-Döring 07</a>) and <a href="#HeunenLandsmanSpitters09">Heunen-Landsman-Spitters 09</a>. The concept is named after <a class="existingWikiWord" href="/nlab/show/Niels+Bohr">Niels Bohr</a>, whose informal ideas about the nature of quantum mechanics (<a href="#interpretation+of+quantum+mechanics#Bohr49">Bohr 1949</a>, cf. <a href="#Scheibe73">Scheibe 1973, p. 24</a>) it is supposed to formalize, see at <em><a href="interpretation+of+quantum+mechanics#BohrStandpoint">interpretation of quantum mechanics – Bohr’s standpoint</a></em>.</p> <p>Sometimes in the literature the discussion of Bohr toposes is referred to as “the topos-theoretic formulation of physics”. But actually Bohr toposes currently formalize but one aspect of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, namely “the quantum mechanical <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>” in the form of the <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> and the <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a>. The plain Bohr topos does not even encode any <a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a>, though in the spirit of <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> a certain presheaf of Bohr toposes on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> does encode dynamics (<a href="#Nuiten11">Nuiten 11</a>). For other and more comprehensive usages of <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> in the formalization of <a class="existingWikiWord" href="/nlab/show/physics">physics</a> see at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em> and at <em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></em>.</p> <h3 id="IdeaInBrief">Brief</h3> <p>To every <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a> is associated its <em>Bohr topos</em>: a <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a> which plays the role of the quantum <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>. The idea of this construction – <em>Bohrification</em> – is that it naturally captures the <a class="existingWikiWord" href="/nlab/show/geometry">geometric</a> and <a class="existingWikiWord" href="/nlab/show/logic">logical</a> aspects of <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> in terms of <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>/<a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a>.</p> <p>One way to understand Bohrification is as a generalization of the construction of the <a class="existingWikiWord" href="/nlab/show/Gelfand+spectrum">Gelfand spectrum</a> of commutative <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>s to a context of <em>noncommutative</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras. It assigns to a noncommutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebra <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> essentially a system of Gelfand dual spaces to each of its <a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">commutative</a> subalgebras. Together, this system yields a generalized <a class="existingWikiWord" href="/nlab/show/Gelfand+spectrum">Gelfand spectrum</a> in the form of a <a class="existingWikiWord" href="/nlab/show/locale">locale</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><mi>Σ</mi><mo>̲</mo></munder> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\underline{\Sigma}_A</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒯</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{T}_A</annotation></semantics></math> over the <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a> of <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>, or equivalently its externalization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>A</mi></msub><mo>→</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma_A \to \mathcal{C}(A)</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/locale">locale</a> <a class="existingWikiWord" href="/nlab/show/over+category">over</a> the locale of open subalgebras.</p> <p>As a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, the Bohr topos is just a <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a>, the topos of presheaves on this <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a> of <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>, but the point is that it is naturally a <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a> with the original non-commutative algebra <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> appearing as a commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to the Bohr topos. This allows to talk about <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> on <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> much like classical states, but internal to the Bohr topos.</p> <p>In fact, under mild assumptions on <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>, its <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a>, and hence the Bohr topos over it, encodes precisely the <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a> underlying <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>. As discussed at <em><a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></em>, this is precisely that part of <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> which knows about the <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> themselves. In order to have the Bohr topos remember the full non-commutative algebra structure of <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>, it needs to be equipped with the information about <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian</a> <a class="existingWikiWord" href="/nlab/show/flows">flows</a> induced on <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> by automorphisms 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><mi>i</mi><mo stretchy="false">[</mo><mi>H</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(i [H,-])</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/commutator">commutator</a> (that gets discarded as one passes to the <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a>). According to <a class="existingWikiWord" href="/nlab/show/Andreas+D%C3%B6ring">Andreas Döring</a> (private communication at MPI Bonn, April 2013), this can be formulated nicely in topos theory.</p> <h3 id="more_detailed">More detailed</h3> <p>The formalization of the notion <em><a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a></em> (see there for details) with its <em><a class="existingWikiWord" href="/nlab/show/states">states</a></em> and <em><a class="existingWikiWord" href="/nlab/show/observables">observables</a></em> is the following.</p> <ul> <li> <p>The system as such is encoded by a <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a> <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>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math> is thought of as being an <a class="existingWikiWord" href="/nlab/show/observable">observable</a> of the system: a kind of observation that one can make about it;</p> </li> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\rho : A \to \mathbb{C}</annotation></semantics></math> (which is “positive” and “normalized”) is thought of as being a <em><a class="existingWikiWord" href="/nlab/show/state">state</a></em> that the system can be in – a physical configuration (or rather: a <a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a> of such);</p> </li> <li> <p>the pairing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">(a, \rho) \mapsto \rho(a) \in \mathbb{C}</annotation></semantics></math> is thought of as being the value of the observable <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> made on the system in state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>: for instance the total <em><a class="existingWikiWord" href="/nlab/show/energy">energy</a></em> of the system as measured in some chosen <a class="existingWikiWord" href="/nlab/show/unit">unit</a>;</p> </li> <li> <p>a one-parameter <a class="existingWikiWord" href="/nlab/show/group">group</a> of <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R} \to Aut(A)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/inner+automorphisms">inner automorphisms</a> for “localized” systems, see below) is thought of as being an <em>evolution</em> of the system, for instance in <a class="existingWikiWord" href="/nlab/show/time">time</a> or more general in <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>.</p> </li> </ul> <p>(Often in the literature quantum mechanical systems are instead dually conceived of in terms of <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert</a> <a class="existingWikiWord" href="/nlab/show/spaces+of+states">spaces of</a> <a class="existingWikiWord" href="/nlab/show/pure+states">pure states</a>. The relation between these two descriptions is established by the <a class="existingWikiWord" href="/nlab/show/GNS-construction">GNS-construction</a> which allows to pass from one to the other.)</p> <p>Notice that these axioms are naturally thought of as exhibiting the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra <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> as a <a class="existingWikiWord" href="/nlab/show/Isbell+duality">formal dual</a> of the would-be <em>quantum <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></em> of the physical system – not quite an ordinary <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> (which by <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> it would be if <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> were commutative) but still a kind of generalized space. Traditionally it is common to regard this as a space in <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a>. Notice that if we do so – hence if we regard the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>A</mi><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Spec A \in C^\ast Alg^{op}</annotation></semantics></math> that is the incarnation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A \in C^\ast Alg</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> – then the definition of <a class="existingWikiWord" href="/nlab/show/observable">observable</a> above reduces to the evident notion of real-valued functions on quantum phase space: such a function is a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>A</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">Spec A \to \mathbb{R}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^\ast Alg^{op}</annotation></semantics></math>, which dually is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>ℝ</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C(\mathbb{R})_0 \to A</annotation></semantics></math> (where on the left we have the <a class="existingWikiWord" href="/nlab/show/unitization">unitization</a> of functions with <a class="existingWikiWord" href="/nlab/show/compact+support">compact support</a>). That such correspond to <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint operator</a>s of <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> is the statement of <a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebras.</p> <p>More subtle is the interpretation of the axiom for <a class="existingWikiWord" href="/nlab/show/state">state</a>s. Historically this had been been subject to some discussion: by the <a class="existingWikiWord" href="/nlab/show/spectral+theorem">spectral theorem</a> two different <a class="existingWikiWord" href="/nlab/show/observable">observable</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a_1, a_2 \in A</annotation></semantics></math> have a compatible set of observable values if and only if these elements commute with each other in <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>. Generally, a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>a</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\{a_i\}_i</annotation></semantics></math> of observables has a jointly consistent set of observable values if and only if the sub-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><msub><mi>a</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">⟩</mo><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\langle \{a_i\}_i\rangle \subset A</annotation></semantics></math> generated by them is commutative. Therefore for the phenomenological interpretation of the axioms it seems to make no sense to demand that a <a class="existingWikiWord" href="/nlab/show/state">state</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\rho : A \to \mathbb{C}</annotation></semantics></math> be linear on non-commuting observables: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">a_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">a_2</annotation></semantics></math> do not commute, it is not a-priori clear that it makes sense to require that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(a_1 + a_2) = \rho(a_1) + \rho(a_2)</annotation></semantics></math>. This might experimentally fail, and hold only for commuting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">a_1, a_2</annotation></semantics></math>.</p> <p>Therefore the notion of <em><a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a></em> was introduced: a quasi-state on <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> is defined to be a (positive and normalized) <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\rho : A \to \mathbb{C}</annotation></semantics></math> which is required to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+function">linear</a> only on all commutative subalgebras of <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>. Operationally, quasi-states should be the genuine states!</p> <p>One would therefore tend to think that the terminology has been chosen in an unfortunate way. While maybe true, it turns out – non-trivially – that in a major class of cases of interest the distinction does not matter: namely <em><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></em> states that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> a separable complex <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>H</mi><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim H \gt 2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = B(H)</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra of <a class="existingWikiWord" href="/nlab/show/bounded+operators">bounded operators</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">H</mo></mrow><annotation encoding="application/x-tex">\H</annotation></semantics></math>, all quasi-states on <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> are automatically states: a function that is linear on all commutative subalgebras is automatically also linear on all of <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>.</p> <p>While this means that the distinction between states and quasi-states disappears in a major case of interest, it does not disappear in all cases of interest. In particular, other foundational theorems about quantum mechanics concern the collection of commutative subalgebras, too.</p> <p>Notably, one may wonder about the evident strengthening of the notion of quasi-states to that of a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\rho : A \to \mathbb{C}</annotation></semantics></math> which is not just linear but also an <a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> on each commuting subalgebra. Notice that by <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> every commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the algebra of continuous functions on some <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sp</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sp(C)</annotation></semantics></math>. Under this duality a <a class="existingWikiWord" href="/nlab/show/state">state</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sp</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sp(C)</annotation></semantics></math>, while an algebra homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">C \to \mathbb{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/point">point</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">X_C</annotation></semantics></math>. Therefore a quasi-state which is commutative-subalgebra-wise an algebra <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> may be thought of as encoding a collection of precise numerical values (as opposed to just expectation values) of <em>all</em> possible observables. Such a hypothetical quasi-state is sometimes called a collection of <em><a class="existingWikiWord" href="/nlab/show/hidden+variable">hidden variable</a>s</em> of the <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>: its existence would mean that despite the apparent probabilistic nature of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, there are “hidden” non-probabilistic states. But there are not. This is the statement of the <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a>: under precisely the assumptions that make <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> work, there is <em>no</em> quasi-state which is commutative-subalgebra-wise an algebra homomorphism.</p> <p>In summary, this means:</p> <div class="standout"> <p>The foundational characteristics of <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> are encoded in notions of <a class="existingWikiWord" href="/nlab/show/functions">functions</a> on the <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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> which are linear and positive only <em>commutative-subalgebra-wise</em> .</p> </div> <p>Since the notion of <strong>commutative-subalgebra-wise homomorphism</strong> is at the heart of quantum physics, it seems worthwhile to consider natural formalizations of this notion. There is indeed a very natural and <a class="existingWikiWord" href="/nlab/show/category+theory">general abstract</a> one: whenever any notion of <a class="existingWikiWord" href="/nlab/show/function">function</a> is defined only <em>locally</em> it is natural to consider the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of such functions over all possible local patches.</p> <p>The historically motivating example, and possibly still the most widely familiar one, is that of <a class="existingWikiWord" href="/nlab/show/holomorphic+function">holomorphic function</a>s on a <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>: there are in general very few holomorphic functions defined over all of a complex manifold, but plenty of them defined over any small enough subset. And it is of fundamental interest to consider the collections of holomorphic functions over each such subset, and how these restrict to each other under restriction of subsets. This collection of local data is a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of functions on the complex manifold.</p> <p>There is an evident analog setup of this situation that applies in the present case of interest, that of functions defined on commutative subalgebras:</p> <p>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> any <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(A)</annotation></semantics></math> for the set of all its commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-subalgebras. This is naturally a <a class="existingWikiWord" href="/nlab/show/poset">poset</a> under inclusion of subalgebras. A (co)<a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> of this set is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}(A) \to Set</annotation></semantics></math>. Any such functor we may think of as a collection of commutative-subalgebra-wise data on <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>, consistent with restriction of subalgebras. The collection of all such functors – which we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}(A), Set]</annotation></semantics></math> – is a <a class="existingWikiWord" href="/nlab/show/category">category</a> called a <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a>.</p> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">Inside</a> this <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, all the above discussion of foundations of quantum mechanics finds a natural simple equivalent reformulation:</p> <p>First of all, the non-commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra <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> naturally induces a <em>commutative</em> <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a> object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{A}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}(A), Set]</annotation></semantics></math>: namely the <a class="existingWikiWord" href="/nlab/show/copresheaf">copresheaf</a> defined by the tautological assignment</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder><mo>:</mo><mo stretchy="false">(</mo><mi>C</mi><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↦</mo><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underline{A} : (C \in \mathcal{C}(A)) \mapsto C \,. </annotation></semantics></math></div> <p>In words this means nothing but that the collection of all commutative subalgebras of <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> may naturally be regarded as a <em>single</em> commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}(A), Set]</annotation></semantics></math>.</p> <p>Below we shall discuss (<a href="...">here</a>) that in a precise sense this commutative internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{A}</annotation></semantics></math> captures precisely all the <a class="existingWikiWord" href="/nlab/show/kinematics">kinematical</a> information encoded in the <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a> of <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> – everything related to <a class="existingWikiWord" href="/nlab/show/states">states</a> and <a class="existingWikiWord" href="/nlab/show/observables">observables</a> but not information about (time) evolution. So everything we have discussed so far.</p> <p>The pair of these two ingredients</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>,</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Bohr(A) := ([\mathcal{C}(A), Set], \underline{A}) </annotation></semantics></math></div> <p>constitutes what is called a <em><a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a></em> – a special case of the notion of a <a class="existingWikiWord" href="/nlab/show/locally+ringed+topos">locally ringed topos</a>. This notion is a fundamental notion for generalized <a class="existingWikiWord" href="/nlab/show/space">space</a>s in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>. The most advanced general theory of <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> (<a class="existingWikiWord" href="/nlab/show/Structured+Spaces">Lurie09</a>) is based on modelling spaces as ringed toposes.</p> <p>We shall call this ringed topos the <strong>Bohr topos</strong> of <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>.</p> <p>This terminology is meant to indicate that one may think of this construction as formalizing faithfully and usefully a heuristic that has been emphasized by <a class="existingWikiWord" href="/nlab/show/Niels+Bohr">Niels Bohr</a> – one of the founding fathers of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> – and is known as the <em>doctrine of classical concepts</em> (<a href="#Scheibe73">Scheibe 1973</a>) in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>. This states that nonclassical/noncommutative as the <a class="existingWikiWord" href="/nlab/show/logic">logic</a>/<a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> of quantum mechanics may be, it is to be probed and detected by classical/commutative logic/geometry.</p> <p>Namely in terms of the Bohr topos we have the following equivalent reformulations of the foundational facts about quantum physics discussed above, now internally in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(A)</annotation></semantics></math>.</p> <div class="standout"> <p><strong>Consistent quantum mechanical states.</strong> A <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a> on <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> is precisely an ordinary <a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{A}</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(A)</annotation></semantics></math>.</p> <p>In particular (<a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a>): if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = B(H)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>H</mi><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim H \gt 2</annotation></semantics></math> then a <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> on the external <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> is precisely a <a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a> on the internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{A}</annotation></semantics></math>.</p> </div> <p>and</p> <div class="standout"> <p><strong>Non-existence of hidden quantum variables.</strong> The <a class="existingWikiWord" href="/nlab/show/Gelfand+spectrum">Gelfand spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sp</mi><mo stretchy="false">(</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sp(\underline{A})</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{A}</annotation></semantics></math> internal to the Bohr topos has no <a class="existingWikiWord" href="/nlab/show/global+element">global point</a>. (<a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker</a>).</p> </div> <p>These two statements might be taken as suggesting that a quantum mechanical system <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> is naturally regarded in terms of its Bohr topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(A)</annotation></semantics></math> – somewhat more naturally than as a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra <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>. (The second, in a slightly different setup, was emphasized in <a href="#ButterfieldHamiltonIsham">IshamHamiltonButterfield</a>, which inspired all of the following discussion, the first in <a href="">Spitters</a>). In fact, thinking of <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a>es as generalized <a class="existingWikiWord" href="/nlab/show/space">space</a>s in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>, it suggests that the Bohr topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(A)</annotation></semantics></math> itself <em>is</em> the <em>quantum <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></em> of the quantum mechanical system in question.</p> <p>To which extent this perspective is genuinely useful is maybe still to be established. For pointers to the literature see the <a href="#References">references</a> below. Discussion along the above lines may suggest that this perspective is indeed useful, but what is probably still missing is a statement about <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> that can be formulated and proven in terms of Bohr toposes, while being hardly conceivable or at least more unnatural without. It is probably currently not clear if such statements have been found.</p> <p>One potential such statement has been suggested in (<a href="#Nuiten11">Nuiten 11</a>) after discussion with <a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Spitters</a>:</p> <p>In the formalization of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> by the <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> – called <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> – every quantum field theory is entirely encoded in terms of its <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> over <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. This is a <a class="existingWikiWord" href="/nlab/show/copresheaf">copresheaf</a> of <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>s</p> <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>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex"> A : Op(X) \to C^\ast Alg </annotation></semantics></math></div> <p>which assigns to every <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>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> the quantum <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex">A_U</annotation></semantics></math> of quantum fields supported in that region. By the above, we may consider for each of these quantum systems their associated quantum <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>s given by the correspondong Bohr toposes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>U</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(A_U)</annotation></semantics></math>. This yields a presheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><msubsup><mi>C</mi> <mi>com</mi> <mo>*</mo></msubsup><mi>TopSpace</mi><mo>↪</mo><msubsup><mi>C</mi> <mi>com</mi> <mo>*</mo></msubsup><mi>Topos</mi></mrow><annotation encoding="application/x-tex"> Bohr(A) : Op(X)^{op} \to C^\ast_{com} TopSpace \hookrightarrow C^\ast_{com}Topos </annotation></semantics></math></div> <p>of ringed toposes whose internal ring object has the structure of a commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra. With the <a class="existingWikiWord" href="/nlab/show/copresheaf">copresheaf</a> thus turned into a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> it is natural to ask under which conditions this is a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>: under which conditions this presheaf satisfies <a class="existingWikiWord" href="/nlab/show/descent">descent</a>.</p> <p>In (<a href="#Nuiten11">Nuiten 11</a>) the following is observed: if <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> satisfies what is called the <em>split property</em> (a strong form of the <a class="existingWikiWord" href="/nlab/show/time+slice+axiom">time slice axiom</a>) then the Bohr-presheaf of quantum phase spaces satisfies spatial descent by <a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a>s precisely if the original copresheaf of observables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A : Op(X) \to C^\ast Alg</annotation></semantics></math> is indeed <em><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local</a></em> – spatially and causally. So this means that a natural property of quantum physics – spatial and <a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a> – corresponds from the perspective of Bohr toposes to a natural property of presheaves of quantum phase spaces: <a class="existingWikiWord" href="/nlab/show/descent">descent</a>.</p> <p>One can probably view this as further suggestive evidence that indeed quantum physics is naturally regarded from the point of view of the Bohr topos. But for seeing where this perspective is headed, it seems that more insights along these lines would be useful.</p> <h2 id="outline">Outline</h2> <p>The discussion below proceeds in the following steps (following (<a href="#Nuiten11">Nuiten 11</a>))</p> <ol> <li> <p><a href="#BohrToposOfQMSystem">Bohr topos of a quantum mechanics system</a></p> <p>This discusses the Bohr topos incarnation of a <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a> – the topos-theoretic quantum <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> – and its <a class="existingWikiWord" href="/nlab/show/functor">functoriality</a>.</p> </li> <li> <p><a href="#KinematicsOnBohrTopos">Kinematics in a Bohr topos</a></p> <p>This discusses how the classical <a class="existingWikiWord" href="/nlab/show/kinematics">kinematics</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> a Bohr topos is the external quantum kinematics of the underlying quantum mechanical system.</p> </li> <li> <p><a href="#SheafOfBohrToposesOfQFT">(Pre-)Sheaf of Bohr toposes of a quantum field theory</a></p> <p>This discusses how the <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> of Bohr toposes obtained by applying Bohrification to a <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> of a <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>.</p> </li> </ol> <h2 id="BohrToposOfQMSystem">Bohr topos of a quantum mechanical system</h2> <p>We discuss the definitions and some basic properties of Bohr toposes: certain <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a>es – in fact <a class="existingWikiWord" href="/nlab/show/ringed+spaces">ringed spaces</a> – associated with any (possibly non-commutative) algebra. We formulate the construction for <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>s, since this is the standard model for <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>s, but actually much of it does not depend on either the <a class="existingWikiWord" href="/nlab/show/topology">topology</a> or the <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> structure until we come to the discussion of the <a href="KinematicsOnBohrTopos">kinematics in a Bohr topos</a> below.</p> <h3 id="algebras"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras</h3> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a> is (…)</p> </div> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>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> a <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a>, an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/normal+operator">normal element</a></strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi><mo>=</mo><mi>a</mi><msup><mi>a</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">a^* a = a a^*</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Every element of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebra is the sum of two normal elements, because</p> <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><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><msup><mi>a</mi> <mo>*</mo></msup><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mi>a</mi><mo>−</mo><msup><mi>a</mi> <mo>*</mo></msup><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a = \frac{1}{2} \left( \left( a + a^* \right) + \left( a - a^* \right) \right) \,. </annotation></semantics></math></div> <p>This means whenever a linear morphism between the vector spaces underlying two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras is defined on normal elements, it is already defined on all elements. This will be used in several of the arguments below.</p> </div> <div class="num_defn" id="PartialCStar"> <h6 id="definition_3">Definition</h6> <p>A <strong>partial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra is a <a class="existingWikiWord" href="/nlab/show/set">set</a> <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> equipped with</strong></p> <ul> <li> <p>a symmetric and reflexive binary <a class="existingWikiWord" href="/nlab/show/relation">relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>A</mi><mo>×</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C \subset A \times A</annotation></semantics></math>;</p> </li> <li> <p>elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">0,1 \in A</annotation></semantics></math>;</p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/involution">involution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>:</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\ast : A \to A</annotation></semantics></math>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</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 stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>ℂ</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(-)\cdot (-) : \mathbb{C} \times A \to A</annotation></semantics></math>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mo>−</mo><mo stretchy="false">‖</mo><mo>:</mo><mi>A</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\Vert-\Vert : A \to \mathbb{R}</annotation></semantics></math>;</p> </li> <li> <p>(partial) binary operations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>,</mo><mo>×</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">+, \times : C \to A</annotation></semantics></math></p> </li> </ul> <p>such that every set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">S \subset A</annotation></semantics></math> of elements that are pairwise in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is contained in a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">T \subset A</annotation></semantics></math> whose elements are also pairwise in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and on which the above operations yield the structure of a commutative <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of partial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra is a function preserving this structure. This defines a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PCstar</mi></mrow><annotation encoding="application/x-tex">PCstar</annotation></semantics></math> of partial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math> algebras.</p> </div> <p>This appears as (<a href="#vdBergHeunen">vdBergHeunen, def. 11,12</a>).</p> <div class="num_defn" id="PartialAlgebraOfNormalOperators"> <h6 id="definition_4">Definition</h6> <p>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> a <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">|</mo><mi>a</mi><msup><mi>a</mi> <mo>*</mo></msup><mo>=</mo><msup><mi>a</mi> <mo>*</mo></msup><mi>a</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> N(A) := \{a \in A | a a^* = a^* a\} </annotation></semantics></math></div> <p>for its set of <a class="existingWikiWord" href="/nlab/show/normal+operator">normal operator</a>s. This is naturally a <a href="#PartialCStar">partial C-star algebra</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>N</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>×</mo><mi>N</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \subset N(A) \times N(A)</annotation></semantics></math> the set of pairs of elements that commute in <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>.</p> </div> <h3 id="BohrSite">The Bohr site</h3> <div class="num_defn" id="PosetOfCommutativeSubalgebras"> <h6 id="definition_5">Definition</h6> <p>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> a <a href="#PartialCStar">partial C-star algebra</a> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(A)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> of total (not partial) commutative sub <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>s. We call this the <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a>.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>This construction extends to a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>:</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi><mo>→</mo><mi>Poset</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C} : C^* Alg \to Poset \,. </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Poset">Poset</a> of <a class="existingWikiWord" href="/nlab/show/posets">posets</a>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> a homomorphism we let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(f)</annotation></semantics></math> be over any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \in \mathcal{C}(A)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/image">image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><msub><mo stretchy="false">|</mo> <mi>C</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">im(f|_C) \in C^\ast Alg</annotation></semantics></math> of the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>By standard properties of <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>s (see <a href="http://ncatlab.org/nlab/show/C-star-algebra#General">here</a>), this image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><msub><mo stretchy="false">|</mo> <mi>C</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(f|_C)</annotation></semantics></math> is simply the set-theoretic image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(C)</annotation></semantics></math>.</p> </div> <p>Notice the following fact about <a class="existingWikiWord" href="/nlab/show/Alexandroff+space">Alexandroff space</a>s:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Alex</mi><mo>:</mo><mi>Poset</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> Alex : Poset \to Top </annotation></semantics></math></div> <p>from <a class="existingWikiWord" href="/nlab/show/posets">posets</a> to <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> that sends a poset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> to the topological space whose underlying set is the underlying set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and whose <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> are the upward closed subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Up</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Up(P)</annotation></semantics></math> exhibits an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Poset</mi><munderover><mo>→</mo><mo>≃</mo><mi>Alex</mi></munderover><mi>AlexTop</mi><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> Poset \underoverset{\simeq}{Alex}{\to} AlexTop \hookrightarrow Top </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/Poset">Poset</a> with the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Alexandroff+space">Alexandroff space</a>s.</p> </div> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A \in C^* Alg</annotation></semantics></math> we call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Alex \mathcal{C}(A)</annotation></semantics></math> the <strong>Bohr <a class="existingWikiWord" href="/nlab/show/site">site</a></strong> of <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>.</p> </div> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">C^* Alg</annotation></semantics></math> is called <strong>commutativity reflecting</strong> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a_1, a_2 \in A</annotation></semantics></math> we have that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a_1)</annotation></semantics></math> commutes with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a_2)</annotation></semantics></math> in <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> then already <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">a_1</annotation></semantics></math> commutes with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">a_2</annotation></semantics></math> in <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>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><msub><mi>Alg</mi> <mi>cr</mi></msub><mo>⊂</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex"> C^* Alg_{cr} \subset C^* Alg </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>s on the commutativity-reflecting morphisms.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> <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 \hookrightarrow B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">C^* Alg</annotation></semantics></math> is commutativity reflecting.</p> </div> <div class="num_prop" id="CommutativityReflectionByAdjoints"> <h6 id="proposition_3">Proposition</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">C^* Alg</annotation></semantics></math> is commutativity reflecting precisely if the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(f)</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⊣</mo><msub><mi>R</mi> <mi>f</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mover><mo>←</mo><mrow><msub><mi>R</mi> <mi>f</mi></msub></mrow></mover></mover><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathcal{C}(f) \dashv R_f) \;\colon\; \mathcal{C}(A) \stackrel{\overset{R_f}{\leftarrow}}{\underset{\mathcal{C}(f)}{\to}} \mathcal{C}(B) \,. </annotation></semantics></math></div></div> <p>This appears as (<a href="#Nuiten11">Nuiten 11, lemma 2.6</a>).</p> <div class="num_defn"> <h6 id="definition_8">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><msub><mi>Alg</mi> <mi>inc</mi></msub><mo>⊂</mo><msup><mi>C</mi> <mo>*</mo></msup><msub><mi>Alg</mi> <mi>cr</mi></msub><mo>⊂</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex"> C^* Alg_{inc} \subset C^* Alg_{cr} \subset C^* Alg </annotation></semantics></math></div> <p>for the subcategories of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">C^* Alg</annotation></semantics></math> on the monomorphisms and on the commutativity-reflecting morphisms, respectively.</p> </div> <h3 id="BohrTopos">The Bohr topos</h3> <div class="num_defn" id="CStarTopos"> <h6 id="definition_9">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>C</mi> <mi>com</mi> <mo>*</mo></msubsup><mi>Top</mi><mo>↪</mo><msubsup><mi>C</mi> <mi>com</mi> <mo>*</mo></msubsup><mi>Topos</mi></mrow><annotation encoding="application/x-tex"> C^*_{com} Top \hookrightarrow C^*_{com} Topos </annotation></semantics></math></div> <p>for the categories of <a class="existingWikiWord" href="/nlab/show/ringed+spaces">ringed spaces</a> and <a class="existingWikiWord" href="/nlab/show/ringed+toposes">ringed toposes</a>, where the <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> ring object is equipped with the structure of an internal <em>commutative</em> <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a> and the morphisms respect the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebra structure.</p> <p>Hence a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℰ</mi><mo>,</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>ℱ</mi><mo>,</mo><munder><mi>B</mi><mo>̲</mo></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{E}, \underline{A}) \to (\mathcal{F}, \underline{B})</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>C</mi> <mi>com</mi> <mo>*</mo></msubsup><mi>Topos</mi></mrow><annotation encoding="application/x-tex">C^\ast_{com} Topos</annotation></semantics></math> is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>ℰ</mi><mo>→</mo><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">f : \mathcal{E} \to \mathcal{F}</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><munder><mi>B</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">f^* \underline{B}</annotation></semantics></math> is still an internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>;</p> </li> <li> <p>and a morphism of internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebras</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><munder><mi>B</mi><mo>̲</mo></munder><mo>→</mo><munder><mi>A</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex"> f^\ast \underline{B} \to \underline{A} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_defn" id="TheSheafTopos"> <h6 id="definition_10">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A \in C^* Alg</annotation></semantics></math> the <strong>Bohr topos</strong> of <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> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-space/topos</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Alex</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">)</mo><mo>∈</mo><msubsup><mi>C</mi> <mi>com</mi> <mo>*</mo></msubsup><mi>Top</mi><mo>↪</mo><msubsup><mi>C</mi> <mi>com</mi> <mo>*</mo></msubsup><mi>Topos</mi></mrow><annotation encoding="application/x-tex"> Bohr(A) := ( Sh(Alex(\mathcal{C}(A))), \underline{A}) \in C^\ast_{com} Top \hookrightarrow C^\ast_{com} Topos </annotation></semantics></math></div> <p>whose underlying <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> is (that corresponding to) the <a href="#BohrSite">Bohr site</a>, and whose internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra is the tautological <a class="existingWikiWord" href="/nlab/show/copresheaf">copresheaf</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder><mo>:</mo><mo stretchy="false">(</mo><mi>C</mi><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↦</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> \underline{A} : (C \in \mathcal{C}(A)) \mapsto C </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Alex</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}(A), Set] \simeq Sh(Alex(\mathcal{C}(A)))</annotation></semantics></math> equipped with the evident objectwise commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra structure.</p> <p>Moreover, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Bohr</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>Sh</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub><mo stretchy="false">(</mo><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Bohr_{\not \not}(A) := (Sh_{\not \not}(Alex \mathcal{C}(A)), \underline{A}) </annotation></semantics></math></div> <p>for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-topos whose underlying <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> is that for the <a class="existingWikiWord" href="/nlab/show/double+negation+topology">double negation topology</a> on the plain Bohr topos.</p> </div> <p>The general notion of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s between <a class="existingWikiWord" href="/nlab/show/topos">topos</a>es are <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>s. But those that remember the morphisms of Bohr sites are <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a>s.</p> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : \mathcal{C}(A) \to \mathcal{C}(B)</annotation></semantics></math> induces an <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mover><mover><munder><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mo>=</mo><msub><mi>Ran</mi> <mi>f</mi></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>f</mi></mrow></mover></mover><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>!</mo></msub><mo>:</mo><mo>=</mo><msub><mi>Lan</mi> <mi>f</mi></msub></mrow></mover></mover><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> (f_! \dashv f^* \dashv f_*) : [\mathcal{C}(A), Set] \stackrel{\overset{f_! := Lan_f}{\longrightarrow}}{\stackrel{\overset{f^* := (-) \circ f}{\leftarrow}}{\underset{f_* := Ran_f }{\longrightarrow}}} [\mathcal{C}(B), Set] </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">Lan_f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ran</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">Ran_f</annotation></semantics></math> are left and right <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, respectively.</p> <p>We also write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>:</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[f,Set] : [\mathcal{C}(A), Set] \to [\mathcal{C}(B), Set]</annotation></semantics></math> for this. Notice that by the equivalence of <a class="existingWikiWord" href="/nlab/show/copresheaves">copresheaves</a> on <a class="existingWikiWord" href="/nlab/show/posets">posets</a> and <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> on the corresponding <a class="existingWikiWord" href="/nlab/show/Alexandroff+locales">Alexandroff locales</a> (see there for details) this is equivalently</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>Alex</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sh (Alex(f))) : Sh(Alex \mathcal{C}(A)) \to Sh(Alex \mathcal{C}(B)) \,. </annotation></semantics></math></div></div> <p>The next proposition asserts that all essential geometric morphisms between Bohr toposes arise this way:</p> <div class="num_prop" id="EssentialGeomMorphismsAndPosetMorphisms"> <h6 id="proposition_4">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>:</mo><mi>Poset</mi><mo>↪</mo><msub><mi>Topos</mi> <mi>ess</mi></msub></mrow><annotation encoding="application/x-tex"> [-,Set] : Poset \hookrightarrow Topos_{ess} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+2-functor">full and faithful 2-functor</a>.</p> <p>Analogously, <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a>s of the underlying toposes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(A) \to Bohr(B)</annotation></semantics></math> are precisely those in image under the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo>∘</mo><mi>Alex</mi></mrow><annotation encoding="application/x-tex">Sh \circ Alex</annotation></semantics></math> of functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(A) \to \mathcal{C}(B)</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the discussion in the section <em><a href="http://ncatlab.org/nlab/show/Cauchy+complete+category#InOrdinaryCatTheoryByEssGeomMorphisms">In terms of essential geometric morphisms</a></em> at <em><a class="existingWikiWord" href="/nlab/show/Cauchy+complete+category">Cauchy complete category</a></em> we have a full and faithful embedding of Cauchy-complete catgeories <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><mi>Set</mi><mo stretchy="false">]</mo><mo>:</mo><msub><mi>Cat</mi> <mi>Cauchy</mi></msub><mo>↪</mo><msub><mi>Topos</mi> <mi>ess</mi></msub></mrow><annotation encoding="application/x-tex">[-,Set] : Cat_{Cauchy} \hookrightarrow Topos_{ess}</annotation></semantics></math>. But posets are trivially Cauchy, complete, hence this restricts to an embedding <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><mi>Set</mi><mo stretchy="false">]</mo><mo>:</mo><mi>Poset</mi><mo>↪</mo><msub><mi>Cat</mi> <mi>Cauchy</mi></msub><mo>↪</mo><msub><mi>Topos</mi> <mi>ess</mi></msub></mrow><annotation encoding="application/x-tex">[-,Set] : Poset \hookrightarrow Cat_{Cauchy} \hookrightarrow Topos_{ess}</annotation></semantics></math>.</p> <p>In terms of Alexandroff topologies: by the discussion of <a href="http://ncatlab.org/nlab/show/specialization+topology#AlexandrovLocales">Alexandroff locales</a> (in the entry <em><a class="existingWikiWord" href="/nlab/show/Alexandroff+space">Alexandroff space</a></em>) we have that the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alex</mi><mo lspace="verythinmathspace">:</mo><mi>Poset</mi><mo>→</mo><mi>Locale</mi></mrow><annotation encoding="application/x-tex">Alex\colon Poset \to Locale</annotation></semantics></math> takes values precisely on those morphisms of locales whose inverse image has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>. The statement then follows using the properties of <a class="existingWikiWord" href="/nlab/show/localic+reflection">localic reflection</a>, which says that the <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo>:</mo><mi>Locale</mi><mo>→</mo><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Sh : Locale \to Topos</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+2-functor">full and faithful 2-functor</a>.</p> </div> <p>For such essential geometric morphisms to be parts of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Topos</mi></mrow><annotation encoding="application/x-tex">C^\ast Topos</annotation></semantics></math>, def. <a class="maruku-ref" href="#CStarTopos"></a> we need that their <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a>s respect internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebras:</p> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : Bohr(A) \to Bohr(B)</annotation></semantics></math> an essential geometric morphism, the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><munder><mi>B</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">f^\ast \underline{B}</annotation></semantics></math> is naturally a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}(A), Set]</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>According to (<a href="#HLSDeep">HLS09, 4.8</a>) every functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}(A) \to C^\ast Alg</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}(A), Set]</annotation></semantics></math>. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><munder><mi>B</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">f^* \underline{B}</annotation></semantics></math> is such a functor, sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↦</mo><msub><mi>im</mi> <mi>f</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C \in \mathcal{C}(A)) \mapsto im_f(C)</annotation></semantics></math>.</p> </div> <p>Using this we now discuss morphisms of Bohr toposes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Topos</mi></mrow><annotation encoding="application/x-tex">C^\ast Topos</annotation></semantics></math>.</p> <div class="num_prop" id="BohrFunctoriality"> <h6 id="proposition_6">Proposition</h6> <p>The construction of Bohr toposes from def. <a class="maruku-ref" href="#TheSheafTopos"></a> extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo>:</mo><msup><mi>C</mi> <mo>*</mo></msup><msubsup><mi>Alg</mi> <mi>cr</mi> <mi>op</mi></msubsup><mo>→</mo><msubsup><mi>C</mi> <mi>com</mi> <mo>*</mo></msubsup><mi>TopSpace</mi><mo>↪</mo><msubsup><mi>C</mi> <mi>com</mi> <mo>*</mo></msubsup><mi>Topos</mi></mrow><annotation encoding="application/x-tex"> Bohr : C^\ast Alg_{cr}^{op} \to C^\ast_{com} TopSpace \hookrightarrow C^\ast_{com} Topos </annotation></semantics></math></div> <p>with the special property that any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> is sent to</p> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphisms">essential geometric morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f^* \dashv f_*) : Bohr(B) \to Bohr(A)</annotation></semantics></math> with an extra <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mover><mover><mover><munder><mo>←</mo><mrow><msup><mi>f</mi> <mo>!</mo></msup></mrow></munder><mover><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></mover></mover><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>→</mo><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow></mover></mover><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Bohr(B) \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\stackrel{\overset{f_*}{\to}}{\underset{f^!}{\leftarrow}}}} Bohr(A) </annotation></semantics></math></div></li> <li> <p>such that the corresponding internal <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of internal algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder><mo>→</mo><msub><mi>f</mi> <mo>*</mo></msub><munder><mi>B</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{A} \to f_* \underline{B}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>.</p> </li> </ul> </div> <p>This is (<a href="#Nuiten11">Nuiten 11, lemma 2.7</a>). (Essentially this argument also appears as (<a href="#vdBergHeunen">vdBergHeunen, prop. 33</a>), where however the extra right adjoint is not made use of and instead the variances of the morphisms involved in the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Topos</mi></mrow><annotation encoding="application/x-tex">C^\ast Topos</annotation></semantics></math> are redefined in order to make the statement come out.)</p> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>To a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><msub><mi>Alg</mi> <mi>cr</mi></msub></mrow><annotation encoding="application/x-tex">C^\ast Alg_{cr}</annotation></semantics></math> which by prop. <a class="maruku-ref" href="#CommutativityReflectionByAdjoints"></a> corresponds to an <a class="existingWikiWord" href="/nlab/show/adjoint+pair">adjoint pair</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⊣</mo><msub><mi>R</mi> <mi>f</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mover><mo>←</mo><mrow><msub><mi>R</mi> <mi>f</mi></msub></mrow></mover></mover><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ( \mathcal{C}(f) \dashv R_f) : \mathcal{C}(A) \stackrel{\overset{R_f}{\leftarrow}}{\underset{\mathcal{C}(f)}{\to}} \mathcal{C}(B) </annotation></semantics></math></div> <p>we assign the <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>!</mo></msup><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mover><mover><mover><munder><mo>←</mo><mrow><msup><mi>f</mi> <mo>!</mo></msup><mo>:</mo><mo>=</mo><msub><mi>Ran</mi> <mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mover><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>R</mi> <mi>f</mi></msub></mrow></mover></mover><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>!</mo></msub><mo>:</mo><mo>=</mo><msub><mi>Lan</mi> <mrow><msub><mi>R</mi> <mi>f</mi></msub></mrow></msub></mrow></mover></mover><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> (f_! \dashv f^* \dashv f_* \dashv f^!) := [\mathcal{C}(B), Set] \stackrel{\overset{f_! := Lan_{R_f}}{\longrightarrow}}{\stackrel{\overset{f^* := (-)\circ R_f}{\leftarrow}}{\stackrel{\overset{f_* := (-)\circ \mathcal{C}(f)}{\longrightarrow}}{\underset{f^! := Ran_{\mathcal{C}(f)}}{\leftarrow}}}} [\mathcal{C}(A), Set] </annotation></semantics></math></div> <p>(wher <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lan</mi></mrow><annotation encoding="application/x-tex">Lan</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ran</mi></mrow><annotation encoding="application/x-tex">Ran</annotation></semantics></math> denote left and right <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a>, respectively) equipped with the morphism of internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>f</mi></msub><mo>:</mo><munder><mi>A</mi><mo>̲</mo></munder><mo>→</mo><msub><mi>f</mi> <mo>*</mo></msub><munder><mi>B</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex"> \eta_f : \underline{A} \to f_* \underline{B} </annotation></semantics></math></div> <p>which over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \in \mathcal{C}(A)</annotation></semantics></math> is the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/corestriction">corestriction</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(f)(C)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo stretchy="false">|</mo> <mi>C</mi></msub><mo>:</mo><mi>C</mi><mo>→</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f|_C : C \to \mathcal{C}(f)(C) </annotation></semantics></math></div> <p>(to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-completion of the algebraic image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo stretchy="false">|</mo> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">f|_C</annotation></semantics></math>).</p> </div> <p>Using prop. <a class="maruku-ref" href="#EssentialGeomMorphismsAndPosetMorphisms"></a> the above prop. <a class="maruku-ref" href="#BohrFunctoriality"></a> has the following partial converse.</p> <div class="num_prop" id="ToposCharacterizationOfAlgebraHomomorphisms"> <h6 id="proposition_7">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A, B \in C^\ast Alg</annotation></semantics></math>, morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : Bohr(B) \to Bohr(A)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Top</mi></mrow><annotation encoding="application/x-tex">C^\ast Top</annotation></semantics></math> for which</p> <ol> <li> <p>the underlying geometric morphism has an extra right adjoint</p> </li> <li> <p>the morphism of internal algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder><mo>→</mo><msub><mi>f</mi> <mo>*</mo></msub><munder><mi>B</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{A} \to f_* \underline{B}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a></p> </li> </ol> <p>are in bijection with <a class="existingWikiWord" href="/nlab/show/function">function</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex"> f : A \to B </annotation></semantics></math></div> <p>that are <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a>s on all commutative subalgebras and reflect commutativity.</p> <p>In particular when <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> is already commutative, morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(B) \to Bohr(A)</annotation></semantics></math> with an extra right adjoint and epi ring homomorphism are in bijection with algebra homomorphisms <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 \to B</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By prop <a class="maruku-ref" href="#EssentialGeomMorphismsAndPosetMorphisms"></a> every <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(Alex \mathcal{C}(A)) \to Sh(Alex \mathcal{C}(B))</annotation></semantics></math> comes from a morphism of locales <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alex</mi><mi>𝒞</mi><mi>A</mi><mo>→</mo><mi>Alex</mi><mi>𝒞</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">Alex \mathcal{C}A \to Alex \mathcal{C}A</annotation></semantics></math>, which by the discussion at <a class="existingWikiWord" href="/nlab/show/Alexandroff+space">Alexandroff space</a> is equivalently a morphism of posets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(f) : \mathcal{C}(A) \to \mathcal{C}(B)</annotation></semantics></math>. By the assumption of the extra right adjoint we also have a geometric morphism the other way round, and hence, again by prop. <a class="maruku-ref" href="#EssentialGeomMorphismsAndPosetMorphisms"></a>, an <a class="existingWikiWord" href="/nlab/show/adjoint+pair">adjoint pair</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⊣</mo><msub><mi>R</mi> <mi>f</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>↔</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\mathcal{C}(f) \dashv R_f) : \mathcal{C}(A) \leftrightarrow \mathcal{C}(B) </annotation></semantics></math></div> <p>that induces functors between toposes as in prop. <a class="maruku-ref" href="#BohrFunctoriality"></a>. Then the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-toposes implies algebra homomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>C</mi></msub><mo>:</mo><mi>C</mi><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f_C : C \to f(C) </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \in \mathcal{C}(A)</annotation></semantics></math>.</p> <p>By the assumption that this are the components of an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> of <a class="existingWikiWord" href="/nlab/show/copresheaves">copresheaves</a> all these component morphisms are themselves epimorphisms and hence we have that indeed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>image</mi> <mrow><msub><mi>f</mi> <mi>C</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(C) = image_{f_C}(C)</annotation></semantics></math>.</p> </div> <h2 id="KinematicsOnBohrTopos">Kinematics in a Bohr topos</h2> <p>A key aspect of the Bohr topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(A)</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a> (as defined <a href="#BohrToposOfQMSystem">above</a>) is that that <em>classical</em> <a class="existingWikiWord" href="/nlab/show/kinematics">kinematics</a> and classical <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a> of the commutative <em>internal</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder><mo>∈</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underline{A} \in Bohr(A)</annotation></semantics></math> is the <em>quantum</em> <a class="existingWikiWord" href="/nlab/show/kinematics">kinematics</a> and <a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> of <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>.</p> <p>In fact, the very definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(A)</annotation></semantics></math> provides a formal context in which <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> has a natural formulation:</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Gleason’s theorem)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>H</mi><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim H \gt 2</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A = B(H) \in C^\ast Alg</annotation></semantics></math> its algebra of <a class="existingWikiWord" href="/nlab/show/bounded+operators">bounded operators</a>, a <a class="existingWikiWord" href="/nlab/show/state">state</a> on <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> is a <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \rho : A \to \mathbb{C} </annotation></semantics></math></div> <p>which is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> when restricted to each commutative subalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C \subset A</annotation></semantics></math>.</p> </div> <p>A function that preserves certain <a class="existingWikiWord" href="/nlab/show/structure">structure</a> locally – here: over each commutative subalgebra – is precisely an <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> fully structure preserving <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> in the <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> over these local objects – here: over commutative subalgebras. Hence we have the following direct topos-theoretic equivalent reformulation of Gleason’s theorem.</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A = B(H) \in C^\ast Alg</annotation></semantics></math> as above, we have a natural bijection between the quantum <a class="existingWikiWord" href="/nlab/show/state">state</a>s on <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 the (classical) states of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{A}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(H)</annotation></semantics></math>.</p> </div> <h3 id="ThePhaseSpace">The phase space</h3> <p>The idea is that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A \in C^* Alg</annotation></semantics></math>, the Bohr topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Alex</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>TopSpace</mi><mo>⊂</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Bohr(A) = (Sh(Alex(\mathcal{C}(A))), \underline{A}) \in C^* TopSpace \subset C^* Topos</annotation></semantics></math> <em>is</em> the corresponding quantum <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>. More precisely, we may think of the internal commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder><mo>∈</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underline{A} \in Bohr(A)</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/Isbell+duality">formal dual</a> to the quantum phase space.</p> <p>The following discussion makes this precise by exhibiting this formal dual as an <a class="existingWikiWord" href="/nlab/show/internal+locale">internal locale</a>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(A)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/spatial+topos">spatial topos</a>, this internal locale is in fact an ordinary <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma \to Alex \mathcal{C}(A)</annotation></semantics></math> over the <a class="existingWikiWord" href="/nlab/show/Alexandroff+space">Alexandroff space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Alex \mathcal{C}(A)</annotation></semantics></math>.</p> <div class="num_defn" id="TheInternalLocale"> <h6 id="definition_11">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><mi>Σ</mi><mo>̲</mo></munder> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\underline{\Sigma}_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Σ</mi> <mi>A</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Sigma_A^{\not \not}</annotation></semantics></math>, respectively for the corresponding <a class="existingWikiWord" href="/nlab/show/internal+locale">internal locale</a>s associated to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{A}</annotation></semantics></math> by internal <a class="existingWikiWord" href="/nlab/show/constructive+Gelfand+duality">constructive Gelfand duality</a>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>A</mi></msub><mo>→</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma_A \to \mathcal{C}(A) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Σ</mi> <mi>A</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msubsup><mo>→</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma^{\not \not}_A \to \mathcal{C}(A) </annotation></semantics></math></div> <p>for the corresponding external locale, given under the <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Loc</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Loc</mi><mo stretchy="false">/</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Loc(Sh(\mathcal{C}(A))) \simeq Loc/\mathcal{C}(A) </annotation></semantics></math></div> <p>discussed at <a class="existingWikiWord" href="/nlab/show/internal+locale">internal locale</a>.</p> </div> <div class="num_defn" id="Bohrification"> <h6 id="definition_12">Definition</h6> <p>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> a (noncommutative) <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>, the assignment</p> <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><msub><mi>Σ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex"> A \mapsto \Sigma_{A} </annotation></semantics></math></div> <p>or</p> <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><msubsup><mi>Σ</mi> <mi>A</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> A \mapsto \Sigma^{\not \not}_{A} </annotation></semantics></math></div> <p>is called the <strong>Bohrification</strong> of <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>.</p> </div> <div class="num_prop" id="TheInternalLocale"> <h6 id="proposition_8">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>Σ</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><munder><mo stretchy="false">(</mo><mo>̲</mo></munder><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underline{\Sigma}(\underline(A))</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/internal+locale">internal locale</a> from def. <a class="maruku-ref" href="#TheInternalLocale"></a>.</p> <p>Regarded as an object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mi>A</mi></msub><mo>→</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><mi>Loc</mi><mo stretchy="false">/</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\Sigma_A \to \mathcal{C}(A)) \in Loc/\mathcal{C}(A) </annotation></semantics></math></div> <p>of external locales over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(A)</annotation></semantics></math>, this is the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> whose underlying set is given by the disjoint union</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>A</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>C</mi><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></munder><mi>Σ</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma_A = \coprod_{C \in \mathcal{C}(A)} \Sigma(C) </annotation></semantics></math></div> <p>over all commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-subalgebras of <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> of the ordinary <a class="existingWikiWord" href="/nlab/show/Gelfand+spectrum">Gelfand spectra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma(C)</annotation></semantics></math> of these commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras, and whose <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>s are defined to be those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mo>⊂</mo><msub><mi>Σ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{U} \subset \Sigma_A</annotation></semantics></math> for which</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><msub><mo stretchy="false">|</mo> <mi>C</mi></msub><mo>∈</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{U}|_C \in \mathcal{O}(\Sigma(C))</annotation></semantics></math> for all commutative subalgebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>;</p> </li> <li> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>↪</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C \hookrightarrow D</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>∈</mo><mi>𝒰</mi><msub><mo stretchy="false">|</mo> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\lambda \in \mathcal{U}|_C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>′</mo><mo>∈</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda' \in \Sigma(D)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>′</mo><msub><mo stretchy="false">|</mo> <mi>C</mi></msub><mo>=</mo><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda'|_C = \lambda</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>′</mo><mo>∈</mo><mi>𝒰</mi><msub><mo stretchy="false">|</mo> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">\lambda' \in \mathcal{U}|_D</annotation></semantics></math>.</p> </li> </ol> <p>Regarded equivalently as an <a class="existingWikiWord" href="/nlab/show/internal+locale">internal locale</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(\mathcal{C}(A))</annotation></semantics></math> this</p> <p>As a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on the poset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(A)</annotation></semantics></math> this is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Σ</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">)</mo><mo>:</mo><mi>U</mi><mo>↦</mo><msub><mi>Σ</mi> <mi>U</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \underline{\Sigma}(\underline{A}) : U \mapsto \Sigma_U \,, </annotation></semantics></math></div> <p>where for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \in \mathcal{O}(\mathcal{C}(A))</annotation></semantics></math> we set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>U</mi></msub><mo>:</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>C</mi><mo>∈</mo><mi>U</mi></mrow></munder><mi>Σ</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma_U := \coprod_{C \in U} \Sigma(C) </annotation></semantics></math></div> <p>with the relative topology inherited from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_A</annotation></semantics></math>.</p> </div> <p>This appears as (<a href="#HLSW">HLSW, theorem 1</a>).</p> <div class="num_prop" id="BohrificationDependsOnNormalElementsOnly"> <h6 id="proposition_9">Proposition</h6> <p>The Bohrification of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>ncCStar</mi></mrow><annotation encoding="application/x-tex">A \in ncCStar</annotation></semantics></math> only depends on its <a href="#PartialCStar">partial C-star algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(A)</annotation></semantics></math> of normal elements</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>A</mi></msub><mo>≃</mo><msub><mi>Σ</mi> <mrow><mi>N</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Sigma_A \simeq \Sigma_{N(A)} \,. </annotation></semantics></math></div></div> <p>This is highlighted in (<a href="#vdBergHeunen">vdBergHeunen</a>).</p> <div class="num_prop" id="ReproducingOrdinaryGelfandDuality"> <h6 id="proposition_10">Proposition</h6> <p>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> a commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Σ</mi> <mi>A</mi> <mi>Gelf</mi></msubsup><mo>∈</mo></mrow><annotation encoding="application/x-tex">\Sigma_A^{Gelf} \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Loc">Loc</a> its ordinary <a class="existingWikiWord" href="/nlab/show/Gelfand+spectrum">Gelfand spectrum</a>, we have that Bohrification in the double negation topology reproduces this ordinary Gelfand spectrum:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Σ</mi> <mi>A</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msubsup><mo>≃</mo><msubsup><mi>Σ</mi> <mi>A</mi> <mi>Gel</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Sigma^{\not \not}_A \simeq \Sigma_A^{Gel} \,. </annotation></semantics></math></div></div> <p>This is (<a href="#Spitters06">Spitters06, theorem 9, corollary 10</a>).</p> <div class="num_prop" id="FunctorToRingedToposes"> <h6 id="proposition_11">Proposition</h6> <p>Then the construction of the <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a> over the <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↦</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>,</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A \mapsto ([\mathcal{C}(A), Set], \underline{A}) </annotation></semantics></math></div> <p>extends to a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><msub><mi>Alg</mi> <mi>incl</mi></msub><mo>→</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>TopSpace</mi><mo>↪</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Topos</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> C^* Alg_{incl} \to C^*TopSpace \hookrightarrow C^* Topos \,, </annotation></semantics></math></div> <p>where on the right the morphisms of internal rings are even morphisms of internal <a class="existingWikiWord" href="/nlab/show/C%2A+algebras">C* algebras</a>.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>With the components of the morphism of internal rings the evident objectwise inclusions, this is directly checked.</p> </div> <div class="num_prop" id="BohrificationFunctorToLoc"> <h6 id="proposition_12">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cstar</mi> <mi>inc</mi></msub></mrow><annotation encoding="application/x-tex">Cstar_{inc}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>s and inclusions. Then <a href="#Bohrification">Bohrification</a> extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><msubsup><mi>CStar</mi> <mi>inc</mi> <mi>op</mi></msubsup><mo>→</mo><mi>Loc</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Sigma_{(-)} : CStar_{inc}^{op} \to Loc \,. </annotation></semantics></math></div></div> <p>This is effectively the functoriality of the internal <a class="existingWikiWord" href="/nlab/show/constructive+Gelfand+duality">constructive Gelfand duality</a> applied to the <a href="#FunctorToRingedToposes">above observation</a>. The statement appears as (<a href="#vdBergHeunen">vdBergHeunen, theorem 35</a>).</p> <h3 id="TheObservables">The observables</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A \in C^\ast Alg</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, then in <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> a <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math> is a <em>quantum <a class="existingWikiWord" href="/nlab/show/observable">observable</a></em>. The following statement asserts that quantum observables on <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> are in a precise sense the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-valued “functions” on the Bohr topos of <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>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\mathbb{R})</annotation></semantics></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra of continuous complex functions on the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a>. We think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bohr(C(\mathbb{R}))</annotation></semantics></math> as the incarnation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> in the context of Bohr toposes.</p> <div class="num_prop"> <h6 id="proposition_13">Proposition</h6> <p>Morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>ℝ</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : Bohr(A) \to Bohr(C(\mathbb{R})_0)</annotation></semantics></math> with an extra right adjoint and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>ℝ</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>→</mo><msub><mi>f</mi> <mo>*</mo></msub><munder><mi>A</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">C(\mathbb{R})_0 \to f_*\underline{A}</annotation></semantics></math> epi are in bijection to the observables on <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>.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>By prop. <a class="maruku-ref" href="#ToposCharacterizationOfAlgebraHomomorphisms"></a> such morphisms are in bijection to algebra homomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>ℝ</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>→</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C(\mathbb{R})_0 \to A \,. </annotation></semantics></math></div> <p>By <a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a>: every <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math> provides such a homomorphism by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>↦</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \mapsto f(a)</annotation></semantics></math>. Conversely, given such an algebra homomorphism, its image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>x</mi><mo>↦</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">i : x \mapsto x</annotation></semantics></math> is a self-adjoint operator in <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 these two constructions are clearly inverses of each other.</p> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>In the different but related context of the <a class="existingWikiWord" href="/nlab/show/spectral+presheaf">spectral presheaf</a> (<a href="#IshamDoering07">Isham-Döring 07</a>) the identification of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> with a topos-theoretic construction, as far as possible, has been called “<a class="existingWikiWord" href="/nlab/show/daseinisation">daseinisation</a>”. This is a bit more involved than the above direct characterization in terms of maps of <a class="existingWikiWord" href="/nlab/show/ringed+toposes">ringed toposes</a>.</p> </div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow></msub><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x_{t_1} \in A</annotation></semantics></math> be an observable and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>x</mi> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle x_{t_1} \rangle</annotation></semantics></math> for the subalgebra generated by it. Then (by general properties of <a href="#http://ncatlab.org/nlab/show/over-topos#PresheafOverTopos">presheaf over-toposes</a>) the <a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo stretchy="false">⟨</mo><msub><mi>x</mi> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">⟩</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Bohr(A)_{/\langle x_{t_1}\rangle}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo stretchy="false">⟨</mo><msub><mi>x</mi> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">⟩</mo></mrow></msub><mo>≃</mo><mo stretchy="false">[</mo><mo stretchy="false">⟨</mo><msub><mi>x</mi> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow></msub><msub><mo stretchy="false">⟩</mo> <mrow><mo stretchy="false">/</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Bohr(A)_{/\langle x_{t_1}\rangle} \simeq [\langle x_{t_1}\rangle_{/\mathcal{C}(A)}, Set] \,, </annotation></semantics></math></div> <p>where on the right we have the <a class="existingWikiWord" href="/nlab/show/copresheaves">copresheaves</a> over the <a class="existingWikiWord" href="/nlab/show/under-category">under-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>x</mi> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow></msub><msub><mo stretchy="false">⟩</mo> <mrow><mo stretchy="false">/</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\langle x_{t_1}\rangle_{/\mathcal{C}(A)}</annotation></semantics></math>. This is precisely the sub-<a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a> on those commutative subalgebras that contain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">x_{t}</annotation></semantics></math>. This means that a classically consistent observation in the slice topos is one that is consistent with the observation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">x_{t_1}</annotation></semantics></math>.</p> </div> <h3 id="the_states">The states</h3> <div class="num_defn"> <h6 id="definition_13">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A \in C^\ast Alg</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Sh(Alex \mathcal{C}(A)), \mathbb{R})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a> as indicated, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> denotes the copresheaf constant on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>.</p> <p>The internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder><mo>∈</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underline{A} \in Bohr(A)</annotation></semantics></math> is an internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-module. Forgetting the algebra structure and only remembering the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-module structure, we get a category of “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-module toposes”.</p> </div> <div class="num_prop"> <h6 id="observation">Observation</h6> <p>There is a canonical morphism of ringed toposes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>:</mo><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi : Bohr(A) \to (Sh(Alex \mathcal{C}(A)), \mathbb{R}) </annotation></semantics></math></div> <p>whose underlying geometric morphism is the identity (and whose morphism of internal ring objects is the unique one).</p> </div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>This <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-topos incarnation of the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>Alex</mi><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma \to Alex \mathcal{C}(A)</annotation></semantics></math> of locales discussed <a href="ThePhaseSpace">above</a>.</p> </div> <div class="num_prop"> <h6 id="observation_2">Observation</h6> <p>A <a class="existingWikiWord" href="/nlab/show/state">state</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> on <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> is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> in the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-moduled toposes that is positive and normalized.</p> </div> <p>(…)</p> <h2 id="SheafOfBohrToposesOfQFT">(Pre-)Sheaf of Bohr toposes of a quantum field theory</h2> <p>Notice that in the context of <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> a <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is encoded by a <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> of <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a>s on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>.</p> <div class="num_note" id="PresheavesFromLocalNets"> <h6 id="note">Note</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian manifold</a> and</p> <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>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mo>*</mo></msup><msub><mi>Alg</mi> <mi>inc</mi></msub></mrow><annotation encoding="application/x-tex"> A : \mathcal{O}(X) \to C^* Alg_{inc} </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> of algebras. Notice that by definition this indeed takes values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebras and <em>inclusions</em> . Then postcomposition with the <a href="#BohrTopos">Bohr topos</a>-functor yields a presheaf of ringed spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><msup><mi>C</mi> <mo>*</mo></msup><msubsup><mi>Alg</mi> <mi>inc</mi> <mi>op</mi></msubsup><mo>↪</mo><msup><mi>C</mi> <mo>*</mo></msup><msubsup><mi>Alg</mi> <mi>cr</mi> <mi>op</mi></msubsup><mover><mo>→</mo><mi>Bohr</mi></mover><msup><mi>C</mi> <mo>*</mo></msup><mi>TopSpace</mi><mo>↪</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Topos</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Bohr(A) : \mathcal{O}(X)^{op} \to C^* Alg_{inc}^{op} \hookrightarrow C^* Alg_{cr}^{op} \stackrel{Bohr}{\to} C^* TopSpace \hookrightarrow C^* Topos \,. </annotation></semantics></math></div></div> <p>This appears as (<a href="#Nuiten11">Nuiten 11, def. 17</a>).</p> <p>Assume that a net of observables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mo>*</mo></msup><msub><mi>Alg</mi> <mi>inc</mi></msub></mrow><annotation encoding="application/x-tex">A : Op(X) \to C^\ast Alg_{inc}</annotation></semantics></math> satisfies the <a class="existingWikiWord" href="/nlab/show/split+property">split property</a>. Then it is <a href=""></a> precisely if the corresponding presheaf of Bohr toposes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mover><mo>→</mo><mi>A</mi></mover><msup><mi>C</mi> <mo>*</mo></msup><msub><mi>Alg</mi> <mi>inc</mi></msub><mover><mo>→</mo><mi>Bohr</mi></mover><msup><mi>C</mi> <mo>*</mo></msup><mi>Top</mi></mrow><annotation encoding="application/x-tex">Bohr(A) : Op(X)^{op} \stackrel{A}{\to} C^\ast Alg_{inc} \stackrel{Bohr}{\to} C^\ast Top</annotation></semantics></math> satisfies <em>spatial</em> <a class="existingWikiWord" href="/nlab/show/descent">descent</a> by <a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a>s (meaning that for every spatial hyperslice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \subset X</annotation></semantics></math> the induced presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bohr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mi>Σ</mi></msub><mo>:</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>Σ</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Top</mi></mrow><annotation encoding="application/x-tex">Bohr(A)|_\Sigma : Op(\Sigma)^{op} \to C^\ast Top</annotation></semantics></math> satisfies <a class="existingWikiWord" href="/nlab/show/descent">descent</a> by <a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a>s.)</p> <p>(…)</p> <p>This appears as (<a href="#Nuiten11">Nuiten 11, theorem 4.2</a>).</p> <h2 id="contravariant_functors_on_open_subsets">Contravariant functors on open subsets</h2> <p>Above is discussed the notion of Bohr topos given by <a class="existingWikiWord" href="/nlab/show/covariant+functors">covariant functors</a> on the <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a> of a <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>. The fact that the functors here are covariant is related to the fact that the algebra itself naturally exists inside the presheaf topos.</p> <p>Alternatively one can explore the situation for <a class="existingWikiWord" href="/nlab/show/contravariant+functors">contravariant functors</a> on the <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a> (<a href="#IshamDoering07">Isham-Döring 07</a>). The resulting <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> then does not directly contain the given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra, but by <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a>, does directly contain an internal <a class="existingWikiWord" href="/nlab/show/locale">locale</a> which is its <a class="existingWikiWord" href="/nlab/show/Gelfand+spectrum">Gelfand spectrum</a>. This is called the “<a class="existingWikiWord" href="/nlab/show/spectral+presheaf">spectral presheaf</a>”.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a></li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> <p><a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a></p> <p><a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a></p> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/observables">observables</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a></p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorems">theorems</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> </ul> </li> </ul> </div> <h2 id="References">References</h2> <p>The assertion by Bohr that all experiments in quantum mechanics must be possible to describe in “classical terms” is in</p> <ul> <li id="Bohr49"> <p><a class="existingWikiWord" href="/nlab/show/Niels+Bohr">Niels Bohr</a>, <em>Discussion with Einstein on Epistemological Problems in Atomic Physics</em>, in: P. A. Schilpp (ed.) <em>Albert Einstein, Philosopher-Scientist</em> (Evanston: Library of Living Philosophers) (1949) 201-241 [<a href="https://doi.org/10.1016/S1876-0503(08)70379-7">doi:10.1016/S1876-0503(08)70379-7</a>]</p> <blockquote> <p>“however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms.”</p> </blockquote> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Niels+Bohr">Niels Bohr</a>‘s views on quantum mechanics that give the construction of <em>Bohrification</em> its name are reviewed further in</p> <ul> <li id="Scheibe73"><a class="existingWikiWord" href="/nlab/show/Erhard+Scheibe">Erhard Scheibe</a>, <em>Bohr’s interpretation of quantum mechanics</em>, Chapter I in: <em>The logical analysis of quantum mechanics</em>, Oxford: Pergamon Press (1973)</li> </ul> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/interpretation+of+quantum+mechanics">interpretation of quantum mechanics</a></em> the section <em><a href="interpretation+of+quantum+mechanics#BohrStandpoint">Bohr’s standpoint</a></em>.</p> <p>Maybe the first article to propose to use <a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic logic</a>/<a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> for the description of quantum physics is</p> <ul> <li>Murray Adelman, John Corbett, <em>A Sheaf Model for Intuitionistic Quantum Mechanics</em>, Appl. Cat. Struct. <strong>3</strong> (1995) 79-104 [<a href="https://doi.org/10.1007/BF00872949">doi:10.1007/BF00872949</a>]</li> </ul> <p>The term <em>Bohrification</em> and the investigations associated with it were initiated in</p> <ul> <li id="HeunenLandsmanSpitters09"><a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, <a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Bas Spitters</a>, <em>Bohrification of operator algebras and quantum logic</em>, Synthese <strong>186</strong> 3 (2012) 719-752 [<a href="http://arxiv.org/abs/0905.2275">arXiv:0905.2275</a>, <a href="https://doi.org/10.1007/s11229-011-9918-4">doi;10.1007/s11229-011-9918-4</a>]</li> </ul> <p>see also related comments in disucssion of the <a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a> here:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, <em>The Born rule and its interpretation</em>, in: <em>Compendium of Quantum Physics</em>, Springer (2009) 64-70 [<a href="https://doi.org/10.1007/978-3-540-70626-7_20">doi:10.1007/978-3-540-70626-7_20</a>, <a href="https://www.math.ru.nl/~landsman/Born.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Landsman-BornRule.pdf" title="pdf">pdf</a>]</li> </ul> <p>See also:</p> <ul> <li id="HLSDeep"><a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, <a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Bas Spitters</a>, <em>Bohrification</em>, in: <em>Deep Beauty – Understanding the Quantum World through Mathematical Innovation</em>, Cambridge University Press (2009) 271-314 [<a href="http://arxiv.org/abs/0909.3468">arXiv:0909.3468</a>, <a href="https://doi.org/10.1017/CBO9780511976971.008">doi:10.1017/CBO9780511976971.008</a>]</li> </ul> <p>The computation of the internal Gelfand spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>Σ</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{\Sigma}</annotation></semantics></math> was initiated in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, <a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Bas Spitters</a> <em>A topos for algebraic quantum theory</em>, Comm. Math. Phys. <strong>291</strong> (2009) 63-110 [<a href="http://arxiv.org/abs/0709.4364">arXiv:0709.4364</a>, <a href="http://dx.doi.org/10.1007/s00220-009-0865-6">doi:10.1007/s00220-009-0865-6</a>]</li> </ul> <p>with some results in section 5 and 6 of</p> <ul> <li>Martijn Caspers, <em>Gelfand spectra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras in topos theory</em>, MSc thesis, Nijmegen (2008) [<a href="http://www.math.ru.nl/~landsman/scriptieMartijn.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Caspers-GelfandSpectra.pdf" title="pdf">pdf</a>]</li> </ul> <p>and completed in</p> <ul> <li>Sander Wolters, <em>Contravariant vs covariant quantum logic: A comparison of two topos-theoretic approaches to quantum theory</em>, Comm. Math. Phys. <strong>317</strong> (2013) 3–53 [<a href="http://arxiv.org/abs/1010.2031">arXiv:1010.2031</a>, <a href="https://doi.org/10.1007/s00220-012-1652-3">doi:10.1007/s00220-012-1652-3</a>]</li> </ul> <p>An outline of the full proof is given in</p> <ul> <li id="HLSW"><a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, <a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Bas Spitters</a>, Sander Wolters, <em>The Gelfand spectrum of a noncommutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra: a topos-theoretic approach</em>, Journal of the Australian Mathematical Society <strong>90</strong> (2011) 39-52 [<a href="http://arxiv.org/abs/1010.2050">arxiv:1010.2050</a>, <a href="https://doi.org/10.1017/S1446788711001157">doi:10.1017/S1446788711001157</a>]</li> </ul> <p>Applications and examples 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> a <a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> algebra are discussed in</p> <ul> <li>Martijn Caspers, <a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, <a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Bas Spitters</a>, <em>Intuitionistic Quantum Logic of an <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>-level System</em>, Foundations of Physics <strong>39</strong> 7 (2009) 731-759 [<a href="https://arxiv.org/abs/0902.3201">arXiv:0902.3201</a>, <a href="https://doi.org/10.1007/s10701-009-9308-7">doi:10.1007/s10701-009-9308-7</a>]</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/functor">functoriality</a> of Bohrification is observed in</p> <ul> <li id="vdBergHeunen"><a class="existingWikiWord" href="/nlab/show/Benno+van+den+Berg">Benno van den Berg</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <em>Noncommutativity as a colimit</em>, Applied Categorical Structures <strong>20</strong> 4 (2012) 393-414 [<a href="http://arxiv.org/abs/1003.3618">arXiv:1003.3618</a>, <a href="https://doi.org/10.1007/s10485-011-9246-3">doi:10.1007/s10485-011-9246-3</a>]</li> </ul> <p>The application of the <a class="existingWikiWord" href="/nlab/show/double+negation+topology">double negation topology</a> to make Bohrification coincide with ordinary <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> on commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras is discussed in</p> <ul> <li id="Spitters06"><a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Bas Spitters</a>, <em>The space of measurement outcomes as a spectrum for non-commutative algebras</em>, EPTCS <strong>26</strong> (2010) 127-133 [<a href="http://arxiv.org/abs/1006.1432">arXiv:1006.1432</a>, <a href="https://doi.org/10.4204/EPTCS.26.12">doi:10.4204/EPTCS.26.12</a>]</li> </ul> <p>The generalization of Bohrification 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> (<a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>):</p> <ul> <li id="Nuiten11"><a class="existingWikiWord" href="/nlab/show/Joost+Nuiten">Joost Nuiten</a>, <em><a class="existingWikiWord" href="/schreiber/show/bachelor+thesis+Nuiten">Bohrification of local nets of observables</a></em>, in: <em>Proceedings of <a href="http://qpl.science.ru.nl/">QPL 2011</a></em>, <a href="http://rvg.web.cse.unsw.edu.au/eptcs/content.cgi?QPL2011">EPTCS 95, 2012</a> (2012) 211-218 [<a href="http://arxiv.org/abs/1109.1397">arXiv:1109.1397</a>, <a href="https://dx.doi.org/10.4204/EPTCS.95.15">doi:10.4204/EPTCS.95.15</a>]</li> </ul> <p>Review:</p> <ul> <li id="Landsman17"><a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, <em>Topos theory and quantum logic</em>, Section 12 of: <em>Foundations of quantum theory – From classical concepts to Operator algebras</em>, Springer Open (2017) [<a href="https://link.springer.com/book/10.1007/978-3-319-51777-3">doi:10.1007/978-3-319-51777-3</a>, <a href="https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf">pdf</a>]</li> </ul> <p>The original suggestion to interpret the <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> in the topos over the <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a> (there taken to be <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> instead of <a class="existingWikiWord" href="/nlab/show/copresheaves">copresheaves</a>) is due to</p> <ul> <li id="ButterfieldHamiltonIsham"> <p><a class="existingWikiWord" href="/nlab/show/Jeremy+Butterfield">Jeremy Butterfield</a>, John Hamilton, <a class="existingWikiWord" href="/nlab/show/Chris+Isham">Chris Isham</a>, <em>A topos perspective on the Kochen-Specker theorem</em>, <em>I. quantum states as generalized valuations</em>, Internat. J. Theoret. Phys. <strong>37</strong> 11 (1998) 2669-2733 [<a href="http://dx.doi.org/10.1023/A:1026680806775">doi:10.1023/A:1026680806775</a>, <a href="https://arxiv.org/abs/quant-ph/9803055">arXiv:quant-ph/9803055</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=1669557">MR2000c:81027</a>]</p> <p><em>II. conceptual aspects and classical analogues</em> Int. J. of Theor. Phys. <strong>38</strong> 3 (1999) 827-859 [<a href="http://www.ams.org/mathscinet-getitem?mr=1697983">MR2000f:81012</a>, <a href="http://dx.doi.org/10.1023/A:1026652817988">doi:10.1023/A:1026652817988</a>]</p> <p><em>III. Von Neumann algebras as the base category</em>, Int. J. of Theor. Phys. <strong>39</strong> 6 (2000) 1413-1436 [<a href="http://arxiv.org/abs/quant-ph/9911020">arXiv:quant-ph/9911020</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=1788498">MR2001k:81016</a>,<a href="http://dx.doi.org/10.1023/A:1003667607842">doi:10.1023/A:1003667607842</a>]</p> <p><em>IV. Interval valuations</em>, Internat. J. Theoret. Phys. <strong>41</strong> 4 (2002) 613-639 [<a href="http://www.ams.org/mathscinet-getitem?mr=1902067">MR2003g:81009</a>, <a href="http://dx.doi.org/10.1023/A:1015276209768">doi</a>]</p> </li> </ul> <p>with some review and outlook in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Isham">Chris Isham</a>, <a class="existingWikiWord" href="/nlab/show/Jeremy+Butterfield">Jeremy Butterfield</a>, <em>Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity</em>, Found. Phys. <strong>30</strong> (2000) 1707-1735 [<a href="https://arxiv.org/abs/gr-qc/9910005">arXiv:gr-qc/9910005</a>, <a href="https://doi.org/10.1023/A:1026406502316">doi:10.1023/A:1026406502316</a>]</li> </ul> <p>and then</p> <ul> <li id="IshamDoering07"> <p><a class="existingWikiWord" href="/nlab/show/Andreas+D%C3%B6ring">Andreas Döring</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Isham">Chris Isham</a>, <em>A Topos Foundation for Theories of Physics</em></p> <p><em>I. Formal Languages for Physics</em>, J. Math. Phys. <strong>49</strong> (2008) 053515 [<a href="http://arxiv.org/abs/quant-ph/0703060">arXiv:quant-ph/0703060</a>, <a href="https://doi.org/10.1063/1.2883740">doi:10.1063/1.2883740</a>]</p> <p><em>II. Daseinisation and the Liberation of Quantum Theory</em>, J. Math. Phys. <strong>49</strong> (2008) 053516 [<a href="http://arxiv.org/abs/quant-ph/0703062">arXiv:quant-ph/0703062</a>, <a href="https://doi.org/10.1063/1.2883742">doi:10.1063/1.2883742</a>]</p> <p><em>III. The Representation of Physical Quantities With Arrows</em>, J. Math. Phys. <strong>49</strong> (2008) 053517 [<a href="https://arxiv.org/abs/quant-ph/0703064">arXiv:quant-ph/0703064</a>, <a href="https://doi.org/10.1063/1.2883777">doi:10.1063/1.2883777</a>]</p> <p><em>IV. Categories of Systems</em>, J. Math. Phys. <strong>49</strong> (2008) 053518 [<a href="http://arxiv.org/abs/quant-ph/0703066">arXiv:quant-ph/0703066</a>, <a href="https://doi.org/10.1063/1.2883826">doi:10.1063/1.2883826</a>]</p> </li> </ul> <p>Further development in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Harding">John Harding</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <em>Topos quantum theory with short posets</em>, Order <strong>38</strong> 111–125 (2021) [<a href="https://arxiv.org/abs/1903.01897">arXiv:1903.01897</a>, <a href="https://doi.org/10.1007/s11083-020-09531-6">doi:10.1007/s11083-020-09531-6</a>]</li> </ul> <p>Discussion of aspects of the process of <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> in terms of Bohr toposes is in</p> <ul> <li>Kunji Nakayama, <em>Sheaves in Quantum Topos Induced by Quantization</em> [<a href="http://arxiv.org/abs/1109.1192">arXiv:1109.1192</a>]</li> </ul> <p>A variant of the Bohr topos construction meant to take more of the topology of the underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra into account has been suggested for finite-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebra in</p> <ul> <li>Guillaume Raynaud, <em>Fibred contextual quantum physics</em>, PhD thesis, University of Birmingham, 2014. [<a href="http://www.cs.bham.ac.uk/~sjv/grpage.php">web</a>]</li> </ul> <p>and generalized to arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebras in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Simon+Henry">Simon Henry</a>, <em>A Geometric Bohr topos</em> [<a href="https://arxiv.org/abs/1502.01896">arXiv:1502.01896</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 19, 2023 at 11:22:54. See the <a href="/nlab/history/Bohr+topos" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Bohr+topos" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4895/#Item_14">Discuss</a><span class="backintime"><a href="/nlab/revision/Bohr+topos/79" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Bohr+topos" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Bohr+topos" accesskey="S" class="navlink" id="history" rel="nofollow">History (79 revisions)</a> <a href="/nlab/show/Bohr+topos/cite" style="color: black">Cite</a> <a href="/nlab/print/Bohr+topos" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Bohr+topos" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>