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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/8182/#Item_16" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry describes a concrete formalization of the general notion of <a class="existingWikiWord" href="/nlab/show/state">state</a> in the context of <a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> and <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> and <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>. For other conceptualizations of <a class="existingWikiWord" href="/nlab/show/states">states</a> see there.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="measure_and_probability_theory">Measure and probability theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a></strong></p> <p>(<a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a>)</p> <h2 id="measure_theory">Measure theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a></p> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state">state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a></p> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a></p> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </div></div> <h4 id="functional_analysis">Functional analysis</h4> <div class="hide"><div> <ul> <li><strong><a class="existingWikiWord" href="/nlab/show/functional+analysis">Functional Analysis</a></strong></li> </ul> <h2 id="overview_diagrams">Overview diagrams</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TVS+relationships">topological vector spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diagram+of+LCTVS+properties">locally convex topological vector spaces</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+convex+topological+vector+space">locally convex topological vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Banach+space">Banach Spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflexive+Banach+space">reflexive</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Smith+space+%28functional+analysis%29">Smith Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+space">Fréchet Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Sobolev+space">Sobolev spaces</a>, <a class="existingWikiWord" href="/nlab/show/Lebesgue+space">Lebesgue Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bornological+vector+space">Bornological Vector Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/barrelled+topological+vector+space">Barrelled Vector Spaces</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+linear+operator">bounded</a>, <a class="existingWikiWord" href="/nlab/show/unbounded+linear+operator">unbounded</a>, <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint</a>, <a class="existingWikiWord" href="/nlab/show/compact+operator">compact</a>, <a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+of+an+operator">spectrum of an operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-Weierstrass+theorem">Stone-Weierstrass theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theory">spectral theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theorem">spectral theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></p> </li> </ul> <h2 id="topics_in_functional_analysis">Topics in Functional Analysis</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/basis+in+functional+analysis">Bases</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theories+in+functional+analysis">Algebraic Theories in Functional Analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/an+elementary+treatment+of+Hilbert+spaces">An Elementary Treatment of Hilbert Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism+classes+of+Banach+spaces">When are two Banach spaces isomorphic?</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/functional+analysis+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="algeraic_quantum_field_theory">Algeraic Quantum Field Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a></p> <h2 id="concepts">Concepts</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a></strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+history">field history</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/observables">observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+net">field net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> <p><a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a></p> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a></p> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a></p> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>)</p> </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#DefinitionForUnitalStarAlgebras'>For unital star algebras</a></li> <li><a href='#for_algebras'>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</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#closure_properties'>Closure properties</a></li> <li><a href='#MultiplicativeStates'>Multiplicative states</a></li> <li><a href='#fells_theorem'>Fell’s theorem</a></li> <li><a href='#gleasons_theorem'>Gleason’s theorem</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#exposition'>Exposition</a></li> <li><a href='#original_articles'>Original articles</a></li> <li><a href='#the_case_of_group_algebras'>The case of group algebras</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>The concept of <em>state on a star-algebra</em> is the formalization of the general idea of <em><a class="existingWikiWord" href="/nlab/show/states">states</a></em> from the point of view of <a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> and <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic quantum theory</a>.</p> <p>In order to motivate the definition from more traditional formulations in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, recall that there a <em><a class="existingWikiWord" href="/nlab/show/state">state</a></em> <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></mrow><annotation encoding="application/x-tex">\langle - \rangle</annotation></semantics></math> is the information that allows to assign to each <a class="existingWikiWord" href="/nlab/show/observable">observable</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> the <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>A</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle A\rangle</annotation></semantics></math> that this observable has when the <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> is assumed to be in that state.</p> <p>Often this is formalized in the <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a> where a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert</a> <a class="existingWikiWord" href="/nlab/show/space+of+states">space of states</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> is taken as primary, and the <a class="existingWikiWord" href="/nlab/show/observables">observables</a> are <a class="existingWikiWord" href="/nlab/show/representation">represented</a> as suitable <a class="existingWikiWord" href="/nlab/show/linear+operators">linear operators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\psi \in \mathcal{H}</annotation></semantics></math> a state (<a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a>) the <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</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> in this state is the <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>ψ</mi><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo><mi>ψ</mi><mo stretchy="false">⟩</mo><mo>≔</mo><mo stretchy="false">(</mo><mi>ψ</mi><mo>,</mo><mi>A</mi><mi>ψ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\langle \psi \vert A \vert \psi \rangle \coloneqq (\psi, A \psi)</annotation></semantics></math>. This defines a <a class="existingWikiWord" href="/nlab/show/linear+function">linear function</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>ψ</mi><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mi>ψ</mi><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>⟶</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \langle \psi \vert - \vert \psi \rangle \;\colon\; \mathcal{A} \longrightarrow \mathbb{C} </annotation></semantics></math></div> <p>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">\mathcal{A}</annotation></semantics></math>, satisfying some extra properties.</p> <p>Conversely, in the <a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a> one may take the “abstract” <a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> <a class="existingWikiWord" href="/nlab/show/algebra+of+quantum+observables">of observables</a> as primary (i.e. not necessarily manifested as an <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>), and declare that a state is any linear functional</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 stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>⟶</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \langle - \rangle \;\colon\; \mathcal{A} \longrightarrow \mathbb{C} </annotation></semantics></math></div> <p>which is <em>positive</em> in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msup><mi>A</mi> <mo>*</mo></msup><mi>A</mi><mo stretchy="false">⟩</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\langle A^\ast A\rangle \geq 0</annotation></semantics></math> and <em>normalized</em> in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mn>1</mn><mo stretchy="false">⟩</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\langle 1\rangle = 1</annotation></semantics></math>. Under suitable conditions a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert</a> <a class="existingWikiWord" href="/nlab/show/space+of+states">space of states</a> may be (re-)constructed from this <em>a posteriori</em> via the <em><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></em>.</p> <p>Traditionally this definition is considered for <a class="existingWikiWord" href="/nlab/show/algebras+of+observables">algebras of observables</a> which are <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> (as usually required for <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative quantum field theory</a>, see e.g. <a href="#Fredenhagen03">Fredenhagen (2003) Sec. 2</a>, <a href="#Landsman17">Landsman (2017) Def. 2.4</a>), but the definition makes sense generally for plain <a class="existingWikiWord" href="/nlab/show/star-algebras">star-algebras</a> (<a href="#Meyer95">Meyer (1995), I.1.1</a>), such as for instance for the <a class="existingWikiWord" href="/nlab/show/formal+power+series+algebras">formal power series algebras</a> that appear in <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> (e.g. <a href="#BordemannWaldmann96">Bordemann-Waldmann (1996) Def. 1</a>, <a href="#FredenhagenRejzner12">Fredenhagen & Rejzner (2012) Def. 2.4</a>, <a href="#KhavkineMoretti15">Khavkine & Moretti (2015) Def. 6</a>, <a href="#Duetsch18">Dütsch (2018) Def. 2.11</a>).</p> <p id="AsPositiveLinearFunctionals"> Of course, the notion of states depends only on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math>-algebra’s <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/partial+order">partially ordered</a> <a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex vector space</a> (see <a href="C-star-algebra#PartialOrderAndPositiveElements">here</a>) and hence makes sense in the generality of any <a class="existingWikiWord" href="/nlab/show/partial+order">partially ordered</a> <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, in which case one refers to them also as <em>positive linear functionals</em> or similar (e.g. <a href="#Murphy90">Murphy (1990) §3.3</a>).</p> <p>The perspective that states are normalized positive linear functionals on the algebra of observables is implicit in traditional <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, where it is encoded in the <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a> corresponding to a <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> or more generally a <a class="existingWikiWord" href="/nlab/show/quasi-free+quantum+state">quasi-free quantum state</a> (the <em><a class="existingWikiWord" href="/nlab/show/Hadamard+propagator">Hadamard propagator</a></em>). The perspective is made explicit in <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (see e.g. <a href="#Fredenhagen03">Fredenhagen 03, section 2</a>) and for <a class="existingWikiWord" href="/nlab/show/star-algebras">star-algebras</a> of observables that are not necessarily <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> in <a class="existingWikiWord" href="/nlab/show/perturbative+algebraic+quantum+field+theory">perturbative algebraic quantum field theory</a> (e.g. <a href="#BordemannWaldmann96">Bordemann-Waldmann 96</a>, <a href="#FredenhagenRejzner12">Fredenhagen-Rejzner 12, def. 2.4</a>, <a href="#KhavkineMoretti15">Khavkine-Moretti 15, def. 6</a>, <a href="#Duetsch18">Dütsch 18, def. 2.11</a>).</p> <h2 id="definition">Definition</h2> <h3 id="DefinitionForUnitalStarAlgebras">For unital star algebras</h3> <div class="num_defn" id="StateOnAStarAlgebra"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/state">state</a> on a <a class="existingWikiWord" href="/nlab/show/unital+algebra">unital</a> <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>)</strong></p> <p>Let <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">\mathcal{A}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/unital">unital</a> <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</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 <em>state</em> 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">\mathcal{A}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/linear+function">linear 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><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>⟶</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \rho \;\colon\; \mathcal{A} \longrightarrow \mathbb{C} </annotation></semantics></math></div> <p>such that</p> <ol> <li id="PositivityCondition"> <p>(positivity) for all <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 \mathcal{A}</annotation></semantics></math> the value of <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 the product <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></mrow><annotation encoding="application/x-tex">A^\ast A</annotation></semantics></math> is</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/real+part">real</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>*</mo></msup><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℝ</mi><mo>↪</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\,\rho(A^\ast A) \in \mathbb{R} \hookrightarrow \mathbb{C}</annotation></semantics></math></p> </li> <li> <p>as such <a class="existingWikiWord" href="/nlab/show/non-negative+real+number">non-negative</a>:</p> </li> </ol> <div class="maruku-equation" id="eq:Positivity"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>*</mo></msup><mi>A</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \rho(A^\ast A) \geq 0 </annotation></semantics></math></div></li> <li> <p>(normalization)</p> <div class="maruku-equation" id="eq:Normalization"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> \rho(\mathbf{1}) = 1 </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathbf{1} \in \mathcal{A}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/unit">unit</a> in the algebra.</p> </li> </ol> </div> <p>(e.g. <a href="#Meyer95">Meyer 95, I.1.1</a>, <a href="#BordemannWaldmann96">Bordemann-Waldmann 96</a>, <a href="#FredenhagenRejzner12">Fredenhagen-Rejzner 12, def. 2.4</a>, <a href="#KhavkineMoretti15">Khavkine-Moretti 15, def. 6</a>, <a href="#Landsman17">Landsman 2017, Def. 2.4</a>)</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/probability+theory">probability theoretic</a> interpretation of <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state</a> is also called a <em><a class="existingWikiWord" href="/nlab/show/quantum+probability+space">quantum probability space</a></em>, at least 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">\mathcal{A}</annotation></semantics></math> is in fact a <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a>.</p> </div> <div class="num_remark" id="StatesFormAConvexSet"> <h6 id="remark_2">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">states</a> form a <a class="existingWikiWord" href="/nlab/show/convex+set">convex set</a>)</strong></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">\mathcal{A}</annotation></semantics></math> a unital <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a>, the <a class="existingWikiWord" href="/nlab/show/set">set</a> of 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">\mathcal{A}</annotation></semantics></math> according to def. <a class="maruku-ref" href="#StateOnAStarAlgebra"></a> is naturally a <a class="existingWikiWord" href="/nlab/show/convex+set">convex set</a>: For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\rho_1, \rho_2 \colon \mathcal{A} \to \mathbb{C}</annotation></semantics></math> two states then for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">p \in [0,1] \subset \mathbb{R}</annotation></semantics></math> also the <a class="existingWikiWord" href="/nlab/show/linear+combination">linear combination</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒜</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>p</mi><msub><mi>ρ</mi> <mn>1</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>ℂ</mi></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>p</mi><msub><mi>ρ</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mi>ρ</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{A} &\overset{p \rho_1 + (1-p) \rho_2}{\longrightarrow}& \mathbb{C} \\ A &\mapsto& p \rho_1(A) + (1-p) \rho_2(A) } </annotation></semantics></math></div> <p>is a state.</p> </div> <div class="num_defn" id="PureStateOnAStarAlgebra"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a>)</strong></p> <p>A state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\rho \colon \mathcal{A} \to \mathbb{C}</annotation></semantics></math> on a unital <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> (def. <a class="maruku-ref" href="#StateOnAStarAlgebra"></a>) is called a <em><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></em> if it is extremal in the <a class="existingWikiWord" href="/nlab/show/convex+set">convex set</a> of all states (remark <a class="maruku-ref" href="#StatesFormAConvexSet"></a>) in that an identification</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>p</mi><msub><mi>ρ</mi> <mn>1</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mi>ρ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> \rho = p \rho_1 + (1-p) \rho_2 </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \in (0,1)</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>ρ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\rho_1 = \rho_2</annotation></semantics></math> (hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">= \rho</annotation></semantics></math>).</p> </div> <h3 id="for_algebras">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</h3> <p>The following discusses states specifically on <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a>.</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>An element <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 an (abstract) <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 called <strong><a class="existingWikiWord" href="/nlab/show/positive+operator">positive</a></strong> if it is <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint</a> and its <a class="existingWikiWord" href="/nlab/show/spectrum+of+an+operator">spectrum</a> is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[0, \infinity)</annotation></semantics></math>. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">A \ge 0</annotation></semantics></math> and say that the set of all <a class="existingWikiWord" href="/nlab/show/positive+operators">positive operators</a> is the positive cone (of a 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^*</annotation></semantics></math>-algebra).</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>This definition is motivated by the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> situation, where an operator <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 \in \mathcal{B} (\mathcal{H})</annotation></semantics></math> is called <a class="existingWikiWord" href="/nlab/show/positive+operator">positive</a> if for every vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">x \in \mathcal{H}</annotation></semantics></math> the inequality <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>A</mi><mi>x</mi><mo stretchy="false">⟩</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \langle x, A x \rangle \ge 0</annotation></semantics></math> holds. If the abstract <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 of the definition above is represented on a Hilbert space, then we see that by <a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a> we can define a self adjoint operator <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> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>≔</mo><mi>f</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \coloneqq f(A)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msup><mi>t</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f(t) := t^{1/2}</annotation></semantics></math> and get <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>A</mi><mi>x</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>B</mi><mi>x</mi><mo>,</mo><mi>B</mi><mi>x</mi><mo stretchy="false">⟩</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \langle x, A x \rangle = \langle B x, B x \rangle \ge 0</annotation></semantics></math>. This shows that the positive elements of the abstract algebra, if represented on a Hilbert space, become positive operators as defined here in the Hilbert space setting.</p> </div> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/linear+functional">linear functional</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 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^*</annotation></semantics></math>-algebra is <strong>positive</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">A \ge 0</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\rho(A) \ge 0</annotation></semantics></math>.</p> <p>A <strong>state</strong> of a unital <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 <a class="existingWikiWord" href="/nlab/show/linear+functional">linear functional</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> such that <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> is positive and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\rho(1) = 1</annotation></semantics></math>.</p> </div> <p>Though the mathematical notion of state is already close to what physicists have in mind, they usually restrict the set of states further and consider normal states only. We let <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">\mathcal{R}</annotation></semantics></math> be 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^*</annotation></semantics></math>-algebra and <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> an representation 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">\mathcal{R}</annotation></semantics></math> on a Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math>.</p> <div class="num_theorem"> <h6 id="definitiontheorem">Definition/Theorem</h6> <p>A <strong>normal state</strong> <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> is a state that satisfies one of the following equivalent conditions:</p> <ul> <li> <p><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> is weak-operator continuous on the unit ball of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>ℛ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi(\mathcal{R})</annotation></semantics></math>.</p> </li> <li> <p><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> is strong-operator continuous on the unit ball <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>ℛ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi(\mathcal{R})</annotation></semantics></math>.</p> </li> <li> <p><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> is ultra-weak continuous.</p> </li> <li> <p>There is an operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/trace+class">trace class</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>tr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">tr(A) = 1</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>tr</mi><mo stretchy="false">(</mo><mi>A</mi><mi>π</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(\pi(R)) = tr(A \pi(R))</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">R \in \mathcal{R}</annotation></semantics></math>.</p> </li> </ul> </div> <p>This appears as <a href="#KadisonRingrose">KadisonRingrose, def. 7.1.11, theorem 7.1.12</a></p> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>This list is not complete, there are more commonly used equivalent characterizations of normal states.</p> <p>The last one is most frequently used by physicists, in that context the operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is also called a <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> or density operator.</p> </div> <p>Sometimes the observables of a system are described by an abstract <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, in this case an important notion is the folium:</p> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p>The <strong>folium</strong> of a representation <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> of 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^*</annotation></semantics></math>-algebra <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">\mathcal{R}</annotation></semantics></math> on a Hilbert space is the set of normal states of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>ℛ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi(\mathcal{R})</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>A 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> of a representation is called a <strong>vector state</strong> if there is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">x \in \mathcal{H}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>π</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\rho(\pi(R)) = \langle \pi(R)x, x \rangle</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">R \in \mathcal{R}</annotation></semantics></math>.</p> </div> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Normal states are vector states if <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">\mathcal{R}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> with a separating vector. More precisely: Let <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">\mathcal{R}</annotation></semantics></math> be a von Neumman algebra acting on a Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math>, let <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> be a normal state 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">\mathcal{R}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">x \in \mathcal{H}</annotation></semantics></math> be a separating vector for <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">\mathcal{R}</annotation></semantics></math>, then there is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">y \in \mathcal{H}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>Ry</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\rho(R) = \langle Ry, y \rangle</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">R \in \mathcal{R}</annotation></semantics></math>.</p> </div> <p>This appears as <a href="#KadisonRingrose">KadisonRingrose, theorem 7.2.3</a>.</p> <p>The set of states of 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^*</annotation></semantics></math>-algebra is sometimes called the <strong><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">state space</a></strong>.</p> <p>The state space is non-empty (define a state on the subalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">\mathbb{C} 1</annotation></semantics></math> and extend it to the whole <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 via the <a class="existingWikiWord" href="/nlab/show/Hahn-Banach+theorem">Hahn-Banach theorem</a>), convex and weak<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo></mo><mo>*</mo></msup></mrow><annotation encoding="application/x-tex">^*</annotation></semantics></math>-compact, so it has extreme points. By the <span class="newWikiWord">Krein-Milman theorem<a href="/nlab/new/Krein-Milman+theorem">?</a></span> (see Wikipedia: <a href="http://en.wikipedia.org/wiki/Krein%E2%80%93Milman_theorem">Krein-Milman theorem</a>) it is the weak<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo></mo><mo>*</mo></msup></mrow><annotation encoding="application/x-tex">^*</annotation></semantics></math>-closure of its extreme points.</p> <div class="num_defn"> <h6 id="definition_8">Definition</h6> <p>A <strong>pure state</strong> is a state that is an extreme point of the state space.</p> </div> <p>The term “pure” originates from the notion of <a class="existingWikiWord" href="/nlab/show/entanglement">entanglement</a>, a pure state is not a mixture of two distinct other states.</p> <h2 id="Examples">Examples</h2> <p> <div class='num_remark' id='ClassicalProbabilityMeasureAsStateOnMeasurableFunctions'> <h6>Example</h6> <p><strong>(classical <a class="existingWikiWord" href="/nlab/show/probability+measure">probability measure</a> as state on <a class="existingWikiWord" href="/nlab/show/measurable+functions">measurable functions</a>)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a> equipped with a compatible structure of a classical <a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a>, hence a <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a> which normalized total measure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>Ω</mi></msub><mi>d</mi><mi>μ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\int_\Omega d\mu = 1</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>≔</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{A} \coloneqq C_0(\Omega)</annotation></semantics></math> be the algebra of <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> with values in the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> and <a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanishing at infinity</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a> by pointwise <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a>. Then forming the <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> defines a <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state</a> (def. <a class="maruku-ref" href="#StateOnAStarAlgebra"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Ω</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mi>μ</mi></msub></mrow></mover></mtd> <mtd><mi>ℂ</mi></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mo>∫</mo> <mi>Ω</mi></msub><mi>A</mi><mspace width="thinmathspace"></mspace><mi>d</mi><mi>μ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C_0(\Omega) &\overset{\langle - \rangle_\mu}{\longrightarrow}& \mathbb{C} \\ A &\mapsto& \int_\Omega A \, d\mu } </annotation></semantics></math></div> <p></p> </div> </p> <p>(e.g. <a href="#Landsman17">Landsman 2017, p. 16-17</a>)</p> <div class="num_example" id="ElementsOfHilbertSpaceAsPureStates"> <h6 id="example">Example</h6> <p><strong>(elements of a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> as <a class="existingWikiWord" href="/nlab/show/pure+states">pure states</a> on <a class="existingWikiWord" href="/nlab/show/bounded+operators">bounded operators</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex</a> <a class="existingWikiWord" href="/nlab/show/separable+Hilbert+space">separable</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> with <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,-\rangle</annotation></semantics></math> and let <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">\mathcal{A} \coloneqq \mathcal{B}(\mathcal{H})</annotation></semantics></math> be the algebra of <a class="existingWikiWord" href="/nlab/show/bounded+operators">bounded operators</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a> under forming <a class="existingWikiWord" href="/nlab/show/adjoint+operators">adjoint operators</a>. Then for every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\psi \in \mathcal{H}</annotation></semantics></math> of unit <a class="existingWikiWord" href="/nlab/show/norm">norm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>ψ</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">⟩</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\langle \psi,\psi\rangle = 1</annotation></semantics></math> there is the <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state</a> (def. <a class="maruku-ref" href="#StateOnAStarAlgebra"></a>) given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℬ</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mi>ψ</mi></msub></mrow></mover></mtd> <mtd><mi>ℂ</mi></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">⟨</mo><mi>ψ</mi><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mi>ψ</mi><mo stretchy="false">⟩</mo></mtd> <mtd><mo>≔</mo></mtd> <mtd><mo stretchy="false">⟨</mo><mi>ψ</mi><mo>,</mo><mi>A</mi><mi>ψ</mi><mo stretchy="false">⟩</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{B}(\mathcal{H}) &\overset{\langle -\rangle_\psi}{\longrightarrow}& \mathbb{C} \\ A &\mapsto& \langle \psi \vert\, A \, \vert \psi \rangle &\coloneqq& \langle \psi, A \psi \rangle } </annotation></semantics></math></div> <p>These are <a class="existingWikiWord" href="/nlab/show/pure+states">pure states</a> (def. <a class="maruku-ref" href="#PureStateOnAStarAlgebra"></a>).</p> <p>More general states in this case are given by <a class="existingWikiWord" href="/nlab/show/density+matrices">density matrices</a>.</p> </div> <p> <div class='num_remark' id='StatesOnGroupAlgebrasAreUnitaryRepresentations'> <h6>Example</h6> <p><strong>(states on group algebras are unitary representations)</strong> <br /> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> 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>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}[G]</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a> regarded as a complex <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> under the combined operation of <a class="existingWikiWord" href="/nlab/show/inverse+element">inversion</a> of group elements and <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a> of the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>.</p> <p>Then each state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo>⟶</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\rho \,\colon\,\mathbb{C}[G] \longrightarrow \mathbb{C}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#StateOnAStarAlgebra"></a>) arises from a <a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mo>⟶</mo><mi>U</mi><mo stretchy="false">(</mo><mi class="mathscript">ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widehat{(\text{-})} \,\colon\, \longrightarrow U(\mathscr{H})</annotation></semantics></math> on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi class="mathscript">ℋ</mi><mo>,</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathscr{H}, \langle -\vert-\rangle)</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>∈</mo><mi class="mathscript">ℋ</mi></mrow><annotation encoding="application/x-tex">\psi \in \mathscr{H}</annotation></semantics></math> as</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 stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">⟨</mo><mi>ψ</mi><mo maxsize="1.2em" minsize="1.2em">|</mo><mover><mi>g</mi><mo>^</mo></mover><mo>⋅</mo><mi>ψ</mi><mo maxsize="1.2em" minsize="1.2em">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho(g) \;=\; \big\langle \psi \big\vert \widehat{g} \cdot \psi \big\rangle \,. </annotation></semantics></math></div> <p>(With suitable adjustments, this statement generalizes from discrete to <a class="existingWikiWord" href="/nlab/show/topological+groups">topological groups</a>.)</p> </div> </p> <p>This is due to <a href="#GelfandRaikov43">Gelfand & Raikov (1943) (2)</a>, <a href="#Naimark56">Naimark (1956) §30 Thm. 1</a> (reviewed at <a href="https://encyclopediaofmath.org/wiki/Positive-definite_function">eom</a>) further generalization in <a href="#Saworotnow70">Saworotnow (1970)</a>, <a href="Saworotnow72">(1972)</a>, textbook account in <a href="#Dixmier77">Dixmier (1977) Thm. 13.4.5 (ii)</a>.</p> <h2 id="properties">Properties</h2> <h3 id="closure_properties">Closure properties</h3> <p> <div class='num_remark' id='Mixtures'> <h6>Example</h6> <p><strong>(mixtures, convex combinations)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><msub><mi>ℕ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}_+</annotation></semantics></math>, let</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>ρ</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℂ</mi><msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">\big( \rho_i \colon \mathcal{A} \to \mathbb{C} \big)_{i = 1}^k</annotation></semantics></math></p> <p>be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/tuple">tuple</a> of states;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">\big( p_i \in \mathbb{R}_{\geq 0} \big)_{i = 1}^k</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><msub><mi>p</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>1</mn></mrow><annotation encoding="application/x-tex">\underoverset{i = 1}{k}{\sum} p_i \;=\; 1</annotation></semantics></math></p> <p>be a <a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a> on the <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>k</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{1, \cdots, k\}</annotation></semantics></math></p> </li> </ul> <p>then the <a class="existingWikiWord" href="/nlab/show/convex+combination">convex combination</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><msub><mi>p</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>ρ</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msup><mi>𝒜</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> \underoverset{i = 1}{k}{\sum} p_i \cdot \rho_i \;\;\; \in \; \mathcal{A}^\ast </annotation></semantics></math></div> <p>is another state on the star-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">\mathcal{A}</annotation></semantics></math>.</p> <p></p> </div> </p> <p> <div class='num_prop' id='OperatorStateCorrespondence'> <h6>Proposition</h6> <p><strong>(operator-state correspondence)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\rho \;\colon\; \mathcal{A} \to \mathbb{C}</annotation></semantics></math> a state, with a non-null observable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">O \in \mathcal{A}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>O</mi> <mo>*</mo></msup><mi>O</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\rho(O^\ast O) \neq 0</annotation></semantics></math>, then also</p> <div class="maruku-equation" id="eq:OperatorImageOfState"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>O</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>O</mi> <mo>*</mo></msup><mi>O</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>⋅</mo><mi>ρ</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>O</mi> <mo>*</mo></msup><mo>⋅</mo><mi>A</mi><mo>⋅</mo><mi>O</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \rho_O \;\colon\; A \;\mapsto\; \tfrac{1}{ \rho(O^\ast O) } \cdot \rho\big( O^\ast \cdot A \cdot O \big) </annotation></semantics></math></div> <p>is a state.</p> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p></p> <p>To check positivity <a class="maruku-eqref" href="#eq:Positivity">(1)</a>, we compute for any <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 \mathcal{A}</annotation></semantics></math> as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>ρ</mi> <mi>O</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>A</mi> <mo>*</mo></msup><mo>⋅</mo><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>O</mi> <mo>*</mo></msup><mi>O</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>⋅</mo><mi>ρ</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>O</mi> <mo>*</mo></msup><mo>⋅</mo><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>*</mo></msup><mo>⋅</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>O</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>O</mi> <mo>*</mo></msup><mi>O</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>⋅</mo><mi>ρ</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⋅</mo><mi>O</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>⋅</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⋅</mo><mi>O</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≥</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \rho_O \big( A^\ast \cdot A \big) & = \tfrac{1}{\rho(O^\ast O)} \cdot \rho \big( O^\ast \cdot ( A^\ast \cdot A ) \cdot O \big) \\ & = \tfrac{1}{\rho(O^\ast O)} \cdot \rho \big( ( A \cdot O )^\ast \cdot ( A \cdot O ) \big) \\ & \geq 0 \,, \end{aligned} </annotation></semantics></math></div> <p>where the first step is the definition <a class="maruku-eqref" href="#eq:OperatorImageOfState">(3)</a> the second step uses the <a class="existingWikiWord" href="/nlab/show/anti-homomorphisms">anti-homomorphisms</a>-property of the star-involution, and the last step follows by the assumed positivivity of <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>.</p> <p>To check normalization <a class="maruku-eqref" href="#eq:Normalization">(2)</a>, we observe that:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>ρ</mi> <mi>O</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>O</mi> <mo>*</mo></msup><mi>O</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>⋅</mo><mi>ρ</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>O</mi> <mo>*</mo></msup><mo>⋅</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo>⋅</mo><mi>O</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>O</mi> <mo>*</mo></msup><mi>O</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>⋅</mo><mi>ρ</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>O</mi> <mo>*</mo></msup><mo>⋅</mo><mi>O</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>1</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \rho_O(\mathbf{1}) & = \tfrac{1}{\rho(O^\ast O)} \cdot \rho \big( O^\ast \cdot \mathbf{1} \cdot O \big) \\ & = \tfrac{1}{\rho(O^\ast O)} \cdot \rho \big( O^\ast \cdot O \big) \\ & = 1 \,. \end{aligned} </annotation></semantics></math></div> <p></p> </div> </p> <h3 id="MultiplicativeStates">Multiplicative states</h3> <p>Recall that a state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>𝒜</mi><mover><mo>→</mo><mspace width="thickmathspace"></mspace></mover><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\rho \,\colon\, \mathcal{A} \xrightarrow{\;} \mathbb{C}</annotation></semantics></math> is generally only required to be a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> on the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> of the 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">\mathcal{A}</annotation></semantics></math> (subject only to the positivity constraint), not necessarily an <a class="existingWikiWord" href="/nlab/show/algebra+homomorphism">algebra homomorphism</a>.</p> <p>Those states which do respect the algebra product, in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>A</mi><mo>⋅</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>ρ</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(A \cdot B) = \rho(A)\, \rho(B)</annotation></semantics></math> 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><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">A,B \in \mathcal{A}</annotation></semantics></math>, are called <em>multiplicative</em>.</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>A multiplicative state is necessarily a <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a>.</p> </div> (e.g. <a href="#Zhu93">Zhu 1993, Ex. 13.3</a>; <a href="#Warner10">Warner 2010, Lem. 7.20</a>)</p> <p> <div class='num_prop' id='OnCommutativeAlgebraPureStatesAreTheMultiplicativeStates'> <h6>Proposition</h6> <p>If the star-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">\mathcal{A}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative</a>, then the <a class="existingWikiWord" href="/nlab/show/pure+states">pure states</a> on it are precisely the multiplicative states.</p> </div> (e.g. <a href="#Zhu93">Zhu 1993, Ex. 13.4</a>; <a href="#Warner10">Warner 2010, Lem. 7.21</a>) <div class='proof'> <h6>Proof</h6> <p>The idea is that for commutative <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">\mathcal{A}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> applies and shows that a multiplicative state is equivalently the operation of point-evaluation on the <a class="existingWikiWord" href="/nlab/show/Gelfand+spectrum">Gelfand spectrum</a>.</p> </div> </p> <h3 id="fells_theorem">Fell’s theorem</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></em>.</p> <h3 id="gleasons_theorem">Gleason’s theorem</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</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> <h3 id="exposition">Exposition</h3> <ul> <li id="Meyer95"> <p><a class="existingWikiWord" href="/nlab/show/Paul-Andr%C3%A9+Meyer">Paul-André Meyer</a>, Section I.1.1 in: <em>Quantum Probability for Probabilists</em>, Lecture Notes in Mathematics <strong>1538</strong>, Springer (1995) [<a href="https://link.springer.com/book/10.1007/BFb0084701">doi:10.1007/BFb0084701</a>]</p> </li> <li id="Gleason09"> <p><a class="existingWikiWord" href="/nlab/show/Jonathan+Gleason">Jonathan Gleason</a>, <em>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>-algebraic formalism of quantum mechanics</em> (2009) [<a class="existingWikiWord" href="/nlab/files/Gleason09.pdf" title="pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GleasonAlgebraic.pdf" title="pdf">pdf</a>]</p> </li> <li id="Gleason11"> <p><a class="existingWikiWord" href="/nlab/show/Jonathan+Gleason">Jonathan Gleason</a>, <em>From Classical to Quantum: The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">F^\ast</annotation></semantics></math>-Algebraic Approach</em>, contribution to <em><a href="http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2011.html">VIGRE REU 2011</a></em>, Chicago (2011) [<a href="https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GleasonFAlgebraic.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>Within monographs:</p> <ul> <li id="BratteliRobinson79"> <p><a class="existingWikiWord" href="/nlab/show/Ola+Bratteli">Ola Bratteli</a>, <a class="existingWikiWord" href="/nlab/show/Derek+W.+Robinson">Derek W. Robinson</a>, §2.3 in: <em>Operator Algebras and Quantum Statistical Mechanics</em> – vol 1: <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^\ast</annotation></semantics></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^\ast</annotation></semantics></math>-Algebras. Symmetry Groups. Decomposition of States.</em>, Springer (1979, 1987, 2002) [<a href="https://doi.org/10.1007/978-3-662-02520-8">doi:10.1007/978-3-662-02520-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nikolay+Bogolyubov">Nikolay Bogolyubov</a>, A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, p. 234 in: <em>Algebra of Observables and State Space</em> [<a href="https://doi.org/10.1007/978-94-009-0491-0_6">doi:10.1007/978-94-009-0491-0_6</a>], Chapter in: <em>General principles of quantum field theory</em>, Mathematical Physics and Applied Mathematics <strong>10</strong>, Kluwer (1990) [<a href="https://doi.org/10.1007/978-94-009-0491-0">doi:10.1007/978-94-009-0491-0</a>]</p> </li> <li id="Zhu93"> <p><a class="existingWikiWord" href="/nlab/show/Kehe+Zhu">Kehe Zhu</a>, Section 13 of: <em>An Introduction to Operator Algebras</em>, CRC Press (1993) [<a href="https://www.routledge.com/An-Introduction-to-Operator-Algebras/Zhu/p/book/9780849378751">ISBN:9780849378751</a>]</p> </li> <li id="Warner10"> <p><a class="existingWikiWord" href="/nlab/show/Garth+Warner">Garth Warner</a>, <em>States</em>, §7 in: <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^\ast</annotation></semantics></math>-Algebras</em>, EPrint Collection, University of Washington (2010) [<a href="http://hdl.handle.net/1773/16302">hdl:1773/16302</a>, <a href="https://sites.math.washington.edu//~warner/C-star.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Waner-CStarAlgebras.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>Discussion under the name “positive linear functionals”:</p> <ul> <li id="Steward76"> <p><a class="existingWikiWord" href="/nlab/show/James+D.+Steward">James D. Steward</a>, <em>Positive definite functions and generalizations, an historical survey</em>, The Rocky Mountain Journal of Mathematics <strong>6</strong> 3 (1976) 409-434 [<a href="https://www.jstor.org/stable/44236118">jstor:44236118</a>]</p> </li> <li id="Dixmier77"> <p><a class="existingWikiWord" href="/nlab/show/Jacques+Dixmier">Jacques Dixmier</a>, Chapter 2 of: <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^\ast</annotation></semantics></math>-algebras</em>, North Holland (1977) [ch2:<a class="existingWikiWord" href="/nlab/files/Dixmier-CStarAlgebras-PosForms.pdf" title="pdf">pdf</a>, ch13:<a class="existingWikiWord" href="/nlab/files/Dixmier-CStarAlgebras-UnitaryReps.pdf" title="pdf">pdf</a>]</p> </li> <li id="Murphy90"> <p><a class="existingWikiWord" href="/nlab/show/Gerard+Murphy">Gerard Murphy</a>, §3.3 in: <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^\ast</annotation></semantics></math>-algebras and Operator Theory</em>, Academic Press (1990) [<a href="https://doi.org/10.1016/C2009-0-22289-6">doi:10.1016/C2009-0-22289-6</a>]</p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Positive_linear_functional">Positive linear functional</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eom">eom</a>, <em><a href="https://encyclopediaofmath.org/wiki/Positive-definite_function">Positive-definite function</a></em></p> </li> </ul> <p>With an eye towards <a class="existingWikiWord" href="/nlab/show/density+matrices">density matrices</a> and their <a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>:</p> <ul> <li>Paolo Facchi, Giovanni Gramegna, Arturo Konderak, around (6) in: <em>Entropy of quantum states</em> (<a href="https://arxiv.org/abs/2104.12611">arXiv:2104.12611</a>, <a href="https://inspirehep.net/literature/1860877">spire:1860877</a>)</li> </ul> <p>With an eye towards <a class="existingWikiWord" href="/nlab/show/Gelfand-Tsetlin+algebra">Gelfand-Tsetlin algebras</a> and in the generality of <a class="existingWikiWord" href="/nlab/show/conditional+expectation+values">conditional expectation values</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anatoly+Vershik">Anatoly Vershik</a>, Def. 1 in: <em>Gel’fand-Tsetlin algebras, expectations, inverse limits, Fourier analysis</em>, pp. 619 in: <a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+S.+Retakh">Vladimir S. Retakh</a>, <a class="existingWikiWord" href="/nlab/show/Isadore+Singer">Isadore Singer</a> (eds.) <em>The Unity of Mathematics</em> In Honor of the Ninetieth Birthday of <a class="existingWikiWord" href="/nlab/show/I.+M.+Gelfand">I. M. Gelfand</a>, (<a href="https://arxiv.org/abs/math/0503140">arXiv:math/0503140</a>, <a href="https://www.springer.com/gp/book/9780817640767">ISBN:978-0-8176-4467-3</a>)</li> </ul> <h3 id="original_articles">Original articles</h3> <ul> <li id="HaagKastler64"> <p><a class="existingWikiWord" href="/nlab/show/Rudolf+Haag">Rudolf Haag</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kastler">Daniel Kastler</a>, equation (3) in: <em>An algebraic approach to quantum field theory</em>, Journal of Mathematical Physics, <strong>5</strong> (1964) 848-861 [<a href="https://doi.org/10.1063/1.1704187">doi:10.1063/1.1704187</a>, <a href="https://inspirehep.net/literature/9124">spire:9124</a>]</p> </li> <li id="KadisonRingrose"> <p>Richard Kadison, John Ringrose, <em>Fundamentals of the theory of operator algebras</em>, AMS (1991)</p> </li> <li id="BordemannWaldmann96"> <p><a class="existingWikiWord" href="/nlab/show/Martin+Bordemann">Martin Bordemann</a>, <a class="existingWikiWord" href="/nlab/show/Stefan+Waldmann">Stefan Waldmann</a>, <em>Formal GNS Construction and States in Deformation Quantization</em>, Commun. Math. Phys. (1998) 195: 549. (<a href="https://arxiv.org/abs/q-alg/9607019">arXiv:q-alg/9607019</a>, <a href="https://doi.org/10.1007/s002200050402">doi:10.1007/s002200050402</a>)</p> </li> <li id="Fredenhagen03"> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, section 2 of <em>Algebraische Quantenfeldtheorie</em>, lecture notes, 2003 (<a class="existingWikiWord" href="/nlab/files/FredenhagenAQFT2003.pdf" title="pdf">pdf</a>)</p> </li> <li id="HalvorsonMueger06"> <p><a class="existingWikiWord" href="/nlab/show/Hans+Halvorson">Hans Halvorson</a>, <a class="existingWikiWord" href="/nlab/show/Michael+M%C3%BCger">Michael Müger</a>, def. 1.11 in <em>Algebraic Quantum Field Theory</em> (<a href="https://arxiv.org/abs/math-ph/0602036">arXiv:math-ph/0602036</a>)</p> </li> <li id="FredenhagenRejzner12"> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, definition 2.4 in <em>Perturbative algebraic quantum field theory</em>, In <em>Mathematical Aspects of Quantum Field Theories</em>, Springer 2016 (<a href="https://arxiv.org/abs/1208.1428">arXiv:1208.1428</a>)</p> </li> <li id="KhavkineMoretti15"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Khavkine">Igor Khavkine</a>, <a class="existingWikiWord" href="/nlab/show/Valter+Moretti">Valter Moretti</a>, <em>Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction</em>, Chapter 5 in <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a> et al. (eds.) <em>Advances in Algebraic Quantum Field Theory</em>, Springer, 2015 (<a href="https://arxiv.org/abs/1412.5945">arXiv:1412.5945</a>)</p> </li> <li id="Rejzner16"> <p><a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, section 2.1.2 of <em>Perturbative Algebraic Quantum Field Theory</em>, Mathematical Physics Studies, Springer 2016 (<a href="https://link.springer.com/book/10.1007%2F978-3-319-25901-7">web</a>)</p> </li> <li id="FredenhagenLindner13"> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Falk Lindner, <em>Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics</em>, Communications in Mathematical Physics Volume 332, Issue 3, pp 895-932, 2014-12-01 (<a href="https://arxiv.org/abs/1306.6519">arXiv:1306.6519</a>)</p> </li> <li id="Landsman17"> <p><a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, around def. 2.4 in: <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>]</p> </li> <li id="Duetsch18"> <p><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, section 2.5 of <em><a class="existingWikiWord" href="/nlab/show/From+classical+field+theory+to+perturbative+quantum+field+theory">From classical field theory to perturbative quantum field theory</a></em>, 2018</p> </li> <li> <p>Nicolò Drago, <a class="existingWikiWord" href="/nlab/show/Valter+Moretti">Valter Moretti</a>, <em>The notion of observable and the moment problem for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math>-algebras and their GNS representations</em> (<a href="https://arxiv.org/abs/1903.07496">doi:1903.07496</a>, <a href="http://inspirehep.net/record/1725528">spire:1725528</a>)</p> </li> </ul> <p>For more references see at <em><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></em>.</p> <h3 id="the_case_of_group_algebras">The case of group algebras</h3> <p>The characterization of states on <a class="existingWikiWord" href="/nlab/show/group+algebras">group algebras</a>:</p> <ul> <li id="GelfandRaikov43"> <p>И. М. Гельфанд, Д. А. Райков, <em>Неприводимые унитарные представления локально бикомпактных групп</em>, Матем. сб., 13(55):2–3 (1943) 301–316 [<a href="https://www.mathnet.ru/links/728d630fdf2220b1ea8bdb2a5f6a5488/sm6181.pdf">mathnet pdf</a>]</p> <p><br /></p> <p><a class="existingWikiWord" href="/nlab/show/Israel+Gelfand">Israel Gelfand</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Raikov">Dmitri Raikov</a>, <em>Irreducible unitary representations of locally bicompact groups</em>, Recueil Mathématique. N.S., 13(55) 2–3 (1943) 301–316 [<a href="https://www.mathnet.ru/eng/sm6181">mathnet:eng/sm6181</a>]</p> </li> <li id="Naimark56"> <p><a class="existingWikiWord" href="/nlab/show/Mark+A.+Naimark">Mark A. Naimark</a>, <em>Normed Rings</em>, Moscow (1956)</p> </li> <li id="Saworotnow70"> <p>Parfeny P. Saworotnow, <em>Representation of a topological group on a Hilbert module</em>, Duke Math. J. <strong>37</strong> 1 (1970) 145-150 [<a href="https://projecteuclid.org/journals/duke-mathematical-journal/volume-37/issue-1/Representation-of-a-topological-group-on-a-Hilbert-module/10.1215/S0012-7094-70-03720-8.short">doi:10.1215/S0012-7094-70-03720-8</a>]</p> </li> <li id="Saworotnow72"> <p>Parfeny P. Saworotnow, <em>Generalized Positive Linear Functionals on a Banach Algebra with an Involution</em>, Proceedings of the AMS, <strong>31</strong> 1 (1972) 299-304 [<a href="https://www.jstor.org/stable/2038564">jstor:2038564</a>]</p> </li> <li> <p><a href="#Steward76">Steward (1976) §4</a></p> </li> <li> <p><a href="#Dixmier77">Dixmier (1977) Thm. 13.4.6</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eom">eom</a>, <em><a href="https://encyclopediaofmath.org/wiki/Positive-definite_function">Positive-definite function</a></em></p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Positive-definite_function_on_a_group">Positive-definite function on a group</a></em></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 27, 2024 at 14:33:25. 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