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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #7 to #8: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='group_theory'>Group Theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/group+theory'>group theory</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/group'>group</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/group+object'>group object</a>, <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>group object in an (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a>, <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/super+abelian+group'>super abelian group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/action'>group action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/progroup'>progroup</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/homogeneous+space'>homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unitary+group'>unitary group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+unitary+group'>special unitary group</a>. <a class='existingWikiWord' href='/nlab/show/diff/projective+unitary+group'>projective unitary group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orthogonal+group'>orthogonal group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+orthogonal+group'>special orthogonal group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symplectic+group'>symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+group'>finite group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+group'>symmetric group</a>, <a class='existingWikiWord' href='/nlab/show/diff/cyclic+group'>cyclic group</a>, <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classification+of+finite+simple+groups'>classification of finite simple groups</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sporadic+finite+simple+group'>sporadic finite simple groups</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Monster+group'>Monster group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mathieu+group'>Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+group'>algebraic group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+variety'>abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+topological+group'>compact topological group</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+group'>locally compact topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/maximal+compact+subgroup'>maximal compact subgroup</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/string+group'>string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+Lie+group'>compact Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kac-Moody+group'>Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/supergroup'>super Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/super+Euclidean+group'>super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/crossed+module'>crossed module</a>, <a class='existingWikiWord' href='/nlab/show/diff/strict+2-group'>strict 2-group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-group'>n-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+group'>simplicial group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/crossed+complex'>crossed complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/k-tuply+groupal+n-groupoid'>k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle+n-group'>circle n-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+2-group'>string 2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/fivebrane+6-group'>fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+cohomology'>group cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+extension'>group extension</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group+extension'>∞-group extension</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/quantum+group'>quantum group</a></li> </ul> </div> <h4 id='operator_algebra'>Operator algebra</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/AQFT'>algebraic quantum field theory</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/perturbative+algebraic+quantum+field+theory'>perturbative</a>, <a class='existingWikiWord' href='/nlab/show/diff/AQFT+on+curved+spacetimes'>on curved spacetimes</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopical+algebraic+quantum+field+theory'>homotopical</a>)</p> <p><a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+perturbative+quantum+field+theory'>Introduction</a></p> <h2 id='concepts'>Concepts</h2> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/field+theory'>field theory</a></strong>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+field+theory'>classical</a>, <a class='existingWikiWord' href='/nlab/show/diff/prequantum+field+theory'>pre-quantum</a>, <a class='existingWikiWord' href='/nlab/show/diff/quantum+field+theory'>quantum</a>, <a class='existingWikiWord' href='/nlab/show/diff/perturbative+quantum+field+theory'>perturbative quantum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relativistic+quantum+field+theory'>relativistic</a>, <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+field+theory'>Euclidean</a>, <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+field+theory'>thermal</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/Lagrangian+quantum+field+theory'>Lagrangian field theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/field+%28physics%29'>field (physics)</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/field+bundle'>field bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/field+history'>field history</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/space+of+field+histories'>space of field histories</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lagrangian+density'>Lagrangian density</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+form'>Euler-Lagrange form</a>, <a class='existingWikiWord' href='/nlab/show/diff/pre-symplectic+current'>presymplectic current</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+equation'>Euler-Lagrange</a> <a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+equation'>equations of motion</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+variational+field+theory'>locally variational field theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/phase+space'>covariant phase space</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peierls+bracket'>Peierls-Poisson bracket</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/advanced+and+retarded+causal+propagators'>advanced and retarded propagator</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/causal+propagator'>causal propagator</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/quantization'>quantization</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometric+quantization'>geometric quantization</a> <a class='existingWikiWord' href='/nlab/show/diff/geometric+quantization+of+symplectic+groupoids'>of symplectic groupoids</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formal+deformation+quantization'>algebraic deformation quantization</a>, <a class='existingWikiWord' href='/nlab/show/diff/star-algebra'>star algebra</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/quantum+mechanics'>quantum mechanical system</a></strong>, <strong><a class='existingWikiWord' href='/nlab/show/diff/quantum+probability+theory'>quantum probability</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subsystem'>subsystem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/observable'>observables</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/field+observable'>field observables</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/local+observable'>local observables</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/polynomial+observable'>polynomial observables</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/microcausal+polynomial+observable'>microcausal observables</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/operator+algebra'>operator algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/C-star-algebra'>C*-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/von+Neumann+algebra'>von Neumann algebra</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/causally+local+net+of+observables'>local net of observables</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/causal+locality'>causal locality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Haag-Kastler+axioms'>Haag-Kastler axioms</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Wightman+axioms'>Wightman axioms</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/field+net'>field net</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/conformal+net'>conformal net</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/state+on+a+star-algebra'>state on a star-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/expectation+value'>expectation value</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pure+state'>pure state</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/space+of+states+%28in+geometric+quantization%29'>wave function</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/wave+function+collapse'>collapse of the wave function</a>/<a class='existingWikiWord' href='/nlab/show/diff/conditional+expectation'>conditional expectation value</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/density+matrix'>mixed state</a>, <a class='existingWikiWord' href='/nlab/show/diff/density+matrix'>density matrix</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/space+of+states+%28in+geometric+quantization%29'>space of quantum states</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/vacuum+state'>vacuum state</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quasi-free+quantum+state'>quasi-free state</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hadamard+distribution'>Hadamard state</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hadamard+distribution'>Wightman propagator</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/picture+of+mechanics'>picture of quantum mechanics</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/free+field'>free field</a> <a class='existingWikiWord' href='/nlab/show/diff/quantization'>quantization</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/star-algebra'>star algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/Moyal+deformation+quantization'>Moyal deformation quantization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Wick+algebra'>Wick algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/canonical+commutation+relation'>canonical commutation relations</a>, <a class='existingWikiWord' href='/nlab/show/diff/Weyl+relation'>Weyl relations</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Wick+algebra'>normal ordered product</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Fock+space'>Fock space</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theories</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/gauge+transformation'>gauge symmetry</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/BRST+complex'>BRST complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/BV-BRST+formalism'>BV-BRST formalism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/local+BRST+cohomology'>local BV-BRST complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/BV-operator'>BV-operator</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quantum+master+equation'>quantum master equation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Ward+identity'>master Ward identity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quantum+anomaly'>gauge anomaly</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/interacting+field+theory'>interacting field</a> <a class='existingWikiWord' href='/nlab/show/diff/quantization'>quantization</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/causal+perturbation+theory'>causal perturbation theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/perturbative+algebraic+quantum+field+theory'>perturbative AQFT</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interaction'>interaction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/S-matrix'>S-matrix</a>, <a class='existingWikiWord' href='/nlab/show/diff/scattering+amplitude'>scattering amplitude</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/causal+additivity'>causal additivity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/time-ordered+product'>time-ordered product</a>, <a class='existingWikiWord' href='/nlab/show/diff/Feynman+propagator'>Feynman propagator</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Feynman+diagram'>Feynman diagram</a>, <a class='existingWikiWord' href='/nlab/show/diff/S-matrix'>Feynman perturbation series</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/effective+action'>effective action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/vacuum+stability'>vacuum stability</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interacting+field+algebra+of+observables'>interacting field algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bogoliubov%27s+formula'>Bogoliubov's formula</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/M%C3%B8ller+operator'>quantum Møller operator</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adiabatic+switching'>adiabatic limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infrared+divergence'>infrared divergence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interacting+vacuum'>interacting vacuum</a></p> </li> </ul> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/renormalization'>renormalization</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/renormalization+scheme'>("re-")normalization scheme</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extension+of+distributions'>extension of distributions</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/renormalization+condition'>("re"-)normalization condition</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quantum+anomaly'>quantum anomaly</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/renormalization+group'>renormalization group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interaction+vertex+redefinition'>interaction vertex redefinition</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/St%C3%BCckelberg-Petermann+renormalization+group'>Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/renormalization+group+flow'>renormalization group flow</a>/<a class='existingWikiWord' href='/nlab/show/diff/renormalization+group+flow'>running coupling constants</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/effective+quantum+field+theory'>effective quantum field theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/effective+quantum+field+theory'>UV cutoff</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/counterterm'>counterterms</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/effective+action+functional'>relative effective action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Polchinski%27s+flow+equation'>Wilsonian RG</a>, <a class='existingWikiWord' href='/nlab/show/diff/Polchinski%27s+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/diff/order-theoretic+structure+in+quantum+mechanics'>order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Alfsen-Shultz+theorem'>Alfsen-Shultz theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/D%C3%B6ring-Harding-Hamhalter+theorem'>Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kochen-Specker+theorem'>Kochen-Specker theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bell%27s+inequality'>Bell's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Fell%27s+theorem'>Fell's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gleason%27s+theorem'>Gleason's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Wigner+theorem'>Wigner theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bub-Clifton+theorem'>Bub-Clifton theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kadison-Singer+problem'>Kadison-Singer problem</a></p> </li> </ul> <h3 id='operator_algebra_2'>Operator algebra</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Wick%27s+lemma'>Wick's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gelfand-Naimark-Segal+construction'>GNS construction</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cyclic+vector'>cyclic vector</a>, <a class='existingWikiWord' href='/nlab/show/diff/separating+vector'>separating vector</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/modular+theory'>modular theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Fell%27s+theorem'>Fell's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Stone-von+Neumann+theorem'>Stone-von Neumann theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Reeh-Schlieder+theorem'>Reeh-Schlieder theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bisognano-Wichmann+theorem'>Bisognano-Wichmann theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/PCT+theorem'>PCT theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spin-statistics+theorem'>spin-statistics theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/DHR+superselection+theory'>DHR superselection theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Osterwalder-Schrader+theorem'>Osterwalder-Schrader theorem</a> (<a class='existingWikiWord' href='/nlab/show/diff/Wick+rotation'>Wick rotation</a>)</p> </li> </ul> <h3 id='perturbative_qft'>Perturbative QFT</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Schwinger-Dyson+equation'>Schwinger-Dyson equation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/main+theorem+of+perturbative+renormalization'>main theorem of perturbative renormalization</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#general'>General</a></li><li><a href='#homomorphisms_into_'>Homomorphisms into <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>PU</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>PU(\mathcal{H})</annotation></semantics></math></a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#literature'>Literature</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The <a class='existingWikiWord' href='/nlab/show/diff/projective+unitary+group'>projective unitary group</a> on an infinite-dimensional <a class='existingWikiWord' href='/nlab/show/diff/separable+Hilbert+space'>separable</a> <a class='existingWikiWord' href='/nlab/show/diff/complex+vector+space'>complex</a> <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a> <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{H}</annotation></semantics></math> is traditionally denoted <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>PU</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>PU(\mathcal{H})</annotation></semantics></math>, being the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient</a> of the <a class='existingWikiWord' href='/nlab/show/diff/unitary+group'>unitary group</a> <a class='existingWikiWord' href='/nlab/show/diff/U%28%E2%84%8B%29'>U(ℋ)</a> by its <a class='existingWikiWord' href='/nlab/show/diff/circle+group'>circle</a> <a class='existingWikiWord' href='/nlab/show/diff/subgroup'>subgroup</a> <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>≃</mo><mi>S</mi><mo stretchy='false'>(</mo><mi>ℂ</mi><mo stretchy='false'>)</mo><mo>≃</mo></mrow><annotation encoding='application/x-tex'>S^1 \simeq S(\mathbb{C}) \simeq</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/circle+group'>U(1)</a>.</p> <h2 id='properties'>Properties</h2> <h3 id='general'>General</h3> <p>\begin{proposition}\label{UHCircleFiberBundle} The <a class='existingWikiWord' href='/nlab/show/diff/circle+group'>U(1)</a>-<a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a> <a class='existingWikiWord' href='/nlab/show/diff/coprojection'>coprojection</a> of <a class='existingWikiWord' href='/nlab/show/diff/U%28%E2%84%8B%29'>U(ℋ)</a> over <a class='existingWikiWord' href='/nlab/show/diff/PU%28%E2%84%8B%29'>PU(ℋ)</a> – both in their strong <a class='existingWikiWord' href='/nlab/show/diff/operator+topology'>operator topology</a> – is a <a class='existingWikiWord' href='/nlab/show/diff/circle+group'>circle</a>-<a class='existingWikiWord' href='/nlab/show/diff/principal+bundle'>principal bundle</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>1</mn></msup></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi mathvariant='normal'>U</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>PU</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo><mpadded width='0'><mrow><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mi mathvariant='normal'>U</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ S^1 &\hookrightarrow& \mathrm{U}(\mathcal{H}) \\ && \big\downarrow \\ && PU(\mathcal{H}) \mathrlap{ \; \simeq \; \mathrm{U}(\mathcal{H})/S^1 } } </annotation></semantics></math></div> <p>\end{proposition} (<a href='#Simms70'>Simms 1970, Thm. 1</a>) \begin{remark} Prop. \ref{UHCircleFiberBundle} means in particular that <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi mathvariant='normal'>U</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mspace width='thickmathspace' /></mover><mi>PU</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{U}(\mathcal{H}) \xrightarrow{\;} PU(\mathcal{H})</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>locally trivial</a>, hence that the <a class='existingWikiWord' href='/nlab/show/diff/coset'>coset space</a> <a class='existingWikiWord' href='/nlab/show/diff/coprojection'>coprojection</a> <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi mathvariant='normal'>U</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mspace width='thickmathspace' /></mover><mi mathvariant='normal'>U</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\mathrm{U}(\mathcal{H}) \xrightarrow{\;} \mathrm{U}(\mathcal{H})/S^1</annotation></semantics></math> admits <a class='existingWikiWord' href='/nlab/show/diff/local+section'>local sections</a>. See also at <em><a class='existingWikiWord' href='/nlab/show/diff/coset+space+coprojection+admitting+local+sections'>coset space coprojection admitting local sections</a></em>. \end{remark}</p> <p>\begin{prop}\label{WellPointedInOperatorTopology} In its <a class='existingWikiWord' href='/nlab/show/diff/operator+topology'>operator topology</a> (<a class='existingWikiWord' href='/nlab/show/diff/U%28%E2%84%8B%29#WithOperatorTopology'>here</a>), <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>PU</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>PU(\mathcal{H})</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/well-pointed+topological+space'>well-pointed topological group</a>. \end{prop} (<a href='#HebestreitSagave19'>Hebestreit & Sagave 2020, p. 23</a>, using <a href='twisted+K-theory#DardalatPennig16'>Dardalat & Pennig 2016, Prop. 2.26</a>)</p> <h3 id='homomorphisms_into_'>Homomorphisms into <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>PU</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>PU(\mathcal{H})</annotation></semantics></math></h3> <p>Since <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi><mspace width='thinmathspace' /><mo>≔</mo><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\Gamma \,\coloneqq\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/PU%28%E2%84%8B%29'>PU(ℋ)</a> is not a (finite-dimensional) <a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a>, it falls outside the applicability of the general theorem that <a class='existingWikiWord' href='/nlab/show/diff/nearby+homomorphisms+from+compact+Lie+groups+are+conjugate'>nearby homomorphisms from compact Lie groups are conjugate</a>. Nevertheless, the conclusion still holds, at least for <a class='existingWikiWord' href='/nlab/show/diff/domain'>domain</a> <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/discrete+group'>discrete</a>, hence <a class='existingWikiWord' href='/nlab/show/diff/finite+group'>finite group</a>:</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/PU%28%E2%84%8B%29'>PU(ℋ)</a><a class='existingWikiWord' href='/nlab/show/diff/topological+G-space'>-space</a> of homomorphisms <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Hom</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>G</mi><mo>,</mo><mspace width='thinmathspace' /><mi>PU</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>Hom\big(G,\, PU(\mathcal{H})\big)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> (<a href='nearby+homomorphisms+from+compact+Lie+groups+are+conjugate#eq:DisjointUnionOfOrbits'>as here</a>) of <a class='existingWikiWord' href='/nlab/show/diff/orbit'>orbits</a> of the <a class='existingWikiWord' href='/nlab/show/diff/conjugation+action'>conjugation action</a>.</p> <p>This is established in <a href='#UribeLueck13'>Uribe & Lück 2013, Sec. 15</a>, <a href='https://arxiv.org/pdf/1304.4862.pdf#page=38'>p. 38</a>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/U%28%E2%84%8B%29'>U(ℋ)</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+equivariant+PU%28H%29-bundle'>universal equivariant $PU(\mathcal{H})$-bundle</a></p> </li> </ul> <h2 id='literature'>Literature</h2> <p>General discussion:</p> <ul> <li id='Simms70'> <p><a class='existingWikiWord' href='/nlab/show/diff/David+John+Simms'>David John Simms</a>, <em>Topological aspects of the projective unitary group</em>, Math. Proc. Camb. Phil. Soc. <strong>68</strong> 1 (1970) 57-60 (<a href='https://doi.org/10.1017/S0305004100001043'>doi:10.1017/S0305004100001043</a>)</p> </li> <li id='UribeLueck13'> <p><a class='existingWikiWord' href='/nlab/show/diff/Bernardo+Uribe'>Bernardo Uribe</a>, <a class='existingWikiWord' href='/nlab/show/diff/Wolfgang+Lueck'>Wolfgang Lück</a>, Section 15 of: <em>Equivariant principal bundles and their classifying spaces</em>, Algebr. Geom. Topol. 14 (2014) 1925-1995 (<a href='https://arxiv.org/abs/1304.4862'>arXiv:1304.4862</a>, <a href='http://dx.doi.org/10.2140/agt.2014.14.1925'>doi:10.2140/agt.2014.14.1925</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jes%C3%BAs+Francisco+Espinoza+Fierro'>Jesus Espinoza</a>, <a class='existingWikiWord' href='/nlab/show/diff/Bernardo+Uribe'>Bernardo Uribe</a>, <em>Topological properties of spaces of projective unitary representations</em>, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 40 (2016), no. 155, 337-352 (<a href='https://arxiv.org/abs/1511.06785'>arXiv:1511.06785</a>, <a href='http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S0370-39082016000200013'>scielo:S0370-39082016000200013</a>, <a href='https://doi.org/10.18257/raccefyn.317'>doi:10.18257/raccefyn.317</a>)</p> </li> </ul> <p>Concerning well-pointedness:</p> <ul> <li id='HebestreitSagave19'><a class='existingWikiWord' href='/nlab/show/diff/Fabian+Hebestreit'>Fabian Hebestreit</a>, <a class='existingWikiWord' href='/nlab/show/diff/Steffen+Sagave'>Steffen Sagave</a>, p. 23 of: <em>Homotopical and operator algebraic twisted K-theory</em>, Mathematische Annalen <strong>378</strong> (2020) 1021-1059 (<a href='https://arxiv.org/abs/1904.01872'>arXiv:1904.01872</a>, <a href='https://doi.org/10.1007/s00208-020-02066-6'>doi:10.1007/s00208-020-02066-6</a>)</li> </ul> <p>The <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2</annotation></semantics></math>-graded projective unitary group, (reflecting the twists of <a class='existingWikiWord' href='/nlab/show/diff/twisted+K-theory'>twisted K-theory</a> not just in degree 3 but also in degree 1):</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ellen+Maycock+Parker'>Ellen Maycock Parker</a>, Prop. 2.2 of: <em>The Brauer Group of Graded Continuous Trace <math class='maruku-mathml' display='inline' id='mathml_b3cfb63160ae8d975045dcad8f6ec0bcf35ea668_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-Algebras</em>, Transactions of the American Mathematical Society <strong>308</strong> 1 (1988) (<a href='https://www.jstor.org/stable/2000953'>jstor:2000953</a>)</li> </ul> <p>reviewed in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Alan+Carey'>Alan Carey</a>, <a class='existingWikiWord' href='/nlab/show/diff/Bai-Ling+Wang'>Bai-Ling Wang</a>, p. 5 of: <em>Thom isomorphism and Push-forward map in twisted K-theory</em>, Journal of K-Theory <strong>1</strong> 2 (2008) 357-393 (<a href='https://arxiv.org/abs/math/0507414'>arXiv:math/0507414</a>, <a href='https://doi.org/10.1017/is007011015jkt011'>doi:10.1017/is007011015jkt011</a>)</li> </ul> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on September 15, 2022 at 16:26:35. 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