Topological boundary maps in physics: General theory and applications

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Niveau: Supérieur, Doctorat, Bac+8
Topological boundary maps in physics: General theory and applications Johannes Kellendonk and Serge Richard Institut Camille Jordan, Batiment Braconnier, Universite Claude Bernard Lyon 1, 43 avenue du 11 novembre 1918, 69622 Villeurbanne cedex, France E-mails: and April 2006 Abstract The material presented here covers two talks given by the authors at the conference Op- erator Algebras and Mathematical Physics organised in Bucharest in August 2005. The first one was a review given by J. Kellendonk on the relation between bulk and boundary topolog- ical invariants in physical systems. In the second talk S. Richard described an application of these ideas to scattering theory. It leads to a topological version of the so-called Levinson's theorem. Introduction The natural language for quantum physics is linear operators on Hilbert spaces and underlying operator algebras. These algebras are fundamentally non-commutative. Topological properties of quantum systems should hence be connected with the topology of these algebras, which is what one calls non-commutative topology. An important first question to be answered is therefore: what is the correct operator algebra related to a physical system? Since we are looking for topological effects this algebra should be a separable C?-algebra and a good starting point is to look for the C?-version of the observables algebra.

  • c?

  • algebra cp

  • crossed product

  • relation between

  • c?-algebras related

  • index map

  • between numerical

  • self- adjoint operator


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Published 01 November 1918
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Topologicalboundarymapsinphysics:GeneraltheoryandapplicationsJohannesKellendonkandSergeRichardInstitutCamilleJordan,BaˆtimentBraconnier,Universite´ClaudeBernardLyon1,43avenuedu11novembre1918,69622Villeurbannecedex,FranceE-mails:kellendonk@math.univ-lyon1.frandsrichard@math.univ-lyon1.frApril2006AbstractThematerialpresentedherecoverstwotalksgivenbytheauthorsattheconferenceOp-eratorAlgebrasandMathematicalPhysicsorganisedinBucharestinAugust2005.ThefirstonewasareviewgivenbyJ.Kellendonkontherelationbetweenbulkandboundarytopolog-icalinvariantsinphysicalsystems.InthesecondtalkS.Richarddescribedanapplicationoftheseideastoscatteringtheory.Itleadstoatopologicalversionoftheso-calledLevinson’stheorem.IntroductionThenaturallanguageforquantumphysicsislinearoperatorsonHilbertspacesandunderlyingoperatoralgebras.Thesealgebrasarefundamentallynon-commutative.Topologicalpropertiesofquantumsystemsshouldhencebeconnectedwiththetopologyofthesealgebras,whichiswhatonecallsnon-commutativetopology.Animportantfirstquestiontobeansweredistherefore:whatisthecorrectoperatoralgebrarelatedtoaphysicalsystem?SincewearelookingfortopologicaleffectsthisalgebrashouldbeaseparableC-algebraandagoodstartingpointistolookfortheC-versionoftheobservablesalgebra.Butwewillhavemoretosayonthisbelow.Oncethequestionabouttherightalgebraissettledweareinterestedinstudyingitsinvariantsaskingaboveall:whichofthemhaveaphysicalinterpretation?Finally,whenwehaveidentifiedtheinvariants,wewanttoderiverelationsbetweenthem,typicallyequationsbetweentopologicalquantisedtransportcoefficientsor,asinLevinson’stheorem,betweeninvariantsoftheboundedpartandthescatteringpartofthephysicalsystem.Suchrelationscanbeobtainedfromtopologicalboundarymapswhichdonotexistonthealgebraiclevel.Thepurposeofthispaperistwofold:Explainwithmoredetailsthegeneraltheoryoutlinedinthepreviousparagraph,andshowitsrelevanceinvariousapplicationsinmathematicalphysics.1