Topological boundary maps in physics: General theory and applications

Description

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

Sujets

• Jordan
• Richard
• Crossed product
• Index map

Informations

• Publié par: mijec
• Publié le: 01 novembre 1918
• Nombre de lectures: 44
• Langue: English

Extrait

Topologicalboundarymapsinphysics: Generaltheoryandapplications JohannesKellendonkandSergeRichard InstitutCamilleJordan,BaˆtimentBraconnier,Universite´ClaudeBernardLyon1, 43avenuedu11novembre1918,69622Villeurbannecedex,France E-mails:[e-mail]@math.univ-lyon1.fr April2006 Abstract ThematerialpresentedherecoverstwotalksgivenbytheauthorsattheconferenceOp- eratorAlgebrasandMathematicalPhysicsorganisedinBucharestinAugust2005.Thefirst onewasareviewgivenbyJ.Kellendonkontherelationbetweenbulkandboundarytopolog- icalinvariantsinphysicalsystems.InthesecondtalkS.Richarddescribedanapplicationof theseideastoscatteringtheory.Itleadstoatopologicalversionoftheso-calledLevinson’s theorem. Introduction ThenaturallanguageforquantumphysicsislinearoperatorsonHilbertspacesandunderlying operatoralgebras.Thesealgebrasarefundamentallynon-commutative.Topologicalproperties ofquantumsystemsshouldhencebeconnectedwiththetopologyofthesealgebras,whichiswhat onecallsnon-commutativetopology.Animportantfirstquestiontobeansweredistherefore: whatisthecorrectoperatoralgebrarelatedtoaphysicalsystem?Sincewearelookingfor topologicaleffectsthisalgebrashouldbeaseparableC ∗ -algebraandagoodstartingpointis tolookfortheC ∗ -versionoftheobservablesalgebra.Butwewillhavemoretosayonthis below.Oncethequestionabouttherightalgebraissettledweareinterestedinstudyingits invariantsaskingaboveall:whichofthemhaveaphysicalinterpretation?Finally,whenwehave identifiedtheinvariants,wewanttoderiverelationsbetweenthem,typicallyequationsbetween topologicalquantisedtransportcoefficientsor,asinLevinson’stheorem,betweeninvariantsof theboundedpartandthescatteringpartofthephysicalsystem.Suchrelationscanbeobtained fromtopologicalboundarymapswhichdonotexistonthealgebraiclevel. Thepurposeofthispaperistwofold:Explainwithmoredetailsthegeneraltheoryoutlinedin thepreviousparagraph,andshowitsrelevanceinvariousapplicationsinmathematicalphysics. 1