Quantum Mathematical Physics: A Bridge between Mathematics and Physics PDF
Preview Quantum Mathematical Physics: A Bridge between Mathematics and Physics
Felix Finster Johannes Kleiner Christian Röken Jürgen Tolksdorf Editors Quantum Mathematical Physics A Bridge between Mathematics and Physics Quantum Mathematical Physics Felix Finster • Johannes Kleiner (cid:129) Christian RoRken (cid:129) JuRrgen Tolksdorf Editors Quantum Mathematical Physics A Bridge between Mathematics and Physics Editors FelixFinster JohannesKleiner FakultätfürMathematik FakultätfürMathematik UniversitätRegensburg UniversitätRegensburg Regensburg,Germany Regensburg,Germany ChristianRoRken JuRrgenTolksdorf FakultätfürMathematik MPIfürMathematikinden UniversitätRegensburg Naturwissenschaften Regensburg,Germany Leipzig,Germany ISBN978-3-319-26900-9 ISBN978-3-319-26902-3 (eBook) DOI10.1007/978-3-319-26902-3 LibraryofCongressControlNumber:2015957955 MathematicsSubjectClassification(2010):81-06,83-06,81T20,81T70,81T75,81T15,81T60,35Q75, 35Q40,83C45,35L10 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Coverdesign:deblik,Berlin Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com) In honorofEberhardZeidler’s 75thbirthday. Preface ThepresentvolumeisbasedontheinternationalconferenceQuantumMathematical Physics – A Bridge between Mathematics and Physics that was held at the UniversityofRegensburg(Germany)fromSeptember29toOctober2,2014.This conferencewas a successor of similar internationalconferenceswhich took place at the Heinrich-FabriInstitute (Blaubeuren)in 2003and 2005,at the Max Planck Institute for Mathematics in the Sciences (Leipzig) in 2007 and at the University ofRegensburgin2010.Thebasicintentionofthisseriesofconferencesistobring together mathematicians and physicists to discuss profoundquestions in quantum field theory and gravity. More specifically, the series aims at discussing concepts which underpin different mathematical and physical approaches to quantum field theoryandgravity. Sincetheinventionofgeneralrelativityandquantummechanicsatthebeginning ofthetwentiethcentury,physicistsmadeanenormousefforttoincorporategravity andquantumphysicsintoaunifiedframework.Indoingso,manyapproacheshave been developed to overcome the basic conceptual and mathematical differences betweenquantumtheoryandgeneralrelativity.Moreover,bothquantumtheoryand generalrelativityhavetheirownproblemsandshortcomings.Itturnsoutthatmany oftheseproblemsarerelatedtoeachotherandtotheproblemoftheunificationof quantumtheoryandgravity.Theaim of theconferencewasto shedlightonthese problemsandtoindicatepossiblesolutions. Ononehand,generalrelativitydescribessystemsonlargescales(likethesolar system, galaxies, and cosmological phenomena). This is reflected in the fact that in general relativity, space-time has locally the simple structure of Minkowski space, whereas gravitational effects usually show up in the large-scale geometry. Undergenericassumptions,therearephenomenalikeblackholesandcosmological singularitieswhicharenotyetunderstoodinaphysicallysatisfyingway.Quantum theory, on the other hand, usually describes systems on small scales (like atoms, nuclei,orelementaryparticles).Indeed,onsmallscalestheHeisenberguncertainty principle becomes relevant and quantum effects come into play. One of the open problems is that there is no satisfying mathematical description of interacting quantumfields. vii viii Preface Oneofthefundamentaldifficultiesincombininggravitywith quantumphysics lies in the fact that general relativity is a theory on the dynamics of space-time itself, whereas quantum theory usually aims to describe the dynamics of matter withina givenspace-timebackground(inthe simplestcase byMinkowskispace). Moreover, the geometric description of general relativity makes it necessary to describeobjectslocallyinanarbitrarysmallneighborhoodofapoint.Butlocalizing quantummechanicalwavefunctionstosuchasmallneighborhood,theHeisenberg uncertainty principle gives rise to large energy fluctuations. Considering these energy fluctuations as a gravitational source, one obtains a contradiction to the abovepicturethatgravitycomesintoplayonlyonlargescales.Thus,althoughboth theories are experimentally well confirmed, they seem to conceptually contradict eachother.Thisincompatibilityalsobecomesapparentinthemathematicalformu- lation:Fromamathematicalperspective,generalrelativityisusuallyregardedasa purelygeometrictheory.However,quantumphysicsisdescribedmathematicallyin analgebraicandfunctionalanalyticlanguage. There are various approaches to overcome these issues. For instance, in string theory one replaces point-like particles by one-dimensional objects. Other ap- proaches, like loop quantum gravity, causal fermion systems, or noncommutative geometry,relyontheassumptionthatthemacroscopicsmoothspace-timestructure shouldemergefrommore fundamentalstructuresonthe microscopicscale. Alter- natively, one tries to treat interacting theories as “effective theories” or considers quantumtheoryfromanaxiomaticandcategoricalviewpointinawaythatallows to incorporatethe conceptoflocalobservers.Mostof these modernmathematical approachestounifyquantumphysicswithgeneralrelativityhavetheadvantageto combinegeometricstructureswithalgebraicandfunctionalanalyticmethods.Some ofthese“quantummathematicalconcepts”arediscussedinthepresentconference volume. The carefully selected and refereed articles in this volume either give a survey or focus on specific issues. They explain the state of the art of various rigorous approachesto quantum field theory and gravity. Most of the articles are based on talksattheabovementionedconference.Alltalksoftheconferencewererecorded, andmostareavailableonlineat http://www.ur.de/qft2014. For the first time, the conference included two evening talks devoted to new experimentaldevelopments(darkmatter/energyandtheHiggsparticle).Itwasagain amainpurposeoftheconferencetosetthestageforstimulatingdiscussions.Tothis end,extratimeslotswerereservedforpanelandplenarydiscussions.Hereisalist ofsomeofthequestionsraisedinthediscussions: 1. Quantum gravity: What should a physically convincing theory of quantum gravity accomplish? Which are the most promising directions to find such a theoryofquantumgravity?Whydoesoneneedto“quantize”gravity–isitnot sufficienttodescribeitclassically?Howimportantismathematicalconsistency? 2. Quantization: Do quantum field theories necessarily arise by quantizing a classical field theory? Is such a quantization procedure necessary in order to Preface ix haveaphysicalinterpretationoftheresultingquantumfieldtheory?Doesitmake physicalsensetoquantizepuregravitywithoutmatter? 3. Future perspectives: Which directions in mathematical physics seem most promising for young researchers to work on? Is it recommendable for young researchersto study new topics or should they rather work on well-established problems? Which are the big challenges for mathematical physics in the next years? 4. Axiomatic frameworks: Do the various axiomatic frameworks (such as alge- braicquantumfieldtheory,causalfermionsystems,noncommutativegeometry, etc.) offer a suitable frameworkfor unifyinggravity and quantum theory? Can causalitybeexpectedtohold? 5. Darkenergyanddarkmatter:Isdarkenergyrelatedtoquantumfieldtheoretic effectslikevacuumfluctuations?Ordotheexplanationsofdarkenergyanddark matter require new physical concepts? Should dark matter and dark energy be consideredassomekindof“matter”or“field”inspace-time? 6. Mathematics of future theories: Which contemporarymathematicaldevelop- mentsmightplayanimportantroleintheformulationofnewphysicaltheories? We are grateful to Klaus Fredenhagen (Hamburg), José Maria Gracia-Bondia (Madrid),Gerhard Börner (München),and Harald Grosse (Wien) for contributing to the discussions as members of the panel. The discussions were moderated by JohannesKleiner. Regensburg,Germany FelixFinster JohannesKleiner ChristianRöken Leipzig,Germany JürgenTolksdorf July2015
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