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Lecture Notes in Physics Volume 868 FoundingEditors W.Beiglböck J.Ehlers K.Hepp H.Weidenmüller EditorialBoard B.-G.Englert,Singapore,Singapore U.Frisch,Nice,France P.Hänggi,Augsburg,Germany W.Hillebrandt,Garching,Germany M.Hjort-Jensen,Oslo,Norway R.A.L.Jones,Sheffield,UK H.vonLöhneysen,Karlsruhe,Germany M.S.Longair,Cambridge,UK M.L.Mangano,Geneva,Switzerland J.-F.Pinton,Lyon,France J.-M.Raimond,Paris,France A.Rubio,Donostia,SanSebastian,Spain M.Salmhofer,Heidelberg,Germany D.Sornette,Zurich,Switzerland S.Theisen,Potsdam,Germany D.Vollhardt,Augsburg,Germany W.Weise,Garching,GermanyandTrento,Italy Forfurthervolumes: www.springer.com/series/5304 The Lecture Notes in Physics TheseriesLectureNotesinPhysics(LNP),foundedin1969,reportsnewdevelop- ments in physics research and teaching—quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging mate- rialbetweenadvancedgraduatetextbooksandtheforefrontofresearchandtoserve threepurposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialistresearchersfromrelatedareas • to be a source of advanced teaching material for specialized seminars, courses andschools Both monographs and multi-author volumes will be considered for publication. Editedvolumesshould,however,consistofaverylimitednumberofcontributions only.ProceedingswillnotbeconsideredforLNP. Volumes published in LNP are disseminated both in print and in electronic for- mats,theelectronicarchivebeingavailableatspringerlink.com.Theseriescontent isindexed,abstractedandreferencedbymanyabstractingandinformationservices, bibliographicnetworks,subscriptionagencies,librarynetworks,andconsortia. Proposals should be sent to a member of the Editorial Board, or directly to the managingeditoratSpringer: ChristianCaron SpringerHeidelberg PhysicsEditorialDepartmentI Tiergartenstrasse17 69121Heidelberg/Germany [email protected] Cecilia Flori A First Course in Topos Quantum Theory CeciliaFlori PerimeterInstituteforTheoreticalStudies Waterloo,Ontario, Canada ISSN0075-8450 ISSN1616-6361(electronic) LectureNotesinPhysics ISBN978-3-642-35712-1 ISBN978-3-642-35713-8(eBook) DOI10.1007/978-3-642-35713-8 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013933014 ©Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub- lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect tothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To my parents Elena Romani and Luciano Flori,forbelievinginme This page intentionally left blank Acknowledgement TheAuthorwouldliketothankPerimeterInstitute.ResearchatPerimeterInstitute is supported by the Government of Canada through Industry Canada and by the ProvinceofOntariothroughtheMinistryofResearchandInnovation. vii This page intentionally left blank Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 PhilosophicalMotivations . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 WhatIsaTheoryofPhysicsandWhatIsItTryingtoAchieve? . . 7 2.2 PhilosophicalPositionofClassicalTheory . . . . . . . . . . . . . 8 2.3 PhilosophyBehindQuantumTheory . . . . . . . . . . . . . . . . 10 2.4 ConceptualProblemsofQuantumTheory . . . . . . . . . . . . . 12 3 Kochen-SpeckerTheorem . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1 ValuationFunctionsinClassicalTheory . . . . . . . . . . . . . . 13 3.2 ValuationFunctionsinQuantumTheory . . . . . . . . . . . . . . 14 3.2.1 DerivingtheFUNCCondition. . . . . . . . . . . . . . . . 16 3.2.2 ImplicationsoftheFUNCCondition . . . . . . . . . . . . 18 3.3 KochenSpeckerTheorem . . . . . . . . . . . . . . . . . . . . . . 20 3.4 ProofoftheKochen-SpeckerTheorem . . . . . . . . . . . . . . . 21 3.5 ConsequencesoftheKochen-SpeckerTheorem . . . . . . . . . . 22 4 IntroducingCategoryTheory . . . . . . . . . . . . . . . . . . . . . . 25 4.1 ChangeofPerspective . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 AxiomaticDefinitionofaCategory . . . . . . . . . . . . . . . . . 32 4.2.1 ExamplesofCategories . . . . . . . . . . . . . . . . . . . 34 4.3 TheDualityPrinciple . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4 ArrowsinaCategory . . . . . . . . . . . . . . . . . . . . . . . . 39 4.4.1 MonicArrows . . . . . . . . . . . . . . . . . . . . . . . . 40 4.4.2 EpicArrows . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.4.3 IsoArrows . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.5 ElementsandTheirRelationsinaCategory. . . . . . . . . . . . . 43 4.5.1 InitialObjects . . . . . . . . . . . . . . . . . . . . . . . . 43 4.5.2 TerminalObjects. . . . . . . . . . . . . . . . . . . . . . . 43 4.5.3 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.5.4 Coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5.5 Equalisers . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ix