A First Course in Topos Quantum Theory

464 Pages·2013·3.54 MB·English

by  Cecilia Flori

Lecture Notes in Physics
Volume 868
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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
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To my parents Elena Romani and Luciano
Flori,forbelievinginme
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Acknowledgement
TheAuthorwouldliketothankPerimeterInstitute.ResearchatPerimeterInstitute
is supported by the Government of Canada through Industry Canada and by the
ProvinceofOntariothroughtheMinistryofResearchandInnovation.
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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