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A Mathematically Coherent Quantum Gravity A Mathematically Coherent Quantum Gravity James Moffat Kings College, University of Aberdeen, Aberdeen, UK IOP Publishing, Bristol, UK ªIOPPublishingLtd2020 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording orotherwise,withoutthepriorpermissionofthepublisher,orasexpresslypermittedbylawor undertermsagreedwiththeappropriaterightsorganization.Multiplecopyingispermittedin accordancewiththetermsoflicencesissuedbytheCopyrightLicensingAgency,theCopyright ClearanceCentreandotherreproductionrightsorganizations. PermissiontomakeuseofIOPPublishingcontentotherthanassetoutabovemaybesought [email protected]. JamesMoffathasassertedhisrighttobeidentifiedastheauthorofthisworkinaccordancewith sections77and78oftheCopyright,DesignsandPatentsAct1988. ISBN 978-0-7503-2580-6(ebook) ISBN 978-0-7503-2578-3(print) ISBN 978-0-7503-2581-3(myPrint) ISBN 978-0-7503-2579-0(mobi) DOI 10.1088/978-0-7503-2580-6 Version:20200901 IOPebooks BritishLibraryCataloguing-in-PublicationData:Acataloguerecordforthisbookisavailable fromtheBritishLibrary. PublishedbyIOPPublishing,whollyownedbyTheInstituteofPhysics,London IOPPublishing,TempleCircus,TempleWay,Bristol,BS16HG,UK USOffice:IOPPublishing,Inc.,190NorthIndependenceMallWest,Suite601,Philadelphia, PA19106,USA Thecoverdesignisbasedonapaintingbytheauthor.Itiscalled‘Logos’whichisGreekforLogic. I dedicate this book jointly to Frank Bonsall and John Ringrose. Contents Preface: why you should read this book x Acknowledgements xiii Author biography xiv Symbols xv 1 Quantum theory 1-1 1.1 Basic notions 1-1 1.2 Non-relativistic quantum theory 1-2 1.2.1 The measurement process 1-4 1.2.2 The Schrödinger equation 1-5 1.3 Exploiting quantum theory embedded in classical gravity 1-6 1.4 Special relativity 1-9 1.4.1 Relativistic waves 1-10 1.4.2 Jones vectors to describe classical polarization states 1-11 1.4.3 The relativistic quantum photon 1-12 1.5 Dirac relativistic spinor theory 1-14 1.5.1 The relativistic spinor 1-15 1.6 von Neumann algebras 1-17 1.7 A higher level of abstraction: quantum W*-algebras 1-19 1.8 A brief comparison with the approach of Rovelli and Penrose 1-20 References 1-21 2 Computational spin networks and quantum paths in space–time 2-1 2.1 Introduction 2-1 2.2 The measurement of space and time 2-2 2.3 Computational spin networks 2-5 2.4 The homology invariants of space–time 2-7 2.5 Quantum paths in space–time 2-10 2.5.1 Fiber bundle structure of classical phase space 2-11 2.5.2 An example of the Weyl form 2-12 2.6 Fractal paths in classical space–time 2-16 2.7 Supersymmetry and the spinor calculus 2-17 2.7.1 2-Spinors 2-17 2.7.2 4-Spinors and the Lorentz and Poincaré groups 2-19 2.7.3 4-Spinors and the spin groups 2-21 vii AMathematicallyCoherentQuantumGravity 2.8 Irreducible representations of the Poincaré Lie algebra 2-22 2.8.1 The supersymmetric extension of the standard model (SSM) 2-22 2.9 Dirac spinors and the spinor calculus 2-23 2.9.1 The supersymmetry algebra 2-25 References 2-26 3 Particles in algebraic quantum gravity 3-1 3.1 Introduction 3-1 3.2 Lie groups, fibre bundles and quantum fields in loop quantum gravity 3-1 3.2.1 A coherent approach to quantum fields 3-3 3.3 A remarkable theorem 3-4 3.4 Properties of the projection onto the base space B of a Stonean 3-6 fibre bundle K 3.4.1 Lifting from the base space 3-6 3.5 Quantum connections 3-7 3.6 The supersymmetric extension of the standard model 3-8 3.7 Factorial representations of the graded Lie algebra 3-11 3.8 Does supersymmetry exist? ATLAS results for Run 2 of 3-13 the LHC at 13 TeV 3.9 Adding fermions and bosons to the mix 3-15 3.10 The 10-dimensional pure gravity action 3-16 3.11 Adding bosons to the theory 3-17 3.12 Adding fermions 3-17 3.13 Symmetry breaking to create mass 3-19 References 3-21 4 The algebraic nature of reality 4-1 4.1 Introduction 4-1 4.2 Symmetry invariance and symmetry breaking in Yang–Mills 4-2 quantum fields 4.2.1 Wigner sets and symmetry invariance in the local algebra O(D) 4-2 4.2.2 Symmetric Yang–Mills quantum states 4-4 4.2.3 Superspace 4-5 4.3 The structure of local algebras of Yang–Mills quantum fields 4-8 4.4 Developing a diffeomorphism invariant theory for quantum states 4-9 4.4.1 Wandering projections and diffeomorphism invariant quantum 4-11 states viii AMathematicallyCoherentQuantumGravity 4.5 The information dynamics of black holes 4-13 4.6 Summary of chapter 4 4-15 Further reading 4-16 References 4-16 5 Implications 5-1 5.1 Implications for mathematics 5-1 5.1.1 Clay Mathematics Institute Millennial question: Yang–Mills 5-1 quantum theory and the mass gap 5.2 Further implications for Physics 5-4 5.2.1 Discrete closed strings in the early Universe 5-4 5.2.2 The continuum limit beyond the Planck regime 5-7 5.2.3 The continuum limit as a quantum field 5-7 5.2.4 A brief non-technical overview 5-10 References 5-11 ix