A Mathematically Coherent Quantum Gravity

The development of a successful theory of quantum gravity in the context of the early universe is the key next step in theoretical physics. This book takes that step by describing a coherent mathematical framework for both the evolution of discrete space-time and the quantum graviton in the Planck regime. In doing so it successfully blends developments in both loop quantum gravity and superstring theory with data from the Large Hadron Collider. The result is a coherent mathematical framework that encapsulates newer developments on both the evolution of discrete space-time and the quantum graviton in the early universe. In doing so it addresses a key issue concerning loop quantum gravity raised by Ashtekar: describing conditions for the existence of locally translation-invariant quantum states; a quantum form of the diffeomorphisms of general relativity **Key Features** * Addresses one of the key questions in theoretical physics: quantum gravity * Accessible to graduate students * Presents a coherent mathematical framework * Author is winner of the President's Medal of the Operational Research Society, the Napier medal in mathematics and the Einstein Medal of the US Marine Corps PRELIMS.pdf 1 Acknowledgements 14 Author biography 15 James Moffat 15 Symbols 16 CH001.pdf 1 Chapter 1 Quantum theory 17 1.1 Basic notions 17 1.2 Non-relativistic quantum theory 18 1.2.1 The measurement process 20 1.2.2 The Schrödinger equation 21 1.3 Exploiting quantum theory embedded in classical gravity 22 1.4 Special relativity 25 1.4.1 Relativistic waves 26 1.4.2 Jones vectors to describe classical polarization states 27 1.4.3 The relativistic quantum photon 28 1.5 Dirac relativistic spinor theory 30 1.5.1 The relativistic spinor 31 1.6 von Neumann algebras 33 1.7 A higher level of abstraction: quantum W*-algebras 35 1.8 A brief comparison with the approach of Rovelli and Penrose11Rovelli, Penrose and Ashtekar were three of the founding fathers of loop quantum gravity along with Smolin and Sen. We discuss Ashtekar’s perspective in chapter 4 as a key motivation for that chapter. 36 References 37 CH002.pdf 1 Chapter 2 Computational spin networks and quantum paths in space–time 39 2.1 Introduction 39 2.2 The measurement of space and time 40 2.3 Computational spin networks 43 2.4 The homology invariants of space–time 45 2.5 Quantum paths in space–time 48 2.5.1 Fibre bundle structure of classical phase space 49 2.5.2 An example of the Weyl form 50 2.6 Fractal paths in classical space–time 54 2.7 Supersymmetry and the spinor calculus 55 2.7.1 2-Spinors 55 2.7.2 4-Spinors and the Lorentz and Poincaré groups 57 2.7.3 4-Spinors and the spin groups 59 2.8 Irreducible representations of the Poincaré Lie algebra 60 2.8.1 The supersymmetric extension of the standard model (SSM) 60 2.9 Dirac spinors and the spinor calculus 61 2.9.1 The supersymmetry algebra 63 References 64 CH003.pdf 1 Chapter 3 Particles in algebraic quantum gravity 66 3.1 Introduction 66 3.2 Lie groups, fibre bundles and quantum fields in loop quantum gravity 66 3.2.1 A coherent approach to quantum fields 68 3.3 A remarkable theorem 69 3.4 Properties of the projection onto the base space B of a Stonean fibre bundle K 71 3.4.1 Lifting from the base space 71 3.5 Quantum connections 72 3.6 The supersymmetric extension of the standard model 73 3.7 Factorial representations of the graded Lie algebra 76 3.8 Does supersymmetry exist? ATLAS results for Run 2 of the LHC at 13 TeV 78 3.9 Adding fermions and bosons to the mix 80 3.10 The 10-dimensional pure gravity action 81 3.11 Adding bosons to the theory 82 3.12 Adding fermions 82 3.13 Symmetry breaking to create mass 84 References 86 CH004.pdf 1 Chapter 4 The algebraic nature of reality 88 4.1 Introduction 88 4.2 Symmetry invariance and symmetry breaking in Yang–Mills quantum fields 89 4.2.1 Wigner sets and symmetry invariance in the local algebra O(D) 89 4.2.2 Symmetric Yang–Mills quantum states 91 4.2.3 Superspace 92 4.3 The structure of local algebras of Yang–Mills quantum fields 95 4.4 Developing a diffeomorphism invariant theory for quantum states 96 4.4.1 Wandering projections and diffeomorphism invariant quantum states 98 4.5 The information dynamics of black holes 100 4.6 Summary of chapter 4 102 Further reading 103 References 103 CH005.pdf 1 Chapter 5 Implications 104 5.1 Implications for mathematics 104 5.1.1 Clay Mathematics Institute Millennial question: Yang–Mills quantum theory and the mass gap 104 5.2 Further implications for Physics 107 5.2.1 Discrete closed strings in the early Universe 107 5.2.2 The continuum limit beyond the Planck regime 110 5.2.3 The continuum limit as a quantum field 110 5.2.4 A brief non-technical overview 113 References 114 "The development of a successful theory of quantum gravity in the context of the early universe is the key next step in theoretical physics. This book takes that step by describing a coherent mathematical framework for both the evolution of discrete space-time and the quantum graviton in the Planck regime. In doing so it successfully blends developments in both loop quantum gravity and superstring theory with data from the Large Hadron Collider. The result is a coherent mathematical framework that encapsulates newer developments on both the evolution of discrete space-time and the quantum graviton in the early universe. In doing so it addresses a key issue concerning loop quantum gravity raised by Ashtekar: describing conditions for the existence of locally translation-invariant quantum states; a quantum form of the diffeomorphisms of General Relativity." -- Prové de l'editor