A Mathematically Coherent Quantum Gravity

✍ Scribed by James Moffat

• Publisher: Iop Publishing
• Year: 2020
• Tongue: English
• Leaves: 114
• Category: Library

✦ Synopsis

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

✦ Table of Contents

PRELIMS.pdf
Acknowledgements
Author biography
James Moffat
Symbols
CH001.pdf
Chapter 1 Quantum theory
1.1 Basic notions
1.2 Non-relativistic quantum theory
1.2.1 The measurement process
1.2.2 The Schrödinger equation
1.3 Exploiting quantum theory embedded in classical gravity
1.4 Special relativity
1.4.1 Relativistic waves
1.4.2 Jones vectors to describe classical polarization states
1.4.3 The relativistic quantum photon
1.5 Dirac relativistic spinor theory
1.5.1 The relativistic spinor
1.6 von Neumann algebras
1.7 A higher level of abstraction: quantum W*-algebras
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.
References
CH002.pdf
Chapter 2 Computational spin networks and quantum paths in space–time
2.1 Introduction
2.2 The measurement of space and time
2.3 Computational spin networks
2.4 The homology invariants of space–time
2.5 Quantum paths in space–time
2.5.1 Fibre bundle structure of classical phase space
2.5.2 An example of the Weyl form
2.6 Fractal paths in classical space–time
2.7 Supersymmetry and the spinor calculus
2.7.1 2-Spinors
2.7.2 4-Spinors and the Lorentz and Poincaré groups
2.7.3 4-Spinors and the spin groups
2.8 Irreducible representations of the Poincaré Lie algebra
2.8.1 The supersymmetric extension of the standard model (SSM)
2.9 Dirac spinors and the spinor calculus
2.9.1 The supersymmetry algebra
References
CH003.pdf
Chapter 3 Particles in algebraic quantum gravity
3.1 Introduction
3.2 Lie groups, fibre bundles and quantum fields in loop quantum gravity
3.2.1 A coherent approach to quantum fields
3.3 A remarkable theorem
3.4 Properties of the projection onto the base space B of a Stonean fibre bundle K
3.4.1 Lifting from the base space
3.5 Quantum connections
3.6 The supersymmetric extension of the standard model
3.7 Factorial representations of the graded Lie algebra
3.8 Does supersymmetry exist? ATLAS results for Run 2 of the LHC at 13 TeV
3.9 Adding fermions and bosons to the mix
3.10 The 10-dimensional pure gravity action
3.11 Adding bosons to the theory
3.12 Adding fermions
3.13 Symmetry breaking to create mass
References
CH004.pdf
Chapter 4 The algebraic nature of reality
4.1 Introduction
4.2 Symmetry invariance and symmetry breaking in Yang–Mills quantum fields
4.2.1 Wigner sets and symmetry invariance in the local algebra O(D)
4.2.2 Symmetric Yang–Mills quantum states
4.2.3 Superspace
4.3 The structure of local algebras of Yang–Mills quantum fields
4.4 Developing a diffeomorphism invariant theory for quantum states
4.4.1 Wandering projections and diffeomorphism invariant quantum states
4.5 The information dynamics of black holes
4.6 Summary of chapter 4
Further reading
References
CH005.pdf
Chapter 5 Implications
5.1 Implications for mathematics
5.1.1 Clay Mathematics Institute Millennial question: Yang–Mills quantum theory and the mass gap
5.2 Further implications for Physics
5.2.1 Discrete closed strings in the early Universe
5.2.2 The continuum limit beyond the Planck regime
5.2.3 The continuum limit as a quantum field
5.2.4 A brief non-technical overview
References