Progress and Visions in Quantum Theory in View of Gravity : Bridging Foundations of Physics and Mathematics

This book focuses on a critical discussion of the status and prospects of current approaches in quantum mechanics and quantum field theory, in particular concerning gravity. It contains a carefully selected cross-section of lectures and discussions at the seventh conference "Progress and Visions in Quantum Theory in View of Gravity" which took place in fall 2018 at the Max Planck Institute for Mathematics in the Sciences in Leipzig. In contrast to usual proceeding volumes, instead of reporting on the most recent technical results, contributors were asked to discuss visions and new ideas in foundational physics, in particular concerning foundations of quantum field theory. A special focus has been put on the question of which physical principles of quantum (field) theory can be considered fundamental in view of gravity. The book is mainly addressed to mathematicians and physicists who are interested in fundamental questions of mathematical physics. It allows the reader to obtain a broad and up-to-date overview of a fascinating active research area. Preface Contents Algebraic Quantum Field Theory 1 Introduction 2 Algebraic Quantum Mechanics 2.1 Postulates of Quantum Mechanics 2.2 Algebraic Approach 2.3 States and Representations 3 Case Study: The van Hove Model 4 Algebraic QFT 4.1 Basic Requirements 4.2 Examples 4.2.1 Free Real Scalar Field 4.2.2 Real Scalar Field with External Source (van Hove Encore) 4.2.3 Weyl Algebra 4.2.4 Complex Scalar Field 4.2.5 Dirac Field 4.3 Quasifree States for the Free Scalar Field 4.3.1 Examples 5 The Spectrum Condition and Reeh–Schlieder Theorem 5.1 The Spectrum Condition 5.2 The Reeh–Schlieder Theorem 6 Local von Neumann Algebras and Their Universal Type 7 The Split Property 8 Superselection Sectors 8.1 Representations of Interest in Particle Physics 8.2 Localized Endomorphisms 8.3 Intertwiners and Permutation Symmetry 9 Conclusions Appendix 1: Some Basic Functional Analysis Appendix 2: Construction of an Algebra from Generators and Relations Appendix 3: Fock Space References Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts 1 Unifying Quantum Field Theory and General Relativity 2 Overview of Concepts and Mathematical Structures in Theoretical Physics 2.1 The Fabric of Spacetime 2.1.1 Topological Manifolds as Models of Spacetime 2.1.2 Establishing Smooth Structures in Spacetime 2.1.3 Encoding the Lorentzian Geometry of Spacetime 2.2 The Einstein Field Equations 2.3 Quantum Theory in a Classical Spacetime 2.4 Incompatibility of General Relativity and Quantum Field Theory 2.5 A Step Back: Quantum Mechanics in Curved Spacetime 2.5.1 The Dirac Equation in Minkowski Space 2.5.2 The Dirac Equation in Curved Spacetime 3 Conceptual and Mathematical Foundations of Causal Fermion Systems 3.1 Guiding Principles of the Theory of Causal Fermion Systems 3.2 Unified Description of Spacetime and the Objects Therein 3.2.1 The Measure Space (F, B, ρ) 3.2.2 The Causal Action Principle 3.3 The Equivalence Principle 3.4 Principle of Causality 3.5 Local Gauge Principle 3.6 Fermionic Building Blocks 3.7 Microscopic Spacetime Structure 4 Modelling a Lorentzian Spacetime by a Causal Fermion System 4.1 General Construction in Curved Spacetimes 4.2 Physical Significance of the Regularization Operator 4.3 Concrete Example: The Minkowski Vacuum 5 Results of the Theory and Further Reading References Quantum Spacetime and the Renormalization Group: Progress and Visions 1 Introduction 2 Functional Renormalization Group: Brief Overview 3 The Asymptotic Safety Scenario for Quantum Gravity 4 Background-Independent Renormalization Group Flows 5 Visions: Bridging the Gap Between Different Approaches to Quantum Gravity References Proposal 42: A New Storyline for the Universe Based on the Causal Fermion Systems Framework 1 Introduction 2 Causal Fermion Systems 2.1 Continuum Limit 3 Dynamical 3.1 Thermodynamic Interpretation 3.2 Dynamical Gravitational Coupling 4 Mechanism of Matter Creation 5 Reheating Uncertainty 6 The Story Line 6.1 Slow Roll to Instability 6.2 Dynamical Gravitational Coupling 7 Discussion of the Causal Fermion Systems Framework 7.1 Relevant Results 7.2 Open Questions 8 Conclusion References Energy Inequalities in Interacting Quantum Field Theories 1 Introduction 2 QEIs in Integrable Systems at One-Particle Level 3 Special Example: The Ising Model 4 Conclusions References Snyder-de Sitter Meets the Grosse-Wulkenhaar Model 1 Introduction 2 Snyder-de Sitter Model 3 Two-Point Function Renormalization 4 Conclusions References Fakeons, Quantum Gravity and the Correspondence Principle 1 Introduction 2 Fakeons 3 Quantum Gravity 4 The Dressed Propagators 5 Projection and Classicization 6 The Upgraded Correspondence Principle 6.1 Uniqueness 6.2 Causality 7 Conclusions References Implementation of the Quantum Equivalence Principle 1 Introduction 2 Basic Idea in This Paper 3 Background 4 Path from the Path Integral to Extended States for a Particle 5 From the Path Integral to Extended States in Quantum Gravity 6 Quantum Diffeomorphisms 7 Beables 8 Quantum Coordinate Systems 9 Quantum Coordinate Transformations 10 Implementing the QEP 11 General Relativity 11.1 The Problem of Relativistic Gravity 11.2 How Einstein Solved the Problem of Relativistic Gravity 12 Quantum Gravity 12.1 The Problem of Quantum Gravity 12.2 A Proposed Path to a Theory of Quantum Gravity 12.3 How Can We Use the QEP? 12.4 Quantum Manifolds 12.5 Quantum Tensor Fields 13 Questions, Comments and Conclusions References The D-CTC Condition in Quantum Field Theory 1 The D-CTC Condition: Bipartite Quantum Systems 2 The D-CTC Condition and Dynamics on CTC Spacetimes 3 Relativistic Quantum Field Theory 4 The D-CTC Condition in QFT on Globally Hyperbolic Spacetimes 5 Discussion References Remarks on Matter-Gravity Entanglement, Entropy, Information Loss and Events 1 Introduction 2 A Classic Thought Experiment in Nonrelativistic Quantum Mechanics 3 Environment Decoherence: Pros and Cons 4 Mathematical Interlude 5 The Matter-Gravity Entanglement Hypothesis 6 The Thermal Atmosphere Puzzle and Its Resolution 7 Resolution of the Black Hole Information Loss Puzzle—and of the Second Law Puzzle 8 Open Systems (Cups of Coffee) 9 Events and Time 9.1 Unravelings 9.2 A Different Possible Approach to Events Which Happen (With or Without Resets) and the Symmetry Puzzle 9.3 An Alternative Different Possible Approach to Events Which Happen (Again With or Without Resets) 9.4 Entropy 9.5 Time References A Generally Covariant Measurement Scheme for Quantum Field Theory in Curved Spacetimes 1 Introduction 2 System, Probe, and Coupling 3 Induced System Observables 4 Instruments and Change of State 5 A Specific Model 6 Conclusion References Understanding ``Understanding'' 1 Introduction 2 Modes of Understanding 3 The Unreasonable Effectiveness of Physics in Mathematics 4 The Donaldson Polynomial Invariants 5 Another Perspective on γ0(M) 6 Witten's Topological Quantum Field Theory 7 Seiberg–Witten 8 More Questions References