Elementary Introduction to Quantum Geometry

Description

This graduate textbook provides an introduction to quantum gravity, when spacetime is two-dimensional. The quantization of gravity is the main missing piece of theoretical physics, but in two dimensions it can be done explicitly with elementary mathematical tools, but it still has most of the conceptional riddles present in higher dimensional (not yet known) quantum gravity. It provides an introduction to a very interdisciplinary field, uniting physics (quantum geometry) and mathematics (combinatorics) in a non-technical way, requiring no prior knowledge of quantum field theory or general relativity. Using the path integral, the chapters provide self-contained descriptions of random walks, random trees and random surfaces as statistical systems where the free relativistic particle, the relativistic bosonic string and two-dimensional quantum gravity are obtained as scaling limits at phase transition points of these statistical systems. The geometric nature of the theories allows one to perform the path integral by counting geometries. In this way the quantization of geometry becomes closely linked to the mathematical fields of combinatorics and probability theory. By counting the geometries, it is shown that the two-dimensional quantum world is fractal at all scales unless one imposes restrictions on the geometries. It is also discussed in simple terms how quantum geometry and quantum matter can interact strongly and change the properties both of the geometries and of the matter systems. It requires only basic undergraduate knowledge of classical mechanics, statistical mechanics and quantum mechanics, as well as some basic knowledge of mathematics at undergraduate level. It will be an ideal textbook for graduate students in theoretical and statistical physics and mathematics studying quantum gravity and quantum geometry. Key features: Presents the first elementary introduction to quantum geometry Explores how to understand quantum geometry without prior knowledge beyond bachelor level physics and mathematics. Contains exercises, problems and solutions to supplement and enhance learning Cover Half Title Title Page Copyright Page Contents Preface Author Chapter 1: Preliminary Material Part 1: The Path Integral 1.1. The Classical Action 1.2. Statistical Mechanics 1.3. Classical to Quantum 1.4. The Feynman Path Integral in Quantum Mechanics 1.5. The Feynman-Kac Path Integral and Imaginary Time 1.6. Problem Sets and Further Reading Chapter 2: The Free Relativistic Particle 2.1. The Propagator 2.2. The Path Integral 2.3. Random Walks and Universality 2.4. ProblemSets and Further Reading Chapter 3: One-Dimensional Quantum Gravity 3.1. Scalar Fields in One Dimension 3.2. Hausdorff Dimension and Scaling Relations 3.3. Problem Sets and Further Reading Chapter 4: Branched Polymers 4.1. Definitions and Generalities 4.2. Rooted Branched Polymers and Universality 4.3. The Two-Point Function 4.4. Intrinsic Properties of Branched Polymers 4.5. Multicritical Branched Polymers 4.6. Global and Local Hausdorff Dimensions 4.7. Problem Sets and Further Reading Chapter 5: Random Surfaces and Bosonic Strings 5.1. The Action, Green Functions and Critical Exponents 5.2. Regularizing the Integration over Geometries 5.3. Digression: Summation over Topologies 5.4. Scaling of the Mass 5.5. Scaling of the String Tension 5.6. Problem Sets and Further Reading Chapter 6: Two-Dimensional Quantum Gravity 6.1. Solving 2D Quantum Gravity by Counting Geometries 6.2. Counting Triangulations of the Disk 6.3. Multiloops and the Loop-Insertion Operator 6.4. Explicit Solution for Bipartite Graphs 6.5. The Number of Large Triangulations 6.6. The Continuum Limit 6.7. Other Universality Classes 6.8. Appendix 6.9. Problem Sets and Further Reading Chapter 7: The Fractal Structure of 2D Gravity 7.1. Universality and the Missing Correlation Length 7.2. The Two-Loop Propagator 7.3. The Two-Point Function 7.4. The Local Hausdorff Dimension in 2D Gravity 7.5. Problem Sets and Further Reading Chapter 8: The Causal Dynamical Triangulation model 8.1. Lorentzian Versus Euclidean Set Up 8.2. Defining and Solving the CDT Model 8.3. GCDT: Showcasing Quantum Geometry 8.4. GCDT Defined as a Scaling Limit of Graphs 8.5. The Classical Continuum Theory Related to 2D CDT 8.6. Problem Sets and Further Reading Appendix A: Preliminary Material Part 2: Green Functions Appendix B: Problem Sets 1–13 B.1. Problem Set 1 B.2. Problem Set 2 B.3. Problem Set 3 B.4. Problem Set 4 B.5. Problem Set 5 B.6. Problem Set 6 B.7. Problem Set 7 B.8. Problem Set 8 B.9. Problem Set 9 B.10. Problem Set 10 B.11. Problem Set 11 B.12. Problem Set 12 B.13. Problem Set 13 Appendix C: Solutions to Problem Sets 1–13 C.1. Solutions to Problem Set 1 C.2. Solutions to Problem Set 2 C.3. Solutions to Problem Set 3 C.4. Solutions to Problem Set 4 C.5. Solutions to problem set 5 C.6. Solutions to Problem Set 6 C.7. Solutions to Problem Set 7 C.8. Solutions to Problem Set 8 C.9. Solutions to Problem Set 9 C.10. Solutions to Problem Set 10 C.11. Solutions to Problem Set 11 C.12. Solutions to Problem Set 12 C.13. Solutions to Problem Set 13 References Index