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Elementary Introduction to Quantum Geometry [Hardback]

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  • Formāts: Hardback, 280 pages, height x width: 234x156 mm, weight: 640 g, 71 Line drawings, black and white; 71 Illustrations, black and white
  • Izdošanas datums: 02-Nov-2022
  • Izdevniecība: CRC Press
  • ISBN-10: 1032335556
  • ISBN-13: 9781032335551
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Elementary Introduction to Quantum GeometryPalielināt
  • Formāts: Hardback, 280 pages, height x width: 234x156 mm, weight: 640 g, 71 Line drawings, black and white; 71 Illustrations, black and white
  • Izdošanas datums: 02-Nov-2022
  • Izdevniecība: CRC Press
  • ISBN-10: 1032335556
  • ISBN-13: 9781032335551
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781032335551.html
Keywords:

This graduate textbook provides an introduction to quantum gravity, when spacetime is two-dimensional.



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

 

 

Preface ix
Author xi
Chapter 1 Preliminary Material Part 1: The Path Integral
1(14)
1.1 The Classical Action
1(1)
1.2 Statistical Mechanics
2(3)
1.3 Classical to Quantum
5(2)
1.4 The Feynman Path Integral in Quantum Mechanics
7(5)
1.5 The Feynman-Kac Path Integral and Imaginary Time
12(2)
1.6 Problem Sets and Further Reading
14(1)
Chapter 2 The Free Relativistic Particle
15(10)
2.1 The Propagator
15(2)
2.2 The Path Integral
17(4)
2.3 Random Walks and Universality
21(2)
2.4 ProblemSets and Further Reading
23(2)
Chapter 3 One-Dimensional Quantum Gravity
25(10)
3.1 Scalar Fields in One Dimension
25(5)
3.2 Hausdorff Dimension and Scaling Relations
30(4)
3.3 Problem Sets and Further Reading
34(1)
Chapter 4 Branched Polymers
35(14)
4.1 Definitions and Generalities
35(2)
4.2 Rooted Branched Polymers and Universality
37(2)
4.3 The Two-Point Function
39(3)
4.4 Intrinsic Properties of Branched Polymers
42(2)
4.5 Multicritical Branched Polymers
44(2)
4.6 Global and Local Hausdorff Dimensions
46(1)
4.7 Problem Sets and Further Reading
47(2)
Chapter 5 Random Surfaces and Bosonic Strings
49(32)
5.1 The Action, Green Functions and Critical Exponents
49(6)
5.2 Regularizing the Integration over Geometries
55(8)
5.3 Digression: Summation over Topologies
63(5)
5.4 Scaling of the Mass
68(8)
5.5 Scaling of the String Tension
76(3)
5.6 Problem Sets and Further Reading
79(2)
Chapter 6 Two-Dimensional Quantum Gravity
81(26)
6.1 Solving 2D Quantum Gravity by Counting Geometries
81(2)
6.2 Counting Triangulations of the Disk
83(9)
6.3 Multiloops and the Loop-Insertion Operator
92(1)
6.4 Explicit Solution for Bipartite Graphs
93(3)
6.5 The Number of Large Triangulations
96(3)
6.6 The Continuum Limit
99(3)
6.7 Other Universality Classes
102(2)
6.8 Appendix
104(1)
6.9 Problem Sets and Further Reading
105(2)
Chapter 7 The Fractal Structure of 2D Gravity
107(14)
7.1 Universality and the Missing Correlation Length
107(1)
7.2 The Two-Loop Propagator
107(7)
7.3 The Two-Point Function
114(3)
7.4 The Local Hausdorff Dimension in 2D Gravity
117(3)
7.5 Problem Sets and Further Reading
120(1)
Chapter 8 The Causal Dynamical Triangulation model
121(24)
8.1 Lorentzian Versus Euclidean Set Up
121(1)
8.2 Denning and Solving the CDT Model
122(9)
8.3 GCDT: Showcasing Quantum Geometry
131(6)
8.4 GCDT Defined as a Scaling Limit of Graphs
137(3)
8.5 The Classical Continuum Theory Related to 2D CDT
140(2)
8.6 Problem Sets and Further Reading
142(3)
Appendix A Preliminary Material Part 2: Green Functions
145(16)
Appendix B Problem Sets 1--13
161(64)
B.1 Problem Set 1
161(3)
B.2 Problem Set 2
164(3)
B.3 Problem Set 3
167(3)
B.4 Problem Set 4
170(5)
B.5 Problem Set 5
175(3)
B.6 Problem Set 6
178(7)
B.7 Problem Set 7
185(6)
B.8 Problem Set 8
191(4)
B.9 Problem Set 9
195(9)
B.10 Problem Set 10
204(6)
B.11 Problem Set 11
210(7)
B.12 Problem Set 12
217(2)
B.13 Problem Set 13
219(6)
Appendix C Solutions to Problem Sets 1--13
225(48)
C.1 Solutions to Problem Set 1
225(4)
C.2 Solutions to Problem Set 2
229(3)
C.3 Solutions to Problem Set 3
232(3)
C.4 Solutions to Problem Set 4
235(3)
C.5 Solutions to problem set 5
238(5)
C.6 Solutions to Problem Set 6
243(4)
C.7 Solutions to Problem Set 7
247(3)
C.8 Solutions to Problem Set 8
250(2)
C.9 Solutions to Problem Set 9
252(5)
C.10 Solutions to Problem Set 10
257(3)
C.11 Solutions to Problem Set 11
260(3)
C.12 Solutions to Problem Set 12
263(4)
C.13 Solutions to Problem Set 13
267(6)
References 273(2)
Index 275
Jan Ambjųrn is a Danish physicist regarded as one of the founders of the statistical theory of geometries. The formalism has been applied to bosonic strings and to quantum gravity in two and higher dimensions, and it was developed as a tool to study string theory and quantum gravity non-perturbatively. A later development, especially designed to study quantum gravity, is known as Causal Dynamical Triangulation Theory. During his career Ambjųrn has done research in numerous other areas, including quantum field theory and QCD, lattice gauge theories, the baryon asymmetry of the universe, matrix models, non-commutative field theory, string theory as well as the statistical theories of random paths and random surfaces. He is currently professor at the Niels Bohr Institute, Copenhagen and Radboud University, Nijmegen.