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Mathematical Methods for Physics: An Introduction to Group Theory, Topology and Geometry [Kõva köide]

5.00/5 (4 hinnangut Goodreads-ist)
(University of Helsinki), (Università degli Studi di Torino, Italy), (University of Helsinki)
  • Formaat: Hardback, 400 pages, kõrgus x laius x paksus: 262x184x26 mm, kaal: 890 g, Worked examples or Exercises
  • Ilmumisaeg: 22-Dec-2022
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107191130
  • ISBN-13: 9781107191136
Teised raamatud teemal:
Mathematical Methods for Physics: An Introduction to Group Theory, Topology and GeometrySuurem pilt
  • Formaat: Hardback, 400 pages, kõrgus x laius x paksus: 262x184x26 mm, kaal: 890 g, Worked examples or Exercises
  • Ilmumisaeg: 22-Dec-2022
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107191130
  • ISBN-13: 9781107191136
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781107191136.html
Märksõnad:
This detailed yet accessible text provides an essential introduction to the advanced mathematical methods at the core of theoretical physics. The book steadily develops the key concepts required for an understanding of symmetry principles and topological structures, such as group theory, differentiable manifolds, Riemannian geometry, and Lie algebras. Based on a course for senior undergraduate students of physics, it is written in a clear, pedagogical style and would also be valuable to students in other areas of science and engineering. The material has been subject to more than twenty years of feedback from students, ensuring that explanations and examples are lucid and considered, and numerous worked examples and exercises reinforce key concepts and further strengthen readers' understanding. This text unites a wide variety of important topics that are often scattered across different books, and provides a solid platform for more specialized study or research.

Arvustused

'The recent explosive development of topological quantum matter requires a deep systematic understanding of modern mathematics. Quantum many-body entanglement in topological quantum matter is a new phenomenon that requires new mathematical language to describe. This is a rare book that provides systematic and in-depth coverage of some of the most important mathematical concepts, such as groups, geometry, topology and algebra, among others. Many abstract mathematical notions are explained in an easy, explicit fashion. This is an in-depth, friendly book on modern mathematics. Very timely and highly recommended.' Xiao-Gang Wen, Massachusetts Institute of Technology

Muu info

This detailed yet accessible text introduces the advanced mathematical methods at the core of theoretical physics.
1 Introduction
1(3)
2 Group Theory
4(48)
2.1 Groups
4(4)
2.1.1 Definitions: Groups, Abelian Groups, and Related Concepts
4(3)
2.1.2 Examples of Groups
7(1)
2.1.3 Examples of Sets That Are Not Groups
8(1)
2.2 Subgroups
8(1)
2.3 Group Homomorphisms and Isomorphisms
9(2)
2.4 The Smallest Finite Groups
11(4)
2.5 Permutations, the Symmetric Group, and Cayley's Theorem
15(5)
2.6 Partitions, Young Diagrams, and Multisets
20(3)
2.6.1 Partitions
20(1)
2.6.2 Young Diagrams
21(1)
2.6.3 Multisets
22(1)
2.7 Free Groups, Presentations of Groups, and Braid Groups
23(6)
2.7.1 Free Groups and Presentations
23(2)
2.7.2 Braids
25(4)
2.8 Continuous Groups and Lie Groups
29(8)
2.8.1 Matrix Lie Groups
31(6)
2.9 Groups Acting on a Set
37(15)
Problems
48(4)
3 Representation Theory of Groups
52(47)
3.1 Complex Vector Spaces and Representations
52(6)
3.1.1 Symmetry Transformations in Quantum Mechanics
56(2)
3.2 Reducibility of Representations
58(3)
3.3 Irreducible Representations
61(4)
3.4 Characters
65(5)
3.5 The Regular Representation
70(3)
3.6 Dual Vectors and Tensors
73(22)
3.6.1 Visualizing Contractions by Tensor Diagrams
81(2)
3.6.2 Tensor Products of Vector Spaces and Dual Vector Spaces
83(4)
3.6.3 Tensor Products of Linear Operators
87(2)
3.6.4 Kronecker Product and Outer Product
89(4)
3.6.5 Traces, Partial Traces, and Determinants
93(2)
3.7 A Spin-Chain Example
95(4)
Problems
97(2)
4 Differentiable Manifolds
99(71)
4.1 Topology
99(8)
4.1.1 Topological Spaces
99(5)
4.1.2 Manifolds
104(3)
4.2 Homotopy
107(13)
4.2.1 Homotopy Groups
107(3)
4.2.2 Properties of the Fundamental Group
110(1)
4.2.3 Examples of Fundamental Groups
111(3)
4.2.4 Retracts
114(3)
4.2.5 Higher Homotopy Groups
117(2)
4.2.6 Homotopy Groups and Exact Sequences
119(1)
4.3 Differentiable Manifolds
120(22)
4.3.1 Calculus on Manifolds
124(2)
4.3.2 Tangent Vectors
126(2)
4.3.3 One-Forms, or Cotangent Vectors
128(1)
4.3.4 Tensors
129(1)
4.3.5 Vector Fields and Tensor Fields
130(5)
4.3.6 Differential Map and Pullback
135(2)
4.3.7 Flow Generated by a Vector Field
137(2)
4.3.8 Lie Derivative
139(3)
4.4 Differential Forms
142(19)
4.4.1 Exterior Derivative
146(2)
4.4.2 Closed and Exact Differential Forms
148(1)
4.4.3 Interior Product, Exterior Derivative, and Lie Derivative
149(3)
4.4.4 Integration of Differential Forms
152(9)
4.5 Classical Mechanics and Symplectic Geometry
161(9)
Problems
164(6)
5 Riemannian Geometry
170(63)
5.1 The Metric Tensor
170(7)
5.1.1 Lengths of Curves, Distance between Points
172(4)
5.1.2 Raising and Lowering Indices (A Musical Interlude)
176(1)
5.2 The Induced Metric
177(2)
5.3 Affine Connection
179(3)
5.4 Coordinate Transformation Properties of Connection Coefficients
182(2)
5.5 Parallel Transport and Holonomy
184(4)
5.5.1 Holonomy
185(3)
5.5.2 The Geodesic Equation
188(1)
5.6 Covariant Derivative of Tensor Fields
188(4)
5.6.1 Lie Derivative and Covariant Derivative
192(1)
5.7 Metric Connection and Levi-Civita Connection
192(4)
5.8 Covariant Derivative with the Levi-Civita Connection
196(2)
5.9 Geodesies of Levi-Civita Connections
198(3)
5.10 Curvature and Torsion
201(9)
5.11 Non-coordinate Basis
210(4)
5.12 Hypersurfaces and Extrinsic Curvature
214(11)
5.12.1 Hypersurfaces and the First Fundamental Form
215(5)
5.12.2 Extrinsic Curvature and the Second Fundamental Form
220(5)
5.13 Isometries
225(1)
5.14 Killing Vector Fields
226(7)
Problems
228(5)
6 Semisimple Lie Algebras and their Unitary Representations
233(81)
6.1 Lie Groups and Algebras
233(1)
6.2 Lie Algebra of a Lie Group
234(6)
6.2.1 The Exponential Map
237(3)
6.3 Irreducible Representations of Lie Algebras
240(2)
6.4 The Defining Representation and the Adjoint Representation
242(10)
6.4.1 The Defining Representation
242(1)
6.4.2 The Adjoint Representation
243(2)
6.4.3 Dynkin Index, Casimir Operators, and Anomalies
245(4)
6.4.4 The Algebra of the SU(2) Group
249(3)
6.5 Roots and Weights
252(14)
6.5.1 An su(2) Subalgebra
258(2)
6.5.2 The Algebra of the SU(3) Group
260(2)
6.5.3 Ordering the Weight Vectors
262(1)
6.5.4 Complex Conjugate Representations
263(3)
6.5.5 The Antifundamental Representation of su(3)
266(1)
6.6 Simple Roots
266(8)
6.6.1 Cartan Matrix and Dynkin Diagrams
269(2)
6.6.2 Fundamental Weights and Fundamental Representations
271(1)
6.6.3 Irreducible Representations of su(3)
272(2)
6.7 Building Representations of su(2) with Tensor Products
274(3)
6.8 Tensor Methods to Build su(3) Representations
277(9)
6.8.1 Young Tableaux for su(3)
281(5)
6.9 The su(N) Lie Algebras
286(5)
6.10 Young Tableaux for su(N)
291(3)
6.11 Representations of SU(N) × SU(M) × U(1)
294(6)
6.11.1 Subalgebras, Embeddings, and Branching Rules
298(2)
6.12 The Lorentz-Poincare Group
300(14)
6.12.1 The Lorentz Group
300(7)
6.12.2 The Lorentz-Poincare Group
307(2)
Problems
309(5)
Appendix A Problem Solutions
314(35)
A.1 A Note on the Problem Solutions
314(1)
A.2 Solutions to Problems of
Chapter 2
314(7)
A.3 Solutions to Problems of
Chapter 3
321(2)
A.4 Solutions to Problems of
Chapter 4
323(4)
A.5 Solutions to Problems of
Chapter 5
327(5)
A.6 Solutions to Problems of
Chapter 6
332(17)
Notes 349(1)
References 350(1)
Index 351
Esko Keski-Vakkuri received his Ph.D. in physics from Massachusetts Institute of Technology in 1995, and is currently a senior faculty member at the University of Helsinki. He has previously held positions at the California Institute of Technology and Uppsala University. His research is focused on string theory, black holes, holographic duality, and quantum information. Claus K. Montonen received his Ph.D. from the University of Cambridge in 1974 and later held positions at the Université de Paris XI (CNRS), the Research Institute for Theoretical Physics, Helsinki, and CERN. He held various senior faculty positions in the Department of Physics at the University of Helsinki from 1978 until his retirement in 2011, where he was responsible for curriculum design in theoretical physics. His research interests are in S-matrix theory, string theory and quantum field theory, having made major contributions to early string theory and duality in field and string theory. Marco Panero received his Ph.D. in physics from the University of Turin in 2003, after which he held postdoctoral positions at the Dublin Institute for Advanced Studies, the University of Regensburg, ETH Zurich, the University of Helsinki, and the Autonomous University of Madrid. Since 2014 he has been an Associate Professor in physics at the University of Turin. His main research interests are in lattice field theory and in theoretical high-energy physics.