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Symmetry, Broken Symmetry, and Topology in Modern Physics: A First Course [Hardback]

(Shanghai Jiao Tong University, China), (University of Tennessee, Knoxville)
  • Formāts: Hardback, 664 pages, height x width x depth: 253x194x33 mm, weight: 1560 g, Worked examples or Exercises
  • Izdošanas datums: 31-Mar-2022
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 1316518612
  • ISBN-13: 9781316518618
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Symmetry, Broken Symmetry, and Topology in Modern Physics: A First CoursePalielināt
  • Formāts: Hardback, 664 pages, height x width x depth: 253x194x33 mm, weight: 1560 g, Worked examples or Exercises
  • Izdošanas datums: 31-Mar-2022
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 1316518612
  • ISBN-13: 9781316518618
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781316518618.html
Keywords:
Written for use in teaching and for self-study, this book provides a comprehensive and pedagogical introduction to groups, algebras, geometry, and topology. It assimilates modern applications of these concepts, assuming only an advanced undergraduate preparation in physics. It provides a balanced view of group theory, Lie algebras, and topological concepts, while emphasizing a broad range of modern applications such as Lorentz and Poincaré invariance, coherent states, quantum phase transitions, the quantum Hall effect, topological matter, and Chern numbers, among many others. An example based approach is adopted from the outset, and the book includes worked examples and informational boxes to illustrate and expand on key concepts. 344 homework problems are included, with full solutions available to instructors, and a subset of 172 of these problems have full solutions available to students.

Recenzijas

'The whole of theoretical physics, and our general picture of the world, are based on symmetries. This book is devoted to symmetries and their manifestations in nature, and it allows students to develop a theoretical and experimental understanding of the fundamental properties of the Universe. This path is carefully paved by the authors.' Professor Vladimir Zelevinsky, Michigan State University 'What sets this book apart from the majority of popular books covering similar subject matter is its remarkable combination of in-depth theory and practical applications. It covers a wide range of special topics, including electrons on periodic lattices, harmonic oscillators, the Lorentz group, the Higgs mechanism and quantum phase transitions, ensuring its appeal to both mathematicians and physicists. This book is highly esteemed for its pedagogical approach.' Firdous Ahmad Mala, Acta Crystallographica

Papildus informācija

A pedagogical introduction to the modern applications of groups, algebras, and topology for undergraduate and graduate students in physics.
Preface xxix
Part I Symmetry Groups and Algebras
1 Introduction
3(2)
2 Some Properties of Groups
5(35)
2.1 Invariance and Conservation Laws
5(2)
2.2 Definition of a Group
7(1)
2.3 Examples of Groups
8(4)
2.3.1 Additive Group of Integers
8(1)
2.3.2 Rotation and Translation Groups
9(1)
2.3.3 Parameterization of Continuous Groups
10(1)
2.3.4 Permutation Groups
11(1)
2.4 Subgroups
12(1)
2.5 Homomorphism and Isomorphism
13(2)
2.6 Matrix Representations
15(3)
2.6.1 A Matrix Representation of S3
16(1)
2.6.2 Dimensionality of Matrix Representations
16(1)
2.6.3 Linear Operators and Matrix Representations
17(1)
2.7 Reducible and Irreducible Representations
18(1)
2.8 Degenerate Multiplet Structure
18(2)
2.9 Some Examples of Matrix Groups
20(1)
2.9.1 General Linear Groups
20(1)
2.9.2 Unitary Groups
20(1)
2.9.3 Orthogonal Groups
21(1)
2.9.4 Symplectic Groups
21(1)
2.10 Group Generators
21(1)
2.11 Conjugate Classes
22(2)
2.12 Invariant Subgroups
24(1)
2.13 Simple and Semisimple Groups
25(1)
2.14 Cosets and Factor Groups
26(3)
2.14.1 Left and Right Coset Decompositions
26(2)
2.14.2 Factor Groups
28(1)
2.15 Direct Product Groups
29(2)
2.16 Direct Product of Representations
31(1)
2.17 Characters of Representations
31(9)
2.17.1 Character Theorems
32(2)
2.17.2 Character Tables
34(1)
Background and Further Reading
35(1)
Problems
35(5)
3 Introduction to Lie Groups
40(25)
3.1 Lie Groups
40(2)
3.2 Lie Algebras
42(3)
3.2.1 Invariant Subalgebras
43(1)
3.2.2 Adjoint Representation of the Algebra
44(1)
3.3 Angular Momentum and the Group SU(2)
45(8)
3.3.1 Fundamental Representation of SU(2)
45(2)
3.3.2 The Cartan--Dynkin Method
47(1)
3.3.3 Cartan--Dynkin Analysis of SU(2)
48(2)
3.3.4 The Clebsch--Gordan Series for SU(2)
50(2)
3.3.5 SU(2) Adjoint Representation
52(1)
3.4 Isospin
53(5)
3.4.1 The Neutron--Proton System
53(1)
3.4.2 Algebraic Structure for Isospin
54(1)
3.4.3 The U(1) and SU(2) Subgroups of U(2)
55(1)
3.4.4 Analogy between Angular Momentum and Isospin
56(2)
3.4.5 The Adjoint Representation of Isospin
58(1)
3.5 The Importance of Lie Groups in Physics
58(1)
3.6 Symmetry and Dynamics
59(6)
3.6.1 Local Gauge Theories
59(1)
3.6.2 Dynamical Symmetries
60(1)
Background and Further Reading
60(1)
Problems
60(5)
4 Permutation Groups
65(10)
4.1 Young Diagrams
65(2)
4.1.1 Two-Particle Young Diagrams
66(1)
4.1.2 Many-Particle Young Diagrams
66(1)
4.1.3 A Compact Notation
66(1)
4.2 Standard Arrangement of Young Tableaux
67(1)
4.3 Irreducible Representations
68(1)
4.3.1 Counting Standard Arrangements
68(1)
4.3.2 The Hook Rule
69(1)
4.4 Basis Vectors
69(1)
4.5 Products of Representations
70(5)
4.5.1 Direct Products
71(1)
4.5.2 Outer Products
72(1)
Background and Further Reading
73(1)
Problems
73(2)
5 Electrons on Periodic Lattices
75(22)
5.1 The Direct Lattice
75(1)
5.1.1 Brevais Lattices
75(1)
5.1.2 Wigner--Seitz Cells
76(1)
5.2 The Reciprocal Lattice
76(1)
5.3 Brillouin Zones
77(2)
5.4 Bloch's Theorem
79(1)
5.5 Electronic Band Structure
80(2)
5.6 Point Groups
82(2)
5.6.1 Point Group Operations
83(1)
5.6.2 The Crystallographic Point Groups
83(1)
5.7 Example: The Ammonia Molecule
84(8)
5.7.1 Symmetry Operations
84(3)
5.7.2 A Matrix Representation
87(3)
5.7.3 Class Structure
90(1)
5.7.4 Other Irreducible Representations
91(1)
5.8 General Lattice Symmetry Classifications
92(1)
5.9 Space Groups
93(4)
5.9.1 Elements of the Space Group
93(1)
5.9.2 Symmorphic Space Groups
93(1)
Background and Further Reading
94(1)
Problems
94(3)
6 The Rotation Group
97(29)
6.1 Three-Dimensional Rotations
97(1)
6.2 The SO(2) Group
98(7)
6.2.1 Generators of SO(2) Rotations
98(1)
6.2.2 SO(2) Irreducible Representations
99(2)
6.2.3 Connectedness of the Manifold
101(1)
6.2.4 Compactness of the Manifold
102(1)
6.2.5 Invariant Group Integration
103(2)
6.3 The SO(3) Group
105(7)
6.3.1 Generators of SO(3)
105(1)
6.3.2 Matrix Elements of the Rotation Operator
105(2)
6.3.3 Properties of D-Matrices
107(1)
6.3.4 Characters for SO(3)
108(1)
6.3.5 Direct Products of SO(3) Representations
109(1)
6.3.6 SO(3) Vector-Coupling Coefficients
109(1)
6.3.7 Properties of SO(3) Clebsch--Gordan Coefficients
110(1)
6.3.8 37 Symbols
111(1)
6.3.9 Construction of SO(3) Irreducible Multiplets
111(1)
6.4 Tensor Operators under Group Transformations
112(2)
6.5 Tensors for the Rotation Group
114(1)
6.6 SO(3) Tensor Products
115(1)
6.7 The Wigner--Eckart Theorem
116(1)
6.8 The Wigner--Eckart Theorem for SO(3)
117(2)
6.8.1 Reduced Matrix Elements
117(1)
6.8.2 Selection Rules
118(1)
6.9 Relationship of SO(3) and SU(2)
119(7)
6.9.1 SO(3) and SU(2) Group Manifolds
119(1)
6.9.2 Universal Covering Group of the SU(2) Algebra
120(1)
Background and Further Reading
121(1)
Problems
121(5)
7 Classification of Lie Algebras
126(17)
7.1 Adjoint Representations
126(3)
7.1.1 The Cartan Subalgebra
126(1)
7.1.2 Raising and Lowering Operators
127(2)
7.2 The Cartan--Weyl Basis
129(1)
7.2.1 Semisimple Algebras
129(1)
7.2.2 Metric Tensor, Semisimplicity, and Compactness
129(1)
7.3 Structure of the Root Space
130(2)
7.3.1 Root Space Restrictions
130(1)
7.3.2 Lengths and Angles for Root Vectors
131(1)
7.4 Construction of Root Diagrams
132(4)
7.4.1 Rank-1 and Rank-2 Compact Lie Algebras
133(2)
7.4.2 An Ordering Prescription for Weights
135(1)
7.5 Simple Roots
136(1)
7.6 Dynkin Diagrams
137(4)
7.6.1 The Cartan Matrix
138(1)
7.6.2 Constructing All Roots from Dynkin Diagrams
139(1)
7.6.3 Constructing the Algebra from the Roots
139(2)
7.7 Dynkin Diagrams and the Simple Algebras
141(2)
Background and Further Reading
142(1)
Problems
142(1)
8 Unitary and Special Unitary Groups
143(18)
8.1 Generators and Commutators for SU(3)
143(2)
8.2 SU(3) Casimir Operators
145(1)
8.3 SU(3) Weight Space
145(4)
8.3.1 SU(3) Raising and Lowering Operators
145(1)
8.3.2 SU(3) Irreducible Representations
146(1)
8.3.3 Dimensionality of SU(3) Irreps
146(2)
8.3.4 Construction of SU(3) Weight Diagrams
148(1)
8.4 Complex Conjugate Representations
149(1)
8.5 Real and Complex Representations
149(1)
8.6 Unitary Symmetry and Young Diagrams
150(1)
8.7 Young Diagrams for SU(N)
151(4)
8.7.1 Two Particles in Two States
153(1)
8.7.2 Two Particles in Three States
154(1)
8.7.3 Fundamental and Conjugate Representations
154(1)
8.8 Dimensionality of SU(N) Representations
155(1)
8.9 Direct Products of SU(N) Representations
156(1)
8.10 Weights from Young Diagrams
157(1)
8.11 Graphical Construction of Direct Products
158(3)
Background and Further Reading
158(1)
Problems
159(2)
9 SU(3) Flavor Symmetry
161(13)
9.1 Symmetry in Particle Physics
161(1)
9.1.1 SU(3) Phenomenology and Quarks
161(1)
9.1.2 Non-Abelian Gauge Symmetries
162(1)
9.2 Fundamental SU(3) Quark Representations
162(1)
9.3 SU(3) Flavor Multiplets
163(3)
9.3.1 Mass Splittings in SU(3) Multiplets
164(1)
9.3.2 Quark Structure for Mesons and Baryons
165(1)
9.4 Isospin Subgroups of SU(3)
166(4)
9.4.1 Subgroup Analysis Using Weight Diagrams
167(1)
9.4.2 Subgroup Analysis Using Young Diagrams
168(2)
9.5 Extensions of Flavor SU(3) Symmetry
170(4)
9.5.1 Higher-Rank Flavor Symmetries
170(1)
9.5.2 SU(6) Flavor--Spin Symmetry
171(1)
9.5.3 Baryons and Mesons under SU(6) Symmetry
171(1)
Background and Further Reading
172(1)
Problems
172(2)
10 Harmonic Oscillators and SU(3)
174(17)
10.1 The 3D Quantum Oscillator
174(4)
10.1.1 Eigenvalues
174(1)
10.1.2 Wavefunctions
175(1)
10.1.3 Unitary Symmetry
175(1)
10.1.4 Angular Momentum Subgroup
176(1)
10.1.5 SO(3) Transformation Properties
177(1)
10.1.6 Group Structure
178(1)
10.1.7 Many-Body Operators
178(1)
10.2 SU(3) and the Nuclear Shell Model
178(1)
10.3 SU(3) Classification of SD Shell States
179(6)
10.3.1 Classification Strategy
180(1)
10.3.2 Orbital and Spin--Isospin Symmetry
181(1)
10.3.3 Permutation Symmetry
182(1)
10.3.4 Example: Two Particles in the SD Shell
182(3)
10.4 SU(2) Subgroups and Intrinsic States
185(2)
10.4.1 Weight Space Operators and Diagrams
185(1)
10.4.2 Angular Momentum Content of Multiplets
186(1)
10.5 Collective Motion in the Nuclear SD Shell
187(4)
10.5.1 Hamiltonian
187(1)
10.5.2 Group-Theoretical Solution
188(1)
10.5.3 The Theoretical Spectrum
189(1)
Background and Further Reading
190(1)
Problems
190(1)
11 SU(3) Matrix Elements
191(18)
11.1 Clebsch--Gordan Coefficients for SU(3)
191(1)
11.2 Constructing SU(3) Clebsch--Gordan Coefficients
192(3)
11.3 Matrix Elements of Generators
195(1)
11.4 Isoscalar Factors
195(2)
11.4.1 Racah Factorization Lemma
197(1)
11.4.2 Evaluating and Using Isoscalar Factors
197(1)
11.5 SU(3) ⊃ SO(3) Tensor Operators
197(1)
11.6 The SU(3) Wigner--Eckart Theorem
198(1)
11.7 Structure of SU(3) Matrix Elements
199(1)
11.8 The Gell--Mann, Okubo Mass Formula
200(2)
11.9 SU(3) Oscillator Reduced Matrix Elements
202(4)
11.9.1 Spherical Operators
202(1)
11.9.2 Matrix Elements for Creation and Annihilation Operators
203(1)
11.9.3 Electromagnetic Transitions in the SD Shell
204(2)
11.10 Lie Algebras and Many-Body Systems
206(3)
Background and Further Reading
206(1)
Problems
206(3)
12 Introduction to Non-Compact Groups
209(11)
12.1 Review of the Compact Group SU(n)
209(1)
12.2 The Non-Compact Group SU(l, m)
210(1)
12.2.1 Signature of the Metric
210(1)
12.2.2 Parameter Space for SU(1, 1)
211(1)
12.3 The Non-Compact Group SO(l, m)
211(1)
12.4 Euclidean Groups
212(4)
12.4.1 The Euclidean Group E3 for 3D Space
212(1)
12.4.2 The Euclidean Group E2 for 2D Space
212(2)
12.4.3 Semidirect Product Groups
214(1)
12.4.4 Algebraic Properties of E2
214(1)
12.4.5 Invariant Subgroup of Translations
215(1)
12.5 Method of Induced Representations for E2
216(4)
12.5.1 Generating the Representation
216(2)
12.5.2 Significance of the Abelian Invariant Subgroup
218(1)
Background and Further Reading
218(1)
Problems
218(2)
13 The Lorentz Group
220(20)
13.1 Spacetime Tensors
220(4)
13.1.1 A Covariant Notation
220(2)
13.1.2 Tensor Transformation Laws
222(2)
13.2 Lorentz Transformations
224(3)
13.2.1 Lorentz Boosts as Minkowski Rotations
224(1)
13.2.2 Generators of Boosts and Rotations
225(1)
13.2.3 Commutation Algebra for the Lorentz Group
226(1)
13.3 Classification of Lorentz Transformations
227(2)
13.3.1 The Four Pieces of the Full Lorentz Group
227(1)
13.3.2 Improper Lorentz Transformations
228(1)
13.3.3 Lightcone Classification of Minkowski Vectors
228(1)
13.4 Properties of the Lorentz Group
229(1)
13.5 The Lorentz Group and SL(2,C)
230(1)
13.5.1 A Mapping between 4-Vectors and Matrices
230(1)
13.5.2 The Universal Covering Group of SO(3, 1)
231(1)
13.6 Spinors and Lorentz Transformations
231(2)
13.6.1 SU(2) × SU(2) Representations of the Lorentz Group
232(1)
13.6.2 Two Inequivalent Spinor Representations
232(1)
13.7 Space Inversion for the Lorentz Group
233(3)
13.7.1 Action of Parity on Generators and Representations
234(1)
13.7.2 General and Self-Conjugate Representations
235(1)
13.8 Parity and 4-Spinors
236(1)
13.9 Higher-Dimensional Lorentz Representations
237(1)
13.10 Non-Unitarity of Representations
238(1)
13.11 Meaning of Non-Unitary Representations
238(2)
Background and Further Reading
239(1)
Problems
239(1)
14 Lorentz-Covariant Fields
240(24)
14.1 Lorentz Covariance of Maxwell's Equations
240(3)
14.1.1 Scalar and Vector Potentials
241(1)
14.1.2 Gauge Transformations
241(1)
14.1.3 Manifestly Covariant Form of the Maxwell Equations
242(1)
14.2 The Dirac Equation
243(2)
14.2.1 Lorentz-Boosted Spinors
244(1)
14.2.2 A Lorentz-Covariant Notation
244(1)
14.3 Dirac Bilinear Covariants
245(3)
14.3.1 Covariance of the Dirac Equation
246(1)
14.3.2 Transformation Properties of Bilinear Products
247(1)
14.4 Weyl Equations and Massless Fermions
248(2)
14.5 Chiral Invariance
250(4)
14.5.1 Helicity States for Fermions
250(1)
14.5.2 Dirac Equation in Pauli--Dirac Representation
251(1)
14.5.3 Helicity and Chirality for Dirac Fermions
252(1)
14.5.4 Projection Operators for Chiral Fermions
252(2)
14.5.5 Interactions and Chiral Symmetry
254(1)
14.6 The Majorana Equation
254(3)
14.6.1 Dirac and Majorana Masses
255(1)
14.6.2 Neutrinoless Double β-Decay
256(1)
14.7 Summary: Possible Spinor Types
257(2)
14.8 Spinor Symmetry in the Weak Interactions
259(5)
14.8.1 The Left Hand of the Neutrino
259(1)
14.8.2 Violation of Parity P
259(1)
14.8.3 C, CP, and T Symmetries
260(1)
14.8.4 A More Complete Picture
260(1)
Background and Further Reading
261(1)
Problems
261(3)
15 Poincare Invariance
264(16)
15.1 The Poincare Multiplication Rule
264(1)
15.2 Generators of Poincare Transformations
265(1)
15.2.1 Proper Lorentz Transformations
265(1)
15.2.2 Four-Dimensional Spacetime Translations
266(1)
15.2.3 Commutators for Poincare Generators
266(1)
15.3 Representation Theory of the Poincare Group
266(3)
15.3.1 Casimir Operators for the Poincare Group
266(2)
15.3.2 Classification of Poincare States
268(1)
15.3.3 Method of Induced Representations
269(1)
15.4 Massive Representations of the Poincare Group
269(3)
15.4.1 Quantum Numbers for Massive States
270(1)
15.4.2 Action of the Poincare Group on Massive States
270(1)
15.4.3 Summary: Representations for Massive States
271(1)
15.5 Massless Representations
272(2)
15.5.1 The Standard Lightlike Vector
272(1)
15.5.2 Lie Algebra of the Little Group
273(1)
15.5.3 Quantum Numbers for Massless States
273(1)
15.6 Mass and Spin for Poincare Representations
274(1)
15.7 Lorentz and Poincare Representations
274(6)
15.7.1 Operators for Relativistic Quantum Fields
275(1)
15.7.2 Wave Equations for Quantum Fields
276(1)
15.7.3 Plane-Wave Expansion of the Fields
276(1)
15.7.4 The Relationship of Fields and Particles
277(1)
15.7.5 Symmetry and the Wave Equation
277(1)
Background and Further Reading
278(1)
Problems
278(2)
16 Gauge Invariance
280(21)
16.1 Relativistic Quantum Field Theory
280(4)
16.1.1 Quantization of Classical Fields
280(1)
16.1.2 Symmetries of the Classical Action
281(1)
16.1.3 Lagrangian Densities for Free Fields
281(2)
16.1.4 Euler--Lagrange Field Equations
283(1)
16.2 Conserved Currents and Charges
284(3)
16.2.1 Noether's Theorem
284(1)
16.2.2 Conserved Charges
285(1)
16.2.3 Symmetries for Interacting Fields
286(1)
16.2.4 Partially Conserved Currents
287(1)
16.3 Gauge Invariance in Quantum Mechanics
287(2)
16.4 Gauge Invariance and the Photon Mass
289(1)
16.5 Quantum Electrodynamics
289(2)
16.5.1 Global U(1) Gauge Invariance
289(1)
16.5.2 Local U(1) Gauge Invariance
289(1)
16.5.3 Gauging the U(1) Symmetry
290(1)
16.6 Yang--Mills Fields
291(10)
16.6.1 Non-Abelian Gauge Invariance
291(1)
16.6.2 Covariant Derivatives
292(1)
16.6.3 Non-Abelian Generalization of QED
293(1)
16.6.4 Properties of Non-Abelian Gauge Fields
293(2)
Background and Further Reading
295(1)
Problems
295(6)
Part II Broken Symmetry
17 Spontaneous Symmetry Breaking
301(10)
17.1 Modes of Symmetry Breaking
301(1)
17.2 Explicit Symmetry Breaking
301(1)
17.3 The Vacuum and Hidden Symmetry
302(1)
17.4 Spontaneously Broken Discrete Symmetry
303(2)
17.4.1 Symmetry in the Wigner Mode
303(1)
17.4.2 Spontaneously Broken Symmetry
304(1)
17.4.3 Summary of Spontaneously Broken Discrete Symmetry
305(1)
17.5 Spontaneously Broken Continuous Symmetry
305(6)
17.5.1 Symmetric Classical Vacuum
306(1)
17.5.2 Hidden Continuous Symmetry
306(1)
17.5.3 The Goldstone Theorem
307(2)
17.5.4 The Stability Subgroup
309(1)
Background and Further Reading
309(1)
Problems
310(1)
18 The Higgs Mechanism
311(13)
18.1 Photons and the Higgs Loophole
311(1)
18.2 The Abelian Higgs Model
312(3)
18.2.1 Lagrangian Density
312(1)
18.2.2 Symmetry Breaking
313(1)
18.2.3 Understanding the Higgs Mechanism
314(1)
18.3 Vacuum Screening Currents
315(6)
18.3.1 Gauge Invariance and Mass
315(1)
18.3.2 Screening Currents and Effective Mass
316(1)
18.3.3 Atomic Screening Currents
316(2)
18.3.4 The Meissner Effect and Massive Photons
318(2)
18.3.5 Gauge Invariance and Longitudinal Polarization
320(1)
18.4 The Higgs Boson
321(3)
Background and Further Reading
322(1)
Problems
322(2)
19 The Standard Model
324(16)
19.1 The Standard Electroweak Model
324(8)
19.1.1 Guidance from Data
324(2)
19.1.2 The Gauge Group
326(1)
19.1.3 Electroweak Lagrangian Density
327(2)
19.1.4 The Electroweak Higgs Mechanism
329(1)
19.1.5 Particle Spectrum
330(2)
19.2 Quantum Chromodynamics
332(6)
19.2.1 A Color Gauge Theory
332(1)
19.2.2 The QCD Lagrangian Density
333(1)
19.2.3 Symmetries of the QCD Lagrangian Density
334(1)
19.2.4 Asymptotic Freedom and Confinement
335(1)
19.2.5 Exotic Hadrons and Glueballs
336(2)
19.3 The Gauge Theory of Fundamental Interactions
338(2)
Background and Further Reading
338(1)
Problems
338(2)
20 Dynamical Symmetry
340(28)
20.1 The Microscopic Dynamical Symmetry Method
340(3)
20.1.1 Solution Algorithm
341(1)
20.1.2 Validity and Utility of the Approach
342(1)
20.1.3 Spontaneously Broken Symmetry and Dynamical Symmetry
343(1)
20.1.4 Kinematics and Dynamics
343(1)
20.2 Monolayer Graphene in a Strong Magnetic Field
343(18)
20.2.1 Electronic Dispersion in Monolayer Graphene
344(1)
20.2.2 Landau Levels for Massless Dirac Electrons
345(3)
20.2.3 SU(4) Quantum Hall Ferromagnetism
348(1)
20.2.4 Fermion Dynamical Symmetries for Graphene
349(4)
20.2.5 Graphene SO(8) Dynamical Symmetries
353(2)
20.2.6 Generalized Coherent States for Graphene
355(3)
20.2.7 Physical Interpretation of the Energy Surfaces
358(1)
20.2.8 Quantum Phase Transitions in Graphene
359(2)
20.3 Universality of Emergent States
361(7)
20.3.1 Topological and Algebraic Constraints
362(3)
20.3.2 Analogy with General Relativity
365(1)
20.3.3 Analogy with Renormalization Group Flow
365(1)
Background and Further Reading
365(1)
Problems
366(2)
21 Generalized Coherent States
368(9)
21.1 Glauber Coherent States
368(1)
21.2 Symmetry and Coherent Electromagnetic States
369(2)
21.2.1 Quantum Optics Hamiltonian
369(1)
21.2.2 Symmetry of the Hamiltonian
370(1)
21.2.3 Hilbert Space
370(1)
21.2.4 Stability Subgroup
370(1)
21.2.5 Coset Space
371(1)
21.2.6 The Coherent State
371(1)
21.3 Construction of Generalized Coherent States
371(2)
21.4 Atoms Interacting with Classical Radiation
373(2)
21.5 Fermion Coherent States
375(2)
Background and Further Reading
376(1)
Problems
376(1)
22 Restoring Symmetry by Projection
377(23)
22.1 Rotational Symmetry in Atomic Nuclei
377(1)
22.2 The Method of Generator Coordinates
378(3)
22.2.1 Generator Coordinates and Generating Functions
378(1)
22.2.2 The Hill--Wheeler Equation
379(2)
22.3 Angular Momentum Projection
381(4)
22.3.1 The Rotation Operator and its Representations
381(2)
22.3.2 The Angular Momentum Projection Operator
383(1)
22.3.3 Solving the Eigenvalue Equation
384(1)
22.4 Particle Number Projection
385(7)
22.4.1 Violation of Particle Number in BCS Theory
386(1)
22.4.2 Bogoliubov Quasiparticles
387(3)
22.4.3 The Particle Number Projection Operator
390(2)
22.5 Parity Projection
392(3)
22.5.1 The Parity Transformation
392(1)
22.5.2 Breaking Parity Spontaneously
393(1)
22.5.3 The Parity Projection Operator
394(1)
22.6 Spin and Momentum Projection for Electrons
395(5)
22.6.1 Hartree--Fock Approximation for the Hubbard Model
395(1)
22.6.2 Spin and Momentum Projection in the Hubbard Model
396(1)
Background and Further Reading
397(1)
Problems
398(2)
23 Quantum Phase Transitions
400(19)
23.1 Classical and Quantum Phases
400(1)
23.1.1 Thermal and Quantum Fluctuations
400(1)
23.1.2 Quantum Critical Behavior
401(1)
23.2 Classification of Phase Transitions
401(1)
23.3 Classical Second-Order Phase Transitions
402(2)
23.3.1 Critical Exponents
402(1)
23.3.2 Universality
403(1)
23.4 Continuous Quantum Phase Transitions
404(2)
23.4.1 Order Only at Zero Temperature
405(1)
23.4.2 Order Also at Finite Temperature
406(1)
23.5 Quantum to Classical Crossover
406(2)
23.5.1 The Classical-Quantum Mapping
406(1)
23.5.2 Optimal Dimensionality
407(1)
23.5.3 Quantum versus Classical Phase Transitions
407(1)
23.6 Example: Ising Spins in a Transverse Field
408(4)
23.6.1 Hamiltonian
409(1)
23.6.2 Ground States and Quasiparticle States for g → 0
409(1)
23.6.3 Ground States and Quasiparticle States for g → ∞
410(1)
23.6.4 Competing Ground States
410(1)
23.6.5 The Quantum Critical Region
411(1)
23.6.6 Phase Diagram
411(1)
23.7 Dynamical Symmetry and Quantum Phases
412(7)
23.7.1 Quantum Phases in Superconductors
412(1)
23.7.2 Unique Perspective of Dynamical Symmetries
413(1)
23.7.3 Quantum Phases and Insights from Symmetry
414(1)
Background and Further Reading
415(1)
Problems
416(3)
Part III Topology and Geometry
24 Topology, Manifolds, and Metrics
419(20)
24.1 Basic Concepts of Topology
419(2)
24.1.1 Discrete Categories Distinguished Qualitatively
419(1)
24.1.2 The Nature of Topological Proofs
420(1)
24.1.3 Neighborhoods
421(1)
24.2 Topology and Topological Spaces
421(6)
24.2.1 Formal Definition of a Topology
423(1)
24.2.2 Continuity
423(1)
24.2.3 Compactness
424(2)
24.2.4 Connectedness
426(1)
24.2.5 Homeomorphism
426(1)
24.3 Topological Invariants
427(2)
24.3.1 Compactness Is a Topological Invariant
427(1)
24.3.2 Connectedness Is a Topological Invariant
427(1)
24.3.3 Dimensionality Is a Topological Invariant
427(2)
24.4 Homotopies
429(5)
24.4.1 Homotopic Equivalence Classes
429(1)
24.4.2 Homotopy Classes Are Topological Invariants
429(1)
24.4.3 The First Homotopy Group
430(3)
24.4.4 Higher Homotopy Groups
433(1)
24.5 Manifolds and Metric Spaces
434(5)
24.5.1 Differentiable Manifolds
434(3)
24.5.2 Metric Spaces
437(1)
Background and Further Reading
437(1)
Problems
437(2)
25 Topological Solitons
439(11)
25.1 Models in (1+1) Dimensions
439(4)
25.1.1 Equations of Motion
439(1)
25.1.2 Vacuum States and Boundary Conditions
440(1)
25.1.3 Topological Charges
441(1)
25.1.4 Soliton Solutions in (1+1) Dimensions
442(1)
25.2 Solitons in (2+1) and (3+1) Dimensions
443(2)
25.2.1 Homotopy Groups
443(1)
25.2.2 Mapping Spheres to Spheres
444(1)
25.3 Yang-Mills Fields and Instantons
445(5)
25.3.1 Solitons in the Euclidean Yang-Mills Field
445(1)
25.3.2 Boundary Conditions
446(1)
25.3.3 Topological Classification of Solutions
447(1)
25.3.4 Physical Interpretation of Instantons
447(2)
Background and Further Reading
449(1)
Problems
449(1)
26 Geometry and Gauge Theories
450(14)
26.1 Parallel Transport
450(5)
26.1.1 Flat and Curved Manifolds
450(3)
26.1.2 Connections and Covariant Derivatives
453(1)
26.1.3 Curvature and Parallel Transport
454(1)
26.2 Absolute Derivatives
455(1)
26.3 Parallel Transport in Charge Space
455(1)
26.4 Fiber Bundles and Gauge Manifolds
456(2)
26.4.1 Tangent Spaces and Tangent Bundles
456(1)
26.4.2 Fiber Bundles
457(1)
26.5 Gauge Symmetry on a Spacetime Lattice
458(6)
26.5.1 Path-Dependent Gauge Representations
458(1)
26.5.2 Lattice Gauge Symmetries
459(2)
Background and Further Reading
461(1)
Problems
462(2)
27 Geometrical Phases
464(14)
27.1 The Aharonov--Bohm Effect
464(4)
27.1.1 Experimental Setup
464(1)
27.1.2 Analysis of Magnetic Fields
465(1)
27.1.3 Phase of the Electron Wavefunction
465(1)
27.1.4 Topological Origin of the Aharonov--Bohm Effect
466(2)
27.2 The Berry Phase
468(6)
27.2.1 Fast and Slow Degrees of Freedom
468(2)
27.2.2 The Berry Connection
470(1)
27.2.3 Trading the Connection for a Phase
471(1)
27.2.4 Berry Phases
471(1)
27.2.5 Berry Curvature
472(2)
27.3 An Electron in a Magnetic Field
474(1)
27.4 Topological Implications of Berry Phases
475(3)
Background and Further Reading
476(1)
Problems
477(1)
28 Topology of the Quantum Hall Effect
478(26)
28.1 The Classical Hall Effect
478(2)
28.1.1 Hall Effect Measurements
478(1)
28.1.2 Quantization of the Hall Effect
479(1)
28.2 Landau Levels for Non-Relativistic Electrons
480(2)
28.2.1 Hamiltonian and Schrodinger Equation
481(1)
28.2.2 Landau Levels and Density of States
482(1)
28.3 The Integer Quantum Hall Effect
482(7)
28.3.1 Understanding the Integer Quantum Hall Effect
483(2)
28.3.2 Disorder and the Integer Quantum Hall State
485(2)
28.3.3 Edge States and Conduction
487(2)
28.4 Topology and Integer Quantum Hall Effects
489(9)
28.4.1 Berry Phases and Adiabatic Curvature
490(1)
28.4.2 Chern Numbers
491(7)
28.5 The Fractional Quantum Hall Effect
498(6)
28.5.1 Properties of the Fractional Quantum Hall State
498(2)
28.5.2 Fractionally Charged Quasiparticles
500(1)
28.5.3 Nature of the Edge States
500(1)
28.5.4 Topology and Fractional Quantum Hall States
500(1)
Background and Further Reading
501(1)
Problems
502(2)
29 Topological Matter
504(31)
29.1 Topology and the Many-Body Paradigm
504(3)
29.1.1 Adiabatic Continuity
504(1)
29.1.2 Spontaneous Symmetry Breaking
505(1)
29.1.3 Beyond the Landau Picture
506(1)
29.2 Berry Phases and Brillouin Zones
507(1)
29.3 Topological States and Symmetry
508(1)
29.4 Topological Insulators
509(9)
29.4.1 The Quantum Spin Hall Effect
510(2)
29.4.2 The Z2 Topological Index
512(6)
29.5 Weyl Semimetals
518(4)
29.5.1 A Topological Conservation Law
519(1)
29.5.2 Realization of a Weyl Semimetal
520(2)
29.6 Majorana Modes
522(2)
29.6.1 The Dirac Equation in Condensed Matter
523(1)
29.6.2 Quasiparticles and Anti-Quasiparticles
523(1)
29.7 Topological Superconductors
524(1)
29.7.1 Topological Majorana Fermions
524(1)
29.7.2 Fractionalization of Electrons
524(1)
29.8 Fractional Statistics
525(3)
29.8.1 Anyon Statistics
525(1)
29.8.2 The Braid Group
526(2)
29.8.3 Abelian and Non-Abelian Anyons
528(1)
29.9 Quantum Computers and Topological Matter
528(7)
29.9.1 Qubits and Quantum Information
529(1)
29.9.2 The Problem of Decoherence
530(1)
29.9.3 Topological Quantum Computation
531(1)
Background and Further Reading
531(1)
Problems
532(3)
Part IV A Variety of Physical Applications
30 Angular Momentum Recoupling
535(8)
30.1 Recoupling of Three Angular Momenta
535(2)
30.1.1 67 Coefficients
536(1)
30.1.2 Racah Coefficients
536(1)
30.2 Matrix Elements of Tensor Products
537(1)
30.3 Recoupling of Four Angular Momenta
538(5)
30.3.1 97 Coefficients
538(1)
30.3.2 Transformation Between L-S and J-JCoupling
539(1)
30.3.3 Matrix Element of an Independent Tensor Product
540(1)
30.3.4 Matrix Element of a Scalar Product
540(1)
Background and Further Reading
541(1)
Problems
541(2)
31 Nuclear Fermion Dynamical Symmetry
543(14)
31.1 The Ginocchio Model
543(3)
31.2 The Fermion Dynamical Symmetry Model
546(8)
31.2.1 Dynamical Symmetry Generators
547(1)
31.2.2 The FDSM Dynamical Symmetries
548(1)
31.2.3 FDSM Irreducible Representations
549(2)
31.2.4 Quantitative FDSM Calculations
551(3)
31.3 The Interacting Boson Model
554(3)
Background and Further Reading
555(1)
Problems
555(2)
32 Superconductivity and Superfluidity
557(21)
32.1 Conventional Superconductors
557(1)
32.2 Unconventional Superconductors
557(3)
32.3 The SU(4) Model of Non-Abelian Superconductors
560(9)
32.3.1 The SU(4) Algebra
561(3)
32.3.2 The SU(4) Collective Subspace
564(1)
32.3.3 The Dynamical Symmetry Hamiltonian
565(1)
32.3.4 The SU(4) Dynamical Symmetry Limits
566(1)
32.3.5 The SO(4) Dynamical Symmetry Limit
567(1)
32.3.6 The SU(2) Dynamical Symmetry Limit
568(1)
32.3.7 The SO(5) Dynamical Symmetry Limit
568(1)
32.3.8 Conventional and Unconventional Superconductors
569(1)
32.4 Some Implications of SU(4) Symmetry
569(9)
32.4.1 No Double Occupancy
569(1)
32.4.2 Quantitative Gap and Phase Diagrams
570(1)
32.4.3 Coherent State Energy Surfaces
571(2)
32.4.4 Fundamental SU(4) Instabilities
573(1)
32.4.5 Origin of High Critical Temperatures
574(2)
32.4.6 Universality of Dynamical Symmetry States
576(1)
Background and Further Reading
576(1)
Problems
577(1)
33 Current Algebra
578(6)
33.1 The CVC and PCAC Hypotheses
578(2)
33.1.1 Current Algebra and Chiral Symmetry
578(1)
33.1.2 The Partially Conserved Axial Current
579(1)
33.2 The Linear a-Model
580(4)
33.2.1 The Particle Spectrum
580(2)
33.2.2 Explicit Breaking of Chiral Symmetry
582(1)
Background and Further Reading
582(1)
Problems
582(2)
34 Grand Unified Theories
584(7)
34.1 Evolution of Fundamental Coupling Constants
584(1)
34.2 Minimal Criteria for a Grand Unified Group
585(1)
34.3 The SU(5) Grand Unified Theory
586(2)
34.4 Beyond Simple GUTs
588(3)
Background and Further Reading
590(1)
Problems
590(1)
Appendix A Second Quantization
591(12)
A.1 Symmetrized Many-Particle Wavefunctions
591(2)
A.1.1 Bosonic and Fermionic Wavefunctions
592(1)
A.1.2 Slater Determinants
593(1)
A.2 Dirac Notation
593(2)
A.2.1 Bras, Kets, and Bra-Ket Pairs
594(1)
A.2.2 Bras and Kets as Row and Column Vectors
594(1)
A.2.3 Linear Operators Acting on Bras and Kets
595(1)
A.3 Occupation Number Representation
595(8)
A.3.1 Creation and Annihilation Operators
597(1)
A.3.2 Basis Transformations
598(1)
A.3.3 Many-Particle Vector States
599(1)
A.3.4 One-Body and Two-Body Operators
600(3)
Appendix B Natural Units
603(2)
B.1 The Advantage of Natural Units
603(1)
B.2 Natural Units in Quantum Field Theory
603(2)
Appendix C Angular Momentum Tables
605(3)
Appendix D Lie Algebras
608(1)
References 609(8)
Index 617
Mike Guidry is Professor in Physics and Astronomy at the University of Tennessee. He is the author of more than 125 journal articles and six published textbooks. He has been the Lead Educational Technology Developer for several major college textbooks in introductory physics, astronomy, biology, genetics, and microbiology. During his career, he has won multiple teaching awards and has taken the lead in a variety of science outreach initiatives. Yang Sun gained his Ph.D. at the Technical University of Munich and has many years of experience teaching undergraduate courses, ranging from introductory physics to quantum mechanics. He is the author of more than 250 journal articles, mainly in the field of nuclear many-body theory, but also in other correlated fermionic systems. He was awarded the Wu Youxun Prize by the Chinese Physical Society for his research achievements.