Symmetry, Broken Symmetry, and Topology in Modern Physics: A First Course (the book itself and the solution manual for instructors) (book, solutions)

Cover
Half-title
Title page
Copyright information
Dedication
Brief Contents
Contents
Preface
Part I Symmetry Groups and Algebras
	1 Introduction
	2 Some Properties of Groups
		2.1 Invariance and Conservation Laws
		2.2 Definition of a Group
		2.3 Examples of Groups
			2.3.1 Additive Group of Integers
			2.3.2 Rotation and Translation Groups
			2.3.3 Parameterization of Continuous Groups
			2.3.4 Permutation Groups
		2.4 Subgroups
		2.5 Homomorphism and Isomorphism
		2.6 Matrix Representations
			2.6.1 A Matrix Representation of S[sub(3)]
			2.6.2 Dimensionality of Matrix Representations
			2.6.3 Linear Operators and Matrix Representations
		2.7 Reducible and Irreducible Representations
		2.8 Degenerate Multiplet Structure
		2.9 Some Examples of Matrix Groups
			2.9.1 General Linear Groups
			2.9.2 Unitary Groups
			2.9.3 Orthogonal Groups
			2.9.4 Symplectic Groups
		2.10 Group Generators
		2.11 Conjugate Classes
		2.12 Invariant Subgroups
		2.13 Simple and Semisimple Groups
		2.14 Cosets and Factor Groups
			2.14.1 Left and Right Coset Decompositions
			2.14.2 Factor Groups
		2.15 Direct Product Groups
		2.16 Direct Product of Representations
		2.17 Characters of Representations
			2.17.1 Character Theorems
			2.17.2 Character Tables
		Background and Further Reading
		Problems
	3 Introduction to Lie Groups
		3.1 Lie Groups
		3.2 Lie Algebras
			3.2.1 Invariant Subalgebras
			3.2.2 Adjoint Representation of the Algebra
		3.3 Angular Momentum and the Group SU(2)
			3.3.1 Fundamental Representation of SU(2)
			3.3.2 The Cartan–Dynkin Method
			3.3.3 Cartan–Dynkin Analysis of SU(2)
			3.3.4 The Clebsch–Gordan Series for SU(2)
			3.3.5 SU(2) Adjoint Representation
		3.4 Isospin
			3.4.1 The Neutron–Proton System
			3.4.2 Algebraic Structure for Isospin
			3.4.3 The U(1) and SU(2) Subgroups of U(2)
			3.4.4 Analogy between Angular Momentum and Isospin
			3.4.5 The Adjoint Representation of Isospin
		3.5 The Importance of Lie Groups in Physics
		3.6 Symmetry and Dynamics
			3.6.1 Local Gauge Theories
			3.6.2 Dynamical Symmetries
		Background and Further Reading
		Problems
	4 Permutation Groups
		4.1 Young Diagrams
			4.1.1 Two-Particle Young Diagrams
			4.1.2 Many-Particle Young Diagrams
			4.1.3 A Compact Notation
		4.2 Standard Arrangement of Young Tableaux
		4.3 Irreducible Representations
			4.3.1 Counting Standard Arrangements
			4.3.2 The Hook Rule
		4.4 Basis Vectors
		4.5 Products of Representations
			4.5.1 Direct Products
			4.5.2 Outer Products
		Background and Further Reading
		Problems
	5 Electrons on Periodic Lattices
		5.1 The Direct Lattice
			5.1.1 Brevais Lattices
			5.1.2 Wigner–Seitz Cells
		5.2 The Reciprocal Lattice
		5.3 Brillouin Zones
		5.4 Bloch’s Theorem
		5.5 Electronic Band Structure
		5.6 Point Groups
			5.6.1 Point Group Operations
			5.6.2 The Crystallographic Point Groups
		5.7 Example: The Ammonia Molecule
			5.7.1 Symmetry Operations
			5.7.2 A Matrix Representation
			5.7.3 Class Structure
			5.7.4 Other Irreducible Representations
		5.8 General Lattice Symmetry Classifications
		5.9 Space Groups
			5.9.1 Elements of the Space Group
			5.9.2 Symmorphic Space Groups
		Background and Further Reading
		Problems
	6 The Rotation Group
		6.1 Three-Dimensional Rotations
		6.2 The SO(2) Group
			6.2.1 Generators of SO(2) Rotations
			6.2.2 SO(2) Irreducible Representations
			6.2.3 Connectedness of the Manifold
			6.2.4 Compactness of the Manifold
			6.2.5 Invariant Group Integration
		6.3 The SO(3) Group
			6.3.1 Generators of SO(3)
			6.3.2 Matrix Elements of the Rotation Operator
			6.3.3 Properties of D-Matrices
			6.3.4 Characters for SO(3)
			6.3.5 Direct Products of SO(3) Representations
			6.3.6 SO(3) Vector-Coupling Coefficients
			6.3.7 Properties of SO(3) Clebsch–Gordan Coefficients
			6.3.8 3J Symbols
			6.3.9 Construction of SO(3) Irreducible Multiplets
		6.4 Tensor Operators under Group Transformations
		6.5 Tensors for the Rotation Group
		6.6 SO(3) Tensor Products
		6.7 The Wigner–Eckart Theorem
		6.8 The Wigner–Eckart Theorem for SO(3)
			6.8.1 Reduced Matrix Elements
			6.8.2 Selection Rules
		6.9 Relationship of SO(3) and SU(2)
			6.9.1 SO(3) and SU(2) Group Manifolds
			6.9.2 Universal Covering Group of the SU(2) Algebra
		Background and Further Reading
		Problems
	7 Classification of Lie Algebras
		7.1 Adjoint Representations
			7.1.1 The Cartan Subalgebra
			7.1.2 Raising and Lowering Operators
		7.2 The Cartan–Weyl Basis
			7.2.1 Semisimple Algebras
			7.2.2 Metric Tensor, Semisimplicity, and Compactness
		7.3 Structure of the Root Space
			7.3.1 Root Space Restrictions
			7.3.2 Lengths and Angles for Root Vectors
		7.4 Construction of Root Diagrams
			7.4.1 Rank-1 and Rank-2 Compact Lie Algebras
			7.4.2 An Ordering Prescription for Weights
		7.5 Simple Roots
		7.6 Dynkin Diagrams
			7.6.1 The Cartan Matrix
			7.6.2 Constructing All Roots from Dynkin Diagrams
			7.6.3 Constructing the Algebra from the Roots
		7.7 Dynkin Diagrams and the Simple Algebras
		Background and Further Reading
		Problems
	8 Unitary and Special Unitary Groups
		8.1 Generators and Commutators for SU(3)
		8.2 SU(3) Casimir Operators
		8.3 SU(3) Weight Space
			8.3.1 SU(3) Raising and Lowering Operators
			8.3.2 SU(3) Irreducible Representations
			8.3.3 Dimensionality of SU(3) Irreps
			8.3.4 Construction of SU(3) Weight Diagrams
		8.4 Complex Conjugate Representations
		8.5 Real and Complex Representations
		8.6 Unitary Symmetry and Young Diagrams
		8.7 Young Diagrams for SU(N)
			8.7.1 Two Particles in Two States
			8.7.2 Two Particles in Three States
			8.7.3 Fundamental and Conjugate Representations
		8.8 Dimensionality of SU(N) Representations
		8.9 Direct Products of SU(N) Representations
		8.10 Weights from Young Diagrams
		8.11 Graphical Construction of Direct Products
		Background and Further Reading
		Problems
	9 SU(3) Flavor Symmetry
		9.1 Symmetry in Particle Physics
			9.1.1 SU(3) Phenomenology and Quarks
			9.1.2 Non-Abelian Gauge Symmetries
		9.2 Fundamental SU(3) Quark Representations
		9.3 SU(3) Flavor Multiplets
			9.3.1 Mass Splittings in SU(3) Multiplets
			9.3.2 Quark Structure for Mesons and Baryons
		9.4 Isospin Subgroups of SU(3)
			9.4.1 Subgroup Analysis Using Weight Diagrams
			9.4.2 Subgroup Analysis Using Young Diagrams
		9.5 Extensions of Flavor SU(3) Symmetry
			9.5.1 Higher-Rank Flavor Symmetries
			9.5.2 SU(6) Flavor–Spin Symmetry
			9.5.3 Baryons and Mesons under SU(6) Symmetry
		Background and Further Reading
		Problems
	10 Harmonic Oscillators and SU(3)
		10.1 The 3D Quantum Oscillator
			10.1.1 Eigenvalues
			10.1.2 Wavefunctions
			10.1.3 Unitary Symmetry
			10.1.4 Angular Momentum Subgroup
			10.1.5 SO(3) Transformation Properties
			10.1.6 Group Structure
			10.1.7 Many-Body Operators
		10.2 SU(3) and the Nuclear Shell Model
		10.3 SU(3) Classification of SD Shell States
			10.3.1 Classification Strategy
			10.3.2 Orbital and Spin–Isospin Symmetry
			10.3.3 Permutation Symmetry
			10.3.4 Example: Two Particles in the SD Shell
		10.4 SU(2) Subgroups and Intrinsic States
			10.4.1 Weight Space Operators and Diagrams
			10.4.2 Angular Momentum Content of Multiplets
		10.5 Collective Motion in the Nuclear SD Shell
			10.5.1 Hamiltonian
			10.5.2 Group-Theoretical Solution
			10.5.3 The Theoretical Spectrum
		Background and Further Reading
		Problems
	11 SU(3) Matrix Elements
		11.1 Clebsch–Gordan Coefficients for SU(3)
		11.2 Constructing SU(3) Clebsch–Gordan Coefficients
		11.3 Matrix Elements of Generators
		11.4 Isoscalar Factors
			11.4.1 Racah Factorization Lemma
			11.4.2 Evaluating and Using Isoscalar Factors
		11.5 SU(3)⊃SO(3) Tensor Operators
		11.6 The SU(3) Wigner–Eckart Theorem
		11.7 Structure of SU(3) Matrix Elements
		11.8 The Gell-Mann, Okubo Mass Formula
		11.9 SU(3) Oscillator Reduced Matrix Elements
			11.9.1 Spherical Operators
			11.9.2  Matrix Elements for Creation and Annihilation Operators
			11.9.3  Electromagnetic Transitions in the SD Shell
		11.10 Lie Algebras and Many-Body Systems
		Background and Further Reading
		Problems
	12 Introduction to Non-Compact Groups
		12.1 Review of the Compact Group SU(n)
		12.2 The Non-Compact Group SU(l,m)
			12.2.1 Signature of the Metric
			12.2.2 Parameter Space for SU(1,1)
		12.3 The Non-Compact Group SO(l,m)
		12.4 Euclidean Groups
			12.4.1 The Euclidean Group E[sub(3)] for 3D Space
			12.4.2 The Euclidean Group E[sub(2)] for 2D Space
			12.4.3 Semidirect Product Groups
			12.4.4 Algebraic Properties of E[sub(2)]
			12.4.5 Invariant Subgroup of Translations
		12.5 Method of Induced Representations for E[sub(2)]
			12.5.1 Generating the Representation
			12.5.2 Significance of the Abelian Invariant Subgroup
		Background and Further Reading
		Problems
	13 The Lorentz Group
		13.1 Spacetime Tensors
			13.1.1 A Covariant Notation
			13.1.2 Tensor Transformation Laws
		13.2 Lorentz Transformations
			13.2.1 Lorentz Boosts as Minkowski Rotations
			13.2.2 Generators of Boosts and Rotations
			13.2.3 Commutation Algebra for the Lorentz Group
		13.3 Classification of Lorentz Transformations
			13.3.1 The Four Pieces of the Full Lorentz Group
			13.3.2 Improper Lorentz Transformations
			13.3.3 Lightcone Classification of Minkowski Vectors
		13.4 Properties of the Lorentz Group
		13.5 The Lorentz Group and SL(2,C)
			13.5.1 A Mapping between 4-Vectors and Matrices
			13.5.2 The Universal Covering Group of SO(3,1)
		13.6 Spinors and Lorentz Transformations
			13.6.1 SU(2)×SU(2) Representations of the Lorentz Group
			13.6.2 Two Inequivalent Spinor Representations
		13.7 Space Inversion for the Lorentz Group
			13.7.1 Action of Parity on Generators and Representations
			13.7.2 General and Self-Conjugate Representations
		13.8 Parity and 4-Spinors
		13.9 Higher-Dimensional Lorentz Representations
		13.10 Non-Unitarity of Representations
		13.11 Meaning of Non-Unitary Representations
		Background and Further Reading
		Problems
	14 Lorentz-Covariant Fields
		14.1 Lorentz Covariance of Maxwell’s Equations
			14.1.1 Scalar and Vector Potentials
			14.1.2 Gauge Transformations
			14.1.3 Manifestly Covariant Form of the Maxwell Equations
		14.2 The Dirac Equation
			14.2.1 Lorentz-Boosted Spinors
			14.2.2 A Lorentz-Covariant Notation
		14.3 Dirac Bilinear Covariants
			14.3.1 Covariance of the Dirac Equation
			14.3.2 Transformation Properties of Bilinear Products
		14.4 Weyl Equations and Massless Fermions
		14.5 Chiral Invariance
			14.5.1 Helicity States for Fermions
			14.5.2 Dirac Equation in Pauli–Dirac Representation
			14.5.3 Helicity and Chirality for Dirac Fermions
			14.5.4 Projection Operators for Chiral Fermions
			14.5.5 Interactions and Chiral Symmetry
		14.6 The Majorana Equation
			14.6.1 Dirac and Majorana Masses
			14.6.2 Neutrinoless Double β-Decay
		14.7 Summary: Possible Spinor Types
		14.8 Spinor Symmetry in the Weak Interactions
			14.8.1 The Left Hand of the Neutrino
			14.8.2 Violation of Parity P
			14.8.3 C, CP, and T Symmetries
			14.8.4 A More Complete Picture
		Background and Further Reading
		Problems
	15 Poincaré Invariance
		15.1 The Poincaré Multiplication Rule
		15.2 Generators of Poincaré Transformations
			15.2.1 Proper Lorentz Transformations
			15.2.2 Four-Dimensional Spacetime Translations
			15.2.3 Commutators for Poincaré Generators
		15.3 Representation Theory of the Poincaré Group
			15.3.1 Casimir Operators for the Poincaré Group
			15.3.2 Classification of Poincaré States
			15.3.3 Method of Induced Representations
		15.4 Massive Representations of the Poincaré Group
			15.4.1 Quantum Numbers for Massive States
			15.4.2 Action of the Poincaré Group on Massive States
			15.4.3 Summary: Representations for Massive States
		15.5 Massless Representations
			15.5.1 The Standard Lightlike Vector
			15.5.2 Lie Algebra of the Little Group
			15.5.3 Quantum Numbers for Massless States
		15.6 Mass and Spin for Poincaré Representations
		15.7 Lorentz and Poincaré Representations
			15.7.1 Operators for Relativistic Quantum Fields
			15.7.2 Wave Equations for Quantum Fields
			15.7.3 Plane-Wave Expansion of the Fields
			15.7.4 The Relationship of Fields and Particles
			15.7.5 Symmetry and the Wave Equation
		Background and Further Reading
		Problems
	16 Gauge Invariance
		16.1 Relativistic Quantum Field Theory
			16.1.1 Quantization of Classical Fields
			16.1.2 Symmetries of the Classical Action
			16.1.3 Lagrangian Densities for Free Fields
			16.1.4 Euler–Lagrange Field Equations
		16.2 Conserved Currents and Charges
			16.2.1 Noether’s Theorem
			16.2.2 Conserved Charges
			16.2.3 Symmetries for Interacting Fields
			16.2.4 Partially Conserved Currents
		16.3 Gauge Invariance in Quantum Mechanics
		16.4 Gauge Invariance and the Photon Mass
		16.5 Quantum Electrodynamics
			16.5.1 Global U(1) Gauge Invariance
			16.5.2 Local U(1) Gauge Invariance
			16.5.3 Gauging the U(1) Symmetry
		16.6 Yang–Mills Fields
			16.6.1 Non-Abelian Gauge Invariance
			16.6.2 Covariant Derivatives
			16.6.3 Non-Abelian Generalization of QED
			16.6.4 Properties of Non-Abelian Gauge Fields
		Background and Further Reading
		Problems
Part II Broken Symmetry
	17 Spontaneous Symmetry Breaking
		17.1 Modes of Symmetry Breaking
		17.2 Explicit Symmetry Breaking
		17.3 The Vacuum and Hidden Symmetry
		17.4 Spontaneously Broken Discrete Symmetry
			17.4.1 Symmetry in the Wigner Mode
			17.4.2 Spontaneously Broken Symmetry
			17.4.3 Summary of Spontaneously Broken Discrete Symmetry
		17.5 Spontaneously Broken Continuous Symmetry
			17.5.1 Symmetric Classical Vacuum
			17.5.2 Hidden Continuous Symmetry
			17.5.3 The Goldstone Theorem
			17.5.4 The Stability Subgroup
		Background and Further Reading
		Problems
	18 The Higgs Mechanism
		18.1 Photons and the Higgs Loophole
		18.2 The Abelian Higgs Model
			18.2.1 Lagrangian Density
			18.2.2 Symmetry Breaking
			18.2.3 Understanding the Higgs Mechanism
		18.3 Vacuum Screening Currents
			18.3.1 Gauge Invariance and Mass
			18.3.2 Screening Currents and Effective Mass
			18.3.3 Atomic Screening Currents
			18.3.4 The Meissner Effect and Massive Photons
			18.3.5 Gauge Invariance and Longitudinal Polarization
		18.4 The Higgs Boson
		Background and Further Reading
		Problems
	19 The Standard Model
		19.1 The Standard Electroweak Model
			19.1.1 Guidance from Data
			19.1.2 The Gauge Group
			19.1.3 Electroweak Lagrangian Density
			19.1.4 The Electroweak Higgs Mechanism
			19.1.5 Particle Spectrum
		19.2 Quantum Chromodynamics
			19.2.1 A Color Gauge Theory
			19.2.2 The QCD Lagrangian Density
			19.2.3 Symmetries of the QCD Lagrangian Density
			19.2.4 Asymptotic Freedom and Confinement
			19.2.5 Exotic Hadrons and Glueballs
		19.3 The Gauge Theory of Fundamental Interactions
		Background and Further Reading
		Problems
	20 Dynamical Symmetry
		20.1 The Microscopic Dynamical Symmetry Method
			20.1.1 Solution Algorithm
			20.1.2 Validity and Utility of the Approach
			20.1.3 Spontaneously Broken Symmetry and Dynamical Symmetry
			20.1.4 Kinematics and Dynamics
		20.2 Monolayer Graphene in a Strong Magnetic Field
			20.2.1 Electronic Dispersion in Monolayer Graphene
			20.2.2 Landau Levels for Massless Dirac Electrons
			20.2.3 SU(4) Quantum Hall Ferromagnetism
			20.2.4 Fermion Dynamical Symmetries for Graphene
			20.2.5 Graphene SO(8) Dynamical Symmetries
			20.2.6 Generalized Coherent States for Graphene
			20.2.7 Physical Interpretation of the Energy Surfaces
			20.2.8 Quantum Phase Transitions in Graphene
		20.3 Universality of Emergent States
			20.3.1 Topological and Algebraic Constraints
			20.3.2 Analogy with General Relativity
			20.3.3 Analogy with Renormalization Group Flow
		Background and Further Reading
		Problems
	21 Generalized Coherent States
		21.1 Glauber Coherent States
		21.2 Symmetry and Coherent Electromagnetic States
			21.2.1 Quantum Optics Hamiltonian
			21.2.2 Symmetry of the Hamiltonian
			21.2.3 Hilbert Space
			21.2.4 Stability Subgroup
			21.2.5 Coset Space
			21.2.6 The Coherent State
		21.3 Construction of Generalized Coherent States
		21.4 Atoms Interacting with Classical Radiation
		21.5 Fermion Coherent States
		Background and Further Reading
		Problems
	22 Restoring Symmetry by Projection
		22.1 Rotational Symmetry in Atomic Nuclei
		22.2 The Method of Generator Coordinates
			22.2.1 Generator Coordinates and Generating Functions
			22.2.2 The Hill–Wheeler Equation
		22.3 Angular Momentum Projection
			22.3.1 The Rotation Operator and its Representations
			22.3.2 The Angular Momentum Projection Operator
			22.3.3 Solving the Eigenvalue Equation
		22.4 Particle Number Projection
			22.4.1 Violation of Particle Number in BCS Theory
			22.4.2 Bogoliubov Quasiparticles
			22.4.3 The Particle Number Projection Operator
		22.5 Parity Projection
			22.5.1 The Parity Transformation
			22.5.2 Breaking Parity Spontaneously
			22.5.3 The Parity Projection Operator
		22.6 Spin and Momentum Projection for Electrons
			22.6.1 Hartree–Fock Approximation for the Hubbard Model
			22.6.2 Spin and Momentum Projection in the Hubbard Model
		Background and Further Reading
		Problems
	23 Quantum Phase Transitions
		23.1 Classical and Quantum Phases
			23.1.1 Thermal and Quantum Fluctuations
			23.1.2 Quantum Critical Behavior
		23.2 Classification of Phase Transitions
		23.3 Classical Second-Order Phase Transitions
			23.3.1 Critical Exponents
			23.3.2 Universality
		23.4 Continuous Quantum Phase Transitions
			23.4.1 Order Only at Zero Temperature
			23.4.2 Order Also at Finite Temperature
		23.5 Quantum to Classical Crossover
			23.5.1 The Classical–Quantum Mapping
			23.5.2 Optimal Dimensionality
			23.5.3 Quantum versus Classical Phase Transitions
		23.6 Example: Ising Spins in a Transverse Field
			23.6.1 Hamiltonian
			23.6.2 Ground States and Quasiparticle States for g → 0
			23.6.3 Ground States and Quasiparticle States for g → ∞
			23.6.4 Competing Ground States
			23.6.5 The Quantum Critical Region
			23.6.6 Phase Diagram
		23.7 Dynamical Symmetry and Quantum Phases
			23.7.1 Quantum Phases in Superconductors
			23.7.2 Unique Perspective of Dynamical Symmetries
			23.7.3 Quantum Phases and Insights from Symmetry
		Background and Further Reading
		Problems
Part III Topology and Geometry
	24 Topology, Manifolds, and Metrics
		24.1 Basic Concepts of Topology
			24.1.1 Discrete Categories Distinguished Qualitatively
			24.1.2 The Nature of Topological Proofs
			24.1.3 Neighborhoods
		24.2 Topology and Topological Spaces
			24.2.1 Formal Definition of a Topology
			24.2.2 Continuity
			24.2.3 Compactness
			24.2.4 Connectedness
			24.2.5 Homeomorphism
		24.3 Topological Invariants
			24.3.1 Compactness Is a Topological Invariant
			24.3.2 Connectedness Is a Topological Invariant
			24.3.3 Dimensionality Is a Topological Invariant
		24.4 Homotopies
			24.4.1 Homotopic Equivalence Classes
			24.4.2 Homotopy Classes Are Topological Invariants
			24.4.3 The First Homotopy Group
			24.4.4 Higher Homotopy Groups
		24.5 Manifolds and Metric Spaces
			24.5.1 Differentiable Manifolds
			24.5.2 Metric Spaces
		Background and Further Reading
		Problems
	25 Topological Solitons
		25.1 Models in (1+1) Dimensions
			25.1.1 Equations of Motion
			25.1.2 Vacuum States and Boundary Conditions
			25.1.3 Topological Charges
			25.1.4 Soliton Solutions in (1+1) Dimensions
		25.2 Solitons in (2+1) and (3+1) Dimensions
			25.2.1 Homotopy Groups
			25.2.2 Mapping Spheres to Spheres
		25.3 Yang–Mills Fields and Instantons
			25.3.1 Solitons in the Euclidean Yang–Mills Field
			25.3.2 Boundary Conditions
			25.3.3 Topological Classification of Solutions
			25.3.4 Physical Interpretation of Instantons
		Background and Further Reading
		Problems
	26 Geometry and Gauge Theories
		26.1 Parallel Transport
			26.1.1 Flat and Curved Manifolds
			26.1.2 Connections and Covariant Derivatives
			26.1.3 Curvature and Parallel Transport
		26.2 Absolute Derivatives
		26.3 Parallel Transport in Charge Space
		26.4 Fiber Bundles and Gauge Manifolds
			26.4.1 Tangent Spaces and Tangent Bundles
			26.4.2 Fiber Bundles
		26.5 Gauge Symmetry on a Spacetime Lattice
			26.5.1 Path-Dependent Gauge Representations
			26.5.2 Lattice Gauge Symmetries
		Background and Further Reading
		Problems
	27 Geometrical Phases
		27.1 The Aharonov–Bohm Effect
			27.1.1 Experimental Setup
			27.1.2 Analysis of Magnetic Fields
			27.1.3 Phase of the Electron Wavefunction
			27.1.4 Topological Origin of the Aharonov–Bohm Effect
		27.2 The Berry Phase
			27.2.1 Fast and Slow Degrees of Freedom
			27.2.2 The Berry Connection
			27.2.3 Trading the Connection for a Phase
			27.2.4 Berry Phases
			27.2.5 Berry Curvature
		27.3 An Electron in a Magnetic Field
		27.4 Topological Implications of Berry Phases
		Background and Further Reading
		Problems
	28 Topology of the Quantum Hall Effect
		28.1 The Classical Hall Effect
			28.1.1 Hall Effect Measurements
			28.1.2 Quantization of the Hall Effect
		28.2 Landau Levels for Non-Relativistic Electrons
			28.2.1 Hamiltonian and Schrödinger Equation
			28.2.2 Landau Levels and Density of States
		28.3 The Integer Quantum Hall Effect
			28.3.1 Understanding the Integer Quantum Hall Effect
			28.3.2 Disorder and the Integer Quantum Hall State
			28.3.3 Edge States and Conduction
		28.4 Topology and Integer Quantum Hall Effects
			28.4.1 Berry Phases and Adiabatic Curvature
			28.4.2 Chern Numbers
		28.5 The Fractional Quantum Hall Effect
			28.5.1 Properties of the Fractional Quantum Hall State
			28.5.2 Fractionally Charged Quasiparticles
			28.5.3 Nature of the Edge States
			28.5.4 Topology and Fractional Quantum Hall States
		Background and Further Reading
		Problems
	29 Topological Matter
		29.1 Topology and the Many-Body Paradigm
			29.1.1 Adiabatic Continuity
			29.1.2 Spontaneous Symmetry Breaking
			29.1.3 Beyond the Landau Picture
		29.2 Berry Phases and Brillouin Zones
		29.3 Topological States and Symmetry
		29.4 Topological Insulators
			29.4.1 The Quantum Spin Hall Effect
			29.4.2 The Z[sub(2)] Topological Index
		29.5 Weyl Semimetals
			29.5.1 A Topological Conservation Law
			29.5.2 Realization of a Weyl Semimetal
		29.6 Majorana Modes
			29.6.1 The Dirac Equation in Condensed Matter
			29.6.2 Quasiparticles and Anti-Quasiparticles
		29.7 Topological Superconductors
			29.7.1 Topological Majorana Fermions
			29.7.2 Fractionalization of Electrons
		29.8 Fractional Statistics
			29.8.1 Anyon Statistics
			29.8.2 The Braid Group
			29.8.3 Abelian and Non-Abelian Anyons
		29.9 Quantum Computers and Topological Matter
			29.9.1 Qubits and Quantum Information
			29.9.2 The Problem of Decoherence
			29.9.3 Topological Quantum Computation
		Background and Further Reading
		Problems
Part IV A Variety of Physical Applications
	30 Angular Momentum Recoupling
		30.1 Recoupling of Three Angular Momenta
			30.1.1 6J Coefficients
			30.1.2 Racah Coefficients
		30.2 Matrix Elements of Tensor Products
		30.3 Recoupling of Four Angular Momenta
			30.3.1 9J Coefficients
			30.3.2 Transformation Between L–S and J–J Coupling
			30.3.3 Matrix Element of an Independent Tensor Product
			30.3.4 Matrix Element of a Scalar Product
		Background and Further Reading
		Problems
	31 Nuclear Fermion Dynamical Symmetry
		31.1 The Ginocchio Model
		31.2 The Fermion Dynamical Symmetry Model
			31.2.1 Dynamical Symmetry Generators
			31.2.2 The FDSM Dynamical Symmetries
			31.2.3 FDSM Irreducible Representations
			31.2.4 Quantitative FDSM Calculations
		31.3 The Interacting Boson Model
		Background and Further Reading
		Problems
	32 Superconductivity and Superfluidity
		32.1 Conventional Superconductors
		32.2 Unconventional Superconductors
		32.3 The SU(4) Model of Non-Abelian Superconductors
			32.3.1 The SU(4) Algebra
			32.3.2 The SU(4) Collective Subspace
			32.3.3 The Dynamical Symmetry Hamiltonian
			32.3.4 The SU(4) Dynamical Symmetry Limits
			32.3.5 The SO(4) Dynamical Symmetry Limit
			32.3.6 The SU(2) Dynamical Symmetry Limit
			32.3.7 The SO(5) Dynamical Symmetry Limit
			32.3.8 Conventional and Unconventional Superconductors
		32.4 Some Implications of SU(4) Symmetry
			32.4.1 No Double Occupancy
			32.4.2 Quantitative Gap and Phase Diagrams
			32.4.3 Coherent State Energy Surfaces
			32.4.4 Fundamental SU(4) Instabilities
			32.4.5 Origin of High Critical Temperatures
			32.4.6 Universality of Dynamical Symmetry States
		Background and Further Reading
		Problems
	33 Current Algebra
		33.1 The CVC and PCAC Hypotheses
			33.1.1 Current Algebra and Chiral Symmetry
			33.1.2 The Partially Conserved Axial Current
		33.2 The Linear σ-Model
			33.2.1 The Particle Spectrum
			33.2.2 Explicit Breaking of Chiral Symmetry
		Background and Further Reading
		Problems
	34 Grand Unified Theories
		34.1 Evolution of Fundamental Coupling Constants
		34.2 Minimal Criteria for a Grand Unified Group
		34.3 The SU(5) Grand Unified Theory
		34.4 Beyond Simple GUTs
		Background and Further Reading
		Problems
Appendix A Second Quantization
	A.1 Symmetrized Many-Particle Wavefunctions
		A.1.1 Bosonic and Fermionic Wavefunctions
		A.1.2 Slater Determinants
	A.2 Dirac Notation
		A.2.1 Bras, Kets, and Bra-Ket Pairs
		A.2.2 Bras and Kets as Row and Column Vectors
		A.2.3 Linear Operators Acting on Bras and Kets
	A.3 Occupation Number Representation
		A.3.1 Creation and Annihilation Operators
		A.3.2 Basis Transformations
		A.3.3 Many-Particle Vector States
		A.3.4 One-Body and Two-Body Operators
Appendix B Natural Units
	B.1 The Advantage of Natural Units
	B.2 Natural Units in Quantum Field Theory
Appendix C Angular Momentum Tables
Appendix D Lie Algebras
References
Index
Student Solutions Manual
	Front Page
	Chapter 01 Introduction
	Chapter 02 Some Properties of Groups
	Chapter 03 Introduction to Lie Groups
	Chapter 04 Permutation Groups
	Chapter 05 Electrons on Periodic Lattices
	Chapter 06 The Rotation Group
	Chapter 07 Classification of Lie Groups
	Chapter 08 Unitary and Special Unitary Groups
	Chapter 09 SU(3) Flavor Symmetry
	Chapter 10 Harmonic Oscillators and SU(3)
	Chapter 11 SU(3) Matrix Elements
	Chapter 12 Non-Compact Groups
	Chapter 13 Lorentz Group
	Chapter 14 Lorentz Covariant Fields
	Chapter 15 Poincaré Invariance
	Chapter 16 Gauge Invariance
	Chapter 17 Spontaneous Symmetry Breaking
	Chapter 18 The Higgs Mechanism
	Chapter 19 The Standard Model
	Chapter 20 Dynamical Symmetry
	Chapter 21 Generalized Coherent States
	Chapter 22 Restoring Symmetry by Projection
	Chapter 23 Quantum Phase Transitions
	Chapter 24 Topology, Manifolds, and Metrics
	Chapter 25 Topological Solitons
	Chapter 26 Geometry and Gauge Theories
	Chapter 27 Geometrical Phases
	Chapter 28 Topology of the Quantum Hall Effect
	Chapter 29 Topological Matter
	Chapter 30 Angular Momentum Recoupling
	Chapter 31 Nuclear Fermion Dynamical Symmetry
	Chapter 32 Superconductivity and Superfluidity
	Chapter 33 Current Algebra
	Chapter 34 Grand Unified Theories