PREFACE/7,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 1: INTRODUCTION/21,Black,notBold,notItalic,open,FitWidth,-1
1.1 Particle on a One-Dimensional Lattice/22,Black,notBold,notItalic,open,FitWidth,-1
1.2 Representations of the Discrete Translation Operators/24,Black,notBold,notItalic,open,FitWidth,-1
1.3 Physical Consequences of Translational Symmetry/26,Black,notBold,notItalic,open,FitWidth,-1
1.4 The Representation Functions and Fourier Analysis/28,Black,notBold,notItalic,open,FitWidth,-1
1.5 Symmetry Groups of Physics/29,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 2: BASIC GROUP THEORY/32,Black,notBold,notItalic,open,FitWidth,-1
2.1 Basic Definitions and Simple Examples/32,Black,notBold,notItalic,open,FitWidth,-1
2.2 Further Examples, Subgroups/34,Black,notBold,notItalic,open,FitWidth,-1
2.3 The Rearrangement Lemma and the Symmetric Permutation) Group/36,Black,notBold,notItalic,open,FitWidth,-1
2.4 Classes and Invariant Subgroups/39,Black,notBold,notItalic,open,FitWidth,-1
2.5 Cosets and Factor (Quotient) Groups/41,Black,notBold,notItalic,open,FitWidth,-1
2.6 Homomorphisms/43,Black,notBold,notItalic,open,FitWidth,-1
2.7 Direct Products/44,Black,notBold,notItalic,open,FitWidth,-1
Problems/45,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 3: GROUP REPRESENTATIONS/47,Black,notBold,notItalic,open,FitWidth,-1
3.1 Representations/47,Black,notBold,notItalic,open,FitWidth,-1
3.2 Irreducible, Inequivalent Representations/52,Black,notBold,notItalic,open,FitWidth,-1
3.3 Unitary Representations/55,Black,notBold,notItalic,open,FitWidth,-1
3.4 Schur's Lemmas/57,Black,notBold,notItalic,open,FitWidth,-1
3.5 Orthonormality and Completeness Relations of Irreducible Representation Matrices/59,Black,notBold,notItalic,open,FitWidth,-1
3.6 Orthonormality and Completeness Relations of Irreducible Characters/62,Black,notBold,notItalic,open,FitWidth,-1
3.7 The Regular Representation/65,Black,notBold,notItalic,open,FitWidth,-1
3.8 Direct Product Representations, Clebsch-Gordan Coefficients/68,Black,notBold,notItalic,open,FitWidth,-1
Problems/72,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 4: GENERAL PROPERTIES OF IRREDUCIBLE VECORS AND OPERATORS/74,Black,notBold,notItalic,open,FitWidth,-1
4.1 Irreducible Basis Vectors/74,Black,notBold,notItalic,open,FitWidth,-1
4.2 The Reduction of Vectors — Projection Operators for Irreducible Components/76,Black,notBold,notItalic,open,FitWidth,-1
4.3 Irreducible Operators and the Wigner-Eckart Theorem/79,Black,notBold,notItalic,open,FitWidth,576
Problems/82,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 5: REPRESENTATIONS OF THE SYMMETRIC GROUPS/84,Black,notBold,notItalic,open,FitWidth,-1
5.1 One-Dimensional Representations/85,Black,notBold,notItalic,open,FitWidth,-1
5.2 Partitions and Young Diagrams/85,Black,notBold,notItalic,open,FitWidth,-1
5.3 Symmetrizers and Anti-Symmetrizers of Young Tableaux/87,Black,notBold,notItalic,open,FitWidth,-1
5.4 Irreducible Representations of Sₙ/88,Black,notBold,notItalic,open,TopLeftZoom,684,191,0.0
5.5 Symmetry Classes of Tensors/90,Black,notBold,notItalic,open,FitWidth,-1
Problems/98,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 6: ONE-DIMENSIONAL CONTINUOUS GROUPS/100,Black,notBold,notItalic,open,FitWidth,-1
6.1 The Rotation Group SO(2)/101,Black,notBold,notItalic,open,FitWidth,-1
6.2 The Generator of SO(2)/103,Black,notBold,notItalic,open,FitWidth,-1
6.3 Irreducible Representations of SO(2)/104,Black,notBold,notItalic,open,FitWidth,-1
6.4 Invariant Integration Measure, Orthonorrnality and Completeness Relations/106,Black,notBold,notItalic,open,FitWidth,-1
6.5 Multi-Valued Representations/108,Black,notBold,notItalic,open,FitWidth,-1
6.6 Continuous Translational Group in One Dimension/109,Black,notBold,notItalic,open,FitWidth,-1
6.7 Conjugate Basis Vectors/111,Black,notBold,notItalic,open,FitWidth,-1
Problems/113,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 7: ROTATIONS IN THREE-DIMENSIONAL SPACE-THE GROUP SO(3)/114,Black,notBold,notItalic,open,FitWidth,-1
7.1 Description of the Group SO(3)/114,Black,notBold,notItalic,open,FitWidth,-1
7.1.1 The Angle-and-Axis Parameterization/116,Black,notBold,notItalic,open,FitWidth,-1
7.1.2 The Euler Angles/117,Black,notBold,notItalic,open,FitWidth,-1
7.2 One Parameter Subgroups, Generators, and the Lie Algebra/119,Black,notBold,notItalic,open,FitWidth,-1
7.3 Irreducible Representations of the SO(3) Lie Algebra/122,Black,notBold,notItalic,open,FitWidth,-1
7.4 Properties of the Rotational Matrices Dʲ(α,β,γ)/127,Black,notBold,notItalic,open,FitWidth,-1
7.5 Application to Particle in a Central Potential/129,Black,notBold,notItalic,open,FitWidth,-1
7.5.1 Characterization of States/130,Black,notBold,notItalic,open,FitWidth,-1
7.5.2 Asymptotic Plane Wave States/131,Black,notBold,notItalic,open,FitWidth,-1
7.5.3 Partial Wave Decomposition/131,Black,notBold,notItalic,open,FitWidth,-1
7.5.4 Summary/132,Black,notBold,notItalic,open,FitWidth,-1
7.6 Transformation Properties of Wave Functions and Operators/132,Black,notBold,notItalic,open,FitWidth,-1
7.7 Direct Product Representations and Their Reduction/137,Black,notBold,notItalic,open,FitWidth,-1
7.8 Irreducible Tensors and the Wigner-Eckart Theorem/142,Black,notBold,notItalic,open,FitWidth,-1
Problems/143,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 8: THE GROUP SU(2) AND MORE ABOUT SO(3)/145,Black,notBold,notItalic,open,FitWidth,-1
8.1 The Relationship between SO(3) and SU(2)/145,Black,notBold,notItalic,open,FitWidth,-1
8.2 Invariant Integration/149,Black,notBold,notItalic,open,FitWidth,-1
8.3 Orthonormality and Completeness Relations of Dʲ/153,Black,notBold,notItalic,open,FitWidth,-1
8.4 Projection Operators and Their Physical Applications/155,Black,notBold,notItalic,open,FitWidth,-1
8.4.1 Single Particle State with Spin/156,Black,notBold,notItalic,open,FitWidth,-1
8.4.2 Two Particle States with Spin/158,Black,notBold,notItalic,open,FitWidth,-1
8.4.3 Partial Wave Expansion for Two Particle Scattering with Spin/160,Black,notBold,notItalic,open,FitWidth,-1
8.5 Differential Equations Satisfied by the Dʲ-Functions/161,Black,notBold,notItalic,open,FitWidth,-1
8.6 Group Theoretical Interpretation of Spherical Harmonics/163,Black,notBold,notItalic,open,FitWidth,-1
8.6.1 Transformation under Rotation/164,Black,notBold,notItalic,open,FitWidth,-1
8.6.2 Addition Theorem/165,Black,notBold,notItalic,open,FitWidth,-1
8.6.3 Decomposition of Products of Yₗₘ With the Same Arguments/165,Black,notBold,notItalic,open,TopLeftZoom,433,191,0.0
8.6.4 Recursion Formulas/165,Black,notBold,notItalic,open,FitWidth,-1
8.6.5 Symmetry in m/166,Black,notBold,notItalic,open,TopLeftZoom,342,191,0.0
8.6.6 Orthonormality and Completeness/166,Black,notBold,notItalic,open,FitWidth,-1
8.6.7 Summary Remarks/166,Black,notBold,notItalic,open,TopLeftZoom,593,191,0.0
8.7 Multipole Radiation of the Electromagnetic Field/167,Black,notBold,notItalic,open,FitWidth,-1
Problems/170,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 9: EUCLIDEAN GROUPS IN TWO- AND THREE-DIMENSIONAL SPACE/172,Black,notBold,notItalic,open,FitWidth,-1
9.1 The Euclidean Group in Two-Dimensional Space E₂/174,Black,notBold,notItalic,open,TopLeftZoom,297,191,0.0
9.2 Unitary Irreducible Representations of E₂ — the Angular-Momentum Basis/176,Black,notBold,notItalic,open,FitWidth,-1
9.3 The Induced Representation Method and the Plane-Wave Basis/180,Black,notBold,notItalic,open,FitWidth,-1
9.4 Differential Equations, Recursion Formulas, and Addition Theorem of the Bessel Function/183,Black,notBold,notItalic,open,FitWidth,-1
9.5 Group Contraction — SO(3) and E₂/185,Black,notBold,notItalic,open,FitWidth,-1
9.6 The Euclidean Group in Three Dimensions: E₃/186,Black,notBold,notItalic,open,FitWidth,-1
9.7 Unitary Irreducible Representations of E₃ by the Induced Representation Method/188,Black,notBold,notItalic,open,FitWidth,-1
9.8 Angular Momentum Basis and the Spherical Bessel Function/190,Black,notBold,notItalic,open,FitWidth,-1
Problems/191,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 10: THE LORENTZ AND POINCARÉ GROUPS, AND SPACE-TIME SYMMETRIES/193,Black,notBold,notItalic,open,FitWidth,-1
10.1 The Lorentz and Poincaré Groups/193,Black,notBold,notItalic,open,FitWidth,-1
10.1.1 Homogeneous Lorentz Transformations/194,Black,notBold,notItalic,open,FitWidth,-1
10.1.2 The Proper Lorentz Group/197,Black,notBold,notItalic,open,FitWidth,-1
10.1.3 Decomposition of Lorentz Transformations/199,Black,notBold,notItalic,open,FitWidth,-1
10.1.4 Relation of the Proper Lorentz Group to SL(2)/200,Black,notBold,notItalic,open,FitWidth,-1
10.1.5 Four-Dimensional Translations and the Poincaré Group/201,Black,notBold,notItalic,open,FitWidth,-1
10.2 Generators and the Lie Algebra/202,Black,notBold,notItalic,open,FitWidth,-1
10.3 Irreducible Representations of the Proper Lorentz Group/207,Black,notBold,notItalic,open,FitWidth,-1
10.3.1 Equivalence of the Lie Algebra to SU(2)×SU(2)/207,Black,notBold,notItalic,open,FitWidth,-1
10.3.2 Finite Dimensional Representations/208,Black,notBold,notItalic,open,FitWidth,-1
10.3.3 Unitary Representations/209,Black,notBold,notItalic,open,FitWidth,-1
10.4 Unitary Irreducible Representations of the Poincaré Group/211,Black,notBold,notItalic,open,FitWidth,-1
10.4.1 Null Vector case (Pµ = 0)/212,Black,notBold,notItalic,open,FitWidth,-1
10.4.2 Time-Like Vector Case (c₁ > 0)/212,Black,notBold,notItalic,open,FitWidth,-1
10.4.3 The Second Casimir Operator/215,Black,notBold,notItalic,open,FitWidth,-1
10.4.4 Light-Like Case (c₁ = 0)/216,Black,notBold,notItalic,open,FitWidth,-1
10.4.5 Space-Like Case (c₁ < 0)/219,Black,notBold,notItalic,open,FitWidth,-1
10.4.6 Covariant Normalization of Basis States and Integration Measure/220,Black,notBold,notItalic,open,FitWidth,-1
10.5 Relation Between Representations of the Lorentz and Poincaré Groups - Relativistic Wave Functions, Fields, and Wave Equations/222,Black,notBold,notItalic,open,FitWidth,-1
10.5.1 Wave Functions and Field Operators/222,Black,notBold,notItalic,open,FitWidth,-1
10.5.2 Relativistic Wave Equations and the Plane Wave Expansion/223,Black,notBold,notItalic,open,FitWidth,-1
10.5.3 The Lorentz-Poincaré Connection/226,Black,notBold,notItalic,open,FitWidth,-1
10.5.4 "Deriving" Relativistic Wave Equations/228,Black,notBold,notItalic,open,FitWidth,-1
Problems/230,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 11: SPACE INVERSION INVARIANCE/232,Black,notBold,notItalic,open,FitWidth,-1
11.1 Space Inversion in Two-Dimensional Euclidean Space/232,Black,notBold,notItalic,open,FitWidth,-1
11.1.1 The Group O(2)/233,Black,notBold,notItalic,open,FitWidth,-1
11.1.2 Irreducible Representations of O(2)/19,Black,notBold,notItalic,open,FitWidth,-1
11.1.3 The Extended Euclidean Group Ẽ₂ and its Irreducible Representations/238,Black,notBold,notItalic,open,FitWidth,-1
11.2 Space Inversion in Three-Dimensional Euclidean Space/241,Black,notBold,notItalic,open,FitWidth,-1
11.2.1 The Group O(3) and its Irreducible Representations/241,Black,notBold,notItalic,open,FitWidth,-1
11.2.2 The Extended Euclidean Group Ẽ₃ and its Irreducible Representations/243,Black,notBold,notItalic,open,FitWidth,-1
11.3 Space Inversion in Four-Dimensional Minkowski Space/247,Black,notBold,notItalic,open,FitWidth,-1
11.3.1 The Complete Lorentz Group and its Irreducible Representations/247,Black,notBold,notItalic,open,FitWidth,-1
11.3.2 The Extended Poincaré Group and its Irreducible Representations/251,Black,notBold,notItalic,open,FitWidth,-1
11.4 General Physical Consequences of Space Inversion/257,Black,notBold,notItalic,open,FitWidth,-1
11.4.1 Eigenstates of Angular Momentum and Parity/258,Black,notBold,notItalic,open,FitWidth,-1
11.4.2 Scattering Amplitudes and Electromagnetic Multipole Transitions/260,Black,notBold,notItalic,open,FitWidth,-1
Problems/263,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 12: TIME REVERSAL INVARIANCE/265,Black,notBold,notItalic,open,FitWidth,-1
12.1 Preliminary Discussion/265,Black,notBold,notItalic,open,FitWidth,-1
12.2 Time Reversal Invariance in Classical Physics/266,Black,notBold,notItalic,open,FitWidth,-1
12.3 Problems with Linear Realization of Time Reversal Transformation/267,Black,notBold,notItalic,open,FitWidth,-1
12.4 The Anti-Unitary Time Reversal Operator/270,Black,notBold,notItalic,open,FitWidth,-1
12.5 Irreducible Representations of the Full Poincaré Group in the Time-Like Case/271,Black,notBold,notItalic,open,FitWidth,-1
12.6 Irreducible Representations in the Light-Like Case (c₁ = c₂ = 0)/274,Black,notBold,notItalic,open,FitWidth,-1
12.7 Physical Consequences of Time Reversal Invariance/276,Black,notBold,notItalic,open,FitWidth,-1
12.7.1 Time Reversal and Angular Momentum Eigenstates/276,Black,notBold,notItalic,open,FitWidth,-1
12.7.2 Time-Reversal Symmetry of Transition Amplitudes/277,Black,notBold,notItalic,open,FitWidth,-1
12.7.3 Time Reversal Invariance and Perturbation Amplitudes/279,Black,notBold,notItalic,open,FitWidth,-1
Problems/281,Black,notBold,notItalic,open,FitWidth,-1
CHAPTER 13: FINITE-DIMENSIONAL REPRESENTATIONS OF THE CLASSICAL GROUPS/282,Black,notBold,notItalic,open,FitWidth,-1
13.1 GL(m): Fundamental Representations and The Associated Vector Spaces/283,Black,notBold,notItalic,open,FitWidth,-1
13.2 Tensors in V×Ṽ, Contraction, and GL(m) Transformations/285,Black,notBold,notItalic,open,FitWidth,-1
13.3 Irreducible Representations of GL(m) on the Space Of General Tensors/289,Black,notBold,notItalic,open,FitWidth,-1
13.4 Irreducible Representations of Other Classical Linear Groups/297,Black,notBold,notItalic,open,FitWidth,-1
13.4.1 Unitary Groups U(m) and U(m₊,m₋)/297,Black,notBold,notItalic,open,FitWidth,-1
13.4.2 Special Linear Groups SL(m) and the Special Unitary Groups SU(m₊,m₋)/300,Black,notBold,notItalic,open,FitWidth,-1
13.4.3 The Real Orthogonal Group O(m₊,m₋; R) and the Special Real Orthogonal Group SO(m₊,m₋; R)/303,Black,notBold,notItalic,open,TopLeftZoom,593,191,0.0
13.5 Concluding Remarks/309,Black,notBold,notItalic,open,FitWidth,-1
Problems/310,Black,notBold,notItalic,open,FitWidth,-1
APPENDIX I: NOTATIONS AND SYMBOLS/312,Black,notBold,notItalic,open,FitWidth,-1
I.1 Summation Convention/312,Black,notBold,notItalic,open,FitWidth,-1
I.2 Vectors and Vector Indices/312,Black,notBold,notItalic,open,FitWidth,-1
I.3 Matrix Indices/313,Black,notBold,notItalic,open,FitWidth,-1
APPENDIX II: SUMMARY OF LINEAR VECTOR SPACES/315,Black,notBold,notItalic,open,FitWidth,-1
II.1 Linear Vector Space/19,Black,notBold,notItalic,open,FitWidth,-1
II.2 Linear Transformations (Operators) on Vector Spaces/317,Black,notBold,notItalic,open,FitWidth,-1
II.3 Matrix Representation of Linear Operators/319,Black,notBold,notItalic,open,FitWidth,-1
II.4 Dual Space, Adjoint Operators/321,Black,notBold,notItalic,open,FitWidth,-1
II.5 Inner (Scalar) Product and Inner Product Space/322,Black,notBold,notItalic,open,FitWidth,-1
II.6 Linear Transformations (Operators) on Inner Product Spaces/324,Black,notBold,notItalic,open,FitWidth,-1
APPENDIX III: GROUP ALGEBRA AND THE REDUCTION OF REGULAR REPRESENTATION/327,Black,notBold,notItalic,open,FitWidth,-1
III.1 Group Algebra/327,Black,notBold,notItalic,open,FitWidth,-1
III.2 Left Ideals, Projection Operators/328,Black,notBold,notItalic,open,FitWidth,-1
III.3 Idempotents/329,Black,notBold,notItalic,open,FitWidth,-1
III.4 Complete Reduction of the Regular Representation/332,Black,notBold,notItalic,open,FitWidth,-1
APPENDIX IV: SUPPLEMENTS TO THE THEORY OF SYMMETRIC GROUPS Sₙ/334,Black,notBold,notItalic,open,FitWidth,-1
APPENDIX V: CLEBSCH-GORDAN COEFFICIENTS AND SPHERICAL HARMONICS/338,Black,notBold,notItalic,open,FitWidth,-1
APPENDIX VI: ROTATIONAL AND LORENTZ SPINORS/340,Black,notBold,notItalic,open,FitWidth,-1
APPENDIX VII: UNITARY REPRESENTATIONS OF THE PROPER LORENTZ GROUP/348,Black,notBold,notItalic,open,FitWidth,-1
APPENDIX VIII: ANTI-LINEAR OPERATORS/351,Black,notBold,notItalic,open,FitWidth,-1
REFERENCES AND BIBLIOGRAPHY/355,Black,notBold,notItalic,open,FitWidth,-1
INDEX/358,Black,notBold,notItalic,open,FitWidth,-1
