1. Symmetries in Quantum Mechanics 1.1 Symmetries in Classical Physics 1.2 Spatial Translations in Quantum Mechanics 1.3 The Unitary Translation Operator 1.4 The Equation of Motion for States Shifted in Space 1.5 Symmetry and Degeneracy of States 1.6 Time Displacements in Quantum Mechanics 1.7 Mathematical Supplement: Definition of a Group 1.8 Mathematical Supplement:Rotations and their Group Theoretical Properties 1.9 An Isomorphism of the Rotation Group 1.10 The Rotation Operator for Many-Particle States 1.11 Biographical Notes 2. Angular Momentum Algebra Representation of Angular Momentum Operators: Generators of SO(3) 2.1 Irreducible Representations of the Rotation Group 2.2 Matrix Representations of Angular Momentum Operators 2.3 Addition of Two Angular Momenta 2.4 Evaluation of Clebsch-Gordan Coefficients 2.5 Recursion Relations for Clebsch-Gordan Coefficients 2.6 Explicit Calculation of Clebsch-Gordan Coefficients 2.7 Biographical Notes 3. Mathematical Supplement: Fundamental Properties of Lie Groups 3.1 General Structure of Lie Groups 3.2 Interpretation of Commutators as Generalized Vector Products, Lie's Theorem, Rank of Lie Group 3.3 Invariant Subgroups, Simple and Semisimple Lie Groups, Ideals 3.4 Compact Lie Groups and Lie Algebras 3.5 Invariant Operators (Casimir Operators) 3.6 Theorem of Racah 3.7 Comments on Multiplets 3.8 Invariance Under a Symmetry Group 3.9 Construction of the Invariant Operators 3.10 Remark on Casimir Operators of Abelian Lie Groups 3.11 Completeness Relation for Casimir Operators 3.12 Review of Some Groups and Their Propertiesand Transformations of Functions 3.14 Biographical Notes 4.Symmetry Groups and Their Physical Meaning:General Considerations 4.1 Biographical Notes 5.The lsospin Group (Isobaric Spin) 5.1 Isospin Operators for a Multi-Nucleon System 5.2 General Properties of Representations of a Lie Algebra 5.3 Regular (or Adjoint) Representation of a Lie Algebra 5.4 Transformation Law for Isospin Vectors 5.5 Experimental Test of Isospin lnvariance 5.6 Biographical Notes 6.The Hypercharge 6.1 Biographical Notes 7. The SU(3) Symmetry 7.1 The Groups U(n) and SU(n) 7.2 The Generators of SU(3) 7.3 The Lie Algebra of SU(3) 7.4 The Subalgebras of the SU(3) Lie Algebra and the Shift Operators 7.5 Coupling of T, U and V Multiplets 7.6 Quantitative Analysis of Our Reasoning 7.7 Further Remarks About the Geometric Form of an SU(3) Multiplet 7.8 The Number of States on Mesh Points on Inner Shells 8.Quarks and SU(3) 8.1 Searching for Quarks 8.2 The Transformation Properties of Quark States 8.3 Construction of all SU(3) Multiplets from the Elementary Representations [3] and 8.4 Construction of the Representation D(p, q) from Quarks and Antiquarks 8.5 Meson Multiplets 8.6 Rules for the Reduction of Direct Products of SU(3) Multiplets 8.7 U-Spin Invariance 8.8 Test of U-Spin Invariance …… 9.Representations of the Permutation Group and Young Tableanx 10.Mathematical Excursion Group Characters 11.Charm and SU(4) 12.Mathematical Supplement 13.Special Discrdts Symmetries 14.Dynamical Symmetries 15.Mathematical Excursion:Non-compact Lie Gruoups 16.Proff of Racah s Theorem Subject Index
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