Group Theory and Its Applications

List of ContributorsPrefaceGlossary of Symbols and AbbreviationsThe Algebras of Lie Groups and Their Representations I. Introduction II. Preliminary Survey III. Lie's Theorem, the Rank Theorem, and the First Criterion of Solvability IV. The Cartan Subalgebra and Root Systems V. The Classification of Semisimple Lie Algebras in Terms of Their Root Systems VI. Representations and Weights for Semisimple Lie Algebras ReferencesInduced and Subdued Representations I. Introduction II. Group, Topological, Borel, and Quotient Structures III. The Generalized Schur Lemma and Type I Representations IV. Direct Integrals of Representations V. Murray-von Neumann Typology VI. Induced Representations of Finite Groups VII. Orthogonality Relations for Square-Integrable Representations VIII. Functions of Positive Type and Compact Groups IX. Inducing for Locally Compact Groups X. Applications ReferencesOn a Generalization of Euler's Angles I. Origin of the Problem II. Summary of Results III. Proof IV. Corollary ReferencesProjective Representation of the Poincare Group in a Quaternionic Hilbert Space I. Introduction II. The Lattice Structure of General Quantum Mechanics III. The Group of Automorphisms in a Proposition System IV. Projective Representation of the Poincare Group in Quaternionic Hilbert Space V. Conclusion ReferencesGroup Theory in Atomic Spectroscopy I. Introduction II. Shell Structure III. Coupled Tensors IV. Representations V. The Wigner-Eckart Theorem VI. Conclusion ReferencesGroup Lattices and Homomorphisms I. Introduction II. Groups III. Symmetry Adaption of Vector Spaces IV. The Lattice of the Quasi-Relativistic Dirac Hamiltonian V. Applications ReferencesGroup Theory in Solid State Physics I. Introduction II. Stationary States in the Quantum Theory of Matter III. The Group of the Hamiltonian IV. Symmetry Groups of Solids V. Lattice Vibrations in Solids VI. Band Theory of Solids VII. Electromagnetic Fields in Solids ReferencesGroup Theory of Harmonic Oscillators and Nuclear Structure I. Introduction and Summary II. The Symmetry Group U (3n); The Subgroup U(3) X U(n); Gelfand States III. The Central Problem: Permutational Symmetry of the Orbital States IV. Orbital Fractional Parentage Coefficients V. Group Theory and n-Particle States in Spin-Isospin Space VI. Spin-Isospin Fractional Parentage Coefficients VII. Evaluation of Matrix Elements of One-Body and Two-Body Operators VIII. The Few-Nucleon Problem IX. The Elliott Model in Nuclear Shell Theory X. Clustering Properties and Interactions ReferencesBroken Symmetry I. Introduction II. Wigner-Eckart Theorem III. Some Relevant Group Theory IV. Particle Physics SU(3) from the Point of View of the Wigner-Eckart Theorem V. Foils to SU(3) and the Eightfold Way VI. Broken Symmetry in Nuclear and Atomic Physics VII. General Questions concerning Broken Symmetry VIII. A Note on SU(6) ReferencesBroken SU(3) as a Particle Symmetry I. Introduction II. Perturbative Approach III. Algebra of SU(3) IV. Representations V. Tensor and Wigner Operators VI. Particle Classification, Masses, and Form Factors VII. Some Remarks on R and SU(3)/Z3 VIII. Couplings and Decay Widths IX. Weak Interactions X. Appendix ReferencesDe Sitter Space and Positive Energy I. Introduction and Summary II. Ambivalent Nature of the Classes of de Sitter Groups III. The Infinitesimal Elements of Unitary Representations of the de Sitter Group IV. Finite Elements of the Unitary Representations of Section III V. Spatial and Time Reflections VI. The Position Operators VII. General Remarks about Contraction of Groups and Their Representations VIII.