Group Theory and Its Applications

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Contributors

Group Theory and Its Applications

COPYRIGHT © 1968, BY ACADEMIC PRESS
     LCC 67023166

Dedication

List of Contributors

Preface

Contents

Glossary of Symbols and Abbreviations

The 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
     REFERENCES

Induced and Subduced Representations
     I. Introduction
     II. Group, Topological, Borel, and Quotient Structures
     Ill. 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-I ntegrable Representations
     VIII. Functions of Positive Type and Compact Groups
     IX. Inducing for Locally Compact Groups
     X. Applications
          A. GALILEI AND POINCARE GROUPS
               1. Rigid Motions in Euclidean n Space, E
               2. Extended Poincare Group
               3. Galilei Group
          B. PRODUCTS OF REPRESENTATIONS AND BRANCHING LAWS
               1. The Poincare Group
               2. Representations of SN
          C. IRREDUCIBLE REPRESENTATIONS OF COMPACT LIE GROUPS
          D. SPACE GROUPS
          E. EXAMPLES OF TYPE II REPRESENTATIONS
          F. MAGNETIC TRANSLATION GROUP
          G. REPRESENTATIONS OF NONCOMPACT LIE GROUPS
          REFERENCES

On a Generalization of Euler's Angles
     I. Origin of the Problem
     II. Summary of Results
     III. Proof
     IV. Corollary
     REFERENCES

Projective Representation of the Poincare Group in a Quaternionic Hilbert Space
     I. Introduction
          A. RELATIVISTIC QUANTUM MECHANICS
          B. GENERAL QUANTUM MECHANICS
          C. INTERVENTION OF GROUP THEORY
     II. The Lattice Structure of General Quantum Mechanics
          A. THE PROPOSITION SYSTEM
               1. The Elementary Propositions (Yes-No Experiments)
               2. The Partial Ordering of Propositions
               3. Intersection, Union, and Orthocomplement of Proposition
               4. The States of a Physical System
          B. DISTRIBUTIVITY, MODULARITY, AND ATOMIC'ITY
               1. Distributirity
               2. Modularity and Weak Modularity
               3. Atomicity
          C. SUPERPOSITION PRINCIPLE AND SUPERSELECTION RULES
               1. Reducible and Irreducible Lattices
               2. The Superposition Principle
     III. The Group of Automorphisms in a Proposition System
          A. MORPHISMS
               1. Definition of Morphisms
               2. Various Invariance Properties
               3. A utomorphisms
          B. THE SYMMETRY GROUP OF A PROPOSITION SYSTEM
               1. Topology in a Group of Automorphisms
               2. The Connected Component and Superselection Rules
               3. Representations of Symmetry Groups
          C. IRREDUCIBLE PROPOSITION SYSTEMS AS SUBSPACES OF A HILBERT SPACE
               1. Proposition Systems and Projective Geometries
               2. The Representation Theorem for Proposition Systems
          D. PROJECTIVE REPRESENTATIONS OF SYMMETRY GROUPS
               1. The Semilinear Transformations
               2. A utomorphisms of Subspaces
               3. Wigner's Theorem
               4. Unitary Projective Representations of Symmetry Groups
     IV. Projective Representation of the Poincare Group in Quaternionic Hilbert Space
          A. QUATERNIONIC HILBERT SPACE
               1. Quaternions
               2. Elementary Properties of Quaternionic Hilbert Space
               3. Linear and Semilinear Operators
               4. Ray Transformations
          B. PROJECTIVE REPRESENTATIONS OF SYMMETRY GROUPS IN QUATERNIONIC HILBERT SPACE
               1. Local Lifting of Factors
               2. Global Lifting of Factors
               3. Schur's Lemma and Its Corollary
               4. The Symplectic Decomposition of D
               5. Restriction and Extension of Representations
               6. Representation of Abelian Groups
          C. REPRESENTATION THEORY OF THE POINCARE GROUP
               1. The Poincare Group
               2. Physical Heuristics
               3. The Physical Representations of the Connected Component
               4. Induced Representations (Discrete Case)
               5. Induced Representations (Continuous Case)
               6. Semidirect Products
     V. Conclusion
     REFERENCES

Group Theory in Atomic Spectroscopy
     I. Introduction
     II. Shell Structure
          A. ROOT FIGURES
          B. ANNIHILATION AND CREATION OPERATORS
          C. REPRESENTATIONS
          D. SUBGROUPS
          E. UNITARY GROUPS
     III. Coupled Tensors
          A. THE GROUP O+(3)
          B. COMMUTATORS
          C. SUBGROUPS OF U(41 + 2)
          D. THE CONFIGURATIONS f^N
     IV. Representations
          A. BRANCHING RULES
          B. SENIORITY
          C. ALTERNATIVE DECOMPOSITIONS
          D. INNER KRONECKER PRODUCTS
     V. The Wigner-Eckart Theorem
          A. MATRIX ELEMENTS
          B. SINGLE-PARTICLE OPERATORS
          C. EXAMPLES
          D. QUASISPIN
          E. THE COULOMB INTERACTION
     VI. Conclusion
     REFERENCES

Group Lattices and Homomorphisms
     I. Introduction
     II. Groups
          A. DEFINITIONS AND NOTATION
          B. LATTICES OF SUBGROUPS
          C. DIRECT PRODUCT GROUPS
          D. THE LATTICE OF A HAMILTONIAN
     III. Symmetry Adaptation of Vector Spaces
          A. INTRODUCTION
          B. THE EIGENVECTOR PROBLEM; PERTURBATION THEORY
          C. SYMMETRY ADAPTATION OF PRODUCT SPACES
     IV. The Lattice of the Quasi-Relativistic Dirac Hamiltonian
          A. THE DIRAC HAMILTONIAN
          B. THE FOLDY-WOUTHUYSEN TRANSFORMATION
          C. THE LATTICE OF THE QUASI-RELATIVISTIC DIRAC HAMILTONIAN
          D. APPENDIX: DOUBLE GROUP MATRICES
     V. Applications
          A. AN ELECTRON IN A CENTRAL FIELD
          B. N ELECTRONS IN A CENTRAL FIELD
          C. AN ELECTRON IN A NONCENTRAL FIELD
          D. NUCLEAR STATES
     Acknowledgments
     REFERENCES

Group Theory in Solid State Physics
     I. Introduction
     II. Stationary States in the Quantum Theory of Matter
          A. GASEOUS STATES
          B. FLUID AND SOLID STATES
          C. THE ROLE OF SYMMETRY
     III. The Group of the Ham i lton ian
          A. REPRESENTATION THEORY
          B. IRREDUCIBLE SUBSPACES
          C. EXPECTATION VALUES
          D. TRANSITION PROBABILITIES AND SELECTION RULES
          E. PROJECTION OPERATORS
          F. REDUCTION OF BASIS SETS
     IV. Symmetry Groups of Solids
          A. THE GROUP OF PRIMITIVE TRANSLATIONS
          B. POINT GROUPS
          C. SYMMORPHIC CRYSTALLOGRAPHIC GROUPS
          D. NONSYMMORPHIC CRYSTALLOGRAPHIC GROUPS
          E. DOUBLE SPACE GROUPS
          F. TIME-REVERSAL SYMMETRY
          G. MAGNETIC GROUPS
          H. PERMUTATION SYMMETRY FOR PARTICLES IN SOLIDS
     V. Lattice Vibrations in Solids
          A. CLASSICAL TREATMENT
               1. One Atom per Unit Cell
               2. Translational Symmetry
               3. The Case of Several Atoms per Unit Cell
          B. QUANTUM-MECHANICAL TREATMENT
          C. BOSE STATISTICS
     V1. Band Theory of Solids
          A. FERMI STATISTICS
          B. THE HARTREE-FOCK EQUATIONS
          C. BRILLOUIN ZONES
          D. DEGENERACY IN k SPACE
          E. THE PLANE WAVE (PW) METHOD
          F. THE ORTHOGONALIZED PLANE WAVE (OPW) METHOD
          G. THE AUGMENTED PLANE WAVE (APW) AND RELATED METHODS
          H. THE TIGHT-BINDING METHOD
          1. SYMMETRY PROPERTIES OF THE IRREDUCIBLE CRYSTAL HAMILTONIAN
     VII. Electromagnetic Fields in Solids
          A. WAN N IER STATES
          B. QUASI-CLASSICAL BAND MECHANICS
          C. BAND ELECTRONS IN ELECTRIC FIELDS
          D. BAND ELECTRONS IN MAGNETIC FIELDS
     REFERENCES

Group Theory of Harmonic Oscillators and Nuclear Structure
     I. Introduction and Summary
     II. The Symmetry Group U(3n); the Subgroup QI(3) X U(n); Gelfand States
          A. THE HARMONIC OSCILLATOR HAMILTONIAN AND ITS UNITARY SYMMETRY GROUPS
          B. n-PARTICLE STATES AS BASES FOR IRREDUCIBLE REPRESENTATIONS OF THE GROUPS U(3n) QI(3) X U(n)
               1. State of Highest Weight
               2. Lowering Operators
               3. The Physical Chain of Groups £ (3) (91(3):D6+(2)
          C. APPENDIX: GENERATORS OF THE UNITARY GROUP IN r DIMENSIONS
     III. The Central Problem: Permutational Symmetry of  theOrbital States
          A. SHELL MODEL STATES IN THE X U (n) SCHEME
               1. Three-Particle Shell Model States in the ?(3) X U(n) Scheme
               2. Irreducible Representations of the Groups K(3) and K(n)
               3. Irreducible Representations of K(3) Contained in an Irreducible Representation of U(3)
               4. Construction of Three-Particle Shell Model States
               5. n-Particle Shell Model States
          B. TRANSLATIONAL-INVARIANT STATES
               1. The Chain U(n) U (n - 1) O(n - 1) S (n)
               2. Translational-Invariant Four-Particle States
     IV. Orbital Fractional Parentage Coefficients
          A. ONE-PARTICLE FRACTIONAL PARENTAGE COEFFICIENTS
          B. TWO-PARTICLE FRACTIONAL PARENTAGE COEFFICIENTS
          C. PAIR FRACTIONAL PARENTAGE COEFFICIENTS
          D. FRACTIONAL PARENTAGE COEFFICIENTS FOR THREE-PARTICLE SHELL MODEL STATES
          F. ONE-Row WIGNER COEFFICENTS OF QI(3)
     V. Group Theory and n-Particle States in Spin-Isospin Space
          A. SPIN-ISOSPIN STATES WITH PERMUTATIONAL SYMMETRY
          B. BASES FOR IRREDUCIBLE REPRESENTATIONS OF THE U(4n) GROUP IN THE QIl(4) X U(n) CHAIN
          C. STATES WITH DEFINITE TOTAL SPIN AND ISOSPIN
          D. THE SPECIAL GELFAND STATES AS BASES FOR IRREDUCIBLE REPRESENTATIONS OF THE SYMMETRIC GROUP
     VI. Spin-Isospin Fractional Parentage Coefficients
          A. EQUIVALENCE OF THE FRACTIONAL PARENTAGE COEFFICIENTS AND THE WIGNER COEFFICIENTS OF QI(4)
          B. ONE-BLOCK WIGNER COEFFICIENTS OF U(n) IN THE CANONICAL CHAIN
          C. THE ONE-PARTICLE SPIN-ISOSPIN FRACTIONAL PARENTAGE COEFFICIENTS
          D. THE TWO-PARTICLE SPIN-ISOSPIN FRACTIONAL PARENTAGE COEFFICIENTS
     VII. Evaluation of Matrix Elements of One-Body and Two-Body Operators
          A. ONE-BODY AND TWO-BODY OPERATORS
          B. GENERAL PROCEDURE FOR DERIVING MATRIX ELEMENTS OF ONE-BODY AND TWO-BODY OPERATORS
               1. One-Body Operators in Shell Model States
               2. Matrix Elements of Two-Body Interactions
          C. MATRIX ELEMENTS FOR THREE-PARTICLE AND FOUR-PARTICLE STATES
               1. Matrix Elements of One-Body and Two-Body Operators for Three-Particle Shell Model States
               2. Matrix Elements of Two-Body Interactions for Translational-Invariant Four-Particle States
     VIII. The Few-Nucleon Problem
          A. THE INTRINSIC HAMILTONIAN
          B. THE FOUR-NUCLEON PROBLEM
     IX. The El l iott Model in Nuclear Shell Theory
          A. THE ELLIOTT MODEL FOR A SINGLE SHELL
          B. EXTENSION OF THE ELLIOTT MODEL TO MULTISHELL CONFIGURATIONS
          C. THE QUADRUPOLE-QUADRUPOLE INTERACTION
          D. SINGLE-SHELL APPLICATIONS
     X. Clustering Properties and Interactions
          A. DEFINITION OF CLUSTERING; STATES OF MAXIMUM CLUSTERING
          B. PERMUTATIONAL LIMITS ON CLUSTERING; WHEELER OPERATORS
          C. CLUSTERING OF FOUR-PARTICLE STATES; WILDERMUTH STATES
          D. CLUSTERING INTERACTION
          E. QUADRUPOLE-QUADRUPOLE INTERACTION AND CLUSTERING INTERACTION IN THE l S- lp SHELL
          F. APPENDIX: EIGENVALUES OF WHEELER OPERATORS
     Acknowledgments
     REFERENCES

Broken Symmetry
     I. Introduction
     II. Wigner-Eckart Theorem
     Ill. 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)
     Acknowledgments
     REFERENCES

Broken SU(3) as a Particle Symmetry
     I. Introduction
     II. Perturbative Approach
     III. Algebra of SU(3)
     IV. Representations
          A. WEIGHTS AND LABELING OF BASES
          B. ACTION OF GENERATORS ON BASES
          C. MULTIPLICITIES AND DIRECT PRODUCT DECOMPOSITION
     V. Tensor and Wigner Operators
     VI. Particle Classification, Masses, and Form Factors
          A. THE BARYON STATES B /2+
          B. THE BARYON STATES 8312+
          C. THE BARYON STATES B!,(1405) AND B
          D. THE MESON STATES Mp
          E. THE MESON STATES M
          F. THE MESON STATES M2+
     VII. Some Remarks on R and SU(3)/Z3
     VIII. Couplings and Decay Widths
          A. BARYON DECAYS
          B. BOSON DECAYS
     IX. Weak Interactions
          A. SEMILEPTONIC DECAYS
          B. NONLEPTONIC DECAYS
     X. Appendix
     Acknowledgments
     REFERENCES

De Sitter Space and Positive Energy
     I. Introduction and Summary
     II. Ambivalent Nature of the Classes of de Sitter Groups
     Ill. 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. Contraction of the Representations of the 2 + I de Sitter Group
     Acknowledgment
     REFERENCES

Author Index

Subject Index

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