Group theory and its applications / 1

From the Preface: The importance of group theory and its utility in applications to various branches of physics and chemistry is now so well established and universally recognized that its explicit use needs neither apology nor justification. Matters have moved a long way since the time, just thirty years ago, when Condon and Shortley, in the introduction to their famous book, "The Theory of Atomic Spectra", justified their doing "group theory without group theory" by the statement that "... the theory of groups ... is not . . . part of the ordinary mathematical equipment of physicists." The somewhat adverse, or at least sceptical, attitude toward group theory illustrated by the telling there of the well-known anecdote concerning the Weyl-Dirac exchange,\* has been replaced by an uninhibited and enthusiastic espousal. This is apparent from the steadily increasing number of excellent textbooks published in this field that seek to instruct ever widening audiences in the nature and use of this tool. There is, however, a gap between the material treated there and the research literature and it is this gap that the present treatise is designed to fill. The articles, by noted workers in the various areas of group theory, each review a substantial field and bring the reader from the level of a general understanding of the subject to that of the more advanced literature. The serious student and beginning research worker in a particular branch should find the article or articles in his specialty very helpful in acquainting him with the background and literature and bringing him up to the frontiers of current research; indeed, even the seasoned specialist in a particular branch will still learn something new. The editor hopes also to have the treatise serve another useful function: to entice the specialist in one area into becoming acquainted with another. Such ventures into novel fields might be facilitated by the recognition that similar basic techniques are applied throughout; e.g., the use of the Wigner-Eckart theorem can be recognized as a unifying thread running through much of the treatise. The applications of group theory can be subdivided generally into two broad areas: one, where the underlying dynamical laws (of interactions) and therefore all the resulting symmetries are known exactly; the other, where these are as yet unknown and only the kinematical symmetries (i.e. those of the underlying space-time continuum) can serve as a certain guide. In the first area, group theoretical techniques are used essentially to exploit the known symmetries, either to simplify numerical calculations or to draw exact, qualitative conclusions. All (extra-nuclear) atomic and molecular phenomena are believed to belong to this category; the central chapters in this book deal with such applications, which, until relatively recently, formed the bulk of all uses of group theory. In the second major area, application of group theory proceeds essentially in the opposite direction: It is used to discover as much as possible of the underlying symmetries and, through them, learn about the physical laws of interaction. This area, which includes all aspects of nuclear structure and elementary particle theory, has mushroomed in importance and volume of research to an extraordinary degree in recent times; the articles in the second half of the treatise are devoted to it. I n part as a consequence of these developments, physical scientists have been forced to concern themselves more profoundly with mathematical aspects of the theory of groups that previously could be left aside; questions of topology, representations of noncompact groups, more powerful methods for generating representations as well as a systematic study of Lie groups and their algebras in general belong in this category. They are treated in the earlier chapters of this book. Considerations of both space and timing have forced omission from this volume of articles dealing with several important areas of applied group theory like molecular spectra, hidden symmetry and "accidental" degeneracy, group theory and computers, and others. These will be included in a second volume, currently in preparation. Complete uniformity and consistency of notation is an ideal to be striven for but difficult to attain; it is especially hard to achieve when, as in the present case, many different and widely separated specialities are discussed, each of which usually has a well-established notational system of its own which may not be reconcilable with an equally well-established one in another area. In the present book uniformity has been carried as far as possible, subject to these restrictions, except where it would impair clarity. Cover 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 Back Cover