Group Theory and Its Applications, Volume II

Contributors

Group Theory and Its Applications, VOLUME II

COPYRIGHT (C) 1971, BY ACADEMIC PRESS
LCCN 67023166

Contents

List of Contributors

Preface

Contents of Volume I

The Representations and Tensor Operators of the Unitary Groups U(n)
I. Introduction: The Connection between the Representation Theory of S(n) and That of U(n), and Other Preliminaries
II. The Group SU(2) and Its Representations
Ill. The Matrix Elements for the Generators of U(n)
IV. Tensor Operators and Wigner Coefficients on the Unitary Groups
REFERENCES

Symmetry and Degeneracy
I. Introduction
II. Symmetry of the Hydrogen Atom
Ill. Symmetry of the Harmonic Oscillator
IV. Symmetry of Tops and Rotators
V. Bertrand's Theorem
VI. Non-Bertrandian Systems
VII. Cyclotron Motion
VIII. The Magnetic Monopole
IX Two Coulomb Centers
X. Relativistic Systems
Xl. Zitterbewegung
XII. Dirac Equation for the Hydrogen Atom
XIII. Other Possible Systems and Symmetries
XIV. Universal Symmetry Groups
XV. Summary
Acknowledgments
REFERENCES

Dynamical Groups in Atomic and Molecular Physics
I. Introduction
II. The Second Vector Constant of Motion in Kepler Systems
Ill. The Four-Dimensional Orthogonal Group and the Hydrogen Atom
IV. Generalization of Fock's Equation: 0(5) as a Dynamical Noninvariance Group
V. Symmetry Breaking in Helium
VI. Symmetry Breaking in First-Row Atoms
VII. The Conformal Group and One-Electron Systems
VIII. Conclusion
Acknowledgments
REFERENCES

Symmetry Adaptation of Physical States by Means of Computers
I. Introduction
II. Constants of Motion and the Unitary Group of the Hamiltonian
III. Separation of Hilbert Space with Respect to the Constants of Motion
IV. Dixon's Method for Computing Irreducible Characters
V. Computation of Irreducible Matrix Representatives
VI. Group Theory and Computers
REFERENCES

Galilel Group and Galilean Invariance
I. Introduction
A. HISTORICAL BACKGROUND AND MOTIVATIONS
B. CONTENTS
II. The Galilei Group and Its Lie Algebra
A. DEFINITION AND SPACE-TIME PROPERTIES
1. Definition and Invariants
2. Generalizations and Relations
B. STRUCTURE OF THE GALILEI GROUP AND ITS SUBGROUPS
1. The Group Law
2. Subgroups
3. Homogeneous Spaces
4. Connectivity Properties
C. THE GALILEI LIE ALGEBRA
D. DISCRETE TRANSFORMATIONS AND AUTOMORPHISMS
III. The Extended Galilei Group and Lie Algebra
A. EXTENSIONS OF GROUPS AND LIE ALGEBRAS
1. Extensions of Groups
2. Extensions of Lie Algebras
3. Examples
B. THE EXTENDED GALILEI LIE ALGEBRA
C. THE EXTENDED GALILEI GROUP
IV. Representations of the Gal i lei Groups
A. UNITARY REPRESENTATIONS OF SEMIDIRECT PRODUCTS
1. Induced Representations
2. Example: The Two-Dimensional Eculidean Group d12)
B. UNITARY REPRESENTATIONS OF THE GALILEI GROUP
1. Representations of the Group
2. Representations of the Lie Algebra
C. UNITARY REPRESENTATIONS OF THE EXTENDED GALILEI GROUP
D. PROJECTIVE REPRESENTATIONS OF THE GALILEI GROUP*
E. NONUNITARY REPRESENTATIONS OF THE GALILEI GROUP
V. Applications to Classical Physics
A. FOUNDATIONS OF CLASSICAL MECHANICS: FREE PARTICLES
1. Lagrangian Formalism
2. Conservation Laws
3. Hamiltonian Formalism
B. FOUNDATIONS OF CLASSICAL MECHANICS: INTERACTING PARTICLES
1. External Forces
2. Mutually Interacting Particles
C. GALILEAN ELECTROMAGNETISM AND FIELD THEORIES
1. Formalism
2. Physical Discussion
VI. Applications to Quantum Physics
A. LoCALIZABILITY AND PHYSICAL PARTICLES
1. Position Operator and Representations of G
2. Physical Representation in a Configuration Space
3. Zero-Mass "Particles" and the Nonrelativistic Limit
B. KINEMATICS OF MANY-PARTICLE SYSTEMS
1. The Role of Mass and Internal Energy: Compound Systems
2. Decomposition of the Tensor Product of Two Physical Representations*
3. Clebsch-Gordan Coefficients and Partial- Wave Analysis
C. WAVE EQUATIONS
1. Galilean Invariance and External Forces
2. Wave Equations for Nonzero Spin
D. QUANTUM FIELD THEORIES AND PARTICLES PHYSICS
1 . Galilean Quantum Field Theory and Many-Body Problems
2. Internal and Space-Time Symmetries
REFERENCES

Author Index

Subject Index