The application 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 onlt 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 symmetrics, either to simplify numerical calculations or to draw exact, qualitative conclusions. In the second major area, application of group theory proceeds essentially in the opposite direction. In 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 noncompact groups, more powerful methods for generating representations, as well as a systematic study of Lie groups and the algebras, in general belong in this category. This volume, as did the earlier ones, contains contributions in all these areas.The coverage of subjects of applied group theory is still neither complete nor completely balanced, though it is more so than it was in Volume I and II. To a large extent this is inevitable in a filed growing and evolving as rapidly as this one.
Author(s): Ernest Moshe Loebl
Publisher: Academic Press
Year: 1975
Language: English
Commentary: Cover, OCR, Enhanced, Optimized, Single Pages
Pages: C, xvi, 480
Cover
Contributors
Group Theory and Its Applications
COPYRIGHT © 1975, BY ACADEMIC PRESS
ISBN 0-12-455153-X (v. 3)
QA171.L79 512'.2
LCCN 67023166
Contents
List of Contributors
Preface
Contents of Other Volumes
Finite Groups and Semisimple Algebras in Quantum Mechanics
I. Introduction
II. Linear Associative Algebras
Ill. Semisimple Algebras
IV. Semisimple Algebras in Quantum Mechanics
V. Group Algebras
VI. Fundamental Representation Theory
VII. Sequence Adaptation
VIII. Induced and Subduced Representations
IX. Approximate Symmetries in Quantum Mechanics
A. DESCENT IN SYMMETRY
B. ASCENT IN SYMMETRY
C. MIXED DESCENT AND ASCENT IN SYMMETRY
X. Weakly Interacting Sites
(A) The Zero-Order Group
(B) The Perturbed Group
(C) The Intersection Group
(D) The Join Group
XI. Double Sequence Adaptation and Recoupling Coefficients
XII. Recoupling Coefficients in Quantum Mechanics
XIII. Point Group Symmetry Adaptation
XIV. Branching Rules
A. THE SYMMETRIC GROUP
B. PERMUTATION STATES FROM AN ORBITAL PRODUCT
C. EQUIVALENT ELECTRONS
D. ANGULAR MOMENTUM STATES FOR EQUIVALENT ELECTRONS
E. DIHEDRAL GROUP STATES FOR EQUIVALENT PARTICLES
F. STATES FROM MOLECULAR ORBITAL CONFIGURATIONS
G. WEAKLY INTERACTING SITES
XV. Double Cosets
XVI. Effective Hamiltonians for Weakly Interacting Sites
A. EXCITON THEORY
B. EXCHANGE HAMILTONIAN THEORY
XVII. Conclusion
Acknowledgments
REFERENCES
Semisimple Subalgebras of Semisimple Lie Algebras: The Algebra As (SU(6)) as a Physically Significant Example
I. Introduction
II. Definitions
III. Embedding of Subalgebras
IV. Regular Subalgebras
V. S-Subalgebras
V1. Classification of Subalgebras of the Algebra As
1. Maximal Subalgebras of Q4
2. Maximal Subalgebras of Q3
3. Maximal Subalgebras of Q2+Q1
4. Maximal Subalgebras of Q2
5. Maximal Subalgebras of Q2+Q2
6. Maximal Subalgebras of Q3
7. Maximal Subalgebras of Q1+Q1
8. Completion of the Classification
VII. Inclusion Relations
VIll. Physically Significant Chains of Subalgebras of Q5
A. ATOMIC PHYSICS
B. NUCLEAR PHYSICS
C. MOLECULAR PHYSICS
D. PARTICLE PHYSICS
E. CHIRALITY GROUP SU(3) x SU(3) (22)
Acknowledgments
REFERENCES
Frobenius Algebras and the Symmetric Group*
I. Introduction
II. The Frobenius Algebra and Its Centrum
A. THE ALGEBRA
B. THE CENTRUM
III. The Matric Basis and Symmetry Adaptation
A. INTRODUCTION
B. SYMMETRY ADAPTATION
IV. The Algebra of the Symmetric Group*
V. Isospin-Free Nuclear Theory*
VI. Spin- Free (Supermultiplet) Nuclear Theory
VII. Spin-Free Atomic Theory
VIII. Summary
REFERENCES
The Heisenberg-Weyl Ring in Quantum Mechanics
I. Introduction
II. The Heisenberg-Weyl Group
A. THE ALGEBRA W AND COVERING ALGEBRA W
B. CONSTRUCTION OF THE GROUP W
C. OTHER VERSIONS OF THE HEISENBERG-WEYL GROUP
1. An n-Dimensional Version of W
2. A Group of Translations in a Magnetic Field
D. REPRESENTATIONS OF THE GROUP W
1. A Multiplier Representation
2. Representations with P Diagonal
3. Representations with C Diagonal
4. Mixed Representations
5. Orthogonality and Completeness Relations
6. The Coordinate Basis
7. The Harmonic Oscillator Basis
E. DISCUSSION
III. The Heisenberg-Weyl Ring D
A. CONSTRUCTION AND PROPERTIES
B. REPRESENTATIONS OF THE RING
C. PHASE SPACE REPRESENTATIVE FUNCTIONS
D. COMMUTATORS AND POISSON BRACKETS
E. DISCUSSION
IV. The Quantization Process
A. THE CLASSICAL LIMIT
B. THE QUANTIZATION SCHEME PROBLEM
1. Statement of the Problem
2. The Born-Jordan Rule
3. The Dirac-von Neumann Construction
4. The Weyl-McCoy Scheme
5. The Symmetrization Rule
6. Normal Ordering
7. The Feynman Formulation
8. Cohen's Scheme Function
9. Quantization-Scheme-Independent Statements
C. DISCUSSION
V. Canonical Transformations
A. CLASSICAL CANONICAL TRANSFORMATIONS
1. Definition
2. The Classical Group and Its Generators
3. The Inhomogeneous Linear Subgroup
4. Point Transformations and Their Generators
B. QUANTUM CANONICAL TRANSFORMATIONS
1. Definition
2. Unitary Quantum Canonical Transformations
C. CORRESPONDENCE PRESERVED AND BROKEN
1. The Question of Isomorphism
2. Extended Quantum Linear Transformations
3. Quantum Point Transformations
4. Example of Correspondence Broken
D. DISCUSSION
VI. Quantum Mechanics on a Compact Space
A. THE MIXED GROUP W*
1. Definition
2. Representations of W
B. THE RING B*
C. CHARACTERISTICS OF "COMPACT" QUANTUM MECHANICS
1. Infinite-Radius Limit
2. The Classical Limit
3. The Quantization Process
4. Canonical Transformations
D. DISCUSSION
REFERENCES
Complex Extensions of Canonical Transformations and Quantum Mechanics
I. Introduction and Summary
ll. Groups of Classical Canonical Transformations
Ill. Unitary Representations of Canonical Transformations in Quantum Mechanics
IV. Complex Phase Space and Bargmann Hilbert Space
A. CLASSICAL COMPLEX PHASE SPACE
B. BARGMANN HILBERT SPACE
C. MAPPING OF HILBERT SPACES
D. REPRESENTATIONS OF LINEAR CANONICAL TRANSFORMATIONS IN BARGMANN HILBERT SPACE
V. Complex Extensions of Canonical Transformations
A. EXTENSION OF REAL LINEAR CANONICAL TRANSFORMATIONS TO A COMPLEXSEMI GROUP
B. CONFORMAL TRANSFORMATIONS IN BARGMANN HILBERT SPACE
C. GENERATING FUNCTIONS FOR MATRIX ELEMENTS
VI. Barut Hilbert Space and Angular Momentum Projection in Bargmann Hilbert Space
A. ANGULAR-MOMENTUM-PROJECTED SUBSPACES OF BARGMANN HILBERT SPACE
B. BARUT HILBERT SPACE AND THE RADIAL OSCILLATOR WITH A CENTRIFUG AL BARRIER
C. CANONICAL TRANSFORMATIONS IN RADIAL SPACE AND THEIR REPRESENTATIONS IN BARUT HILBERT SPACE
VIl. Applications to Problems of Accidental Degeneracy in Quantum Mechanics
A. THE TWO-DIMENSIONAL ANISOTROPIC OSCILLATOR WHEN THE RATIO OF THE FREQUENCIES Is RATIONAL
B. THE OSCILLATOR IN A SECTOR OF ANGLE 2 $\pi/k§
C. THE TWO-DIMENSIONAL OSCILLATOR WITH CENTRIFUGAL FORCES
Vlll. The Three-Body Problem
A. THE ONE-DIMENSIONAL THREE-BODY SYSTEM
B. THE THREE-BODY PROBLEM IN BARGMANN HILBERT SPACE
C. THREE PARTICLES IN THREE DIMENSIONS
IX. Applications to the Clustering Theory of Nuclei
A. CLUSTERING IN THE THREE-BODY SYSTEM
B. CONFIGURATIONS OF k CLUSTERS
C. CALCULATION OF OPERATORS FOR k-CLUSTER CONFIGURATIONS
X. Conclusion
Acknowledgment
REFERENCES
Quantization as an Eigenvalue Problem
I. Quantization
II. Operators on Hilbert Space
Ill. Differential Equation Theory
IV. Symplectic Boundary Form
V. Spectral Density
VI. Continuation in the Complex Eigenvalue Plane
VII. One-Dimensional Relativistic Harmonic Oscillator
VIII. Survey
Acknowledgments
REFERENCES
Elementary Particle Reactions and the Lorentz and Galilei Groups
I. Introduction
II. Single-Variable Expansions for Four-Body Scattering
A. NONRELATIVISTIC EXPANSIONS
1. Partial- Wave Expansion
2. The Eikonal Expansion
B. RELATIVISTIC EXPANSIONS
1. The Poincare Group
2. Single- Variable Expansions of Spinless Relativistic Amplitudes
Ill. Lorentz Group Two-Variable Expansions for Spinless Particles and the Lorentz Amplitudes
A. GENERAL OUTLINE OF THE METHOD
B. MATHEMATICAL PRELIMINARIES
1. The Group 0(3, 1) and Its Subgroups
2. Representations of the Group 0(3, 1)
3. Basis Functions for Representations of 0(3, 1)
C. EXPLICIT FORM OF Two-VARIABLE EXPANSIONS
1. Scattering Amplitude as Function on an 0(3, 1) Manifold
2. Expansion Formulas and Lorentz Amplitudes
D. PHYSICAL ASPECTS OF TWO-VARIABLE 0(3, 1) EXPANSIONS
1. Relation to Little-Group Expansions
2. Threshold Behavior, Asymptotic Behavior, Resonances, and Regge Poles
3. The Crossing Transformation and Analyticity Properties of the Lorentz Amplitudes
IV. Two-Variable Expansions Based on the 0(4) Group for Three-Body Decays
A. DECAY KINEMATICS AND THE O (4) EXPANSIONS
B. PROPERTIES OF THE O (4) EXPANSIONS
1. Comparison with 0(3, 1) Expansions. Possibility of Truncation
2. Relation to 0(3) Expansions, Threshold Behavior, and Identical Particles
C. COMPARISON WITH THE DALITZ-FABRI EXPANSIONS
D. FURTHER SYMMETRIES AND APPLICATION TO K-->3 $\pi$ AND \eta ---> 3$\pi$ DECAYS
V. 0(3, 1) and 0(4) Expansions for Particles with Arbitrary Spins
A. 0(3, 1) EXPANSIONS OF THE SCATTERING AMPLITUDES
B. 0(4) EXPANSIONS OF DECAY AMPLITUDES
VI. Explicitly Crossing Symmetric Expansions Based on the 0(2, 1) Group
A. NONSUBGROUP-TYPE BASES FOR 0(2, 1), ELLIPTIC COORDINATES, AND LAME FUNCTIONS
B. MAPPING OF MANDELSTAM PLANE ONTO AN Q(2, 1) HYPERBOLOID
C. TWO-VARIABLE 0(2, 1) EXPANSION IN TERMS OF LAME FUNCTIONS
D. CROSSING SYMMETRY
E. THRESHOLD AND ASYMPTOTIC BEHAVIOR
1. Threshold Behavior
2. Asymptotic Behavior
VII. Two-Variable Expansions of Nonrelativistic Scattering Amplitudes Based on the E(3) Group
A. THE GALILEI GROUP E(3) AND ITS REPRESENTATIONS
B. EXPANSIONS OF NONRELATIVISTIC SCATTERING AMPLITUDES IN TERMS OF UNITARY AND NONUNITARY REPRESENTATIONS OF THE GALILEI GROUP
C. PHYSICAL FEATURES OF THE EXPANSIONS
1. Low-Energy Limit
2. High-Energy Limit
3. Dynamical Singularities
D. GALILEI AMPLITUDES AND POTENTIAL SCATTERING
VIII. Two-Variable Expansions Based on the Group SU(3) and Their Generalizations
A. CROSSING SYMMETRY AND THE SU(3) EXPANSIONS
B. SU(3) ANALYSIS OF THREE-BODY FINAL STATES AND DALITZ PLOTS
IX. Conclusions
REFERENCES
Author Index
Subject Index
