THEORY OF GROUP REPRESENTATIONS AND APPLICATIONS

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1 THEORY OF GROUP REPRESENTATIONS AND APPLICATIONS ASIM 0. BARUT Institute for Theoretical Physics, University of Colorado, Boulder, Colo., U.S.A. RYSZARD RATJZKA Institute for Nuclear Research, Warszawa, Polska Second revised edition > World Scientific

2 Contents PREFACE OUTLINE OF THE BOOK NOTATIONS VII XV XIX CHAPTER 1 LIE ALGEBRAS 1. Basic Concepts and General Properties 1 2. Solvable, Nilpotent, Semisimple and Simple Lie Algebras The Structure of Lie Algebras Classification of Simple, Complex Lie Algebras Classification of Simple, Real Lie Algebras The Gauss, Cartan and Iwasawa Decompositions An Application. On Unification of the Poincare Algebra and Internal Symmetry Algebra Contraction of Lie Algebras Comments and Supplements Exercises 48 CHAPTER 2 TOPOLOGICAL GROUPS 1. Topological Spaces Topological Groups The Haar Measure Comments and Supplements Exercises 71 CHAPTER 3 LIE GROUPS 1. Differentiable Manifolds Lie Groups The Lie Algebra of a Lie Group The Direct and Semidirect Products Levi-Malcev Decomposition Gauss, Cartan, Iwasawa and Bruhat Global Decompositions Classification of Simple Lie Groups Structure of Compact Lie Groups 108 IX

3 X 9. Invariant Metrie and Invariant Measure on Lie Groups Comments and Supplements Exercises 114 CHAPTER 4 HOMOGENEOUS AND SYMMETRIC SPACES 1. Homogeneous Spaces Symmetrie Spaces 124 3, Invariant and Quasi-Invariant Measures on Homogeneous Spaces Comments and Supplements Exercises 132 CHAPTER 5 GROUP REPRESENTATIONS 1. Basic Concepts Equivalence of Representations Irreducibility and Reducibility Cyclic Representations Tensor Product of Representations Direct Integral Decomposition of Unitary Representations Comments and Supplements Exercises CHAPTER 6 REPRESENTATIONS OF COMMUTATIVE GROUPS 1. Irreducible Representations and Characters Stone and SNAG Theorems Comments and Supplements Exercises 164 CHAPTER 7 REPRESENTATIONS OF COMPACT GROUPS 1. Basic Properties of Representations of Compact Groups Peter-Weyl and Weyl Approximation Theorems Projection Operators and Irreducible Representations Applications Representations of Finite Groups Comments and Supplements Exercises 197 CHAPTER 8 FINITE-DIMENSIONAL REPRESENTATIONS OF LIE GROUPS 1. General Properties of Representations of Solvable and Semisimple Lie Groups 199

4 XI 2. Induced Representations of Lie Groups The Representations of GL(«, C), GL(/7, R), U(p, q), U(n), SL(n, C), SL(/7, R), SU(p, q), and SU(«) The Representations of the Symplectic Groups Sp(«, C), Sp(«, R) and Sp(«) The Representations of Orthogonal Groups SO(«, C), SO(p, q), SO*(«), and SO(w) The Fundamental Representations Representations of Arbitrary Lie Groups Further Results and Comments Exercises 238 CHAPTER 9 TENSOR OPERATORS, ENVELOPING ALGEBRAS AND ENVELOPING FIELDS 1. The Tensor Operators The Enveloping Algebra The Invariant Operators Casimir Operators for Classical Lie Group The Enveloping Field Further Results and Comments Exercises 275 CHAPTER 10 THE EXPLICIT CONSTRUCTION OF FINITE-DIMENSIONAL IRREDUCIBLE REPRESENTATIONS 1. The Gel'fand-Zetlin Method The Tensor Method The Method of Harmonie Functions The Method of Creation and Annihilation Operators Comments and Supplements Exercises 314 CHAPTER 11 REPRESENTATION THEORY OF LIE AND ENVELOPING ALGEBRAS BY UNBOUNDED OPERATORS: ANALYTIC VECTORS AND INTEGRABILITY 1. Representations of Lie Algebras by Unbounded Operators Representations of Enveloping Algebras by Unbounded Operators Analytic Vectors and Analytic Dominance Analytic Vectors for Unitary Representations of Lie Groups Integrability of Representations of Lie Algebras FS 3 -Theory of Integrability of Lie Algebras Representations The 'Heat Equation' on a Lie Group and Analytic Vectors

5 XII 8. Algebraic Construction of Irreducible Representations Comments and Supplements Exercises 373 CHAPTER 12 QUANTUM DYNAMICAL APPLICATIONS OF LIE ALGEBRA REPRESENTATIONS 1. Symmetry Algebras in Hamiltonian Formulation Dynamical Lie Algebras Exercises 386 CHAPTER 13 GROUP THEORY AND GROUP REPRESENTATIONS IN QUANTUM THEORY 1. Group Representations in Physics Kinematical Postulates of Quantum Theory Symmetries of Physical Systems Dynamical Symmetries of Relativistic and Non-Relativistic Systems Comments and Supplements Exercises 418 CHAPTER 14 HARMONIC ANALYSIS ON LIE GROUPS. SPECIAL FUNCTIONS AND GROUP REPRESENTATIONS 1. Harmonic Analysis on Abelian and Compact Lie Groups Harmonic Analysis on Unimodular Lie Groups Harmonic Analysis on Semidirect Product of Groups Comments and Supplements Exercises CHAPTER 15 HARMONIC ANALYSIS ON HOMOGENEOUS SPACES 1. Invariant Operators on Homogeneous Spaces Harmonic Analysis on Homogeneous Spaces Harmonic Analysis on Symmetrie Spaces Associated with Pseudo- Orthogonal Groups SO(p, q) Generalized Projection Operators Comments and Supplements Exercises 470 CHAPTER 16 INDUCED REPRESENTATIONS 1. The Concept of Induced Representations Basic Properties of Induced Representation 487

6 XJJJ 3. Systems of Imprimitivity Comments and Supplements Exercises 493 CHAPTER 17 INDUCED REPRESENTATIONS OF SEMIDIRECT PRODUCTS 1. Representation Theory of Semidirect Products Induced Unitary Representations of the Poincare Group Representation of the Extended Poincare Group Indecomposable Representations of Poincare Group Comments and Supplements Exercises 537 CHAPTER 18 FUNDAMENTAL THEOREMS OF INDUCED REPRESENTATIONS 1. The Induction-Reduction Theorem Tensor-Product Theorem The Frobenius Reciprocity Theorem Comments and Supplements Exercises 553 CHAPTER 19 INDUCED REPRESENTATIONS OF SEMISIMPLE LIE GROUPS 1. Induced Representations of Semisimple Lie Groups Properties of the Group SL(n, C) and Its Subgroups The Principal Nondegenerate Series of Unitary Representations of SL(/?, C) Principal Degenerate Series of SL(«, C) Supplementary Nondegenerate and Degenerate Series Comments and Supplements Exercises 578 CHAPTER 20 APPLICATIONS OF INDUCED REPRESENTATIONS 1. The Relativistic Position Operator The Representations of the Heisenberg Commutation Relations Comments and Supplements Exercises 593 CHAPTER 21 GROUP REPRESENTATIONS IN RELATIVISTIC QUANTUM THEORY 1. Relativistic Wave Equations and Induced Representations Finite Component Relativistic Wave Equations 601

7 XIV 3. Infinite Component Wave Equations Group Extensions and Applications Space-Time and Internal Symmetries Comments and Supplements Exercises 636 APPENDIX A ALGEBRA, TOPOLOGY, MEASURE AND INTEGRATION THEORY 637 APPENDIX B FUNCTIONAL ANALYSIS 1. Closed, Symmetrie and Self-Adjoint Operators in Hubert Space Integration of Vector and Operator Functions Spectral Theory of Operators Functions of Self-Adjoint Operators Essentially Self-Adjoint Operators 663 BIBLIOGRAPHY 667 LIST OF IMPORTANT SYMBOLS 703 AUTHOR INDEX 706 SUBJECT INDEX 710

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