Applications of unitary symmetry and combinatorics

• Main Author: Louck, James D
• Format: eBook Book
• Language: English
• Published: Singapore World Scientific Publishing Co. Pte. Ltd 2011; World Scientific; World Scientific Publishing Company; WORLD SCIENTIFIC; WSPC
• Edition: 1
• Subjects: Combinatorial analysis; Eightfold way (Nuclear physics); Symmetry (Physics); Unitary Symmetry; Combinatorics Binary Tree; Angular Momentum Theory
• ISBN: 9789814350716; 9814350710
• DOI: 10.1142/8161

This monograph is a synthesis of the theory of the pairwise coupling of the angular momenta of arbitrarily many independent systems to the total angular momentum in which the universal role of doubly stochastic matrices and their quantum-mechanical probabilistic interpretation is a major theme. A un...

Table of Contents:

• Applications of unitary symmetry and combinatorics -- Preface and Prelude -- Contents -- Notation -- Chapter 1: Composite Quantum Systems -- Chapter 2: Algebra of Permutation Matrices -- Chapter 3: Coordinates of A in Basis PΣn(e,p) -- Chapter 4: Further Applications of Permutation Matrices -- Chapter 5: Doubly Stochastic Matrices in Angular Momentum Theory -- Chapter 6: Magic Squares -- Chapter 7: Alternating Sign Matrices -- Chapter 8: The Heisenberg Magnetic Ring -- Appendix A: Counting Formulas for Compositions and Partitions -- Appendix B: No Single Coupling Scheme for n ≥ 5 -- Appendix C: Generalization of Binary Coupling Schemes -- Bibliography -- Index -- Errata and Related Notes
• 5.5.1 Pair of Spin-1/2 Systems -- 5.5.2 Pair of Spin-1/2 Systems as a Composite System -- 5.6 Binary Coupling of Angular Momenta -- 5.6.1 Complete Sets of Commuting Hermitian Observables -- 5.6.2 Domain of Definition RT (j) -- 5.6.3 Binary Bracketings, Shapes, and Binary Trees -- 5.7 State Vectors: Uncoupled and Coupled -- 5.8 General Binary Tree Couplings and Doubly Stochastic Matrices -- 5.8.1 Overview -- 5.8.2 Uncoupled States -- 5.8.3 Generalized WCG Coefficients -- 5.8.4 Binary Tree Coupled State Vectors -- 5.8.5 Racah Sum-Rule and Biedenharn-Elliott Identity as Transition Probability Amplitude Relations -- 5.8.6 Symmetries of the 6 - j and 9 - j Coefficients -- 5.8.7 General Binary Tree Shape Transformations -- 5.8.8 Summary -- 5.8.9 Expansion of Doubly Stochastic Matrices into Permutation Matrices -- 6 Magic Squares -- 6.1 Review -- 6.2 Magic Squares and Addition of Angular Momenta -- 6.3 Rational Generating Function of Hn(r) -- 7 Alternating Sign Matrices -- 7.1 Introduction -- 7.2 Standard Gelfand-Tsetlin Patterns -- 7.2.1 A-Matrix Arrays -- 7.2.2 Strict Gelfand-Tsetlin Patterns -- 7.3 Strict Gelfand-Tsetlin Patterns for λ = (n n . 1 · · · 2 1) -- 7.3.1 Symmetries -- 7.4 Sign-Reversal-Shift Invariant Polynomials -- 7.5 The Requirement of Zeros -- 7.6 The Incidence Matrix Formulation -- 8 The Heisenberg Magnetic Ring -- 8.1 Introduction -- 8.2 Matrix Elements of H in the Uncoupled and Coupled Bases -- 8.3 Exact Solution of the Heisenberg Ring Magnet for n = 2, 3, 4 -- 8.4 The Heisenberg Ring Hamiltonian: Even n -- 8.4.1 Summary of Properties of Recoupling Matrices -- 8.4.2 Maximal Angular Momentum Eigenvalues -- 8.4.3 Shapes and Paths for Coupling Schemes I and II -- 8.4.4 Determination of the Shape Transformations -- 8.4.5 The Transformation Method for n = 4 -- 8.4.6 The General 3(2f - 1) - j Coefficients
• Intro -- Contents -- Preface and Prelude -- OVERVIEW AND SYNTHESIS OF BINARY COUPLING THEORY -- TOPICAL CONTENTS -- MATTERS OF STYLE, READERSHIP, AND RECOGNITION -- Notation -- 1 Composite Quantum Systems -- 1.1 Introduction -- 1.2 Angular Momentum State Vectors of a Composite System -- 1.2.1 Group Actions in a Composite System -- 1.3 Standard Form of the Kronecker Direct Sum -- 1.3.1 Reduction of Kronecker Products -- 1.4 Recoupling Matrices -- 1.5 Preliminary Results on Doubly Stochastic Matrices and Permutation Matrices -- 1.6 Relationship between Doubly Stochastic Matrices and Density Matrices in Angular Momentum Theory -- 2 Algebra of Permutation Matrices -- 2.1 Introduction -- 2.2 Basis Sets of Permutation Matrices -- 2.2.1 Summary -- 3 Coordinates of A in Basis P n(e,p) -- 3.1 Notations -- 3.2 The A-Expansion Rule in the Basis P n(e,p) -- 3.3 Dual Matrices in the Basis Set Σn(e, p) -- 3.3.1 Dual Matrices for Σ3(e, p) -- 3.3.2 Dual Matrices for Σ4(e, p) -- 3.4 The General Dual Matrices in the Basis Σn(e, p) -- 3.4.1 Relation between the A-Expansion and Dual Matrices -- 4 Further Applications of Permutation Matrices -- 4.1 Introduction -- 4.2 An Algebra of Young Operators -- 4.3 Matrix Schur Functions -- 4.4 Real Orthogonal Irreducible Representations of Sn -- 4.4.1 Matrix Schur Function Real Orthogonal Irreducible Representations -- 4.4.2 Jucys-Murphy Real Orthogonal Representations -- 4.5 Left and Right Regular Representations of Finite Groups -- 5 Doubly Stochastic Matrices in Angular Momentum Theory -- 5.1 Introduction -- 5.2 Abstractions and Interpretations -- 5.3 Permutation Matrices as Doubly Stochastic -- 5.4 The Doubly Stochastic Matrix for a Single System with Angular Momentum J -- 5.4.1 Spin-1/2 System -- 5.4.2 Angular Momentum-j System -- 5.5 Doubly Stochastic Matrices for Composite Angular Momentum Systems
• 8.4.7 The General 3(2f - 1) - j Coefficients Continued -- 8.5 The Heisenberg Ring Hamiltonian: Odd n -- 8.5.1 Matrix Representations of H -- 8.5.2 Matrix Elements of Rj2 -- j1 : The 6f - j Coefficients -- 8.5.3 Matrix Elements of Rj3 -- j1 : The 3(f + 1) - j Coefficients -- 8.5.4 Properties of Normal Matrices -- 8.6 Recount, Synthesis, and Critique -- 8.7 Action of the Cyclic Group -- 8.7.1 Representations of the Cyclic Group -- 8.7.2 The Action of the Cyclic Group on Coupled State Vectors -- 8.8 Concluding Remarks -- A Counting Formulas for Compositions and Partitions -- A.1 Compositions -- A.2 Partitions -- B No Single Coupling Scheme for n ≥ 5 -- B.1 No Single Coupling Scheme Diagonalizing H for n ≥ 5 -- C Generalization of Binary Coupling Schemes -- C.1 Generalized Systems -- C.2 The Composite U(n) System Problem -- Bibliography -- Index -- Errata and Related Notes