Applications of unitary symmetry and combinatorics
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...
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| Main Author | |
|---|---|
| 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 | |
| Online Access | Get full text |
| ISBN | 9789814350716 9814350710 |
| DOI | 10.1142/8161 |
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| Abstract | 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 uniform viewpoint is presented based on the structure of binary trees. This includes a systematic method for the evaluation of all 3n–j coefficients and their relationship to cubic graphs. A number of topical subjects that emerge naturally are also developed, such as the algebra of permutation matrices, the properties of magic squares and an associated generalized Regge form, the Zeilberger counting formula for alternating sign matrices, and the Heisenberg ring problem, viewed as a composite system in which the total angular momentum is conserved. |
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| AbstractList | 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 uniform viewpoint is presented based on the structure of binary trees. This includes a systematic method for the evaluation of all 3n–j coefficients and their relationship to cubic graphs. A number of topical subjects that emerge naturally are also developed, such as the algebra of permutation matrices, the properties of magic squares and an associated generalized Regge form, the Zeilberger counting formula for alternating sign matrices, and the Heisenberg ring problem, viewed as a composite system in which the total angular momentum is conserved. 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 uniform viewpoint is presented based on the structure of binary trees. This includes a systematic method for the evaluation of all 3n-j coefficients and their relationship to cubic graphs. A number of topical subjects that emerge naturally are also developed, such as the algebra of permutation matrices, the properties of magic squares and an associated generalized Regge form, the Zeilberger counting formula for alternating sign matrices, and the Heisenberg ring problem, viewed as a composite system in which the total angular momentum is conserved.The readership is intended to be advanced graduate students and researchers interested in learning about the relationship between unitary symmetry and combinatorics and challenging unsolved problems. The many examples serve partially as exercises, but this monograph is not a textbook. It is hoped that the topics presented promote further and more rigorous developments that lead to a deeper understanding of the angular momentum properties of complex systems viewed as composite wholes.Sample Chapter(s)Chapter 1: Composite Quantum Systems (312 KB)Contents: Composite Quantum SystemsAlgebra of Permutation MatricesCoordinates of A in Basis ℙΣn(e,p)Further Applications of Permutation MatricesDoubly Stochastic Matrices in Angular Momentum TheoryMagic SquaresAlternating Sign MatricesThe Heisenberg Magnetic RingReadership: Graduate students and researchers in physics and mathematics who wish to learn about the relationships between symmetry and combinatorics. |
| Author | Louck, James D |
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| Keywords | Unitary Symmetry Combinatorics Binary Tree Angular Momentum Theory |
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| Publisher | World Scientific Publishing Co. Pte. Ltd World Scientific World Scientific Publishing Company WORLD SCIENTIFIC WSPC |
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| SubjectTerms | Combinatorial analysis Eightfold way (Nuclear physics) Symmetry (Physics) |
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| TableOfContents | 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 |
| Title | Applications of unitary symmetry and combinatorics |
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