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Group Theory For Physicists Second Edition [Mīkstie vāki]

(Chinese Academy Of Sciences, China)
  • Formāts: Paperback / softback, 656 pages
  • Izdošanas datums: 18-Sep-2019
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 9813277963
  • ISBN-13: 9789813277960
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Group Theory For Physicists Second EditionPalielināt
  • Formāts: Paperback / softback, 656 pages
  • Izdošanas datums: 18-Sep-2019
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 9813277963
  • ISBN-13: 9789813277960
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9789813277960.html
Keywords:

This textbook explains the fundamental concepts and techniques of group theory by making use of language familiar to physicists. Calculation methods in the context of physics are emphasized. New materials drawn from the teaching and research experience of the author are included. The generalized Gel'fand's method is presented to calculate the matrices of irreducible representations of the simple Lie algebra and its Clebsch–Gordan coefficients. This book is for graduate students and young researchers in physics, especially theoretical physics. It is also for graduate students in theoretical chemistry.

Preface v
1 Review on Linear Algebras
1(16)
1.1 Linear Space and Basis Vectors
1(2)
1.2 Matrices
3(2)
1.3 Linear Transformations and Linear Operators
5(2)
1.4 Similarity Transformation
7(2)
1.5 Eigenvectors and Diagonalization of a Matrix
9(2)
1.6 Inner Product of Vectors
11(2)
1.7 The Direct Product of Matrices
13(2)
1.8 Exercises
15(2)
2 Group and Its Subsets
17(38)
2.1 Symmetry
17(2)
2.2 Group and its Multiplication Table
19(12)
2.2.1 Definition of a Group
19(4)
2.2.2 Subgroup
23(1)
2.2.3 Symmetry Group of a Regular Polygon
24(3)
2.2.4 Symmetry of System of Identical Particles
27(4)
2.3 Subsets in a Group
31(7)
2.3.1 Cosets and Invariant Subgroup
31(2)
2.3.2 Conjugate Elements and Class
33(5)
2.4 Homomorphism
38(2)
2.5 Symmetry of a Regular Polyhedron
40(9)
2.5.1 Regular Polyhedrons
40(1)
2.5.2 Symmetry of a Cube
41(4)
2.5.3 Symmetry of an Icosahedron
45(4)
2.6 Direct Product of Groups and Improper Point Groups
49(3)
2.6.1 Direct Product of Two Groups
49(1)
2.6.2 Improper Point Groups
50(2)
2.7 Exercises
52(3)
3 Theory of Representations
55(56)
3.1 Linear Representations of a Group
55(5)
3.1.1 Definition of Linear Representation
55(1)
3.1.2 Group Algebra and Regular Representation
56(3)
3.1.3 Class Operator
59(1)
3.2 Transformation Operators for Scalar Functions
60(5)
3.3 Equivalent Representations
65(3)
3.4 Inequivalent Irreducible Representations
68(12)
3.4.1 Irreducible Representations
68(2)
3.4.2 Schur Theorem
70(1)
3.4.3 Orthogonal Relation
71(2)
3.4.4 Completeness of Representations
73(3)
3.4.5 Character Tables of Finite Groups
76(3)
3.4.6 Self-conjugate Representations
79(1)
3.5 Subduced and Induced Representations
80(7)
3.5.1 Frobenius Theorem
80(3)
3.5.2 Irreducible Representations of DN
83(4)
3.6 Applications in Physics
87(11)
3.6.1 Classification of Static Wave Functions
87(3)
3.6.2 Clebsch--Gordan Series and Coefficients
90(1)
3.6.3 Wigner--Eckart Theorem
91(3)
3.6.4 Normal Degeneracy and Accidental Degeneracy
94(1)
3.6.5 Example of Physical Application
95(3)
3.7 Irreducible Bases in Group Algebra
98(10)
3.7.1 Decomposition of Regular Representation
98(2)
3.7.2 Irreducible Bases of CN and DN
100(1)
3.7.3 Irreducible Bases of T and O
101(7)
3.8 Exercises
108(3)
4 Permutation Groups
111(48)
4.1 Ideals and Idempotents
111(7)
4.1.1 Ideals
111(2)
4.1.2 Idempotents
113(2)
4.1.3 Primitive Idempotents
115(3)
4.2 Primitive Idempotents of Sn
118(11)
4.2.1 Young Patterns
118(1)
4.2.2 Young Tableaux
119(2)
4.2.3 Young Operators
121(2)
4.2.4 Symmetry of Young Operators
123(2)
4.2.5 Products of Young Operators
125(2)
4.2.6 Young Operators as Primitive Idempotents
127(2)
4.3 Irreducible Representations of Sn
129(14)
4.3.1 Orthogonal Young Operators
129(4)
4.3.2 Irreducible Bases
133(1)
4.3.3 Representation Matrices
134(3)
4.3.4 Characters by Graphic Method
137(2)
4.3.5 The Permutation Group S3
139(2)
4.3.6 Inner Product of Irreducible Representations
141(1)
4.3.7 Character Table of the Group I
142(1)
4.4 Real Orthogonal Representation of Sn
143(6)
4.5 Outer Product of Irreducible Representations of Sn
149(7)
4.5.1 Group Sn+m and Its Subgroup Sn ⊗ Sm
149(3)
4.5.2 Littlewood--Richardson Rule
152(4)
4.6 Exercises
156(3)
5 Three-Dimensional Rotation Group
159(88)
5.1 Rotations in Three-dimensional Space
159(3)
5.2 Fundamental Concept of a Lie Group
162(7)
5.2.1 The Composition Functions of a Lie Group
162(2)
5.2.2 The Local Property of a Lie Group
164(1)
5.2.3 Generators and Differential Operators
165(1)
5.2.4 The Global Property of a Lie Group
166(3)
5.3 The Covering Group of SO(3)
169(8)
5.3.1 The Group SU(2)
169(1)
5.3.2 Homomorphism of SU(2) onto SO(3)
170(3)
5.3.3 Group Integral for Parameters ω
173(4)
5.4 Irreducible Representations of SU(2)
177(23)
5.4.1 Euler Angles
177(2)
5.4.2 Group Integral for the Euler Angles
179(2)
5.4.3 Linear Representations of SU(2)
181(5)
5.4.4 Spherical Harmonic Functions
186(3)
5.4.5 Harmonic Polynomials
189(2)
5.4.6 Irreducible Bases of the Group I
191(9)
5.5 The Lie Theorems
200(7)
5.5.1 The First Lie Theorem
200(2)
5.5.2 The Second Lie Theorem
202(3)
5.5.3 The Third Lie Theorem
205(1)
5.5.4 Adjoint Representation of a Lie Group
205(2)
5.6 Clebsch--Gordan Coefficients of SU(2)
207(10)
5.6.1 Direct Product of Representations
207(3)
5.6.2 Calculation of Clebsch--Gordan Coefficients
210(2)
5.6.3 Applications
212(2)
5.6.4 Sum of Three Angular Momentums
214(3)
5.7 Tensors and Spinors
217(9)
5.7.1 Vector Fields
218(2)
5.7.2 Tensor Fields
220(1)
5.7.3 Spinor Fields
221(1)
5.7.4 Total Angular Momentum Operator
222(4)
5.8 Irreducible Tensor Operators and Their Application
226(8)
5.8.1 Irreducible Tensor Operators
226(2)
5.8.2 Wigner-Eckart Theorem
228(2)
5.8.3 Selection Rule and Relative Intensity of Radiation
230(2)
5.8.4 Lande Factor and Zeeman Effects
232(2)
5.9 An Isolated Quantum n-body System
234(9)
5.9.1 Separation of the Motion of Center-of-Mass
235(2)
5.9.2 Separation of the Rotational Degree of Freedoms
237(6)
5.10 Exercises
243(4)
6 Symmetry of Crystals
247(60)
6.1 Symmetry Group of Crystals
247(2)
6.2 Crystallography Point Groups
249(12)
6.2.1 Elements in a Crystallographic Point Group
249(2)
6.2.2 Proper Crystallographic Point Groups
251(4)
6.2.3 Improper Crystallographic Point Group
255(2)
6.2.4 Double-Valued Representations of the Point Groups
257(4)
6.3 Crystal Systems and Bravais Lattice
261(12)
6.3.1 Restrictions on Vectors of Crystal Lattice
261(3)
6.3.2 Triclinic Crystal System
264(1)
6.3.3 Monoclinic Crystal System
264(1)
6.3.4 Orthorhombic Crystal System
265(1)
6.3.5 Trigonal and Hexagonal Crystal System
266(3)
6.3.6 Tetragonal Crystal System
269(2)
6.3.7 Cubic Crystal System
271(2)
6.4 Space Group
273(16)
6.4.1 Symmetry Elements
273(4)
6.4.2 Symbols of a Space Group
277(5)
6.4.3 Method for Determining the Space Groups
282(1)
6.4.4 Example for the Space Groups in Type A
283(1)
6.4.5 Example for the Space Groups in Type B
284(3)
6.4.6 Analysis of the Symmetry of a Crystal
287(2)
6.5 Linear Representations of Space Groups
289(16)
6.5.1 Irreducible Representations of T
289(1)
6.5.2 The Bloch Theorem
290(1)
6.5.3 Star of Wave Vectors and the Little Group
290(3)
6.5.4 General Form of Irreducible Representation of S
293(2)
6.5.5 Irreducible Representation of A Little Group
295(5)
6.5.6 Study on Graphene by Group Theory
300(4)
6.5.7 Energy Band in a Crystal
304(1)
6.6 Exercises
305(2)
7 Lie Groups and Lie Algebras
307(56)
7.1 Lie Algebras and its Structure Constants
307(8)
7.1.1 Review on the Concepts of Lie Groups
307(4)
7.1.2 The Lie Algebra
311(3)
7.1.3 The Killing Form and the Cartan Criteria
314(1)
7.2 The Regular Form of a Semisimple Lie Algebra
315(10)
7.2.1 The Inner Product in a Semisimple Lie Algebra
315(1)
7.2.2 The Cartan Subalgebra
316(1)
7.2.3 Regular Commutative Relations of Generators
317(2)
7.2.4 The Inner Product of Roots
319(3)
7.2.5 Positive Roots and Simple Roots
322(3)
7.3 Classification of Simple Lie Algebras
325(8)
7.3.1 Angle between Two Simple Roots
325(1)
7.3.2 Dynkin Diagrams
326(6)
7.3.3 The Cartan Matrix
332(1)
7.4 Classical Simple Lie Algebras
333(10)
7.4.1 The SU(N) Group and its Lie Algebra
333(4)
7.4.2 The SO(N) Group and its Lie Algebra
337(2)
7.4.3 The USp(2) Group and its Lie Algebra
339(4)
7.5 Representations of a Simple Lie Algebra
343(18)
7.5.1 Representations and Weights
343(3)
7.5.2 Weyl Reflection and Equivalent Weights
346(3)
7.5.3 Mathematical Property of Representations
349(1)
7.5.4 Fundamental Dominant Weights
350(1)
7.5.5 The Casimir Operator of Order 2
351(1)
7.5.6 Main Data of Simple Lie Algebras
352(9)
7.6 Exercises
361(2)
8 Gel'Fand's Method and Its Generalization
363(82)
8.1 Method of Block weight diagram
363(5)
8.1.1 Chevalley Bases
363(1)
8.1.2 Block Weight Diagram
364(4)
8.2 Gel'fand's Method
368(14)
8.2.1 The Gel'fand's Bases
368(4)
8.2.2 Some Representations of A2
372(5)
8.2.3 Some Representations of A3
377(5)
8.3 Generalized Gel'fand's Method
382(25)
8.3.1 Generalized Gel'fand's Bases
382(4)
8.3.2 Some Representations of C
386(11)
8.3.3 Some Representations of B
397(6)
8.3.4 Some Representations of D
403(2)
8.3.5 Some Representations of G2
405(2)
8.4 Planar Weight Diagrams
407(3)
8.5 Clebsch--Gordan Coefficients
410(33)
8.5.1 Representations in the CG Series
411(2)
8.5.2 Method of Dominant Weight Diagram
413(2)
8.5.3 The CG series for At
415(21)
8.5.4 The CG series for C3
436(7)
8.6 Exercises
443(2)
9 Unitary Groups
445(66)
9.1 Irreducible Representations of SU(N)
445(18)
9.1.1 Symmetry of the Tensor Space of SU(N)
446(2)
9.1.2 Basis Tensors in the Tensor Subspace
448(6)
9.1.3 Chevalley Bases of Generators in SU(N)
454(1)
9.1.4 Irreducible Representations of SU(N)
455(1)
9.1.5 Dimension of the Representation of SU(N)
456(1)
9.1.6 Antisymmetric Wave Functions of Fermions
457(5)
9.1.7 Subduced Representations
462(1)
9.2 Orthonormal Irreducible Basis Tensors
463(22)
9.2.1 Orthonormal Basis Tensors in T[ λ]μ of A2
464(2)
9.2.2 Some Examples on Orthonormal Basis Tensors
466(18)
9.2.3 Orthonormal Basis Tensors in Sn
484(1)
9.3 Direct Product of Tensor Representations
485(10)
9.3.1 Direct Product of Tensors
485(3)
9.3.2 Covariant and Contravariant Tensors
488(2)
9.3.3 Traceless Mixed Tensors
490(4)
9.3.4 Adjoint Representation of SU(N)
494(1)
9.4 SU(3) Symmetry and Wave Functions of Hadrons
495(14)
9.4.1 Quantum Numbers of Quarks
495(2)
9.4.2 Planar Weight Diagrams of Mesons and Baryons
497(5)
9.4.3 Mass Formulas
502(2)
9.4.4 Wave Functions of Mesons
504(2)
9.4.5 Wave Functions of Baryons
506(3)
9.5 Exercises
509(2)
10 Real Orthogonal Groups
511(74)
10.1 Tensor Representations of SO(N)
511(38)
10.1.1 Tensors of SO(N)
511(3)
10.1.2 Irreducible Basis Tensors of SO(2l +1)
514(19)
10.1.3 Irreducible Basis Tensors of SO(2l)
533(11)
10.1.4 Dimensions of Tensor Representations of SO(N)
544(2)
10.1.5 Adjoint Representation of SO(N)
546(1)
10.1.6 Tensor Representations of O(N)
547(2)
10.2 Γ Matrix Groups
549(5)
10.2.1 Property of Γ Matrix Groups
549(1)
10.2.2 The Case N = 2l
550(3)
10.2.3 The Case N = 2l + 1
553(1)
10.3 Spinor Representations of SO(N)
554(20)
10.3.1 Covering Groups of SO(N)
554(3)
10.3.2 Fundamental Spinors of SO(N)
557(1)
10.3.3 Direct Products of Spinor Representations
558(2)
10.3.4 Spinor Representations of Higher Ranks
560(12)
10.3.5 Dimensions of Spinor Representations of SO(N)
572(2)
10.4 Rotational Symmetry in N-Dimensional Space
574(8)
10.4.1 Orbital Angular Momentum Operators
574(1)
10.4.2 Spherical Harmonic Functions
575(2)
10.4.3 Schrodinger Equation for a Two-body System
577(1)
10.4.4 Schrodinger Equation for a Three-body System
578(4)
10.5 Exercises
582(3)
11 Lorentz Groups
585(24)
11.1 The SO(4) Group
585(6)
11.1.1 Irreducible Representations of SO(4)
585(4)
11.1.2 Single-valued Representations of O(4)
589(2)
11.2 The Lorentz Groups
591(10)
11.2.1 The Proper Lorentz Group and its Cosets
591(1)
11.2.2 Irreducible Representations of Lp
592(3)
11.2.3 The Covering Group of Lp
595(2)
11.2.4 Classes of Lp
597(1)
11.2.5 Irreducible Representations of Lh
598(3)
11.3 Dirac Equation in (N + 1)-dimensional Space-time
601(6)
11.4 Exercises
607(2)
12 Symplectic Groups
609(16)
12.1 Irreducible Representations of USp(2l)
609(14)
12.1.1 Decomposition of the Tensor Space of USp(2l)
609(2)
12.1.2 Orthonormal Irreducible Basis Tensors of USp(2l)
611(9)
12.1.3 Dimensions of Irreducible Representations
620(3)
12.2 Physical Application
623(1)
12.3 Exercises
624(1)
Bibliography 625(6)
Index 631