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E-raamat: Lie Algebras and Applications

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  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Physics 891
  • Ilmumisaeg: 13-Oct-2014
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783662444948
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Lie Algebras and ApplicationsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Physics 891
  • Ilmumisaeg: 13-Oct-2014
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783662444948
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This course-based primer provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, it concisely presents the basic concepts of Lie algebras, their representations and their invariants. The second part includes a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book highlights a number of examples that help to illustrate the abstract algebraic definitions and includes a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators, and the dimensions of the representations of all classical Lie algebras.

For this new edition, the text has been carefully revised and expanded; in particular, a new chapter has been added on the deformation and contraction of Lie algebras.

From the reviews of the first edition:

"Iachello has written a pedagogical and straightforward presentation of Lie algebras [ ...]. It is a great text to accompany a course on Lie algebras and their physical applications."

(Marc de Montigny, Mathematical Reviews, Issue, 2007 i)

"This book [ ...] written by one of the leading experts in the field [ ...] will certainly be of great use for students or specialists that want to refresh their knowledge on Lie algebras applied to physics. [ ...] An excellent reference for those interested in acquiring practical experience [ ...] and leaving the embarrassing theoretical presentations aside."

(Rutwig Campoamor-Stursberg, Zentralblatt MATH, Vol. 1156, 2009)

Arvustused

Its intention is to provide an introduction to Lie algebras at the level of a one-semester course in physics. these lecture notes are of high interest for physicists. But a mathematician will also find it interesting to see Lie algebras in action in physics. There are a lot of detailed examples available in the book. (Martin Schlichenmaier, Mathematical Reviews, January, 2016)

1 Basic Concepts
1(20)
1.1 Definitions
1(1)
1.2 Lie Algebras
2(1)
1.3 Change of Basis
3(1)
1.4 Complex Extensions
4(1)
1.5 Lie Subalgebras
5(1)
1.6 Abelian Algebras
6(1)
1.7 Direct Sum
6(2)
1.8 Ideals (Invariant Subalgebras)
8(1)
1.9 Semisimple Algebras
8(1)
1.10 Semidirect Sum
9(1)
1.11 Metric Tensor
9(2)
1.12 Compact and Non-Compact Algebras
11(1)
1.13 Derived Algebras
11(1)
1.14 Nilpotent Algebras
12(1)
1.15 Invariant Casimir Operators
13(1)
1.16 Invariant Operators for Non-Semisimple Algebras
14(1)
1.17 Contractions of Lie Algebras
15(2)
1.18 Structure of Lie Algebras
17(4)
1.18.1 Algebras with One Element
17(1)
1.18.2 Algebras with Two Elements
18(1)
1.18.3 Algebras with Three Elements
18(3)
2 Semisimple Lie Algebras
21(16)
2.1 Cartan--Weyl Form of a (Complex) Semisimple Lie Algebra
21(1)
2.2 Graphical Representation of Root Vectors
22(2)
2.3 Explicit Construction of the Cartan--Weyl Form
24(1)
2.4 Dynkin Diagrams
25(2)
2.5 Classification of (Complex) Semisimple Lie Algebras
27(1)
2.6 Rules for Constructing the Root Vector Diagrams of Classical Lie Algebras
27(2)
2.7 Rules for Constructing Root Vector Diagrams of Exceptional Lie Algebras
29(1)
2.8 Dynkin Diagrams of Classical Lie Algebras
30(1)
2.9 Dynkin Diagrams of Exceptional Lie Algebras
30(1)
2.10 Cartan Matrices
30(1)
2.11 Cartan Matrices of Classical Lie Algebras
31(1)
2.12 Cartan Matrices of Exceptional Lie Algebras
32(1)
2.13 Real Forms of Complex Semisimple Lie Algebras
32(1)
2.14 Isomorphisms of Complex Semisimple Lie Algebras
33(1)
2.15 Isomorphisms of Real Lie Algebras
33(1)
2.16 Enveloping Algebra
33(1)
2.17 Realizations of Lie Algebras
34(1)
2.18 Other Realizations of Lie Algebras
35(2)
3 Lie Groups
37(16)
3.1 Groups of Transformations
37(1)
3.2 Groups of Matrices
38(1)
3.3 Properties of Matrices
38(1)
3.4 Continuous Matrix Groups
39(4)
3.5 Examples of Groups of Transformations
43(5)
3.5.1 The Rotation Group in Two Dimensions, SO(2)
43(1)
3.5.2 The Lorentz Group in One Plus One Dimension, SO(1, 1)
44(1)
3.5.3 The Rotation Group in Three Dimensions, SO(3)
45(1)
3.5.4 The Special Unitary Group in Two Dimensions, SU(2)
46(1)
3.5.5 Relation Between SO(3) and SU(2)
47(1)
3.6 Other Important Groups of Transformations
48(5)
3.6.1 Translation Group, T(n)
48(1)
3.6.2 Affine Group, A(n)
49(1)
3.6.3 Euclidean Group, E(n)
49(1)
3.6.4 Poincare' Group, P(n)
50(1)
3.6.5 Dilatation Group, D(1)
50(1)
3.6.6 Special Conformai Group, C(n)
51(1)
3.6.7 General Conformai Group, GC(n)
51(2)
4 Lie Algebras and Lie Groups
53(4)
4.1 The Exponential Map
53(1)
4.2 Definition of Exp
53(1)
4.3 Matrix Exponentials
54(1)
4.4 More on Exponential Maps
55(1)
4.5 Infinitesimal Transformations
56(1)
5 Homogeneous and Symmetric Spaces (Coset Spaces)
57(4)
5.1 Definitions
57(1)
5.2 Cartan Classification
58(1)
5.3 How to Construct Coset Spaces
58(3)
6 Irreducible Bases (Representations)
61(32)
6.1 Definitions
61(1)
6.2 Abstract Characterization
61(1)
6.3 Irreducible Tensors
61(5)
6.3.1 Irreducible Tensors with Respect to GL(n)
61(2)
6.3.2 Construction of Irreducible Tensors with Respect to GL(n). Young Method
63(2)
6.3.3 Irreducible Tensors with Respect to SU(n)
65(1)
6.3.4 Irreducible Tensors with Respect to SO(n)
65(1)
6.4 Tensor Representations of Classical Compact Algebras
66(2)
6.4.1 Unitary Algebras u(n)
66(1)
6.4.2 Special Unitary Algebras su(n)
66(1)
6.4.3 Orthogonal Algebras so(n), N = Odd
67(1)
6.4.4 Orthogonal Algebras so(n), N = Even
67(1)
6.4.5 Symplectic Algebras sp(n), N = Even
67(1)
6.5 Spinor Representations
68(1)
6.5.1 Orthogonal Algebras so(n), N = Odd
68(1)
6.5.2 Orthogonal Algebras so(n), N = Even
69(1)
6.6 Fundamental Representations
69(2)
6.6.1 Unitary Algebras
69(1)
6.6.2 Special Unitary Algebras
70(1)
6.6.3 Orthogonal Algebras, N = Odd
70(1)
6.6.4 Orthogonal Algebras, N = Even
70(1)
6.6.5 Symplectic Algebras
71(1)
6.7 Realization of Bases
71(1)
6.8 Chains of Algebras
72(1)
6.9 Canonical Chains
72(3)
6.9.1 Unitary Algebras
72(2)
6.9.2 Orthogonal Algebras
74(1)
6.10 Isomorphisms of Spinor Algebras
75(1)
6.11 Nomenclature for u(n)
76(1)
6.12 Dimensions of the Representations
77(3)
6.12.1 Dimensions of the Representations of u(n)
77(1)
6.12.2 Dimensions of the Representations of su(n)
78(1)
6.12.3 Dimensions of the Representations of An ≡ su(n + 1)
78(1)
6.12.4 Dimensions of the Representations of Bn ≡ so(2n + 1)
79(1)
6.12.5 Dimensions of the Representations of Cn ≡ sp(2n)
79(1)
6.12.6 Dimensions of the Representations of Dn ≡ so(2n)
80(1)
6.13 Action of the Elements of G on the Basis β
80(3)
6.14 Tensor Products
83(4)
6.15 Non-canonical Chains
87(6)
7 Casimir Operators and Their Eigenvalues
93(14)
7.1 Definitions
93(1)
7.2 Independent Casimir Operators
93(3)
7.2.1 Casimir Operators of u(n)
93(1)
7.2.2 Casimir Operators of su(n)
94(1)
7.2.3 Casimir Operators of so(n), N = Odd
95(1)
7.2.4 Casimir Operators of so(n), N = Even
95(1)
7.2.5 Casimir Operators of sp(n), N = Even
95(1)
7.2.6 Casimir Operators of the Exceptional Algebras
96(1)
7.3 Complete set of Commuting Operators
96(1)
7.3.1 The Unitary Algebra u(n)
96(1)
7.3.2 The Orthogonal Algebra so(n), N = Odd
97(1)
7.3.3 The Orthogonal Algebra so(n), N = Even
97(1)
7.4 Eigenvalues of Casimir Operators
97(8)
7.4.1 The Algebras u(n) and su(n)
97(3)
7.4.2 The Orthogonal Algebra so(2n + 1)
100(2)
7.4.3 The Symplectic Algebra sp(2n)
102(1)
7.4.4 The Orthogonal Algebra so(2n)
103(2)
7.5 Eigenvalues of Casimir Operators of Order One and Two
105(2)
8 Tensor Operators
107(18)
8.1 Definitions
107(1)
8.2 Coupling Coefficients
108(1)
8.3 Wigner--Eckart Theorem
109(2)
8.4 Nested Algebras: Racah's Factorization Lemma
111(3)
8.5 Adjoint Operators
114(1)
8.6 Recoupling Coefficients
115(2)
8.7 Symmetry Properties of Coupling Coefficients
117(1)
8.8 How to Compute Coupling Coefficients
117(1)
8.9 How to Compute Recoupling Coefficients
118(1)
8.10 Properties of Recoupling Coefficients
119(1)
8.11 Double Recoupling Coefficients
120(1)
8.12 Coupled Tensor Operators
121(1)
8.13 Reduction Formula of the First Kind
122(1)
8.14 Reduction Formula of the Second Kind
123(2)
9 Boson Realizations
125(50)
9.1 Boson Operators
125(1)
9.2 The Unitary Algebra u(1)
126(2)
9.3 The Algebras u(2) and su(2)
128(4)
9.3.1 Subalgebra Chains
128(4)
9.4 The Algebras u(n), N ≥ 3
132(3)
9.4.1 Racah Form
132(1)
9.4.2 Tensor Coupled Form of the Commutators
133(1)
9.4.3 Subalgebra Chains Containing so(3)
134(1)
9.5 The Algebras u(3) and su(3)
135(10)
9.5.1 Subalgebra Chains
135(4)
9.5.2 Lattice of Algebras
139(1)
9.5.3 Boson Calculus of u(3) ⊃ so(3)
139(2)
9.5.4 Matrix Elements of Operators in u(3) ⊃ so(3)
141(1)
9.5.5 Tensor Calculus of u(3) ⊃ so(3)
142(1)
9.5.6 Other Boson Constructions of u(3)
143(2)
9.6 The Unitary Algebra u(4)
145(7)
9.6.1 Subalgebra Chains not Containing so(3)
145(1)
9.6.2 Subalgebra Chains Containing so(3)
146(6)
9.7 The Unitary Algebra u(6)
152(10)
9.7.1 Subalgebra Chains not Containing so(3)
152(1)
9.7.2 Subalgebra Chains Containing so(3)
152(10)
9.8 The Unitary Algebra u(7)
162(8)
9.8.1 Subalgebra Chain Containing g2
162(2)
9.8.2 The Triplet Chains
164(6)
9.9 Contractions of Bosonic Algebras
170(5)
9.9.1 The Heisenberg Algebra h(2)
170(1)
9.9.2 The Heisenberg Algebra h(n), N = Even
171(4)
10 Fermion Realizations
175(18)
10.1 Fermion Operators
175(1)
10.2 Lie Algebras Constructed with Fermion Operators
176(1)
10.3 Racah Form
177(1)
10.4 The Algebras u(2j + 1)
178(5)
10.4.1 Subalgebra Chain Containing Spin(3)
178(1)
10.4.2 The Algebras u(2) and su(2): Spinors
179(1)
10.4.3 The Algebra u(4)
180(2)
10.4.4 The Algebra u(6)
182(1)
10.4.5 Branchings of u(2j + 1)
183(1)
10.5 The Algebra U (Σi (2ji + 1))
183(1)
10.6 Internal Degrees of Freedom (Different Spaces)
184(3)
10.6.1 The Algebras u(4) and su(4)
184(1)
10.6.2 The Algebras u(6) and su(b)
185(2)
10.7 Internal Degrees of Freedom (Same Space)
187(6)
10.7.1 The Algebra u((2l + 1)(2s + 1)): L-S Coupling
187(3)
10.7.2 The Algebra U (Σj (2j + 1)): j-j Coupling
190(1)
10.7.3 The Algebra U ((Σl (2l+ 1)) (2s + 1)): Mixed L-S Configurations
191(2)
11 Differential Realizations
193(8)
11.1 Differential Operators
193(1)
11.2 Unitary Algebras u(n)
193(1)
11.3 Orthogonal Algebras so(n)
194(4)
11.3.1 Casimir Operators: Laplace-Beltrami Form
197(1)
11.3.2 Basis for the Representations
197(1)
11.4 Orthogonal Algebras so(n, m)
198(1)
11.5 Symplectic Algebras sp(2n)
199(2)
12 Matrix Realizations
201(10)
12.1 Matrices
201(1)
12.2 Unitary Algebras u(n)
202(3)
12.3 Orthogonal Algebras so(n)
205(1)
12.4 Symplectic Algebras sp(2n)
205(2)
12.5 Basis for the Representation
207(1)
12.6 Casimir Operators
208(3)
13 Coset Spaces
211(24)
13.1 Coset Spaces U(n)/U(n -- 1) ⊗ U(1)
211(1)
13.2 Coherent (or Intrinsic) States
212(3)
13.2.1 'Algebraic' Coherent States
212(1)
13.2.2 'Group' Coherent States
213(1)
13.2.3 'Projective' Coherent States
214(1)
13.2.4 Coset Spaces
215(1)
13.3 Coherent States of u(n), N = Even
215(5)
13.3.1 Coherent States of u(2)
216(1)
13.3.2 Coherent States of u(4)
216(2)
13.3.3 Coherent States of u(6)
218(2)
13.4 Coherent States of u(n), N = Odd
220(1)
13.4.1 Coherent States of u(3)
220(1)
13.5 Generalized Coherent States
221(1)
13.6 The Geometry of Algebraic Models
221(1)
13.7 Coset Spaces SO(n + m)/SO(n) ⊗ SO(m)
222(7)
13.7.1 The Coset Space SO(3)/SO(2): The Sphere S2
222(2)
13.7.2 The Coset Space SO(4)/SO(3)
224(2)
13.7.3 The Coset Spaces SO(n + 2)/SO(n) ⊗ SO(2)
226(3)
13.8 Coset Spaces SO(n, m)/SO(n) ⊗ SO(m)
229(2)
13.8.1 The Coset Space SO(2, l)/SO(2): The Hyperboloid H2
229(1)
13.8.2 The Coset Space SO(3, l)/SO(3)
230(1)
13.9 Action of the Coset P = G/H
231(4)
13.9.1 Cosets SO(n + 1)/SO(n)
231(2)
13.9.2 Cosets SO(u, 1)/SO(n)
233(2)
14 Spectrum Generating Algebras and Dynamic Symmetries
235(12)
14.1 Spectrum Generating Algebras
235(1)
14.2 Dynamic Symmetries
236(1)
14.3 Bosonic Systems
236(6)
14.3.1 Dynamic Symmetries of u(4)
237(3)
14.3.2 Dynamic Symmetries of u(6)
240(2)
14.4 Fermionic Systems
242(5)
14.4.1 Dynamic Symmetry of u(4)
243(1)
14.4.2 Dynamic Symmetry of u(6)
244(3)
15 Degeneracy Algebras and Dynamical Algebras
247(18)
15.1 Degeneracy Algebras
247(1)
15.2 Degeneracy Algebras in V ≥ 2 Dimensions
248(8)
15.2.1 The Isotropic Harmonic Oscillator
248(3)
15.2.2 The Coulomb Problem
251(5)
15.3 Degeneracy Algebra in V = 1 Dimension
256(1)
15.4 Dynamical Algebras
256(1)
15.5 Dynamical Algebras in V ≥ 2 Dimensions
257(1)
15.5.1 Harmonic Oscillator
257(1)
15.5.2 Coulomb Problem
257(1)
15.6 Dynamical Algebra in V = 1 Dimension
258(7)
15.6.1 Poschl-Teller Potential
258(2)
15.6.2 Morse Potential
260(3)
15.6.3 Lattice of Algebras
263(2)
References 265(4)
Index 269
Iachello is a renowned physicist. Honors: Chiaudano Prize, 1964 Fulbright Fellow, 1968 AKZO Prize of the Netherlands Society of Sciences, 1981 Wigner Medal, 1990 Taormina Prize, 1991 Dr. Hon., University of Ferrara, Italy, 1992 Bonner Prize of the American Physical Society, 1993 Dr. Hon., University of Seville, Spain, 1993 Ph.D. Hon. Chung Yuan University, Republic of China, 1993 Honorary Professor Nanjing University, China, 1995 Foreign Member Royal Netherlands Academy of Arts and Sciences, 1996 Honorary Fellow Eotvos Physical Society, Hungary, 1996 Centennial Prize of the Italian Physical Society, 1997 Foreign Member Croatian Academy of Arts and Sciences, 1997 Zernike Professor University of Groningen, The Netherlands, 1997 Eminent Scientist Award, RIKEN, Tokyo, Japan, 2000 Meitner Prize of the European Physical Society 2002 Dr. Hon., University of Bucharest, Romania, 2005 Italian National Medal of Science, 2007 Majorana Prize, 2007 Commemorative Medal, University of Prague, Czech Republic, 2008
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