In this book, the author convinces that Sir Arthur Stanley Eddington had things a little bit wrong, as least as far as physics is concerned. He explores the theory of groups and Lie algebras and their representations to use group representations as labor-saving tools.
Author(s): Howard Georgi
Edition: 2
Publisher: CRC Press
Year: 1999
Language: English
Pages: 331
Tags: Group Theory
Cover
Title page
Date-line
Dedication
Preface to the Revised Edition
Contents
Why Group Theory?
1 Finite Groups
1.1 Groups and representations
1.2 Example - $Z_3$
1.3 The regular representation
1.4 Irreducible representations
1.5 Transformation groups
1.6 Application: parity in quantum mechanics
1.7 Example: $S_3$
1.8 Example: addition of integers
1.9 Useful theorems
1.10 Subgroups
1.11 Schur's lemma
1.12 * Orthogonality relations
1.13 Characters
1.14 Eigenstates
1.15 Tensor products
1.16 Example of tensor products
1.17 * Finding the normal modes
1.18 * Symmetries of $2n+1$-gons
1.19 Pennutation group on $n$ objects
1.20 Conjugacy classes
1.21 Young tableaux
1.22 Example our old friend $S_3$
1.23 Another example - $S_4$
1.24 * Young tableaux and representations of $S_n$
2 Lie Groups
2.1 Generators
2.2 Lie algebras
2.3 The Jacobi identity
2.4 The adjoint representation
2.5 Simple algebras and groups
2.6 States and operators
2.7 Fun with exponentials
3 $SU(2)$
3.1 $J_3$ eigenstates
3.2 Raising and lowering operators
3.3 The standard notation
3.4 Tensor products
3.5 $J_3$ values add
4 Tensor Operators
4.1 Orbital angular momentum
4.2 Using tensor operators
4.3 The Wigner-Eckart theorem
4.4 Example
4.5 * Making tensor operators
4.6 Products of operators
5 Isospin
5.1 Charge independence
5.2 Creation operators
5.3 Number operators
5.4 Isospin generators
5.5 Symmetry of tensor products
5.6 The deuteron
5.7 Superselection rules
5.8 Other particles
5.9 Approximate isospin symmetry
5.10 Perturbation theory
6 Roots and Weights
6.1 Weights
6.2 More on the adjoint representation
6.3 Roots
6.4 Raising and lowering
6.5 Lots of $SU(2)$s
6.6 Watch carefully - this is important!
7 $SU(3)$
7.1 The Gell-Mann matrices
7.2 Weights and roots of $SU(3)$
8 Simple Roots
8.1 Positive weights
8.2 Simple roots
8.3 Constructing the algebra
8.4 Dynkin diagrams
8.5 Example: $G_2$
8.6 The roots of $G_2$
8.7 The Cartan matrix
8.8 Finding all the roots
8.9 The $SU(2)$s
8.10 Constructing the $G_2$ algebra
8.11 Another example: the algebra $G_3$
8.12 Fundamental weights
8.13 The trace of a generator
9 More $SU(3)$
9.1 Fundamental representations of $SU(3)$
9.2 Constructing the states
9.3 The Weyl group
9.4 Complex conjugation
9.5 Examples of other representations
10 Tensor Methods
10.1 lower and upper indices
10.2 Tensor components and wave functions
10.3 Irreducible representations and symmetry
10.4 Invariant tensors
10.5 Clebsch-Gordan decomposition
10.6 Triality
10.7 Matrix elements and operators
10.8 Normalization
10.9 Tensor operators
1O.10 The dimension of $(n, m)$
10.11 * The weights of $(n, m)$
10.12 Generalization of Wigner-Eckart
10.13 * Tensors for $SU(2)$
10.14 * Clebsch-Gordan coefficients from tensors
10.15 * Spin $s_1 + s_2 - 1$
10.16 * Spin $s_1 + s_2 - k$
11 Hypercharge and Strangeness
11.1 The eight-fold way
11.2 The Gell-Mann Okubo formula
11.3 Hadron resonances
11.4 Quarks
12 Young Tableaux
12.1 Raising the indices
12.2 Clebsch-Gordan decomposition
12.3 $SU(3) to SU(2) times U(1)$
13 $SU(N)$
13.1 Generalized Gell-Mann matrices
13.2 $SU(N)$ tensors
13.3 Dimensions
13.4 Complex representations
13.5 $SU(N) otimes SU(M) in SU(N+M)$
14 3-D Harmonic Oscillator
14.1 Raising and lowering operators
14.2 Angular momentum
14.3 A more complicated example
15 $SU(6)$ and the Quark Model
15.1 Including the spin
15.2 $SU(N) otimes SU(M) in SU(NM)$
15.3 The baryon states
15.4 Magnetic moments
16 Color
16.1 Colored quarks
16.2 Quantum Chromodynamics
16.3 Heavy quarks
16.4 Flavor $SU(4)$ is useless!
17 Constituent Quarks
17.1 The nonrelativistic limit
18 Unified Theories and $SU(5)$
18.1 Grand unification
18.2 Parity violation, helicity and handedness
18.3 Spontaneously broken symmetry
18.4 Physics of spontaneous symmetry breaking
18.5 Is the Higgs real?
18.6 Unification and $SU(5)$
18.7 Breaking $SU(5)$
18.8 Proton decay
19 The Classical Groups
19.1 The $SO(2n)$ algebras
19.2 The $SO(2n+1)$ algebras
19.3 The $Sp(2n)$ algebras
19.4 Quaternions
20 The Classification Theorem
20.1 $Pi$-systems
20.2 Regular subalgebras
20.3 Other Subalgebras
21 $SO(2n+1)$ and Spinors
21.1 Fundamental weight of $SO(2n+1)$
21.2 Real and pseudo-real
21.3 Real representations
21.4 Pseudo-real representations
21.5 $R$ is an invariant tensor
21.6 The explicit form for $R$
22 $SO(2n+2)$ Spinors
22.1 Fundamental weights of $SO(2n+2)$
23 $SU(n) subset SO(2n)$
23.1 Clifford algebras
23.2 $Gamma_m$ and $R$ as invariant tensors
23.3 Products of $Gamma$s
23.4 Self-duality
23.5 Example: $SO(10)$
23.6 The $SU(n)$ subalgebra
24 $SO(10)$
24.1 $SO(10)$ and $SU(4) times SU(2) times SU(2)$
24.2 * Spontaneous breaking of $SO(10)$
24.3 * Breaking $SO(10) to SU(5)$
24.4 * Breaking $SO(10) to SU(3) times SU(2) times U(1)$
24.5 * Breaking $SO(10) to SU(3) times U(1)$
24.6 * Lepton number as a fourth color
25 Automorphisms
25.1 Outer automorphisms
25.2 Fun with $SO(8)$
26 $Sp(2n)$
26.1 Weights of $SU(n)$
26.2 Tensors for $Sp(2n)$
27 Odds and Ends
27.1 Exceptional algebras and octonians
27.2 $E_6$ unification
27.3 Breaking $E_6$
27.4 $SU(3) times SU(3) times SU(3)$ unification
27.5 Anomalies
Epilogue
Index
