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Lie Algebras In Particle Physics: from Isospin To Unified Theories

, by Georgi,Howard

Lie Algebras In Particle Physics: from Isospin To Unified Theories by Georgi,Howard, 9780738202334

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9780738202334 | 0738202339
Cover: Nonspecific Binding
Copyright: 10/22/1999

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Summary

Author Biography

Table of Contents

Supplemental Materials

Howard Georgi is the co-inventor (with Sheldon Glashow) of the SU(5) theory. This extensively revised and updated edition of his classic text makes the theory of Lie groups accessible to graduate students, while offering a perspective on the way in which knowledge of such groups can provide an insight into the development of unified theories of strong, weak, and electromagnetic interactions.

Why Group Theory?

1

(1)

Finite Groups

2

(41)

Groups and representations

2

(1)

Example - Z3

3

(1)

The regular representation

4

(1)

Irreducible representations

5

(1)

Transformation groups

6

(1)

Application: parity in quantum mechanics

7

(1)

Example: S3

8

(1)

Example: addition of integers

9

(1)

Useful theorems

10

(1)

Subgroups

11

(2)

Schur's lemma

13

(4)

Orthogonality relations

17

(3)

Characters

20

(5)

Eigenstates

25

(1)

Tensor products

26

(1)

Example of tensor products

27

(2)

Finding the normal modes

29

(4)

Symmetries of 2n+1-gons

33

(1)

Permutation group on n objects

34

(1)

Conjugacy classes

35

(2)

Young tableaux

37

(1)

Example---our old friend S3

38

(1)

Another example---S4

38

(1)

Young tableaux and representations of Sn

38

(5)

Lie Groups

43

(13)

Generators

43

(2)

Lie algebras

45

(2)

The Jacobi identity

47

(1)

The adjoint representation

48

(3)

Simple algebras and groups

51

(1)

States and operators

52

(1)

Fun with exponentials

53

(3)

SU(2)

56

(12)

J3 eigenstates

56

(1)

Raising and lowering operators

57

(3)

The standard notation

60

(3)

Tensor products

63

(1)

J3 values add

64

(4)

Tensor Operators

68

(11)

Orbital angular momentum

68

(1)

Using tensor operators

69

(1)

The Wigner-Eckart theorem

70

(2)

Example

72

(3)

Making tensor operators

75

(2)

Products of operators

77

(2)

Isospin

79

(11)

Charge independence

79

(1)

Creation operators

80

(2)

Number operators

82

(1)

Isospin generators

82

(1)

Symmetry of tensor products

83

(1)

The deuteron

84

(1)

Superselection rules

85

(1)

Other particles

86

(2)

Approximate isospin symmetry

88

(1)

Perturbation theory

88

(2)

Roots and Weights

90

(8)

Weights

90

(1)

More on the adjoint representation

91

(1)

Roots

92

(1)

Raising and lowering

93

(1)

Lots of SU(2)s

93

(2)

Watch carefully-this is important!

95

(3)

SU(3)

98

(5)

The Gell-Mann matrices

98

(2)

Weights and roots of SU(3)

100

(3)

Simple Roots

103

(22)

Positive weights

103

(2)

Simple roots

105

(3)

Constructing the algebra

108

(3)

Dynkin diagrams

111

(1)

Example: G2

112

(1)

The roots of G2

112

(2)

The Cartan matrix

114

(1)

Finding all the roots

115

(2)

The SU(2)s

117

(1)

Constructing the G2 algebra

118

(2)

Another example: the algebra C3

120

(1)

Fundamental weights

121

(2)

The trace of a generator

123

(2)

More SU(3)

125

(13)

Fundamental representations of SU(3)

125

(2)

Constructing the states

127

(3)

The Weyl group

130

(1)

Complex conjugation

131

(1)

Examples of other representations

132

(6)

Tensor Methods

138

(28)

lower and upper indices

138

(1)

Tensor components and wave functions

139

(1)

Irreducible representations and symmetry

140

(1)

Invariant tensors

141

(1)

Clebsch-Gordan decomposition

141

(2)

Triality

143

(1)

Matrix elements and operators

143

(1)

Normalization

144

(1)

Tensor operators

145

(1)

The dimension of (n, m)

145

(1)

The weights of (n, m)

146

(6)

Generalization of Wigner-Eckart

152

(2)

Tensors for SU(2)

154

(2)

Clebsch-Gordan coefficients from tensors

156

(1)

Spin s1 + s2 - 1

157

(3)

Spin s1 + s2 - k

160

(6)

Hypercharge and Strangeness

166

(12)

The eight-fold way

166

(3)

The Gell-Mann Okubo formula

169

(4)

Hadron resonances

173

(1)

Quarks

174

(4)

Young Tableaux

178

(9)

Raising the indices

178

(2)

Clebsch-Gordan decomposition

180

(3)

SU(3) → SU(2) x U(1)

183

(4)

SU(N)

187

(11)

Generalized Gell-Mann matrices

187

(3)

SU(N) tensors

190

(3)

Dimensions

193

(1)

Complex representations

194

(1)

SU(N) ⊗ SU(M) ∈ SU(N + M)

195

(3)

3-D Harmonic Oscillator

198

(7)

Raising and lowering operators

198

(2)

Angular momentum

200

(1)

A more complicated example

200

(5)

SU(6) and the Quark Model

205

(9)

Including the spin

205

(1)

SU(N) ⊗ SU(M) ∈ SU(NM)

206

(2)

The baryon states

208

(2)

Magnetic moments

210

(4)

Color

214

(7)

Colored quarks

214

(4)

Quantum Chromodynamics

218

(1)

Heavy quarks

219

(1)

Flavor SU(4) is useless!

219

(2)

Constriuent Quarks

221

(4)

The nonrelativistic limit

221

(4)

Unified Theories and SU(5)

225

(12)

Grand unification

225

(1)

Parity violation, helicity and handedness

226

(2)

Spontaneously broken symmetry

228

(1)

Physics of spontaneous symmetry breaking

229

(1)

Is the Higgs real?

230

(1)

Unification and SU(5)

231

(3)

Breaking SU(5)

234

(1)

Proton decay

235

(2)

The Classical Groups

237

(7)

The SO(2n) algebras

237

(1)

The SO(2n + 1) algebras

238

(1)

The Sp(2n) algebras

239

(1)

Quaternions

240

(4)

The Classification Theorem

244

(11)

II-systems

244

(7)

Regular subalgebras

251

(2)

Other Subalgebras

253

(2)

SO(2n + 1) and Spinors

255

(10)

Fundamental weight of SO(2n + 1)

255

(4)

Real and pseudo-real

259

(2)

Real representations

261

(1)

Pseudo-real representations

262

(1)

R is an invariant tensor

262

(1)

The explicit from for R

262

(3)

SO(2n + 2) Spinors

265

(5)

Fundamental weights of SO(2n + 2)

265

(5)

SU(n) ⊂ SO(2n)

270

(12)

Clifford algebras

270

(2)

Γm and R as invariant tensors

272

(2)

Products of Γs

274

(3)

Self-duality

277

(2)

Example: SO(10)

279

(1)

The SU(n) subalgebra

279

(3)

SO(10)

282

(9)

SO(10) and SU(4) x SU(2) x SU(2)

282

(3)

Spontaneous breaking of SO(10)

285

(1)

Breaking SO(10) → SU(5)

285

(2)

Breaking SO(10) → SU(3) x SU(2) x U(1)

287

(2)

Breaking SO(10) → SU(3) x U(1)

289

(1)

Lepton number as a fourth color

289

(2)

Automorphisms

291

(6)

Outer automorphisms

291

(2)

Fun with SO(8)

293

(4)

Sp(2n)

297

(5)

Weights of SU(n)

297

(2)

Tensors for Sp(2n)

299

(3)

Odds and Ends

302

(9)

Exceptional algebras and octonians

302

(2)

E6 unification

304

(4)

Breaking E6

308

(1)

SU(3) x SU(3) x SU(3) unification

308

(1)

Anomalies

309

(2)

Epilogue

311

(1)

Index

312

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