## Lie Algebras in Particle Physics: From Isospin to Unified Theories (Google eBook)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. |

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### Contents

103 Irreducible representations and symmetry | 140 |

104 Invariant tensors | 141 |

106 Triality | 143 |

108 Normalization | 144 |

109 Tensor operators | 145 |

1011 The weights of n m | 146 |

1012 Generalization of WignerEckart | 152 |

1013 Tensors for SU2 | 154 |

11 | |

12 | |

14 | |

18 | |

113 Characters | 21 |

114 Eigenstates | 26 |

115 Tensor products | 27 |

116 Example of tensor products | 28 |

117 Finding the normal modes | 30 |

118 Symmetries of 2n+1gons | 34 |

119 Permutation group on n objects | 35 |

120 Conjugacy classes | 36 |

121 Young tableaux | 38 |

122 Example our old friend | 39 |

Lie Groups | 44 |

22 Lie algebras | 46 |

23 The Jacobi identity | 48 |

24 The adjoint representation | 49 |

25 Simple algebras and groups | 52 |

26 States and operators | 53 |

27 Fun with exponentials | 54 |

SU2 | 57 |

32 Raising and lowering operators | 58 |

33 The standard notation | 61 |

34 Tensor products | 64 |

35 J₃ values add | 65 |

Tensor Operators | 69 |

42 Using tensor operators | 70 |

43 The WignerEckart theorem | 71 |

44 Example | 73 |

45 Making tensor operators | 76 |

46 Products of operators | 78 |

Isospin | 80 |

52 Creation operators | 81 |

53 Number operators | 83 |

55 Symmetry of tensor products | 84 |

56 The deuteron | 85 |

57 Superselection rules | 86 |

58 Other particles | 87 |

59 Approximate isospin symmetry | 89 |

Roots and Weights | 91 |

62 More on the adjoint representation | 92 |

63 Roots | 93 |

64 Raising and lowering | 94 |

66 Watch carefully this is important | 96 |

SU3 | 99 |

72 Weights and roots of SU3 | 101 |

Simple Roots | 103 |

82 Simple roots | 105 |

83 Constructing the algebra | 108 |

84 Dynkin diagrams | 111 |

G₂ | 112 |

87 The Cartan matrix | 114 |

88 Finding all the roots | 115 |

89 The SU 2s | 117 |

810 Constructing the G₂ algebra | 118 |

the algebra C₃ | 120 |

812 Fundamental weights | 121 |

813 The trace of a generator | 123 |

More SU3 | 125 |

92 Constructing the states | 127 |

93 The Weyl group | 130 |

94 Complex conjugation | 131 |

95 Examples of other representations | 132 |

Tensor Methods | 138 |

102 Tensor components and wave functions | 139 |

1014 ClebschGordan coefficients from tensors | 156 |

1015 Spin 𝑠₁ + 𝑠₂ 1 | 157 |

1016 Spin 𝑠₁+ 𝑠₂ _ k | 160 |

Hypercharge and Strangeness | 166 |

112 The GellMann Okubo formula | 169 |

113 Hadron resonances | 173 |

114 Quarks | 174 |

Young Tableaux | 178 |

122 ClebschGordan decomposition | 180 |

123 SU3 SU2 x U1 | 183 |

SUN | 187 |

132 SUN tensors | 190 |

133 Dimensions | 193 |

134 Complex representations | 194 |

135 SUN SUM SUN + M | 195 |

3D Harmonic Oscillator | 198 |

142 Angular momentum | 200 |

SU6 and the Quark Model | 205 |

152 SUN SUM SUNM | 206 |

153 The baryon states | 208 |

154 Magnetic moments | 210 |

Color | 214 |

162 Quantum Chromodynamics | 218 |

163 Heavy quarks | 219 |

Constituent Quarks | 221 |

Unified Theories and SU5 | 225 |

182 Parity violation helicity and handedness | 226 |

183 Spontaneously broken symmetry | 228 |

184 Physics of spontaneous symmetry breaking | 229 |

185 Is the Higgs real? | 230 |

186 Unification and SUb | 231 |

187 Breaking SU5 | 234 |

188 Proton decay | 235 |

The Classical Groups | 237 |

192 The S02n + 1 algebras | 238 |

193 The Sp2n algebras | 239 |

194 Quaternions | 240 |

The Classification Theorem | 244 |

202 Regular subalgebras | 251 |

203 Other Subalgebras | 253 |

21 SO2n+ and Spinors | 255 |

212 Real and pseudoreal | 259 |

213 Real representations | 261 |

214 Pseudoreal representations | 262 |

22 SO2n + 2 Spinors | 265 |

SUn SO2n | 270 |

232 𝛤𝒎 and R as invariant tensors | 272 |

233 Products of 𝛤s | 274 |

234 Selfduality | 277 |

SO10 | 279 |

241 SO10 and SU4 x SU2 x SU2 | 282 |

242 Spontaneous breaking of SO10 | 285 |

244 Breaking SO10 SU3 x SU2 x U1 | 287 |

245 Breaking SO 10 SU3 x U1 | 289 |

Automorphisms | 291 |

252 Fun with SO8 | 293 |

26 Sp2n | 297 |

262 Tensors for Sp2n | 299 |

27 Odds and Ends | 302 |

272 E₆ unification | 304 |

273 Breaking E₆ | 308 |

275 Anomalies | 309 |

Epilogue | 311 |

Index | 312 |

### Common terms and phrases

adjoint representation angular momentum annihilation operators anticommute automorphism baryon boxes breaking called Cartan Chapter Clebsch-Gordan Clifford algebra color commutation relations complex conjugate conjugacy classes construct corresponding creation operators decompose defining representation diagonal dimensional space doublet Dynkin diagram eigenstates equivalent example explicitly finite group Gell-Mann group elements hermitian Higgs field highest weight Hilbert space hypercharge II-system integers invariant tensor irreducible representations isospin J3 value labels linear combination lower indices lowering operators mass matrix elements mesons multiplication law neutrino neutron non-zero nontrivial notation nucleon orthogonal pair particles Pauli matrices permutation physics positive roots problem properties proton quarks raising and lowering regular representation representation of SU(3 right-handed satisfy Schur's lemma simple roots singlet spin spinor representations SU(N subalgebra subgroup subspace symmetry tensor components tensor operators tensor product theorem theory tion traceless trivial unitary upper indices vacuum value vector Young tableau zero

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