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Why Group Theory?p. 1
Finite Groupsp. 2
Groups and representationsp. 2
Example - Z[subscript 3]p. 3
The regular representationp. 4
Irreducible representationsp. 5
Transformation groupsp. 6
Application: parity in quantum mechanicsp. 7
Example: S[subscript 3]p. 8
Example: addition of integersp. 9
Useful theoremsp. 10
Subgroupsp. 11
Schur's lemmap. 13
* Orthogonality relationsp. 17
Charactersp. 20
Eigenstatesp. 25
Tensor productsp. 26
Example of tensor productsp. 27
* Finding the normal modesp. 29
* Symmetries of 2n+1-gonsp. 33
Permutation group on n objectsp. 34
Conjugacy classesp. 35
Young tableauxp. 37
Example -- our old friend S[subscript 3]p. 38
Another example -- S[subscript 4]p. 38
* Young tableaux and representations of S[subscript n]p. 38
Lie Groupsp. 43
Generatorsp. 43
Lie algebrasp. 45
The Jacobi identityp. 47
The adjoint representationp. 48
Simple algebras and groupsp. 51
States and operatorsp. 52
Fun with exponentialsp. 53
SU(2)p. 56
J[subscript 3] eigenstatesp. 56
Raising and lowering operatorsp. 57
The standard notationp. 60
Tensor productsp. 63
J[subscript 3] values addp. 64
Tensor Operatorsp. 68
Orbital angular momentump. 68
Using tensor operatorsp. 69
The Wigner-Eckart theoremp. 70
Examplep. 72
* Making tensor operatorsp. 75
Products of operatorsp. 77
Isospinp. 79
Charge independencep. 79
Creation operatorsp. 80
Number operatorsp. 82
Isospin generatorsp. 82
Symmetry of tensor productsp. 83
The deuteronp. 84
Superselection rulesp. 85
Other particlesp. 86
Approximate isospin symmetryp. 88
Perturbation theoryp. 88
Roots and Weightsp. 90
Weightsp. 90
More on the adjoint representationp. 91
Rootsp. 92
Raising and loweringp. 93
Lots of SU(2)sp. 93
Watch carefully - this is important!p. 95
SU(3)p. 98
The Gell-Mann matricesp. 98
Weights and roots of SU(3)p. 100
Simple Rootsp. 103
Positive weightsp. 103
Simple rootsp. 105
Constructing the algebrap. 108
Dynkin diagramsp. 111
Example: G[subscript 2]p. 112
The roots of G[subscript 2]p. 112
The Cartan matrixp. 114
Finding all the rootsp. 115
The SU(2)sp. 117
Constructing the G[subscript 2] algebrap. 118
Another example: the algebra C[subscript 3]p. 120
Fundamental weightsp. 121
The trace of a generatorp. 123
More SU(3)p. 125
Fundamental representations of SU(3)p. 125
Constructing the statesp. 127
The Weyl groupp. 130
Complex conjugationp. 131
Examples of other representationsp. 132
Tensor Methodsp. 138
Lower and upper indicesp. 138
Tensor components and wave functionsp. 139
Irreducible representations and symmetryp. 140
Invariant tensorsp. 141
Clebsch-Gordan decompositionp. 141
Trialityp. 143
Matrix elements and operatorsp. 143
Normalizationp. 144
Tensor operatorsp. 145
The dimension of (n,m)p. 145
* The weights of (n,m)p. 146
Generalization of Wigner-Eckartp. 152
* Tensors for SU(2)p. 154
* Clebsch-Gordan coefficients from tensorsp. 156
* Spin s[subscript 1] + s[subscript 2] - 1p. 157
* Spin s[subscript 1] + s[subscript 2] - kp. 160
Hypercharge and Strangenessp. 166
The eight-fold wayp. 166
The Gell-Mann Okubo formulap. 169
Hadron resonancesp. 173
Quarksp. 174
Young Tableauxp. 178
Raising the indicesp. 178
Clebsch-Gordan decompositionp. 180
SU(3) [right arrow] SU(2) [times] U(1)p. 183
SU(N)p. 187
Generalized Gell-Mann matricesp. 187
SU(N) tensorsp. 190
Dimensionsp. 193
Complex representationsp. 194
SU(N) [multiply sign in circle] SU(M) [set membership] SU(N +M)p. 195
3-D Harmonic Oscillatorp. 198
Raising and lowering operatorsp. 198
Angular momentump. 200
A more complicated examplep. 200
SU(6) and the Quark Modelp. 205
Including the spinp. 205
SU(N) [multiply sign in circle] SU(M) [set membership] SU(NM)p. 206
The baryon statesp. 208
Magnetic momentsp. 210
Colorp. 214
Colored quarksp. 214
Quantum Chromodynamicsp. 218
Heavy quarksp. 219
Flavor SU(4) is useless!p. 219
Constituent Quarksp. 221
The nonrelativistic limitp. 221
Unified Theories and SU(5)p. 225
Grand unificationp. 225
Parity violation, helicity and handednessp. 226
Spontaneously broken symmetryp. 228
Physics of spontaneous symmetry breakingp. 229
Is the Higgs real?p. 230
Unification and SU(5)p. 231
Breaking SU(5)p. 234
Proton decayp. 235
The Classical Groupsp. 237
The SO(2n) algebrasp. 237
The SO(2n + 1) algebrasp. 238
The Sp(2n) algebrasp. 239
Quaternionsp. 240
The Classification Theoremp. 244
II-systemsp. 244
Regular subalgebrasp. 251
Other Subalgebrasp. 253
SO(2n + 1) and Spinorsp. 255
Fundamental weight of SO(2n + 1)p. 255
Real and pseudo-realp. 259
Real representationsp. 261
Pseudo-real representationsp. 262
R is an invariant tensorp. 262
The explicit form for Rp. 262
SO(2n + 2) Spinorsp. 265
Fundamental weights of SO(2n + 2)p. 265
SU(n) [subset or is implied by] SO(2n)p. 270
Clifford algebrasp. 270
[Gamma][subscript m] and R as invariant tensorsp. 272
Products of [Gamma][subscript s]p. 274
Self-dualityp. 277
Example: SO(10)p. 279
The SU(n) subalgebrap. 279
SO(10)p. 282
SO(10) and SU(4) [times] SU(2) [times] SU(2)p. 282
* Spontaneous breaking of SO(10)p. 285
* Breaking SO(10) [right arrow] SU(5)p. 285
* Breaking SO(10) [right arrow] SU(3) [times] SU(2) [times] U(1)p. 287
* Breaking SO(10) [right arrow] SU(3) [times] U(1)p. 289
* Lepton number as a fourth colorp. 289
Automorphismsp. 291
Outer automorphismsp. 291
Fun with SO(8)p. 293
Sp(2n)p. 297
Weights of SU(n)p. 297
Tensors for Sp(2n)p. 299
Odds and Endsp. 302
Exceptional algebras and octoniansp. 302
E[subscript 6] unificationp. 304
Breaking E[subscript 6]p. 308
SU(3) [times] SU(3) [times] SU(3) unificationp. 308
Anomaliesp. 309
Epiloguep. 311
Indexp. 312
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