Lie Algebras In Particle Physics

from Isospin To Unified Theories

By: Howard Georgi

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Paperback | 22 October 1999 | Edition Number 2

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340 Pages

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An exciting new edition of a classic text. 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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Why Group Theory?	p. 1
Finite Groups	p. 2
Groups and representations	p. 2
Example - Z[subscript 3]	p. 3
The regular representation	p. 4
Irreducible representations	p. 5
Transformation groups	p. 6
Application: parity in quantum mechanics	p. 7
Example: S[subscript 3]	p. 8
Example: addition of integers	p. 9
Useful theorems	p. 10
Subgroups	p. 11
Schur's lemma	p. 13
* Orthogonality relations	p. 17
Characters	p. 20
Eigenstates	p. 25
Tensor products	p. 26
Example of tensor products	p. 27
* Finding the normal modes	p. 29
* Symmetries of 2n+1-gons	p. 33
Permutation group on n objects	p. 34
Conjugacy classes	p. 35
Young tableaux	p. 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 Groups	p. 43
Generators	p. 43
Lie algebras	p. 45
The Jacobi identity	p. 47
The adjoint representation	p. 48
Simple algebras and groups	p. 51
States and operators	p. 52
Fun with exponentials	p. 53
SU(2)	p. 56
J[subscript 3] eigenstates	p. 56
Raising and lowering operators	p. 57
The standard notation	p. 60
Tensor products	p. 63
J[subscript 3] values add	p. 64
Tensor Operators	p. 68
Orbital angular momentum	p. 68
Using tensor operators	p. 69
The Wigner-Eckart theorem	p. 70
Example	p. 72
* Making tensor operators	p. 75
Products of operators	p. 77
Isospin	p. 79
Charge independence	p. 79
Creation operators	p. 80
Number operators	p. 82
Isospin generators	p. 82
Symmetry of tensor products	p. 83
The deuteron	p. 84
Superselection rules	p. 85
Other particles	p. 86
Approximate isospin symmetry	p. 88
Perturbation theory	p. 88
Roots and Weights	p. 90
Weights	p. 90
More on the adjoint representation	p. 91
Roots	p. 92
Raising and lowering	p. 93
Lots of SU(2)s	p. 93
Watch carefully - this is important!	p. 95
SU(3)	p. 98
The Gell-Mann matrices	p. 98
Weights and roots of SU(3)	p. 100
Simple Roots	p. 103
Positive weights	p. 103
Simple roots	p. 105
Constructing the algebra	p. 108
Dynkin diagrams	p. 111
Example: G[subscript 2]	p. 112
The roots of G[subscript 2]	p. 112
The Cartan matrix	p. 114
Finding all the roots	p. 115
The SU(2)s	p. 117
Constructing the G[subscript 2] algebra	p. 118
Another example: the algebra C[subscript 3]	p. 120
Fundamental weights	p. 121
The trace of a generator	p. 123
More SU(3)	p. 125
Fundamental representations of SU(3)	p. 125
Constructing the states	p. 127
The Weyl group	p. 130
Complex conjugation	p. 131
Examples of other representations	p. 132
Tensor Methods	p. 138
Lower and upper indices	p. 138
Tensor components and wave functions	p. 139
Irreducible representations and symmetry	p. 140
Invariant tensors	p. 141
Clebsch-Gordan decomposition	p. 141
Triality	p. 143
Matrix elements and operators	p. 143
Normalization	p. 144
Tensor operators	p. 145
The dimension of (n,m)	p. 145
* The weights of (n,m)	p. 146
Generalization of Wigner-Eckart	p. 152
* Tensors for SU(2)	p. 154
* Clebsch-Gordan coefficients from tensors	p. 156
* Spin s[subscript 1] + s[subscript 2] - 1	p. 157
* Spin s[subscript 1] + s[subscript 2] - k	p. 160
Hypercharge and Strangeness	p. 166
The eight-fold way	p. 166
The Gell-Mann Okubo formula	p. 169
Hadron resonances	p. 173
Quarks	p. 174
Young Tableaux	p. 178
Raising the indices	p. 178
Clebsch-Gordan decomposition	p. 180
SU(3) [right arrow] SU(2) [times] U(1)	p. 183
SU(N)	p. 187
Generalized Gell-Mann matrices	p. 187
SU(N) tensors	p. 190
Dimensions	p. 193
Complex representations	p. 194
SU(N) [multiply sign in circle] SU(M) [set membership] SU(N +M)	p. 195
3-D Harmonic Oscillator	p. 198
Raising and lowering operators	p. 198
Angular momentum	p. 200
A more complicated example	p. 200
SU(6) and the Quark Model	p. 205
Including the spin	p. 205
SU(N) [multiply sign in circle] SU(M) [set membership] SU(NM)	p. 206
The baryon states	p. 208
Magnetic moments	p. 210
Color	p. 214
Colored quarks	p. 214
Quantum Chromodynamics	p. 218
Heavy quarks	p. 219
Flavor SU(4) is useless!	p. 219
Constituent Quarks	p. 221
The nonrelativistic limit	p. 221
Unified Theories and SU(5)	p. 225
Grand unification	p. 225
Parity violation, helicity and handedness	p. 226
Spontaneously broken symmetry	p. 228
Physics of spontaneous symmetry breaking	p. 229
Is the Higgs real?	p. 230
Unification and SU(5)	p. 231
Breaking SU(5)	p. 234
Proton decay	p. 235
The Classical Groups	p. 237
The SO(2n) algebras	p. 237
The SO(2n + 1) algebras	p. 238
The Sp(2n) algebras	p. 239
Quaternions	p. 240
The Classification Theorem	p. 244
II-systems	p. 244
Regular subalgebras	p. 251
Other Subalgebras	p. 253
SO(2n + 1) and Spinors	p. 255
Fundamental weight of SO(2n + 1)	p. 255
Real and pseudo-real	p. 259
Real representations	p. 261
Pseudo-real representations	p. 262
R is an invariant tensor	p. 262
The explicit form for R	p. 262
SO(2n + 2) Spinors	p. 265
Fundamental weights of SO(2n + 2)	p. 265
SU(n) [subset or is implied by] SO(2n)	p. 270
Clifford algebras	p. 270
[Gamma][subscript m] and R as invariant tensors	p. 272
Products of [Gamma][subscript s]	p. 274
Self-duality	p. 277
Example: SO(10)	p. 279
The SU(n) subalgebra	p. 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 color	p. 289
Automorphisms	p. 291
Outer automorphisms	p. 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 Ends	p. 302
Exceptional algebras and octonians	p. 302
E[subscript 6] unification	p. 304
Breaking E[subscript 6]	p. 308
SU(3) [times] SU(3) [times] SU(3) unification	p. 308
Anomalies	p. 309
Epilogue	p. 311
Index	p. 312
Table of Contents provided by Syndetics. All Rights Reserved.

Lie Algebras In Particle Physics

from Isospin To Unified Theories

At a Glance

Paperback

Shipping

How to return your order

Defective items

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