Aspects of Symmetry: Selected Erice Lectures Book

	Preface	xiii
	Acknowledgements	xiv
1	An introduction to unitary symmetry
1	The search for higher symmetries	1
1.1	The eight-baryon puzzle	1
1.2	The elimination of G[subscript 0]	4
2	SU(3) and its representations	5
2.1	The representations of SU(n)	5
2.2	The representations of SU(2)	6
2.3	The representations of SU(3)	7
2.4	Dimensions of the IRs	8
2.5	Isospin and hypercharge	9
2.6	Isospin-hypercharge decompositions	10
2.7	The Clebsch-Gordan series	12
2.8	Some theorems	15
2.9	Invariant couplings	17
2.10	The problem of Cartesian components	17
2.11	SU(2) again	18
2.12	SU(3) octets: trilinear couplings	19
2.13	SU(3) OCTETS: QUADRILINEAR COUPLINGS	20
2.14	A mixed notation	21
3	Applications	23
3.1	Electromagnetism	23
3.2	Magnetic moments: baryons	24
3.3	Electromagnetic mass splittings	25
3.4	Electromagnetic properties of the decuplet	26
3.5	The medium-strong interactions	26
4	Ideas of octet enhancement	28
	Bibliography	35
2	Soft pions
1	The reduction formula	36
2	The weak interactions: first principles	40
3	The Goldberger-Treiman relation and a first glance at PCAC	41
4	A hard look at PCAC	42
5	The gradient-coupling model	45
6	Adler's rule for the emission of one soft pion	47
7	Current commutators	50
7.1	Vector-vector commutators	50
7.2	Vector-axial commutators	51
7.3	Axial-axial commutators	51
8	The Weinberg-Tomozawa formula and the Adler-Weisberger relation	52
9	Pion-pion scattering a la Weinberg	57
10	Kaon decays	60
Appendix 1	Notational conventions	63
Appendix 2	No-renormalization theorem	63
Appendix 3	Threshold S-matrix and threshold scattering lengths	64
	Bibliography	65
3	Dilatations
1	Introduction	67
2	The formal theory of broken scale invariance	68
2.1	Symmetries, currents, and Ward identities	68
2.2	Scale transformations and scale dimensions	70
2.3	More about the scale current and a quick look at the conformal group	71
2.4	Hidden scale invariance	76
3	The death of scale invariance	79
3.1	Some definitions and technical details	79
3.2	A disaster in the deep Euclidean region	80
3.3	Anomalous dimensions and other anomalies	82
3.4	The last anomalies: the Callan-Symanzik equations	84
4	The resurrection of scale invariance	88
4.1	The renormalization group equations and their solution	88
4.2	The return of scaling in the deep Euclidean region	90
4.3	Scaling and the operator product expansion	93
5	Conclusions and questions	96
	Notes and references	97
4	Renormalization and symmetry: a review for non-specialists
1	Introduction	99
2	Bogoliubov's method and Hepp's theorem	99
3	Renormalizable and non-renormalizable interactions	104
4	Symmetry and symmetry-breaking: Symanzik's rule	106
5	Symmetry and symmetry-breaking: currents	108
	Notes and references	111
5	Secret symmetry: an introduction to spontaneous symmetry breakdown and gauge fields
1	Introduction	113
2	Secret symmetries in classical field theory	115
2.1	The idea of spontaneous symmetry breakdown	115
2.2	Goldstone bosons in an Abelian model	118
2.3	Goldstone bosons in the general case	119
2.4	The Higgs phenomenon in the Abelian model	121
2.5	Yang-Mills fields and the Higgs phenomenon in the general case	124
2.6	Summary and remarks	126
3	Secret renormalizability	128
3.1	The order of the arguments	128
3.2	Renormalization reviewed	128
3.3	Functional methods and the effective potential	132
3.4	The loop expansion	135
3.5	A sample computation	136
3.6	The most important part of this lecture	138
3.7	The physical meaning of the effective potential	139
3.8	Accidental symmetry and related phenomena	142
3.9	An alternative method of computation	144
4	Functional integration (vulgarized)	145
4.1	Integration over infinite-dimensional spaces	145
4.2	Functional integrals and generating functionals	148
4.3	Feynman rules	152
4.4	Derivative interactions	154
4.5	Fermi fields	156
4.6	Ghost fields	158
5	The Feynman rules for gauge field theories	159
5.1	Troubles with gauge invariance	159
5.2	The Faddeev-Popov Ansatz	160
5.3	The application of the Ansatz	163
5.4	Justification of the Ansatz	165
5.5	Concluding remarks	167
6	Asymptotic freedom	169
6.1	Operator products and deep inelastic electroproduction	169
6.2	Massless field theories and the renormalization group	171
6.3	Exact and approximate solutions of the renormalization group equations	174
6.4	Asymptotic freedom	176
6.5	No conclusions	179
Appendix	One-loop effective potential in the general case	180
	Notes and references	182
6	Classical lumps and their quantum descendants
1	Introduction	185
2	Simple examples and their properties	187
2.1	Some time-independent lumps in one space dimension	187
2.2	Small oscillations and stability	191
2.3	Lumps are like particles (almost)	192
2.4	More dimensions and a discouraging theorem	194
3	Topological conservation laws	195
3.1	The basic idea and the main results	195
3.2	Gauge field theories revisited	198
3.3	Topological conservation laws, or, homotopy classes	202
3.4	Three examples in two spatial dimensions	205
3.5	Three examples in three dimensions	208
3.6	Patching together distant solutions, or, homotopy groups	209
3.7	Abelian and non-Abelian magnetic monopoles, or, [Pi][subscript 2](G/H) as a subgroup of [Pi][subscript 1](H)	215
4	Quantum lumps	223
4.1	The nature of the classical limit	223
4.2	Time-independent lumps: power-series expansion	225
4.3	Time-independent lumps: coherent-state variational method	232
4.4	Periodic lumps: the old quantum theory and the DHN formula	239
5	A very special system	246
5.1	A curious equivalence	246
5.2	The secret of the soliton	250
5.3	Qualitative and quantitative knowledge	252
5.4	Some opinions	253
Appendix 1	A three-dimensional scalar theory with non-dissipative solutions	254
Appendix 2	A theorem on gauge fields	256
Appendix 3	A trivial extension	257
Appendix 4	Looking for solutions	257
Appendix 5	Singular and non-singular gauge fields	259
	Notes and references	262
7	The uses of instantons
1	Introduction	265
2	Instantons and bounces in particle mechanics	268
2.1	Euclidean functional integrals	268
2.2	The double well and instantons	270
2.3	Periodic potentials	277
2.4	Unstable states and bounces	278
3	The vacuum structure of gauge field theories	282
3.1	Old stuff	282
3.2	The winding number	284
3.3	Many vacua	291
3.4	Instantons: generalities	295
3.5	Instantons: particulars	297
3.6	The evaluation of the determinant and an infrared embarrassment	300
4	The Abelian Higgs model in 1 + 1 dimensions	302
5	't Hooft's solution of the U(1) problem	307
5.1	The mystery of the missing meson	307
5.2	Preliminaries: Euclidean Fermi fields	311
5.3	Preliminaries: chiral Ward identities	314
5.4	QCD (baby version)	316
5.5	QCD (the real thing)	323
5.6	Miscellany	324
6	The fate of the false vacuum	327
6.1	Unstable vacua	327
6.2	The bounce	329
6.3	The thin-wall approximation	332
6.4	The fate of the false vacuum	334
6.5	Determinants and renormalization	336
6.6	Unanswered questions	339
Appendix 1	How to compute determinants	340
Appendix 2	The double well done doubly well	341
Appendix 3	Finite action is zero measure	344
Appendix 4	Only winding number survives	345
Appendix 5	No wrong-chirality solutions	347
	Notes and references	348
8	1/N
1	Introduction	351
2	Vector representations, or, soluble models	352
2.1	[phi][superscript 4] theory (half-way)	352
2.2	The Gross-Neveu model	358
2.3	The CP[superscript N - 1] model	362
3	Adjoint representations, or, chromodynamics	368
3.1	The double-line representation and the dominance of planar graphs	368
3.2	Topology and phenomenology	373
3.3	The 't Hooft model	378
3.4	Witten's theory of baryons	386
3.5	The master field	391
3.6	Restrospect and prospect	396
Appendix 1	The Euler characteristic	397
Appendix 2	The 't Hooft equations	398
Appendix 3	U(N) as an approximation to SU(N)	400
	Notes and references	401