Acknowledgements An introduction to unitary symmetry The search for higher symmetries p. 1 The eight-baryon puzzle p. 1 The elimination of

Transcription

1 Preface p. xiii Acknowledgements p. xiv An introduction to unitary symmetry The search for higher symmetries p. 1 The eight-baryon puzzle p. 1 The elimination of G[subscript 0] p. 4 SU(3) and its representations p. 5 The representations of SU(n) p. 5 The representations of SU(2) p. 6 The representations of SU(3) p. 7 Dimensions of the IRs p. 8 Isospin and hypercharge p. 9 Isospin-hypercharge decompositions p. 10 The Clebsch-Gordan series p. 12 Some theorems p. 15 Invariant couplings p. 17 The problem of Cartesian components p. 17 SU(2) again p. 18 SU(3) octets: trilinear couplings p. 19 SU(3) OCTETS: QUADRILINEAR COUPLINGS p. 20 A mixed notation p. 21 Applications p. 23 Electromagnetism p. 23 Magnetic moments: baryons p. 24 Electromagnetic mass splittings p. 25 Electromagnetic properties of the decuplet p. 26 The medium-strong interactions p. 26 Ideas of octet enhancement p. 28 Bibliography p. 35 Soft pions The reduction formula p. 36 The weak interactions: first principles p. 40 The Goldberger-Treiman relation and a first glance at PCAC p. 41 A hard look at PCAC p. 42 The gradient-coupling model p. 45 Adler's rule for the emission of one soft pion p. 47 Current commutators p. 50 Vector-vector commutators p. 50 Vector-axial commutators p. 51 Axial-axial commutators p. 51 The Weinberg-Tomozawa formula and the Adler-Weisberger relation p. 52

2 Pion-pion scattering a la Weinberg p. 57 Kaon decays p. 60 Notational conventions p. 63 No-renormalization theorem p. 63 Threshold S-matrix and threshold scattering lengths p. 64 Bibliography p. 65 Dilatations Introduction p. 67 The formal theory of broken scale invariance p. 68 Symmetries, currents, and Ward identities p. 68 Scale transformations and scale dimensions p. 70 More about the scale current and a quick look at the conformal group p. 71 Hidden scale invariance p. 76 The death of scale invariance p. 79 Some definitions and technical details p. 79 A disaster in the deep Euclidean region p. 80 Anomalous dimensions and other anomalies p. 82 The last anomalies: the Callan-Symanzik equations p. 84 The resurrection of scale invariance p. 88 The renormalization group equations and their solution p. 88 The return of scaling in the deep Euclidean region p. 90 Scaling and the operator product expansion p. 93 Conclusions and questions p. 96 Notes and references p. 97 Renormalization and symmetry: a review for non-specialists Introduction p. 99 Bogoliubov's method and Hepp's theorem p. 99 Renormalizable and non-renormalizable interactions p. 104 Symmetry and symmetry-breaking: Symanzik's rule p. 106 Symmetry and symmetry-breaking: currents p. 108 Notes and references p. 111 Secret symmetry: an introduction to spontaneous symmetry breakdown and gauge fields Introduction p. 113 Secret symmetries in classical field theory p. 115 The idea of spontaneous symmetry breakdown p. 115 Goldstone bosons in an Abelian model p. 118 Goldstone bosons in the general case p. 119 The Higgs phenomenon in the Abelian model p. 121 Yang-Mills fields and the Higgs phenomenon in the general case p. 124 Summary and remarks p. 126 Secret renormalizability p. 128

3 The order of the arguments p. 128 Renormalization reviewed p. 128 Functional methods and the effective potential p. 132 The loop expansion p. 135 A sample computation p. 136 The most important part of this lecture p. 138 The physical meaning of the effective potential p. 139 Accidental symmetry and related phenomena p. 142 An alternative method of computation p. 144 Functional integration (vulgarized) p. 145 Integration over infinite-dimensional spaces p. 145 Functional integrals and generating functionals p. 148 Feynman rules p. 152 Derivative interactions p. 154 Fermi fields p. 156 Ghost fields p. 158 The Feynman rules for gauge field theories p. 159 Troubles with gauge invariance p. 159 The Faddeev-Popov Ansatz p. 160 The application of the Ansatz p. 163 Justification of the Ansatz p. 165 Concluding remarks p. 167 Asymptotic freedom p. 169 Operator products and deep inelastic electroproduction p. 169 Massless field theories and the renormalization group p. 171 Exact and approximate solutions of the renormalization group equations p. 174 Asymptotic freedom p. 176 No conclusions p. 179 One-loop effective potential in the general case p. 180 Notes and references p. 182 Classical lumps and their quantum descendants Introduction p. 185 Simple examples and their properties p. 187 Some time-independent lumps in one space dimension p. 187 Small oscillations and stability p. 191 Lumps are like particles (almost) p. 192 More dimensions and a discouraging theorem p. 194 Topological conservation laws p. 195 The basic idea and the main results p. 195 Gauge field theories revisited p. 198 Topological conservation laws, or, homotopy classes p. 202

4 Three examples in two spatial dimensions p. 205 Three examples in three dimensions p. 208 Patching together distant solutions, or, homotopy groups p. 209 Abelian and non-abelian magnetic monopoles, or, [Pi][subscript 2](G/H) as a subgroup of [Pi][subscript 1](H) p. 215 Quantum lumps p. 223 The nature of the classical limit p. 223 Time-independent lumps: power-series expansion p. 225 Time-independent lumps: coherent-state variational method p. 232 Periodic lumps: the old quantum theory and the DHN formula p. 239 A very special system p. 246 A curious equivalence p. 246 The secret of the soliton p. 250 Qualitative and quantitative knowledge p. 252 Some opinions p. 253 A three-dimensional scalar theory with non-dissipative solutions p. 254 A theorem on gauge fields p. 256 A trivial extension p. 257 Looking for solutions p. 257 Singular and non-singular gauge fields p. 259 Notes and references p. 262 The uses of instantons Introduction p. 265 Instantons and bounces in particle mechanics p. 268 Euclidean functional integrals p. 268 The double well and instantons p. 270 Periodic potentials p. 277 Unstable states and bounces p. 278 The vacuum structure of gauge field theories p. 282 Old stuff p. 282 The winding number p. 284 Many vacua p. 291 Instantons: generalities p. 295 Instantons: particulars p. 297 The evaluation of the determinant and an infrared embarrassment p. 300 The Abelian Higgs model in dimensions p. 302 't Hooft's solution of the U(1) problem p. 307 The mystery of the missing meson p. 307 Preliminaries: Euclidean Fermi fields p. 311 Preliminaries: chiral Ward identities p. 314 QCD (baby version) p. 316 QCD (the real thing) p. 323

5 Miscellany p. 324 The fate of the false vacuum p. 327 Unstable vacua p. 327 The bounce p. 329 The thin-wall approximation p. 332 The fate of the false vacuum p. 334 Determinants and renormalization p. 336 Unanswered questions p. 339 How to compute determinants p. 340 The double well done doubly well p. 341 Finite action is zero measure p. 344 Only winding number survives p. 345 No wrong-chirality solutions p. 347 Notes and references p /N Introduction p. 351 Vector representations, or, soluble models p. 352 [phi][superscript 4] theory (half-way) p. 352 The Gross-Neveu model p. 358 The CP[superscript N - 1] model p. 362 Adjoint representations, or, chromodynamics p. 368 The double-line representation and the dominance of planar graphs p. 368 Topology and phenomenology p. 373 The 't Hooft model p. 378 Witten's theory of baryons p. 386 The master field p. 391 Restrospect and prospect p. 396 The Euler characteristic p. 397 The 't Hooft equations p. 398 U(N) as an approximation to SU(N) p. 400 Notes and references p. 401 Table of Contents provided by Blackwell's Book Services and R.R. Bowker. Used with permission.