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Aspects of Symmetry

Book Description

For almost two decades, Sidney Coleman has been giving review lectures on frontier topics in theoretical high-energy physics at the International School of Subnuclear Physics held each year at Erice, Sicily. This volume is a collection of some of the best of these lectures. To this day they have few rivals for clarity of exposition and depth of insight. Although very popular when first published, many of the lectures have been difficult to obtain recently. Graduate students and professionals in high-energy physics will welcome this collection by a master of the field.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. Acknowledgements
  7. 1 An introduction to unitary symmetry
    1. 1 The search for higher symmetries
      1. 1.1 The eight-baryon puzzle
      2. 1.2 The elimination of G0
    2. 2 SU(3) and its representations
      1. 2.1 The representations of SU(n)
      2. 2.2 The representations of SU(2)
      3. 2.3 The representations of SU(3)
      4. 2.4 Dimensions of the IRs
      5. 2.5 Isospin and hypercharge
      6. 2.6 Isospin–hypercharge decompositions
      7. 2.7 The Clebsch–Gordan series
      8. 2.8 Some theorems
      9. 2.9 Invariant couplings
      10. 2.10 The problem of Cartesian components
      11. 2.11 SU(2) again
      12. 2.12 SU(3) octets: trilinear couplings
      13. 2.13 SU(3) octets: quadrilinear couplings
      14. 2.14 A mixed notation
    3. 3 Applications
      1. 3.1 Electromagnetism
      2. 3.2 Magnetic moments: baryons
      3. 3.3 Electromagnetic mass splittings
      4. 3.4 Electromagnetic properties of the decuplet
      5. 3.5 The medium-strong interactions
    4. 4 Ideas of octet enhancement
    5. Bibliography
  8. 2 Soft pions
    1. 1 The reduction formula
    2. 2 The weak interactions: first principles
    3. 3 The Goldberger-Treiman relation and a first glance at PCAC
    4. 4 A hard look at PCAC
    5. 5 The gradient-coupling model
    6. 6 Adler’s rule for the emission of one soft pion
    7. 7 Current commutators
      1. 7.1 Vector-vector commutators
      2. 7.2 Vector-axial commutators
      3. 7.3 Axial-axial commutators
    8. 8 The Weinberg-Tomozawa formula and the Adler-Weisberger relation
    9. 9 Pion–pion scattering à la Weinberg
    10. 10 Kaon decays
    11. Appendix 1: Notational conventions
    12. Appendix 2: No-renormalization theorem
    13. Appendix 3: Threshold S-matrix and threshold scattering lengths
    14. Bibliography
  9. 3 Dilatations
    1. 1 Introduction
    2. 2 The formal theory of broken scale invariance
      1. 2.1 Symmetries, currents, and Ward identities
      2. 2.2 Scale transformations and scale dimensions
      3. 2.3 More about the scale current and a quick look at the conformal group
      4. 2.4 Hidden scale invariance
    3. 3 The death of scale invariance
      1. 3.1 Some definitions and technical details
      2. 3.2 A disaster in the deep Euclidean region
      3. 3.3 Anomalous dimensions and other anomalies
      4. 3.4 The last anomalies: the Callan-Symanzik equations
    4. 4 The resurrection of scale invariance
      1. 4.1 The renormalization group equations and their solution
      2. 4.2 The return of scaling in the deep Euclidean region
      3. 4.3 Scaling and the operator product expansion
    5. 5 Conclusions and questions
    6. Notes and references
  10. 4 Renormalization and symmetry: a review for non-specialists
    1. 1 Introduction
    2. 2 Bogoliubov’s method and Hepp’s theorem
    3. 3 Renormalizable and non-renormalizable interactions
    4. 4 Symmetry and symmetry-breaking: Symanzik’s rule
    5. 5 Symmetry and symmetry-breaking: currents
    6. Notes and references
  11. 5 Secret symmetry: an introduction to spontaneous symmetry breakdown and gauge fields
    1. 1 Introduction
    2. 2 Secret symmetries in classical field theory
      1. 2.1 The idea of spontaneous symmetry breakdown
      2. 2.2 Goldstone bosons in an Abelian model
      3. 2.3 Goldstone bosons in the general case
      4. 2.4 The Higgs phenomenon in the Abelian model
      5. 2.5 Yang-Mills fields and the Higgs phenomenon in the general case
      6. 2.6 Summary and remarks
    3. 3 Secret renormalizability
      1. 3.1 The order of the arguments
      2. 3.2 Renormalization reviewed
      3. 3.3 Functional methods and the effective potential
      4. 3.4 The loop expansion
      5. 3.5 A sample computation
      6. 3.6 The most important part of this lecture
      7. 3.7 The physical meaning of the effective potential
      8. 3.8 Accidental symmetry and related phenomena
      9. 3.9 An alternative method of computation
    4. 4 Functional integration (vulgarized)
      1. 4.1 Integration over infinite-dimensional spaces
      2. 4.2 Functional integrals and generating functionals
      3. 4.3 Feynman rules
      4. 4.4 Derivative interactions
      5. 4.5 Fermi fields
      6. 4.6 Ghost fields
    5. 5 The Feynman rules for gauge field theories
      1. 5.1 Troubles with gauge in variance
      2. 5.2 The Faddeev-Popov Ansatz
      3. 5.3 The application of the Ansatz
      4. 5.4 Justification of the Ansatz
      5. 5.5 Concluding remarks
    6. 6 Asymptotic freedom
      1. 6.1 Operator products and deep inelastic electroproduction
      2. 6.2 Massless field theories and the renormalization group
      3. 6.3 Exact and approximate solutions of the renormalization group equations
      4. 6.4 Asymptotic freedom
      5. 6.5 No conclusions
    7. Appendix: One-loop effective potential in the general case
    8. Notes and references
  12. 6 Classical lumps and their quantum descendants
    1. 1 Introduction
    2. 2 Simple examples and their properties
      1. 2.1 Some time-independent lumps in one space dimension
      2. 2.2 Small oscillations and stability
      3. 2.3 Lumps are like particles (almost)
      4. 2.4 More dimensions and a discouraging theorem
    3. 3 Topological conservation laws
      1. 3.1 The basic idea and the main results
      2. 3.2 Gauge field theories revisited
      3. 3.3 Topological conservation laws, or, homotopy classes
      4. 3.4 Three examples in two spatial dimensions
      5. 3.5 Three examples in three dimensions
      6. 3.6 Patching together distant solutions, or, homotopy groups
      7. 3.7 Abelian and non-Abelian magnetic monopoles, or, π2(G/H) as a subgroup of π1(H)
    4. 4 Quantum lumps
      1. 4.1 The nature of the classical limit
      2. 4.2 Time-independent lumps: power-series expansion
      3. 4.3 Time-independent lumps: coherent-state variational method
      4. 4.4 Periodic lumps: the old quantum theory and the DHN formula
    5. 5 A very special system
      1. 5.1 A curious equivalence
      2. 5.2 The secret of the soliton
      3. 5.3 Qualitative and quantitative knowledge
      4. 5.4 Some opinions
    6. Appendix 1: A three-dimensional scalar theory with non-dissipative solutions
    7. Appendix 2: A theorem on gauge fields
    8. Appendix 3: A trivial extension
    9. Appendix 4: Looking for solutions
    10. Appendix 5: Singular and non-singular gauge fields
    11. Notes and references
  13. 7 The uses of instantons
    1. 1 Introduction
    2. 2 Instantons and bounces in particle mechanics
      1. 2.1 Euclidean functional integrals
      2. 2.2 The double well and instantons
      3. 2.3 Periodic potentials
      4. 2.4 Unstable states and bounces
    3. 3 The vacuum structure of gauge field theories
      1. 3.1 Old stuff
      2. 3.2 The winding number
      3. 3.3 Many vacua
      4. 3.4 Instantons: generalities
      5. 3.5 Instantons: particulars
      6. 3.6 The evaluation of the determinant and an infrared embarrassment
    4. 4 The Abelian Higgs model in 1+1 dimensions
    5. 5 ’t Hooft’s solution of the U(l) problem
      1. 5.1 The mystery of the missing meson
      2. 5.2 Preliminaries: Euclidean Fermi fields
      3. 5.3 Preliminaries: chiral Ward identities
      4. 5.4 QCD (baby version)
      5. 5.5 QCD (the real thing)
      6. 5.6 Miscellany
    6. 6 The fate of the false vacuum
      1. 6.1 Unstable vacua
      2. 6.2 The bounce
      3. 6.3 The thin-wall approximation
      4. 6.4 The fate of the false vacuum
      5. 6.5 Determinants and renormalization
      6. 6.6 Unanswered questions
    7. Appendix 1: How to compute determinants
    8. Appendix 2: The double well done doubly well
    9. Appendix 3: Finite action is zero measure
    10. Appendix 4: Only winding number survives
    11. Appendix 5: No wrong-chirality solutions
    12. Notes and references
  14. 8 1/N
    1. 1 Introduction
    2. 2 Vector representations, or, soluble models
      1. 2.1 ϕ4 theory (half-way)
      2. 2.2 The Gross-Neveu model
      3. 2.3 The CPN-1 model
    3. 3 Adjoint representations, or, chromodynamics
      1. 3.1 The double-line representation and the dominance of planar graphs
      2. 3.2 Topology and phenomenology
      3. 3.3 The ’t Hooft model
      4. 3.4 Witten’s theory of baryons
      5. 3.5 The master field
      6. 3.6 Restrospect and prospect
    4. Appendix 1: The Euler characteristic
    5. Appendix 2: The ’t Hooft equations
    6. Appendix 3: U(N) as an approximation to SU(N)
    7. Notes and references