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Quantum Field Theory, Second Edition

Book Description

This book is a modern pedagogic introduction to the ideas and techniques of quantum field theory. After a brief overview of particle physics and a survey of relativistic wave equations and Lagrangian methods, the quantum theory of scalar and spinor fields, and then of gauge fields, is developed. The emphasis throughout is on functional methods, which have played a large part in modern field theory. The book concludes with a brief survey of 'topological' objects in field theory and, new to this edition, a chapter devoted to supersymmetry.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Dedication
  5. Contents
  6. Preface to the first edition
  7. Preface to the second edition
  8. 1 Introduction: synopsis of particle physics
    1. 1.1 Quantum field theory
    2. 1.2 Gravitation
    3. 1.3 Strong interactions
    4. 1.4 Weak interactions
    5. 1.5 Leptonic quantum numbers
    6. 1.6 Hadronic quantum numbers
    7. 1.7 Resonances
    8. 1.8 The quark model
    9. 1.9 SU(2), SU(3), SU(4), . . .
    10. 1.10 Dynamical evidence for quarks
    11. 1.11 Colour
    12. 1.12 QCD
    13. 1.13 Weak interactions
    14. Guide to further reading
  9. 2 Single-particle relativistic wave equations
    1. 2.1 Relativistic notation
    2. 2.2 Klein–Gordon equation
    3. 2.3 Dirac equation
      1. SU(2) and the rotation group
      2. SU(2, C) and the Lorentz group
    4. 2.4 Prediction of antiparticles
    5. 2.5 Construction of Dirac spinors: algebra of y matrices
    6. 2.6 Non-relativistic limit and the electron magnetic moment
    7. 2.7 The relevance of the Poincare group: spin operators and the zero mass limit
    8. 2.8 Maxwell and Proca equations
    9. 2.9 Maxwell’s equations and differential geometry
    10. Summary
    11. Guide to further reading
  10. 3 Lagrangian formulation, symmetries and gauge fields
    1. 3.1 Lagrangian formulation of particle mechanics
    2. 3.2 The real scalar field: variational principle and Noether’s theorem
    3. 3.3 Complex scalar fields and the electromagnetic field
    4. 3.4 Topology and the vacuum: the Bohm–Aharonov effect
    5. 3.5 The Yang–Mills field
    6. 3.6 The geometry of gauge fields
    7. Summary
    8. Guide to further reading
  11. 4 Canonical quantisation and particle interpretation
    1. 4.1 The real Klein–Gordon field
    2. 4.2 The complex Klein–Gordon field
    3. 4.3 The Dirac field
    4. 4.4 The electromagnetic field
      1. Radiation gauge quantisation
      2. Lorentz gauge quantisation
    5. 4.5 The massive vector field
    6. Summary
    7. Guide to further reading
  12. 5 Path integrals and quantum mechanics
    1. 5.1 Path-integral formulation of quantum mechanics
    2. 5.2 Perturbation theory and the S matrix
    3. 5.3 Coulomb scattering
    4. 5.4 Functional calculus: differentiation
    5. 5.5 Further properties of path integrals
    6. Appendix: some useful integrals
    7. Summary
    8. Guide to further reading
  13. 6 Path-integral quantisation and Feynman rules: scalar and spinor fields
    1. 6.1 Generating functional for scalar fields
    2. 6.2 Functional integration
    3. 6.3 Free particle Green’s functions
    4. 6.4 Generating functionals for interacting fields
    5. 6.5 ϕ4 theory
      1. Generating functional
      2. 2-point function
      3. 4-point function
    6. 6.6 Generating functional for connected diagrams
    7. 6.7 Fermions and functional methods
    8. 6.8 The 5 matrix and reduction formula
    9. 6.9 Pion–nucleon scattering amplitude
    10. 6.10 Scattering cross section
    11. Summary
    12. Guide to further reading
  14. 7 Path-integral quantisation: gauge fields
    1. 7.1 Propagators and gauge conditions in QED
      1. Photon propagator – canonical formalism
      2. Photon propagator – path-integral method
      3. Gauge-fixing terms
      4. Propagator for transverse photons
    2. 7.2 Non-Abelian gauge fields and the Faddeev–Popov method
      1. Feynman rules in the Lorentz gauge
      2. Gauge-field propagator in the axial gauge
    3. 7.3 Self-energy operator and vertex function
      1. Geometrical interpretation of the Legendre transformation
      2. Thermodynamic analogy
    4. 7.4 Ward–Takahashi identities in QED
    5. 7.5 Becchi–Rouet–Stora transformation
    6. 7.6 Slavnov–Taylor identities
    7. 7.7 A note on ghosts and unitarity
    8. Summary
    9. Guide to further reading
  15. 8 Spontaneous symmetry breaking and the Weinberg–Salam model
    1. 8.1 What is the vacuum?
    2. 8.2 The Goldstone theorem
    3. 8.3 Spontaneous breaking of gauge symmetries
    4. 8.4 Superconductivity
    5. 8.5 The Weinberg–Salam model
    6. Summary
    7. Guide to further reading
  16. 9 Renormalisation
    1. 9.1 Divergences in ϕ4 theory
      1. Dimensional analysis
    2. 9.2 Dimensional regularisation of ϕ4 theory
      1. Loop expansion
    3. 9.3 Renormalisation of ϕ4 theory
      1. Counter-terms
    4. 9.4 Renormalisation group
    5. 9.5 Divergences and dimensional regularisation of QED
    6. 9.6 1-loop renormalisation of QED
      1. Anomalous magnetic moment of the electron
      2. Asymptotic behaviour
    7. 9.7 Renormalisability of QED
    8. 9.8 Asymptotic freedom of Yang–Mills theories
    9. 9.9 Renormalisation of pure Yang–Mills theories
    10. 9.10 Chiral anomalies
      1. Cancellation of anomalies
    11. 9.11 Renormalisation of Yang–Mills theories with spontaneous symmetry breakdown
      1. ’t Hooft’s gauges
      2. The effective potential
      3. Loop expansion of the effective potential
    12. Appendix A: integration in d dimensions
    13. Appendix B: the gamma function
    14. Summary
    15. Guide to further reading
  17. 10 Topological objects in field theory
    1. 10.1 The sine–Gordon kink
    2. 10.2 Vortex lines
    3. 10.3 The Dirac monopole
    4. 10.4 The ’t Hooft–Polyakov monopole
    5. 10.5 Instantons
      1. Quantum tunnelling, 0-vacua and symmetry breaking
    6. Summary
    7. Guide to further reading
  18. 11 Supersymmetry
    1. 11.1 Introduction
    2. 11.2 Lorentz transformations: Dirac, Weyl and Majorana spinors Some further relations
    3. 11.3 Simple Lagrangian model
      1. Digression: Fierz rearrangement formula
    4. 11.4 Simple Lagrangian model (cont.): closure of relations
      1. Mass term
    5. 11.5 Towards a super-Poincaré algebra
    6. 11.6 Superspace
    7. 11.7 Superfields
      1. Chiral superfield
    8. 11.8 Recovery of the Wess–Zumino model
    9. Appendix: some 2-spinor conventions
    10. Summary
    11. Guide to further reading
  19. References
  20. Index