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Non-Perturbative Field Theory

Book Description

Providing a new perspective on quantum field theory, this book gives a pedagogical and up-to-date exposition of non-perturbative methods in relativistic quantum field theory and introduces the reader to modern research work in theoretical physics. It describes in detail non-perturbative methods in quantum field theory, and explores two- dimensional and four- dimensional gauge dynamics using those methods. The book concludes with a summary emphasizing the interplay between two- and four- dimensional gauge theories. Aimed at graduate students and researchers, this book covers topics from two-dimensional conformal symmetry, affine Lie algebras, solitons, integrable models, bosonization, and 't Hooft model, to four-dimensional conformal invariance, integrability, large N expansion, Skyrme model, monopoles and instantons. Applications, first to simple field theories and gauge dynamics in two dimensions, and then to gauge theories in four dimensions and quantum chromodynamics (QCD) in particular, are thoroughly described.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Acknowledgements
  9. Part I: Non-Perturbative Methods in Two-Dimensional Field Theory
    1. 1. From massless free scalar field to conformal field theories
      1. 1.1 Complex geometry
      2. 1.2 Free massless scalar field
      3. 1.3 Symmetries of the classical action
      4. 1.4 Mode expansion
      5. 1.5 Noether currents and charges
      6. 1.6 Canonical quantization
      7. 1.7 Radial quantization
      8. 1.8 Operator product expansion
      9. 1.9 Path integral quantization
      10. 1.10 Affine current algebra
      11. 1.11 Virasoro algebra
    2. 2. Conformal field theory
      1. 2.1 Conformal symmetry in two dimensions
      2. 2.2 Primary fields
      3. 2.3 Conformal properties of the energy-momentum tensor
      4. 2.4 Virasoro algebra for CFT
      5. 2.5 Descendant operators
      6. 2.6 Hilbert space of states
      7. 2.7 Unitary CFT and Kac determinant
      8. 2.8 Characters
      9. 2.9 Correlators and the conformal Ward identity
      10. 2.10 Crossing symmetry, duality and bootstrap
      11. 2.11 Verlinde’s formula
      12. 2.12 Free Majorana fermions – an example of a CFT
      13. 2.13 The Ising model – the m = 3 unitary minimal model
    3. 3. Theories invariant under affine current algebras
      1. 3.1 Simple finite-dimensional Lie algebras
      2. 3.2 Affine current algebra
      3. 3.3 Current OPEs and the Sugawara construction
      4. 3.4 Primary fields
      5. 3.5 ALA characters
      6. 3.6 Correlators, null vectors and the Knizhnik–Zamolodchikov equation
      7. 3.7 Free fermion realization
      8. 3.8 Free Dirac fermions and the U(N)
    4. 4. Wess–Zumino–Witten model and coset models
      1. 4.1 From free massless scalar theory to the WZW model
      2. 4.2 Perturbative conformal invariance
      3. 4.3 ALA, Sugawara construction and the Virasoro algebra
      4. 4.4 Correlation functions of primary fields
      5. 4.5 WZW models with boundaries – D branes
      6. 4.6 G/H coset models
      7. 4.7 G/G coset models
    5. 5. Solitons and two-dimensional integrable models
      1. 5.1 Introduction
      2. 5.2 From the theory of a massive free scalar field to integrable models
      3. 5.3 Classical solitons
      4. 5.4 Breathers or “doublets”
      5. 5.5 Quantum solitons
      6. 5.6 Integrability and factorized S-matrix
      7. 5.7 Yang–Baxter equations
      8. 5.8 The general solution of the S-matrix
      9. 5.9 From conformal field theories to integrable models
      10. 5.10 Conserved charges and classical integrability
      11. 5.11 Multilocal conserved charges
      12. 5.12 Quantum integrable charges in the O(N) model
      13. 5.13 Non-local charges and quantum groups
      14. 5.14 Integrable spin chain models and the algebraic Bethe ansatz
      15. 5.15 The continuum thermodynamic Bethe ansatz
    6. 6. Bosonization
      1. 6.1 Abelian bosonization
      2. 6.2 Duality between the Thirring model and the sine-Gordon model
      3. 6.3 Witten’s non-abelian bosonization
      4. 6.4 Chiral bosons
      5. 6.5 Bosonization of systems of operators of high conformal dimension
    7. 7. The large N limit of two-dimensional models
      1. 7.1 Introduction
      2. 7.2 The Gross–Neveu model
      3. 7.3 The CPN–1 model
  10. Part II: Two-Dimensional Non-Perturbative Gauge Dynamics
    1. 8. Gauge theories in two dimensions – basics
      1. 8.1 Pure Maxwell theory
      2. 8.2 QED2 – Schwinger’s model
      3. 8.3 Yang–Mills theory
      4. 8.4 Quantum chromodynamics
    2. 9. Bosonized gauge theories
      1. 9.1 QED2 – The massive Schwinger model
      2. 9.2 Abelian bosonization of flavored QCD2
      3. 9.3 Non-abelian bosonization of QCD2
    3. 10. The ’t Hooft solution of 2d QCD
      1. 10.1 Scattering of mesons
      2. 10.2 Higher 1/N corrections
    4. 11. Mesonic spectrum from current algebra
      1. 11.1 Introduction
      2. 11.2 Universality of conformal field theories coupled to YM2
      3. 11.3 Mesonic spectra of two-current states
      4. 11.4 The adjoint vacuum and its one-current state
    5. 12. DLCQ and the spectra of QCD with fundamental and adjoint fermions
      1. 12.1 Discretized light-cone quantization
      2. 12.2 Application of DLCQ to QCD2 with fundamental fermions
      3. 12.3 The spectrum of QCD2 with adjoint fermions
    6. 13. The baryonic spectrum of multiflavor QCD2 in the strong coupling limit
      1. 13.1 The strong coupling limit
      2. 13.2 Classical soliton solutions
      3. 13.3 Semi-classical quantization and the baryons
      4. 13.4 The baryonic spectrum
      5. 13.5 Quark flavor content of the baryons
      6. 13.6 Multibaryons
      7. 13.7 States, wave functions and binding energies
      8. 13.8 Meson-baryon scattering
    7. 14. Confinement versus screening
      1. 14.1 The string tension of the massive Schwinger model
      2. 14.2 The Schwinger model in bosonic form
      3. 14.3 Beyond the small mass abelian string tension
      4. 14.4 Correction to the leading long distance abelian potential
      5. 14.5 Finite temperature
      6. 14.6 Two-dimensional QCD
      7. 14.7 Symmetric and antisymmetric representations
    8. 15. QCD2, coset models and BRST quantization
      1. 15.1 Introduction
      2. 15.2 The action
      3. 15.3 Two-dimensional Yang–Mills theory
      4. 15.4 Schwinger model revisited
      5. 15.5 Back to the YM theory
      6. 15.6 An alternative formulation
      7. 15.7 The resolution of the puzzle
      8. 15.8 On bosonized QCD2
      9. 15.9 Summary and discussion
    9. 16. Generalized Yang–Mills theory on a Riemann surface
      1. 16.1 Introduction
      2. 16.2 The partition function of the YM2 theory
      3. 16.3 The partition function of gYM2 theories
      4. 16.4 Loop averages in the generalized case
      5. 16.5 Stringy YM2 theory
      6. 16.6 Toward the stringy generalized YM2
      7. 16.7 Examples
      8. 16.8 Summary
  11. Part III: From Two to Four Dimensions
    1. 17. Conformal invariance in four-dimensional field theories and in QCD
      1. 17.1 Conformal symmetry algebra in four dimensions
      2. 17.2 Conformal invariance of fields, Noether currents and conservation laws
      3. 17.3 Collinear and transverse conformal transformations of fields
      4. 17.4 Collinear primary fields and descendants
      5. 17.5 Conformal operator product expansion
      6. 17.6 Conformal Ward identities
      7. 17.7 Conformal invariance and QCD4
    2. 18. Integrability in four-dimensional gauge dynamics
      1. 18.1 Integrability of large N four-dimensional N = 4 SYM
      2. 18.2 High energy scattering and integrability
    3. 19. Large N methods in QCD4
      1. 19.1 Large N QCD in four dimensions
      2. 19.2 Meson phenomenology
      3. 19.3 Baryons in the large N expansion
      4. 19.4 Scattering processes
    4. 20. From 2d bosonized baryons to 4d Skyrmions
      1. 20.1 Introduction
      2. 20.2 The Skyrme action
      3. 20.3 The baryon as a Skyrmion
      4. 20.4 The Skyrme model for Nf = 3
    5. 21. From two-dimensional solitons to four-dimensional magnetic monopoles
      1. 21.1 Introduction
      2. 21.2 The Yang–Mills Higgs theory – basics
      3. 21.3 Topological solitons and magnetic monopoles
      4. 21.4 The ’t Hooft–Polyakov magnetic monopole solution
      5. 21.5 Charge quantization
      6. 21.6 Zero modes, time-dependent solutions and dyons
      7. 21.7 BPS monopoles and dyons
      8. 21.8 Montonen Olive duality
      9. 21.9 Nahm construction of multimonopole solutions
      10. 21.10 Moduli space of monopoles
    6. 22. Instantons of QCD
      1. 22.1 The basic properties of the instanton
      2. 22.2 The ADHM construction of instantons
      3. 22.3 On the moduli space of instantons
      4. 22.4 Instantons and tunneling between the vacua of the YM theory
      5. 22.5 Instantons, theta vacua and the UA (1) anomaly
    7. 23. Summary, conclusions and outlook
      1. 23.1 General
      2. 23.2 Conformal invariance
      3. 23.3 Integrability
      4. 23.4 Bosonization
      5. 23.5 Topological field configurations
      6. 23.6 Confinement versus screening
      7. 23.7 Hadronic phenomenology of two dimensions versus four dimensions
      8. 23.8 Outlook
  12. References
  13. Index