CONTENTS -- Preface -- Introduction -- Chapter 1 Gauge Invariance and Interactions -- Remembering Robert Mills C. N. Yang -- Quantizing Gauge Field Theories -- Chapter 2 The Space of Gauge Fields: Its Structure and Geometry Bryce DeWitt -- Introduction -- Notation -- Gauge invariance -- Gauge transformations. The gauge group -- The structure of -- A natural metric on -- A family of natural connections on -- Fibre-adapted coordinate patches -- A metric for /g. Vilkovisky's connection -- The functional integral for "in-out" amplitudes -- Properties of the Jacobian J[ ] -- A special choice for [ ] -- The ghost -- One-loop approximation for the measure [ ] -- Geodesic normal fields -- References -- Ghosts for Physicists -- PREFACE -- Chapter 3 Perturbation Theory for Gauge-Invariant Fields V. N. Popov and L. D. Faddeev -- 1. Introduction -- 2. Quantization by Path Integral -- 3. Quantum Electrodynamics -- 4. The Yang-Mills Fields -- 5. The Theory of Gravitation -- References -- Breaking the Symmetry -- Chapter 4 Broken Symmetry and Yang-Mills Theory Francois Englert -- 1. Introduction -- 2. Spontaneous Broken Global Symmetry -- 2.1. Broken symmetry in statistical physics -- 2.2. Broken continuous symmetry in field theory -- 3. The Asymmetric Yang-Mills Phase -- 3.1. Global to local symmetry -- 3.2. The field theoretic approach -- α) Breaking by scalar fields -- β) Dynamical symmetry breaking -- 3.3. The equation of motion formulation -- 3.4. The renormalization issue -- 4. The Unification Paradigm -- 5. Further Developments: Conceptual Issues -- 5.1. Monopoles, electromagnetic duality, confinement -- 5.2. Fermions from bosons -- 5.3. A geometrical view on the BEH mechanism -- Acknowledgments -- References -- Towards the Standard Model -- Chapter 5 The Making of the Standard Model Steven Weinberg -- References -- Renormalization.

Chapter 6 Renormalization Gerardus 't Hooft -- 1. An Introduction -- 1.1. The largest time equation -- 1.2. Dressed propagators -- 1.3. Wave functions for in- and outgoing particles -- 1.4. Dispersion relations -- 1.5. Renormalization -- 1.6. Regularization schemes -- 1.7. Pauli-Villars regularization -- 1.8. Dimensional regularization -- 1.9. Equivalence of regularization schemes -- 1.10. Renormalization of gauge theories -- References -- Chapter 7 From Koszul Complexes to Gauge Fixing R. Stora -- 1. Introduction -- 2. The Free Maxwell Field -- 3. Free Massive Antisymmetric Tensor Fields -- 4. Conclusion and Outlook -- Appendix A: Vector Spaces Subspaces Quotient Spaces and Their Graded Symmetric Tensor Algebras -- A.1. Invariance -- A.2. Gauge Fixing -- A.3. Invariance versus Gauge Fixing -- A.4. Invariance and Gauge Fixing -- Appendix B -- Acknowledgements -- References -- Chapter 8 The Renormalization Content of Slavnov-Taylor Identities Carlo Becchi -- 1. Introduction -- 2. The Model -- 3. The Renormalization Conditions -- 4. Conclusions -- Acknowledgments -- References -- Anomalies -- Chapter 9 Anomalies to All Orders Stephen L. Adler -- 1. Introduction -- 2. Non-Renormalization of the Chiral Anomaly -- 2.1. Setting the stage -- 2.2. Anomaly non-renormalization -- 2.3. Point splitting calculations of the anomaly. Did Schwinger discover the anomaly? -- 2.4. The non-Abelian anomaly, its non-renormalization and geometric interpretation -- 3. Trace Anomalies and Their All Orders Calculation -- 4. Epilogue: Chiral and Trace Anomalies in Supersymmetric Yang-Mills Theories -- Acknowledgements -- Appendix A. Some Details of the Zumino and Schwinger Calculations -- References -- Chapter 10 Fifty Years of Yang-Mills Theory and Our Moments of Triumph Roman Jackiw -- 1. Gauge Theory and Quantum Anomalies -- 2. Mathematical Connections.

3. Gauge Field Dynamics Other than Yang-Mills -- 4. Gauge Formalism for General Relativity Variables -- A. Christoffel connection as a gauge potential -- B. Gravitational Chern-Simons term from gauge theory Chern-Simons term -- C. Coordinate transformations in general relativity and gauge theory -- (i) Response to changes in coordinates -- (ii) Invariant fields and constants of motion -- References -- Asymptotic Freedom -- Chapter 11 Yang-Mills Theory In, Beyond, and Behind Observed Reality Frank Wilczek -- 1. Yang-Mills Theory, Directly in the Phenomena -- 2. From Running Coupling to Quantitative Unification to Supersymmetry -- 3. From Dark Momentum to Gluonization to Higgs and Dark Matter -- 4. From Asymptotic Simplicity to Quark-Gluon Plasma to Quark-Hadron Continuity -- 5. Consistency, Minimalism, Explanatory Method -- 5.1. Gauge symmetry and consistency -- 5.2. Minimal coupling, predictivity, and more consistency -- 5.3. Patterns of explanation -- References -- Magnetic Monopoles -- Chapter 12 To Be Or Not to Be? F. Alexander Bais -- 1. Introduction -- 2. Monopoles in Yang-Mills-Higgs Theories -- 2.1. The BPS limit -- 2.2. The 't Hooft-Polyakov monopole -- 3. Charge Quantization in Non-Abelian Theories -- 3.1. Topological charges -- 3.2. Charge instabilities, a mode analysis -- 4. Cheshire Charge and Core Instabilities -- 5. Physics from Moduli Space -- 5.1. Topologically non-trivial gauge transformations -- 5.2. The Witten effect: CP violation in the monopole sector -- 5.3. Isodoublet modes: Spin from isospin -- 5.4. Fermions from bosons -- 5.5. The Rubakov-Callan effect: Monopole induced catalysis of baryon number violation -- 6. Supersymmetric Monopoles -- 6.1. The supersymmetry algebra with central charge -- 6.2. Duality regained -- 6.3. SL(2, Z) duality -- 6.4. The Seiberg-Witten results: The N = 2 vacuum structure.

7. Beyond SU(2): Gauge Groups and Integrable Systems -- 7.1. Spherical symmetry: Toda systems -- 7.2. Axial symmetry: Non-linear sigma models -- 7.3. The Nahm equations -- Conclusions and Outlook -- Acknowledgements -- References -- Quark Confinement and Strings -- Chapter 13 Confinement and Liberation A. M. Polyakov -- References -- Fixing in Gauge Condition Non-Perturbatively -- Chapter 14 Non-Perturbative Aspects of Gauge Fixing Pierre van Baal -- 1. Introduction -- 2. The Hamiltonian Setting -- 3. Gauge Fixing -- 4. Low-Lying Spectrum -- 5. Dimension-Two Condensate -- 6. Epilogue -- Acknowledgments -- References -- The Lattice -- Chapter 15 Yang-Mills Fields and the Lattice Michael Creutz -- 1. Introduction -- 2. Before the Lattice -- 3. The Birth of QCD -- 4. The 70's Revolution -- 5. The Lattice -- 6. Gauge Fields and Phases -- 7. The Wilson Action -- 8. A Paucity of Parameters -- 9. Numerical Simulation -- 10. Quarks and Random Numbers -- 11. Chirality, Anomalies, and the Lattice -- 12. Concluding Remarks -- Acknowledgments -- References -- Fermions on the Lattice -- Chapter 16 Chiral Symmetry on the Lattice P. Hasenfratz -- 1. Introduction -- 2. QCD on the Lattice -- 3. Doublers, Chiral Symmetry and a No-Go Theorem -- 4. The Ginsparg-Wilson Relation -- 5. Improvement Programs -- 5.1. Symanzik improvement -- 5.2. Classically perfect actions -- 6. Domain-Wall and Overlap Fermions -- 7. The Fixed-Point Dirac Operator in QCD Solves the Ginsparg-Wilson Equation -- 8. Index Theorem on the Lattice -- 9. Ginsparg-Wilson Fermions from the Domain-Wall Construction -- 10. Exact Chiral Symmetry Transformation -- 11. Left- and Right-Handed Fermions, the Mass Term and the Parameter -- 12. Chiral Gauge Theories -- 13. Closing Remarks -- Acknowledgements -- References -- Confrontation with Experiment.

Chapter 17 Fifty Years of Yang-Mills Theories: A Phenomenological Point of View Alvaro De Rujula -- 1. Introduction -- 2. Neutral Currents -- 3. QCD and Charm in Their Early Years -- 3.1. A deep inelastic dawning -- 3.2. The November revolution -- 3.3. A first year of lean cows -- 3.4. Good news at last -- 3.5. Yet another year of lank cows -- 3.6. Charm is found -- 3.7. Back to the deep inelastic domain -- 3.8. Seeing is believing -- 4. Grand Unification and Proton Decay -- 5. Gravity -- 6. Afterword -- References -- Supersymmetry and Supergravity -- Chapter 18 Supergravity as a Yang-Mills Theory Peter van Nieuwenhuizen -- 1. Introduction -- 2. Basics of Supergravity -- 3. Simple (N = 1) Supergravity -- 4. Covariant Quantization -- 5. Conformal Simple Supergravity -- 6. Constraints in Superspace -- References -- Physics of the 21st Century -- Chapter 19 Gauge/String Duality for Weak Coupling Edward Witten -- 1. Introduction -- 2. Helicity Amplitudes and Twistor Space -- 2.1. Spinors -- 2.2. Helicity amplitudes -- 2.3. Maximally helicity violating amplitudes -- 2.4. Conformal invariance -- 2.5. Fourier transform to twistor space -- Analog with Euclidean signature -- Pairing with external wavefunctions -- Physical interpretation of twistor space amplitudes -- 2.6. Extension with N = 4 supersymmetry -- Transform to super-twistor space -- 3. MHV Amplitudes in Twistor Space -- 3.1 MHV amplitudes -- 3.2. Supersymmetric extension -- References.