Introduction to Gauge Field Theories (Theoretical and Mathematical Physics)

In recent years, gauge fields have attracted much attention in elementary par ticle physics. The reason is that great progress has been achieved in solving a number of important problems of field theory and elementary particle physics by means of the quantum theory of gauge fields. This refers, in particular, to constructing unified gauge models and theory of strong interactions between the elementary particles. This book expounds the fundamentals of the quantum theory of gauge fields and its application for constructing unified gauge models and the theory of strong interactions. In writing the book, the authors’ aim was three-fold: firstly, to outline the basic ideas underlying the unified gauge models and the theory of strong inter actions; secondly, to discuss the major unified gauge models, the theory of strong interactions and their experimental implications; and, thirdly, to acquaint the reader with a rather special mathematical approach (path-in tegral method) which has proved to be well suited for constructing the quantum theory of gauge fields. Gauge fields are a vigorously developing area. In this book, we have select ed for presentation the more or less traditional and commonly accepted mate rial. There also exist a number of different approaches which are presently being developed. The most important of them are touched upon in the Conclusion. Units and Notation.- I Invariant Lagrangians.- 1. Global Invariance.- 1.1 The Global Lorentz Group. Relativistic Invariance.- 1.1.1 The Lorentz Group.- 1.1.2 The Lie Algebra of the Lorentz Group.- 1.1.3 Representations of the Lorentz Group.- 1.1.4 Reducible and Irreducible Representations.- 1.1.5 Relativistically Invariant Quantities.- 1.1.6 The Lagrangian Formalism.- 1.1.7 The Hamiltonian Formalism.- 1.1.8 The Operator Form of the Quantum Field Theory.- 1.2 Global Groups of Internal Symmetry. Unitary Symmetry.- 1.2.1 Internal Symmetry Properties.- 1.2.2 Condition for the Invariance of the Lagrangian.- 1.2.3 Classification of Groups.- 2. Local (Gauge) Invariance.- 2.1 Locally (Gauge-) Invariant Lagrangians.- 2.1.1 The Group of Local Transformations.- 2.1.2 Gauge Fields.- 2.1.3 Conditions for the Local Invariance of the Lagrangian.- 2.1.4 Connection Between the Globally and the Locally Invariant Lagrangians.- 2.2 Gauge Fields.- 2.2.1 Transformations of the Gauge Fields.- 2.2.2 The Lagrangian for Gauge Fields.- 2.2.3 Conserved Currents.- 2.3 The Abelian Group Ux. The Electromagnetic Field.- 2.4 The Non-Abelian Group SU2. The Yang-Mills Field.- 3. Spontaneous Symmetry-Breaking.- 3.1 Degeneracy of the Vacuum States and Symmetry-Breaking.- 3.2 Spontaneous Breaking of Global Symmetry.- 3.2.1 Exact Symmetry.- 3.2.2 Spontaneous Symmetry-Breaking.- 3.3 Spontaneous Breaking of Local Symmetry.- 3.3.1 Exact Symmetry.- 3.3.2 Spontaneously Broken Symmetry.- 3.4 Residual Symmetry.- II Quantum Theory of Gauge Fields.- 4. Path Integrals and Transition Amplitudes.- 4.1 Unconstrained Fields.- 4.1.1 Systems with One Degree of Freedom.- 4.1.2 Systems with a Finite Number of Degrees of Freedom.- 4.1.3 The Boson Fields.- 4.1.4 The Fermion Fields.- 4.2 Fields with Constraints.- 4.2.1 Systems with a Finite Number of Degrees of Freedom.- 4.2.2 The Electromagnetic Field.- 4.2.3 The Yang-Mills Field.- 5. Covariant Perturbation Theory.- 5.1 Green’s Functions. Generating Functionals.- 5.1.1 Path-Integral Formulation.- 5.1.2 Generating Functional W (J).- 5.1.3 Green’s Functions in Perturbation Theory.- 5.1.4 Types of Diagrams.- 5.1.5 Generating Functional Z (J).- 5.1.6 Generating Functional ? (?).- 5.1.7 The Tree Approximation for ? (?).- 5.1.8 Another Expression for W (J).- 5.1.9 Expression for the Matrix Elements of the S-Matrix in Terms of Green’s Functions.- 5.2 The ?4 Interaction Model.- 5.3 A Model with Non-Abelian Gauge Fields.- 5.4 The 1/TV-Expansion.- III Gauge Theory of Electroweak Interactions.- 6. Lagrangians of the Electroweak Interactions.- 6.1 The Standard Model for the Electroweak Interactions of Leptons.- 6.1.1 The Standard Model.- 6.2 Quark Models of Hadrons.- 6.2.1 Hadrons.- 6.2.2 SU3-Symmetry. Three Quarks.- 6.2.3 SU4-Symmetry. Charm.- 6.2.4 Coloured Quarks.- 6.2.5 Quark-Lepton Symmetry.- 6.2.6 Two Types of Models with Coloured Quarks.- 6.2.7 Heavy Quarks.- 6.3 The Standard Model of Electroweak Interactions of Quarks.- 6.4 Non-Standard Models.- 6.4.1 Ambiguity in the Choice of the Model.- 6.4.2 The SU3 U1-Model.- 7. Quantum Electrodynamics.- 7.1 Covariant Perturbation Theory for Quantum Electrodynamics.- 7.2 Differential Cross-Sections.- 7.2.1 Formula for the Differential Cross-Section.- 7.2.2 Cross-Section of the Compton Scattering.- 8. Weak Interactions.- 8.1 Processes Caused by Neutral Weak Currents.- 8.1.1 Diagonal Terms.- 8.1.2 Non-Diagonal Terms.- 8.1.3 Comparison with Experiment.- 8.1.4 Non-Standard Models.- 8.2 Processes Caused by Charged Weak Currents.- 8.2.1 Charged Weak Quark Current.- 8.2.2 Leptonic Processes.- 8.2.3 Semi-Leptonic Decays.- 8.2.4 Non-Leptonic Hadronic Decays.- 8.3 CP Violation.- 8.3.1 NeutralAT-MesonDecay.- 8.3.2 CP Violation and Gauge Models.- 9. Higher Orders in Perturbation Theory.- 9.1 Divergences of Matrix Elements.- 9.2 Renormalization.- 9.2.1The Renormalization Procedure.- 9.2.2 Dimensional Regularization.- 9.2.3 The R-Operation.- 9.2.4 Generalized Ward Identities.- 9.2.5 Renormalization of Gauge Fields.- 9.2.6 Unitarity of the Amplitude.- 9.2.7 Anomalies.- 9.2.8 Renormalization of the Standard Model.- IV Gauge Theory of Strong Interactions.- 10. Asymptotically Free Theories.- 10.1 Renormalization Group Equations and Their Solutions.- 10.1.1 Multiplicative Renormalizations and Their Groups.- 10.1.2 Dependence of the Factors Zi on the Dimensionless Parameters.- 10.1.3 The Renormalization Group Equations.- 10.1.4 Effective Charge and Asymptotic Freedom.- 10.1.5 Method of Investigation.- 10.2 The Models.- 10.2.1 Models Without Non-Abelian Gauge Fields.- 10.2.2 Models with Non-Abelian Gauge Fields.- 11. Dynamical Structure of Hadrons.- 11.1 Experimental Basis for Scaling.- 11.2 Exact Scaling and the Parton Structure of Hadrons.- 11.2.1 The Quark-Parton Model.- 11.2.2 Sum Rules.- 11.2.3 Relations Between the Structure Functions.- 11.2.4 Comparison with Experiment.- 11.2.5 Quark and Gluon Distribution Functions.- 12. Quantum Chromodynamics. Perturbation Theory.- 12.1 Covariant Perturbation Theory for Quantum Chromodynamics.- 12.1.1 The Lagrangian for Quantum Chromodynamics.- 12.1.2 Covariant Perturbation Theory.- 12.1.3 Renormalizability of Quantum Chromodynamics.- 12.2 Examples of Perturbation-Theory Calculations.- 12.2.1 Basic Processes.- 12.2.2 Cross-Sections for the Sub-Processes.- 12.2.3 Radiative Corrections.- 12.2.4 The Effective (or Running) Coupling Constant.- 12.2.5 Asymptotic Freedom of Quantum Chromodynamics.- 12.2.6 Scaling Violation.- 12.3 The Method of Operator Product Expansion.- 12.4 Evolution Equations.- 12.5 The Summation Method of Feynman Diagrams.- 12.6 Quark and Gluon Fragmentation into Hadrons.- 13. Lattice Gauge Theories. Quantum Chromodynamics on a Lattice.- 13.1 Classical and Quantum Chromodynamics on a Lattice.- 13.1.1 The Lattice and Its Elements.- 13.1.2 Gauge Fields on a Lattice.- 13.1.3 The Spinor (Quark) Field on a Lattice.- 13.1.4 Classical Chromodynamics on a Lattice.- 13.1.5 Quantum Chromodynamics on a Lattice.- 13.1.6 The Continuum Limit.- 13.2 Strong Coupling Expansion for the Gauge Fields.- 13.2.1 The Loop Average.- 13.2.2 The Area Law. The Quark Confinement.- 13.3 Non-Perturbative Calculations in Quantum Chromodynamics by Means of Monte-Carlo Simulations.- 13.3.1 Monte-Carlo Simulations.- 13.3.2 Results of Calculations.- 14. Grand Unification.- 14.1 The SU5-Model.- 14.1.1 The Lagrangian for the Model.- 14.1.2 Energy Dependence of the Coupling Constants.- 14.1.3 Proton Decay.- 14.2 Structure of the Fermion Multiplets in the SUn-Model.- 14.2.1 Structure of SUn-Multiplets with Respect to the Subgroup SU3.- 14.2.2 Structure of SUn-Multiplets with Respect to the Electric Charge.- 14.2.3 Structure of SUn-Multiplets with Respect to the Subgroup SU2 U1.- 14.3 General Requirements and the Choice of Model.- 14.4 The SU8-Model.- 14.4.1 Composition of the Model.- 14.4.2 The Yukawa Terms and the Fermion Masses.- 14.4.3 Spontaneous Breaking of SU8-Symmetry.- 14.5 The Pati-Salam Model.- 15. Topological Solitons and Instantons.- 15.1 One-Dimensional and Two-Dimensional Models.- 15.1.1 One-Dimensional Solitons.- 15.1.2 Vortices.- 15.1.3 The Idea of Topological Analysis.- 15.2 Homotopy Groups.- 15.2.1 Homotopy Classes.- 15.2.2 Homotopy Group.- 15.2.3 Determining the Homotopy Groups.- 15.2.4 Topological Charge.- 15.2.5 Dimension of the Space and Compositio of the Model.- 15.3 Monopole.- 15.3.1 The DiracMonopole.- 15.3.2 Gauge Theory of the Monopole.- 15.4 Instantons.- 15.5 Quantum Theory of Solitons.- 15.5.1 The Generating Functional for the Green’s Functions..- 15.5.2 Perturbation Theory.- 16. Conclusion.- 16.1 Quark and Gluon Confinement.- 16.2 Potential Approach.- 16.3 QCD Sum Rules.- 16.4 Theory of Loop Functional.- 16.5 Unified Gauge Models.- 16.6 Unified Supersymmetric Models.- 16.7 Gauge Theory of Gravitation.- 16.8 Superunification.- 16.9 Composite Models.- 16.10 The Elementary Particles and the Cosmology.- List of Symbols.