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These lecture notes provide a tutorial review of non-Abelian discrete groups and show some applications to issues in physics where discrete symmetries constitute an important principle for model building in particle physics. While Abelian discrete symmetries are often imposed in order to control couplings for particle physics - in particular model building beyond the standard model - non-Abelian discrete symmetries have been applied to understand the three-generation flavor structure in particular. Indeed, non-Abelian discrete symmetries are considered to be the most attractive choice for the flavor sector: model builders have tried to derive experimental values of quark and lepton masses, and mixing angles by assuming non-Abelian discrete flavor symmetries of quarks and leptons, yet, lepton mixing has already been intensively discussed in this context, as well. The possible origins of the non-Abelian discrete symmetry for flavors is another topic of interest, as they can arise from an underlying theory - e.g. the string theory or compactification via orbifolding – thereby providing a possible bridge between the underlying theory and the corresponding low-energy sector of particle physics. This text explicitly introduces and studies the group-theoretical aspects of many concrete groups and shows how to derive conjugacy classes, characters, representations, and tensor products for these groups (with a finite number) when algebraic relations are given, thereby enabling readers to apply this to other groups of interest.
This book provides an introduction to topological quantum field theory as well as discrete gauge theory with quantum groups. In contrast to much of the existing literature, the present approach is at the same time intuitive and mathematically rigorous, making extensive use of suitable diagrammatic methods. It provides a highly unified description of lattice gauge theory, topological quantum field theory and models of quantum (super)gravity. The reader is thus in a unique position to understand the relations between these subjects as well as the underlying groundwork.
Non-Abelian gauge theories, such as quantum chromodynamics (QCD) or electroweak theory, are best studied with the aid of Green's functions that are gauge-invariant off-shell, but unlike for the photon in quantum electrodynamics, conventional graphical constructions fail. The Pinch Technique provides a systematic framework for constructing such Green's functions, and has many useful applications. Beginning with elementary one-loop examples, this book goes on to extend the method to all orders, showing that the Pinch Technique is equivalent to calculations in the background field Feynman gauge. The Pinch Technique Schwinger-Dyson equations are derived, and used to show how a dynamical gluon mass arises in QCD. Applications are given to the center vortex picture of confinement, the gauge-invariant treatment of resonant amplitudes, the definition of non-Abelian effective charges, high-temperature effects, and even supersymmetry. This book is ideal for elementary particle theorists and graduate students.
Almost all theories of fundamental interactions are nowadays based on the gauge concept. Starting with the historical example of quantum electrodynamics, we have been led to the successful unified gauge theory of weak and electromagnetic interactions, and finally to a non abelian gauge theory of strong interactions with the notion of permanently confined quarks. The. early theoretical work on gauge theories was devoted to proofs of renormalizability, investigation of short distance behaviour, the discovery of asymptotic freedom, etc . . , aspects which were accessible to tools extrapolated from renormalised perturbation theory. The second phase of the subject is concerned with the problem of quark confinement which necessitates a non-perturbative understanding of gauge theories. This phase has so far been marked by the introduc tion of ideas from geometry, topology and statistical mechanics in particular the theory of phase transitions. The 1979 Cargese Institute on "Recent Developments on Gauge Theories" was devoted to a thorough discussion of these non-perturbative, global aspects of non-abelian gauge theories. In the lectures and seminars reproduced in this volume the reader wilf find detailed reports on most of the important developments of recent times on non perturbative gauge fields by some of the leading experts and innovators in this field. Aside from lectures on gauge fields proper, there were lectures on gauge field concepts in condensed matter physics and lectures by mathematicians on global aspects of the calculus of variations, its relation to geometry and topology, and related topics.
This is the second volume of the third edition of a successful text, now substantially enlarged and updated to reflect developments over the last decade in the curricula of university courses and in particle physics research. Volume I covered relativistic quantum mechanics, electromagnetism as a gauge theory, and introductory quantum field theory, and ended with the formulation and application of quantum electrodynamics (QED), including renormalization. Building on these foundations, this second volume provides a complete, accessible, and self-contained introduction to the remaining two gauge theories of the standard model of particle physics: quantum chromodynamics (QCD) and the electroweak theory. The treatment significantly extends that of the second edition in several important respects. Simple ideas of group theory are now incorporated into the discussion of non-Abelian symmetries. Two new chapters have been added on QCD, one devoted to the renormalization group and scaling violations in deep inelastic scattering and the other to non-perturbative aspects of QCD using the lattice (path-integral) formulation of quantum field theory; the latter is also used to illuminate various aspects of renormalization theory, via analogies with condensed matter systems. Three chapters treat the fundamental topic of spontaneous symmetry breaking: the (Bogoliubov) superfluid and the (BCS) superconductor are studied in some detail; one chapter is devoted to the implications of global chiral symmetry breaking in QCD; and one to the breaking of local SU(2)xU(1) symmetry in the electroweak theory. Weak interaction phenomenology is extended to include discussion of discrete symmetries and of the possibility that neutrinos are Majorana (rather than Dirac) particles. Most of these topics are normally found only in more advanced texts, and this is the first book to treat them in a manner accessible to the wide readership that the previous editions have attracted.
Contents: Extended Systems in Field Theory :Introduction (J-L Gervais and A Neveu)Vortices and Quark Confinement in Non-Abelian Gauge Theories (S Mandelstam)Magnetic and Electric Confinement of Quarks (Y Nambu)Examples of Four-Dimensional Soliton Solutions and Abnormal Nuclear States (T D Lee)Classical Solution in the Massive Thirring Model (S-J Chang)Semiclassical Quantization Methods in Field Theory (A Neveu)The Quantum Theory of Solitons and Other Non-Linear Classical Waves (R Jackiw)Collective Coordinate Method for Quantization of Extended Systems (J-L Gervais, A Jevicki and B Sakita)Quantum Expansion of Soliton Solutions (N H Christ)Hartree-Type Approximation Applied to a ϕ4 Field Theory (S-J Chang)Soliton Operators for the Quantized Sine–Gordon Equation (S Mandelstam)Classical Aspects and Fluctuation-Behaviour of Two Dimensional Models in Statistical Mechanics and Many Body Physics (B Schroer)Quarks on a Lattice, or, the Colored String Model (K G Wilson)New Ideas about Confinement (L Susskind and J Kogut)Gauge Fields on a Lattice (C Itzykson)Non-Perturbative Aspects in Quantum Field Theory:Self-Dual Solutions to Euclidean Yang–Mills Equations (E Corrigan)An Introduction to the Twistor Programme (J Madore, J L Richard and R Stora)Collective Coordinates with Non-Trivial Dynamics (J-L Gervais)A Theory of the Strong Interactions (D J Gross)Magneticmonopoles (D Olive)Dynamical and Topological Considerations on Quark Confinement (F Englert and P Windey)Difficulties in Fixing the Gauge in Non-Abelian Gauge Theories (S Sciuto)Indeterminate-Mass Particles (B M Mccoy and T T Wu)Duality for Discrete Lattice Gauge Fields (C Itzykson)Large Order Estimates in Perturbation Theory (J Zinn-Justin)The Borel Transform and the Renormalization Group (G Parisi)Planar Diagrams (E Brezin)Exact S-Matrices and Form Factors in 1 + 1 Dimensional Field Theoretic Models with Soliton Behaviour (M Karowski)Topology and Higher Symmetries of the Two-Dimensional Nonlinear σ Model (A D'adda, M Luscher and P Di Vecchia)Two-Dimensional Yang–Mills Theory in the Leading 1/N Expansion (T T Wu)Superfluidity and the Two-Dimensional XY Model' (D R Nelson)Bosonized Fermions in Three Dimensions (A Luther)Symmetry and Topology Concepts for Spin Glasses and Other Glasses (G Toulouse)Common Trends in Particle and Condensed Matter Physics:Introduction to Localization(D J Thouless)Conductivity Scaling and Localization(E Abrahams)Disordered Electronic System as a Model of Interacting Matrices(F Wegner)Status Report on Spin Glasses (Not Included in this Report)(S Kirkpatrick)Mean Field Theory for Spin Glasses(G Parisi)The Random Energy Model(B Derrida)Towards a Mean Field Theory of Spin Glasses: the Tap Route Revisited (C De Dominicis)On the Connection Between Spin Glasses and Gauge Field Theories(G Toulouse, J Vannimenus)Monte Carlo Simulations of Lattice Gauge Theories(C Rebbi)Large Dimension Expansions and Transition Patterns in Lattice Gauge Theories(J-M Drouffe)Progress in Lattice Gauge Theory(J B Kogut)Phase Structure of the Z(2) Gauge and Matter Theory(D Horn)General Introduction to Confinement(S Mandelstam)A Simple Picture of the Weak-to-Strong Coupling Transition in Quantum Chromodynamics(C G Callan Jr.)Quantum Fluctuations in a Multiinstanton Background(B A Berg)Some Comments on the Crossover Between Strong and Weak Coupling in Su(2) Pure Yang–Mills Theory(J Frohlich)String Dynamics in QCD (J-L Gervais, A Neveu)Dual Models and Strings: The Critical Dimension(C B Thorn: )Duality and Finite Size Effects in Six Vertex Models(C.B. Thorn: )Scaling at a Bifurcation Point(M Nauenberg, D Scalapino)Some Implications of a Cosmological Phase Transition(T W B Kibble) Readership: Graduate students and researchers in particle physics andcondensed matter physics.