This book is an introduction to the applications in nonequilibrium statistical mechanics of chaotic dynamics, and also to the use of techniques in statistical mechanics important for an understanding of the chaotic behaviour of fluid systems. The fundamental concepts of dynamical systems theory are reviewed and simple examples are given. Advanced topics including SRB and Gibbs measures, unstable periodic orbit expansions, and applications to billiard-ball systems, are then explained. The text emphasises the connections between transport coefficients, needed to describe macroscopic properties of fluid flows, and quantities, such as Lyapunov exponents and Kolmogorov-Sinai entropies, which describe the microscopic, chaotic behaviour of the fluid. Later chapters consider the roles of the expanding and contracting manifolds of hyperbolic dynamical systems and the large number of particles in macroscopic systems. Exercises, detailed references and suggestions for further reading are included.

The theorems of Hawking and Penrose show that space-times are likely to contain incomplete geodesics. Such geodesics are said to end at a singularity if it is impossible to continue the space-time and geodesic without violating the usual topological and smoothness conditions on the space-time. In this book the different possible singularities are defined, and the mathematical methods needed to extend the space-time are described in detail. The results obtained (many appearing here for the first time) show that singularities are associated with a lack of smoothness in the Riemann tensor. While the Friedmann singularity is analysed as an example, the emphasis is on general theorems and techniques rather than on the classification of particular exact solutions.

This book shows how modern cosmology and astronomy have led to the need to introduce dark matter in the universe. Some of this dark matter is in the familiar form of protons, electrons and neutrons, but most of it must have a more exotic form. The favoured, but not the only, possibility is neutrinos of non-zero rest mass, pair-created in the hot big bang and surviving to the present day. After a review of modern cosmology, this book gives a detailed account of the author's recent theory in which these neutrinos decay into photons which are the main ionising agents in hydrogen and nitrogen in the interstellar and intergalactic medium. This theory, though speculative, explains a number of rather different puzzling phenomena in astronomy and cosmology in a unified way and predicts values of various important quantities such as the mass of the decaying neutrino and the Hubble constant. Written by a cosmologist of the first rank, this topical book will be essential reading to all cosmologists and astrophysicists.

This author provides an easily accessible introduction to quantum field theory via Feynman rules and calculations in particle physics. His aim is to make clear what the physical foundations of present-day field theory are, to clarify the physical content of Feynman rules. The book begins with a brief review of some aspects of Einstein's theory of relativity that are of particular importance for field theory, before going on to consider the relativistic quantum mechanics of free particles, interacting fields, and particles with spin. The techniques learnt in the chapters are then demonstrated in examples that might be encountered in real accelerator physics. Further chapters contain discussions of renormalization, massive and massless vector fields and unitarity. A final chapter presents concluding arguments concerning quantum electrodynamics. The book includes valuable appendices that review some essential mathematics, including complex spaces, matrices, the CBH equation, traces and dimensional regularization. An appendix containing a comprehensive summary of the rules and conventions used is followed by an appendix specifying the full Lagrangian of the Standard Model and the corresponding Feynman rules. To make the book useful for a wide audience a final appendix provides a discussion of the metric used, and an easy-to-use dictionary connecting equations written with different metrics. Written as a textbook, many diagrams, exercises and examples are included. This book will be used by beginning graduate students taking courses in particle physics or quantum field theory, as well as by researchers as a source and reference book on Feynman diagrams and rules.

This book introduces the use of Lie algebra and differential geometry methods to study nonlinear integrable systems of Toda type. Many challenging problems in theoretical physics are related to the solution of nonlinear systems of partial differential equations. One of the most fruitful approaches in recent years has resulted from a merging of group algebraic and geometric techniques. The book gives a comprehensive introduction to this exciting branch of science. Chapters 1 and 2 review basic notions of Lie algebras and differential geometry with an emphasis on further applications to integrable nonlinear systems. Chapter 3 contains a derivation of Toda type systems and their general solutions based on Lie algebra and differential geometry methods. The last chapter examines explicit solutions of the corresponding equations. The book is written in an accessible 'lecture note' style with many examples and exercises to illustrate key points and to reinforce understanding.

Self-organized criticality (SOC) is based upon the idea that complex behavior can develop spontaneously in certain multi-body systems whose dynamics vary abruptly. This book is a clear and concise introduction to the field of self-organized criticality, and contains an overview of the main research results. The author begins with an examination of what is meant by SOC, and the systems in which it can occur. He then presents and analyzes computer models to describe a number of systems, and he explains the different mathematical formalisms developed to understand SOC. The final chapter assesses the impact of this field of study, and highlights some key areas of new research. The author assumes no previous knowledge of the field, and the book contains several exercises. It will be ideal as a textbook for graduate students taking physics, engineering, or mathematical biology courses in nonlinear science or complexity.

This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.

This book provides an introduction to lattice models of polymers. This is an important topic both in the theory of critical phenomena and the modelling of polymers. The first two chapters introduce the basic theory of random, directed and self-avoiding walks. The next two chapters develop and expand this theory to explore the self-avoiding walk in both two and three dimensions. Following chapters describe polymers near a surface, dense polymers, self-interacting polymers and branched polymers. The book closes with discussions of some geometrical and topological properties of polymers, and of self-avoiding surfaces on a lattice. The volume combines results from rigorous analytical and numerical work to give a coherent picture of the properties of lattice models of polymers. This book will be valuable for graduate students and researchers working in statistical mechanics, theoretical physics and polymer physics. It will also be of interest to those working in applied mathematics and theoretical chemistry.

This controversial book examines one of the most fundamental questions of modern cosmology: how much matter is there in the Universe? This issue affects theories of the origin and evolution of the Universe as well as its geometrical structure and ultimate fate. The authors discuss all the relevant cosmological and astrophysical evidence and come to the conclusion that the balance of arguments presently lies with a density of around twenty per cent of the critical density required for the Universe to ultimately recollapse. Because the arguments presented constitute a 'state-of-the-art' analysis of the observational and theoretical arguments surrounding the existence of dark matter, primordial nucleosynthesis, large-scale structure formation and the cosmic microwave background radiation, this study provides the reader with an indispensable introduction to the most exciting recent developments in modern cosmology. Written by two eminent cosmologists, this topical and provocative book will be essential reading for all cosmologists and astrophysicists.

This timely review provides a self-contained introduction to the mathematical theory of stationary black holes and a self-consistent exposition of the corresponding uniqueness theorems. The opening chapters examine the general properties of space-times admitting Killing fields and derive the Kerr-Newman metric. Strong emphasis is given to the geometrical concepts. The general features of stationary black holes and the laws of black hole mechanics are then reviewed. Critical steps towards the proof of the 'no-hair' theorem are then discussed, including the methods used by Israel, the divergence formulae derived by Carter, Robinson and others, and finally the sigma model identities and the positive mass theorem. The book is rounded off with an extension of the electro-vacuum uniqueness theorem to self-gravitating scalar fields and harmonic mappings. This volume provides a rigorous textbook for graduate students in physics and mathematics. It also offers an invaluable, up-to-date reference for researchers in mathematical physics, general relativity and astrophysics.

The Dirac operator has many useful applications in theoretical physics and mathematics. This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or non-local boundary conditions. The theory is introduced at a level suitable for graduate students. Numerous examples are then given to illustrate the peculiar properties of the Dirac operator, and the role of boundary conditions in heat-kernel asymptotics and quantum field theory. Topics covered include the introduction of spin-structures in Riemannian and Lorentzian manifolds; applications of index theory; heat-kernel asymptotics for operators of Laplace type; quark boundary conditions; one-loop quantum cosmology; conformally covariant operators; and the role of the Dirac operator in some recent investigations of four-manifolds. This volume provides graduate students with a rigorous introduction and researchers with a valuable reference to the Dirac operator and its applications in theoretical physics.

This volume describes the Pomeron, an object of crucial importance in very high energy particle physics. Starting with a general description of the Pomeron within the framework of Regge theory, the emergence of the Pomeron within scalar field theory is discussed, providing a natural foundation on which to develop the more realistic case of QCD. The reggeization of the gluon is demonstrated and used to build the Pomeron of perturbative QCD. The dynamical nature of the Pomeron and its role in small-x deep inelastic scattering and in diffractive scattering is also examined in detail. The volume concludes with a study of the colour dipole approach to high energy scattering and the explicit role of unitarity corrections. This book will be of interest to theoretical and experimental particle physicists, and applied mathematicians. First published in 1997, this title has been reissued as an Open Access publication on Cambridge Core.

This volume describes the Pomeron, an object of crucial importance in very high energy particle physics. Starting with a general description of the Pomeron within the framework of Regge theory, the emergence of the Pomeron within scalar field theory is discussed, providing a natural foundation on which to develop the more realistic case of QCD. The reggeization of the gluon is demonstrated and used to build the Pomeron of perturbative QCD. The dynamical nature of the Pomeron and its role in small-x deep inelastic scattering and in diffractive scattering is also examined in detail. The volume concludes with a study of the colour dipole approach to high energy scattering and the explicit role of unitarity corrections. This book will be of interest to theoretical and experimental particle physicists, and applied mathematicians. First published in 1997, this title has been reissued as an Open Access publication on Cambridge Core.

This book provides a concise introduction to quantum fields on a lattice: a precise and non-perturbative definition of quantum field theory obtained by replacing continuous space-time by a discrete set of points on a lattice. The path integral on the lattice is explained in concrete examples using weak and strong coupling expansions. Fundamental concepts such as 'triviality' of Higgs fields and confinement of quarks and gluons into hadrons are described and illustrated with the results of numerical simulations. The book also provides an introduction to chiral symmetry and chiral gauge theory, as well as quantized non-Abelian gauge fields, scaling and universality. Based on the lecture notes of a course given by the author, this book contains many explanatory examples and exercises, and is suitable as a textbook for advanced undergraduate and graduate courses. Originally published in 2002, this title has been reissued as an Open Access publication on Cambridge Core.

This book is an introduction to the applications in nonequilibrium statistical mechanics of chaotic dynamics, and also to the use of techniques in statistical mechanics important for an understanding of the chaotic behaviour of fluid systems. The fundamental concepts of dynamical systems theory are reviewed and simple examples are given. Advanced topics including SRB and Gibbs measures, unstable periodic orbit expansions, and applications to billiard-ball systems, are then explained. The text emphasises the connections between transport coefficients, needed to describe macroscopic properties of fluid flows, and quantities, such as Lyapunov exponents and Kolmogorov-Sinai entropies, which describe the microscopic, chaotic behaviour of the fluid. Later chapters consider the roles of the expanding and contracting manifolds of hyperbolic dynamical systems and the large number of particles in macroscopic systems. Exercises, detailed references and suggestions for further reading are included.

The theorems of Hawking and Penrose show that space-times are likely to contain incomplete geodesics. Such geodesics are said to end at a singularity if it is impossible to continue the space-time and geodesic without violating the usual topological and smoothness conditions on the space-time. In this book the different possible singularities are defined, and the mathematical methods needed to extend the space-time are described in detail. The results obtained (many appearing here for the first time) show that singularities are associated with a lack of smoothness in the Riemann tensor. While the Friedmann singularity is analysed as an example, the emphasis is on general theorems and techniques rather than on the classification of particular exact solutions.

This book shows how modern cosmology and astronomy have led to the need to introduce dark matter in the universe. Some of this dark matter is in the familiar form of protons, electrons and neutrons, but most of it must have a more exotic form. The favoured, but not the only, possibility is neutrinos of non-zero rest mass, pair-created in the hot big bang and surviving to the present day. After a review of modern cosmology, this book gives a detailed account of the author's recent theory in which these neutrinos decay into photons which are the main ionising agents in hydrogen and nitrogen in the interstellar and intergalactic medium. This theory, though speculative, explains a number of rather different puzzling phenomena in astronomy and cosmology in a unified way and predicts values of various important quantities such as the mass of the decaying neutrino and the Hubble constant. Written by a cosmologist of the first rank, this topical book will be essential reading to all cosmologists and astrophysicists.

This author provides an easily accessible introduction to quantum field theory via Feynman rules and calculations in particle physics. His aim is to make clear what the physical foundations of present-day field theory are, to clarify the physical content of Feynman rules. The book begins with a brief review of some aspects of Einstein's theory of relativity that are of particular importance for field theory, before going on to consider the relativistic quantum mechanics of free particles, interacting fields, and particles with spin. The techniques learnt in the chapters are then demonstrated in examples that might be encountered in real accelerator physics. Further chapters contain discussions of renormalization, massive and massless vector fields and unitarity. A final chapter presents concluding arguments concerning quantum electrodynamics. The book includes valuable appendices that review some essential mathematics, including complex spaces, matrices, the CBH equation, traces and dimensional regularization. An appendix containing a comprehensive summary of the rules and conventions used is followed by an appendix specifying the full Lagrangian of the Standard Model and the corresponding Feynman rules. To make the book useful for a wide audience a final appendix provides a discussion of the metric used, and an easy-to-use dictionary connecting equations written with different metrics. Written as a textbook, many diagrams, exercises and examples are included. This book will be used by beginning graduate students taking courses in particle physics or quantum field theory, as well as by researchers as a source and reference book on Feynman diagrams and rules.

This book introduces the use of Lie algebra and differential geometry methods to study nonlinear integrable systems of Toda type. Many challenging problems in theoretical physics are related to the solution of nonlinear systems of partial differential equations. One of the most fruitful approaches in recent years has resulted from a merging of group algebraic and geometric techniques. The book gives a comprehensive introduction to this exciting branch of science. Chapters 1 and 2 review basic notions of Lie algebras and differential geometry with an emphasis on further applications to integrable nonlinear systems. Chapter 3 contains a derivation of Toda type systems and their general solutions based on Lie algebra and differential geometry methods. The last chapter examines explicit solutions of the corresponding equations. The book is written in an accessible 'lecture note' style with many examples and exercises to illustrate key points and to reinforce understanding.

Self-organized criticality (SOC) is based upon the idea that complex behavior can develop spontaneously in certain multi-body systems whose dynamics vary abruptly. This book is a clear and concise introduction to the field of self-organized criticality, and contains an overview of the main research results. The author begins with an examination of what is meant by SOC, and the systems in which it can occur. He then presents and analyzes computer models to describe a number of systems, and he explains the different mathematical formalisms developed to understand SOC. The final chapter assesses the impact of this field of study, and highlights some key areas of new research. The author assumes no previous knowledge of the field, and the book contains several exercises. It will be ideal as a textbook for graduate students taking physics, engineering, or mathematical biology courses in nonlinear science or complexity.

This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.

This book provides an introduction to lattice models of polymers. This is an important topic both in the theory of critical phenomena and the modelling of polymers. The first two chapters introduce the basic theory of random, directed and self-avoiding walks. The next two chapters develop and expand this theory to explore the self-avoiding walk in both two and three dimensions. Following chapters describe polymers near a surface, dense polymers, self-interacting polymers and branched polymers. The book closes with discussions of some geometrical and topological properties of polymers, and of self-avoiding surfaces on a lattice. The volume combines results from rigorous analytical and numerical work to give a coherent picture of the properties of lattice models of polymers. This book will be valuable for graduate students and researchers working in statistical mechanics, theoretical physics and polymer physics. It will also be of interest to those working in applied mathematics and theoretical chemistry.

This controversial book examines one of the most fundamental questions of modern cosmology: how much matter is there in the Universe? This issue affects theories of the origin and evolution of the Universe as well as its geometrical structure and ultimate fate. The authors discuss all the relevant cosmological and astrophysical evidence and come to the conclusion that the balance of arguments presently lies with a density of around twenty per cent of the critical density required for the Universe to ultimately recollapse. Because the arguments presented constitute a 'state-of-the-art' analysis of the observational and theoretical arguments surrounding the existence of dark matter, primordial nucleosynthesis, large-scale structure formation and the cosmic microwave background radiation, this study provides the reader with an indispensable introduction to the most exciting recent developments in modern cosmology. Written by two eminent cosmologists, this topical and provocative book will be essential reading for all cosmologists and astrophysicists.

This timely review provides a self-contained introduction to the mathematical theory of stationary black holes and a self-consistent exposition of the corresponding uniqueness theorems. The opening chapters examine the general properties of space-times admitting Killing fields and derive the Kerr-Newman metric. Strong emphasis is given to the geometrical concepts. The general features of stationary black holes and the laws of black hole mechanics are then reviewed. Critical steps towards the proof of the 'no-hair' theorem are then discussed, including the methods used by Israel, the divergence formulae derived by Carter, Robinson and others, and finally the sigma model identities and the positive mass theorem. The book is rounded off with an extension of the electro-vacuum uniqueness theorem to self-gravitating scalar fields and harmonic mappings. This volume provides a rigorous textbook for graduate students in physics and mathematics. It also offers an invaluable, up-to-date reference for researchers in mathematical physics, general relativity and astrophysics.

The Dirac operator has many useful applications in theoretical physics and mathematics. This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or non-local boundary conditions. The theory is introduced at a level suitable for graduate students. Numerous examples are then given to illustrate the peculiar properties of the Dirac operator, and the role of boundary conditions in heat-kernel asymptotics and quantum field theory. Topics covered include the introduction of spin-structures in Riemannian and Lorentzian manifolds; applications of index theory; heat-kernel asymptotics for operators of Laplace type; quark boundary conditions; one-loop quantum cosmology; conformally covariant operators; and the role of the Dirac operator in some recent investigations of four-manifolds. This volume provides graduate students with a rigorous introduction and researchers with a valuable reference to the Dirac operator and its applications in theoretical physics.

This volume describes the Pomeron, an object of crucial importance in very high energy particle physics. Starting with a general description of the Pomeron within the framework of Regge theory, the emergence of the Pomeron within scalar field theory is discussed, providing a natural foundation on which to develop the more realistic case of QCD. The reggeization of the gluon is demonstrated and used to build the Pomeron of perturbative QCD. The dynamical nature of the Pomeron and its role in small-x deep inelastic scattering and in diffractive scattering is also examined in detail. The volume concludes with a study of the colour dipole approach to high energy scattering and the explicit role of unitarity corrections. This book will be of interest to theoretical and experimental particle physicists, and applied mathematicians. First published in 1997, this title has been reissued as an Open Access publication on Cambridge Core.

This volume describes the Pomeron, an object of crucial importance in very high energy particle physics. Starting with a general description of the Pomeron within the framework of Regge theory, the emergence of the Pomeron within scalar field theory is discussed, providing a natural foundation on which to develop the more realistic case of QCD. The reggeization of the gluon is demonstrated and used to build the Pomeron of perturbative QCD. The dynamical nature of the Pomeron and its role in small-x deep inelastic scattering and in diffractive scattering is also examined in detail. The volume concludes with a study of the colour dipole approach to high energy scattering and the explicit role of unitarity corrections. This book will be of interest to theoretical and experimental particle physicists, and applied mathematicians. First published in 1997, this title has been reissued as an Open Access publication on Cambridge Core.

This book provides a concise introduction to quantum fields on a lattice: a precise and non-perturbative definition of quantum field theory obtained by replacing continuous space-time by a discrete set of points on a lattice. The path integral on the lattice is explained in concrete examples using weak and strong coupling expansions. Fundamental concepts such as 'triviality' of Higgs fields and confinement of quarks and gluons into hadrons are described and illustrated with the results of numerical simulations. The book also provides an introduction to chiral symmetry and chiral gauge theory, as well as quantized non-Abelian gauge fields, scaling and universality. Based on the lecture notes of a course given by the author, this book contains many explanatory examples and exercises, and is suitable as a textbook for advanced undergraduate and graduate courses. Originally published in 2002, this title has been reissued as an Open Access publication on Cambridge Core.