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This volume summarizes the many alternatives and extensions to Einstein's General Theory of Relativity, and shows how symmetry principles can be applied to identify physically viable models. The first part of the book establishes the foundations of classical field theory, providing an introduction to symmetry groups and the Noether theorems. A quick overview of general relativity is provided, including discussion of its successes and shortcomings, then several theories of gravity are presented and their main features are summarized. In the second part, the 'Noether Symmetry Approach' is applied to theories of gravity to identify those which contain symmetries. In the third part of the book these selected models are tested through comparison with the latest experiments and observations. This constrains the free parameters in the selected models to fit the current data, demonstrating a useful approach that will allow researchers to construct and constrain modified gravity models for further applications.

This 2002 book introduces the quantum theory of gauge fields. Emphasis is placed on four non-perturbative methods: path integrals, lattice gauge theories, the 1/N expansion, and reduced matrix models, all of which have important contemporary applications. Written as a textbook, it assumes a knowledge of quantum mechanics and elements of perturbation theory, while many relevant concepts are pedagogically introduced at a basic level in the first half of the book. The second half comprehensively covers large-N Yang-Mills theory. The book uses an approach to gauge theories based on path-dependent phase factors known as the Wilson loops, and contains problems with detailed solutions to aid understanding. Suitable for advanced graduate courses in quantum field theory, the book will also be of interest to researchers in high energy theory and condensed matter physics as a survey of recent developments in gauge theory.

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.

This book was originally published in 2006. Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory. The theory has evolved to describe the relationship between finite groups, modular forms and vertex operator algebras. Moonshine Beyond the Monster describes the general theory of Moonshine and its underlying concepts, emphasising the interconnections between mathematics and mathematical physics. Written in a clear and pedagogical style, this book is ideal for graduate students and researchers working in areas such as conformal field theory, string theory, algebra, number theory, geometry and functional analysis. Containing over a hundred exercises, it is also a suitable textbook for graduate courses on Moonshine and as supplementary reading for courses on conformal field theory and string theory.

Most of the numerical predictions of experimental phenomena in particle physics over the last decade have been made possible by the discovery and exploitation of the simplifications that can happen when phenomena are investigated on short distance and time scales. This book provides a coherent exposition of the techniques underlying these calculations. After reminding the reader of some basic properties of field theories, examples are used to explain the problems to be treated. Then the technique of dimensional regularization and the renormalization group. Finally a number of key applications are treated, culminating in the treatment of deeply inelastic scattering.

Providing a new perspective on quantum field theory, this book is useful for graduate students and researchers within and outside the field. It describes non-perturbative methods, and explores two-dimensional and four-dimensional gauge dynamics using those methods. Applications are thoroughly described.

This book, originally published in 2006, describes the general theory of Moonshine and its underlying concepts, emphasising the fundamental ideas and examples behind some of the most fascinating topics in mathematics and physics. For graduates and researchers working in areas such as algebra, number theory, geometry, analysis, quantum field theory and conformal field theory.

Scientists have been debating the meaning of quantum mechanics for over a century. This book for graduate students and researchers gets to the root of the problem; the contextual nature of empirical truth, the laws of observation and how these impact on our understanding of quantum physics. Bridging the gap between non-relativistic quantum mechanics and quantum field theory, this novel approach to quantum mechanics extends the standard formalism to cover the observer and their apparatus. The author demystifies some of the aspects of quantum mechanics that have traditionally been regarded as extraordinary, such as wave-particle duality and quantum superposition, by emphasizing the scientific principles rather than the mathematical modelling involved. Including key experiments and worked examples throughout to encourage the reader to focus on empirically sound concepts, this book avoids metaphysical speculation and also alerts the reader to the use of computer algebra to explore quantum experiments of virtually limitless complexity.

This 2002 book provides a thorough introduction to quantum theory of gauge fields, with emphasis on non-perturbative methods. Suitable as a textbook for advanced graduate courses in quantum field theory, it will also be of interest to researchers in high energy theory and condensed matter physics.

The 2006 second edition of this book develops the basic formalism and theoretical techniques for studying relativistic field theory at finite temperature and density. It covers renormalization, linear response theory, spontaneous symmetry breaking, lattice gauge theory, and nucleation theory. Applications to astrophysics, cosmology, and relativistic nucleus-nucleus collisions are also included.

This book will be of great interest to graduate students wishing to learn about semi-classical methods in nuclear physics.

This study deals with the twistor treatment of linear and non-linear partial differential equations. The authors develop the mathematical background, go on to discuss Yang Mills fields and gravitational fields in classical language and finish by solving a number of field-theoretic problems.

Unifying a range of topics that are currently scattered throughout the literature, this book offers a unique and definitive review of mathematical aspects of quantization and quantum field theory. The authors present both basic and more advanced topics of quantum field theory in a mathematically consistent way, focusing on canonical commutation and anti-commutation relations. They begin with a discussion of the mathematical structures underlying free bosonic or fermionic fields, like tensors, algebras, Fock spaces, and CCR and CAR representations (including their symplectic and orthogonal invariance). Applications of these topics to physical problems are discussed in later chapters. Although most of the book is devoted to free quantum fields, it also contains an exposition of two important aspects of interacting fields: diagrammatics and the Euclidean approach to constructive quantum field theory. With its in-depth coverage, this text is essential reading for graduate students and researchers in departments of mathematics and physics.

"First published in 1973, this influential work discusses Einstein's General Theory of Relativity to show how two of its predictions arise: first, that the ultimate fate of many massive stars is to undergo gravitational collapse to form 'black holes'; and second, that there was a singularity in the past at the beginning of the universe. Starting with a precise formulation of the theory, including the necessary differential geometry, the authors discuss the significance of spacetime curvature and examine the properties of a number of exact solutions of Einstein's field equations. They develop the theory of the causal structure of a general spacetime, and use it to prove a number of theorems establishing the inevitability of singularities under certain conditions. A foreword contributed by Abhay Ashtekar and a new preface from George Ellis help put the volume into context of the developments in the field over the past 50 years"--

This is a concise graduate level introduction to analytical functional methods in quantum field theory. Functional integral methods provide relatively simple solutions to a wide range of problems in quantum field theory. After introducing the basic mathematical background, this book goes on to study applications and consequences of the formalism to the study of series expansions, measure, phase transitions, physics on spaces with nontrivial topologies, stochastic quantisation, fermions, QED, non-abelian gauge theories, symmetry breaking, the effective potential, finite temperature field theory, instantons and compositeness. Serious attention is paid to the shortcomings of the conventional formalism (e.g. problems of measure) as well as detailed appraisal of the ambiguities of series summation. This book will be of great use to graduate students in theoretical physics wishing to learn the use of functional integrals in quantum field theory. It will also be a useful reference for researchers in theoretical physics, especially those with an interest in experimental and theoretical particle physics and quantum field theory.

This book introduces the lattice approach to quantum field theory. The spectacular successes of this technique include compelling evidence that exchange of gauge gluons can confine the quarks within subnuclear matter. The lattice framework enables novel schemes for quantitative calculation and has caused considerable cross-disciplinary activity between elementary particle and solid state physicists. The treatment begins with the lattice definition of a path integral and ends on Monte Carlo simulation methods. Other topics include invariant group integration, duality, mean field theory and renormalization group techniques. The reader is assumed to have a basic background in relativistic quantum mechanics and some exposure to gauge theories.

This is an updated and expanded second edition of a successful and well-reviewed text presenting a detailed exposition of the modern theory of supermanifolds, including a rigorous account of the super-analogs of all the basic structures of ordinary manifold theory. The exposition opens with the theory of analysis over supernumbers (Grassman variables), Berezin integration, supervector spaces and the superdeterminant. This basic material is then applied to the theory of supermanifolds, with an account of super-analogs of Lie derivatives, connections, metric, curvature, geodesics, Killing flows, conformal groups, etc. The book goes on to discuss the theory of super Lie groups, super Lie algebras, and invariant geometrical structures on coset spaces. Complete descriptions are given of all the simple super Lie groups. The book then turns to applications. Chapter 5 contains an account of the Peierals bracket for superclassical dynamical systems, super Hilbert spaces, path integration for fermionic quantum systems, and simple models of Bose-Fermi supersymmetry. The sixth and final chapter, which is new in this revised edition, examines dynamical systems for which the topology of the configuration supermanifold is important. A concise but complete account is given of the pathintegral derivation of the Chern-Gauss-Bonnet formula for the Euler-Poincare characteristic of an ordinary manifold, which is based on a simple extension of a point particle moving freely in this manifold to a supersymmetric dynamical system moving in an associated supermanifold. Many exercises are included to complement the text.

This monograph provides an account of the structure of gauge theories from a group theoretical point of view. The first part of the text is devoted to a review of those aspects of compact Lie groups (the Lie algebras, the representation theory, and the global structure) which are necessary for the application of group theory to the physics of particles and fields. The second part describes the way in which compact Lie groups are used to construct gauge theories. Models that describe the known fundamental interactions and the proposed unification of these interactions (grand unified theories) are considered in some detail. The book concludes with an up to date description of the group structure of spontaneous symmetry breakdown, which plays a vital role in these interactions. This book will be of interest to graduate students and to researchers in theoretical physics and applied mathematics, especially those interested in the applications of differential geometry and group theory in physics.

In the two volumes that comprise this work Roger Penrose and Wolfgang Rindler introduce the calculus of 2-spinors and the theory of twistors, and discuss in detail how these powerful and elegant methods may be used to elucidate the structure and properties of space-time. In volume 1, Two-spinor calculus and relativistic fields, the calculus of 2-spinors is introduced and developed. Volume 2, Spinor and twistor methods in space-time geometry, introduces the theory of twistors, and studies in detail how the theory of twistors and 2-spinors can be applied to the study of space-time. This work will be of great value to all those studying relativity, differential geometry, particle physics and quantum field theory from beginning graduate students to experts in these fields.

The two pillars of modern physics are general relativity and quantum field theory, the former describes the large scale structure and dynamics of space-time, the latter, the microscopic constituents of matter. Combining the two yields quantum field theory in curved space-time, which is needed to understand quantum field processes in the early universe and black holes, such as the well-known Hawking effect. This book examines the effects of quantum field processes back-reacting on the background space-time which become important near the Planck time (10-43 sec). It explores the self-consistent description of both space-time and matter via the semiclassical Einstein equation of semiclassical gravity theory, exemplified by the inflationary cosmology, and fluctuations of quantum fields which underpin stochastic gravity, necessary for the description of metric fluctuations (space-time foams). Covering over four decades of thematic development, this book is a valuable resource for researchers interested in quantum field theory, gravitation and cosmology.

This book gives a rigourous discussion of the local effects of curvature on the behaviour of waves. In the course of this discussion many techniques are developed which are also needed for a study of more general problems, in which the gravitational field itself plays a dynamical role.

In this thorough and coherent introduction to tau functions, Harnad and Balogh start with the basics and extend right through to the most recent research results. This monograph is ideal for graduates or researchers in related fields unacquainted with the full range of applications of the theory of tau functions.

Interpreting general relativity relies on a proper description of non-inertial frames and Dirac observables. This book describes global non-inertial frames in special and general relativity and provides a detailed description of mathematical methods. It is ideal for theoretical physicists, researchers and Ph.D. students.