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  3. Cambridge Monographs on Mathematical Physics

This highly acclaimed series of monographs provides introductory accounts of specialised topics in mathematical and theoretical physics for graduate students and research workers. The monographs in this series are of outstanding scholarship and written by those at the very frontiers of research. Subject areas covered include cosmology, astrophysics, relativity theory, particle physics, quantum theory, nuclear physics, statistical mechanics, condensed matter physics, plasma physics, and the theory of chaos.

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111 results in Cambridge Monographs on Mathematical Physics

Page 1 of 6

Causal Fermion Systems
  • An Introduction to Fundamental Structures, Methods and Applications
  • Felix Finster, Sebastian Kindermann, Jan‐Hendrik Treude
  • Coming soon
  • Expected online publication date:
    October 2025
    Print publication:
    31 October 2025
  • The theory of causal fermion systems represents a novel approach to fundamental physics and is a promising candidate for a unified physical theory. This book offers a comprehensive overview of the theory, structured in four parts: the first lays the necessary mathematical and physical foundations; the second offers an introduction to the theory and the causal action principle; the third describes the mathematical tools for analyzing causal fermion systems; and the fourth gives an outlook on the key physical applications. With relevance across mathematical and theoretical physics, the book is aimed at graduate students and researchers interested in novel approaches to the structure of spacetime and alternative perspectives to the more established quantum field theories. It can be used for advanced courses in the subject or as a reference for research and self-guided study. Exercises are included at the end of each chapter to build and develop key concepts.

Non-Perturbative Field Theory
  • Yitzhak Frishman, Jacob Sonnenschein
  • Published online:
    13 July 2023
    Print publication:
    27 July 2023
  • Providing a new perspective on quantum field theory, this book gives a pedagogical exposition of non-perturbative methods in relativistic quantum field theory and introduces the reader to modern research in theoretical physics. After describing non-perturbative methods in detail, it uses these methods to explore two-dimensional and four-dimensional gauge dynamics. 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 in particular, are thoroughly described. Published originally in 2010, this title has been reissued as an Open Access publication on Cambridge Core.

Quantized Detector Networks
  • The Theory of Observation
  • George Jaroszkiewicz
  • Published online:
    13 July 2023
    Print publication:
    27 July 2023
  • Scientists have been debating the meaning of quantum mechanics for more than a century. This book for graduate students and researchers gets to the root of the problem: how the contextual nature of empirical truth and the laws of observation 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, emphasizing the scientific principles rather than the mathematical modelling. Including key experiments and worked examples throughout, the author encourages the reader to focus on empirically sound concepts and avoid metaphysical speculation. Originally released in 2017, this title has been reissued as an Open Access publication on Cambridge Core.

Moonshine beyond the Monster
  • The Bridge Connecting Algebra, Modular Forms and Physics
  • Terry Gannon
  • Published online:
    13 July 2023
    Print publication:
    27 July 2023
  • 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 more than 100 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. Originally published in 2006, this title has been reissued as an Open Access publication on Cambridge Core.

Supersymmetric Solitons
  • M. Shifman, A. Yung
  • Published online:
    13 July 2023
    Print publication:
    27 July 2023
  • In the last decade methods and techniques based on supersymmetry have provided deep insights in quantum chromodynamics and other non-supersymmetric gauge theories at strong coupling. This book summarizes major advances in critical solitons in supersymmetric theories, and their implications for understanding basic dynamical regularities of non-supersymmetric theories. After an extended introduction on the theory of critical solitons, including a historical introduction, the authors focus on three topics: non-Abelian strings and confined monopoles; reducing the level of supersymmetry; and domain walls as D-brane prototypes. They also provide a thorough review of issues at the cutting edge, such as non-Abelian flux tubes. The book presents an extensive summary of the current literature so researchers in this field can understand the background and related issues. First published in 2009, this title has been reissued as an Open Access publication on Cambridge Core.

Methods of Contemporary Gauge Theory
  • Yuri Makeenko
  • Published online:
    13 July 2023
    Print publication:
    27 July 2023
  • This book introduces the quantum theory of gauge fields, emphasising four non-perturbative methods which have important applications: path integrals, lattice gauge theories, the 1/N expansion, and reduced matrix models. Written as a textbook, it assumes a knowledge of quantum mechanics and elements of perturbation theory, while many relevant concepts are 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 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. Originally published in 2002, this title has been reissued as an Open Access publication on Cambridge Core.

D-Branes
  • Clifford V. Johnson
  • Published online:
    13 July 2023
    Print publication:
    27 July 2023
  • D-branes represent a key theoretical tool in the understanding of strongly coupled superstring theory and M-theory. They have led to many striking discoveries, including the precise microphysics underlying the thermodynamic behaviour of certain black holes, and remarkable holographic dualities between large-N gauge theories and gravity. This book provides a self-contained introduction to the technology of D-branes, presenting their development in a pedagogical manner. The introductory material is developed by first starting with the main features of string theory needed to get rapidly to grips with D-branes. Many advanced applications are covered, with discussions of open problems which could form the basis for other avenues of research. Suitable as a textbook in graduate courses on modern string theory and theoretical particle physics, it will also be an indispensable reference for seasoned practitioners. First published in 2003, this title has been reissued as an Open Access publication on Cambridge Core.

Renormalization
  • An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion
  • John C. Collins
  • Published online:
    13 July 2023
    Print publication:
    27 July 2023
  • Many numerical predictions of experimental phenomena in particle physics are made possible by exploiting the discovery that simplifications can happen when phenomena are investigated on short distance and time scales. This book provides a coherent exposition of the renormalization 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. The technique of dimensional regularization and the renormalization group is then shown. Finally a number of key applications are demonstrated, culminating in the treatment of deeply inelastic scattering. Originally published in 1977, this title has been reissued as an Open Access publication on Cambridge Core.

An Introduction to Regge Theory and High Energy Physics
  • P. D. B. Collins
  • Published online:
    13 July 2023
    Print publication:
    27 July 2023
  • This book presents an extended introduction to the theory of hadrons, the elementary particles that occur in the atomic nucleus. The main emphasis is on the theory of the complex angular momentum plane 'Regge theory'. In 1959 Tullio Regge demonstrated that it is useful to regard angular momentum as a complex variable when discussing solutions of the Schrödinger equation for non-relativistic potential scattering. This theory helps to classify the many different particles we know of, to explain the forces between them and to predict the results of high-energy scattering experiments. Regge theory thus serves as a unifying concept drawing together many different features of high-energy physics. This monograph is intended primarily for research students just beginning in particle physics, but experienced practitioners will also find much of interest. Originally published in 1977, this title has been reissued as an Open Access publication on Cambridge Core.

Finite-Temperature Field Theory
  • Principles and Applications
  • 2nd edition
  • Joseph I. Kapusta, Charles Gale
  • Published online:
    13 July 2023
    Print publication:
    27 July 2023
  • This book develops the basic formalism and theoretical techniques for studying relativistic quantum field theory at high temperature and density. Specific physical theories treated include QED, QCD, electroweak theory, and effective nuclear field theories of hadronic and nuclear matter. Topics covered include: functional integral representation of the partition function, diagrammatic expansions, linear response theory, screening and plasma oscillations, spontaneous symmetry breaking, Goldstone theorem, resummation and hard thermal loops, lattice gauge theory, phase transitions, nucleation theory, quark-gluon plasma, and color superconductivity. Applications to astrophysics and cosmology cover white dwarf and neutron stars, neutrino emissivity, baryon number violation in the early universe, and cosmological phase transitions. Applications to relativistic nucleus-nucleus collisions are also included. The book is written for theorists in elementary particle physics, nuclear physics, astrophysics, and cosmology. Released initially in 2006, this title has been reissued as an Open Access publication on Cambridge Core.

The Large Scale Structure of Space-Time
  • 50th Anniversary Edition
  • Stephen W. Hawking, George F. R. Ellis
  • Published online:
    17 February 2023
    Print publication:
    16 February 2023
  • 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 space-time 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 space-time, 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 fifty years.

Noether Symmetries in Theories of Gravity
  • With Applications to Astrophysics and Cosmology
  • Francesco Bajardi, Salvatore Capozziello
  • Published online:
    10 November 2022
    Print publication:
    24 November 2022
  • 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.

Loops, Knots, Gauge Theories and Quantum Gravity
  • Rodolfo Gambini, Jorge Pullin
  • Foreword by Abhay Ashtekar
  • Published online:
    10 November 2022
    Print publication:
    16 February 2023
  • This volume provides a self-contained introduction to applications of loop representations, and the related topic of knot theory, in particle physics and quantum gravity. These topics are of considerable interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. The book begins with a detailed review of loop representation theory and then describes loop representations in Maxwell theory, Yang-Mills theories as well as lattice techniques. Applications in quantum gravity are then discussed, with the following chapters considering knot theories, braid theories and extended loop representations in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research. First published in 1996, this title has been reissued as an Open Access publication on Cambridge Core.

Quarks, Gluons and Lattices
  • Michael Creutz
  • Published online:
    05 November 2022
    Print publication:
    16 February 2023
  • 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.

Nonequilibrium Quantum Field Theory
  • Esteban A. Calzetta, Bei-Lok B. Hu
  • Published online:
    04 November 2022
    Print publication:
    09 February 2023
  • Bringing together the key ideas from nonequilibrium statistical mechanics and powerful methodology from quantum field theory, this 2008 book captures the essence of nonequilibrium quantum field theory. Beginning with the foundational aspects of the theory, the book presents important concepts and useful techniques, discusses issues of basic interest, and shows how thermal field, linear response, kinetic theories and hydrodynamics emerge. It also illustrates how these concepts are applied to research topics including nonequilibrium phase transitions, thermalization in relativistic heavy ion collisions, the nonequilibrium dynamics of Bose-Einstein condensation, and the generation of structures from quantum fluctuations in the early Universe. This self-contained book is a valuable reference for graduate students and researchers in particle physics, gravitation, cosmology, atomic-optical and condensed matter physics. It has been reissued as an Open Access publication on Cambridge Core.

Classical Covariant Fields
  • Mark Burgess
  • Published online:
    04 November 2022
    Print publication:
    09 February 2023
  • This 2002 book discusses the classical foundations of field theory, using the language of variational methods and covariance. It explores the limits of what can be achieved with purely classical notions, and shows how these have a deep and important connection with the second quantized field theory, which follows on from the Schwinger Action Principle. The book takes a pragmatic view of field theory, focusing on issues which are usually omitted from quantum field theory texts and cataloging results which are hard to find in the literature. Care is taken to explain how results arise and how to interpret them physically, for graduate students starting out in the field. An ideal supplementary text for courses on elementary field theory, group theory and dynamical systems, it is also a valuable reference for researchers working in these and related areas. It has been reissued as an Open Access publication on Cambridge Core.

Mathematics of Quantization and Quantum Fields
  • Jan Dereziński, Christian Gérard
  • Published online:
    04 November 2022
    Print publication:
    09 February 2023
  • Unifying topics that are scattered throughout the literature, this book offers a definitive review of mathematical aspects of quantization and quantum field theory. It presents both basic and advanced topics of quantum field theory in a mathematically consistent way, focusing on canonical commutation and anti-commutation relations. It begins with a discussion of the mathematical structures underlying free bosonic or fermionic fields: tensors, algebras, Fock spaces, and CCR and CAR representations. 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 mathematics and physics. This title, first published in 2013, has been reissued as an Open Access publication on Cambridge Core.

The Theory and Applications of Instanton Calculations
  • Manu Paranjape
  • Published online:
    04 November 2022
    Print publication:
    09 February 2023
  • Instantons, or pseudoparticles, are solutions to the equations of motion in classical field theories on a Euclidean spacetime. Instantons are found everywhere in quantum theories as they have many applications in quantum tunnelling. Diverse physical phenomena may be described through quantum tunnelling, for example: the Josephson effect, the decay of meta-stable nuclear states, band formation in tight binding models of crystalline solids, the structure of the gauge theory vacuum, confinement in 2+1 dimensions, and the decay of superheated or supercooled phases. Drawing inspiration from Sidney Coleman's Erice lectures, this volume provides an accessible, detailed introduction to instanton methods, with many applications, making it a valuable resource for graduate students in many areas of physics, from condensed matter, particle and nuclear physics, to string theory. This title, first published in 2017, has been reissued as an Open Access publication on Cambridge Core.

Tau Functions and their Applications
  • John Harnad, Ferenc Balogh
  • Published online:
    21 January 2021
    Print publication:
    04 February 2021
  • Tau functions are a central tool in the modern theory of integrable systems. This volume provides a thorough introduction, starting from the basics and extending to recent research results. It covers a wide range of applications, including generating functions for solutions of integrable hierarchies, correlation functions in the spectral theory of random matrices and combinatorial generating functions for enumerative geometrical and topological invariants. A self-contained summary of more advanced topics needed to understand the material is provided, as are solutions and hints for the various exercises and problems that are included throughout the text to enrich the subject matter and engage the reader. Building on knowledge of standard topics in undergraduate mathematics and basic concepts and methods of classical and quantum mechanics, this monograph is ideal for graduate students and researchers who wish to become acquainted with the full range of applications of the theory of tau functions.

Formulations of General Relativity
  • Gravity, Spinors and Differential Forms
  • Kirill Krasnov
  • Published online:
    06 November 2020
    Print publication:
    26 November 2020
  • This monograph describes the different formulations of Einstein's General Theory of Relativity. Unlike traditional treatments, Cartan's geometry of fibre bundles and differential forms is placed at the forefront, and a detailed review of the relevant differential geometry is presented. Particular emphasis is given to general relativity in 4D space-time, in which the concepts of chirality and self-duality begin to play a key role. Associated chiral formulations are catalogued, and shown to lead to many practical simplifications. The book develops the chiral gravitational perturbation theory, in which the spinor formalism plays a central role. The book also presents in detail the twistor description of gravity, as well as its generalisation based on geometry of 3-forms in seven dimensions. Giving valuable insight into the very nature of gravity, this book joins our highly prestigious Cambridge Monographs in Mathematical Physics series. It will interest graduate students and researchers in the fields of theoretical physics and differential geometry.

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