Series: Cambridge Studies in Advanced Mathematics (No. 127)
Hardback (ISBN-13: 9780521115773)
10 b/w illus. 45 exercises
Page extent: 300 pages
Size: 228 x 152 mm
Over the last century quantum field theory has made a significant impact on the formulation and solution of mathematical problems and inspired powerful advances in pure mathematics. However, most accounts are written by physicists, and mathematicians struggle to find clear definitions and statements of the concepts involved. This graduate-level introduction presents the basic ideas and tools from quantum field theory to a mathematical audience. Topics include classical and quantum mechanics, classical field theory, quantization of classical fields, perturbative quantum field theory, renormalization, and the standard model. The material is also accessible to physicists seeking a better understanding of the mathematical background, providing the necessary tools from differential geometry on such topics as connections and gauge fields, vector and spinor bundles, symmetries and group representations.
* Quantum field theory finally made accessible to mathematicians * A useful introduction for researchers in mathematics and mathematical physics * Provides both the mathematical tools and the necessary physical background, giving a more complete picture of the subjects
Foreword Dennis Sullivan; Preface; 1. Classical mechanics; 2. Quantum mechanics; 3. Relativity, the Lorentz group and Dirac's equation; 4. Fiber bundles, connections and representations; 5. Classical field theory; 6. Quantization of classical fields; 7. Perturbative quantum field theory; 8. Renormalization; 9. The standard model; Appendix A. Hilbert spaces and operators; Appendix B. C* algebras and spectral theory; Bibliography; Index.
Series: Cambridge Monographs on Mathematical Physics
Hardback (ISBN-13: 9780521889308)
14 b/w illus. 1 table 80 exercises
Page extent: 256 pages
Size: 247 x 174 mm
The theory of relativity describes the laws of physics in a given space-time. However, a physical theory must provide observational predictions expressed in terms of measurements, which are the outcome of practical experiments and observations. Ideal for readers with a mathematical background and a basic knowledge of relativity, this book will help readers understand the physics behind the mathematical formalism of the theory of relativity. It explores the informative power of the theory of relativity, and highlights its uses in space physics, astrophysics and cosmology. Readers are given the tools to pick out from the mathematical formalism those quantities that have physical meaning and which can therefore be the result of a measurement. The book considers the complications that arise through the interpretation of a measurement, which is dependent on the observer who performs it. Specific examples of this are given to highlight the awkwardness of the problem.
* Provides a large sample of observers and reference frames in space-times that can be applied to space physics, astrophysics and cosmology * Tackles the problems encountered in interpreting measurements, giving specific examples * Features advice to help readers understand the logic of a given theory and its limitations
1. Introduction; 2. The theory of relativity: a mathematical overview; 3. Space-time splitting; 4. Special frames; 5. The world function; 6. Local measurements; 7. Non-local measurements; 8. Observers in physical relevant space-times; 9. Measurements in physically relevant space-times; 10. Measurements of spinning bodies.
Series: Institute of Mathematical Statistics Textbooks (No. 1)
Hardback (ISBN-13: 9780521197984)
45 b/w illus. 90 exercises
Page extent: 270 pages
Size: 228 x 152 mm
Paperback (ISBN-13: 9780521147354)
This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. Schramm*Lowner evolutions (SLE) arise in various contexts. The choice of topics is strongly motivated by modern applications and focuses on areas that merit further research. Special features include a simple account of Smirnov's proof of Cardy's formula for critical percolation, and a fairly full account of the theory of influence and sharp-thresholds. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.
* Author renowned for his clear, readable style * Probability theory sheds light on everything * Engages your brain and gets your hands dirty
Preface; 1. Random walks on graphs; 2. Uniform spanning tree; 3. Percolation and self-avoiding walk; 4. Association and influence; 5. Further percolation; 6. Contact process; 7. Gibbs states; 8. Random-cluster model; 9. Quantum Ising model; 10. Interacting particle systems; 11. Random graphs; 12. Lorentz gas; References; Index.
Series: New Mathematical Monographs (No. 16)
Hardback (ISBN-13: 9780521118422)
4 b/w illus.
Page extent: 160 pages
Size: 228 x 152 mm
Spencer Bloch's 1979 Duke lectures, a milestone in modern mathematics, have been out of print almost since their first publication in 1980, yet they have remained influential and are still the best place to learn the guiding philosophy of algebraic cycles and motives. This edition, now professionally typeset, has a new preface by the author giving his perspective on developments in the field over the past 30 years. The theory of algebraic cycles encompasses such central problems in mathematics as the Hodge conjecture and the Bloch*Kato conjecture on special values of zeta functions. The book begins with Mumford's example showing that the Chow group of zero-cycles on an algebraic variety can be infinite-dimensional, and explains how Hodge theory and algebraic K-theory give new insights into this and other phenomena.
* Classic treatment of an important area of mathematics by one of its founders * This second edition includes a new, extended preface by the author and an index * Ideal for graduate courses and lecture series
Preface to the second edition; Introduction; 1. 0-cycles on surfaces; Lecture 1 Appendix. On an argument of Mumford in the theory of algebraic cycles; 2. Curves on threefolds and intermediate Jacobians; 3. Curves on threefolds and intermediate Jacobians - the relative case; 4. K-theoretic and cohomological methods; 5. Torsion in the Chow group; 6. Complements on H2(K2); 7. Diophantine questions; 8. Relative cycles and zeta functions; 9. Relative cycles and zeta functions (continued); References; Index.
Series: New Mathematical Monographs (No. 17)
Hardback (ISBN-13: 9780521195607)
Page extent: 500 pages
Size: 228 x 152 mm
Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, of general smooth connected algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book as it develops tools likely to be used in tackling other problems.
* Strong collaboration of authors representing three important areas: number theory, algebraic geometry and algebraic groups * Presents foundational results very useful to mathematicians working in related areas * Appendix supplies Titsf results on unipotent groups
Introduction; Terminology, conventions, and notation; Part I. Constructions, Examples, and Structure Theory: 1. Overview of pseudo-reductivity; 2. Root groups and root systems; 3. Basic structure theory; Part II. Standard Presentations and Their Applications: 4. Variation of (G', k'/k, T', C); 5. Universality of the standard construction; 6. Classification results; Part III. General Classification and Applications: 7. General classification and applications; 8. Preparations for classification in characteristics 2 and 3; 9. The absolutely pseudo-simple case in characteristic 2; 10. General case; 11. Applications; Part IV. Appendices: A. Background in linear algebraic groups; B. Tits' work on unipotent groups in nonzero characteristic; C. Rational conjugacy in connected groups; References; Index.