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Mathematical Problems in Quantum Physics
This volume contains the proceedings of the QMATH13: Mathematical Results in Quantum Physics conference, held from October 8–11, 2016, at the Georgia Institute of Technology, Atlanta, Georgia. In recent years, a number of new frontiers have opened in mathematical physics, such as many-body localization and Schrödinger operators on graphs. There has been progress in developing mathematical techniques as well, notably in renormalization group methods and the use of Lieb–Robinson bounds in various quantum models. The aim of this volume is to provide an overview of some of these developments. Topics include random Schrödinger operators, many-body fermionic systems, atomic systems, effective equations, and applications to quantum field theory. A number of articles are devoted to the very active area of Schrödinger operators on graphs and general spectral theory of Schrödinger operators. Some of the articles are expository and can be read by an advanced graduate student.
Luttinger Model
The Luttinger Model is the only model of many-fermion physics with legitimate claims to be both exactly and completely solvable. In several respects it plays the same role in many-body theory as does the 2D Ising model in statistical physics. Interest in the Luttinger model has increased steadily ever since its introduction half a century ago. The present volume starts with reprints of the seminal papers in which it was originally introduced and solved, and continues with several contributions setting out the landscape of the principal advances of the last fifty years and of prominent new directions.
Non-perturbative Renormalization
The notion of renormalization is at the core of several spectacular achievements of contemporary physics, and in the last years powerful techniques have been developed allowing to put renormalization on a firm mathematical basis. This book provides a self-consistent and accessible introduction to the sophisticated tools used in the modern theory of non-perturbative renormalization, allowing an unified and rigorous treatment of Quantum Field Theory, Statistical Physics and Condensed Matter models. In particular the first part of this book is devoted to Constructive Quantum Field Theory, providing a mathematical construction of models at low dimensions and discussing the removal of the ultravi...
Introduction to a Renormalisation Group Method
This is a primer on a mathematically rigorous renormalisation group theory, presenting mathematical techniques fundamental to renormalisation group analysis such as Gaussian integration, perturbative renormalisation and the stable manifold theorem. It also provides an overview of fundamental models in statistical mechanics with critical behaviour, including the Ising and φ4 models and the self-avoiding walk. The book begins with critical behaviour and its basic discussion in statistical mechanics models, and subsequently explores perturbative and non-perturbative analysis in the renormalisation group. Lastly it discusses the relation of these topics to the self-avoiding walk and supersymmetry. Including exercises in each chapter to help readers deepen their understanding, it is a valuable resource for mathematicians and mathematical physicists wanting to learn renormalisation group theory.
Xvith International Congress On Mathematical Physics (With Dvd-rom)
The International Congress on Mathematical Physics is the flagship conference in this exciting field. Convening every three years, it gives a survey on the progress achieved in all branches of mathematical physics. It also provides a superb platform to discuss challenges and new ideas. The present volume collects material from the XVIth ICMP which was held in Prague, August 2009, and features most of the plenary lectures and invited lectures in topical sessions as well as information on other parts of the congress program.This volume provides a broad coverage of the field of mathematical physics, from dominantly mathematical subjects to particle physics, condensed matter, and application of mathematical physics methods in various areas such as astrophysics and ecology, amongst others.
Recent Progress In Many-body Theories - Proceedings Of The 10th International Conference
Quantum many-body theory as a discipline in its own right dates largely from the 1950's. It has developed since then to its current position as one of the cornerstones of modern theoretical physics. The field remains vibrant and active, vigorous and exciting. Indeed, its successes and importance were vividly illustrated prior to the conference by the sharing of the 1998 Nobel Prizes in both Physics and Chemistry by three many-body theorists. Two of those Nobel laureates, Walter Kohn and Bob Laughlin, delivered invited lectures at this meeting, the tenth in the series of International Conferences on Recent Progress in Many-Body Theories. This series is universally recognized as being the prem...
Perturbation Theory
This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, is devoted to the fundamentals of Perturbation Theory (PT) as well as key applications areas such as Classical and Quantum Mechanics, Celestial Mechanics, and Molecular Dynamics. Less traditional fields of application, such as Biological Evolution, are also discussed. Leading scientists in each area of the field provide a comprehensive picture of the landscape and the state of the art, with the specific goal of combining mathematical rigor, explicit computational methods, and relevance to concrete applications. New to this edition are chapters on Water Waves, Rogue Waves, Multiple Scales methods, legged locomo...
Hamiltonian Systems with Three or More Degrees of Freedom
A survey of current knowledge about Hamiltonian systems with three or more degrees of freedom and related topics. The Hamiltonian systems appearing in most of the applications are non-integrable. Hence methods to prove non-integrability results are presented and the different meaning attributed to non-integrability are discussed. For systems near an integrable one, it can be shown that, under suitable conditions, some parts of the integrable structure, most of the invariant tori, survive. Many of the papers discuss near-integrable systems. From a topological point of view, some singularities must appear in different problems, either caustics, geodesics, moving wavefronts, etc. This is also r...
The Genesis of the Langlands Program
A step-by-step guide to Langlands' early work leading up the Langlands Program for mathematicians and advanced students.
