Factorization Algebras in Quantum Field Theory


Author: Kevin Costello,Owen Gwilliam
Publisher: Cambridge University Press
ISBN: 1107163102
Category: Mathematics
Page: 398
View: 4368

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This first volume develops factorization algebras with a focus upon examples exhibiting their use in field theory, which will be useful for researchers and graduates.

Renormalization and Effective Field Theory


Author: Kevin Costello
Publisher: American Mathematical Soc.
ISBN: 0821852884
Category: Mathematics
Page: 251
View: 9875

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This book tells mathematicians about an amazing subject invented by physicists and it tells physicists how a master mathematician must proceed in order to understand it. Physicists who know quantum field theory can learn the powerful methodology of mathematical structure, while mathematicians can position themselves to use the magical ideas of quantum field theory in ``mathematics'' itself. The retelling of the tale mathematically by Kevin Costello is a beautiful tour de force. --Dennis Sullivan This book is quite a remarkable contribution. It should make perturbative quantum field theory accessible to mathematicians. There is a lot of insight in the way the author uses the renormalization group and effective field theory to analyze perturbative renormalization; this may serve as a springboard to a wider use of those topics, hopefully to an eventual nonperturbative understanding. --Edward Witten Quantum field theory has had a profound influence on mathematics, and on geometry in particular. However, the notorious difficulties of renormalization have made quantum field theory very inaccessible for mathematicians. This book provides complete mathematical foundations for the theory of perturbative quantum field theory, based on Wilson's ideas of low-energy effective field theory and on the Batalin-Vilkovisky formalism. As an example, a cohomological proof of perturbative renormalizability of Yang-Mills theory is presented. An effort has been made to make the book accessible to mathematicians who have had no prior exposure to quantum field theory. Graduate students who have taken classes in basic functional analysis and homological algebra should be able to read this book.

Vertex Algebras and Algebraic Curves: Second Edition


Author: Edward Frenkel,David Ben-Zvi
Publisher: American Mathematical Soc.
ISBN: 0821836749
Category: Mathematics
Page: 400
View: 1621

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Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The notion of a vertex algebra is introduced in a coordinate-independent way, so that vertex operators become well defined on arbitrary smooth algebraic curves, possibly equipped with additional data, such as a vector bundle. Vertex algebras then appear as the algebraic objects encoding the geometric structure of various moduli spaces associated with algebraic curves. Therefore they may be used to give a geometric interpretation of various questions of representation theory. The book contains many original results, introduces important new concepts, and brings new insights into the theory of vertex algebras. The authors have made a great effort to make the book self-contained and accessible to readers of all backgrounds. Reviewers of the first edition anticipated that it would have a long-lasting influence on this exciting field of mathematics and would be very useful for graduate students and researchers interested in the subject. This second edition, substantially improved and expanded, includes several new topics, in particular an introduction to the Beilinson-Drinfeld theory of factorization algebras and the geometric Langlands correspondence.

Mathematical Aspects of Quantum Field Theories


Author: Damien Calaque,Thomas Strobl
Publisher: Springer
ISBN: 3319099493
Category: Science
Page: 556
View: 9718

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Despite its long history and stunning experimental successes, the mathematical foundation of perturbative quantum field theory is still a subject of ongoing research. This book aims at presenting some of the most recent advances in the field, and at reflecting the diversity of approaches and tools invented and currently employed. Both leading experts and comparative newcomers to the field present their latest findings, helping readers to gain a better understanding of not only quantum but also classical field theories. Though the book offers a valuable resource for mathematicians and physicists alike, the focus is more on mathematical developments. This volume consists of four parts: The first Part covers local aspects of perturbative quantum field theory, with an emphasis on the axiomatization of the algebra behind the operator product expansion. The second Part highlights Chern-Simons gauge theories, while the third examines (semi-)classical field theories. In closing, Part 4 addresses factorization homology and factorization algebras.

Towards the Mathematics of Quantum Field Theory


Author: Frederic Paugam
Publisher: Springer Science & Business Media
ISBN: 3319045644
Category: Science
Page: 487
View: 6334

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This ambitious and original book sets out to introduce to mathematicians (even including graduate students ) the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in use. This in turn promotes the interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, though mathematicians are the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature. The first part of the book introduces the mathematical methods needed to work with the physicists' spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications to general gauge theories. The second part presents a large family of examples of classical field theories, both from experimental and theoretical physics, while the third part provides an introduction to quantum field theory, presents various renormalization methods, and discusses the quantization of factorization algebras.

Uncommon Paths in Quantum Physics


Author: Konstantin V. Kazakov
Publisher: Elsevier
ISBN: 0128015985
Category: Science
Page: 206
View: 2502

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Quantum mechanics is one of the most fascinating, and at the same time most controversial, branches of contemporary science. Disputes have accompanied this science since its birth and have not ceased to this day. Uncommon Paths in Quantum Physics allows the reader to contemplate deeply some ideas and methods that are seldom met in the contemporary literature. Instead of widespread recipes of mathematical physics, based on the solutions of integro-differential equations, the book follows logical and partly intuitional derivations of non-commutative algebra. Readers can directly penetrate the abstract world of quantum mechanics. First book in the market that treats this newly developed area of theoretical physics; the book will thus provide a fascinating overview of the prospective applications of this area, strongly founded on the theories and methods that it describes. Provides a solid foundation for the application of quantum theory to current physical problems arising in the interpretation of molecular spectra and important effects in quantum field theory. New insight into the physics of anharmonic vibrations, more feasible calculations with improved precision.

Factorization Method in Quantum Mechanics


Author: Shi-Hai Dong
Publisher: Springer Science & Business Media
ISBN: 1402057962
Category: Science
Page: 289
View: 1049

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This book introduces the factorization method in quantum mechanics at an advanced level, with the aim of putting mathematical and physical concepts and techniques like the factorization method, Lie algebras, matrix elements and quantum control at the reader’s disposal. For this purpose, the text provides a comprehensive description of the factorization method and its wide applications in quantum mechanics which complements the traditional coverage found in quantum mechanics textbooks.

Form Factors in Completely Integrable Models of Quantum Field Theory


Author: F A Smirnov
Publisher: World Scientific
ISBN: 9814506907
Category: Science
Page: 224
View: 5189

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' The monograph summarizes recent achievements in the calculation of matrix elements of local operators (form factors) for completely integrable models. Particularly, it deals with sine-Gordon, chiral Gross-Neven and O(3) nonlinear s models. General requirements on form factors are formulated and explicit formulas for form factors of most fundamental local operators are presented for the above mentioned models. Contents:Completely Integrable Models of Quantum Field TheoryThe Space of Physical StatesThe Necessary Properties of Form FactorsThe Local Commutativity TheoremSoliton Form Factors in SG ModelThe Main Properties of the Soliton Form FactorsBreathers Form Factors in SG ModelProperties of the Operators jμ, Tμν, exp(± iβu/2) in SG ModelForm Factors in SU(2)-Invariant Thirring ModelForm Factors in O(3)-Nonlinear σ-modelAsymptotics of Form FactorsCurrent AlgebrasForm Factors in SU(N) — Invariant Thirring Model (SU(N) Chiral Gross-Neveu Model)Phenomenological Reasonings Readership: Mathematical physicists. Keywords:Integrable;Quantum Field Theory in Two Dimensions;S-Matrix;Existence of Completely Integrable Models;Scattering Operator;Many-Particle Scattering;Yang-Baxter Triangle Equation;Soluable Lattice Models of Classical Statistical Mechanics;Form Factors;Zamolodchikov-Faddeev Approach;SU(2)-Invariant Thirring Model;Kinks;O(3)-Invariant σ-Model “It will be of great help to those who look for a reliable source of the numerous detailed calculations that have been performed over the years by many experts.” Mathematics Abstracts '

Mathematical Foundations of Quantum Field Theory and Perturbative String Theory


Author: Hisham Sati,Urs Schreiber
Publisher: American Mathematical Soc.
ISBN: 0821851950
Category: Mathematics
Page: 354
View: 1532

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Conceptual progress in fundamental theoretical physics is linked with the search for the suitable mathematical structures that model the physical systems. Quantum field theory (QFT) has proven to be a rich source of ideas for mathematics for a long time. However, fundamental questions such as ``What is a QFT?'' did not have satisfactory mathematical answers, especially on spaces with arbitrary topology, fundamental for the formulation of perturbative string theory. This book contains a collection of papers highlighting the mathematical foundations of QFT and its relevance to perturbative string theory as well as the deep techniques that have been emerging in the last few years. The papers are organized under three main chapters: Foundations for Quantum Field Theory, Quantization of Field Theories, and Two-Dimensional Quantum Field Theories. An introduction, written by the editors, provides an overview of the main underlying themes that bind together the papers in the volume.

Quantum Field Theory for Mathematicians


Author: Robin Ticciati,Robin (Maharishi University of Management Ticciati, Iowa)
Publisher: Cambridge University Press
ISBN: 9780521632652
Category: Mathematics
Page: 699
View: 9990

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This should be a useful reference for anybody with an interest in quantum theory.

Quantum Field Theory

A Tourist Guide for Mathematicians
Author: G. B. Folland
Publisher: American Mathematical Soc.
ISBN: 0821847058
Category: Mathematics
Page: 325
View: 1912

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Quantum field theory has been a great success for physics, but it is difficult for mathematicians to learn because it is mathematically incomplete. Folland, who is a mathematician, has spent considerable time digesting the physical theory and sorting out the mathematical issues in it. Fortunately for mathematicians, Folland is a gifted expositor. The purpose of this book is to present the elements of quantum field theory, with the goal of understanding the behavior of elementary particles rather than building formal mathematical structures, in a form that will be comprehensible to mathematicians. Rigorous definitions and arguments are presented as far as they are available, but the text proceeds on a more informal level when necessary, with due care in identifying the difficulties. The book begins with a review of classical physics and quantum mechanics, then proceeds through the construction of free quantum fields to the perturbation-theoretic development of interacting field theory and renormalization theory, with emphasis on quantum electrodynamics. The final two chapters present the functional integral approach and the elements of gauge field theory, including the Salam-Weinberg model of electromagnetic and weak interactions.

An Introduction to Quantum Field Theory


Author: George Sterman
Publisher: Cambridge University Press
ISBN: 9780521311328
Category: Science
Page: 572
View: 8166

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This is a systematic presentation of Quantum Field Theory from first principles, emphasizing both theoretical concepts and experimental applications. Starting from introductory quantum and classical mechanics, this book develops the quantum field theories that make up the "Standard Model" of elementary processes. It derives the basic techniques and theorems that underly theory and experiment, including those that are the subject of theoretical development. Special attention is also given to the derivations of cross sections relevant to current high-energy experiments and to perturbative quantum chromodynamics, with examples drawn from electron-positron annihilation, deeply inelastic scattering and hadron-hadron scattering. The first half of the book introduces the basic ideas of field theory. The discussion of mathematical issues is everywhere pedagogical and self contained. Topics include the role of internal symmetry and relativistic invariance, the path integral, gauge theories and spontaneous symmetry breaking, and cross sections in the Standard Model and the parton model. The material of this half is sufficient for an understanding of the Standard Model and its basic experimental consequences. The second half of the book deals with perturbative field theory beyond the lowest-order approximation. The issues of renormalization and unitarity, the renormalization group and asymptotic freedom, infrared divergences in quantum electrodynamics and infrared safety in quantum chromodynamics, jets, the perturbative basis of factorization at high energy and the operator product expansion are discussed. Exercises are included for each chapter, and several appendices complement the text.

String-Math 2013


Author: Ron Donagi, Michael R. Douglas,Ljudmila Kamenova,Martin Rocek
Publisher: American Mathematical Soc.
ISBN: 1470410516
Category: Mathematics
Page: 370
View: 8461

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This volume contains the proceedings of the conference `String-Math 2013' which was held June 17-21, 2013 at the Simons Center for Geometry and Physics at Stony Brook University. This was the third in a series of annual meetings devoted to the interface of mathematics and string theory. Topics include the latest developments in supersymmetric and topological field theory, localization techniques, the mathematics of quantum field theory, superstring compactification and duality, scattering amplitudes and their relation to Hodge theory, mirror symmetry and two-dimensional conformal field theory, and many more. This book will be important reading for researchers and students in the area, and for all mathematicians and string theorists who want to update themselves on developments in the math-string interface.

Explorations in Quantum Computing


Author: Colin P. Williams
Publisher: Springer Science & Business Media
ISBN: 9781846288876
Category: Computers
Page: 717
View: 4104

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By the year 2020, the basic memory components of a computer will be the size of individual atoms. At such scales, the current theory of computation will become invalid. "Quantum computing" is reinventing the foundations of computer science and information theory in a way that is consistent with quantum physics - the most accurate model of reality currently known. Remarkably, this theory predicts that quantum computers can perform certain tasks breathtakingly faster than classical computers – and, better yet, can accomplish mind-boggling feats such as teleporting information, breaking supposedly "unbreakable" codes, generating true random numbers, and communicating with messages that betray the presence of eavesdropping. This widely anticipated second edition of Explorations in Quantum Computing explains these burgeoning developments in simple terms, and describes the key technological hurdles that must be overcome to make quantum computers a reality. This easy-to-read, time-tested, and comprehensive textbook provides a fresh perspective on the capabilities of quantum computers, and supplies readers with the tools necessary to make their own foray into this exciting field. Topics and features: concludes each chapter with exercises and a summary of the material covered; provides an introduction to the basic mathematical formalism of quantum computing, and the quantum effects that can be harnessed for non-classical computation; discusses the concepts of quantum gates, entangling power, quantum circuits, quantum Fourier, wavelet, and cosine transforms, and quantum universality, computability, and complexity; examines the potential applications of quantum computers in areas such as search, code-breaking, solving NP-Complete problems, quantum simulation, quantum chemistry, and mathematics; investigates the uses of quantum information, including quantum teleportation, superdense coding, quantum data compression, quantum cloning, quantum negation, and quantum cryptography; reviews the advancements made towards practical quantum computers, covering developments in quantum error correction and avoidance, and alternative models of quantum computation. This text/reference is ideal for anyone wishing to learn more about this incredible, perhaps "ultimate," computer revolution. Dr. Colin P. Williams is Program Manager for Advanced Computing Paradigms at the NASA Jet Propulsion Laboratory, California Institute of Technology, and CEO of Xtreme Energetics, Inc. an advanced solar energy company. Dr. Williams has taught quantum computing and quantum information theory as an acting Associate Professor of Computer Science at Stanford University. He has spent over a decade inspiring and leading high technology teams and building business relationships with and Silicon Valley companies. Today his interests include terrestrial and Space-based power generation, quantum computing, cognitive computing, computational material design, visualization, artificial intelligence, evolutionary computing, and remote olfaction. He was formerly a Research Scientist at Xerox PARC and a Research Assistant to Prof. Stephen W. Hawking, Cambridge University.

The Large N Expansion in Quantum Field Theory and Statistical Physics

From Spin Systems to 2-Dimensional Gravity
Author: E Brézin,S R Wadia
Publisher: World Scientific
ISBN: 981450663X
Category: Science
Page: 1114
View: 7550

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Readership: High energy physicists and condensed matter physicists. keywords:Quantum Field Theory;Spontaneous Symmetry Breaking;Phase Transitions;Gauge Theories;Topological Expansions;Matrix Models

Mathematical Methods in Quantum Mechanics

With Applications to Schrödinger Operators
Author: Gerald Teschl
Publisher: American Mathematical Soc.
ISBN: 0821846604
Category: Mathematics
Page: 305
View: 8148

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Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. It is well suited for self-study and includes numerous exercises (many with hints).

Classical and Quantum Computation


Author: Alexei Yu. Kitaev,Alexander Shen,Mikhail N. Vyalyi
Publisher: American Mathematical Soc.
ISBN: 0821832298
Category: Mathematics
Page: 257
View: 4723

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This book presents a concise introduction to an emerging and increasingly important topic, the theory of quantum computing. The development of quantum computing exploded in 1994 with the discovery of its use in factoring large numbers--an extremely difficult and time-consuming problem when using a conventional computer. In less than 300 pages, the authors set forth a solid foundation to the theory, including results that have not appeared elsewhere and improvements on existing works. The book starts with the basics of classical theory of computation, including NP-complete problems and the idea of complexity of an algorithm. Then the authors introduce general principles of quantum computing and pass to the study of main quantum computation algorithms: Grover's algorithm, Shor's factoring algorithm, and the Abelian hidden subgroup problem. In concluding sections, several related topics are discussed (parallel quantum computation, a quantum analog of NP-completeness, and quantum error-correcting codes). This is a suitable textbook for a graduate course in quantum computing. Prerequisites are very modest and include linear algebra, elements of group theory and probability, and the notion of an algorithm (on a formal or an intuitive level). The book is complete with problems, solutions, and an appendix summarizing the necessary results from number theory.