Deformation Quantization
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Deformation Quantization
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Author : Gilles Halbout
language : en
Publisher: Walter de Gruyter
Release Date : 2012-10-25
Deformation Quantization written by Gilles Halbout and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-10-25 with Mathematics categories.
This book contains eleven refereed research papers on deformation quantization by leading experts in the respective fields. These contributions are based on talks presented on the occasion of the meeting between mathematicians and theoretical physicists held in Strasbourg in May 2001. Topics covered are: star-products over Poisson manifolds, quantization of Hopf algebras, index theorems, globalization and cohomological problems. Both the mathematical and the physical approach ranging from asymptotic quantum electrodynamics to operads and prop theory will be presented. Historical remarks and surveys set the results presented in perspective. Directed at research mathematicians and theoretical physicists as well as graduate students, the volume will give an overview of a field of research that has seen enourmous acticity in the last years, with new ties to many other areas of mathematics and physics.
Kontsevich S Deformation Quantization And Quantum Field Theory
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Author : Nima Moshayedi
language : en
Publisher: Springer Nature
Release Date : 2022-08-11
Kontsevich S Deformation Quantization And Quantum Field Theory written by Nima Moshayedi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-08-11 with Mathematics categories.
This book provides an introduction to deformation quantization and its relation to quantum field theory, with a focus on the constructions of Kontsevich and Cattaneo & Felder. This subject originated from an attempt to understand the mathematical structure when passing from a commutative classical algebra of observables to a non-commutative quantum algebra of observables. Developing deformation quantization as a semi-classical limit of the expectation value for a certain observable with respect to a special sigma model, the book carefully describes the relationship between the involved algebraic and field-theoretic methods. The connection to quantum field theory leads to the study of important new field theories and to insights in other parts of mathematics such as symplectic and Poisson geometry, and integrable systems. Based on lectures given by the author at the University of Zurich, the book will be of interest to graduate students in mathematics or theoretical physics. Readers will be able to begin the first chapter after a basic course in Analysis, Linear Algebra and Topology, and references are provided for more advanced prerequisites.
Poisson Geometry Deformation Quantisation And Group Representations
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Author : Simone Gutt
language : en
Publisher: Cambridge University Press
Release Date : 2005-06-21
Poisson Geometry Deformation Quantisation And Group Representations written by Simone Gutt and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-06-21 with Mathematics categories.
An accessible introduction to Poisson geometry suitable for graduate students.
Lectures On Deformation Quantisation From Moyal Product To Kontsevich S Formality Theorem
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Author : Georgy Igorevich Sharygin
language : en
Publisher: World Scientific
Release Date : 2025-05-20
Lectures On Deformation Quantisation From Moyal Product To Kontsevich S Formality Theorem written by Georgy Igorevich Sharygin and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-05-20 with Science categories.
Principles of classical Hamiltonian mechanics say that the evolution of a dynamical system is determined by the Poisson bracket of observable functions with the given Hamiltonian function of the system. In Quantum Mechanics, these principles are modified so that the algebra of observable functions should be replaced by a noncommutative algebra of operators and the Poisson bracket by their commutator so that the canonical commutation relations hold. Thus, working with quantum systems, we must determine the 'quantisation' of our observables, i.e. to choose a noncommutative algebra whose elements would play the role of the observables. With some modifications, this question is the main content of the Deformation Quantisation problem formulated in 1978 by Flato and others.This book is based on the course that the author taught in the Fall semester of 2019 at Peking University. The main purpose of that course and of this book is to acquaint the reader with the vast scope of ideas related to the Deformation Quantisation of Poisson manifolds. The book begins with Quantum Mechanics and Moyal product formula and covers the three main constructions that solve the Deformation Quantisation problem: Lecomte and de Wilde deformation of symplectic manifolds, Fedosov's Quantisation theory and Kontsevich's formality theorem. In the appendices, the Tamarkin's proof of formality theorem is outlined.The book is written in a reader-friendly manner and is as self-contained as possible. It includes several sets of problems and exercises that will help the reader to master the material.
Deformation Quantization For Actions Of R D
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Author : Marc Aristide Rieffel
language : en
Publisher: American Mathematical Soc.
Release Date : 1993
Deformation Quantization For Actions Of R D written by Marc Aristide Rieffel and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with Mathematics categories.
This work describes a general construction of a deformation quantization for any Poisson bracket on a manifold which comes from an action of R ]d on that manifold. These deformation quantizations are strict, in the sense that the deformed product of any two functions is again a function and that there are corresponding involutions and operator norms. Many of the techniques involved are adapted from the theory of pseudo-differential operators. The construction is shown to have many favorable properties. A number of specific examples are described, ranging from basic ones such as quantum disks, quantum tori, and quantum spheres, to aspects of quantum groups.
Morita Equivalence In Deformation Quantization
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Author : Henrique Bursztyn
language : en
Publisher:
Release Date : 2001
Morita Equivalence In Deformation Quantization written by Henrique Bursztyn and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with categories.
Quantization Geometry And Noncommutative Structures In Mathematics And Physics
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Author : Alexander Cardona
language : en
Publisher: Springer
Release Date : 2017-10-26
Quantization Geometry And Noncommutative Structures In Mathematics And Physics written by Alexander Cardona and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-10-26 with Science categories.
This monograph presents various ongoing approaches to the vast topic of quantization, which is the process of forming a quantum mechanical system starting from a classical one, and discusses their numerous fruitful interactions with mathematics.The opening chapter introduces the various forms of quantization and their interactions with each other and with mathematics.A first approach to quantization, called deformation quantization, consists of viewing the Planck constant as a small parameter. This approach provides a deformation of the structure of the algebra of classical observables rather than a radical change in the nature of the observables. When symmetries come into play, deformation quantization needs to be merged with group actions, which is presented in chapter 2, by Simone Gutt.The noncommutativity arising from quantization is the main concern of noncommutative geometry. Allowing for the presence of symmetries requires working with principal fiber bundles in a non-commutative setup, where Hopf algebras appear naturally. This is the topic of chapter 3, by Christian Kassel. Nichols algebras, a special type of Hopf algebras, are the subject of chapter 4, by Nicolás Andruskiewitsch. The purely algebraic approaches given in the previous chapters do not take the geometry of space-time into account. For this purpose a special treatment using a more geometric point of view is required. An approach to field quantization on curved space-time, with applications to cosmology, is presented in chapter 5 in an account of the lectures of Abhay Ashtekar that brings a complementary point of view to non-commutativity.An alternative quantization procedure is known under the name of string theory. In chapter 6 its supersymmetric version is presented. Superstrings have drawn the attention of many mathematicians, due to its various fruitful interactions with algebraic geometry, some of which are described here. The remaining chapters discuss further topics, as the Batalin-Vilkovisky formalism and direct products of spectral triples.This volume addresses both physicists and mathematicians and serves as an introduction to ongoing research in very active areas of mathematics and physics at the border line between geometry, topology, algebra and quantum field theory.
Deformation Quantization Of Some Non Compact Solvable Lie Groups And Their Representation Theory
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Author : Byung-Jay Kahng
language : en
Publisher:
Release Date : 1997
Deformation Quantization Of Some Non Compact Solvable Lie Groups And Their Representation Theory written by Byung-Jay Kahng and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with categories.
Formality Theory
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Author : Chiara Esposito
language : en
Publisher: Springer
Release Date : 2014-09-04
Formality Theory written by Chiara Esposito and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-04 with Science categories.
This book is a survey of the theory of formal deformation quantization of Poisson manifolds, in the formalism developed by Kontsevich. It is intended as an educational introduction for mathematical physicists who are dealing with the subject for the first time. The main topics covered are the theory of Poisson manifolds, star products and their classification, deformations of associative algebras and the formality theorem. Readers will also be familiarized with the relevant physical motivations underlying the purely mathematical construction.
Mathematical Aspects Of Quantization
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Author : Sam Evens
language : en
Publisher: American Mathematical Soc.
Release Date : 2012
Mathematical Aspects Of Quantization written by Sam Evens and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
This book is a collection of expository articles from the Center of Mathematics at Notre Dame's 2011 program on quantization. Included are lecture notes from a summer school on quantization on topics such as the Cherednik algebra, geometric quantization, detailed proofs of Willwacher's results on the Kontsevich graph complex, and group-valued moment maps. This book also includes expository articles on quantization and automorphic forms, renormalization, Berezin-Toeplitz quantization in the complex setting, and the commutation of quantization with reduction, as well as an original article on derived Poisson brackets. The primary goal of this volume is to make topics in quantization more accessible to graduate students and researchers.