Integrability Quantization And Geometry Ii Quantum Theories And Algebraic Geometry

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Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry

Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman
Integrability Quantization and Geometry II Quantum Theories and Algebraic Geometry Book PDF

Author : Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman
Publisher : American Mathematical Soc.
Page : 480 pages
File Size : 46,5 Mb
Release : 2021-04-12
Category : Education
ISBN : 9781470455927

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Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry by Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman Pdf

This book is a collection of articles written in memory of Boris Dubrovin (1950–2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions to this collection of papers are split into two parts: “Integrable Systems” and “Quantum Theories and Algebraic Geometry”, reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.

Integrability, Quantization, and Geometry

Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman
Integrability Quantization and Geometry Book PDF

Author : Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman
Publisher : Unknown
Page : 983 pages
File Size : 44,7 Mb
Release : 2024-06-16
Category : Electronic
ISBN : 1470455900

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Integrability, Quantization, and Geometry by Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman Pdf

This two-volume set containts parts I and II. Each volume is a collection of articles written in memory of Boris Dubrovin (1950-2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions are split into two parts: ``Integrable Systems'' and ``Quantum Theories and Algebraic Geometry'', reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.

Integrability, Quantization, and Geometry: I. Integrable Systems

Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman
Integrability Quantization and Geometry I Integrable Systems Book PDF

Author : Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman
Publisher : American Mathematical Soc.
Page : 0 pages
File Size : 51,9 Mb
Release : 2021-04-12
Category : Education
ISBN : 1470455919

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Integrability, Quantization, and Geometry: I. Integrable Systems by Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman Pdf

This book is a collection of articles written in memory of Boris Dubrovin (1950–2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions to this collection of papers are split into two parts: “Integrable Systems” and “Quantum Theories and Algebraic Geometry”, reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.

Quantum Theory, Deformation and Integrability

R. Carroll
Quantum Theory Deformation and Integrability Book PDF

Author : R. Carroll
Publisher : Elsevier
Page : 420 pages
File Size : 49,5 Mb
Release : 2000-11-09
Category : Science
ISBN : 0080540082

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Quantum Theory, Deformation and Integrability by R. Carroll Pdf

About four years ago a prominent string theorist was quoted as saying that it might be possible to understand quantum mechanics by the year 2000. Sometimes new mathematical developments make such understanding appear possible and even close, but on the other hand, increasing lack of experimental verification make it seem to be further distant. In any event one seems to arrive at new revolutions in physics and mathematics every year. This book hopes to convey some of the excitment of this period, but will adopt a relatively pedestrian approach designed to illuminate the relations between quantum and classical. There will be some discussion of philosophical matters such as measurement, uncertainty, decoherence, etc. but philosophy will not be emphasized; generally we want to enjoy the fruits of computation based on the operator formulation of QM and quantum field theory. In Chapter 1 connections of QM to deterministic behavior are exhibited in the trajectory representations of Faraggi-Matone. Chapter 1 also includes a review of KP theory and some preliminary remarks on coherent states, density matrices, etc. and more on deterministic theory. We develop in Chapter 4 relations between quantization and integrability based on Moyal brackets, discretizations, KP, strings and Hirota formulas, and in Chapter 2 we study the QM of embedded curves and surfaces illustrating some QM effects of geometry. Chapter 3 is on quantum integrable systems, quantum groups, and modern deformation quantization. Chapter 5 involves the Whitham equations in various roles mediating between QM and classical behavior. In particular, connections to Seiberg-Witten theory (arising in N = 2 supersymmetric (susy) Yang-Mills (YM) theory) are discussed and we would still like to understand more deeply what is going on. Thus in Chapter 5 we will try to give some conceptual background for susy, gauge theories, renormalization, etc. from both a physical and mathematical point of view. In Chapter 6 we continue the deformation quantization then by exhibiting material based on and related to noncommutative geometry and gauge theory.

Quantization, Geometry and Noncommutative Structures in Mathematics and Physics

Alexander Cardona,Pedro Morales,Hernán Ocampo,Sylvie Paycha,Andrés F. Reyes Lega
Quantization Geometry and Noncommutative Structures in Mathematics and Physics Book PDF

Author : Alexander Cardona,Pedro Morales,Hernán Ocampo,Sylvie Paycha,Andrés F. Reyes Lega
Publisher : Springer
Page : 341 pages
File Size : 52,6 Mb
Release : 2017-11-06
Category : Science
ISBN : 3319654268

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Quantization, Geometry and Noncommutative Structures in Mathematics and Physics by Alexander Cardona,Pedro Morales,Hernán Ocampo,Sylvie Paycha,Andrés F. Reyes Lega Pdf

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.

Arithmetic and Geometry Around Quantization

Özgür Ceyhan,Yu. I. Manin,Matilde Marcolli
Arithmetic and Geometry Around Quantization Book PDF

Author : Özgür Ceyhan,Yu. I. Manin,Matilde Marcolli
Publisher : Birkhäuser
Page : 292 pages
File Size : 51,6 Mb
Release : 2010-06-04
Category : Mathematics
ISBN : 0817648305

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Arithmetic and Geometry Around Quantization by Özgür Ceyhan,Yu. I. Manin,Matilde Marcolli Pdf

This volume comprises both research and survey articles originating from the conference on Arithmetic and Geometry around Quantization held in Istanbul in 2006. A wide range of topics related to quantization are covered, thus aiming to give a glimpse of a broad subject in very different perspectives.

Non-commutative Geometry in Mathematics and Physics

Giuseppe Dito
Non commutative Geometry in Mathematics and Physics Book PDF

Author : Giuseppe Dito
Publisher : American Mathematical Soc.
Page : 143 pages
File Size : 44,8 Mb
Release : 2008
Category : Mathematics
ISBN : 9780821841471

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Non-commutative Geometry in Mathematics and Physics by Giuseppe Dito Pdf

This volume represents the proceedings of the conference on Topics in Deformation Quantization and Non-Commutative Structures held in Mexico City in September 2005. It contains survey papers and original contributions by various experts in the fields of deformation quantization and non-commutative derived algebraic geometry in the interface between mathematics and physics. It also contains an article based on the XI Memorial Lectures given by M. Kontsevich, which were delivered as part of the conference. This is an excellent introductory volume for readers interested in learning about quantization as deformation, Hopf algebras, and Hodge structures in the framework of non-commutative algebraic geometry.

Quantization, Classical and Quantum Field Theory and Theta Functions

Andrej Tyurin
Quantization Classical and Quantum Field Theory and Theta Functions Book PDF

Author : Andrej Tyurin
Publisher : American Mathematical Soc.
Page : 150 pages
File Size : 42,8 Mb
Release : 2003
Category : Functions, Theta
ISBN : 9780821832400

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Quantization, Classical and Quantum Field Theory and Theta Functions by Andrej Tyurin Pdf

This book is written by a well-known expert in classical algebraic geometry. Tyurin's research was specifically in explicit computations to vector bundles on algebraic varieties. This is the only available monograph written from his unique viewpoint. Ordinary (abelian) theta functions describe properties of moduli spaces of one-dimensional vector bundles on algebraic curves. Non-abelian theta functions, which are the main topic of this book, play a similar role in the study of higher-dimensional vector bundles. The book presents various aspects of the theory of non-abelian theta functions and the moduli spaces of vector bundles, including their applications to problems of quantization and to classical and quantum conformal field theories. The book is an important source of information for specialists in algebraic geometry and its applications to mathematical aspects of quantum field theory.

Differential Geometry, Group Representations, and Quantization

Jörg Dieter Hennig,Wolfgang Lücke,Jiří Tolar
Differential Geometry Group Representations and Quantization Book PDF

Author : Jörg Dieter Hennig,Wolfgang Lücke,Jiří Tolar
Publisher : Springer
Page : 300 pages
File Size : 40,5 Mb
Release : 1991
Category : Mathematics
ISBN : UOM:39015021996056

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Differential Geometry, Group Representations, and Quantization by Jörg Dieter Hennig,Wolfgang Lücke,Jiří Tolar Pdf

Differential geometry and analytic group theory are among the most powerful tools in mathematical physics. This volume presents review articles on a wide variety of applications of these techniques in classical continuum physics, gauge theories, quantization procedures, and the foundations of quantum theory. The articles, written by leading scientists, address both researchers and grad- uate students in mathematics, physics, and philosophy of science.

Geometric And Algebraic Topological Methods In Quantum Mechanics

Luigi Mangiarotti,Gennadi A Sardanashvily,Giovanni Giachetta
Geometric And Algebraic Topological Methods In Quantum Mechanics Book PDF

Author : Luigi Mangiarotti,Gennadi A Sardanashvily,Giovanni Giachetta
Publisher : World Scientific
Page : 715 pages
File Size : 43,7 Mb
Release : 2005-01-27
Category : Science
ISBN : 9789814481144

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Geometric And Algebraic Topological Methods In Quantum Mechanics by Luigi Mangiarotti,Gennadi A Sardanashvily,Giovanni Giachetta Pdf

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry's geometric factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization.

Integrability, Quantization, and Geometry: I. Integrable Systems

Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman
Integrability Quantization and Geometry I Integrable Systems Book PDF

Author : Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman
Publisher : American Mathematical Soc.
Page : 516 pages
File Size : 48,8 Mb
Release : 2021-04-12
Category : Education
ISBN : 9781470455910

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Integrability, Quantization, and Geometry: I. Integrable Systems by Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman Pdf

This book is a collection of articles written in memory of Boris Dubrovin (1950–2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions to this collection of papers are split into two parts: “Integrable Systems” and “Quantum Theories and Algebraic Geometry”, reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.

From Quantum Cohomology to Integrable Systems

Martin A. Guest
From Quantum Cohomology to Integrable Systems Book PDF

Author : Martin A. Guest
Publisher : OUP Oxford
Page : 336 pages
File Size : 49,7 Mb
Release : 2008-03-13
Category : Mathematics
ISBN : 9780191606960

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From Quantum Cohomology to Integrable Systems by Martin A. Guest Pdf

Quantum cohomology has its origins in symplectic geometry and algebraic geometry, but is deeply related to differential equations and integrable systems. This text explains what is behind the extraordinary success of quantum cohomology, leading to its connections with many existing areas of mathematics as well as its appearance in new areas such as mirror symmetry. Certain kinds of differential equations (or D-modules) provide the key links between quantum cohomology and traditional mathematics; these links are the main focus of the book, and quantum cohomology and other integrable PDEs such as the KdV equation and the harmonic map equation are discussed within this unified framework. Aimed at graduate students in mathematics who want to learn about quantum cohomology in a broad context, and theoretical physicists who are interested in the mathematical setting, the text assumes basic familiarity with differential equations and cohomology.

Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds

A.K. Prykarpatsky,I.V. Mykytiuk
Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds Book PDF

Author : A.K. Prykarpatsky,I.V. Mykytiuk
Publisher : Springer Science & Business Media
Page : 555 pages
File Size : 55,8 Mb
Release : 2013-04-09
Category : Science
ISBN : 9789401149945

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Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds by A.K. Prykarpatsky,I.V. Mykytiuk Pdf

In recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets. Very often such Poisson structures on corresponding manifolds are canonical, which gives rise to the possibility of producing their hidden group theoretical essence for many completely integrable dynamical systems. It is a well understood fact that great part of comprehensive integrability theories of nonlinear dynamical systems on manifolds is based on Lie-algebraic ideas, by means of which, in particular, the classification of such compatibly bi Hamiltonian and isospectrally Lax type integrable systems has been carried out. Many chapters of this book are devoted to their description, but to our regret so far the work has not been completed. Hereby our main goal in each analysed case consists in separating the basic algebraic essence responsible for the complete integrability, and which is, at the same time, in some sense universal, i. e. , characteristic for all of them. Integrability analysis in the framework of a gradient-holonomic algorithm, devised in this book, is fulfilled through three stages: 1) finding a symplectic structure (Poisson bracket) transforming an original dynamical system into a Hamiltonian form; 2) finding first integrals (action variables or conservation laws); 3) defining an additional set of variables and some functional operator quantities with completely controlled evolutions (for instance, as Lax type representation).

Geometric Quantization and Quantum Mechanics

Jedrzej Sniatycki
Geometric Quantization and Quantum Mechanics Book PDF

Author : Jedrzej Sniatycki
Publisher : Springer Science & Business Media
Page : 241 pages
File Size : 48,6 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781461260660

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Geometric Quantization and Quantum Mechanics by Jedrzej Sniatycki Pdf

This book contains a revised and expanded version of the lecture notes of two seminar series given during the academic year 1976/77 at the Department of Mathematics and Statistics of the University of Calgary, and in the summer of 1978 at the Institute of Theoretical Physics of the Technical University Clausthal. The aim of the seminars was to present geometric quantization from the point of view· of its applica tions to quantum mechanics, and to introduce the quantum dynamics of various physical systems as the result of the geometric quantization of the classical dynamics of these systems. The group representation aspects of geometric quantiza tion as well as proofs of the existence and the uniqueness of the introduced structures can be found in the expository papers of Blattner, Kostant, Sternberg and Wolf, and also in the references quoted in these papers. The books of Souriau (1970) and Simms and Woodhouse (1976) present the theory of geometric quantization and its relationship to quantum mech anics. The purpose of the present book is to complement the preceding ones by including new developments of the theory and emphasizing the computations leading to results in quantum mechanics.

Quantum Groups and Noncommutative Spaces

Matilde Marcolli,Deepak Parashar
Quantum Groups and Noncommutative Spaces Book PDF

Author : Matilde Marcolli,Deepak Parashar
Publisher : Springer Science & Business Media
Page : 240 pages
File Size : 51,8 Mb
Release : 2010-11-02
Category : Mathematics
ISBN : 9783834898319

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Quantum Groups and Noncommutative Spaces by Matilde Marcolli,Deepak Parashar Pdf

This book is aimed at presenting different methods and perspectives in the theory of Quantum Groups, bridging between the algebraic, representation theoretic, analytic, and differential-geometric approaches. It also covers recent developments in Noncommutative Geometry, which have close relations to quantization and quantum group symmetries. The volume collects surveys by experts which originate from an acitvity at the Max-Planck-Institute for Mathematics in Bonn.