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Author: Mircea Puta
Publisher: Springer Science & Business Media
ISBN: 9401119929
Category : Mathematics
Languages : en
Pages : 289
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Book Description
This volume presents various aspects of the geometry of symplectic and Poisson manifolds, and applications in Hamiltonian mechanics and geometric quantization are indicated. Chapter 1 presents some general facts about symplectic vector space, symplectic manifolds and symplectic reduction. Chapter 2 deals with the study of Hamiltonian mechanics. Chapter 3 considers some standard facts concerning Lie groups and algebras which lead to the theory of momentum mappings and the Marsden--Weinstein reduction. Chapters 4 and 5 consider the theory and the stability of equilibrium solutions of Hamilton--Poisson mechanical systems. Chapters 6 and 7 are devoted to the theory of geometric quantization. This leads, in Chapter 8, to topics such as foliated cohomology, the theory of the Dolbeault--Kostant complex, and their applications. A discussion of the relation between geometric quantization and the Marsden--Weinstein reduction is presented in Chapter 9. The final chapter considers extending the theory of geometric quantization to Poisson manifolds, via the theory of symplectic groupoids. Each chapter concludes with problems and solutions, many of which present significant applications and, in some cases, major theorems. For graduate students and researchers whose interests and work involve symplectic geometry and Hamiltonian mechanics.
Author: Mircea Puta
Publisher: Springer Science & Business Media
ISBN: 9401119929
Category : Mathematics
Languages : en
Pages : 289
Get Book Here
Book Description
This volume presents various aspects of the geometry of symplectic and Poisson manifolds, and applications in Hamiltonian mechanics and geometric quantization are indicated. Chapter 1 presents some general facts about symplectic vector space, symplectic manifolds and symplectic reduction. Chapter 2 deals with the study of Hamiltonian mechanics. Chapter 3 considers some standard facts concerning Lie groups and algebras which lead to the theory of momentum mappings and the Marsden--Weinstein reduction. Chapters 4 and 5 consider the theory and the stability of equilibrium solutions of Hamilton--Poisson mechanical systems. Chapters 6 and 7 are devoted to the theory of geometric quantization. This leads, in Chapter 8, to topics such as foliated cohomology, the theory of the Dolbeault--Kostant complex, and their applications. A discussion of the relation between geometric quantization and the Marsden--Weinstein reduction is presented in Chapter 9. The final chapter considers extending the theory of geometric quantization to Poisson manifolds, via the theory of symplectic groupoids. Each chapter concludes with problems and solutions, many of which present significant applications and, in some cases, major theorems. For graduate students and researchers whose interests and work involve symplectic geometry and Hamiltonian mechanics.
Author: G. Giachetta
Publisher: World Scientific
ISBN: 9814313726
Category : Science
Languages : en
Pages : 405
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Book Description
The geometric formulation of autonomous Hamiltonian mechanics in the terms of symplectic and Poisson manifolds is generally accepted. This book provides the geometric formulation of non-autonomous mechanics in a general setting of time-dependent coordinate and reference frame transformations.
Author: Sean Bates
Publisher: American Mathematical Soc.
ISBN: 9780821807989
Category : Mathematics
Languages : en
Pages : 150
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Book Description
These notes are based on a course entitled ``Symplectic Geometry and Geometric Quantization'' taught by Alan Weinstein at the University of California, Berkeley (fall 1992) and at the Centre Emile Borel (spring 1994). The only prerequisite for the course needed is a knowledge of the basic notions from the theory of differentiable manifolds (differential forms, vector fields, transversality, etc.). The aim is to give students an introduction to the ideas of microlocal analysis and the related symplectic geometry, with an emphasis on the role these ideas play in formalizing the transition between the mathematics of classical dynamics (hamiltonian flows on symplectic manifolds) and quantum mechanics (unitary flows on Hilbert spaces). These notes are meant to function as a guide to the literature. The authors refer to other sources for many details that are omitted and can be bypassed on a first reading.
Author: S Twareque Ali
Publisher: World Scientific
ISBN: 9814522031
Category :
Languages : en
Pages : 256
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Book Description
The aim of the conference was to find common elements between quantization and coherent states, and quantization on Poisson manifolds. Topics included are coherent states, geometric quantization, phase space quantization, deformation and *-products and Berry's phase.
Author: Leon Armenovich Takhtadzhi͡an
Publisher: American Mathematical Soc.
ISBN: 0821846302
Category : Mathematics
Languages : en
Pages : 410
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Book Description
Presents a comprehensive treatment of quantum mechanics from a mathematics perspective. Including traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin.
Author: C. T. J. Dodson
Publisher: Springer Science & Business Media
ISBN: 9780792342939
Category : Mathematics
Languages : en
Pages : 428
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Book Description
This book arose from courses taught by the authors, and is designed for both instructional and reference use during and after a first course in algebraic topology. It is a handbook for users who want to calculate, but whose main interests are in applications using the current literature, rather than in developing the theory. Typical areas of applications are differential geometry and theoretical physics. We start gently, with numerous pictures to illustrate the fundamental ideas and constructions in homotopy theory that are needed in later chapters. We show how to calculate homotopy groups, homology groups and cohomology rings of most of the major theories, exact homotopy sequences of fibrations, some important spectral sequences, and all the obstructions that we can compute from these. Our approach is to mix illustrative examples with those proofs that actually develop transferable calculational aids. We give extensive appendices with notes on background material, extensive tables of data, and a thorough index. Audience: Graduate students and professionals in mathematics and physics.
Author: R. Carroll
Publisher: Elsevier
ISBN: 0080540082
Category : Mathematics
Languages : en
Pages : 421
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Book Description
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.
Author: Stefano Marchiafava
Publisher: World Scientific
ISBN: 9810246307
Category : Mathematics
Languages : en
Pages : 486
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Book Description
During the last five years, after the first meeting on ?Quaternionic Structures in Mathematics and Physics?, interest in quaternionic geometry and its applications has continued to increase. Progress has been made in constructing new classes of manifolds with quaternionic structures (quaternionic Khler, hyper-Khler, hyper-complex, etc.), studying the differential geometry of special classes of such manifolds and their submanifolds, understanding relations between the quaternionic structure and other differential-geometric structures, and also in physical applications of quaternionic geometry. Some generalizations of classical quaternion-like structures (like HKT structures and hyper-Khler manifolds with singularities) appeared naturally and were studied. Some of those results are published in this book.
Author: L. Tamássy
Publisher: Springer Science & Business Media
ISBN: 9400901496
Category : Mathematics
Languages : en
Pages : 427
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Book Description
Proceedings of the Colloquium on Differential Geometry, Debrecen, Hungary, July 26-30, 1994
Author: Nikolai Tarkhanov
Publisher: Springer Science & Business Media
ISBN: 9401103275
Category : Mathematics
Languages : en
Pages : 407
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Book Description
This book gives a systematic account of the facts concerning complexes of differential operators on differentiable manifolds. The central place is occupied by the study of general complexes of differential operators between sections of vector bundles. Although the global situation often contains nothing new as compared with the local one (that is, complexes of partial differential operators on an open subset of ]Rn), the invariant language allows one to simplify the notation and to distinguish better the algebraic nature of some questions. In the last 2 decades within the general theory of complexes of differential operators, the following directions were delineated: 1) the formal theory; 2) the existence theory; 3) the problem of global solvability; 4) overdetermined boundary problems; 5) the generalized Lefschetz theory of fixed points, and 6) the qualitative theory of solutions of overdetermined systems. All of these problems are reflected in this book to some degree. It is superfluous to say that different directions sometimes whimsically intersect. Considerable attention is given to connections and parallels with the theory of functions of several complex variables. One of the reproaches avowed beforehand by the author consists of the shortage of examples. The framework of the book has not permitted their number to be increased significantly. Certain parts of the book consist of results obtained by the author in 1977-1986. They have been presented in seminars in Krasnoyarsk, Moscow, Ekaterinburg, and N ovosi birsk.
