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Geometric Quantization
The geometric approach to quantization was introduced by Konstant and Souriau more than 20 years ago. It has given valuable and lasting insights into the relationship between classical and quantum systems, and continues to be a popular research topic. The ideas have proved useful in pure mathematics, notably in representation theory, as well as in theoretical physics. The most recent applications have been in conformal field theory and in the Jones-Witten theory of knots. The successful original edition of this book was published in 1980. Now it has been completely revised and extensively rewritten. The presentation has been simplified and many new examples have been added. The material on field theory has been expanded.
Geometric Quantization and Quantum Mechanics
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 th...
Geometric Quantization in Action
Approach your problems from the right It isn't that they can't see the solution. It end and begin with the answers. Then, is that they can't see the problem. one day, perhaps you will fmd the final question. G. K. Chesterton, The Scandal of Father Brown 'The Point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. Van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate ar...
Lectures on the Geometry of Quantization
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.
Hamiltonian Mechanical Systems and Geometric Quantization
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. Thi...
Geometry, Topology and Quantization
This is a monograph on geometrical and topological features which arise in various quantization procedures. Quantization schemes consider the feasibility of arriving at a quantum system from a classical one and these involve three major procedures viz. i) geometric quantization, ii) Klauder quantization, and iii) stochastic quanti zation. In geometric quantization we have to incorporate a hermitian line bundle to effectively generate the quantum Hamiltonian operator from a classical Hamil tonian. Klauder quantization also takes into account the role of the connection one-form along with coordinate independence. In stochastic quantization as pro posed by Nelson, Schrodinger equation is derive...
Loop Spaces, Characteristic Classes and Geometric Quantization
This book examines the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kähler geometry of the space of knots, and Cheeger--Chern--Simons secondary characteristics classes. It also covers the Dirac monopole and Dirac’s quantization of the electrical charge.
Loop Spaces, Characteristic Classes and Geometric Quantization
This book examines the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kähler geometry of the space of knots, and Cheeger--Chern--Simons secondary characteristics classes. It also covers the Dirac monopole and Dirac’s quantization of the electrical charge.
The Metaplectic Representation, $Mp^c$ Structures and Geometric Quantization
We present an account of the metaplectic representation in terms of the Bargmann-Segal model rather than the standard Schrödinger model; this allows us to describe explicitly the structure and pairing of vacuum states for positive polarizations. We develop a scheme for geometric spinors and, following Hess, using Mp[superscript italic]c structures in place of metaplectic structures.
