Buy geometric quantization oxford mathematical monographs on. The geometric approach to quantization was introduced by kostant and souriau more that twenty years ago. Loop spaces, characteristic classes and geometric quantization by jeanluc brylinski, 9780817647308, available at book depository with free delivery worldwide. However, rather than elaborating on the relation between quantum and classical mechanics, geometric quantization unearthed a large. He has been one of the principal developers of 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. Buy geometric quantization oxford mathematical monographs second by n. This integral transform can be understood as a geometric quantization of the kostantsekiguchi correspondence. In this spirit, this book develops the differential geometry associated to the topology and obstruction theory of certain fiber bundles more precisely, associated to grebes. Geometric quantization of chern simons gauge theory 789 are associated with the jones polynomial, from the point of view of the three dimensional quantum field theory. Geometric quantization is one formalization of the notion of quantization of a classical mechanical systemclassical field theory to a quantum mechanical systemquantum field theory.
The new theory is a 3dimensional analog of the familiar kostantweil theory of line bundles. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. The word quantization is used both in physical and in mathematical works in many different senses. On geometric quantization and its connections with the. This establishes a geometric quantization of the nilpotent kcorbit okc. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, kahler geometry of the space of knots, cheegerchernsimons. In this theory the additional structure needed for quantization is explicitely defined in terms of the geometry of the phase space, and it consists of a prequantization, a polarization and a metaplectic structure 3. The main function of this chapter, as far as the rest of the book is concerned, is the introduction of metalinear structures, halfforms, and metaplectic. Bertram kostant academic dictionaries and encyclopedias. Czyz institute of mathematics, technical university, warsaw, poland received july 25. A quantization method, generalizing the usual canonical one.
It has given valuable and lasting insights into the. This book contains a revised and expanded version of the lecture notes of two. The geometric approach to quantization was introduced by kostant and souriau more that. Geometric quantization is a marvelous tool for understanding the relation between classical physics and quantum physics. To date, this program has succeeded in unifying various older methods of quantizing. By using this interpretation, we establish a geometric quantization formula for a hamiltonian action of a compact lie group acting on a noncompact symplectic manifold with proper moment map. Bertram kostant born march 24, 1928, united states of. The madelung picture as a foundation of geometric quantum. In this theory the additional structure needed for quantization is explicitely defined in terms of the geometry of the phase space, and it consists of a prequantization, a polarization and a.
Browse the amazon editors picks for the best books of 2019, featuring our favorite reads. Library of congress cataloginginpublication data guillemin, v. The geometric approach to quantization was introduced by konstant and souriau more than 20 years ago. Momentum space quantization for compact lie groups. In comparison to deformation quantization it focuses on spaces of states, hence on the schrodinger picture of quantum mechanics. Lie theory, algebra and geometric quantization symposium in honor of 65th birthday massachusetts institute of technology, 1993, book lie theory and geometry named in his honor, 1994.
In particular, we present a solution to a conjecture of michele vergne in her icm 2006 plenary lecture. The goal is to associate a hubert space to ev ery closed oriented 2 manifold. This volume, dedicated to bertram kostant on the occasion of his 65th birthday, is a collection of 22 invited papers by leading mathematicians working in lie theory, geometry, algebra, and. Geometric quantization for nilpotent coadjoint orbits. If we compare the geometric prequantization of the symplectic manifold r2, dp. The madelung picture as a foundation of geometric quantum theory. Kostant, coadjoint orbits and a new symbol calculus for line bundles, in conference on differential geometric methods in theoretical physics g. On geometric quantization and its connections with the maslov. This leads, in chapter 8, to topics such as foliated cohomology, the theory of the dolbeaultkostant complex, and their applications. Geometric quantization and quantum mechanics ebook, 1980. By using this interpretation, we establish a geometric quantization formula for a hamiltonian action of a compact lie group acting on a non.
Part of the encyclopaedia of mathematical sciences book series ems, volume 4. Kostant bertram kostant am workshop enveloping algebras and. The ideas have proved useful in pure mathematics, notably in representation theory, as well as in theoretical physics. In his book in french, then translated in english he defines elementary particles as some coadjoint orbits of the poincare group, after introducing what is now known as first geometric quantization. The book 29 treats further topics in symplectic geometry and mechanics, with special attention to the role of symmetry groups. Geometric quantization 1 introduction the aim of the geometric quantization program is to describe a quantization procedure in terms of natural geometric structures. The book can be used by researchers and graduate students working in symplectic geometry and its applications. Symplectic manifold cotangent bundle geometric quantization. The kostant souriau method of geometric quantization is applied to homogeneous and isotropic cosmological models with positive intrinsic curvature and a massless kleingordon scalar field.
Introduction in the theory of geometric quantization of b. We establish an analytic interpretation for the index of certain transversally elliptic symbols on noncompact manifolds. Member nas, american academy arts and sciences, american math society colloquium lecturer 1985. In this note, i shall explain a way of looking at halfforms which is derived from recent ideas of axelrod, delia pietra, and witten 1991 and atiyah 1990, and from rather older ones of kostant 1974. However, rather than elaborating on the relation between quantum and classical mechanics, geometric quantization unearthed a large amount of geometric structures 11, x23. Kostant s work has involved representation theory, lie groups, lie algebras, homogeneous spaces, differential geometry and mathematical physics, particularly symplectic geometry. Starting around 1970 kostant did a great deal of work developing the theory of geometric quantization and the idea of quantizing coadjoint orbits to get representations other figures to mention in this context are kirillov and souriau. The classical phase space doesnt have to be a cotangent bundle, and one has a general framework to talk about different. Moment maps, cobordisms, and hamiltonian group actions. With a symplectic manifold regarded as a classical mechanical. I learned geometric quantization from jeanmarie souriau one of the initiators of the subject with bertram kostant in two texts essentially. An invitation to geometric quantization alex fok department of mathematics, cornell university april 2012. An invitation to geometric quantization alex fok department of mathematics, cornell university.
Geometric quantization theory and coadjoint orbits. The classical phase space doesnt have to be a cotangent bundle, and one has a general framework to talk about different representations coordinate, momentum, bargmann. One of the more puzzling features of geometric quantization is the halfform construction. The book is provided with an introductory chapter containing basic notions on infinitedimensional frechet manifolds and frechet lie groups. However, its a bit like a power tool you have to be an expert to operate it without running the risk of seriously injuring your brain.
The typical example of this is the cotangent bundle of a. The theory of geometric quantization has been developed independently by b. It has given valuable and lasting insights into the relationship. A discussion of the relation between geometric quantization and the marsdenweinstein reduction is presented in chapter 9. Other readers will always be interested in your opinion of the books youve read. Kostant, b graded manifolds, graded lie theory, and prequantization. Almost complex structures and geometric quantization. Chapters 6 and 7 are devoted to the theory of geometric quantization. L with curvature rl satisfying the following prequantization. It is in french, thats true, but there is an english version too.
This book contains a revised and expanded version of the lecture notes of two seminar series given during the academic year 197677 at the department of mathematics and statistics of the university of calgary, and in the summer of 1978 at the institute of theoretical. Semiclassical properties of spinc quantization 10 5. On geometric quantization of classical systems sciencedirect. Jedrzej sniatycki geometric quantization and quantum.
The book 29 treats further topics in symplectic geometry and mechanics, with special attention to the role of symmetry groups, a topic pretty much ignored in the present notes. The basic idea the setting of the hamiltonian version of classical newtonian mechanics is the phase space position and momentum, which is a symplectic manifold. The theory is a 3dimensional analog of the familiar kostantweil theory of line bundles. Geometric quantization oxford mathematical monographs. Geometric quantization and quantum mechanics applied. Loop spaces, characteristic classes and geometric quantization. Souriau a key role is played by the concept of polarization.
Hamiltonian mechanical systems and geometric quantization. The book 29 treats further topics in symplectic geometry. Geometric quantization and quantum mechanics jedrzej. The kostantsouriau method of geometric quantization is applied to homogeneous and isotropic cosmological models with positive intrinsic curvature and a massless kleingordon scalar field. Geometric quantization and quantum mechanics jedrzej sniatycki. 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. In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. The motivation for this mathematical is to mimic quantum mechanics, where a manifold the classical phase space, parametrizing position and momentum is replaced by a vector space with an inner product. Geometric quantization in this chapter we discuss the current state of knowledge concerning the geometry of quantization as introduced independently by kostant and souriau. The books of souriau 1970 and simms and woodhouse 1976 present the theory of geometric quantization and its relationship to quantum mech anics. Kirillovs work in representation theory 71, segal, kostant 72 and souriau 26 were able to construct the algorithm of geometric quantization. Geometric quantization pdf download top book edition.
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