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Author: Boris Fedosov Publisher: Wiley-VCH ISBN: 9783055017162 Category : Mathematics Languages : en Pages : 325
Book Description
In the monograph a new approach to deformation quantization on a symplectic manifold is developed. This approach gives rise to an important invariant, the so-called Weyl curvature, which is a formal deformation of the symplectic form. The isomophy classes of the deformed algebras are classified by the cohomology classes of the coefficients of the Weyl curvature. These algebras have many common features with the algebra of complete symbols of pseudodifferential operators except that in general there are no corresponding operator algebras. Nevertheless, the developed calculus allows to define the notion of an elliptic element and its index as well as to prove an index theorem similar to that of Atiyah-Singer for elliptic operators. The corresponding index formula contains the Weyl curvature and the usual ingredients entering the Atiyah-Singer formula. Applications of the index theorem are connected with the so-called asymptotic operator representation of the deformed algebra (the operator quantization), the formal deformation parameter h should be replaced by a numerical one ranging over some admissible set of the unit interval having 0 as its limit point. The fact that the index of any elliptic operator is an integer results in necessary quantization conditions: the index of any elliptic element should be asymptotically integer-valued as h tends to 0 over the admissible set. For a compact manifold a direct construction of the asymptotic operator representation shows that these conditions are also sufficient. Finally, a reduction theorem for deformation quantization is proved generalizing the classical Marsden-Weinstein theorem. In this case the index theorem gives the Bohr-Sommerfeld quantization rule and the multiplicities of eigenvalues.
Author: Boris Fedosov Publisher: Wiley-VCH ISBN: 9783055017162 Category : Mathematics Languages : en Pages : 325
Book Description
In the monograph a new approach to deformation quantization on a symplectic manifold is developed. This approach gives rise to an important invariant, the so-called Weyl curvature, which is a formal deformation of the symplectic form. The isomophy classes of the deformed algebras are classified by the cohomology classes of the coefficients of the Weyl curvature. These algebras have many common features with the algebra of complete symbols of pseudodifferential operators except that in general there are no corresponding operator algebras. Nevertheless, the developed calculus allows to define the notion of an elliptic element and its index as well as to prove an index theorem similar to that of Atiyah-Singer for elliptic operators. The corresponding index formula contains the Weyl curvature and the usual ingredients entering the Atiyah-Singer formula. Applications of the index theorem are connected with the so-called asymptotic operator representation of the deformed algebra (the operator quantization), the formal deformation parameter h should be replaced by a numerical one ranging over some admissible set of the unit interval having 0 as its limit point. The fact that the index of any elliptic operator is an integer results in necessary quantization conditions: the index of any elliptic element should be asymptotically integer-valued as h tends to 0 over the admissible set. For a compact manifold a direct construction of the asymptotic operator representation shows that these conditions are also sufficient. Finally, a reduction theorem for deformation quantization is proved generalizing the classical Marsden-Weinstein theorem. In this case the index theorem gives the Bohr-Sommerfeld quantization rule and the multiplicities of eigenvalues.
Author: Boris Fedosov Publisher: Wiley-VCH ISBN: 9783527400881 Category : Mathematics Languages : en Pages : 325
Book Description
In the monograph a new approach to deformation quantization on a symplectic manifold is developed. This approach gives rise to an important invariant, the so-called Weyl curvature, which is a formal deformation of the symplectic form. The isomophy classes of the deformed algebras are classified by the cohomology classes of the coefficients of the Weyl curvature. These algebras have many common features with the algebra of complete symbols of pseudodifferential operators except that in general there are no corresponding operator algebras. Nevertheless, the developed calculus allows to define the notion of an elliptic element and its index as well as to prove an index theorem similar to that of Atiyah-Singer for elliptic operators. The corresponding index formula contains the Weyl curvature and the usual ingredients entering the Atiyah-Singer formula. Applications of the index theorem are connected with the so-called asymptotic operator representation of the deformed algebra (the operator quantization), the formal deformation parameter h should be replaced by a numerical one ranging over some admissible set of the unit interval having 0 as its limit point. The fact that the index of any elliptic operator is an integer results in necessary quantization conditions: the index of any elliptic element should be asymptotically integer-valued as h tends to 0 over the admissible set. For a compact manifold a direct construction of the asymptotic operator representation shows that these conditions are also sufficient. Finally, a reduction theorem for deformation quantization is proved generalizing the classical Marsden-Weinstein theorem. In this case the index theorem gives the Bohr-Sommerfeld quantization rule and the multiplicities of eigenvalues.
Author: Gilles Halbout Publisher: Walter de Gruyter ISBN: 3110866226 Category : Mathematics Languages : en Pages : 244
Book Description
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.
Author: Marc Aristide Rieffel Publisher: American Mathematical Soc. ISBN: 0821825755 Category : Mathematics Languages : en Pages : 110
Book Description
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.
Author: Martin Markl Publisher: American Mathematical Soc. ISBN: 0821889796 Category : Mathematics Languages : en Pages : 143
Book Description
This book brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was available. The relation between the uniform continuity of algebraic maps and topologized tensor products is explained in detail, however, as this subject does not seem to be commonly known and the literature is scarce. The exposition begins by recalling Gerstenhaber's classical theory for associative algebras. The focus then shifts to a homotopy-invariant setup of Maurer-Cartan moduli spaces. As an application, Kontsevich's approach to deformation quantization of Poisson manifolds is reviewed. Then, after a brief introduction to operads, a strongly homotopy Lie algebra governing deformations of (diagrams of) algebras of a given type is described, followed by examples and generalizations.
Author: Victor W. Guillemin Publisher: Birkhäuser ISBN: 3319279092 Category : Mathematics Languages : en Pages : 855
Book Description
This book features a selection of articles by Louis Boutet de Monvel and presents his contributions to the theory of partial differential equations and analysis. The works selected here reveal his central role in the development of his field, including three cornerstones: firstly, analytic pseudodifferential operators, which have become a fundamental aspect of analytic microlocal analysis, and secondly the Boutet de Monvel calculus for boundary problems for elliptic partial differential operators, which is still an important tool also in index theory. Thirdly, Boutet de Monvel was one of the first people to recognize the importance of the existence of generalized functions, whose singularities are concentrated on a single ray in phase space, which led him to make essential contributions to hypoelliptic operators and to a very successful and influential calculus of Toeplitz operators with applications to spectral and index theory. Other topics treated here include microlocal analysis, star products and deformation quantization as well as problems in several complex variables, index theory and geometric quantization. This book will appeal to both experts in the field and students who are new to this subject.
Author: Alberto S. Cattaneo Publisher: Societe Mathematique de France ISBN: Category : Business & Economics Languages : en Pages : 210
Book Description
In 1997, M. Kontsevich proved that every Poisson manifold admits a formal quantization, canonical up to equivalence. In doing so he solved a longstanding problem in mathematical physics. Through his proof and his interpretation of a later proof given by Tamarkin, he also opened up new research avenues in Lie theory, quantum group theory, deformation theory and the study of operads ... and uncovered fascinating links of these topics with number theory, knot theory and the theory of motives. Without doubt, his work on deformation quantization will continue to influence these fields for many years to come. In the three parts of this volume, we will 1) present the main results of Kontsevich's 1997 preprint and sketch his interpretation of Tamarkin's approach, 2) show the relevance of Kontsevich's theorem for Lie theory and 3) explain the idea from topological string theory which inspired Kontsevich's proof. An appendix is devoted to the geometry of configuration spaces.
Author: Michael Demuth Publisher: De Gruyter Akademie Forschung ISBN: Category : Mathematics Languages : en Pages : 364
Book Description
The analysis of boundary value problems has a long tradition in mathematics. Understanding the criteria for solvability and the structure of the solutions is of central interest both for theory and applications. Boundary value problems on manifolds with singularities present an additional challenge. They exhibit a wealth of analytic and algebraic structures, also under the aspect of index theory. In the first contribution to this volume, boundary value problems without the transmission condition are interpreted as particular problems on manifolds with edges; it deals with the new effects caused by variable and branching asymptotics. In the second paper, a pseudo–differential calculus is constructed for boundary value problems on manifolds with conical singularities. A concept of ellipticity is introduced that allows a parametrix construction and entails the Fredholm property in weighted Sobolev spaces. Moreover, this approach lays the foundations for treating boundary value problems on manifolds with edges. Two further contributions deal with deformation quantization, an important topic of Mathematical Physics. The first one gives a complete proof of the index theorem in deformation quantization, while the other one treats trace densities. The final article in this volume, also from the area of Mathematical Physics, presents new results on the spectrum of perturbed periodic Schrödinger operators.
Author: Araminta Amabel Wilson Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
We study naturally occurring genera (i.e. cobordism invariants) from the deformation theory in- spired by supersymmetric quantum mechanics. First, we construct a canonical deformation quantization for symplectic supermanifolds. This gives a novel proof of the super-analogue of Fedosov quantization. Our proof uses the formalism of Gelfand-Kazhdan descent, whose foundations we establish in the super-symplectic setting. In the second part of this thesis, we prove a super-version of Nest-Tsygan's algebraic index theorem, generalizing work of Engeli. This work is inspired by the appearance of the same genera in three related stories: index theory, trace methods in deformation theory, and partition functions in quantum field theory. Using the trace methodology, we compute the genus appearing in the story for supersymmetric quantum mechanics. This involves investigating supertraces on Weyl-Clifford algebras and deformations of symplectic supermanifolds.
Author: Nima Moshayedi Publisher: Springer Nature ISBN: 303105122X Category : Mathematics Languages : en Pages : 345
Book Description
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.
Author: Boris Fedosov
Author: Boris Fedosov
Author: Boris Fedosov
Author: Gilles Halbout
Author: Marc Aristide Rieffel
Author: Martin Markl
Author: Victor W. Guillemin
Author: Alberto S. Cattaneo
Author: Michael Demuth
Author: Araminta Amabel Wilson
Author: Nima Moshayedi