Deformation Quantization For Actions Of Rd

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Deformation Quantization for Actions of $R^d$

Author : Marc Aristide Rieffel
Publisher : American Mathematical Soc.
ISBN 13 : 0821825755
Total Pages : 93 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Deformation Quantization for Actions of $R^d$ by : Marc Aristide Rieffel

Download or read book Deformation Quantization for Actions of $R^d$ written by Marc Aristide Rieffel and published by American Mathematical Soc.. This book was released on 1993 with total page 93 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Deformation Quantization for Actions of R ]D

Author : Marc A. Rieffel
Publisher : Oxford University Press, USA
ISBN 13 : 9781470400835
Total Pages : 110 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Deformation Quantization for Actions of R ]D by : Marc A. Rieffel

Download or read book Deformation Quantization for Actions of R ]D written by Marc A. Rieffel and published by Oxford University Press, USA. This book was released on 2014-08-31 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Deformation Quantization for Actions of Kahlerian Lie Groups

Author : Pierre Bieliavsky
Publisher : American Mathematical Soc.
ISBN 13 : 1470414910
Total Pages : 154 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Deformation Quantization for Actions of Kahlerian Lie Groups by : Pierre Bieliavsky

Download or read book Deformation Quantization for Actions of Kahlerian Lie Groups written by Pierre Bieliavsky and published by American Mathematical Soc.. This book was released on 2015-06-26 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let B be a Lie group admitting a left-invariant negatively curved Kählerian structure. Consider a strongly continuous action of B on a Fréchet algebra . Denote by the associated Fréchet algebra of smooth vectors for this action. In the Abelian case BR and isometric, Marc Rieffel proved that Weyl's operator symbol composition formula (the so called Moyal product) yields a deformation through Fréchet algebra structures R on . When is a -algebra, every deformed Fréchet algebra admits a compatible pre- -structure, hence yielding a deformation theory at the level of -algebras too. In this memoir, the authors prove both analogous statements for general negatively curved Kählerian groups. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geom,etrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calderón-Vaillancourt Theorem. In particular, the authors give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.

Formality Theory

Author : Chiara Esposito
Publisher : Springer
ISBN 13 : 3319092901
Total Pages : 90 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Formality Theory by : Chiara Esposito

Download or read book Formality Theory written by Chiara Esposito and published by Springer. This book was released on 2014-09-04 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a survey of the theory of formal deformation quantization of Poisson manifolds, in the formalism developed by Kontsevich. It is intended as an educational introduction for mathematical physicists who are dealing with the subject for the first time. The main topics covered are the theory of Poisson manifolds, star products and their classification, deformations of associative algebras and the formality theorem. Readers will also be familiarized with the relevant physical motivations underlying the purely mathematical construction.

Quantization, Geometry and Noncommutative Structures in Mathematics and Physics

Author : Alexander Cardona
Publisher : Springer
ISBN 13 : 3319654276
Total Pages : 341 pages
Book Rating : 4.3/5 (196 download)

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Book Synopsis Quantization, Geometry and Noncommutative Structures in Mathematics and Physics by : Alexander Cardona

Download or read book Quantization, Geometry and Noncommutative Structures in Mathematics and Physics written by Alexander Cardona and published by Springer. This book was released on 2017-10-26 with total page 341 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Geometric Methods in Physics XXXVII

Author : Piotr Kielanowski
Publisher : Springer Nature
ISBN 13 : 3030340724
Total Pages : 260 pages
Book Rating : 4.0/5 (33 download)

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Book Synopsis Geometric Methods in Physics XXXVII by : Piotr Kielanowski

Download or read book Geometric Methods in Physics XXXVII written by Piotr Kielanowski and published by Springer Nature. This book was released on 2019-11-26 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book consists of articles based on the XXXVII Białowieża Workshop on Geometric Methods in Physics, 2018. The series of Białowieża workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. This edition of the workshop featured a special session dedicated to Professor Daniel Sternheimer on the occasion of his 80th birthday. The previously unpublished papers present cutting-edge current research, typically grounded in geometry and analysis, with applications to classical and quantum physics. For the past seven years, the Białowieża Workshops have been complemented by a School on Geometry and Physics comprising a series of advanced lectures for graduate students and early-career researchers. The book also includes abstracts of the five lecture series that were given at the seventh school.

From Classical Field Theory to Perturbative Quantum Field Theory

Author : Michael Dütsch
Publisher : Springer
ISBN 13 : 3030047385
Total Pages : 536 pages
Book Rating : 4.0/5 (3 download)

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Book Synopsis From Classical Field Theory to Perturbative Quantum Field Theory by : Michael Dütsch

Download or read book From Classical Field Theory to Perturbative Quantum Field Theory written by Michael Dütsch and published by Springer. This book was released on 2019-03-18 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops a novel approach to perturbative quantum field theory: starting with a perturbative formulation of classical field theory, quantization is achieved by means of deformation quantization of the underlying free theory and by applying the principle that as much of the classical structure as possible should be maintained. The resulting formulation of perturbative quantum field theory is a version of the Epstein-Glaser renormalization that is conceptually clear, mathematically rigorous and pragmatically useful for physicists. The connection to traditional formulations of perturbative quantum field theory is also elaborated on, and the formalism is illustrated in a wealth of examples and exercises.

Deformation Quantization

Author : Gilles Halbout
Publisher : Walter de Gruyter
ISBN 13 : 3110866226
Total Pages : 244 pages
Book Rating : 4.1/5 (18 download)

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Book Synopsis Deformation Quantization by : Gilles Halbout

Download or read book Deformation Quantization written by Gilles Halbout and published by Walter de Gruyter. This book was released on 2012-10-25 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Deformation Quantization and Index Theory

Author : Boris Fedosov
Publisher : Wiley-VCH
ISBN 13 : 9783055017162
Total Pages : 325 pages
Book Rating : 4.0/5 (171 download)

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Book Synopsis Deformation Quantization and Index Theory by : Boris Fedosov

Download or read book Deformation Quantization and Index Theory written by Boris Fedosov and published by Wiley-VCH. This book was released on 1995-12-28 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Deformation Quantization and Index Theory

Author : Boris Fedosov
Publisher : Wiley-VCH
ISBN 13 : 9783527400881
Total Pages : 325 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Deformation Quantization and Index Theory by : Boris Fedosov

Download or read book Deformation Quantization and Index Theory written by Boris Fedosov and published by Wiley-VCH. This book was released on 1996-02-08 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Superstrings, Geometry, Topology, and $C^*$-algebras

Author : Robert S. Doran
Publisher : American Mathematical Soc.
ISBN 13 : 0821848879
Total Pages : 265 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Superstrings, Geometry, Topology, and $C^*$-algebras by : Robert S. Doran

Download or read book Superstrings, Geometry, Topology, and $C^*$-algebras written by Robert S. Doran and published by American Mathematical Soc.. This book was released on 2010-10-13 with total page 265 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of an NSF-CBMS Conference held at Texas Christian University in Fort Worth, Texas, May 18-22, 2009. The papers, written especially for this volume by well-known mathematicians and mathematical physicists, are an outgrowth of the talks presented at the conference. Topics examined are highly interdisciplinary and include, among many other things, recent results on D-brane charges in $K$-homology and twisted $K$-homology, Yang-Mills gauge theory and connections with non-commutative geometry, Landau-Ginzburg models, $C^*$-algebraic non-commutative geometry and ties to quantum physics and topology, the rational homotopy type of the group of unitary elements in an Azumaya algebra, and functoriality properties in the theory of $C^*$-crossed products and fixed point algebras for proper actions. An introduction, written by Jonathan Rosenberg, provides an instructive overview describing common themes and how the various papers in the volume are interrelated and fit together. The rich diversity of papers appearing in the volume demonstrates the current interplay between superstring theory, geometry/topology, and non-commutative geometry. The book will be of interest to graduate students, mathematicians, mathematical physicists, and researchers working in these areas.

From Geometry to Quantum Mechanics

Author : Yoshiaki Maeda
Publisher : Springer Science & Business Media
ISBN 13 : 0817645306
Total Pages : 324 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis From Geometry to Quantum Mechanics by : Yoshiaki Maeda

Download or read book From Geometry to Quantum Mechanics written by Yoshiaki Maeda and published by Springer Science & Business Media. This book was released on 2007-04-22 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: * Invited articles in differential geometry and mathematical physics in honor of Hideki Omori * Focus on recent trends and future directions in symplectic and Poisson geometry, global analysis, Lie group theory, quantizations and noncommutative geometry, as well as applications of PDEs and variational methods to geometry * Will appeal to graduate students in mathematics and quantum mechanics; also a reference

An Invitation to Noncommutative Geometry

Author : Masoud Khalkhali
Publisher : World Scientific
ISBN 13 : 981270616X
Total Pages : 515 pages
Book Rating : 4.8/5 (127 download)

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Book Synopsis An Invitation to Noncommutative Geometry by : Masoud Khalkhali

Download or read book An Invitation to Noncommutative Geometry written by Masoud Khalkhali and published by World Scientific. This book was released on 2008 with total page 515 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first existing volume that collects lectures on this important and fast developing subject in mathematics. The lectures are given by leading experts in the field and the range of topics is kept as broad as possible by including both the algebraic and the differential aspects of noncommutative geometry as well as recent applications to theoretical physics and number theory.

Poisson Geometry, Deformation Quantisation and Group Representations

Author : Simone Gutt
Publisher : Cambridge University Press
ISBN 13 : 9780521615051
Total Pages : 380 pages
Book Rating : 4.6/5 (15 download)

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Book Synopsis Poisson Geometry, Deformation Quantisation and Group Representations by : Simone Gutt

Download or read book Poisson Geometry, Deformation Quantisation and Group Representations written by Simone Gutt and published by Cambridge University Press. This book was released on 2005-06-21 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible introduction to Poisson geometry suitable for graduate students.

In Search of the Riemann Zeros

Author : Michel Laurent Lapidus
Publisher : American Mathematical Soc.
ISBN 13 : 9780821842225
Total Pages : 594 pages
Book Rating : 4.8/5 (422 download)

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Book Synopsis In Search of the Riemann Zeros by : Michel Laurent Lapidus

Download or read book In Search of the Riemann Zeros written by Michel Laurent Lapidus and published by American Mathematical Soc.. This book was released on 2008 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possible--or, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line.

Deformation Quantization of Some Non-compact Solvable Lie Groups and Their Representation Theory

Author : Byung-Jay Kahng
Publisher :
ISBN 13 :
Total Pages : 308 pages
Book Rating : 4.:/5 (34 download)

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Book Synopsis Deformation Quantization of Some Non-compact Solvable Lie Groups and Their Representation Theory by : Byung-Jay Kahng

Download or read book Deformation Quantization of Some Non-compact Solvable Lie Groups and Their Representation Theory written by Byung-Jay Kahng and published by . This book was released on 1997 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Quanta of Maths

Author : Institut des hautes études scientifiques (Paris, France)
Publisher : American Mathematical Soc.
ISBN 13 : 0821852035
Total Pages : 695 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Quanta of Maths by : Institut des hautes études scientifiques (Paris, France)

Download or read book Quanta of Maths written by Institut des hautes études scientifiques (Paris, France) and published by American Mathematical Soc.. This book was released on 2010 with total page 695 pages. Available in PDF, EPUB and Kindle. Book excerpt: The work of Alain Connes has cut a wide swath across several areas of mathematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics. Specific themes covered by the articles are as follows: entropy in operator algebras, regular $C^*$-algebras of integral domains, properly infinite $C^*$-algebras, representations of free groups and 1-cohomology, Leibniz seminorms and quantum metric spaces; von Neumann algebras, fundamental Group of $\mathrm{II}_1$ factors, subfactors and planar algebras; Baum-Connes conjecture and property T, equivariant K-homology, Hermitian K-theory; cyclic cohomology, local index formula and twisted spectral triples, tangent groupoid and the index theorem; noncommutative geometry and space-time, spectral action principle, quantum gravity, noncommutative ADHM and instantons, non-compact spectral triples of finite volume, noncommutative coordinate algebras; Hopf algebras, Vinberg algebras, renormalization and combinatorics, motivic renormalization and singularities; cyclotomy and analytic geometry over $F_1$, quantum modular forms; differential K-theory, cyclic theory and S-cohomology.