noncommutative geometry and particle physics mathematical physics studies

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Noncommutative Geometry And Particle Physics

Author : Walter van Suijlekom
ISBN : 9789401791625
Genre : Science
File Size : 65. 58 MB
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This book provides an introduction to noncommutative geometry and presents a number of its recent applications to particle physics. It is intended for graduate students in mathematics/theoretical physics who are new to the field of noncommutative geometry, as well as for researchers in mathematics/theoretical physics with an interest in the physical applications of noncommutative geometry. In the first part, we introduce the main concepts and techniques by studying finite noncommutative spaces, providing a “light” approach to noncommutative geometry. We then proceed with the general framework by defining and analyzing noncommutative spin manifolds and deriving some main results on them, such as the local index formula. In the second part, we show how noncommutative spin manifolds naturally give rise to gauge theories, applying this principle to specific examples. We subsequently geometrically derive abelian and non-abelian Yang-Mills gauge theories, and eventually the full Standard Model of particle physics, and conclude by explaining how noncommutative geometry might indicate how to proceed beyond the Standard Model.

Noncommutative Geometry

Author : Alain Connes
ISBN : 3540203575
Genre : Mathematics
File Size : 32. 37 MB
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Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.

An Introduction To Noncommutative Geometry

Author : Joseph C. Várilly
ISBN : 3037190248
Genre : Mathematics
File Size : 48. 90 MB
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This introduction is aimed at graduate students of both mathematics and theoretical physics. It deals with Dirac operators on spin manifolds, noncommutative tori, Moyal quantization and tangent groupoids, action functionals, and isospectral deformations. The structural framework is the concept of a noncommutative spin geometry; the conditions on spectral triples which determine this concept are developed in detail. The emphasis throughout is on gaining understanding by computing the details of specific examples. The book provides a middle ground between a comprehensive text and a narrowly focused research monograph. It is intended for self-study, enabling the reader to gain access to the essentials of noncommutative geometry. New features since the original course are an expanded bibliography and a survey of more recent examples and applications of spectral triples.

Noncommutative Geometry Quantum Fields And Motives

Author : Alain Connes
ISBN : 0821874780
Genre : Mathematics
File Size : 57. 84 MB
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Noncommutative Cosmology

Author : Marcolli Matilde
ISBN : 9789813202863
Genre : Science
File Size : 26. 94 MB
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Modified gravity models play an important role in contemporary theoretical cosmology. The present book proposes a novel approach to the topic based on techniques from noncommutative geometry, especially the spectral action functional as a gravity model. The book discusses applications to early universe models and slow-roll inflation models, to the problem of cosmic topology, to non-isotropic cosmologies like mixmaster universes and Bianchi IX gravitational instantons, and to multifractal structures in cosmology. Relations between noncommutative and algebro-geometric methods in cosmology is also discussed, including the occurrence of motives, periods, and modular forms in spectral models of gravity. Contents: PrefaceAcknowledgmentsGravity and Matter in Noncommutative GeometryRenormalization Group Flows and Early Universe ModelsCosmic TopologyAlgebro-geometric Models in CosmologyMixmaster CosmologiesThe Spectral Action on Bianchi IX CosmologiesFractal and Multifractal Structures in CosmologyNoncommutative Quantum CosmologyBibliography Readership: Graduate students and researchers in mathematical physics and theoretical cosmology. Keywords: Cosmology;Gravity;Particle Physics;GeometryReview: Key Features: While there are recent books that focus on applications of noncommutative geometry to particle physics models, there is presently no other book that focuses on applications to cosmology

Clifford Algebras

Author : Rafal Ablamowicz
ISBN : 9781461220442
Genre : Mathematics
File Size : 24. 20 MB
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The invited papers in this volume provide a detailed examination of Clifford algebras and their significance to analysis, geometry, mathematical structures, physics, and applications in engineering. While the papers collected in this volume require that the reader possess a solid knowledge of appropriate background material, they lead to the most current research topics. With its wide range of topics, well-established contributors, and excellent references and index, this book will appeal to graduate students and researchers.

Local Quantum Physics

Author : Rudolf Haag
ISBN : 9783642614583
Genre : Science
File Size : 53. 51 MB
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The new edition provided the opportunity of adding a new chapter entitled "Principles and Lessons of Quantum Physics". It was a tempting challenge to try to sharpen the points at issue in the long lasting debate on the Copenhagen Spirit, to assess the significance of various arguments from our present vantage point, seventy years after the advent of quantum theory, where, after ali, some problems appear in a different light. It includes a section on the assumptions leading to the specific mathematical formalism of quantum theory and a section entitled "The evolutionary picture" describing my personal conclusions. Alto gether the discussion suggests that the conventional language is too narrow and that neither the mathematical nor the conceptual structure are built for eter nity. Future theories will demand radical changes though not in the direction of a return to determinism. Essential lessons taught by Bohr will persist. This chapter is essentially self-contained. Some new material has been added in the last chapter. It concerns the char acterization of specific theories within the general frame and recent progress in quantum field theory on curved space-time manifolds. A few pages on renor malization have been added in Chapter II and some effort has been invested in the search for mistakes and unclear passages in the first edition. The central objective of the book, expressed in the title "Local Quantum Physics", is the synthesis between special relativity and quantum theory to gether with a few other principles of general nature.

The Interface Of Mathematics And Particle Physics

Author : Daniel G. Quillen
ISBN : UOM:39015018958564
Genre : Psychology
File Size : 70. 39 MB
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This comprehensive volume provides an exciting and up-to-date survey of the current research and advances in particle physics, with an emphasis on the crucial interaction between theoretical physics and mathematics. Internationally known experts including A. Connes, R. Penrose, Peter Goddard, and Alice Rogers give readers the latest information on quantum groups, instantons in Yang-Mills theories, conformal algebras, gauge theories, cyclic cohomology, rational conformal field theories, string theory, and twistors. Their contributions show how questions in physics have stimulated important advances in mathematics, which in turn have facilitated further theoretical progress in physics. The wide scope of the book makes it an invaluable collection for anyone working in either field.

An Introduction To Noncommutative Differential Geometry And Its Physical Applications

Author : J. Madore
ISBN : 0521659914
Genre : Mathematics
File Size : 63. 25 MB
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A thoroughly revised introduction to non-commutative geometry.

Quantization Geometry And Noncommutative Structures In Mathematics And Physics

Author : Alexander Cardona
ISBN : 9783319654270
Genre : Science
File Size : 20. 31 MB
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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.

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