Quantum Geometry

Author : Margaret Prugovecki
Publisher : Springer Science & Business Media
Page : 543 pages
File Size : 44,8 Mb
Release : 2013-03-14
Category : Science
ISBN : 9789401579711

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Quantum Geometry by Margaret Prugovecki Pdf

This monograph presents a review and analysis of the main mathematical, physical and epistomological difficulties encountered at the foundational level by all the conventional formulations of relativistic quantum theories, ranging from relativistic quantum mechanics and quantum field theory in Minkowski space, to the various canonical and covariant approaches to quantum gravity. It is, however, primarily devoted to the systematic presentation of a quantum framework meant to deal effectively with these difficulties by reconsidering the foundations of these subjects, analyzing their epistemic nature, and then developing mathematical tools which are specifically designed for the elimination of all the basic inconsistencies. A carefully documented historical survey is included, and additional extensive notes containing quotations from original sources are incorporated at the end of each chapter, so that the reader will be brought up-to-date with the very latest developments in quantum field theory in curved spacetime, quantum gravity and quantum cosmology. The survey further provides a backdrop against which the new foundational and mathematical ideas of the present approach to these subjects can be brought out in sharper relief.

Author : Edoardo Ballico,Alessandra Bernardi,Iacopo Carusotto,Sonia Mazzucchi,Valter Moretti
Publisher : Springer
Page : 173 pages
File Size : 55,6 Mb
Release : 2019-03-13
Category : Science
ISBN : 9783030061227

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Quantum Physics and Geometry by Edoardo Ballico,Alessandra Bernardi,Iacopo Carusotto,Sonia Mazzucchi,Valter Moretti Pdf

This book collects independent contributions on current developments in quantum information theory, a very interdisciplinary field at the intersection of physics, computer science and mathematics. Making intense use of the most advanced concepts from each discipline, the authors give in each contribution pedagogical introductions to the main concepts underlying their present research and present a personal perspective on some of the most exciting open problems. Keeping this diverse audience in mind, special efforts have been made to ensure that the basic concepts underlying quantum information are covered in an understandable way for mathematical readers, who can find there new open challenges for their research. At the same time, the volume can also be of use to physicists wishing to learn advanced mathematical tools, especially of differential and algebraic geometric nature.

Author : Jan Ambjørn,Bergfinnur Durhuus,Þórður Jónsson
Publisher : Cambridge University Press
Page : 377 pages
File Size : 40,8 Mb
Release : 1997-06-19
Category : Science
ISBN : 9780521461672

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Quantum Geometry by Jan Ambjørn,Bergfinnur Durhuus,Þórður Jónsson Pdf

Describes random geometry and applications to strings, quantum gravity, topological field theory and membrane physics.

Author : Edwin J. Beggs,Shahn Majid
Publisher : Springer Nature
Page : 809 pages
File Size : 43,9 Mb
Release : 2020-01-31
Category : Science
ISBN : 9783030302948

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Quantum Riemannian Geometry by Edwin J. Beggs,Shahn Majid Pdf

This book provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points. Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a `bottom up’ one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum `Levi-Civita’ bimodule connection, geometric Laplacians and, in some cases, Dirac operators. The book also covers elements of Connes’ approach to the subject coming from cyclic cohomology and spectral triples. Other topics include various other cohomology theories, holomorphic structures and noncommutative D-modules. A unique feature of the book is its constructive approach and its wealth of examples drawn from a large body of literature in mathematical physics, now put on a firm algebraic footing. Including exercises with solutions, it can be used as a textbook for advanced courses as well as a reference for researchers.

Author : Taro Kimura
Publisher : Springer Nature
Page : 297 pages
File Size : 53,7 Mb
Release : 2021-07-05
Category : Science
ISBN : 9783030761905

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Instanton Counting, Quantum Geometry and Algebra by Taro Kimura Pdf

This book pedagogically describes recent developments in gauge theory, in particular four-dimensional N = 2 supersymmetric gauge theory, in relation to various fields in mathematics, including algebraic geometry, geometric representation theory, vertex operator algebras. The key concept is the instanton, which is a solution to the anti-self-dual Yang–Mills equation in four dimensions. In the first part of the book, starting with the systematic description of the instanton, how to integrate out the instanton moduli space is explained together with the equivariant localization formula. It is then illustrated that this formalism is generalized to various situations, including quiver and fractional quiver gauge theory, supergroup gauge theory. The second part of the book is devoted to the algebraic geometric description of supersymmetric gauge theory, known as the Seiberg–Witten theory, together with string/M-theory point of view. Based on its relation to integrable systems, how to quantize such a geometric structure via the Ω-deformation of gauge theory is addressed. The third part of the book focuses on the quantum algebraic structure of supersymmetric gauge theory. After introducing the free field realization of gauge theory, the underlying infinite dimensional algebraic structure is discussed with emphasis on the connection with representation theory of quiver, which leads to the notion of quiver W-algebra. It is then clarified that such a gauge theory construction of the algebra naturally gives rise to further affinization and elliptic deformation of W-algebra.

Author : Ingemar Bengtsson,Karol Życzkowski
Publisher : Cambridge University Press
Page : 637 pages
File Size : 45,5 Mb
Release : 2017-08-18
Category : Mathematics
ISBN : 9781107026254

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Geometry of Quantum States by Ingemar Bengtsson,Karol Życzkowski Pdf

This new edition describes the space of quantum states and the theory of quantum entanglement from a geometric perspective.

Author : Veeravalli S. Varadarajan
Publisher : Unknown
Page : 255 pages
File Size : 47,8 Mb
Release : 1970
Category : Electronic
ISBN : OCLC:174919096

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Geometry of Quantum Theory by Veeravalli S. Varadarajan Pdf

Author : Maurice A. de Gosson
Publisher : Springer Science & Business Media
Page : 375 pages
File Size : 53,9 Mb
Release : 2006-08-06
Category : Mathematics
ISBN : 9783764375751

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Symplectic Geometry and Quantum Mechanics by Maurice A. de Gosson Pdf

This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.

Author : Lárus Thorlacius
Publisher : Springer Science & Business Media
Page : 482 pages
File Size : 49,8 Mb
Release : 2000-09-30
Category : Mathematics
ISBN : 0792364740

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M-Theory and Quantum Geometry by Lárus Thorlacius Pdf

Proceedings of the NATO Advanced Study Institute on Quantum Geometry, held in Akureyri, Iceland, on August 9-20, 1999

Author : Daniel S. Freed,Karen K. Uhlenbeck,American Mathematical Society,Institute for Advanced Study (Princeton, N.J.)
Publisher : American Mathematical Soc.
Page : 476 pages
File Size : 46,7 Mb
Release : 1995
Category : Science
ISBN : 0821886835

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Geometry and Quantum Field Theory by Daniel S. Freed,Karen K. Uhlenbeck,American Mathematical Society,Institute for Advanced Study (Princeton, N.J.) Pdf

The first title in a new series, this book explores topics from classical and quantum mechanics and field theory. The material is presented at a level between that of a textbook and research papers making it ideal for graduate students. The book provides an entree into a field that promises to remain exciting and important for years to come.

Author : Kieran Finn
Publisher : Springer Nature
Page : 212 pages
File Size : 44,5 Mb
Release : 2021-10-07
Category : Science
ISBN : 9783030852696

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Geometric Approaches to Quantum Field Theory by Kieran Finn Pdf

The ancient Greeks believed that everything in the Universe should be describable in terms of geometry. This thesis takes several steps towards realising this goal by introducing geometric descriptions of systems such as quantum gravity, fermionic particles and the origins of the Universe itself. The author extends the applicability of previous work by Vilkovisky, DeWitt and others to include theories with spin 1⁄2 and spin 2 degrees of freedom. In addition, he introduces a geometric description of the potential term in a quantum field theory through a process known as the Eisenhart lift. Finally, the methods are applied to the theory of inflation, where they show how geometry can help answer a long-standing question about the initial conditions of the Universe. This publication is aimed at graduate and advanced undergraduate students and provides a pedagogical introduction to the exciting topic of field space covariance and the complete geometrization of quantum field theory.

Author : Dariusz Chruscinski,Andrzej Jamiolkowski
Publisher : Springer Science & Business Media
Page : 337 pages
File Size : 42,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9780817681760

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Geometric Phases in Classical and Quantum Mechanics by Dariusz Chruscinski,Andrzej Jamiolkowski Pdf

Several well-established geometric and topological methods are used in this work in an application to a beautiful physical phenomenon known as the geometric phase. This book examines the geometric phase, bringing together different physical phenomena under a unified mathematical scheme. The material is presented so that graduate students and researchers in applied mathematics and physics with an understanding of classical and quantum mechanics can handle the text.

Author : Alain Connes,Matilde Marcolli
Publisher : American Mathematical Soc.
Page : 785 pages
File Size : 46,9 Mb
Release : 2019-03-13
Category : Electronic
ISBN : 9781470450458

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Noncommutative Geometry, Quantum Fields and Motives by Alain Connes,Matilde Marcolli Pdf

The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.

Author : Jedrzej Sniatycki
Publisher : Springer Science & Business Media
Page : 241 pages
File Size : 43,8 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781461260660

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Geometric Quantization and Quantum Mechanics by Jedrzej Sniatycki Pdf

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 the uniqueness of the introduced structures can be found in the expository papers of Blattner, Kostant, Sternberg and Wolf, and also in the references quoted in these papers. The books of Souriau (1970) and Simms and Woodhouse (1976) present the theory of geometric quantization and its relationship to quantum mech anics. 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.

Author : José F. Cariñena,Alberto Ibort,Giuseppe Marmo,Giuseppe Morandi
Publisher : Springer
Page : 739 pages
File Size : 49,6 Mb
Release : 2014-09-23
Category : Science
ISBN : 9789401792202

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Geometry from Dynamics, Classical and Quantum by José F. Cariñena,Alberto Ibort,Giuseppe Marmo,Giuseppe Morandi Pdf

This book describes, by using elementary techniques, how some geometrical structures widely used today in many areas of physics, like symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics. It is assumed that what can be accessed in actual experiences when studying a given system is just its dynamical behavior that is described by using a family of variables ("observables" of the system). The book departs from the principle that ''dynamics is first'' and then tries to answer in what sense the sole dynamics determines the geometrical structures that have proved so useful to describe the dynamics in so many important instances. In this vein it is shown that most of the geometrical structures that are used in the standard presentations of classical dynamics (Jacobi, Poisson, symplectic, Hamiltonian, Lagrangian) are determined, though in general not uniquely, by the dynamics alone. The same program is accomplished for the geometrical structures relevant to describe quantum dynamics. Finally, it is shown that further properties that allow the explicit description of the dynamics of certain dynamical systems, like integrability and super integrability, are deeply related to the previous development and will be covered in the last part of the book. The mathematical framework used to present the previous program is kept to an elementary level throughout the text, indicating where more advanced notions will be needed to proceed further. A family of relevant examples is discussed at length and the necessary ideas from geometry are elaborated along the text. However no effort is made to present an ''all-inclusive'' introduction to differential geometry as many other books already exist on the market doing exactly that. However, the development of the previous program, considered as the posing and solution of a generalized inverse problem for geometry, leads to new ways of thinking and relating some of the most conspicuous geometrical structures appearing in Mathematical and Theoretical Physics.