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Quantum Stochastic Processes And Noncommutative Geometry
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Book Synopsis Quantum Stochastic Processes and Noncommutative Geometry by : Kalyan B. Sinha
Download or read book Quantum Stochastic Processes and Noncommutative Geometry written by Kalyan B. Sinha and published by Cambridge University Press. This book was released on 2007-01-25 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: The classical theory of stochastic processes has important applications arising from the need to describe irreversible evolutions in classical mechanics; analogously quantum stochastic processes can be used to model the dynamics of irreversible quantum systems. Noncommutative, i.e. quantum, geometry provides a framework in which quantum stochastic structures can be explored. This book is the first to describe how these two mathematical constructions are related. In particular, key ideas of semigroups and complete positivity are combined to yield quantum dynamical semigroups (QDS). Sinha and Goswami also develop a general theory of Evans-Hudson dilation for both bounded and unbounded coefficients. The unique features of the book, including the interaction of QDS and quantum stochastic calculus with noncommutative geometry and a thorough discussion of this calculus with unbounded coefficients, will make it of interest to graduate students and researchers in functional analysis, probability and mathematical physics.
Book Synopsis Noncommutative Mathematics for Quantum Systems by : Uwe Franz
Download or read book Noncommutative Mathematics for Quantum Systems written by Uwe Franz and published by Cambridge University Press. This book was released on 2016-01-07 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probability, mathematical physics and operator algebras.
Book Synopsis Quantum Independent Increment Processes I by : David Applebaum
Download or read book Quantum Independent Increment Processes I written by David Applebaum and published by Springer. This book was released on 2005-09-14 with total page 299 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the first of two volumes containing the revised and completed notes lectures given at the school "Quantum Independent Increment Processes: Structure and Applications to Physics". This school was held at the Alfried-Krupp-Wissenschaftskolleg in Greifswald during the period March 9 – 22, 2003, and supported by the Volkswagen Foundation. The school gave an introduction to current research on quantum independent increment processes aimed at graduate students and non-specialists working in classical and quantum probability, operator algebras, and mathematical physics. The present first volume contains the following lectures: "Lévy Processes in Euclidean Spaces and Groups" by David Applebaum, "Locally Compact Quantum Groups" by Johan Kustermans, "Quantum Stochastic Analysis" by J. Martin Lindsay, and "Dilations, Cocycles and Product Systems" by B.V. Rajarama Bhat.
Book Synopsis Quantum Stochastics by : Mou-Hsiung Chang
Download or read book Quantum Stochastics written by Mou-Hsiung Chang and published by Cambridge University Press. This book was released on 2015-02-19 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a systematic, self-contained treatment of the theory of quantum probability and quantum Markov processes for graduate students and researchers. Building a framework that parallels the development of classical probability, it aims to help readers up the steep learning curve of the quantum theory.
Book Synopsis Quantum Potential Theory by : Philippe Biane
Download or read book Quantum Potential Theory written by Philippe Biane and published by Springer Science & Business Media. This book was released on 2008-09-23 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers the revised and completed notes of lectures given at the 2007 conference, "Quantum Potential Theory: Structures and Applications to Physics." These lectures provide an introduction to the theory and discuss various applications.
Book Synopsis Quantum Probability and Related Topics by : L Accardi
Download or read book Quantum Probability and Related Topics written by L Accardi and published by World Scientific. This book was released on 1991-10-31 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains several surveys of important developments in quantum probability. The new type of quantum central limit theorems, based on the notion of free independence rather than the usual Boson or Fermion independence is discussed. A surprising result is that the role of the Gaussian for this new type of independence is played by the Wigner distribution. This motivated the introduction of new type of quantum independent increments noise, the free noise and the corresponding stochastic calculus. A further generalization, the ϖ-noises, is discussed. The free stochastic calculus is shown to be able to fit naturally into the general representation free calculus. The basic free are shown to be realized as non-adapted stochastic integrals with respect to the usual Boson white noises. Quantum noise on the finite difference algebra is expressed in terms of the usual Boson white noises. A new quantum way of looking at classical stochastic flows, in particular diffusions on Riemannian Manifolds is explained. Quantum groups are discussed from the point of view of possible applications to quantum probability. The applications of quantum probability to physics are surveyed. Contents:An Invitation to the Weak Coupling and Low Density Limits (L Accardi et al.)Quantum Markov Chains: The Recurrence Problem (L Accardi & D Koroliuk)Towards a Quantum Theory of Classical Diffusions on Riemannian Manifolds (D Applebaum)Applications of Quantum Stochastic Calculus to Quantum Optics (A Barchielli)On Quantum Stochastic Integration with Respect to “Free” Noises (F Fagnola)Fractal Dimensions of States (M Ohya)Quantum Groups and Quantum Probability (S Majid)Unification of Quantum Noise Processes in Fock Spaces (K R Parthasarathy & K B Sinha)Survey on the Stochastic Integration on the Full Fock Space (R Speicher)A Spectral Property of One Parameter Family of Sampling Functions — From Signal Analysis to Functional Analysis (H Umegaki)Free Noncommutative Random Variables, Random Matrices and II1 Factors of Free Groups (D Voiculescu)Classical and Quantum Intrinsically Random Dynamical SystemsAn Invitation to the Prigogine Theory of Irreversibility (A Weron & K Weron)and other papers Readership: Mathematicians. keywords:Quantum Probability
Book Synopsis An Introduction to Quantum Stochastic Calculus by : K.R. Parthasarathy
Download or read book An Introduction to Quantum Stochastic Calculus written by K.R. Parthasarathy and published by Birkhäuser. This book was released on 2012-12-06 with total page 299 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Elegantly written, with obvious appreciation for fine points of higher mathematics...most notable is [the] author's effort to weave classical probability theory into [a] quantum framework." – The American Mathematical Monthly "This is an excellent volume which will be a valuable companion both for those who are already active in the field and those who are new to it. Furthermore there are a large number of stimulating exercises scattered through the text which will be invaluable to students." – Mathematical Reviews An Introduction to Quantum Stochastic Calculus aims to deepen our understanding of the dynamics of systems subject to the laws of chance both from the classical and the quantum points of view and stimulate further research in their unification. This is probably the first systematic attempt to weave classical probability theory into the quantum framework and provides a wealth of interesting features: The origin of Ito's correction formulae for Brownian motion and the Poisson process can be traced to communication relations or, equivalently, the uncertainty principle. Quantum stochastic interpretation enables the possibility of seeing new relationships between fermion and boson fields. Quantum dynamical semigroups as well as classical Markov semigroups are realized through unitary operator evolutions. The text is almost self-contained and requires only an elementary knowledge of operator theory and probability theory at the graduate level.
Book Synopsis Quantum Groups and Noncommutative Spaces by : Matilde Marcolli
Download or read book Quantum Groups and Noncommutative Spaces written by Matilde Marcolli and published by Springer Science & Business Media. This book was released on 2010-11-02 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is aimed at presenting different methods and perspectives in the theory of Quantum Groups, bridging between the algebraic, representation theoretic, analytic, and differential-geometric approaches. It also covers recent developments in Noncommutative Geometry, which have close relations to quantization and quantum group symmetries. The volume collects surveys by experts which originate from an acitvity at the Max-Planck-Institute for Mathematics in Bonn.
Book Synopsis Non-commutative Analysis by : Jorgensen Palle
Download or read book Non-commutative Analysis written by Jorgensen Palle and published by World Scientific. This book was released on 2017-01-24 with total page 564 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book features new directions in analysis, with an emphasis on Hilbert space, mathematical physics, and stochastic processes. We interpret "non-commutative analysis" broadly to include representations of non-Abelian groups, and non-Abelian algebras; emphasis on Lie groups and operator algebras (C* algebras and von Neumann algebras.) A second theme is commutative and non-commutative harmonic analysis, spectral theory, operator theory and their applications. The list of topics includes shift invariant spaces, group action in differential geometry, and frame theory (over-complete bases) and their applications to engineering (signal processing and multiplexing), projective multi-resolutions, and free probability algebras. The book serves as an accessible introduction, offering a timeless presentation, attractive and accessible to students, both in mathematics and in neighboring fields.
Book Synopsis Quantum Quadratic Operators and Processes by : Farrukh Mukhamedov
Download or read book Quantum Quadratic Operators and Processes written by Farrukh Mukhamedov and published by Springer. This book was released on 2015-10-12 with total page 231 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covering both classical and quantum approaches, this unique and self-contained book presents the most recent developments in the theory of quadratic stochastic operators and their Markov and related processes. The asymptotic behavior of dynamical systems generated by classical and quantum quadratic operators is investigated and various properties of quantum quadratic operators are studied, providing an insight into the construction of quantum channels. This book is suitable as a textbook for an advanced undergraduate/graduate level course or summer school in quantum dynamical systems. It can also be used as a reference book by researchers looking for interesting problems to work on, or useful techniques and discussions of particular problems. Since it includes the latest developments in the fields of quadratic dynamical systems, Markov processes and quantum stochastic processes, researchers at all levels are likely to find the book inspiring and useful.
Book Synopsis Quantum and Stochastic Mathematical Physics by : Astrid Hilbert
Download or read book Quantum and Stochastic Mathematical Physics written by Astrid Hilbert and published by Springer Nature. This book was released on 2023-04-02 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sergio Albeverio gave important contributions to many fields ranging from Physics to Mathematics, while creating new research areas from their interplay. Some of them are presented in this Volume that grew out of the Random Transformations and Invariance in Stochastic Dynamics Workshop held in Verona in 2019. To understand the theory of thermo- and fluid-dynamics, statistical mechanics, quantum mechanics and quantum field theory, Albeverio and his collaborators developed stochastic theories having strong interplays with operator theory and functional analysis. His contribution to the theory of (non Gaussian)-SPDEs, the related theory of (pseudo-)differential operators, and ergodic theory had several impacts to solve problems related, among other topics, to thermo- and fluid dynamics. His scientific works in the theory of interacting particles and its extension to configuration spaces lead, e.g., to the solution of open problems in statistical mechanics and quantum field theory. Together with Raphael Hoegh Krohn he introduced the theory of infinite dimensional Dirichlet forms, which nowadays is used in many different contexts, and new methods in the theory of Feynman path integration. He did not fear to further develop different methods in Mathematics, like, e.g., the theory of non-standard analysis and p-adic numbers.
Book Synopsis Quantum Stochastics by : Mou-Hsiung Chang
Download or read book Quantum Stochastics written by Mou-Hsiung Chang and published by . This book was released on 2014 with total page 412 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The classical probability theory initiated by Kolmogorov and its quantum counterpart, pioneered by von Neumann, were created at about the same time in the 1930s, but development of the quantum theory has trailed far behind. Although highly appealing, the quantum theory has a steep learning curve, requiring tools from both probability and analysis and a facility for combining the two viewpoints. This book is a systematic, self-contained account of the core of quantum probability and quantum stochastic processes for graduate students and researchers. The only assumed background is knowledge of the basic theory of Hilbert spaces, bounded linear operators, and classical Markov processes. From there, the book introduces additional tools from analysis, and then builds the quantum probability framework needed to support applications to quantum control and quantum information and communication. These include quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups and processes, and large-time asymptotic behavior of quantum Markov semigroups"--
Book Synopsis Geometric Science of Information by : Frank Nielsen
Download or read book Geometric Science of Information written by Frank Nielsen and published by Springer. This book was released on 2017-10-30 with total page 877 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the Third International Conference on Geometric Science of Information, GSI 2017, held in Paris, France, in November 2017. The 101 full papers presented were carefully reviewed and selected from 113 submissions and are organized into the following subjects: statistics on non-linear data; shape space; optimal transport and applications: image processing; optimal transport and applications: signal processing; statistical manifold and hessian information geometry; monotone embedding in information geometry; information structure in neuroscience; geometric robotics and tracking; geometric mechanics and robotics; stochastic geometric mechanics and Lie group thermodynamics; probability on Riemannian manifolds; divergence geometry; non-parametric information geometry; optimization on manifold; computational information geometry; probability density estimation; session geometry of tensor-valued data; geodesic methods with constraints; applications of distance geometry.
Book Synopsis Quantum Potential Theory by : Philippe Biane
Download or read book Quantum Potential Theory written by Philippe Biane and published by Springer. This book was released on 2008-10-16 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers the revised and completed notes of lectures given at the 2007 conference, "Quantum Potential Theory: Structures and Applications to Physics." These lectures provide an introduction to the theory and discuss various applications.
Book Synopsis Real and Stochastic Analysis by : M. M. Rao
Download or read book Real and Stochastic Analysis written by M. M. Rao and published by Springer Science & Business Media. This book was released on 2004-09-23 with total page 422 pages. Available in PDF, EPUB and Kindle. Book excerpt: The interplay between functional and stochastic analysis has wide implications for problems in partial differential equations, noncommutative or "free" probability, and Riemannian geometry. Written by active researchers, each of the six independent chapters in this volume is devoted to a particular application of functional analytic methods in stochastic analysis, ranging from work in hypoelliptic operators to quantum field theory. Every chapter contains substantial new results as well as a clear, unified account of the existing theory; relevant references and numerous open problems are also included.Key topics:* Stochastic differential equations (SDEs), hypoelliptic operators, and SDEs based on Lévy processes* Stochastic calculus on Riemannian manifolds and curved Wiener spaces* Noncommutative and quantum probability* The Feynman integral, evolution processes, the Feynman-Kac formula, and applications to quantum field theory* Convolution operators and the amenability of the underlying locally compact groups, with connections among classical random walks, spectral theory, and Beurling and Segal subalgebrasSelf-contained, well-motivated, and replete with suggestions for further investigation, this book will be especially valuable as a seminar text for dissertation-level graduate students. Research mathematicians and physicists will also find it a useful and stimulating reference.Contributors: D.R. Bell; B.K. Driver; S. Gudder; B. Jefferies; H. Kunita; and M.M. Rao.
Book Synopsis Contributions in Mathematical Physics by : S. Twareque Ali
Download or read book Contributions in Mathematical Physics written by S. Twareque Ali and published by Springer. This book was released on 2007-08-15 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: Professor Gerard G. Emch has been one of the pioneers of the C-algebraic approach to quantum and classical statistical mechanics. In a prolific scientific career, spanning nearly five decades, Professor Emch has been one of the creative influences in the general area of mathematical physics. The present volume is a collection of tributes, from former students, colleagues and friends of Professor Emch, on the occasion of his 70th birthday. The articles featured here are a small yet representative sample of the breadth and reach of some of the ideas from mathematical physics.It is also a testimony to the impact that Professor Emch's work has had on several generations of mathematical physicists as well as to the diversity of mathematical methods used to understand them.
Book Synopsis Quantum Isometry Groups by : Debashish Goswami
Download or read book Quantum Isometry Groups written by Debashish Goswami and published by Springer. This book was released on 2017-01-05 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an up-to-date overview of the recently proposed theory of quantum isometry groups. Written by the founders, it is the first book to present the research on the “quantum isometry group”, highlighting the interaction of noncommutative geometry and quantum groups, which is a noncommutative generalization of the notion of group of isometry of a classical Riemannian manifold. The motivation for this generalization is the importance of isometry groups in both mathematics and physics. The framework consists of Alain Connes’ “noncommutative geometry” and the operator-algebraic theory of “quantum groups”. The authors prove the existence of quantum isometry group for noncommutative manifolds given by spectral triples under mild conditions and discuss a number of methods for computing them. One of the most striking and profound findings is the non-existence of non-classical quantum isometry groups for arbitrary classical connected compact manifolds and, by using this, the authors explicitly describe quantum isometry groups of most of the noncommutative manifolds studied in the literature. Some physical motivations and possible applications are also discussed.
