An Introduction To Quantum Stochastic Calculus

Author : K.R. Parthasarathy
Publisher : Birkhäuser
Page : 299 pages
File Size : 55,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034886413

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An Introduction to Quantum Stochastic Calculus by K.R. Parthasarathy Pdf

"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.

Author : David Applebaum,B.V. Rajarama Bhat,Johan Kustermans,J. Martin Lindsay
Publisher : Springer
Page : 299 pages
File Size : 53,6 Mb
Release : 2005-09-14
Category : Mathematics
ISBN : 9783540314509

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Quantum Independent Increment Processes I by David Applebaum,B.V. Rajarama Bhat,Johan Kustermans,J. Martin Lindsay Pdf

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.

Author : K.R. Parthasarathy
Publisher : Springer Science & Business Media
Page : 290 pages
File Size : 49,6 Mb
Release : 2012-12-13
Category : Mathematics
ISBN : 9783034805667

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An Introduction to Quantum Stochastic Calculus by K.R. Parthasarathy Pdf

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 commutation relations or, equivalently, the uncertainty principle. Quantum stochastic integration enables the possibility of seeing new relationships between fermion and boson fields. Many quantum dynamical semigroups as well as classical Markov semigroups are realised 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. - - - This is an excellent volume which will be a valuable companion both to 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) This monograph gives a systematic and self-contained introduction to the Fock space quantum stochastic calculus in its basic form (...) by making emphasis on the mathematical aspects of quantum formalism and its connections with classical probability and by extensive presentation of carefully selected functional analytic material. This makes the book very convenient for a reader with the probability-theoretic orientation, wishing to make acquaintance with wonders of the noncommutative probability, and, more specifcally, for a mathematics student studying this field. (Zentralblatt MATH) 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)

Author : Timothy M.W. Eyre
Publisher : Springer
Page : 142 pages
File Size : 45,9 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540683858

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Quantum Stochastic Calculus and Representations of Lie Superalgebras by Timothy M.W. Eyre Pdf

This book describes the representations of Lie superalgebras that are yielded by a graded version of Hudson-Parthasarathy quantum stochastic calculus. Quantum stochastic calculus and grading theory are given concise introductions, extending readership to mathematicians and physicists with a basic knowledge of algebra and infinite-dimensional Hilbert spaces. The develpment of an explicit formula for the chaotic expansion of a polynomial of quantum stochastic integrals is particularly interesting. The book aims to provide a self-contained exposition of what is known about Z_2-graded quantum stochastic calculus and to provide a framework for future research into this new and fertile area.

Author : David Applebaum,B.V. Rajarama Bhat,Johan Kustermans,J. Martin Lindsay
Publisher : Springer
Page : 299 pages
File Size : 41,9 Mb
Release : 2005-02-18
Category : Mathematics
ISBN : 3540244069

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Quantum Independent Increment Processes I by David Applebaum,B.V. Rajarama Bhat,Johan Kustermans,J. Martin Lindsay Pdf

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.

Author : Timothy M. W. Eyre
Publisher : Unknown
Page : 138 pages
File Size : 47,8 Mb
Release : 1998
Category : Electronic
ISBN : OCLC:1132032607

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Quantum Stochastic Calculus and Representations for Lie Superalgebras by Timothy M. W. Eyre Pdf

Author : Zhi-yuan Huang,Jia-an Yan
Publisher : Springer Science & Business Media
Page : 308 pages
File Size : 51,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401141086

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Introduction to Infinite Dimensional Stochastic Analysis by Zhi-yuan Huang,Jia-an Yan Pdf

The infinite dimensional analysis as a branch of mathematical sciences was formed in the late 19th and early 20th centuries. Motivated by problems in mathematical physics, the first steps in this field were taken by V. Volterra, R. GateallX, P. Levy and M. Frechet, among others (see the preface to Levy[2]). Nevertheless, the most fruitful direction in this field is the infinite dimensional integration theory initiated by N. Wiener and A. N. Kolmogorov which is closely related to the developments of the theory of stochastic processes. It was Wiener who constructed for the first time in 1923 a probability measure on the space of all continuous functions (i. e. the Wiener measure) which provided an ideal math ematical model for Brownian motion. Then some important properties of Wiener integrals, especially the quasi-invariance of Gaussian measures, were discovered by R. Cameron and W. Martin[l, 2, 3]. In 1931, Kolmogorov[l] deduced a second partial differential equation for transition probabilities of Markov processes order with continuous trajectories (i. e. diffusion processes) and thus revealed the deep connection between theories of differential equations and stochastic processes. The stochastic analysis created by K. Ito (also independently by Gihman [1]) in the forties is essentially an infinitesimal analysis for trajectories of stochastic processes. By virtue of Ito's stochastic differential equations one can construct diffusion processes via direct probabilistic methods and treat them as function als of Brownian paths (i. e. the Wiener functionals).

Author : Kalyan B. Sinha,Debashish Goswami
Publisher : Cambridge University Press
Page : 301 pages
File Size : 50,6 Mb
Release : 2007-01-25
Category : Mathematics
ISBN : 9781139461696

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Quantum Stochastic Processes and Noncommutative Geometry by Kalyan B. Sinha,Debashish Goswami Pdf

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.

Author : Mou-Hsiung Chang
Publisher : Unknown
Page : 412 pages
File Size : 42,6 Mb
Release : 2014
Category : MATHEMATICS
ISBN : 1107706548

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Quantum Stochastics by Mou-Hsiung Chang Pdf

"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"--

Author : Ioannis Karatzas,Steven Shreve
Publisher : Springer
Page : 490 pages
File Size : 41,6 Mb
Release : 2014-03-27
Category : Mathematics
ISBN : 9781461209492

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Brownian Motion and Stochastic Calculus by Ioannis Karatzas,Steven Shreve Pdf

A graduate-course text, written for readers familiar with measure-theoretic probability and discrete-time processes, wishing to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed, illustrated by results concerning representations of martingales and change of measure on Wiener space, which in turn permit a presentation of recent advances in financial economics. The book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The whole is backed by a large number of problems and exercises.

Author : Paul A. Meyer
Publisher : Springer
Page : 322 pages
File Size : 51,8 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540369592

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Quantum Probability for Probabilists by Paul A. Meyer Pdf

In recent years, the classical theory of stochastic integration and stochastic differential equations has been extended to a non-commutative set-up to develop models for quantum noises. The author, a specialist of classical stochastic calculus and martingale theory, tries to provide an introduction to this rapidly expanding field in a way which should be accessible to probabilists familiar with the Ito integral. It can also, on the other hand, provide a means of access to the methods of stochastic calculus for physicists familiar with Fock space analysis. For this second edition, the author has added about 30 pages of new material, mostly on quantum stochastic integrals.

Author : Uwe Franz,Adam Skalski
Publisher : Cambridge University Press
Page : 200 pages
File Size : 40,8 Mb
Release : 2016-01-07
Category : Mathematics
ISBN : 9781316674048

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Noncommutative Mathematics for Quantum Systems by Uwe Franz,Adam Skalski Pdf

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.

Author : Paul A. Meyer
Publisher : Springer
Page : 316 pages
File Size : 47,5 Mb
Release : 1995-11-17
Category : Mathematics
ISBN : 3540602704

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Quantum Probability for Probabilists by Paul A. Meyer Pdf

In recent years, the classical theory of stochastic integration and stochastic differential equations has been extended to a non-commutative set-up to develop models for quantum noises. The author, a specialist of classical stochastic calculus and martingale theory, tries to provide an introduction to this rapidly expanding field in a way which should be accessible to probabilists familiar with the Ito integral. It can also, on the other hand, provide a means of access to the methods of stochastic calculus for physicists familiar with Fock space analysis. For this second edition, the author has added about 30 pages of new material, mostly on quantum stochastic integrals.

Author : Stanley P. Gudder
Publisher : Courier Corporation
Page : 242 pages
File Size : 55,6 Mb
Release : 2014-05-05
Category : Science
ISBN : 9780486149189

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Stochastic Methods in Quantum Mechanics by Stanley P. Gudder Pdf

This introductory survey of stochastic methods and techniques in quantum physics, functional analysis, probability theory, communications, and electrical engineering also serves as a useful and comprehensive reference volume. 1979 edition.

Author : Stéphane Attal
Publisher : Springer Science & Business
Page : 318 pages
File Size : 43,7 Mb
Release : 2003
Category : Mathematics
ISBN : 981238975X

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Quantum Probability Communications by Stéphane Attal Pdf

Extensions of quantum stochastic calculus / Stéphane Attal -- Quantum Itô algebras: axioms, representations, decompositions / Viacheslav Belavkin -- Free probability for probabilists / Philippe Biane -- Conditional expectations on von Neumann algebras / Carlo Cecchini -- Classical probability theory: an outline of stochastic integrals and diffusions / Michel Émery -- Quantum stochastic differential equations / Franco Fagnola -- Canonical commutation and anticommutation relations / Mark Fannes -- Quantum and classical stochastic calculus / Alexander Holevo -- An introduction to quantum stochastic calculus and some of its applications / Robin Hudson -- Stationary processes in quantum probability / Burkhard Kümmerer