An Introduction To Quantum Stochastic Calculus

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"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.

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Genre : Mathematics
Author : K.R. Parthasarathy
Publisher : Birkhäuser
Release : 2012-12-06
File : 299 Pages
ISBN-13 : 9783034886413


Quantum Stochastic Calculus And Representations Of Lie Superalgebras

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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.

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Genre : Mathematics
Author : Timothy M.W. Eyre
Publisher : Springer
Release : 2006-11-14
File : 142 Pages
ISBN-13 : 9783540683858


A Measure Theoretical Approach To Quantum Stochastic Processes

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This monograph takes as starting point that abstract quantum stochastic processes can be understood as a quantum field theory in one space and in one time coordinate. As a result it is appropriate to represent operators as power series of creation and annihilation operators in normal-ordered form, which can be achieved using classical measure theory. Considering in detail four basic examples (e.g. a two-level atom coupled to a heat bath of oscillators), in each case the Hamiltonian of the associated one-parameter strongly continuous group is determined and the spectral decomposition is explicitly calculated in the form of generalized eigen-vectors. Advanced topics include the theory of the Hudson-Parthasarathy equation and the amplified oscillator problem. To that end, a chapter on white noise calculus has also been included.

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Genre : Science
Author : Wilhelm Waldenfels
Publisher : Springer
Release : 2013-11-29
File : 241 Pages
ISBN-13 : 9783642450822


Quantum Stochastic Processes And Noncommutative Geometry

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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.

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Genre : Mathematics
Author : Kalyan B. Sinha
Publisher : Cambridge University Press
Release : 2007-01-25
File : 301 Pages
ISBN-13 : 9781139461696


Open Quantum Systems Ii

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Understanding dissipative dynamics of open quantum systems remains a challenge in mathematical physics. This problem is relevant in various areas of fundamental and applied physics. Significant progress in the understanding of such systems has been made recently. These books present the mathematical theories involved in the modeling of such phenomena. They describe physically relevant models, develop their mathematical analysis and derive their physical implications.

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Genre : Mathematics
Author : Stéphane Attal
Publisher : Springer Science & Business Media
Release : 2006-06-07
File : 254 Pages
ISBN-13 : 9783540309925


Quantum Probability Communications

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Lecture notes from a Summer School on Quantum Probability held at the University of Grenoble are collected in these two volumes of the QP-PQ series. The articles have been refereed and extensively revised for publication. It is hoped that both current and future students of quantum probability will be engaged, informed and inspired by the contents of these two volumes. An extensive bibliography containing the references from all the lectures is included in Volume 12.

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Genre : Science
Author : S Attal
Publisher : World Scientific
Release : 2003
File : 314 Pages
ISBN-13 : 9789812389756


Quantum Probability Communications Qp Pq Volumes 11

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Lecture notes from a Summer School on Quantum Probability held at the University of Grenoble are collected in these two volumes of the QP-PQ series. The articles have been refereed and extensively revised for publication. It is hoped that both current and future students of quantum probability will be engaged, informed and inspired by the contents of these two volumes. An extensive bibliography containing the references from all the lectures is included in Volume 12.

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Genre : Mathematics
Author : J Martin Lindsay
Publisher : World Scientific
Release : 2003-06-27
File : 314 Pages
ISBN-13 : 9789814485593


Quantum Probability And Related Topics

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Quantum Probability and Related Topics is a series of volumes based on material discussed at the various QP conferences. It aims to provide an update on the rapidly growing field of classical probability, quantum physics and functional analysis.

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Genre : Science
Author : L. Accardi
Publisher : World Scientific
Release : 1993
File : 390 Pages
ISBN-13 : 9810211406


Quantum Probability And Related Topics Volume Viii

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Quantum Probability and Related Topics is a series of volumes based on material discussed at the various QP conferences. It aims to provide an update on the rapidly growing field of classical probability, quantum physics and functional analysis.

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Genre : Mathematics
Author : Luigi Accardi
Publisher : World Scientific
Release : 1993-09-30
File : 380 Pages
ISBN-13 : 9789814505208


White Noise Analysis And Quantum Information

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This volume is to pique the interest of many researchers in the fields of infinite dimensional analysis and quantum probability. These fields have undergone increasingly significant developments and have found many new applications, in particular, to classical probability and to different branches of physics. These fields are rather wide and are of a strongly interdisciplinary nature. For such a purpose, we strove to bridge among these interdisciplinary fields in our Workshop on IDAQP and their Applications that was held at the Institute for Mathematical Sciences, National University of Singapore from 3-7 March 2014. Readers will find that this volume contains all the exciting contributions by well-known researchers in search of new directions in these fields.

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Genre : Mathematics
Author : Luigi Accardi
Publisher : World Scientific
Release : 2017-08-29
File : 243 Pages
ISBN-13 : 9789813225473