Quantum stochastic processes and noncommutative geometry 1st Edition by Kalyan Sinha, Debashish Goswami ISBN 978-0521834506 0521834503

Quantum stochastic processes and noncommutative geometry 1st Edition by Kalyan B. Sinha,  Debashish Goswami  – Ebook PDF Instant Download/Delivery:  978-0521834506, 0521834503
Full download Quantum stochastic processes and noncommutative geometry 1st Edition  after payment

Product details: 

ISBN 10:  0521834503

ISBN 13: 978-0521834506

Author: Kalyan B. Sinha,  Debashish Goswami 

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.

Table of contents: 

1 Introduction

2 Preliminaries

2.1 C* and von Neumann algebras

2.2 Completely positive maps

2.3 Semigroups of linear maps on locally convex spaces

2.4 Fock spaces and Weyl operators

3 Quantum dynamical semigroups

3.1 Generators of uniformly continuous quantum dynamical semigroups: the theorems of Lindblad and Christensen-Evans

3.2 The case of strongly continuous quantum dynamical semigroups

4 Hilbert modules

4.1 Hilbert C*-modules

4.2 Hilbert von Neumann modules

4.3 Group actions on Hilbert modules

5 Quantum stochastic calculus with bounded coefficients

5.1 Basic processes

5.2 Stochastic integrals and quantum Itô formulae

5.3 Hudson-Parthasarathy (H-P) type equations

5.4 Map-valued, E-H-type quantum stochastic calculus

6 Dilation of quantum dynamical semigroups with bounded generator

6.1 Hudson-Parthasarathy (H-P) dilation

6.2 Existence of structure maps and E-H dilation of 7,

6.3 A duality property

6.4 Appearance of Poisson terms in the dilation

6.5 Implementation of E-H flow

6.6 Dilation on a C*-algebra

6.7 Covariant dilation theory

7 Quantum stochastic calculus with unbounded coefficients

7.1 Notation and preliminary results

7.2 Q.S.D.E. with unbounded coefficients

7.3 Application: quantum damped harmonic oscillator

8 Dilation of quantum dynamical semigroups with unbounded generator

8.1 Dilation of a class of covariant quantum dynamical semigroups

8.2 Dilation of quantum dynamical semigroups on U.H.F. algebra

9 Noncommutative geometry and quantum stochastic processes

9.1 Basics of differential and Riemannian geometry

9.2 Heat semigroup and Brownian motion on classical manifolds

9.3 Noncommutative geometry

9.4 Examples

9.5 Asymptotic analysis of heat semigroups and Laplacians

9.6 Quantum Brownian motion on noncommutative manifolds

People also search for:

quantum stochastic processes and noncommutative geometry
    
is quantum mechanics stochastic
    
stochastic vs non stochastic
    
quantum stochastic processes
    
quantum stochastics

Tags: Kalyan Sinha, Debashish Goswami, Quantum stochastic, noncommutative geometry