Quantum Stochastic Processes and Noncommutative Geometry

Details the interaction between the two vigorous fields of non-commutative geometry and quantum stochastic processes.

Kalyan B. Sinha (Author), Debashish Goswami (Author)

9780521834506, Cambridge University Press

Hardback, published 25 January 2007

300 pages
23.5 x 16.2 x 2 cm, 0.561 kg

"The authors, two of the best-known experts in the field, provide a book which should find its place on the bookshelf of everybody who is interested in applying quantum stochastic calculus to advanced problems. In particular, anyone who wants to take up the challenge launched by the authors to include noncommutative manifolds in the discussion will welcome this book as the only source available."
Michael Skeide, Mathematical Reviews

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.

1. Introduction
2. Preliminaries
3. Quantum dynamical semigroups
4. Hilbert modules
5. Quantum stochastic calculus with bounded coefficients
6. Dilation of quantum dynamical semigroups with bounded generator
7. Quantum stochastic calculus with unbounded coefficients
8. Dilation of quantum dynamical semigroups with unbounded generator
9. Noncommutative geometry and quantum stochastic processes
Bibliography
Index.

Subject Areas: Stochastics [PBWL], Geometry [PBM], Calculus & mathematical analysis [PBK]