Quantum potential theory

Philippe Biane, Luc Bouten, Fabio Cipriani, Norio Konno, Nicolas Privault, Quanhua Xu (auth.), Uwe Franz, Michael Schürmann (eds.)3540693645, 9783540693642

This volume contains the revised and completed notes of lectures given at the school “Quantum Potential Theory: Structure and Applications to Physics,” held at the Alfried-Krupp-Wissenschaftskolleg in Greifswald from February 26 to March 10, 2007.

Quantum potential theory studies noncommutative (or quantum) analogs of classical potential theory. These lectures provide an introduction to this theory, concentrating on probabilistic potential theory and it quantum analogs, i.e. quantum Markov processes and semigroups, quantum random walks, Dirichlet forms on C* and von Neumann algebras, and boundary theory. Applications to quantum physics, in particular the filtering problem in quantum optics, are also presented.

Table of contents :
Front Matter….Pages i-xi
Introduction….Pages 1-2
Potential Theory in Classical Probability….Pages 3-59
Introduction to Random Walks on Noncommutative Spaces….Pages 61-116
Interactions between Quantum Probability and Operator Space Theory….Pages 117-159
Dirichlet Forms on Noncommutative Spaces….Pages 161-276
Applications of Quantum Stochastic Processes in Quantum Optics….Pages 277-307
Quantum Walks….Pages 309-452
Back Matter….Pages 453-463