Overview

This volume contains several surveys of important developments in quantum probability. The new type of quantum central limit theorems, based on the notion of free independence rather than the usual Boson or Fermion independence is discussed. A surprising result is that the role of the Gaussian for this new type of independence is played by the Wigner distribution. This motivated the introduction of new type of quantum independent increments noise, the free noise and the corresponding stochastic calculus. A further generalization, the ϖ-noises, is discussed. The free stochastic calculus is shown to be able to fit naturally into the general representation free calculus. The basic free are shown to be realized as non-adapted stochastic integrals with respect to the usual Boson white noises. Quantum noise on the finite difference algebra is expressed in terms of the usual Boson white noises. A new quantum way of looking at classical stochastic flows, in particular diffusions on Riemannian Manifolds is explained. Quantum groups are discussed from the point of view of possible applications to quantum probability. The applications of quantum probability to physics are surveyed.Contents: An Invitation to the Weak Coupling and Low Density Limits (L Accardi et al.) Quantum Markov Chains: The Recurrence Problem (L Accardi & D Koroliuk) Towards a Quantum Theory of Classical Diffusions on Riemannian Manifolds (D Applebaum) Applications of Quantum Stochastic Calculus to Quantum Optics (A Barchielli) On Quantum Stochastic Integration with Respect to Free Noises (F Fagnola) Fractal Dimensions of States (M Ohya) Quantum Groups and Quantum Probability (S Majid) Unification of Quantum Noise Processes in Fock Spaces (K R Parthasarathy & K B Sinha) Survey on the Stochastic Integration on the Full Fock Space (R Speicher) A Spectral Property of One Parameter Family of Sampling Functions -- From Signal Analysis to Functional Analysis (H Umegaki) Free Noncommutative Random Variables, Random Matrices and II 1 Factors of Free Groups (D Voiculescu) Classical and Quantum Intrinsically Random Dynamical SystemsAn Invitation to the Prigogine Theory of Irreversibility (A Weron & K Weron) and other papersReadership: Mathematicians.

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