L Accardi

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. Contents:Wiener Noise Versus Wigner Noise in Quantum Electrodynamics (L Accardi & Y G Lu)A Note on Processes on Bialgebras, Quantum Flows, and Convolution Semigroups of Instruments (A Barchielli & G Lupieri)A Note on the Cohomology of Quantum Flows (P K Das)Chebotarev's Sufficient Conditions for Conservativity of Quantum Dynamical Semigroups (F Fagnola)On Quantum Resonance (C Fernandez & R Rebolledo)The Statistical Invariants in Quantum Probability and De Finetti's Coherence Principle (A Gilio)Some Remarks on Trace of Operator Logarithm and Relative Entropy (F Hiai)On Feynman-Kac Cocycles (R L Hudson)Phonion Limits and Macroscopic Quasi-Particle Spectrum for the BCS-Model (A Verbeure et al.)and other papers Readership: Mathematicians. keywords:Quantum Probability;Wiener Noise;Quantum Flows;Dynamical Semigroups;White-Noise

Volume VIII

Quantum Probability and Related Topics

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 II1 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 papers Readership: Mathematicians. keywords:Quantum Probability

QP — PQ (Volume VI)

Si Si

H H Kuo

L Streit

N Obata

K Saitô

The topics discussed in this book can be classified into three parts: (i) Gaussian processes. The most general and in fact final representation theory of Gaussian processes is included in this book. This theory is still referred to often and its developments are discussed. (ii) White noise analysis. This book includes the notes of the series of lectures delivered in 1975 at Carleton University in Ottawa. They describe the very original idea of introducing the notion of generalized Brownian functionals (nowadays called “generalized white noise functionals”, and sometimes “Hida distribution”. (iii) Variational calculus for random fields. This topic will certainly represent one of the driving research lines for probability theory in the next century, as can be seen from several papers in this volume. Contents:General Theory of White Noise FunctionalsGaussian and Other ProcessesInfinite Dimensional Harmonic Analysis and Rotation GroupQuantum TheoryFeynman Integrals and Random FieldsVariational Calculus and Random FieldsApplication to Biology Readership: Graduate students and researchers in the fields of probability theory, functional analysis, statistics and theoretical physics. Keywords:White Noise;Gaussian;Brownian Motion;LÃ©vy Process;Canonical Representation;Stochastic Infinitesimal Equation;Generalized Functional;Innovation;Multiple Markov;Random FieldReviews:“This collection of papers is a tribute to one of the great researchers within stochastic analysis, Takeyuki Hida … An interesting appendix, however, is the collection of remarks at the end of the book … These remarks serve to put the various papers into perspective, and represent a valuable contribution.”Mathematical Reviews

Selected Papers of Takeyuki Hida

Quantum Probability and Related Topics is a series of volumes based on materials discussed in the various QP conferences. It aims at providing an update on the rapidly growing field of classical probability, quantum physics and functional analysis. Contents:From Markovian Approximation to a New Type of Quantum Stochastic Calculus (L Accardi & Y G Lu)Unitarity of Solutions to Quantum Stochastic Differential Equations and Conservativity of the Associated Semigroups (F Fagnola)Kernel Representation of ∗-Semigroups Associated with Infinitely Divisible States (V P Belavkin)Minimal Solutions in Classical and Quantum Stochastics (A M Chebotarev)Structure Relations for Fermionic Flows (R L Hudson & P A Shepperson)Fermionic Hypercontractivity (J M Lindsay & P A Meyer)Large Deviations and Mean-Field Quantum Systems (R F Werner)and other papers Readership: Mathematicians and mathematical physicists. keywords:Quantum Probability

L Accardi

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