Quantum Probability Communications

The following sections are included:

• CONTENTS of QPC XII

• PREFACE for Volumes XI & XII

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A simple axiomatic characterisation of the general (infinite dimensional, noncommutative) Itô algebra is given and a fundamental pseudo-Euclidean representation for such an algebra is described. The notion of Itô B*-algebra is defined, generalising C*-algebras to include the infinite dimensional Itô algebras of quantum Brownian and quantum Lévy motion; the B*-algebras of vacuum and thermal quantum noise are characterised. It is proved that every Itô algebra is canonically decomposed into an orthogonal sum of quantum Brownian (Wiener) algebra and quantum Lévy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise, as in the classical stochastic case due to the Lévy-Khinchin Theorem.

This is an introduction to some of the most probabilistic aspects of free probability theory.

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The basic notions of classical (commutative, real) stochastic calculus are sketched, with very few proofs: Brownian motions, filtrations, martingales, stopping times, local martingales, predictable processes, semimartingales with their quadratic variation and their stochastic integrals, stochastic differential equations, Markov processes and diffusions. The last chapter deals with the chaotic predictable property, which can be used to interpret Hudson and Parthasarathy’s bosonic quantum stochastic calculus in terms of some classical martingales.

The aim of these notes is the study of quantum stochastic differential equations of the form with possibly unbounded operators in the initial space whose solution is an isometric or unitary cocycle in Boson Fock space.

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Quantum stochastic calculus is developed with emphasis on exponential vectors and the continuous tensor product structure of Fock space. Multidimensional and ℤ₂-graded versions of the theory are included. Some applications, for example to Boson Fermion unification and to explicit construction of universal solutions of the quantum Yang-Baxter equation are described.

In this course we give an introduction to some aspects of the theory of stationary quantum stochastic processes. A first part introduces and discusses the basic notions. The second part discusses a class of stationary Markov chains. The final part gives an introduction to the structure theory of such processes and discusses its relevance to the theory of stochastic integration and continuous time processes.