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This volume represents a part of the main result obtained by a group of French probabilists, together with the contributions of a number of colleagues, mainly from the USA and Japan. All the papers present new results obtained during the academic year 1991-1992. The main themes of the papers are: quantum probability (P.A. Meyer and S. Attal), stochastic calculus (M. Nagasawa, J.B. Walsh, F. Knight,...
This chapter is the closest in these notes to what is usually called “Quantum Mechanics”. The present version is considerably shorter than the original French. It thus becomes more obvious that its main topic is not really elementary quantum mechanics, but rather elementary Fock space, and the quantum analogue of finite dimensional Gaussian random variables.
Résumé Nous donnons une nouvelle présentation des intégrales stochastiques non commutatives définies par Hudson et Parthasarathy [4]. Cette approche, suggérée la première fois par Meyer [5], se place dans une interprétation probabiliste de l'espace de Fock en définissant ces intégrales d'opérateurs grâce à des équations différentielles stochastiques (classiques). Elle permet de voir explicitement...
These notes are a revised version of notes in French published in successive volumes (XX to XXII) of the Séminaire de Probabilités, and we will not offer again a slow introduction to this topic, with detailed justifications of the basic definitions. Non-commutative probability is the kind of probability that goes with the non-commutative world of quantum physics, and as such it is a natural domain...
In this chapter, we reach the main topic of these notes, non-commutative stochastic calculus for adapted families of operators on Fock space, with respect to the basic operator martingales. This calculus is a direct generalization of the classical Ito integration of adapted stochastic processes w.r.t. Brownian motion, or other martingales. Its physical motivation is quantum mechanical evolution in...
From the algebraic point of view, a multiple Fock space over ℌ is nothing but a standard (symmetric or antisymmetric) Fock space over a direct sum of copies of ℌ,and new definitions are not necessary in principle. However, this chapter contains important new notation, and a few rather interesting constructions, like that of the “finite-temperature” (= extremal universally invariant) representations...
We present here an introduction to some recent work of the Heidelberg school, due to W. von Waldenfels, P. Glockner, and specially M. Schürmann, on processes with independent increments in a non-commutative setup. This chapter is reduced to the bare essentials, and its purpose is only to lead the reader to the much richer original articles. Some very recent and interesting remarks from Belavkin [6]...
This chapter has little to do with the geometric property of elementary particles called spin. Historically spin 1/2 provided the first basic example of a “two-level quantum system”, so that till now those simplest of all quantum systems are introduced to students of physics with the help of “fictitious spins”. The truth is, that two-level systems (or spins) play in quantum probability the role of...
The preceding chapters dealt with the non-commutative analogues of discrete r.v.’s, then of real valued r.v.’s, and we now begin to discuss stochastic processes. We start with the description of Fock space (symmetric and antisymmetric) as it is usually given in physics books. Then we show that boson Fock space is isomorphic to the L2 space of Wiener measure, and...
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