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Elements de probabilités quantiques. (Elements of quantum probability). (French) Zbl 0604.60001
Probabilités XX, Proc. Sémin., Strasbourg 1984/85, Lect. Notes Math. 1204, 186-312 (1986).
[For the entire collection see Zbl 0593.00014.]
This is a detailed exposition of the principal notions and results of quantum (noncommutative) probability theory from the point of view of classical stochastic processes. The final aim is to give an introduction to the recently developed quantum stochastic differential calculus. The exposition addresses to probabilists having standard background on measure theory and Hilbert space; no primary knowledge of quantum physics is assumed.
Contents: I. Fundamental notions (Axiomatics of von Neumann - tribes and laws. Axiomatics of von Neumann random variables. Kernels. Appendix - the language of \(C^*\)-algebras, convergence in law). II. Some discrete examples (Spin. Tribe generated by two events. Finite system of spins). III. Canonical pairs (The Stone-von Neumann theorem. Calculus of canonical pairs. Wigner functions). IV. Probabilities on the Fock space (The Fock space - algebraic definition. Probabilistic interpretation. Complements). V. Noncommutative stochastic calculus (Stochastic integrals of operator processes. Stochastic calculus and quantum stochastic differential equations. Operators defined by kernels).
Reviewer: A.S.Holevo

MSC:
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60A99 Foundations of probability theory
60H05 Stochastic integrals
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