On the early years of quantum stochastic calculus

The origins and early history of quantum stochastic calculus are surveyed, with emphasis on the collaboration between K R Parthasarathy and the author. 1. Introduction. I …rst met Partha in 1971 when he was at She¢ eld University. The occasion was a regional meeting of the UK Royal Statistical Society in Leeds. Partha gave what to me was a brilliantly clear exposition of quantum probability as a new theory of probability in which the -…eld of events was replaced by the non-Boolean lattice of sub-Hilbert spaces of a Hilbert space. Real valued random variables, regarded as lattice homomorphisms from the Borel -…eld to the lattice of events, instead of being the set-mapping inverses of measurable functions as in the classical case, are represented as self-adjoint operators through the spectral theorem. Probability measures are characterised by Gleason’s theorem [7] as density operators. At the end of his lecture Partha mentioned that he had learned that a noncommutative central limit theorem had been proved recently in this context, enabling me to introduce myself as the author, with my student C D Cushen, of that theorem [4]. Thereby began the collaboration which has been the most rewarding of my life. Acknowledgements. The author thanks the referee for a number of suggested improvements and corrections, and Paul Jones for a careful reading of the manuscript and for some valuable criticisms. 2. The canonical central limit theorem. In this central limit theorem real-valued random variables are replaced by canonical pairs, that is, pairs of self-adjoint operators (p; q) satisfying a mathematically rigorous form of the Heisenberg commutation relation (with Planck’s constant set equal to 2 ) [p; q] = i: 2000 Mathematics Subject Classi…cation. 81S 25.