Classical and Quantum Probability

Abstract

We survey the development of probability from 1900, starting with Bachelier's theory of speculation. Fisher information appears in the theory of estimation. We touch on Brownian motion, and the Wiener integral. The Ito calculus, and its relation to to the heat equation, is mentioned. Quantum theory is introduced as a generalisation of probability, rather than of mechanics. The weakness of attempts to describe quantum theory in terms of hidden variables is explained, by a simple proof of Bell's inequality. Quantum versions of the Langevin equation are discussed, and the theory of continuous tensor products is used to give a possible quantum version. The quantum stochastic calculus of Barnett, Wilde and the author, as well as that of Parthasarathy and Hudson, is introduced.