Classical and Quantum Probability

R. F. Streater, Journ. Mathematical Phys., June 2000, 41, 3556-3603, 2000. archive reference math-ph/0002049
raymond.streater@kcl.ac.uk

ABSTRACT

We survey the development of probability since Bachelier's theory of speculation in 1900, emphasising that quantum theory is a generalisation of probability theory. We prove Bell's theorem, that there is no classical probability theory giving the same results as quantum theory for all statistical parameters. There is a brief account of Brownian motion and classical martingales, and the Wiener measure. We concentrate on the construction of quantum martingales using infinitely divisible representations of groups, Lie algebras and Clifford algebras. We briefly mention estimation theory via information theory and information geometry. There is an account of the Ito integral and our own construct, the Ito-Clifford integral. We end with some remarks about quantum stochastic processes and non-equilibrium statistical mechanics.

Preprint number KCL-MTH-00-70.