Michal Gnacik (Lancaster) - "From classical to quantum probability - no return"

Abstract

In the 1930s A. Kolmogorov and J. von Neumann proposed two different sets of axioms for mathematically modelling random phenomena. Kolmogorov's model is a classical probability space, that is a triple (Ω, F, ℙ) where Ω is the set of all possible outcomes of the random phenomenon, F is a σ-algebra of subsets of Ω, called events, and ℙ is a probability measure on F. Von Neumann's model is a quantum probability space, that is a pair (M, ϕ) consisting of (what is now called) a von Neumann algebra M, whose projections are the events, and a normal state ϕ on M giving the probabilities. We show that a self-adjoint operator affiliated with the von Neumann algebra of a quantum probability space may be interpreted as both a classical and a quantum random variable. However, a pair of non-commuting self-adjoint operators cannot be simultaneously represented as multiplication operators on the same Hilbert space L²(Ω, F, ℙ). A quantum probability space thereby 'contains' infinitely many classical probability spaces. The quantum model therefore generalises the classical model.