Schrödinger dynamics as the Hilbert space projection of a realistic contextual probabilistic dynamics

Abstract

We study the problem of consistency of classical and quantum probabilistic models (also known as the problem of existence of hidden variables). We show that the solution of this problem depends crucially on a mathematical model for correspondence between classical (“prequantum”) model and quantum model. We construct a projection of the classical measure theoretic model to the complex Hilbert space, which shares the distinguishing features of the quantum model: interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators. These structures are present in a latent form in the classical Kolmogorov probability model. However, the classical model should be considered as a calculus of contextual probabilities. We constructed the representation of contextual probabilistic dynamics in the complex Hilbert space. Thus dynamics of the wave function can be considered as Hilbert space projections of realistic dynamics in a “prespace”. The basic condition for representing the prespace dynamics is the law of statistical conservation of energy—conservation of probabilities. In general, the Hilbert space projection of the “prespace” dynamics can be nonlinear and even irreversible (but it is always unitary).