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

Abstract

We study the problem of consistency of classical and quantumprobabilistic models (also known as the problem of existence ofhidden variables). We show that the solution of this problemdepends crucially on a mathematical model for correspondencebetween classical (“prequantum”) model and quantum model. Weconstruct a projection of the classical measure theoretic modelto the complex Hilbert space, which shares the distinguishingfeatures of the quantum model: interference of probabilities,Born's rule, complex probabilistic amplitudes, Hilbert statespace, representation of observables by operators. Thesestructures are present in a latent form in the classicalKolmogorov probability model. However, the classical model shouldbe considered as a calculus of contextual probabilities. Weconstructed the representation of contextual probabilisticdynamics in the complex Hilbert space. Thus dynamics of the wavefunction can be considered as Hilbert space projections ofrealistic dynamics in a “prespace”. The basic condition forrepresenting the prespace dynamics is the law of statisticalconservation of energy—conservation of probabilities. Ingeneral, the Hilbert space projection of the “prespace” dynamicscan be nonlinear and even irreversible (but it is always unitary).