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).
We compare the classical Kolmogorov and quantum probability models. We show that the gap between these model is not so huge as it was commonly believed. The main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the Kolmogorov model. In particular, we obtain ``interference of probabilities'' without to appeal to the Hilbert space formalism. We interpret ``interference of probabilities'' as a perturbation (by a
cos-term) of the conventional formula of total probability. Our classical derivation of quantum probabilistic formalism can stimulate applications of quantum methods outside of microworld : in psychology, biology, economy,...
We compare the classical Kolmogorov and quantum probability models. We show that the gap between these models is not so huge as was commonly believed. The main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the Kolmogorov model. In particular, we obtain “interference of probabilities” without appealing to the Hilbert space formalism. We interpret “interference of probabilities” as a perturbation (by a cos-term) of the conventional formula of total probability. Our classical derivation of quantum probabilistic formalism can stimulate applications of quantum methods outside of the microworld, for instance, in psychology, biology, economy, and other domains of science.