Interference in the classical probabilistic model and its representation in complex Hilbert space

The notion of a context (complex of physical conditions, that is to say: specification of the measurement setup) is basic in this paper.We show that the main structures of quantum theory (interference of probabilities, Bornʹs rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present already in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities. In our approach it is forbidden to consider abstract context independent probabilities: “first context and only then probability”. We construct the representation of the general contextual probabilistic dynamics in the complex Hilbert space. Thus dynamics of the wave function (in particular, Schrödingerʹs dynamics) can be considered as Hilbert space projections of a realistic dynamics in a “prespace”. The basic condition for representing of 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). Methods developed in this paper can be applied not only to quantum mechanics, but also to classical statistical mechanics. The main quantum-like structures (e.g., interference of probabilities) might be found in some models of classical statistical mechanics. Quantum-like probabilistic behavior can be demonstrated by biological systems. In particular, it was recently found in some psychological experiments.