Classical and quantum statistical models as projections of information prespace

K. Bruce Winterbon

Andrei Yu. Khrennikov

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).

Andrei Yu. Khrennikov

We consider transition from the classical statistical model to the quantum statistical model through ignorance (of huge volume) of information in process of construction of a wave function { a complex probability amplitude. Our approach clarifles relation between classical and quantum statistical models (and hence relation between classical and quantum information theories). In particular, it can be considered as a step toward demystiflcation of quantum theory. The notion of context (complex of physical conditions) is basic in this paper. We show that 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 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: \flrst context and then probability. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. Our formalism can be applied to various domains outside quantum physics, e.g., cognitive sci- ences, psychology, economics, chemistry: roughly speaking in any domain in that it is impossible to extract the whole information about an ontic (or to say realistic) model and hence the epis- temological (experimental or to say observational) model is constructed through huge reduction of information volume. We introduced a notion of prespace { space of states containing complete information about systems under consideration. Classical phase space and quantum Hilbert space are obtained as projections of such a prespace.

Peter Burton

Andrei Yu. Khrennikov

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 of the prespace-dynamics is the law of statistical conservation of energy -- conservation of probabilities. Construction of the dynamical representation is an important step in the development of contextual statistical viewpoint to quantum processes. But the contextual statistical model is essentially more general than the quantum one. Therefore in general the Hilbert space projection of the ``prespace'' dynamics can be nonlinear and even irreversible (but it is always unitary). There were found conditions of linearity and reversibility of the Hilbert space dynamical projection. We also found conditions for the conventional Schroedinger dynamics (including time-dependent Hamiltonians). We remark that in general even the Schroedinger dynamics is based just on the statistical conservation of energy; for individual systems the law of conservation of energy can be violated (at least in our theoretical model).

Roman S. Ingarden