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Theory of Probability and Its Applications

The formula of total probability with the interference term and the representation of the Kolmogorov model in the Hilbert space

Andrei Yu. Khrennikov

Published Jan 1, 2006• 6 citations
OpenAlex

Abstract

The formula of total probability with the interference term was obtained and the representation of the Kolmogorov model in the Hilbert space was constructed.

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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.
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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: ``first context and then probability.'' We start with the conventional formula of total probability for contextual (conditional) probabilities and then we rewrite it by eliminating combinations of incompatible contexts from consideration. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. However, we did not just reconstruct the probabilistic formalism of conventional quantum mechanics. Our contextual probabilistic model is essentially more general and, besides the projection to the complex Hilbert space, it has other projections. The most important new prediction is the possibility (at least theoretical) of appearance of hyperbolic interference.