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Unification of classical and quantum probabilistic formalisms

Andrei Yu. Khrennikov

Published Feb 27, 2003• 1 citation
OpenAlex

Abstract

We demonstrate that the contextual approach to Kolmogorov probability model gives the possibility to unify this conventional model of probability with the quantum (Hilbert space) probability model. In fact, the Kolmogorov model can exhibit all distinguishing features of the quantum probability model. In particular, by using the contextual (interference) formula of total probability one can construct complex amplitudes of Kolmogorov probabilities. There exists a natural Hilbert space structure on the space of those complex amplitudes. Classical (Kolmogorovian) random variables are represented by in general noncommutative operators in the Hilbert space of complex amplitudes. The existence of such a contextual representation of the Kolmogorovian model looks very surprising in the view of the orthodox quantum tradition. Supported by EU-network “Quantum Probability and Applications”.

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We demonstrate that the contextual approach to Kolmogorov probability model gives the possibility to unify this conventional model of probability with the quantum (Hilbert space) probability model. In fact, the Kolmogorov model can exhibit all distinguishing features of the quantum probability model. In particular, by using the contextual (interference) formula of total probability one can construct complex amplitudes of Kolmogorov probabilities. There exists a natural Hilbert space structure on the space of those complex amplitudes. Classical (Kolmogorovian) random variables are represented by in general noncommutative operators in the Hilbert space of complex amplitudes. The existence of such a contextual representation of the Kolmogorovian model looks very surprising in the view of the orthodox quantum tradition. Supported by EU-network “Quantum Probability and Applications”.
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.
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⁡\coscos-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,...
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.
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. A projection of the classical contextual probabilistic model to the hyperbolic Hilbert space (a module over the commutative two dimensional Clifford algebra) has some similarities with the projection to the complex Hilbert space. However, in the hyperbolic quantum mechanics the principle of superposition is violated. Our realistic (but contextual!) approach to quantum mechanics does not contradict to various “no‐go theorems”, e.g., von Neumann, Bell, Kochen‐Specker. We emphasize that our projection of the classical probabilistic model to the complex Hilbert space is based on two incompatible observables (“reference observables”), e.g., the position and the momentum, or the position and the energy. Only these two observables can be considered as objective properties of quantum systems.
We show that, in spite of a rather common opinion, interference of probabilities can be easily obtained in purely corpuscular statistical model. All distinguishing features of the quantum probabilistic model (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, classical model should be considered as a contextual model (in the sense that all probabilities are determined by contexts-complexes of physical conditions). One of the main consequences of our contextual probabilistic reconsideration of foundations of quantum mechanics is that interference can be explained without wave-particle duality.