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arXiv:quant-ph/0306003 · Journal of Mathematical Physics

Contextual approach to quantum mechanics and the theory of the fundamental prespace

Andrei Yu. Khrennikov

Published Feb 12, 2004• 29 citations
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Abstract

We constructed a Hilbert space representation of a contextual Kolmogorov model. This representation is based on two fundamental observables—in the standard quantum model these are the position and momentum observables. This representation has all distinguishing features of the quantum model. Our representation is not standard model with hidden variables. In particular, this is not a reduction of the quantum model to the classical one.

DOI
10.1063/1.1645650

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We constructed a Hilbert space representation of a contextual Kolmogorov model. This representation is based on two fundamental observables—in the standard quantum model these are the position and momentum observables. This representation has all distinguishing features of the quantum model. Our representation is not standard model with hidden variables. In particular, this is not a reduction of the quantum model to the classical one.
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 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).
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 present critical arguments against individual interpretation of Bohr's complementarity and Heisenberg's uncertainty principles. Statistical interpretation of these principles is discussed in the contextual framework. We support the possibility to use Statistical Contextual Realist Interpretation of quantum formalism. In spite of all {\bf no-go} theorems (e.g., von Neumann, Kochen and Specker,..., Bell,...), recently (quant-ph/0306003 and 0306069) we constructed a realist basis of quantum mechanics. In our model both classical and quantum spaces are rough images of the fundamental {\bf prespace.} Quantum mechanics cannot be reduced to classical one. Both classical and quantum representations induce reductions of prespace information.
We present critical arguments against individual interpretation of Bohr’s complementarity and Heisenberg’s uncertainty principles. Statistical interpretation of these principles is discussed in the contextual framework. We support the possibility to use Statistical Contextual Realist Interpretation of quantum formalism. In spite of all no-go theorems (e.g., von Neumann, Kochen and Specker,..., Bell,...), recently (quant-ph/0306003 and 0306069) we constructed a realist basis of quantum mechanics. In our model both classical and quantum spaces are rough images of the fundamental prespace. Quantum mechanics cannot be reduced to classical one. Both classical and quantum representations induce reductions of prespace information. ”Often we enter the unknown edifice of a new scientific discipline through a lesser gate that leads us into a side passage. It may take us a long while to find our way to the main portal and to view the whole structure in its proper perspective”. E. Schrödinger 1