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Il Nuovo Cimento B

Frequency derivation of the EPR-Bohm correlations

Andrei Yu. Khrennikov

Published Feb 1, 2004• 1 citation
Publisher

Abstract

The probabilistic structure of quantum mechanics is investigated in the frequency framework. By using rather complicated frequency calculations we reproduce the EPR-Bohm correlation function which is typically derived by using the calculus of probabilities in a Hilbert space. Our frequency probabilistic model of the EPR-Bohm experiment is a realist model – physical observables are considered as objective properties of physical systems. It is also local – a measurement over one part of a composite system does not disturb another part of this system. Nevertheless, our result does not contradict to the well known Bell’s “NO-GO ” theorem. J. Bell used the conventional (Kolmogorov) measure-theoretical approach. We use the frequency approach. In the latter approach there are no reasons to assume that the simultaneous probability distribution exists: corresponding frequencies may fluctuate and not approach any definite limit (which Bell would like to use as the probability). The frequency probabilistic derivation demonstrated that incompatibility of observables under consideration plays the crucial role in producing of the EPR-Bohm correlations. 1 1

DOI
10.1393/ncb/i2003-10090-7

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The probabilistic structure of quantum mechanics is investigated in the frequency framework. Such an approach can be interpreted as a contextual approach to quantum probabilities. By using rather complicated frequency calculations we reproduce the EPR-Bohm correlation function which is typically derived by using the calculus of probabilities in a Hilbert space. Our frequency probabilistic model of the EPR-Bohm experiment is a realist model -- physical observables are considered as objective properties of physical systems. It is also local -- a measurement over one part of a composite system does not disturb another part of this system. Nevertheless, our result does not contradict to the well known Bell's ``NO-GO'' theorem. J. Bell used the conventional (Kolmogorov) measure-theoretical approach. We use the frequency approach. In the latter approach there are no reasons to assume that the simultaneous probability distribution exists: corresponding frequencies may fluctuate and not approach any definite limit (which Bell would like to use as the probability). The frequency probabilistic derivation demonstrated that incompatibility of observables under consideration plays the crucial role in producing of the EPR-Bohm correlations.

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•Mar 25, 1996•0 citations
The probabilistic structure of quantum mechanics is investigated in the frequency framework. By using rather complicated frequency calculations we reproduce the EPR-Bohm correlation function which is typically derived by using the calculus of probabilities in a Hilbert space. Our frequency probabilistic model of the EPR-Bohm experiment is a realist model – physical observables are considered as objective properties of physical systems. It is also local – a measurement over one part of a composite system does not disturb another part of this system. Nevertheless, our result does not contradict to the well known Bell’s “NO-GO ” theorem. J. Bell used the conventional (Kolmogorov) measure-theoretical approach. We use the frequency approach. In the latter approach there are no reasons to assume that the simultaneous probability distribution exists: corresponding frequencies may fluctuate and not approach any definite limit (which Bell would like to use as the probability). The frequency probabilistic derivation demonstrated that incompatibility of observables under consideration plays the crucial role in producing of the EPR-Bohm correlations. 1 1
The probabilistic structure of quantum mechanics is investigated in the frequency framework. Such an approach can be interpreted as a contextual approach to quantum probabilities. By using rather complicated frequency calculations we reproduce the EPR-Bohm correlation function which is typically derived by using the calculus of probabilities in a Hilbert space. Our frequency probabilistic model of the EPR-Bohm experiment is a realist model -- physical observables are considered as objective properties of physical systems. It is also local -- a measurement over one part of a composite system does not disturb another part of this system. Nevertheless, our result does not contradict to the well known Bell's ``NO-GO'' theorem. J. Bell used the conventional (Kolmogorov) measure-theoretical approach. We use the frequency approach. In the latter approach there are no reasons to assume that the simultaneous probability distribution exists: corresponding frequencies may fluctuate and not approach any definite limit (which Bell would like to use as the probability). The frequency probabilistic derivation demonstrated that incompatibility of observables under consideration plays the crucial role in producing of the EPR-Bohm correlations.
We demonstrate that the EPR-Bohm probabilities can be easily obtained in the classical (but contextual) probabilistic framework by using the formula of interference of probabilities. From this point of view the EPR-Bell experiment is just an experiment on interference of probabilities. We analyze the time structure of contextuality in the EPR-Bohm experiment. The conclusion is that quantum mechanics does not contradict to a local realistic model in which probabilities are calculated as averages over conditionings/measurements for pairs of instances of time t1 < t2. If we restrict our consideration only to simultaneous measurements at the fixed instance of time t we would get contradiction with Bell’s theorem. One of implications of this fact might be the impossibility to define instances of time with absolute precision on the level of the microscopic realistic model. 1. Introduction.In a series of papers, see, e.g., [1] it was demonstrated
We demonstrate that the EPR-Bohm probabilities can be easily obtained in the classical (but contextual) probabilistic framework by using the formula of interference of probabilities. From this point of view the EPR-Bell experiment is just an experiment on interference of probabilities. We analyse the time structure of contextuality in the EPR-Bohm experiment. The conclusion is that quantum mechanics does not contradict to a local realistic model in which probabilities are calculated as averages over conditionings/measurements for pairs of instances of time t1<t2.t_1< t_2.t1​<t2​. If we restrict our consideration only to simultaneous measurements at the fixed instance of time ttt we would get contradiction with Bell's theorem. One of implications of this fact might be the impossibility to define instances of {\it time with absolute precision} on the level of the contextual microscopic realistic model.
We perform frequency analysis of the EPR-Bell argumentation. One of the main consequences of our investigation is that the existence of probability distributions of the Kolmogorov-type which was supposed by some authors is a mathematical assumption which may not be supported by actual physical quantum processes. In fact, frequencies for hidden variables for quantum particles and measurement devices may fluctuate from run to run of an experiment. These fluctuations of frequencies for micro-parameters need not contradict to the stabilization of frequencies for physical observables. If, nevertheless, micro-parameters are also statistically stable, then violations of Bell's inequality and its generalizations may be a consequence of dependence of collectives corresponding to two different measurement devices. Such a dependence implies the violation of the factorization rule for the simultaneous probability distribution. Formally this rule coincides with the well known BCHS locality condition (or outcome independence condition). However, the frequency approach implies totally different interpretation of dependence. It is not dependence of events, but it is dependence of collectives. Such a dependence may be induced by the same preparation procedure.
The Bell theorem [103] can be seen in several different ways, as the EPR argument. Initially, Bell invented it as a direct logical continuation of the EPR theorem: the idea is to take the existence of the EPR elements of reality seriously, and to push it further by introducing them explicitly into the mathematics with the notation λ; one then proceeds to study all possible kinds of correlations that can be obtained from the fluctuations of one or several variables λ, making the condition of locality explicit in the mathematics (locality was already useful in the EPR theorem, but not used in equations). The reasoning develops within determinism (considered as proved by the EPR reasoning) and classical probabilities; it studies in a completely general way all kinds of correlations that can be predicted from the fluctuations of some classical common cause in the past – if one prefers, from some random choice concerning the initial state of the system. This leads to the famous inequalities. But subsequent studies have shown that the scope of the Bell theorem is not limited to determinism; for instance, the λ variables may determine the probabilities of the results of future experiments, instead of the results themselves (see Appendix B), without canceling the theorem. We postpone the discussion of the various possible generalizations to §4.2.3. For the moment, we just emphasize that the essential condition for the validity of the Bell theorem is locality: all kinds of fluctuations can be assumed, but their effect must affect physics only locally.
Measuring processes of a single spin-1/2 object and of a pair of spin-1/2 objects in the EPR-Bohm state are modeled by systems of differential equations. These models are based on mainly the following two ideas: (i) as a result of the interaction with a measuring apparatus, the state of the object, changes and approaches an attractor, in which the object possesses the measured value in the phase space of states of objects; (ii) a quantum-mechanical state corresponds to an ensemble of states in the corresponding attractor in the phase space. The latter model is a local model with hidden variables of the EPR-Bohm gedanken experiment. Although there is no dynamical interaction between the pair of spin-1/2 objects, the model reproduces approximately the quantum-mechanical correlations by using coincidence counting. Hence the Bell inequality is violated. This result supports the idea that the coincidence counting is the source of the apparent nonlocality in the EPR-Bohm gedankenexperiment.