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arXiv:quant-ph/0106073

`Quantum probabilities' as context depending probabilities

Andrei Yu. Khrennikov

Published Jun 13, 2001• 12 citations
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Abstract

We study transformations of conventional (`classical') probabilities induced by context transitions. It is demonstrated that the transition from one complex of conditions to another induces a perturbation of the classical rule for the addition of probabilistic alternatives. We classify such perturbations. It is shown that there are two classes of perturbations: (a) trigonometric interference; (b) hyperbolic interference. In particular, the well known `quantum interference of probabilistic alternatives' can be obtained in classical (but contextual) probabilistic framework. Therefore we need not apply to wave arguments (or consider superposition principle) to get Quantum Statistics. In particular, interference could be a feature of experiments with purely corpuscular objects.

DOI
10.48550/arxiv.quant-ph/0106073

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We study transformations of conventional (classical') probabilities induced by context transitions. It is demonstrated that the transition from one complex of conditions to another induces a perturbation of the classical rule for the addition of probabilistic alternatives. We classify such perturbations. It is shown that there are two classes of perturbations: (a) trigonometric interference; (b) hyperbolic interference. In particular, the well known quantum interference of probabilistic alternatives' can be obtained in classical (but contextual) probabilistic framework. Therefore we need not apply to wave arguments (or consider superposition principle) to get Quantum Statistics. In particular, interference could be a feature of experiments with purely corpuscular objects.
We study the following problem: Is it possible to explain the quantum interference of probabilities in the purely corpuscular model for elementary particles? We demonstrate that (by taking into account perturbation effects of measurement and preparation procedures) we can obtain cos⁡θ\cosθcosθ-perturbation (interference term) in probabilistic rule connecting preparation procedures for purely corpuscular objects. On one hand, our investigation demonstrated that there is nothing special in so called `quantum probabilities': the right choice of statistical ensembles gives the possibility to escape all `pathologies'. On the other hand, we found that the standard trigonometric interference of alternatives (observed, in particular, in quantum mechanics) is not the unique possibility to extend (disturb) the conventional probabilistic rule for addition of alternatives. There exist two other probabilistic rules that connect three preparation procedures: hyperbolic and hyper-trigonometric interferences.
By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. Each class is characterized by a perturbation of the `classical probabilistic rule': (a) costheta\\cos\\thetacostheta, (b) coshtheta\\cosh \\thetacoshtheta, (c) both costheta\\cos\\thetacostheta and coshtheta\\cosh \\thetacoshtheta. Trigonometric transformations correspond to context-transitions that induce statistical deviations of relatively small magnitudes (in classical physics - negligibly small); hyperbolic - relatively large magnitudes. We found that not only preparation procedures described by conventional quantum formalism can have trigonometric probabilistic behaviour. We propose generalizations of bfC{\\bf C}bfC-linear space probabilistic calculus to describe non quantum (trigonometric and hyperbolic) probabilistic transformations.
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.
By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. It is shown that not only quantum preparation procedures can have trigonometric probabilistic behaviour. We propose generalizations of C -linear space probabilistic calculus to describe non-quantum (trigonometric and hyperbolic) probabilistic transformations. We also analyse the superposition principle in this framework.
We demonstrate the quantum probabilistic rule (which differ from classical Bayes' formula by the cosinus factor) can be obtained on purely classical basis as a consequence of the perturbation effect of preparation procedures. In any case the interference can be easily simulated for macrosystems. Moreover, we obtain the classification of all possible transformations of probablities which can arise via perturbation effects. There are two main transformation classes: with trigonometric and hyperbolic ("interference") perturbations of classical Bayes' formula. Quantum probabilistic rule is just a particular case of trigonometric probablistic behaviour. Hyperbolic probablistic behaviour can be easily simulated. However, the question of its observation for "real physical phenomena" is open.
We study the role of context, complex of physical conditions, in quantum as well as classical experiments. It is shown that by taking into account contextual dependence of experimental probabilities we can derive the quantum rule for the addition of probabilities of alternatives. Thus we obtain quantum interference without applying to wave or Hilbert space approach. The Hilbert space representation of contextual probabilities is obtained as a consequence of the elementary geometric fact: cos\\coscos-theorem. By using another fact from elementary algebra we obtain complex-amplitude representation of probabilities. Finally, we found contextual origin of noncommutativity of incompatible observables.