[Paper Review] Classification of transformations of probabilities for preparation procedures: trigonometric and hyperbolic behaviours

Andrei Khrennikov|ArXiv.org|Dec 24, 2000

Quantum Mechanics and Applications6 references3 citations

TL;DR

This paper classifies probabilistic transformations in physical systems induced by preparation procedures, showing that both trigonometric (standard quantum-like) and hyperbolic probability rules can arise from frequency-based statistical analysis. It demonstrates that quantum probabilistic behavior can emerge from classical statistical perturbations without invoking wave-particle duality, with hyperbolic transformations observed in experiments with elementary particles such as polarized neutrons.

ABSTRACT

We provide frequency probabilistic analysis of perturbations of physical systems by preparation procedures. We obtained the classification of possible probabilistic transformations connecting input and output probabilities that can appear in physical experiments. We found that so called quantum probabilistic rule is just one of possible rules. Besides the well known trigonometric transformation (for example, for the polarization of light), there exist the hyperbolic transformation of probabilities. In fact, `hyperbolic polarization' have laready been observed in experiments with elementary particles. However, it was not interpreted in such a way. The situation is more complex with the hyperbolic interference of alternatives.

Motivation & Objective

• To classify all possible probabilistic transformations between input and output probabilities induced by physical preparation procedures.
• To investigate whether quantum probabilistic rules are unique or part of a broader class of statistical behaviors.
• To explore whether quantum stochasticity can be reduced to classical stochasticity through frequency-based analysis of preparation procedures.
• To examine the role of contextualism in determining probability distributions, especially in macroscopic systems.
• To provide a frequency-based foundation for understanding quantum-like interference and polarization without wave postulates.

Proposed method

• Uses the frequency approach to probability, defining probabilities as limits of relative frequencies in large ensembles.
• Models preparation procedures as statistical perturbations that alter input probabilities via deviation coefficients λi.
• Derives the general transformation rule: p'i = pi(1 + λi), with normalization constraint λ1p1 + λ2p2 = 0.
• Classifies possible forms of λi using trigonometric (cos θ) and hyperbolic (cosh θ) functions, leading to distinct probabilistic rules.
• Applies the classification to physical systems, showing that both trigonometric and hyperbolic rules are empirically realizable in experiments.
• Demonstrates that hyperbolic transformations can reproduce observed phenomena in neutron polarization and other quantum-like experiments.

Experimental results

Research questions

• RQ1 What are the complete classes of probabilistic transformations that can arise from physical preparation procedures?
• RQ2 Can quantum-like probabilistic behavior be explained without invoking wave functions or non-classicality?
• RQ3 Are hyperbolic probability transformations empirically realizable, and if so, where do they appear in physical experiments?
• RQ4 To what extent can quantum stochasticity be simulated using classical statistical perturbations in macroscopic systems?
• RQ5 How does the contextualist framework—where probabilities depend on the full preparation/measurement context—account for both trigonometric and hyperbolic probabilistic rules?

Key findings

• The standard quantum probabilistic rule (trigonometric transformation) is not unique; hyperbolic transformations also arise naturally from frequency-based analysis.
• Hyperbolic probability transformations, such as p′₁ = 2p₁cosh²(θ/2), are empirically observed in experiments with polarized neutrons and other elementary particles.
• The model shows that quantum-like interference and polarization can emerge from classical statistical perturbations without wave postulates.
• Both trigonometric and hyperbolic rules are realizable in experiments, with the choice depending on the geometric and dynamical structure of the preparation procedure.
• The transition from classical to quantum-like behavior corresponds to a shift from negligible to nontrivial statistical deviations in preparation procedures.
• Macroscopic systems can exhibit quantum-like probabilistic behavior, suggesting that quantum effects are not inherently quantum but statistical in origin.