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arXiv:0906.0509 · Il Nuovo Cimento B

p-adic probability prediction of correlations between particles in the two-slit and neutron interferometry experiments

Andrei Yu. Khrennikov

Published Jun 2, 2009• 11 citations
arXivHTMLPDF

Abstract

We start from Feynman`s idea to use negative probabilities to describe the two slit experiment and other quantum interfernce experiments. Formally by using negative probability distributions we can explain the results of the two slit experiment on the basis of the pure corpuscular picture of quantum mechnanics. However, negative probabilities are absurd objects in the framework of the standard Kolmogorov theory of probability. We present a large class of non-Kolmogorovean probability models where negative probabilities are well defined on the frequency basis. These are models with probabilities which belong to the so-called field of $p$-adic numbers. However, these models are characterized by correlations between trails. Therefore, we predict correlations between particles in interference experiments. In fact, our predictions are similar to the predictions of the so-called nonergodic interpretation of quantum mechanics, which was proposed by V. Buonomano. We propose the concrete experiments (in particular, in the framework of the neutron interferometry) to verify our predictions on the correlations.

DOI
10.48550/arxiv.0906.0509

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Section I.- Theoretical Implications of Time-Dependent Double Resonance Neutron Interferometry.- Calculations in the Causal Interpretation of Quantum Mechanics.- Calculation of Amplification Processes Generated by Empty Quantum Waves.- Relative Metrics and Physical Models for Non-local Particles.- A Two-Fluid Vacuum, the Photex, and the Photex-Photon Connection.- Discussion I.- Section II.- Summary of F. De Martini's Paper, Testing of Particle Distinguish-ability with Laser Interference Experiments.- Toward a Causal Interpretation of the Relativistic Quantum Mechanics of a Spinning Particle.- Nonlinear Behaviour of Light at Very Low Intensities: The Photon Clump Model.- Geometry-Dependent Predictions of the Quantum Potential Model for the Anomalous Photoelectric Effect.- EPR-Type Experiments and Enhanced Photon Detection.- Discussion II.- Section III.- Stochastic Optics: Wave-Particle Duality Without Particles.- Neo-Hertzian Electromagnetism.- Effective Photon Hypothesis vs. Quantum Potential Theory: Theoretical Predictions and Experimental Verification.- Compositeness of Photons and Its Implications.- Logical Meanings in Quantum Mechanics for Axioms and for Imaginary and Transfinite Numbers and Exponentials.- The Aharonov-Bohn Effect and the Quantum Interference Phase Shift due to an Electrostatic Field.- Discussion III.- Section IV.- Critical Analysis of the Tests of Local Realism Versus Quantum Mechanics.- Physical Photons: Theory, Experiment and Implications.- The Theoretical Nature of a Photon in a Lattice Vacuum.- Is the Photon Concept Really Necessary?.- Beyond Quantum Mechanics Using the Subvac.- Discussion IV.- Section V.- Does Quantum Electrodynamics Explain the Observed Lamb Shift?.- The Quantum and Rest Frame Vacuum - Absolute Motion with Lorentz Covariance.- The Double Slit Experiment and the Nonergodic Interpretation of Quantum Mechanics.- Einstein's Conception of the Ether and Its Up-To-Date Applications in the Relativistic Wave Mechanics.- Discussion V.- Section VI.- Concluding Remarks.- Concluding Remarks.- Comments.- Photo.- Conference Participants and Observers.

•Jan 1, 1987•19 citations
We start from Feynman`s idea to use negative probabilities to describe the two slit experiment and other quantum interfernce experiments. Formally by using negative probability distributions we can explain the results of the two slit experiment on the basis of the pure corpuscular picture of quantum mechnanics. However, negative probabilities are absurd objects in the framework of the standard Kolmogorov theory of probability. We present a large class of non-Kolmogorovean probability models where negative probabilities are well defined on the frequency basis. These are models with probabilities which belong to the so-called field of ppp-adic numbers. However, these models are characterized by correlations between trails. Therefore, we predict correlations between particles in interference experiments. In fact, our predictions are similar to the predictions of the so-called nonergodic interpretation of quantum mechanics, which was proposed by V. Buonomano. We propose the concrete experiments (in particular, in the framework of the neutron interferometry) to verify our predictions on the correlations.
This paper is devoted to foundations of probability theory. We discuss interpretations of probability, corresponding mathematical formalisms, and applications to quantum physics. One of the aims of this paper is to show that the probability model based on Kolmogorov's axiomatics cannot describe all stochastic phenomena, i.e., that quantum physics induces natural restrictions of the use of Kolmogorov's theory and that we need to develop non-Kolmogorov models for describing some quantum phenomena. The physical motivations are presented in a clear and brief manner. Thus the reader does not need to have preliminary knowledgeof quantum physics. Our main idea is that we cannot develop non-Kolmogorov models by the formal change of Kolmogorov's axiomatics. We begin with interpretations (classical, frequency, and proportional). Then we present a class of non-Kolmogorov models described by so-called p-adic numbers. Here, in particular, we obtain a quite natural realization of negative probabilities. These negative probability distributions might provide a solution of some quantum paradoxes.
A hidden variables model for interference phenomena based on ppp-adic random dynamical systems
We propose a model based on random dynamical systems (RDS) in information spaces (realized as rings of ppp-adic integers) which supports Buonomano's non-ergodic interpretation of quantum mechanics. In this model the memory system of an equipment works as a dynamical system perturbed by noise. Interference patterns correspond to attractors of RDS. There exists a large class of ppp-adic RDS for which interference patterns cannot be disturbed by noise. Therefore, if the equipment is described by such a RDS then the result of statistical experiment does not depend on noise in the equipment. On the one hand, we support the corpuscular model, because a quantum particle can be described as a corpuscular object. On the other hand, our model does not differ strongly from the wave model, because a quantum particle interacts with the whole equipment. Hence the interaction has nonlocal character. For example, in the two slit experiment a quantum particle interacts with both slits (but it passes only one of them).
Inter-relation between quantum and classical probability models is one of the most fundamental problems of quantum foundations. Nowadays this problem also plays an important role in quantum technologies, in quantum cryptography and the theory of quantum random generators. In this letter, we compare the viewpoint of Richard Feynman that the behavior of quantum particles cannot be described by classical probability theory with the viewpoint that quantum–classical inter-relation is more complicated (cf, in particular, with the tomographic model of quantum mechanics developed in detail by Vladimir Man'ko). As a basic example, we consider the two-slit experiment, which played a crucial role in quantum foundational debates at the beginning of quantum mechanics (QM). In particular, its analysis led Niels Bohr to the formulation of the principle of complementarity. First, we demonstrate that in complete accordance with Feynman's viewpoint, the probabilities for the two-slit experiment have the non-Kolmogorovian structure, since they violate one of basic laws of classical probability theory, the law of total probability (the heart of the Bayesian analysis). However, then we show that these probabilities can be embedded in a natural way into the classical (Kolmogorov, 1933) probability model. To do this, one has to take into account the randomness of selection of different experimental contexts, the joint consideration of which led Feynman to a conclusion about the non-classicality of quantum probability. We compare this embedding of non-Kolmogorovian quantum probabilities into the Kolmogorov model with well-known embeddings of non-Euclidean geometries into Euclidean space (e.g., the Poincaré disk model for the Lobachvesky plane).
Probability and interference According to Feynman, single-particle interference is ‘the only mystery’ of quantum theory (Feynman, Leighton and Sands 1965, ch. 1, p. 1). Feynman considered an experiment in which particles are fired, one at a time, towards a screen with two holes labelled 1 and 2. With both holes open, the distribution P 12 of particles at the backstop displays an oscillatory pattern of bright and dark fringes. If P 1 is the distribution with only hole 1 open, and P 2 is the distribution with only hole 2 open, then experimentally it is found that P 12 ≠ P 1 + P 2 . According to the argument given by Feynman (as well as by many other authors), this result is inexplicable by ‘classical’ reasoning. By his presentation of the two-slit experiment (as well as by his development of the path-integral formulation of quantum theory), Feynman popularised the idea that the usual probability calculus breaks down in the presence of quantum interference, where it is probability amplitudes (and not probabilities themselves) that are to be added. As pointed out by Koopman (1955), and by Ballentine (1986), this argument is mistaken: the probability distributions at the backstop – P 12 , P 1 and P 2 – are conditional probabilities with three distinct conditions (both slits open, one or other slit closed), and probability calculus does not imply any relationship between these. Feynman's argument notwithstanding, standard probability calculus is perfectly consistent with the two-slit experiment.
Section I.- Theoretical Implications of Time-Dependent Double Resonance Neutron Interferometry.- Calculations in the Causal Interpretation of Quantum Mechanics.- Calculation of Amplification Processes Generated by Empty Quantum Waves.- Relative Metrics and Physical Models for Non-local Particles.- A Two-Fluid Vacuum, the Photex, and the Photex-Photon Connection.- Discussion I.- Section II.- Summary of F. De Martini's Paper, Testing of Particle Distinguish-ability with Laser Interference Experiments.- Toward a Causal Interpretation of the Relativistic Quantum Mechanics of a Spinning Particle.- Nonlinear Behaviour of Light at Very Low Intensities: The Photon Clump Model.- Geometry-Dependent Predictions of the Quantum Potential Model for the Anomalous Photoelectric Effect.- EPR-Type Experiments and Enhanced Photon Detection.- Discussion II.- Section III.- Stochastic Optics: Wave-Particle Duality Without Particles.- Neo-Hertzian Electromagnetism.- Effective Photon Hypothesis vs. Quantum Potential Theory: Theoretical Predictions and Experimental Verification.- Compositeness of Photons and Its Implications.- Logical Meanings in Quantum Mechanics for Axioms and for Imaginary and Transfinite Numbers and Exponentials.- The Aharonov-Bohn Effect and the Quantum Interference Phase Shift due to an Electrostatic Field.- Discussion III.- Section IV.- Critical Analysis of the Tests of Local Realism Versus Quantum Mechanics.- Physical Photons: Theory, Experiment and Implications.- The Theoretical Nature of a Photon in a Lattice Vacuum.- Is the Photon Concept Really Necessary?.- Beyond Quantum Mechanics Using the Subvac.- Discussion IV.- Section V.- Does Quantum Electrodynamics Explain the Observed Lamb Shift?.- The Quantum and Rest Frame Vacuum - Absolute Motion with Lorentz Covariance.- The Double Slit Experiment and the Nonergodic Interpretation of Quantum Mechanics.- Einstein's Conception of the Ether and Its Up-To-Date Applications in the Relativistic Wave Mechanics.- Discussion V.- Section VI.- Concluding Remarks.- Concluding Remarks.- Comments.- Photo.- Conference Participants and Observers.