Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Bell-Inequality and Two Slit Experiments: Comparing Misapplication of Classical Probability by Feynman and Bell

Andrei Yu. Khrennikov

Published Jun 29, 2021• 1 citation
Publisher

Abstract

We start with the discussion on misapplication of classical probability theory by Feynman in his analysis of the two slit experiment (by following the critical argumentation of Koopman, Ballentine, and the author of this paper). The seed of Feynman's conclusion on the impossibility to apply the classical probabilistic description for the two slit experiment is treatment of conditional probabilities corresponding to different experimental contexts as unconditional ones. Then we move to the Bell type inequalities. Bell applied classical probability theory in the same manner as Feynman and, as can be expected, he also obtained the impossibility statement. In contrast to Feynman, he formulated his no-go statement not in the probabilistic terms, but by appealing to nonlocality. This note can be considered as a part of the author's attempts for getting rid off nonlocality from quantum physics.

DOI
10.20944/preprints202106.0703.v1

Related papers

Bell-Inequality and Two Slit Experiments: Comparing Misapplication of Classical Probability by Feynman and Bell

Andrei Yu. Khrennikov

10.33693/2313-223X-2021-8-3-19-22, Khrennikov A.., (2021), BELL-INEQUALITY AND TWO SLIT EXPERIMENTS: COMPARING MISAPPLICATION OF CLASSICAL PROBABILITY BY FEYNMAN AND BELL. Computational Nanotechnology, 3: 19-22.

Computational nanotechnology•Sep 15, 2021•0 citations

Two-slit experiment: quantum and classical probabilities

Andrei Yu. Khrennikov

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).

Physica Scripta•Jun 1, 2015•8 citations

Classical Probabilities and Bell' s Theorem

Tan Tian-rong

It is proved that: What the two-slit experiments have refuted is the preconceived idea that of the event that a micro particle passing through one slit arrives somewhere on the screen is irrelative to the condition that whether or not the other slit is open. Consideringaddition theorem of probabilities, the above idea can be expressed as follows: (a)Assum that a beam passes through a partition with two slit and arrives a screen. The of the event that an indivedual particle falls on somewhere of the screen under the condition that two slit open simultaneously, is equal to the sum of two of the same event under that two slit open in turn. Defining the experimental condition such as two slit open simultaneously or in turn as passage the proposition (a)is manifested as probabilities is independent of passage conditions. On the other hand, in terms of the above, superposition principle of in quantum mechanics ought to be indicated as probability amplitudes is independent of passage conditions. Correspondingly, (a)is named hereof as superposition presumption of probability. For lack of the conception of passage conditions, (a)is confused with theorem of probabilities. As a result, classical physics tacitly approve (a), and thereby it fails to comprehend the two-slit experiments. In reverse, to explain two-slit experiments, quantum mechanics spurns addition theorem of so that it meets with various difficults about explanation. Any alternative of quantum mechanics must be able to explain two-slit experiments, therefore it must contain passage conditions or amplitudes. But hidden variable theory Bell used dissatisfies such a prerequisite. It comes thus into conflict with experiment facts inevitably. As a result, Lochak's argument that Bell's inequality is irrelative to locality is rediscovered.

Journal of Changzhou Institute of Technology•Jan 1, 2002•0 citations

Demystification of Bell inequality

Andrei Yu. Khrennikov

The main aim of this review is to show that the common conclusion that Bell's argument implies that any attempt to proceed beyond quantum mechanics induces a nonlocal model was not totally justified. Our analysis of Bell's argument demonstrates that violation of Bell's inequality implies neither "death of realism" nor nonlocality. This violation is just a sign of non-Kolmogorovness of statistical data - impossibility to put statistical data collected in a few different experiments (corresponding to incompatible settings of polarization beam splitters) in one probability space. This inequality was well known in theoretical probability since 19th century (from works of Boole). We couple non-Kolmogorovness of data with design of modern detectors of photons.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 20, 2009•2 citations

Bell's inequality: Physics meets Probability

Andrei Yu. Khrennikov

We remind the viewpoint that violation of Bell's inequality might be interpreted not only as an evidence of the alternative -- either nonlocality or ``death of reality'' (under the assumption the quantum mechanics is incomplete). Violation of Bell's type inequalities is a well known sufficient condition of incompatibility of random variables -- impossibility to realize them on a single probability space. Thus, in fact, we should take into account an additional interpretation of violation of Bell's inequality -- a few pairs of random variables (two dimensional vector variables) involved in the EPR-Bohm experiment are incompatible. They could not be realized on a single Kolmogorov probability space. Thus one can choose between: a) completeness of quantum mechanics; b) nonlocality; c) `` death of reality''; d) non-Kolmogorovness. In any event, violation of Bell's inequality has a variety of possible interpretations. Hence, it could not be used to derive the unique conclusion on the relation between quantum and classical models.

arXiv:0709.3909•Sep 25, 2007•32 citations

Two Probability Theories in Physics

Tan Tian-rong

It is proved that in a double slit diffraction experiment, superposition principle of probability amplitude means (the probability amplitude of the event) that an electron passes through a certain slit and arrives somewhere on the screen under the condition that two slits open simultaneously, is equal to the sum of two probability amplitudes of the same event under the condition that two slits open in turn. From this thesis it is concluded that the probabilities do not obey superposition principle, which is just the reason that classical probability theory is inapplicable for the very experiment. Kolmogorov's probability theory is based on two foundations: frequency definition of probabilities and Boolean algebra of event operations. In micro processes, the former holds true while the latter, especially its multiplicative permutation law is violated. So long as we do not deal with event operations, joint probabilities holds good in micro processes. But, the event operation formulae provided are applied and the wrong conclusions present probably. The famous Bell's inequality is a wrong conclusion. (Nature and Science. 2005;3(2):82-91).

•Jan 1, 2005•0 citations
We start with the discussion on misapplication of classical probability theory by Feynman in his analysis of the two slit experiment (by following the critical argumentation of Koopman, Ballentine, and the author of this paper). The seed of Feynman's conclusion on the impossibility to apply the classical probabilistic description for the two slit experiment is treatment of conditional probabilities corresponding to different experimental contexts as unconditional ones. Then we move to the Bell type inequalities. Bell applied classical probability theory in the same manner as Feynman and, as can be expected, he also obtained the impossibility statement. In contrast to Feynman, he formulated his no-go statement not in the probabilistic terms, but by appealing to nonlocality. This note can be considered as a part of the author's attempts for getting rid off nonlocality from quantum physics.
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).
It is proved that: What the two-slit experiments have refuted is the preconceived idea that of the event that a micro particle passing through one slit arrives somewhere on the screen is irrelative to the condition that whether or not the other slit is open. Consideringaddition theorem of probabilities, the above idea can be expressed as follows: (a)Assum that a beam passes through a partition with two slit and arrives a screen. The of the event that an indivedual particle falls on somewhere of the screen under the condition that two slit open simultaneously, is equal to the sum of two of the same event under that two slit open in turn. Defining the experimental condition such as two slit open simultaneously or in turn as passage the proposition (a)is manifested as probabilities is independent of passage conditions. On the other hand, in terms of the above, superposition principle of in quantum mechanics ought to be indicated as probability amplitudes is independent of passage conditions. Correspondingly, (a)is named hereof as superposition presumption of probability. For lack of the conception of passage conditions, (a)is confused with theorem of probabilities. As a result, classical physics tacitly approve (a), and thereby it fails to comprehend the two-slit experiments. In reverse, to explain two-slit experiments, quantum mechanics spurns addition theorem of so that it meets with various difficults about explanation. Any alternative of quantum mechanics must be able to explain two-slit experiments, therefore it must contain passage conditions or amplitudes. But hidden variable theory Bell used dissatisfies such a prerequisite. It comes thus into conflict with experiment facts inevitably. As a result, Lochak's argument that Bell's inequality is irrelative to locality is rediscovered.
The main aim of this review is to show that the common conclusion that Bell's argument implies that any attempt to proceed beyond quantum mechanics induces a nonlocal model was not totally justified. Our analysis of Bell's argument demonstrates that violation of Bell's inequality implies neither "death of realism" nor nonlocality. This violation is just a sign of non-Kolmogorovness of statistical data - impossibility to put statistical data collected in a few different experiments (corresponding to incompatible settings of polarization beam splitters) in one probability space. This inequality was well known in theoretical probability since 19th century (from works of Boole). We couple non-Kolmogorovness of data with design of modern detectors of photons.
We remind the viewpoint that violation of Bell's inequality might be interpreted not only as an evidence of the alternative -- either nonlocality or ``death of reality'' (under the assumption the quantum mechanics is incomplete). Violation of Bell's type inequalities is a well known sufficient condition of incompatibility of random variables -- impossibility to realize them on a single probability space. Thus, in fact, we should take into account an additional interpretation of violation of Bell's inequality -- a few pairs of random variables (two dimensional vector variables) involved in the EPR-Bohm experiment are incompatible. They could not be realized on a single Kolmogorov probability space. Thus one can choose between: a) completeness of quantum mechanics; b) nonlocality; c) `` death of reality''; d) non-Kolmogorovness. In any event, violation of Bell's inequality has a variety of possible interpretations. Hence, it could not be used to derive the unique conclusion on the relation between quantum and classical models.
It is proved that in a double slit diffraction experiment, superposition principle of probability amplitude means (the probability amplitude of the event) that an electron passes through a certain slit and arrives somewhere on the screen under the condition that two slits open simultaneously, is equal to the sum of two probability amplitudes of the same event under the condition that two slits open in turn. From this thesis it is concluded that the probabilities do not obey superposition principle, which is just the reason that classical probability theory is inapplicable for the very experiment. Kolmogorov's probability theory is based on two foundations: frequency definition of probabilities and Boolean algebra of event operations. In micro processes, the former holds true while the latter, especially its multiplicative permutation law is violated. So long as we do not deal with event operations, joint probabilities holds good in micro processes. But, the event operation formulae provided are applied and the wrong conclusions present probably. The famous Bell's inequality is a wrong conclusion. (Nature and Science. 2005;3(2):82-91).