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Journal of Mathematical Physics

p -adic stochastic hidden variable model

Andrew Khrennikov

Published Mar 1, 1998• 5 citations
Publisher

Abstract

We propose stochastic hidden variables model in which hidden variables have a p-adic probability distribution ρ(λ) and at the same time conditional probabilistic distributions P(U,λ), U=A,A′,B,B′, are ordinary probabilities defined on the basis of the Kolmogorov measure-theoretical axiomatics. A frequency definition of p-adic probability is quite similar to the ordinary frequency definition of probability. p-adic frequency probability is defined as the limit of relative frequencies νn but in the p-adic metric. We study a model with p-adic stochastics on the level of the hidden variables description. But, of course, responses of macroapparatuses have to be described by ordinary stochastics. Thus our model describes a mixture of p-adic stochastics of the microworld and ordinary stochastics of macroapparatuses. In this model probabilities for physical observables are the ordinary probabilities. At the same time Bell’s inequality is violated.

DOI
10.1063/1.532386

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We propose stochastic hidden variables model in which hidden variables have a p-adic probability distribution ρ(λ) and at the same time conditional probabilistic distributions P(U,λ), U=A,A′,B,B′, are ordinary probabilities defined on the basis of the Kolmogorov measure-theoretical axiomatics. A frequency definition of p-adic probability is quite similar to the ordinary frequency definition of probability. p-adic frequency probability is defined as the limit of relative frequencies νn but in the p-adic metric. We study a model with p-adic stochastics on the level of the hidden variables description. But, of course, responses of macroapparatuses have to be described by ordinary stochastics. Thus our model describes a mixture of p-adic stochastics of the microworld and ordinary stochastics of macroapparatuses. In this model probabilities for physical observables are the ordinary probabilities. At the same time Bell’s inequality is violated.
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.
We analyze the proof of Bell’s inequality and demonstrate that this inequality is related to one particular model of probability theory, namely Kolmogorov measure-theoretical axiomatics from 1933. We found a (numerical) statistical correction to Bell’s inequality. Such an additional term εφ on the right-hand side of Bell’s inequality can be considered as a probability invariant of a quantum state φ. This is a measure of nonreproducibility of hidden variables in different runs of experiments. Experiments to verify Bell’s inequality can be considered as just experiments to estimate the constant εφ. It seems that Bell’s inequality could not be used as a crucial reason to deny local realism. We consider deterministic as well as stochastic hidden variables models.
It is shown that the following statements about a quantum correlation experiment are mutually equivalent. (1) There is a deterministic hidden-variables model for the experiment. (2) There is a factorizable, stochastic model. (3) There is one joint distribution for all observables of the experiment, returning the experimental probabilities. (4) There are well-defined, compatible joint distributions for all pairs and triples of commuting and noncommuting observables. (5) The Bell inequalities hold.
The situation in probability theory is more or less analogous to the situation in geometry in the 19th century. Many scientists begin to understand that the Kol-mogorov axiomatic approach [1], 1933, to the modern theory of probability cannot describe all probabilistic phenomena observed in nature. Hence, the theory of probability based on this axiomatic approach is not a unique and universal probabilistic formalism. It is only one model, namely, the Kolmogorov probabilistic model. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The conceptually puzzling features of quantum mechanics as a statistical theory all have their source in the impossibility of relating the probability assignments defined by the quantum state to distributions over determinate values of the physical magnitudes. While some interpretations of quantum mechanics, as well as “hidden variable” modifications of the theory, have proposed several constructions for assigning values to magnitudes (beyond the value assignments defined by the quantum state alone), a variety of theorems exist which impose severe restrictions on such theoretical constructions. This note will deal with Bell's proof (in [2]), that any hidden variable theory satisfying a physically reasonable locality condition is characterized by an inequality which is inconsistent with the quantum statistics. Bell's result has initiated a series of experiments designed to refute the entire class of local hidden variable theories. We show that Bell's inequality actually characterizes a feature of hidden variable theories which is much weaker than locality in the sense considered physically motivated.
We formulate stochastic noncontextual hidden-variable theories. In such theories the hidden state \ensuremath{\lambda} specifies not definite values, but expectation values A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}), . . . of observables A, . . . and noncontextuality means that A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}) is independent of which other commuting observables commuting with A are measured together with A. We show via Bell inequalities that such theories conflict with quantum theory and propose a two-photon experimental test. We also show that for a single particle of spin (2n{2}^{\mathit{n}}2n-1)/2, quantum violation of classical noncontextuality grows exponentially with n.