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Teoriya Veroyatnostei i ee Primeneniya, 2001, Volume 46, Issue 2, Pages 311–325
DOI: https://doi.org/10.4213/tvp3920
(Mi tvp3920)
 

This article is cited in 14 scientific papers (total in 14 papers)

Interpretations of Probability and Their $p$-Adic Extensions

A. Yu. Khrennikov

Växjö University
Abstract: 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.
Keywords: $p$-adic, foundations of probability theory, probability model, Bell inequality.
Received: 26.02.1998
English version:
Theory of Probability and its Applications, 2002, Volume 46, Issue 2, Pages 256–273
DOI: https://doi.org/10.1137/S0040585X97978920
Bibliographic databases:
Language: Russian
Citation: A. Yu. Khrennikov, “Interpretations of Probability and Their $p$-Adic Extensions”, Teor. Veroyatnost. i Primenen., 46:2 (2001), 311–325; Theory Probab. Appl., 46:2 (2002), 256–273
Citation in format AMSBIB
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\by A.~Yu.~Khrennikov
\paper Interpretations of Probability and Their $p$-Adic Extensions
\jour Teor. Veroyatnost. i Primenen.
\yr 2001
\vol 46
\issue 2
\pages 311--325
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\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1968688}
\zmath{https://zbmath.org/?q=an:1012.81005}
\transl
\jour Theory Probab. Appl.
\yr 2002
\vol 46
\issue 2
\pages 256--273
\crossref{https://doi.org/10.1137/S0040585X97978920}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000176400600005}
Linking options:
  • https://www.mathnet.ru/eng/tvp3920
  • https://doi.org/10.4213/tvp3920
  • https://www.mathnet.ru/eng/tvp/v46/i2/p311
  • This publication is cited in the following 14 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
     
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