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 Teor. Veroyatnost. i Primenen., 2001, Volume 46, Issue 2, Pages 311–325 (Mi tvp3920)

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.

DOI: https://doi.org/10.4213/tvp3920

Full text: PDF file (1900 kB)

English version:
Theory of Probability and its Applications, 2002, 46:2, 256–273

Bibliographic databases:

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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• http://mi.mathnet.ru/eng/tvp3920
• https://doi.org/10.4213/tvp3920
• http://mi.mathnet.ru/eng/tvp/v46/i2/p311

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Kotovich N.V., Khrennikov A.Y., “Representation and compression of images with the aid of $m$-adic coordinate systems”, Dokl. Math., 66:3 (2002), 330–334
2. Schmitt B.M., “The quantitation of buffering action I. A formal & general approach”, Theoretical Biology and Medical Modelling, 2 (2005), 8
3. Khrennikov A., “$p$-adic probability theory and its generalizations”, p-adic mathematical physics, AIP Conf. Proc., 826, Amer. Inst. Phys., Melville, NY, 2006, 105–120
4. Khrennikov A.Yu., “Generalized probabilities taking values in non-Archimedean fields and in topological groups”, Russ. J. Math. Phys., 14:2 (2007), 142–159
5. Milosevic M., “A Propositional P-Adic Probability Logic”, Publ. Inst. Math.-Beograd, 87:101 (2010), 75–83