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.
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
\Bibitem{Khr01}
\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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\crossref{https://doi.org/10.4213/tvp3920}
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\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:
Paolo Aniello, Stefano Mancini, Vincenzo Parisi, “Quantum mechanics on a p-adic Hilbert space: Foundations and prospects”, Int. J. Geom. Methods Mod. Phys., 21:10 (2024)
Paolo Aniello, Stefano Mancini, Vincenzo Parisi, “Trace class operators and states in p-adic quantum mechanics”, Journal of Mathematical Physics, 64:5 (2023)
Paolo Aniello, “States, observables and symmetries in p-adic quantum mechanics”, J. Phys.: Conf. Ser., 2667:1 (2023), 012055
Paolo Aniello, Stefano Mancini, Vincenzo Parisi, “A p-Adic Model of Quantum States and the p-Adic Qubit”, Entropy, 25:1 (2022), 86
Sergey Yu. Melnikov, Konstantin E. Samouylov, Lecture Notes in Computer Science, 12526, Internet of Things, Smart Spaces, and Next Generation Networks and Systems, 2020, 259
Andrew Schumann, Emergence, Complexity and Computation, 33, Behaviourism in Studying Swarms: Logical Models of Sensing and Motoring, 2019, 1
Andrew Schumann, Emergence, Complexity and Computation, 33, Behaviourism in Studying Swarms: Logical Models of Sensing and Motoring, 2019, 165
Andrew Schumann, “p-Adic valued logical calculi in simulations of the slime mould behaviour”, Journal of Applied Non-Classical Logics, 25:2 (2015), 125
Milosevic M., “A Propositional P-Adic Probability Logic”, Publ. Inst. Math.-Beograd, 87:101 (2010), 75–83
A.N. Gorban, O. Radulescu, Advances in Chemical Engineering, 34, Advances in Chemical Engineering - Mathematics in Chemical Kinetics and Engineering, 2008, 103
Khrennikov A.Yu., “Generalized probabilities taking values in non-Archimedean fields and in topological groups”, Russ. J. Math. Phys., 14:2 (2007), 142–159
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
Schmitt B.M., “The quantitation of buffering action I. A formal & general approach”, Theoretical Biology and Medical Modelling, 2 (2005), 8
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
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