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AIP conference proceedings

p-Adic Probability Theory and Its Generalizations

Andrei Yu. Khrennikov

Published Jan 1, 2006• 0 citations
Publisher

Abstract

This is a review about p‐adic valued probabilities, applications to quantum physics and cognitive sciences. We also do the next step in development of non‐Kolmogorovian models and consider an analogue of probability theory for probabilities taking values in topological groups. The main attention is paid to statistical interpretation of p‐adic probabilities as well as more general probabilities with values in topological groups.

DOI
10.1063/1.2193115

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This is a review about p‐adic valued probabilities, applications to quantum physics and cognitive sciences. We also do the next step in development of non‐Kolmogorovian models and consider an analogue of probability theory for probabilities taking values in topological groups. The main attention is paid to statistical interpretation of p‐adic probabilities as well as more general probabilities with values in topological groups.
The development of p-adic quantum mechanics has made it necessary to construct a probability theory in which the probabilities of events are p-adic numbers. The foundations of this theory are developed here. The frequency definition of probability is used. A general principle of statistical stabilization of relative frequencies is formulated. By virtue of this principle, statistical stabilization of relative frequencies, which are, like all experimental data, rational numbers, can be considered not only in the real topology but also in p-adic topologies.
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.
This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ℚ p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ℚ p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.
The main thesis advocated in the present paper is that some well-established laws of classical probability theory (typically the theorem of composite probabilities) are model dependent assertions, like some theorems of Euclidean geometry. In Sections (1) and (2) we briefly describe how this thesis is framed in the wider context of quantum probability and how it can throw some light on the open debate concerning the interpretation of quantum theory, as well as on the necessity of using a non-Kolmogorovian model in quantum theory. In Sections (3), (4) and (5) we show that the general ideas formulated in Sections (1) and (2), when given a precise mathematical form, allow one to account for all the nonstandard features of the quantum probability model (C -numbers, Hilbert spaces,... ) as well as to exhibit statistical models which are neither Kolmogorovian nor quantum.
p-Adic mathematical physics emerged as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale. One of its main achievements is a successful formulation and development of p-adic and adelic quantum mechanics, which have complex-valued wave functions of p-adic and adelic arguments, respectively. Various aspects of these quantum mechanics are reviewed here. In particular, the corresponding Feynman's path integrals, some minisuperspace cosmological models, and relevant approache to string theory, are presented. As a result of adelic approach, p-adic effects exhibit a spacetime and some other discreteness, which depend on the adelic quantum state of the physical system under consideration. Besides review, this article contains also some new results.
Various versions of noncommutative probability theory are surveyed. It is stressed that the main motivation and applications of these noncommutative theories is quantum mechanics. A review of traditional probability theory and its unsharp version are presented. Sharp and unsharp Hilbert space probability theories are considered next. We then present a general discussion of observables and statistical maps. Finally, we consider sequential effect algebras and show that they unify and generalize all of these types of probability theories.