Toward Theory of P-adic Valued Probabilities

We present a short review on generalization of probability theory in that probabilities take values in the fields of p-adic numbers, Qp. Such probabilities were introduced to serve p-adic theoretical physics. In some quantum physical models a wave function (which is a complex probability amplitude in ordinary QM) takes vales inQp (for some prime number p) or their quadratic extensions. Such a wave function can be interpreted probabilistically in the framework of p-adic probability theory. This theory was developed by using both the frequency approach (by generalizing von Mises) and the measure-theoretic approach (by generalizing Kolmogorov). In particular, some limit theorems were obtained. However, theory of limit theorems for p-adic valued probabilities is far from being completed. Another interesting domain of research is corresponding theory of complexity. We obtained some preliminary results in this direction. However, it is again far from to be completed. Recently p-adic models of classical statistical mechanics were considered and some preliminary results about invariant p-adic valued measures for dynamical systems were obtained.

[1]  Jean-Pierre Vigier,et al.  A review of extended probabilities , 1986 .

[2]  A. Khrennikov.  Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena , 2004 .

[3]  Andrew Khrennikov,et al.  p-Adic stochastics and Dirac quantization with negative probabilities , 1995 .

[4]  Quantum mechanics andp-adic numbers , 1972 .

[5]  E. Prugovec̆ki,et al.  A postulational framework for theories of simultaneous measurement of several observables , 1973 .

[6]  S Albeverio,et al.  Memory retrieval as a p-adic dynamical system. , 1999, Bio Systems.

[7]  P-ADIC PROBABILITY AND STATISTICS , 1992 .

[8]  S. V. Kozyrev,et al.  Replica Symmetry Breaking Related to a General Ultrametric Space Ii: Rsb Solutions and the N → 0 Limit , 2022 .

[9]  Localization in space for a free particle in ultrametric quantum mechanics , 2006 .

[10]  A. Khrennikov.  Limit behaviour of sums of independent random variables with respect to the uniform p-adic distribution , 2001 .

[11]  Andrei Khrennikov,et al.  ATTRACTORS OF RANDOM DYNAMICAL SYSTEMS OVER P-ADIC NUMBERS AND A MODEL OF 'NOISY' COGNITIVE PROCESSES , 1999 .

[12]  S. V. Kozyrev,et al.  Replica Symmetry Breaking Related to a General Ultrametric Space I: Replica Matrices and Functionals , 2022 .

[13]  Andrew Schumann,et al.  Logical approach to p-adic probabilities , 2006 .

[14]  E. Thiran,et al.  p-adic dynamics , 1989 .

[15]  P. Dirac.  On the Analogy Between Classical and Quantum Mechanics , 1945 .

[16]  Andrew Khrennikov,et al.  p-adic probability distributions of hidden variables , 1995 .

[17]  A. Khrennikov,et al.  Interpretations of Probability and Their p-Adic Extensions , 2002 .

[18]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[19]  Igor Volovich,et al.  p-adic string , 1987 .

[20]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .

[21]  Andrei Khrennikov,et al.  p-Adic Valued Distributions in Mathematical Physics , 1994 .

[22]  Andrei Khrennikov,et al.  Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models , 2011 .

[23]  Replica symmetry breaking related to a general ultrametric space III: The case of general measure , 2007 .

[24]  Gunter Ludwig,et al.  CHAPTER IV – The Physical Interpretation of Quantum Mechanics , 1968 .

[25]  B. Dragovic.  On signature change in p-adic space-times , 1991 .