an:01199695
Zbl 0920.11087
Khrennikov, Andrei
Non-archimedean analysis: quantum paradoxes, dynamical systems and biological models
EN
Mathematics and its Applications (Dordrecht). 427. Dordrecht: Kluwer Academic Publishers. xvii, 371 p. (1997).
1997
b
11Z05 81P05 58-02 91-02 92-02 47S10 60A05 11S80 46-02 46S10 60-02 91E10
\(p\)-adic number; \(m\)-adic number; \(p\)-adic valued probability; \(p\)-adic dynamical system; quantum paradox; hidden variable; non-archimedean models; model of quantum mechanics; \(p\)-adic-valued wave functions
The book is devoted to non-archimedean models in physics, biology and social sciences. The author starts (Chapters I and III) from an exposition of elementary \(p\)-adic analysis (sometimes \(m\)-adic numbers are also used, where \(m\) is not a prime number). Simultaneously (Chapter II) a review of the foundations of quantum mechanics is given, including the Einstein-Podolski-Rosen paradox, the Bell inequality and the problem of hidden variables.
Chapters IV, VI and VII are devoted to a model of quantum mechanics with \(p\)-adic-valued wave functions and the ideas proposed by the authors in order to resolve quantum paradoxes on the basis of the \(p\)-adic interpretation of the measurement process. An essential tool is the author's concept of the \(p\)-adic-valued probability (expounded in Chapter V).
Chapter VIII contains an investigation of some simple \(p\)-adic dynamical systems used as toy models for human subconscious, social dynamics, human history, and even God (Section VIII.10 is entitled ``God as a \(p\)-adic Dynamical System'').
At present nobody knows whether the models invented by the author can develop into realistic descriptions of physical, biological or social phenomena. However, the book can help non-mathematicians to obtain some knowledge of non-archimedean analysis and its possible applications.
Anatoly N.Kochubei (Kiev)