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On \(p\)-adic mathematical physics. (English) Zbl 1187.81004

In this paper that opens the first issue of this new journal the author gives a brief but very interesting review of some selected topics in \(p\)-adic mathematical physics.
From the text:“Recently there have been exciting achievements exploring \(p\)-adic, adelic and ultrametric structures in various models of physics: from the spacetime geometry at small scale and strings, via spin glasses and other complex systems, to the universe as a whole. There has been also significant progress in non-Archimedean modeling of some biological, cognitive, information and stochastic phenomena.
Ultrametricity seems to be a generic property of complex systems which contain hierarchy. Moreover, there is some evidence towards much more wide applicability of \(p\)-adic and non-Archimedean methods to various fields of knowledge. To extend \(p\)-adic methods into actual problems in diverse fields of economics, medicine, psychology, sociology, control theory, as well as to many other branches of sciences, is a great challenge and a great opportunity.”
Contents: 1. Numbers: rational, real, \(p\)-adic. 2. \(p\)-adic strings. 3. \(p\)-adic field theory. 4. \(p\)-adic and adelic quantum theory. 5. \(p\)-adic and adelic gravity and cosmology. 6. \(p\)-adic stochastic processes. 7. Vladimirov operator. 8. Dynamics and evolution of complex biological systems. 9. Quantization with \(p\)-adic valued wave functions. 10. \(\mathbb Q_p\)-valued probability. 11. Applications to cognitive science and psychology. 12. Applications to image analysis. 13. \(p\)-adic wavelets. 14. Analysis on general ultrametric spaces. 15. Cascade models of turbulence. 16. Disordered systems and spin glasses. 17. \(p\)-adic dynamical systems. 18. \(p\)-adic models of the genetic code. 19. Applications to economics, finance, data mining.
The article closes with an impressive list of 216 items which are ordered according to the sections treated.

MSC:

81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81Q65 Alternative quantum mechanics (including hidden variables, etc.)
00A69 General applied mathematics
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
12J25 Non-Archimedean valued fields
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References:

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