A brief review of some selected topics in p-adic mathematical physics is presented.1 Numbers: Rational, Real, p-AdicWe present a brief review of some selected topics in p-adic mathematical physics. More details can be found in the references below and the other references are mainly contained therein. We hope that this brief introduction to some aspects of p-adic mathematical physics could be helpful for the readers of the first issue of the journal p-Adic Numbers, Ultrametric Analysis and Applications.The notion of numbers is basic not only in mathematics but also in physics and entire science. Most of modern science is based on mathematical analysis over real and complex numbers. However, it is turned out that for exploring complex hierarchical systems it is sometimes more fruitful to use analysis over p-adic numbers and ultrametric spaces. p-Adic numbers (see, e.g. [1]), introduced by Hensel, are widely used in mathematics: in number theory, algebraic geometry, representation theory, algebraic and arithmetical dynamics, and cryptography.The following view how to do science with numbers has been put forward by Volovich in [2,3]. Suppose we have a physical or any other system and we 1 make measurements. To describe results of the measurements, we can always use rationals. To do mathematical analysis, one needs a completion of the field Q of the rational numbers. According to the Ostrowski theorem there are only two kinds of completions of the rationals. They give real R or p-adic Q p number fields, where p is any prime number with corresponding p-adic norm |x| p , which is non-Archimedean. There is an adelic formula p |x| p = 1 valid for rational x which expresses the real norm |x| ∞ in terms of p-adic ones. Any p-adic number x can be represented as a series x = ∞ i=k a i p i , where k is an integer and a i ∈ {0, 1, 2, ..., p − 1} are digits. To build a mathematical model of the system we use real or p-adic numbers or both, depending on the properties of the system [2,3].Superanalysis over real and p-adic numbers has been considered by Vladimirov and Volovich [4,5]. An adelic approach was emphasized by Manin [6].One can argue that at the very small (Planck) scale the geometry of the spacetime should be non-Archimedean [2,3,7]. There should be quantum fluctuations not only of metrics and geometry but even of the number field. Therefore, it was suggested [2] the following number field invariance principle: Fundamental physical laws should be invariant under the change of the number field.One could start from the ring of integers or the Grothendieck schemes. Then rational, real or p-adic numbers should appear through a mechanism of number field symmetry breaking, similar to the Higgs mechanism [8, 9].Recently (for a review, see [10,11,12,13,14]) 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 al...

p-Adic mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of p-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

Using the Weyl quantization we formulate one-dimensional adelic quantum mechanics, which unifies and treats ordinary and p-adic quantum mechanics on an equal footing. As an illustration the corresponding harmonic oscillator is considered. It is a simple, exact and instructive adelic model. Eigenstates are Schwartz-Bruhat functions. The Mellin transform of a simplest vacuum state leads to the well known functional relation for the Riemann zeta function. Some expectation values are calculated. The existence of adelic matter at very high energies is suggested.

Using basic properties of p-adic numbers, we consider a simple new approach to describe main aspects of DNA sequence and genetic code. Central role in our investigation plays an ultrametric p-adic information space which basic elements are nucleotides, codons and genes. We show that a 5-adic model is appropriate for DNA sequence. This 5-adic model, combined with 2-adic distance, is also suitable for genetic code and for a more advanced employment in genomics. We find that genetic code degeneracy is related to the p-adic distance between codons.

Adelic quantum mechanics is formulated. The corresponding model of the harmonic oscillator is considered. The adelic harmonic oscillator exhibits many interesting features. One of them is a softening of the uncertainty relation.

1. Abstract p-Adic and adelic generalization of ordinary quantum cosmology is considered. In [1], we have calculated p-adic wave functions for some minisuperspace cosmological models according to the "no-boundary" Hartle-Hawking proposal. In this article, applying p-adic and adelic quantum mechanics, we show existence of the corresponding vacuum eigenstates. Adelic wave function contains some information on discrete structure of space-time at the Planck scale.

A new approach to the wave function of the universe is suggested. The key idea is to take into account fluctuating number fields and present the wave function in the form of a Euler product. For this purpose we define a p-adic generalization of both classical and quantum gravitational theory. Elements of p-adic differential geometry are described. The action and gravitation field equations over the p-adic number field are investigated. p-adic analogs of some known solutions to the Einstein equations are presented. It follows that in quantum cosmology one should consider summation only over algebraic manifolds. The correspondence principle with the standard approach is considered.

The Feynman path integral in p-adic quantum mechanics is considered. The probability amplitude ${\cal K}_p (x^{\prime\prime},t^{\prime\prime}; x^\prime,t^\prime)$ for one-dimensional systems with quadratic actions is calculated in an exact form, which is the same as that in ordinary quantum mechanics.Comment: 9 page