On p-adic mathematical physics

Abstract

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...

Towards effective Lagrangians for adelic strings

“…Then this new Lagrangian becomes (20), and (2) g p = |r| p , where r may be any non zero rational number and it gives |r| ∞ p |r| p = 1 (this possibility was considered in [25]). Both these possibilities are consistent with adelic product formula (6). For simplicity, in the sequel we shall take C = ζ(2).…”

“…From (6) follows that the ordinary Veneziano amplitude, which is rather complex, can be expressed as product of all inverse p-adic counterparts, which are much more simpler. Moreover, expression (6) gives rise to consider it as the amplitude for an adelic string, which is composed of the ordinary and p-adic ones.…”

“…Various p-adic structures have been observed and investigated not only in string theory but also in many other models of modern mathematical physics and related topics (see [4,5] for a review of early developments, and [6] for the most recent short overview of p-adic mathematical physics). For instance, in [7] was shown that degeneration of the genetic code has a p-adic structure.…”

Об операторах координаты и импульса в $p$-адической квантовой механике

“…Заметим, что эта идея выска-зывалась в ранних работах по p-адической математической физике (библиографию см. в [2], [6]). …”

On Nonlocal Modified Gravity and Its Cosmological Solutions

“…Our motivation to modify gravity in an analytic nonlocal way comes mainly from string theory, in particular from string field theory (see the very original effort in this direction in [3]) and p-adic string theory [15,38,39,40,65]. Since strings are one-dimensional extended objects, their field theory description contains spacetime nonlocality expressed by some exponential functions of d'Alembert operator .…”