## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Goran S. Djordjević

### Cosmology and Tachyonic Inflation on Non-Archimedean Spaces

Abstract

We review the formulation and some elaboration of $p$-adic and adelic quantum cosmology, i.e. cosmology on the Non-Archimedean spaces and its relation to cosmology on the Archimedean one. $p$-Adic gravity and the wave function of the universe were considered for the first time in 1991 [1].

$p$-Adic quantum cosmology [2] is an application of $p$-adic quantum theory to the universe as a whole. $p$-Adic quantum theory is a $p$-adic generalization of the standard quantum theory [3]. In $p$-adic quantum theory argument of the wave function is a $p$-adic variable. Geometry of $p$-adic spaces has a nonarchimedean structure. Under some restrictions product of the ordinary wave function and its $p$-adic counterparts gives adelic wave function [4]. In adelic Feynman's path integral approach integration over nonarchimedean geometries is also taken into account [5].

The adelic generalization of the Hartle-Hawking proposal does not work in models with matter fields. $p$-Adic and adelic minisuperspace quantum cosmology is well defined as an ordinary application of $p$-adic and adelic quantum mechanics. It is illustrated by a few of cosmological models. As a result of $p$-adic quantum effects and the adelic approach, these models exhibit some discreteness of the minisuperspace.

We end with consideration of a class of tachyon-like potentials, inspired by string theory, $D$-brane dynamics and cosmology in the context of classical and quantum mechanics. Motivated by the trans-Planckian problem in the very early stage of cosmological evolution of the Universe, we consider the theoretical role of DBI-type tachyon scalar field, defined over the field of real as well as of $p$-adic numbers, i.e. archemedean and nonarchimedean spaces. To simplify the equation of motion for the scalar field, canonical transformations are defined and engaged [6]. The corresponding quantum propagators in the Feynman path integral approach are calculated and discussed, as there are possibilities for a quantum adelic generalization and its application in the beginning of inflation [7].

[1] I. Ya. Aref'eva, B. Dragovich, P. H. Frampton and I. V. Volovich, Int. J. Mod. Phys. A6 (1991) 4341.
[2] G.S. Djordjevic, B. Dragovich, Lj.D. Nesic, I.V. Volovich, Int. J. Mod. Phys. A17 (2002) 1413.
[3] V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).
[4] B. Dragovich, Theor. Mat. Phys. 101 (1994) 349.
[5] G.S. Djordjevic, B. Dragovich and Lj. Nesic, Inf. Dim. Analys, Quant. Prob. and Rel. Topics 06(02) (2003) 179.
[6] D.D. Dimitrijevic, G.S. Djordjevic and M. Milosevic, Romanian Rep. Phys. 57, no. 4 (2015).
[7] G.S. Djordjevic, D.D. Dimitrijevic and M. Milosevic, Romanian J. Phys. 61, in press.