Goran S. Djordjević
Cosmology and Tachyonic Inflation on Non-Archimedean Spaces
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 .
$p$-Adic quantum cosmology  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 . 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 . In adelic Feynman's path integral approach integration over nonarchimedean geometries is also taken into account .
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 . 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 .
 I. Ya. Aref'eva, B. Dragovich, P. H. Frampton and I. V. Volovich,
Int. J. Mod. Phys. A6 (1991) 4341.