Branko DragovichOn $p$Adic MatterAbstract
$p$Adic string theory was introduced in 1987 by construction of an analogue of the Veneziano scattering amplitude replacing real worldsheet by the $p$adic one. It was shown that product of ordinary crossing symmetric Veneziano amplitude and its $p$adic counterparts over all primes $p$ is a constant. This convergent infinite product of amplitudes is an example of connection between ordinary and $p$adic strings.
Investigation of $p$adic strings was improved by invention of an effective Lagrangian. This Lagrangian with scalar field describes not only fourpoint scattering amplitude but also all higher ones at the tree level. This line of research has led to new insights on the role of $p$adic strings in string theory. However, $p$adic strings have been mainly treated as an auxiliary tool to better understand ordinary strings. However, if ordinary matter has its origin in ordinary strings, then there should be $p$adic matter related to $p$adic strings.
In this talk, I will start with a brief review of some basic properties of $p$adic strings and their connections with ordinary strings. Then I will consider a slight modification of the Lagrangian of $p$adic open string to get a nontachyonic $p$adic matter. It will be shown that one obtains a new well defined scalar $p$adic field. Equation of motion for this field is nonlocal and nonlinear. Some cosmological consequences are investigated in a weak field approximation. It will be presented
cosmological solution with an exponential expansion (and contraction) of a closed universe.
