  We discuss the relevance of the classical and quantum rolling tachyons
inflation in the frame of the standard, p-adic and adelic minisuperspace
quantum cosmology. The field theory of tachyon matter proposed by Sen in a
zero-dimensional version suggested by Kar leads to a model of a particle moving
in a constant external field with quadratic damping. We calculate the exact
quantum propagator of the model, as well as, the vacuum states and conditions
necessary to construct an adelic generalization.
