  We pursue the view that quantum theory may be an emergent structure related
to large space-time scales. In particular, we consider classical Hamiltonian
systems in which the intrinsic proper time evolution parameter is related
through a probability distribution to the discrete physical time. This is
motivated by studies of ``timeless'' reparametrization invariant models, where
discrete physical time has recently been constructed based on coarse-graining
local observables. Describing such deterministic classical systems with the
help of path-integrals, primordial states can naturally be introduced which
follow unitary quantum mechanical evolution in suitable limits.
