  We study classical Hamiltonian systems in which the intrinsic proper time
evolution parameter is related through a probability distribution to the
physical time, which is assumed to be discrete. - This is motivated by the
``timeless'' reparametrization invariant model of a relativistic particle with
two compactified extradimensions. In this example, discrete physical time is
constructed based on quasi-local observables. - Generally, employing the
path-integral formulation of classical mechanics developed by Gozzi et al., we
show that these deterministic classical systems can be naturally described as
unitary quantum mechanical models. The emergent quantum Hamiltonian is derived
from the underlying classical one. It is closely related to the Liouville
operator. We demonstrate in several examples the necessity of regularization,
in order to arrive at quantum models with bounded spectrum and stable
groundstate.
