  The first goal of this paper is to show that discreteness, locality, and
relativistic covariance can peacefully coexist if the ordinary spacetime (OST)
is replaced with phase spacetime (PST) as a geometric background of a Poisson
process, where PST is a spacetime generalization of a notion of phase space
(this is a 7-dimensional version of the 8-dimensional structure proposed by
Caianiello). Furthermore, Caianiello's idea of finite acceleration is
implemented. After this is done, the paper then goes on to generalize the
geometric notions obtained from the intuition of PST to a general discrete
causal set, without any geometric background. It then takes advantage of the
absence of lightcone singularity to attempt to tackle the definition of
PST-like causal set; that is, a discrete system that approximates the
geometrical properties that we would expect from continuum PST. Finally, the
paper proceeds to introduce quantum field theory on a causal set, and shows
that the locality gained by switching from OST to PST brings us one step closer
to be able to treat quantum field theory on a causal set analytically rather
than numerically.
