  We begin with a description of spacetime by a 4-dimensional cubic lattice
$\sscript$. It follows from this framework that the the speed of light is the
only nonzero instantaneous speed for a particle. The dual space $\sscripthat$
corresponds to a cubic lattice of energy-momentum. This description implies
that there is a discrete set of possible particle masses. We then define
discrete scalar quantum fields on $\sscript$. These fields are employed to
define interaction Hamiltonians and scattering operators. Although the
scattering operator $S$ cannot be computed exactly, approximations are
possible. Whether $S$ is unitary is an unsolved problem. Besides the
definitions of these operators, our main assumption is conservation of
energy-momentum for a scattering process. This article concludes with various
examples of perturbation approximations. These include simplified versions of
electron-electron and electron-proton scattering as well as simple decay
processes. We also define scattering cross-sections, decay rates and lifetimes
within this formalism.
