  Causal set theory is an approach to quantum gravity in which spacetime is
fundamentally discrete at the Planck scale and takes the form of a Lorentzian
lattice, or "causal set", from which continuum spacetime emerges in a
large-scale (low-energy) approximation. In this work, we present new
developments in the framework of interacting quantum field theory on causal
sets. We derive a diagrammatic expansion for in-in correlators in local scalar
field theories with finite polynomial interactions. We outline how these same
correlators can be computed using the double-path integral which acts as a
generating functional for the in-in correlators. We modify the in-in generating
functional to obtain a generating functional for in-out correlators. We define
a notion of scattering amplitudes on causal sets with non-interacting past and
future regions and verify that they are given by S-matrix elements (matrix
elements of the time-evolution operator). We describe how these formal
developments can be implemented to compute early universe observables under the
assumption that spacetime is fundamentally discrete.
