Interacting Scalar Field Theory on Causal Sets

We introduce $\phi^4$ interacting real scalar Quantum Field Theory (QFT) on causal sets. We consider both the canonical framework of causal set free QFT, involving a Hilbert space and operators and so on, and the double path integral framework of causal set QFT outlaid by Sorkin. In both cases we describe how to extend the formalism to include a $\phi^4$ self-interaction, and, to make contact with the continuum, we contrast certain key expressions with their continuum counterparts. We develop a diagram-based algorithm, analogous to Feynman diagrams in the continuum, to compute the interacting 2-point function of our causal set QFT. Notably, causality is manifest in our diagrams in a manner not present in the usual Feynman diagrams of the continuum theory.