Wave Equations for Discrete Quantum Gravity

Abstract: This article is based on the covariant causal set ($c$-causet) approach to discrete quantum gravity. A $c$-causet $x$ is a finite partially ordered set that has a unique labeling of its vertices. A rate of change on $x$ is described by a covariant difference operator and this operator acting on a wave function forms the left side of the wave equation. The right side is given by an energy term acting on the wave function. Solutions to the wave equation corresponding to certain pairs of paths in $x$ are added and normalized to form a unique state. The modulus squared of the state gives probabilities that a pair of interacting particles is at various locations given by pairs of vertices in $x$. We illustrate this model for a few of the simplest nontrivial examples of $c$-causets. Three forces are considered, the attractive and repulsive electric forces and the strong nuclear force. Large models get much more complicated and will probably require a computer to analyze.