A Covariant Causal Set Approach to Discrete Quantum Gravity

A covariant causal set (c-causet) is a causal set that is invariant under labeling. Such causets are well-behaved and have a rigid geometry that is determined by a sequence of positive integers called the shell sequence. We first consider the microscopic picture. In this picture, the vertices of a c-causet have integer labels that are unique up to a label isomorphism. This labeling enables us to define a natural metric d(a, b) between time-like separated vertices a and b. The time metric d(a, b) results in a natural definition of a geodesic from a to b. It turns out that there can be n ≥ 1 such geodesics. Letting a be the origin (the big bang), we define the curvature K(b) of b to be n−1. Assuming that particles tend to move along geodesics, K(b) gives the tendency that vertex b is occupied. In this way, the mass distribution is determined by the geometry of the c-causet. We next consider the macroscopic picture which describes the growth process of c-causets. We propose that this process is governed by a quantum dynamics given by complex amplitudes. At present, these amplitudes are unknown. But if they can be found, they will determine the (approximate) geometry of the c-causet describing our particular universe. As an illustration, we present a simple example of an amplitude process that may have physical relevance. We also give a discrete analogue of Einstein’s field equations.