Covariant causal set approach, Discrete Quantum Gravity
This paper is based on a covariant causal set (c-causet) approach to discrete quantum gravity. A c-causet is a partially ordered set (x, <) that is invariant under labeling. We first consider the microscopic picture which describes the detailed structure of c-causets. The unique labeling of a c-causet x enables us to define a natural metric d(a, b) between comparable vertices a, b of x. The metric is then employed to define geodesics and curvatures on x. We next consider the macroscopic picture which describes the growth process x → y of c-causets. We propose that this process is governed by a quantum dynamics given by complex amplitudes. Denoting the set of c-causets by P we show that the growth process (P, →) can be structured into a discrete 4-manifold. This 4-manifold presents a unified approach to a discrete quantum gravity for which we define discrete analogues of Einstein’s field equations and Dirac’s equation.
Gudder, S., "A Unified Approach to Discrete Quantum Gravity" (2014). Mathematics Preprint Series. Paper 16.