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. 16.