A Unified Approach to Discrete Quantum Gravity

Abstract

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 $x \to y$ 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 $\pscript$ we show that the growth process $(\pscript ,\to)$ 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.