A covariant causal set approach to discrete quantum gravity

Abstract

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 $n\ge 1$ 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.

23 pages, 6 tables; new version corrects some typos in the proof of Theorem 6.1

References in corpus (5)

• Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School)
• Directions in Causal Set Quantum Gravity
• The Universe as a Quantum Computer
• An Einstein equation for discrete quantum gravity
• An Approach to Discrete Quantum Gravity

Cited by in corpus (8)

• Appearing Out of Nowhere: The Emergence of Spacetime in Quantum Gravity
• Curvature and Quantum Mechanics on Covariant Causal Sets
• An Approach to Discrete Quantum Gravity
• A Unified Approach to Discrete Quantum Gravity
• Elementary Particles and the Causet Approach to Discrete Quantum Gravity
• Discrete Quantum Gravity and Quantum Field Theory
• Wave Equations for Discrete Quantum Gravity
• Labeled Causets in Discrete Quantum Gravity