A Dynamics for Discrete Quantum Gravity

TL;DR

This paper develops a quantum sequential growth process for discrete quantum gravity based on causal sets, introducing a Hilbert space framework, a quantum action, and exploring connections to Einstein's field equations.

Contribution

It presents a novel quantum dynamics model for causal set-based discrete quantum gravity, integrating a sum-over-histories approach and discussing links to classical relativity.

Findings

Defined a quantum sequential growth process (QSGP) for causal sets.

Constructed a Hilbert space and positive operators for quantum dynamics.

Proposed a discrete Einstein equation to connect with classical gravity.

Abstract

This paper is based on the causal set approach to discrete quantum gravity. We first describe a classical sequential growth process (CSGP) in which the universe grows one element at a time in discrete steps. At each step the process has the form of a causal set (causet) and the "completed" universe is given by a path through a discretely growing chain of causets. We then quantize the CSGP by forming a Hilbert space $H$ on the set of paths. The quantum dynamics is governed by a sequence of positive operators $\rho_n$ on $H$ that satisfy normalization and consistency conditions. The pair $(H,\brac{\rho_n})$ is called a quantum sequential growth process (QSGP). We next discuss a concrete realization of a QSGP in terms of a natural quantum action. This gives an amplitude process related to the sum over histories" approach to quantum mechanics. Finally, we briefly discuss a discrete form of Einstein's field equation and speculate how this may be employed to compare the present framework with classical general relativity theory.