An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity

TL;DR

This paper develops a covariant causal set model for discrete quantum gravity, introducing an isometric dynamics governed by quantum transition amplitudes that preserve stochastic states and lead to a unitary group representation.

Contribution

It presents a novel isometric quantum dynamics framework for causal set-based discrete quantum gravity, including the construction of a unitary group and energy operator.

Findings

Dynamics is governed by a stochastic, unitary quantum transition amplitude.

The model preserves stochastic states during evolution.

A natural energy operator and canonical position and momentum operators are defined.

Abstract

We consider a covariant causal set approach to discrete quantum gravity. We first review the microscopic picture of this approach. In this picture a universe grows one element at a time and its geometry is determined by a sequence of integers called the shell sequence. We next present the macroscopic picture which is described by a sequential growth process. We introduce a model in which the dynamics is governed by a quantum transition amplitude. The amplitude satisfies a stochastic and unitary condition and the resulting dynamics becomes isometric. We show that the dynamics preserves stochastic states. By "doubling down" on the dynamics we obtain a unitary group representation and a natural energy operator. These unitary operators are employed to define canonical position and momentum operators.