Executive Summary : | Quantum gravity has come centre stage in theoretical and mathematical physics in the past two decades thanks partly to the attention received by string theory and loop quantum gravity, but also because of active observational and experimental searches for quantum gravity phenomenology. While the main stream approaches have had some success, it has been recognised that there is a need for a more diverse approach to the main questions of quantum gravity (see the recent Snowmass 2021 proposal which I have coauthored arXiv: 2207.10618.) The project is based on the ongoing investigation of the causal set approach to quantum gravity, which has garnered increasing interest over the past few years (see the metrics of my 2017 Living Reviews, for example). This is a discrete approach to quantum gravity, where spacetime causality takes centre stage. Its key predictions include the value of the cosmological constant, which appears as a fluctuation about zero, as well as local Lorentz invariance, which has been tested to great accuracy. The aims of the project are multi-pronged. The first is to help establish a stronger connection between quantum geometry and continuum geometry. This is the geometric reconstruction program in causal set theory, where spacetime geometry and topology are obtained from the underlying discrete causal order. One of the new focus directions that I wish to explore in this project is to use recent advances in Lorentzian length spaces to understand the convergence properties, which will help prove the discrete-continuum correspondence. A second aim of the project is to work on the quantum dynamics of causal set theory. Building on my earlier work, one goal is to generalise from the 2 and 3 spacetime dimensional results of statistical causal set dynamics to the 4 dimensional case, to see whether the first order phase transition observed in these cases still manifests itself. This part of the project is numerically intensive and would have significant consequences for early universe physics. The second goal is to ask the broader question of how spacetime dimensions can be emergent in the theory, using the path integral formulation. Recent results in the saddle point approximation suggest that this goal is well within reach. A final goal more ambitious goal in quantum dynamics is to construct more realistic models for quantum sequential growth, which is at the heart of the causal set approach. This project requires the use of results from fields as diverse as measure theory in mathematics and discrete stochastic dynamics in physics. Given the growing interest in this area, the outcome of this project should lead to significant impact in the field of quantum gravity as well as the neighbouring emerging areas of Lorentzian geometry as well as information theory. |