Combinatorial Aspects of Feynman Integrals and Causal Set Theory

Abstract

This thesis consists of two distinct sections, the first highlighting causal set theory (CST) and the second focusing on Feynman period estimation. This work specifically covers how discrete structures from algebraic combinatorics can be applied to problems from physics, and how we can use computational tools and techniques to help solve these problems from a mathematical perspective.

In the first part of the thesis, we introduce CST, an approach to quantum gravity that focuses on how events in spacetime are causally related to one another. In CST, discretized spacetime is given by a locally finite poset. A significant portion of this section focuses on covtree, which allows us to evolve such a discrete spacetime, in a way that does not depend on arbitrary labelling. However recognizing nodes of covtree involve solving a particular downset reconstruction problem. To attack this we define a graph that compares downsets that differ by exactly one element, with a particular focus on the order dimension two case. This section of the thesis sheds light on evolving a spacetime by constructing future elements within the covtree framework. Reconstructing spacetime in this way will allow for researchers to advance our understanding of covtree’s structure and improves causal set theory as an approach to quantum gravity.

In the second part, we provide an introduction to Feynman periods, explaining their significance in quantum field theory (QFT) calculations. We will discuss how machine learning models, such as linear and quadratic regression, and graph neural networks, can be applied to Feynman graphs and their properties, to predict the Feynman period. Even simple techniques like linear regression, on graph parameters only, which do not consider the graph structure directly, can make highly accurate predictions of Feynman periods. The work presented in this section, done jointly with Dr. Paul-Hermann Balduf, is published in the Journal of High Energy Physics. This research has significant implications for QFT computations, which can become computationally infeasible at large scales, and facilitates further exploration in particle physics.