A Mathematical Perspective on 0-Dimensional Quantum Field Theory

Abstract

This thesis explores 0-dimensional quantum field theory from a combinatorial and diagrammatic perspective. In this setting, fields reduce to ordinary variables, and path integrals become standard integrals. This serves as a toy model to develop mathematical tools that are then applicable to higher-dimensional theories.

The central objects of study are partition functions and correlation functions. We focus on their perturbative expansion, which is interpreted combinatorially in terms of Feynman diagrams. This is developed in a rigorous mathematical framework with the use of generating functions, group theory, and graph enumeration techniques. Specifically, we analyse the role of labelled, unlabelled, connected, and bridgeless graphs in encoding the structure of these expansions. The first part of this work establishes this foundation. Then, this theory is applied to certain models in 0-dimensional quantum field theory, showing how the formal power series expansions lead to diagrammatic representations.

In the second part, we acknowledge the divergence of these formal series and address the issue by introducing Padé approximants. These are rational approximants that often yield better convergence properties. We prove their convergence in a specific interacting model and show their efficiency in recovering accurate numerical results.