At the edges of infinity and the finite: Charting a path to UV completion from number theory to quantum fields and strings
Publication Date
July 30, 2026
Creator
Abstract
This thesis explores the foundational presence of infinity at the nexus of theoretical physics and mathematics, developing a framework that connects number theory, quantum field theory, and string theory. Central to this investigation is the introduction and application of η regularisation, a novel technique inspired by smoothed asymptotics and the summation methods of Ramanujan. Starting with a rigorous treatment of divergent series, the thesis constructs a mathematical apparatus capable of extracting finite, physically meaningful values from divergent expressions.
Building on this, the η framework is extended to perturbative quantum field theory, offering deeper insight into a general theory of regularisation based on the preservation of gauge invariance. The approach is further generalised to the path integral formalism, extending η regularisation to operator language.
Tying together previous discussion, the culmination of this work is formulated in the context of ultraviolet finiteness in string theory, where the modular structure of perturbative string amplitudes at one-loop and their tropical limits are shown to induce a natural renormalisation of field theory integrals. We then discuss the hopeful idea of deriving classes of string theory induced regulators as multiplicative weights in the descendent low-energy field theory integration.
Overall, the bottom-up and top-down approach of this thesis suggests that the mechanisms by which quantum gravity might tame infinities may also, in turn, be fundamentally associated with number-theoretic phenomena. By tracing a conceptual arc from the infinite to the finite (and vice versa)---from divergent series to ultraviolet-complete theories---a fresh perspective on regularisation, modularity, and the unification of mathematics and physics is offered.
Item Type
ethesis
Thesis Type
PhD
Supervisors
Keywords
Associated Schools / Departments
School of Physics and Astronomy
eprints ID
82895
UoN Repository URI
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Thesis-Final-Corrections.pdf
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Full-text
Description
Examined. PhD thesis, examiner corrections included following viva
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1.32 MB
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