Geometric Approaches to Quantum Field Theory

Abstract

The ancient Greeks thought that all of creation should be describable in terms of geometry. In this thesis we take a step towards realising this dream by applying the methods of differential geometry to modern ideas about particle physics and cosmology in the form of quantum field theory. We shall achieve this using the formalism of field space covariance, in which the degrees of freedom in a quantum field theory are treated as coordinates on a Riemannian manifold, known as the field space manifold. This formalism allows us to describe such theories geometrically, in a way that is manifestly invariant under arbitrary choices such as the units, spacetime coordinates or field variables used. In this thesis we extend the applicability of field space covariance to quantum field theories with gravitational and fermionic degrees of freedom. We show how to construct the field space manifold for such theories and how to equip it with a natural metric. Thus we are now able to apply this formalism to all realistic theories of particle physics, including the Standard Model. In addition, we show that the potential term in a quantum field theory can also be described geometrically through a process known as the Eisenhart lift. We show how, by introducing new degrees of freedom into the theory, a potential term can be recast as the curvature of field space. Finally, we apply our geometric methods to the theory of inflation. We construct a manifold that describes the evolution of inflation geometrically as a geodesic. We show that the tangent bundle of this manifold, equipped with a natural metric, provides a finite measure on the initial conditions of inflation, which we can use to study fine tuning in these models.

Date of Award	1 Aug 2021
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Apostolos Pilaftsis (Supervisor) & Fedor Bezrukov (Supervisor)