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Geometric Approaches to Quantum Field Theory GEOMETRIC APPROACHES TO QUANTUM FIELD THEORY A thesis submitted to The University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2020 Kieran T. O. Finn School of Physics and Astronomy Supervised by Professor Apostolos Pilaftsis BLANK PAGE 2 Contents Abstract 7 Declaration 9 Copyright 11 Acknowledgements 13 Publications by the Author 15 1 Introduction 19 1.1 Unit Independence . 20 1.2 Reparametrisation Invariance in Quantum Field Theories . 24 1.3 Example: Complex Scalar Field . 25 1.4 Outline . 31 1.5 Conventions . 34 2 Field Space Covariance 35 2.1 Riemannian Geometry . 35 2.1.1 Manifolds . 35 2.1.2 Tensors . 36 2.1.3 Connections and the Covariant Derivative . 37 2.1.4 Distances on the Manifold . 38 2.1.5 Curvature of a Manifold . 39 2.1.6 Local Normal Coordinates and the Vielbein Formalism 41 2.1.7 Submanifolds and Induced Metrics . 42 2.1.8 The Geodesic Equation . 42 2.1.9 Isometries . 43 2.2 The Field Space . 44 2.2.1 Interpretation of the Field Space . 48 3 2.3 The Configuration Space . 50 2.4 Parametrisation Dependence of Standard Approaches to Quan- tum Field Theory . 52 2.4.1 Feynman Diagrams . 53 2.4.2 The Effective Action . 56 2.5 Covariant Approaches to Quantum Field Theory . 59 2.5.1 Covariant Feynman Diagrams . 59 2.5.2 The Vilkovisky–DeWitt Effective Action . 62 2.6 Example: Complex Scalar Field . 66 3 Frame Covariance in Quantum Gravity 69 3.1 The Cosmological Frame Problem . 70 3.2 The Invariant Spacetime Metric . 74 3.3 The Field and Configuration Spaces for Gravity . 78 3.4 Example: Einstein Hilbert Action . 83 3.5 Unique Frame Invariant Effective Action for Quantum Gravity 85 4 Field Space Covariance for Fermionic Theories 89 4.1 Grassmann Numbers and Supermanifolds . 89 4.2 The Fermionic Field Space . 96 4.3 Field Space Covariant Lagrangians for Fermionic Theories . 98 4.4 Tensors on the Field Space . 99 4.5 The Field Space Metric . 103 4.6 Unique Frame Invariant Effective Action for Fermions . 107 4.7 Example: Single Fermion Field . 110 5 The Eisenhart Lift 115 5.1 The Eisenhart Lift in Classical Mechanics . 115 5.2 Example: Simple Harmonic Oscillator . 119 5.3 One-Dimensional Field Theories . 123 5.4 Higher-Dimensional Field Theories . 126 5.5 Further Generalisations of the Eisenhart Lift . 131 5.5.1 Example:Scalar Fermion Theory . 133 6 Cosmic Inflation 135 6.1 Hot Big Bang Cosmology . 135 6.2 The Classic Cosmological Puzzles . 141 6.2.1 The Flatness Problem . 141 4 6.2.2 The Horizon Problem . 143 6.2.3 The Monopole and Primordial Perturbation Problems . 145 6.3 Inflation . 146 6.3.1 The Inflaton . 148 6.3.2 Slow Roll Inflation . 149 6.4 The Initial Conditions of Inflation . 152 6.4.1 Initial Inhomogeneities . 153 6.4.2 Initial Field Values . 155 7 Geometric Initial Conditions for Inflation 163 7.1 The Phase Space Manifold . 163 7.2 The Phase Space Manifold for Inflation . 167 7.3 Symmetries and Constraints . 172 7.3.1 Symmetries of the Inflationary Phase Space . 172 7.3.2 The Hamiltonian Constraint . 174 7.4 Finite Measure for the Initial Conditions of Inflation . 176 7.5 Effects of Anisotropy . 179 7.6 Example: φ4 Inflation . 184 8 Conclusions 193 Appendices 205 A Example Covariant Calculations 207 A.1 Complex Scalar Field . 207 A.2 Curved Field-Space Example . 219 A.3 Example with Hidden Interactions . 227 B A Failed Attempt to Construct a Field Space Metric for Fermions 229 Bibliography 233 This thesis contains 40,251 words. 5 List of Figures 5.1 Motion of a free particle in a two-dimensional lifted space that recreates the simple harmonic oscillator . 122 6.1 The trajectory of the inflaton field in phase space for a range of different initial conditions, showing the attractor nature of the slow roll regime . 156 6.2 Schematic illustration of the potentials for the α-attractor and quintessential models of inflation. 157 7.1 Measure on the initial anisotropy of the Universe . 185 7.2 Distribution of φ and φ˙ on the phase-space manifold . 188 7.3 Probability of achieving N > 60 e-foldings of inflation as a function of the cosmological constant Λ . 190 8.1 Manifold of initial conditions for inflation . 201 List of Tables 6.1 Cosmic evolution of different forms of matter . 139 7.1 The phase space metric for inflation . 171 6 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. 7 BLANK PAGE 8 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. Lancaster, December 8, 2020 Kieran Finn 9 BLANK PAGE 10 Copyright i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copy- right”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, pub- lication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents. manchester.ac.uk/DocuInfo.aspx?DocID=24420), in any relevant The- sis restriction declarations deposited in the University Library, The Uni- versity Library’s regulations (see http://www.library.manchester.ac. uk/about/regulations/) and in The University’s policy on presenta- tion of Theses 11 BLANK PAGE 12 Acknowledgements I would like to start by thanking my supervisor, Apostolos Pilaftsis. His guidance and insight throughout the last three years has been invaluable in completing the research in this thesis. Also Sotirios Karamitsos, who has been a great friend, as well as a collaborator. I would like to thank him for the countless hours spent discussing field space metrics, probability measures, supermanifolds, time-dependent units, and the correct use of the Oxford comma. I would also like to thank the rest of the Manchester theory office: Jack Hol- guin, Chris Shepherd, Matthew DeAngelis, Kiran Ostrolenk, Baptiste Cabouat, Abbie Keats, Mulham Hijazi, Pablo Candia da Silva and Jack Helliwell for providing a fun and productive working environment with just the right balance between getting work done and going to the pub. Throughout my Ph.D. I have been fortunate to attend many interesting con- ferences and workshops, which have allowed me to meet and discuss physics with amazing people from across the world. I would like to thank them for helping me develop my ideas, and for pointing out issues that I would other- wise have missed. In particular, I would like to thank Matthew Kellet, Owen Goodwin and Sam Brady for sharing their knowledge of supermanifolds, Daniel Martin for teaching me the finer points of Mathematica and Josu Aurrokoetxea and David Sloan for helping me understand how to approach the initial conditions problem in inflation.
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