Geometric Aspects of Gauge and Spacetime Symmetries Steffen Christian Martin Gielen Trinity College A dissertation submitted to the University of Cambridge for the degree of Doctor of Philosophy arXiv:1103.0731v1 [gr-qc] 3 Mar 2011 October 2011 2 Declaration This dissertation is my own work and contains nothing which is the outcome of work done in collaboration with others, except as specified in the text (section 1.2). It is not substantially the same as any other thesis that I have submitted or will submit for a degree at this or any other university. This thesis is based on the research presented in the following refereed publica- tions: (1) Gary W. Gibbons and Steffen Gielen. The Petrov and Kaigorodov- Ozsv´ath Solutions: Spacetime as a Group Manifold. Classical and Quantum Gravity 25 (2008), 165009. (2) Gary W. Gibbons and Steffen Gielen. Deformed General Relativity and Torsion. Classical and Quantum Gravity 26 (2009), 135005. (3) Gary W. Gibbons, Steffen Gielen, C. N. Pope, and Neil Turok. Measures on Mixing Angles. Physical Review D79 (2009), 013009. (4) Steffen Gielen. CP Violation Makes Left-Right Symmetric Extensions With Non-Hermitian Mass Matrices Appear Unnatural. Physical Review D81 (2010), 076003. (5) Steffen Gielen and Daniele Oriti. Classical general relativity as BF- Plebanski theory with linear constraints. Classical and Quantum Gravity 27 (2010), 185017. These papers appear in the text as references [65], [66], [67], [68], and [69] respec- tively. The results of section 5 of (3) were also separately published in (6) Gary W. Gibbons, Steffen Gielen, C. N. Pope, and Neil Turok. Naturalness of CP Violation in the Standard Model. Physical Review Letters 102 (2009), 121802. Steffen Gielen Monday, 22 October 2018 3 Acknowledgments My first and foremost thanks go to my supervisor, Prof. Gary Gibbons, for teaching me so much about physics and mathematics, for many fruitful discussions and sug- gestions, for his encouragement and for his support of all of my plans over the last three years. I have also greatly enjoyed working with, and learning from, Dr. Daniele Oriti, Prof. C. N. Pope and Dr. Neil Turok on different projects that are part of this thesis. Invitations by Dr. Turok and Dr. Oriti that allowed me to spend an intensive week at Perimeter Institute in October 2008 and several months at AEI Golm in 2009/10 deserve special thanks. I am grateful to Trinity College for all its support, both financial and otherwise, over the course of the last four years, without which I would not have been able to complete a PhD at Cambridge. My work has also been supported by an EPSRC studentship. I would like to thank my friends in Cambridge, who have hopefully improved my understanding of physics and made my time more enjoyable, not least through countless hours of ‘foosball’, and my friends in Germany who have kept in touch while I was away for so long, and gave support in difficult times. Going back a bit further in time, I would not have ended up as a PhD student in Cambridge (and certainly not at Trinity) without the encouragement of Dipl.-Phys. Igor Pikovski and the strong support of Profs. Ingo Peschel and J¨urgen Bosse. The latter two together with Hermann Schulz (Hochschuldozent) were also influential in shaping my interest in theoretical physics early on. Last but not least, I would like to thank my aunt Odina and uncle Uwe for their support, especially for the first year at Cambridge. 4 Geometric Aspects of Gauge and Spacetime Symmetries Steffen Christian Martin Gielen Summary We investigate several problems in relativity and particle physics where symmetries play a central role; in all cases geometric properties of Lie groups and their quotients are related to physical effects. The first part is concerned with symmetries in gravity. We apply the theory of Lie group deformations to isometry groups of exact solutions in general relativity, relating the algebraic properties of these groups to physical properties of the spacetimes. We then make group deformation local, generalising deformed special relativity (DSR) by describing gravity as a gauge theory of the de Sitter group. We find that in our construction Minkowski space has a connection with torsion; physical effects of torsion seem to rule out the proposed framework as a viable theory. A third chapter discusses a formulation of gravity as a topological BF theory with added linear constraints that reduce the symmetries of the topological theory to those of general relativity. We discretise our constructions and compare to a similar construction by Plebanski which uses quadratic constraints. In the second part we study CP violation in the electroweak sector of the standard model and certain extensions of it. We quantify fine-tuning in the observed magnitude of CP violation by determining a natural measure on the space of CKM matrices, a double quotient of SU(3), introducing different possible choices and comparing their predictions for CP violation. While one generically faces a fine-tuning problem, in the standard model the problem is removed by a measure that incorporates the observed quark masses, which suggests a close relation between a mass hierarchy and suppression of CP violation. Going beyond the standard model by adding a left-right symmetry spoils the result, leaving us to conclude that such additional symmetries appear less natural. CONTENTS 5 Contents 1 Introduction 7 1.1 LieGroupsandHomogeneousSpaces . 11 1.2 StructureofthisThesisandConventions . 15 I Deformed and Constrained Symmetries in Gravity 17 2 Deformations of Spacetime Symmetries 18 2.1 Overview.................................. 18 2.2 DeformationTheory ........................... 21 2.3 The Petrov and Kaigorodov-Ozsv´ath Solutions . 25 2.4 The Kaigorodov Solution and Lobatchevski Plane Gravitational Waves 42 2.5 Summary ................................. 49 3 Deformed General Relativity and Torsion 50 3.1 Introduction................................ 50 3.2 CurvedMomentumSpace ........................ 55 3.3 DeformedSpecialRelativity . 58 3.4 AdeSitterGaugeTheoryofGravity . 60 3.5 Cartan Connections on Homogeneous Bundles . 67 3.6 Synthesis.................................. 71 3.7 SummaryandDiscussion......................... 74 3.8 GaugeInvarianceBroken? . 77 3.9 SymmetricAffineTheory......................... 78 4 General Relativity From a Constrained Topological Theory 80 4.1 Overview.................................. 80 4.2 ActionsforGravity............................ 82 CONTENTS 6 4.3 ConstrainingBFTheory ......................... 85 4.4 LinearConstraintsforBF-PlebanskiTheory . 90 4.5 Lagrangian and Hamiltonian Formulation . 101 4.6 SummaryandOutlook .......................... 103 II Gauge Symmetries and CP Violation 106 5 SU(3) and its Quotients 107 5.1 Introduction................................ 107 5.2 The Homogeneous Space SU(3)/U(1)2 ................. 111 5.3 Metrics on U(1)2 SU(3)/U(1)2 ..................... 112 \ 5.4 HermitianandComplexMatrices . 114 6 Naturalness of CP Violation 120 6.1 IntroductionandMotivation . 120 6.2 CP ViolationintheStandardModelandBeyond . 124 6.3 StatisticsontheSpaceofCKMMatrices . 129 6.4 StatisticsontheSpaceofMassMatrices . 136 6.5 ExtensiontoNeutrinos . .. .. .. 151 6.6 SummaryandOutlook .......................... 154 Summary and Outlook 157 References 161 List of Figures 176 List of Tables 177 Index 178 Introduction 7 Chapter 1 Introduction “Ich kann es nun einmal nicht lassen, in diesem Drama von Mathematik und Physik – die sich im Dunkeln befruchten, aber von Angesicht zu Angesicht so gerne einander verkennen und verleugnen – die Rolle des (wie ich genugsam erfuhr, oft unerw¨unschten) Boten zu spielen.” [164] Hermann Weyl It is impossible to overemphasise the importance of symmetry as a fundamental concept in both mathematics and theoretical physics. In mathematics, the study of groups arose as an abstraction of “transformation groups”, invertible maps of a given “point field” into itself [164]. Groups are the mathematical realisation of the concept of symmetry. Felix Klein’s Erlangen Pro- gramme [96] aimed to describe any geometry by its transformation group; as we will detail shortly, the geometries considered in the Erlangen Programme are given by coset spaces of the form G/H = gH,g G , where G is a continuous (Lie) group { ∈ } and H a closed subgroup of G. Klein himself found the concept of groups most char- acteristic of 19th century mathematics [164]. The study of continuous symmetries is, in mathematical terms, the study of Lie groups and homogeneous spaces which nicely combines aspects of group theory and differential geometry. Beyond the direct study of groups, symmetry is an important concept in other parts of mathematics, such as the study of differential equations. In physics, the underlying symmetries of any theory are arguably its most fun- damental property. As an example, the first physical theory in the modern sense, Newtonian mechanics, fundamentally rests on assumptions about the symmetries of Introduction 8 the physical world: A configuration of bodies in (gravitational) interaction will behave in the same way if it is rotated as a whole (space is isotropic); a common motion of all bodies at constant velocity is not observable (relativity of inertial frames). Con- sequently, the laws of Newtonian mechanics are invariant under the action of the semidirect product of the rotation group SO(3) and commutative Galileian boosts, the Euclidean group E(3). When Einstein understood in 1905, in the wake of the Michelson-Morley experiment [118], that