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Deformation and Contraction of Symmetries in Special Relativity DEFORMATION AND CONTRACTION OF SYMMETRIES IN SPECIAL RELATIVITY ADISSERTATION BY LUKAS KONSTANTIN BRUNKHORST DEFORMATION AND CONTRACTION OF SYMMETRIES IN SPECIAL RELATIVITY Am 28. Februar 2017 dem Fachbereich 1 der Universitat¨ Bremen zur Erlangung des Grades Doktor der Naturwissenschaften (Dr. rer. nat) vorgelegte Dissertation von LUKAS K. BRUNKHORST, geboren am 19.05.1989 in Bremen. Gutachter: Prof. Dr. Domenico Giulini Prof. Dr. Peter Schupp Datum des Dissertationskolloquiums: Montag, der 8. Mai 2017 iii ABSTRACT This dissertation gives an account of the fundamental principles underlying two concep- tionally different ways of embedding Special Relativity into a wider context. Both of them root in the attempt to explore the full scope of the Relativity Postulate. The frst approach uses Lie algebraic analysis alone, but already yields a whole range of alternative kinematics that are all in a quantifable sense near to those in Special Relativity, while being rather far away in a qualitative way. The corresponding models for spacetime are seen to be four-dimensional versions of the prototypical planar geometries associated with the work of Cayley and Klein. The close relationship between algebraic and geometric methods displayed by these considerations is being substantialized in terms of light-like spacetime extensions. The second direction of departures from Special Relativity stresses and develops the algebraic view on spacetime by considering Hopf instead of Lie algebras as candidates for the description of kinematical transformations and hence spacetime symmetry. This approach is motivated by the belief in the existence of a quantum theory of gravity, and the assumption that such manifests itself in nonlinear modifcations of the laws of Special Relativity at length scales comparable to the Planck length. The twofold character of this work, and the presentation of an example for the fully geometric character of a specifc Hopf algebraic deformation of the Poincare´ algebra, enable a conclusion that speculates on a possible relationship between the two developed viewpoints via the technique of nonlinear realizations. A non-perturbative approach to the latter is given which generalizes to all the considered geometries. v ABSTRACT (GERMAN) Die vorliegende Arbeit stellt grundlegende Prinzipien zweier konzeptionell verschie- dener Wege dar, die Poincare-Symmetrie´ der Spezielle Relativitatstheorie¨ durch eine allgemeinere Interpretation des Relativitatspostulats¨ in einen breiteren Rahmen zu rucken.¨ Der erste dieser Ansatze¨ benutzt allein Lie-algebraische Methoden, fuhrt¨ jedoch bereits auf eine Reihe alternativer kinematischer Modelle. Diese sind alle in quantifzierbarer Weise jenem der Speziellen Relativitatstheorie¨ nahe, weichen in qualitativem Sinne jedoch drastisch ab. Die entsprechenden Raumzeitmodelle werden vorgestellt und im Sinne der Klassifzerung moglicher¨ Geometrien nach Cayley und Klein eingeordnet. Es wird erlautert,¨ wie sich in speziellen lichtartigen Raumzeiterweiterungen der enge Zusammenhang zwischen algebraischen und geometrischen Methoden manifestiert. Die zweite Klasse moglicher¨ Abweichungen von der Speziellen Relativitatstheorie¨ erweitert auf Kosten der unmittelbaren geometrischen Anschauung den Symmetrie- begriff hin zu Hopfalgebren. Dies wird motiviert durch die Annahme der Existenz einer Quantentheorie der Gravitation, die sich in nichtlinearen Modifkationen speziell- relativistischer Transformationsgesetze niederschlagt.¨ Der zweiteilige Aufbau der vorliegenden Arbeit ermoglicht¨ die Ruckf¨ uhrung¨ einer besonders viel beachteten Hopf-algebraischen Verallgemeinerung der Poincare-Algebra´ auf rein geometrische Begriffe. Dies fuhrt¨ zur Formulierung einer Annahme uber¨ die Anwendbarkeit ahnlicher¨ Methoden auf eine ganze Klasse von Hopf-Algebren. Als Schlussel¨ dazu wird die Theorie der nichtlinearen Realisierungen von Lie-Gruppen vorgestellt und uber¨ die storungstheoretische¨ Beschreibung hinaus erweitert. vii CONTENTS ABSTRACT v 1. INTRODUCTION 1 2. LIE ALGEBRAIC NEIGHBOURHOOD 7 2.1.Liealgebradeformations............................ 7 2.2. Wigner-Inon¨ ucontractions...........................¨ 15 2.3.Spontaneousbreaking............................. 18 3. CAYLEY-KLEIN MODEL SPACETIMES 21 3.1.Contractionsofgeometry........................... 22 3.2.Automorphisms................................. 29 3.3. Rethinking constant curvature . 51 3.4. Symmetry contraction from spacetime extensions . 63 4. HOPF ALGEBRA BEFORE SPACETIME GEOMETRY 73 4.1.SurpassingLie.................................. 75 4.2. Spacetime, curvature and momentum space . 82 4.3. κ-Minkowskispacetime............................ 83 4.4.Nonlinearrealizations............................. 102 5. RELATED TOPICS 111 5.1. Hamilton geometry and observations . 111 5.2. Gauge theory of gravitation and Horava-Lifshitzˇ gravity . 113 6. SUMMARY AND CONCLUSIONS 117 ACKNOWLEDGEMENTS a A. APPENDIX c A.1.Notation..................................... c A.2.MathematicalGlossary............................. c A.3.Coordinatesystems............................... e BIBLIOGRAPHY h ix 1. INTRODUCTION Nature presents itself to us in uncountable different shapes and guises. Underlying, however, there are certain invariable structures which to determine is the subject of natural sciences, and it belongs to its assumptions that the basic rules by which Nature abides exist independently of those who seek to discover them. In physics, this becomes manifest in the existence of symmetries, and the application of relativity theory. Fundamental physics occupies a singular status as a subdiscipline of natural sciences, in that its aim is to understand only the most elementary constituents of nature, and ask only the most fundamental questions. For a long time, and tightly connected with the names of Galilei and Newton, its great achievement was to provide a unifying framework for the description of moving bodies. In the 19th and 20th century then, with the discovery of electromagnetism and the nuclear forces, new phenomena came into focus which were less accessible to the human eye alone, and whose observation necessitated the use of ever more ingenious setups in the laboratory. Along with this evolution a change in what was understood as an observing system took place. While Galilei was still concerned with the physical experiences made by actual sailors under deck of a ship [GD53], the more general conception of an observer includes his usage of measurement devices capable of assigning to all recorded measurement values defnitive coordinates in space and time, thereby defning a frame of reference. In order to infer physical laws from experiments, it almost appears as a necessity that all reference frames are in principle suitable for this task. This means that one should expect the existence of well-defned operations that translate physical quantities as observed in one laboratory system into their form as observed in any other system—and with them the laws describing their mutual dependence. The (Galilean) Relativity Postulate goes further. It assumes the existence of a privileged class of reference frames, which are distinguished from all others by the property that observed from these, the fundamental physical laws assume their simplest form, which, in addition, stays the same in all such frames. Since they are operationally distinguished as being free of forces, so that their motion is determined by their property of inertia alone, these reference frames have been called ‘inertial frames’. The fact that the fundamental laws are found to share the same form in all such frames does not mean that the quantities determined by these laws have the same value. On the contrary, since inertial frames are a subset of all possible references frames, there exist nontrivial transformations between them that relate all physical quantities. It appears adequate to call this subset of all transformations between reference frames inertial transformations. It is distinguished by the property of leaving the fundamental laws invariant, which means nothing but that inertial transformations constitute a symmetry. To Galilei and Newton it would appear that the invariance under inertial transform- 1 1. Introduction ations is a property of the fundamental physical laws. The conceptual leap that is primarily attributed to Einstein is to instead understand this invariance property as belonging to space and time itself. This becomes plausible when realizing how inertial frames are always characterized with respect to one another in terms of spatio-temporal notions, namely by their location (in space and time), their orientation, and their velocity. Adopting this view, the basic physical laws are seen to merely inherit the symmetry properties that spacetime dictates. Given the Relativity Postulate, an actual theory of relativity, and hence of spacetime as we argued, consists in specifying the set of inertial transformations. This is equivalent with fxing the allowable kinematics for a system, which could be a moving point particle or a feld in spacetime. For his construction of the Special Theory of Relativity, Einstein added the assumption (well-motivated by the famous thought experiments as well as Maxwell’s formulation of electrodynamics) that the speed of light be measured to have the same numerical value in all inertial frames. This allowed him to deduce the Poincare´ group as the proper structure implementing inertial transformations, and at the same time, suggest Minkowski spacetime
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