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The Chiral Structure of Loop Quantum Gravity Wolfgang Martin Wieland

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The Chiral Structure of Loop Quantum Gravity Wolfgang Martin Wieland The Chiral Structure of Loop Quantum Gravity Wolfgang Martin Wieland To cite this version: Wolfgang Martin Wieland. The Chiral Structure of Loop Quantum Gravity. Relativité Générale et Cosmologie Quantique [gr-qc]. Aix-Marseille Université, 2013. Français. tel-00952498 HAL Id: tel-00952498 https://tel.archives-ouvertes.fr/tel-00952498 Submitted on 3 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Aix-Marseille Université École Doctorale ED352 Physique et Sciences de la Matière Thèse de doctorat présentée par WOLFGANG MARTIN WIELAND Pour obtenir le grade de DocTEUR D’AIX-MARSEILLE UnIVERSIté Specialité : Physique théorique et mathématiques STRUCTURE CHIRALE DE LA GRAVITé QUANTIQUE à BOUCLES Soutenue le 12 decembre 2013 Thèse préparée au Centre de Physique Théorique (CPT) Directeur de thèse : Carlo Rovelli CPT (Aix-Marseille Université) Codirecteur : Simone Speziale CPT (Aix-Marseille Université) Rapporteurs : Jerzy Lewandowski IFT (Uniwersytet Warszawski) Karim Noui LMPT (Université François Rabelais) Membres de jury : Kristina Giesel IQG (FAU Erlangen-Nürnberg) Daniel Grumiller ITP (Technische Universität Wien) Ted Jacobson Department of Physics (University of Maryland) Etera Livine Laboratoire de Physique ÉNS de Lyon Karim Noui LMPT (Université François Rabelais) Alejandro Perez CPT (Aix-Marseille Université) Carlo Rovelli CPT (Aix-Marseille Université) WOLFGANG MARTIN WIELAND THE CHIRAL STRUCTURE OF LOOP QUANTUM GRAVITY Meinen Eltern ABSTRACT Loop gravity is a tentative theory to describe what happens at the Planck scale, the scale at which both general relativity and quantum theory become equally important. It comes in two versions. The canonical approach seeks to solve the Wheeler–DeWitt equation and find the physical states of the theory. Spinfoam gravity, on the other hand, takes a covariant path integral representation to define the transition amplitudes of the theory. Both approaches use the same Hilbert space, but we do not know whether they actually define the same theory. In this thesis, I will present four results, all of which lie in between the two ap- proaches. We start with the classical theory. When Ashtekar first formulated Hamil- tonian general relativity in terms of self-dual (complex Ashtekar) variables the ADM constraint equations turned into neat polynomials of the elementary fields. This was a huge simplification and eventually initiated the program of loop quantum gravity. For a number of technical reasons the complex variables have later been abandoned in favour of the SU(2) Ashtekar–Barbero variables, and the simplification of the Hamil- tonian constraint was lost again. These SU(2) variables are usually derived from the Holst action, which contains the Barbero–Immirzi parameter as an additional coupling constant. After the first introductory chapter, we will use the original self-dual connection to repeat the canonical analysis for the Holst action, while leaving the Barbero–Immirzi parameter untouched. The resulting constraint equations depend on this parameter, yet maintain a polynomial form. To guarantee that the metric is real, we have to introduce additional constraints. These reality conditions match the linear simplicity constraints of spinfoam gravity. They are preserved in time only if the spatial spin connection is torsionless, which appears as a secondary constraint in the canonical analysis. This is our first complex of results. The next chapter is about the classical theory, and studies how to discretise gravity in terms of first-order holonomy-flux variables. The corresponding phase space has a non-linear structure. Twistors allow to handle this non-linearity while working on a linear phase space with canonical Darboux coordinates. This framework was originally introduced by Freidel and Speziale, but only for the case of SU(2) Ashtekar–Barbero variables. Here, we develop the generalisation to SL(2, C), that is we use twistors to parametrise the phase space of self-dual holonomy-flux variables. This is the second result. We will then discuss the spinfoam dynamics in terms of these twistorial variables, and arrive at our third result: A new Hamiltonian formulation of discretised gravity. The Hamiltonian comes with a continuum action adapted to a fixed simplicial discretisation of spacetime. The action is a sum of the spinorial analogue of the topological “BF ”- action and the reality conditions that guarantee the existence of a metric. Chapter four studies the resulting quantum theory. Since the action is a polynomial in the spinors, canonical quantisation is straightforward. Transition amplitudes reproduce the EPRL (Engle–Pereira–Rovelli–Livine) spinfoam model. This is our final result. It shows that spinfoam gravity can be derived from a classical action, with spinors as the fundamental configuration variables. iv VORWORT Von allen Grundkräften der Physik passt nur die Schwerkraft nicht zur Quantentheo- rie. Als schwächste aller vier Wechselwirkungen (das sind die beiden Kernkräfte, die elektromagnetische Kraft und eben die Gravitation) spielt sie für die Physik des Mi- krokosmos keine Rolle; die Gravitation beherrscht die Welt im Großen. Die allgemeine Relativitätstheorie liefert den mathematischen Rahmen. Einstein erklärt die Schwer- kraft aus der geometrischen Struktur von Raum und Zeit: Genauso wie die Krümmung der Erdoberfläche den Kurs eines Flugzeugs bestimmt (von Wien nach Washington folgt der Pilot der kürzesten Verbindung, einem Großkreis, keiner Geraden), genauso zwingt die Krümmung der Raumzeit die Erde auf ihre Bahn um die Sonne. Die Quantenmechanik spielt für die Umlaufbahnen der Planeten keine Rolle. Sie beherrscht die Physik im Kleinen. Ort und Impuls eines Teilchens lassen sich als kom- plementäre Variable nicht gleichzeitig scharf messen, sind als Zufallsgrößen unscharf verschmiert. Die Schrödingergleichung beschreibt diese Unschärfen als Wellenfeld in Raum und Zeit. „Das Elektron trifft in zehn Minuten am Ort x ein.“ So ein Satz ist der Quantentheorie ganz unbekannt, wir sagen stattdessen: „In zehn Minuten ist das Elektron mit p(x)-prozentiger Wahrscheinlichkeit am Orte x.“ Den alten Streit um die Frage, ob es kleinste Teilchen gebe, oder die Welt aus einem stofflichen Kontinuum bestehe, beendet die Quantenmechanik mit einem salomonischen Urteil. Beides ist gleichermaßen wahr, und hängt von der Fragestellung ab. In dem einen Experiment enthüllt sich die Quantennatur der Welt: Angeregte Atome senden Licht- teilchen nur ganz bestimmter Farbe aus – das charakteristische Orange der Straßenla- ternen kommt vom Natrium. Ein anderer Versuch zeigt die Kontinuumseigenschaften der Materie: Bei aufmerksamem Blick in eine Straßenlaterne kann man Beugungsringe sehen, wenn im Augapfel die Wellen des Natriumlichts an kleinen Hindernissen streuen. Was hat das nun alles mit der Gravitation zu tun? Nach Einsteins allgemeiner Re- lativitätstheorie hat das Schwerefeld der Erde stets überall einen fest vorhersagbaren Wert. In der Quantenmechanik ist das nicht mehr so, hier gibt es nur mehr Wahrschein- lichkeitsaussagen. Es kann nicht beides stimmen; entweder es gibt echten Zufall, oder die Welt ist streng deterministisch. Außerdem folgt alle Materie der Quantentheorie, doch koppelt in immer gleicher Weise an die Gravitation. Damit müssen sich die Ge- setze der Quantentheorie auch auf die Schwerkraft übertragen. Wie die Quantentheorie mit der Relativitätstheorie zu versöhnen sei, das weiß freilich niemand so genau. Trotz jahrzehntelanger, teils recht phantastischer Bemühungen, fehlt uns noch immer eine Theorie der Quantengravitation. Was können wir von einer Quantentheorie der Gravitation erwarten? Zunächst müss- te sie alle bisherigen experimentellen Tests bestehen. Sie muss uns aber auch Fragen beantworten, die über unser bisheriges Verständnis weit hinausgehen: Was geschah beim Urknall? Was sind die Quanten des Gravitationsfeldes? Ist vielleicht die Geometrie der Raumzeit selbst gequantelt, gibt es gleichsam kleinste Raumatome? Was geschieht im Inneren eines schwarzen Lochs? Meine Doktorarbeit beschäftigt sich mit nur einem Ansatz, diese Fragen zu beant- worten, mit der loop quantum gravity wie die Theorie auf Englisch heißt. An erster Stelle steht die Frage: Gelingt der Übergang zur bekannten Physik? Dafür braucht es geeignetes mathematisches Handwerkszeug. Meine Doktorarbeit entwickelt solches v Werkzeug, und untersucht den klassischen Grenzfall der Theorie. Ich kann zeigen, dass die Schleifentheorie im klassischen Limes sich als Vielteilchentheorie deuten lässt. Die zugehörigen Punktteilchen bewegen sich allerdings nicht in Raum und Zeit, sondern leben in einem zweidimensionalen komplexen Vektorraum, im Raume der Spinoren. RÉSUMÉ La gravité quantique à boucles est une théorie candidate à la description unifiée de la relativité générale et de la mécanique quantique à l’échelle de Planck. Cette théorie peut être formulée de deux manières. L’approche canonique, d’une part, cherche à ré- soudre l’équation de Wheeler–DeWitt et à définir les états physiques. L’approche par les écumes de spins, d’autre
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