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Fermions in Curved Spacetimes fermions in curved spacetimes Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt dem Rat der Physikalisch-Astronomischen Fakultät der Friedrich-Schiller-Universität Jena von M. Sc. Stefan Lippoldt geboren am 31.07.1990 in Jena Gutachter 1. Prof. Dr. Holger Gies (Friedrich-Schiller-Universität Jena) 2. Prof. Dr. Martin Reuter (Johannes Gutenberg-Universität Mainz) 3. Prof. Dr. Roberto Percacci (SISSA, Trieste) Tag der Disputation: 21. Januar 2016 Zusammenfassung In dieser Promotionsschrift untersuchen wir eine Formulierung von Dirac-Fermionen in gekrüm- mten Raumzeiten, die sowohl eine allgemeine Koordinateninvarianz, sowie eine lokale Spinbasen- Invarianz besitzt. Wir beleuchten die Vorteile der Spinbasen-invarianten Formulierung aus konzeptioneller und praktischer Sicht. Dies legt nahe, dass die lokale Spinbasen-Invarianz auf die Liste der (effektiven) Eigenschaften der (Quanten-) Gravitation gehört. Wir finden Argumente für diesen Standpunkt durch die Konstruktion einer globalen Realisierung der Clifford-Algebra auf der 2-Sphäre, eine Struktur die es im (nicht Spinbasen-invarianten) Vielbein-Formalismus nicht gibt. Die natürlichen Variablen für diese Formulierung sind die raumzeitabhängigen Dirac-Matrizen, welche die Clifford-Algebra erfüllen müssen. Insbesondere benötigt man kein “Achsenkreuz”, d. h. ein Vielbein. Wir decken die versteckte Spinbasen-Invarianz des Vielbein- Formalismus auf. Explizite Formeln für den Spin-Zusammenhang in Abhängigkeit der Dirac- Matrizen werden angegeben. Dieser Zusammenhang besteht aus einem kanonischen Teil, der vollständig durch die Dirac-Matrizen fixiert ist, und einem freien Teil, der als Spin-Torsion in- terpretiert werden kann. Die übliche Lorentz-symmetrische Eichung für das Vielbein wird auf die Dirac-Matrizen verallgemeinert, auch für Metriken die nicht linear verbunden sind. Nach bestimmten Kriterien bildet sie die einfachste Eichung – das zeigt warum diese Eichung so nützlich ist. Mit Hilfe des Spinbasen-Formalismus konstruieren wir eine Feldtheorie für quan- tisierte Gravitation and Materiefelder und zeigen, dass die Quantisierung der Metrik und der Materiefelder genügt. Für feldtheoretische Zugänge zur Quantengravitation ist diese Beobach- tung von besonderer Relevanz, da sie ein rein metrikbasiertes Quantisierungsschema, auch in der Anwesenheit von Fermionen, nahelegt. Daher untersuchen wir im zweiten Teil dieser Arbeit die Eich- und Feldparametrisierungsab- hängigkeit des Renormierungsgruppenflusses in der Nähe von nicht-Gaußschen Fixpunkten in Quantengravitationstheorien. Während physikalische Observablen unabhängig von solchen De- tails der Rechnung sind, benötigt man für die Konstruktion von Quantengravitationstheorien typischerweise “off-shell” Größen wie Betafunktionen und erzeugende Funktionale. Daher begeg- net man potenziellen Stabilitätsproblemen bei diesen verallgemeinerten Parametrisierungen. Wir analysieren eine zwei-Parameter Klasse von kovarianten Eichbedingungen, die Rolle der im- pulsabhängigen Feldreskalierung und eine Klasse von Feldparametrisierungen. Die Familie der Eichungen beinhaltet eine (nicht-harmonische) Verallgemeinerung der harmonischen Eichung (De-Donder-Eichung), dabei ist die letztere besonders nützlich für die Analyse von Gravitation- swellen, die vermutlich die asymptotischen Zustände der Quantengravitation sind. Mit Hilfe des Prinzips der kleinsten Sensitivität identifizieren wir über das Produkt von Gravitations- und kosmologischer Konstante stationäre Punkte in diesem Parameterraum, diese zeigen eine beachtliche Insensitivität gegenüber den Details der Parametrisierung. In den insensitivsten Fällen zeigt das quantisierte Gravitationssystem einen nicht-Gaußschen ultravioletten, stabilen Fixpunkt. Das liefert weiteren Rückhalt für asymptotisch sichere Quantengravitation. Im stationären Bereich der Parametrisierung des exponentiellen Splits weist der Renormierungs- gruppenfluss einige bemerkenswerte Eigenschaften auf: (1) eine mögliche Abhängigkeit vom übrigen Eichparameter fällt heraus, dies impliziert einen vergrößerten Grad an Eichinvarianz, (2) der Renormierungsgruppenfluss wird besonders einfach, sodass das Phasendiagramm in der Ebene der Gravitations- und kosmologischen Konstante analytisch berechnet werden kann, (3) im Fluss tauchen keine Singularitäten auf, sodass eine große Klasse von Renormierungs- gruppentrajektorien (inklusive derer mit klassischem Bereich) auf beliebig große und kleine Skalen ausgedehnt werden kann, (4) die ultravioletten, kritischen Exponenten sind reell und nah bei ihren kanonischen Gegenstücken, und (5) es zeigen sich Anzeichen, dass das Szenario der asymptotischen Sicherheit sich nicht direkt auf Dimensionen viel größer als d = 4 fortsetzen lässt. Das letzte Kapitel dieser Promotion widmet sich den Fermionen in gekrümmten Hinter- grundraumzeiten und im Besonderen der katalysierten Symmetriebrechung. Dieses Phänomen entsteht durch eine parametrische Vergrößerung der kritischen Fluktuationen unabhängig von der Kopplungsstärke. Aus Molekularfeldtheoriestudien ist bekannt, dass symmetriebrechende fermionische langreichweitige Fluktuationen so eine Vergrößerung auf negativ gekrümmten Räumen aufweisen. Wir untersuchen gravitative Katalyse aus der Sicht der funktionalen Renormierungsgruppe mit Hilfe des 3d Gross-Neveu-Modells als ein spezielles Beispiel. Wir beobachten gravitative Katalyse hin zu einer Phase gebrochener diskreter chiraler Symmetrie, sowohl auf einer maximal symmetrischen Raumzeit (AdS) als auch auf einer rein räumlich gekrümmten Mannigfaltigkeit (Lobachevsky Ebene) mit konstanter negativer Krümmung. Das resultierende Bild der gravitativen Katalyse, das wir aus dem Renormierungsgruppenfluss er- halten haben, ist eng verbunden mit dem der magnetischen Katalyse. Als eine Anwendung schätzen wir die nötige Krümmung für ein subkritisches System endlicher Länge ab, um eine gravitativ katalysierte Massenlücke auszubilden. Summary In this thesis we study a formulation of Dirac fermions in curved spacetime that respects general coordinate invariance as well as invariance under local spin base transformations. We emphasize the advantages of the spin base invariant formalism both from a conceptual as well as from a practical viewpoint. This suggests that local spin base invariance should be added to the list of (effective) properties of (quantum) gravity theories. We find support for this viewpoint by the explicit construction of a global realization of the Clifford algebra on a 2-sphere which is impossible in the spin-base non-invariant vielbein formalism. The natural variables for this formulation are spacetime-dependent Dirac matrices subject to the Clifford-algebra constraint. In particular, a coframe, i.e. vielbein field is not required. We disclose the hidden spinbase invariance of the vielbein formalism. Explicit formulas for the spin connection as a function of the Dirac matrices are found. This connection consists of a canonical part that is completely fixed in terms of the Dirac matrices and a free part that can be interpreted as spin torsion.The common Lorentz symmetric gauge for the vielbein is constructed for the Dirac matrices, even for metrics which are not linearly connected. Under certain criteria, it constitutes the simplest possible gauge, demonstrating why this gauge is so useful. Using the spin base formulation for building a field theory of quantized gravity and matter fields, we show that it sufficesto quantize the metric and the matter fields. This observation is of particular relevance forfield theory approaches to quantum gravity, as it can serve for a purely metric-based quantization scheme for gravity even in the presence of fermions. Hence, in the second part of this thesis we critically examine the gauge, and the field- parametrization dependence of renormalization group flows in the vicinity of non-Gaußian fixed points in quantum gravity. While physical observables are independent of such calculational specifications, the construction of quantum gravity field theories typically relies onoff-shell quantities such as beta functions and generating functionals. Thus one faces potential stability issues with regard to such generalized parametrizations. We analyze a two-parameter class of covariant gauge conditions, the role of momentum-dependent field rescalings and a class of field parametrizations. The family of gauges includes a (non-harmonic) generalization of the harmonic gauge (De-Donder gauge), the latter being particularly useful for the analysis of gravitational waves which presumably are the asymptotic states of quantum gravity. Using the product of Newton and cosmological constant as an indicator, the principle of minimum sensitivity identifies stationary points in this parametrization space which show a remarkable insensitivity to the details of the parametrization choices. In the most insensitive cases, the quantized gravity system exhibits a non-Gaußian ultraviolet stable fixed point, lending further support to asymptotically free quantum gravity. In the stationary regime of the parametriza- tion based on the exponential split, the resulting renormalization group flow exhibits several remarkable properties: (1) a possible dependence on the residual gauge parameter drops out implying an enhanced degree of gauge invariance, (2) the renormalization group flow becomes particularly simple, so that the phase diagram in the plane of Newton and cosmological con- stant can be computed analytically, (3) no singularities arise
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