Quantum Einstein gravity - advancements of heat kernel-based renormalization group studies

The asymptotic safety scenario allows to define a consistent theory of quantized gravity within the framework of quantum field theory. The central conjecture of this scenario is the existence of a non-Gaussian fixed point of the theory's renormalization group flow, that allows to formulate renormalization conditions that render the theory fully predictive. Investigations of this possibility use an exact functional renormalization group equation as a primary non-perturbative tool. This equation implements Wilsonian renormalization group transformations, and is demonstrated to represent a reformulation of the functional integral approach to quantum field theory.rnAs its main result, this thesis develops an algebraic algorithm which allows to systematically construct the renormalization group flow of gauge theories as well as gravity in arbitrary expansion schemes. In particular, it uses off-diagonal heat kernel techniques to efficiently handle the non-minimal differential operators which appear due to gauge symmetries. The central virtue of the algorithm is that no additional simplifications need to be employed, opening the possibility for more systematic investigations of the emergence of non-perturbative phenomena. As a by-product several novel results on the heat kernel expansion of the Laplace operator acting on general gauge bundles are obtained.rnThe constructed algorithm is used to re-derive the renormalization group flow of gravity in the Einstein-Hilbert truncation, showing the manifest background independence of the results. The well-studied Einstein-Hilbert case is further advanced by taking the effect of a running ghost field renormalization on the gravitational coupling constants into account. A detailed numerical analysis reveals a further stabilization of the found non-Gaussian fixed point.rnFinally, the proposed algorithm is applied to the case of higher derivative gravity including all curvature squared interactions. This establishes an improvement of existing computations, taking the independent running of the Euler topological term into account. Known perturbative results are reproduced in this case from the renormalization group equation, identifying however a unique non-Gaussian fixed point.rn

Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

EPR spectroscopic investigation of membrane protein structure and folding on light harvesting complex LHCIIb

Structure and folding of membrane proteins are important issues in molecular and cell biology. In this work new approaches are developed to characterize the structure of folded, unfolded and partially folded membrane proteins. These approaches combine site-directed spin labeling and pulse EPR techniques. The major plant light harvesting complex LHCIIb was used as a model system. Measurements of longitudinal and transversal relaxation times of electron spins and of hyperfine couplings to neighboring nuclei by electron spin echo envelope modulation(ESEEM) provide complementary information about the local environment of a single spin label. By double electron electron resonance (DEER) distances in the nanometer range between two spin labels can be determined. The results are analyzed in terms of relative water accessibilities of different sites in LHCIIb and its geometry. They reveal conformational changes as a function of micelle composition. This arsenal of methods is used to study protein folding during the LHCIIb self assembly and a spatially and temporally resolved folding model is proposed. The approaches developed here are potentially applicable for studying structure and folding of any protein or other self-assembling structure if site-directed spin labeling is feasible and the time scale of folding is accessible to freeze-quench techniques.

Structural analysis of the major lightharvesting complex II by electron paramagnetic resonance (EPR) : comparison between monomers and trimers

Der Haupt-Lichtsammenkomplex II (LHCII) höherer Pflanzen ist das häufigsternMembranprotein der Welt und in die chloroplastidäre Thylakoidmembran integriert. DerrnLHCII kann als Modellsystem genutzt werden, um die Funktionsweise vonrnMembranproteinen besser zu verstehen, da 96 % seiner Struktur kristallografisch aufgelöstrnist und er in rekombinanter Form in vitro rückgefaltet werden kann. Hierbei entsteht einrnvoll funktionaler Protein-Pigment.Komplex, der nahezu identisch mit der in vivo Varianternist.rnElektronenparamagnetischen Resonanz (EPR) Spektroskopie ist eine hoch sensitive undrnideal geeignete Methode, um die Strukturdynamik von Proteinen zu untersuchen. Hierzurnist eine ortsspezifische Markierung mit Spinsonden notwendig, die kovalent an Cysteinernbinden. Möglich wird dies, indem sorgfältig ausgewählte Aminosäuren gegen Cysteinerngetauscht werden, ohne dass die Funktionsweise des LHCII beeinträchtigt wird.rnIm Rahmen dieser Arbeit wurden die Stabilität des verwendeten Spinmarkers und diernProbenqualität verbessert, indem alle Schritte der Probenpräparation untersucht wurden.rnMithilfe dieser Erkenntnisse konnte sowohl die Gefahr einer Proteinaggregation als auchrnein Verlust des EPR Signals deutlich vermindert werden. In Kombination mit derrngleichzeitigen Etablierung des Q-Band EPR können nun deutlich geringer konzentrierternProben zuverlässig vermessen werden. Darüber hinaus wurde eine reproduzierbarernMethode entwickelt, um heterogene Trimere herzustellen. Diese bestehen aus einemrndoppelt markierten Monomer und zwei unmarkierten Monomeren und erlauben es, diernkristallografisch unvollständig aufgelöste N-terminale Domäne im monomeren undrntrimeren Assemblierungsgrad zu untersuchen. Die Ergebnisse konnten einerseits diernVermutung bestätigen, dass diese Domäne im Vergleich zum starren Proteinkern sehrrnflexibel ist und andererseits, dass sie in Monomeren noch mobiler ist als in Trimeren.rnZudem wurde die lumenale Schleifenregion bei unterschiedlichen pH Werten undrnvariierender Pigmentzusammensetzung untersucht, da dieser Bereich sehr kontroversrndiskutiert wird. Die Messergebnisse offenbarten, dass diese Region starre und flexiblerernSektionen aufweist. Während der pH Wert keinen Einfluss auf die Konformation hatte,rnzeigte sich, dass die Abwesenheit von Neoxanthin zu einer Änderung der Konformationrnführt. Weiterführende Analysen der strukturellen Dynamik des LHCII in einerrnLipidmembran konnten hingegen nicht durchgeführt werden, da dies eine gerichteternInsertion des rückgefalteten Proteins in Liposomen erfordert, was trotz intensiverrnVersuche nicht zum Erfolg führte.