Quantum gravity effects in the cosmic microwave background
One of the most exciting recent developments in physics is the detailed measurement and theoretical description of the cosmic microwave background. As it turns out quantum gravitational effects might be relevant for a complete understanding of this background radiation. A prediction of these possibly measurable quantum gravitational effects within a variety of models has been one focus of my research.
Spectral (zeta) functions:
Many properties of physical systems or Riemannian manifolds are encoded in the spectrum of certain interesting, mostly Laplace-type, differential operators. These properties are analyzed by considering suitable functions of the spectrum. The most relevant examples of such functions are partition sums, functional determinants and the heat-kernel. The most prominent spectral function is the zeta function, which can be related to all the above-mentioned spectral functions and which represents a very intelligent organization of the spectrum. My research focus is on the development and the application of techniques for the analysis of these functions.
The properties of spectral functions strongly depend on the differential operators considered. My main interests are located around the following subjects.
Quantum field theory under the influence of external conditions:
A wide field where spectral functions make their appearance is quantum field theory or quantum mechanics under external conditions. External refers to the fact that the condition is assumed to be known as a function of space and time and that it only appears in the equation of motion of other fields. External conditions considered originate from dielectric media and different kinds of background fields (electric, magnetic and gravitational). Of great importance is the evaluation of effective actions which, in a certain approximation, are given by a functional determinant.
In case the external conditions originate from boundaries present the term Casimir effect is used. In this context the dependence of spectral functions on boundary conditions and the geometry of boundaries is of great relevance. A deeper understanding of very fundamental issues like the question about the sign of the Casimir energy, indicating whether the Casimir effect tends to contract or expand a physical system, are still lacking.
The case where external fields are used to confine a gas of atoms or molecules and to cool them down to very low temperatures of the order of fractions of microkelvins has become extremely relevant in the context of Bose-Einstein condensation. Thermodynamical properties of the gas are of basic interest and conveniently analyzed using partition sums. The so-called microcanonical approach nicely relates to number theory problems, more specifically to problems in the theory of partitions.
Another typical differential operator considered is the Laplace-Beltrami or the Dirac operator on Riemannian manifolds. The structure of spectral functions for strongly elliptic second-order differential operators on a smooth compact Riemannian manifold with a smooth boundary are well understood. However, if the manifold has conical singularities this structure is dramatically altered and remains an area of recent research.