Abstract:
The way one quantizes the kinematical structure of Loop Quantum Gravity is deeply connected to category theory. Fundamental is the notion of the path groupoid, i.e. the collection of all paths in a manifold. In this language e.g. connections, gauge transformations and diffeomorphisms arise as functors, natural transformations and (path groupoid) automorphisms. In this talk it will be shown how similar constructions can be carried out for any sufficiently well-behaved groupoid, and that the choice of groupoid, which only contains combinatorial information about paths, already determines analytical properties such as a topology and a measure (the analogue of the Ashtekar-Isham-Lewandowski measure) on quantum configuration space. Both are automatically invariant under the action of gauge transformations and groupoid automorphisms (which play the role of "diffeomorphisms" in this context). The uniqueness of measure and topology will be discussed. Examples for these structures are, besides LQG, also lattice gauge theory and Chern-Simons theory.

Abstract:
For the quantized linear scalar field on Friedman-Robertson-Walker spacetimes, states of low energy are a well-motivated class of reference states. The low-energy property is localized near some value of the cosmological time-parameter. We present calculations of the relative particle production between a state of low energy at early time and another such state at later time. In an exponentially expanding universe, we find that the particle production may show oscillations with respect to the energy modes. The basis of the method for calculating the relative particle production is, in contrast to previously investigated approaches, completely rigorous. Approximations are only used at the level of numerical calculation.

Abstract:
We propose a procedure to obtain a well-defined expectation value for the stress tensor of quantized Dirac fields on globally hyperbolic curved backgrounds. The result is obtained by both using a modified version of the classical stress tensor and employing point-splitting regularization by means of an Hadamard bidistribution, thereby obtaining finite (i.e. smooth) expectation values for all Hadamard states. The hereby obtained expectation values are shown to fulfill the first four of Wald's axioms, thus providing an adequate source term for the semiclassical Einstein equation. The quantum trace anomaly is computed, reproducing known results in a conceptually clear, rigorous and swift way.