This series consists of talks in the area of Quantum Gravity.
There is a deep relation between Loop Quantum Gravity and notions from category theory, which have been pointed out by many researchers, such as Baez or Velhinho. Concepts like holonomies, connections and gauge transformations can be naturally formulated in that language. In this formulation, the (spatial) diffeomorphisms appear as the path grouopid automorphisms. We investigate the effect of extending the diffeomorphisms to all such automorphisms, which can be viewed as \"distributional diffeomorphisms\".
In 3d quantum gravity, Planck's constant, the Planck length and the cosmological constant control the lack of (co)-commutativity of quantities like angular momenta, momenta and postion coordinates. I will explain this statement, using the quantum groups which arise in the 3d quantum gravity but avoiding technical details. The non-commutative structures in 3d quantum gravity are quite different from those in the deformed version of special relativity desribed by the kappa-Poincare group, but can be related to the latter by an operation called semi-dualisation.
An ingredient in recent discussions of curvature singularity avoidance in quantum gravity is the "inverse scale factor" operator and its generalizations. I describe a general lattice origin of this idea, and show how it applies to the Coulomb singularity in quantum mechanics, and more generally to lattice formulations of quantum gravity. The example also demonstrates that a lattice discretized Schrodinger or Wheeler-DeWitt equation is computationally equivalent to the so called "polymer"
quantization derived from loop quantum gravity.
It is a standard axiom of quantum mechanics that the Hamiltonian H must be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary. In this talk we examine an alternative formulation of quantum mechanics in which the conventional requirement of Hermiticity is replaced by the more general and physical condition of space- time reflection (PT) symmetry. We show that if the PT symmetry of H is unbroken, Then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are $H=p^2+ix^3$ and $H=p^2-x^4$.
Space-time measurements and gravitational experiments are made by the mutual relations between objects, fields, particles etc... Any operationally meaningful assertion about spacetime is therefore intrinsic to the degrees of freedom of the matter (i.e. non-gravitational) fields and concepts such as ``locality'' and ``proximity'' should, at least in principle, be definible entirely within the dynamics of the matter fields. We propose to consider the regions of space just as general ``subsystems''.
We calculate analytically the highly damped quasinormal mode spectra of generic single-horizon black holes using the rigorous WKB techniques of Andersson and Howls. We thereby provide a firm foundation for previous analysis, and point out some of their possible limitations. The numerical coefficient in the real part of the highly damped frequency is generically determined by the behavior of coupling of the perturbation to the gravitational field near the origin, as expressed in tortoise coordinates.
The phenomenology of quantum gravity can be examined even though the underlying theory is not yet fully understood. Effective extensions of the standard model allow us to study specific features, such as the existence of extra dimensions or a minimal length scale. I will talk about some applications of this approach which can be used to make predictions for particle- and astrophysics, and fill in some blanks in the puzzle of quantum gravity. A central point of this investigations is the physics of black holes.