General Relativity & Quantum Cosmology

At low energy and small curvature, general relativity has the form of an effective field theory. I will describe the structure of the effective field theory, and show how it can be used to calculate low energy quantum effects.

Category theory is a general language for describing things and processes - called "objects" and "morphisms". In this language, many counterintuitive features of quantum theory turn out to be properties shared by the category of Hilbert spaces and the category of cobordisms, in which objects are choices of "space" and emorphisms are choices of "spacetime". This striking fact suggests that "n-categories with duals" are a promising language for a quantum theory of spacetime. We sketch the historical development of these ideas from Feynman diagrams to string theory, topological quantum field theory, spin networks and spin foams, and especially recent work on open-closed string theory, 3d quantum gravity coupled to point particles, and 4d BF theory coupled to strings.

While modern theories lavishly invoke several spatial dimensions within models that seek to unify relativity theory and quantum mechanics, none seems to consider the possibility that a yet-unfamiliar aspect of time may do the work. I introduce the notion of Becoming and then consider its consequences for physical theory. Becoming portrays a possible aspect of time that is "curled" very much like the extra spatial dimensions in superstring theories. Within the resulting picture of spacetime, some fundamental aspects of quantum mechanics, special and general relativity, thermodynamics and modern cosmology fit in very naturally. The proposed model is not yet a scientific theory as it still lacks a rigorous formalism and experimental predictions, yet it points out an entire family of possible theories that merit serious consideration.

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''. By writing the Hilbert space of the matter fields as a generic tensor product of subsystems we analyse the evolution of a state vector on an information theoretical basis and discuss general principles to recover a posteriori the usual space-time relations. We apply such principles to generic interacting second quantized models with a finite number of fermionic degrees of freedom. Finally, we discuss the possible role of gravity in this framework.

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. This fact makes it difficult to understand how the (in)famous ln(3) could be related to the quantum gravitational microstates near the horizon.

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. I will comment on possible ways to proceed and on the missing pieces I find most important to look for.

Preon models enjoyed considerable popularity during the early 1980s, but have seen little progress since then. I will describe a correspondence between one of the more successful preon models and a simple game involving the twisting and braiding of ribbons, subject to straightforward topological conditions. This reproduces the fermions and gauge bosons of the standard model, as well as the electromagnetic, weak and colour interactions. The prospect that such structures may occur naturally within Loop Quantum Gravity will be discussed

Noncommutative geometry is a more general formulation of geometry that does not require coordinates to commute. As such it unifies quantum theory and geometry and should appear in any effective theory of quantum gravity. In this general talk we present quantum groups as a microcosm of this unification in the same way that Lie groups are a microcosm of usual geometry, and give a flavour of some of the deeper insights they provide. One of them is the ability to interchange the roles of quantum theory and gravity by `arrow reversal'. Another is that noncommutative spaces typically carry a canonical 1-parameter evolution or intrinsic time created from the fundamental conflict between noncommuting coordinates and differential calculus. In physical terms one could say that quantising space typically has an anomaly for the spatial translation group and this forces the system to evolve. We give an example where we derive Schroedinger's equation in this way.

The causal set -- mathematically a finitary partial order -- is a candidate discrete substratum for spacetime. I will introduce this idea and describe some aspects of causal set kinematics, dynamics, and phenomenology, including, as time permits, a notion of fractal dimension, a (classical) dynamics of stochastic growth, and an idea for explaining some of the puzzling large numbers of cosmology. I will also mention some general insights that have emerged from the study of causal sets, the most recent one concerning the role of intermediate length-scales in discrete spacetime theories.