In this plenary talk I will review the attempts to formulate 4d quantum gravity as a sum over histories in such a way that computer simulations can be performed. I will report on computer simulations of a quantum universe with a positive cosmological constant as well as a quantum universe where test matter is included.

In the last couple of years, several simple but physically interesting models were solved. Solutions led to concrete and detailed realizations of a number of ideas that have been heuristically expected for decades. There were also some surprises. These analyses suggest viewpoints and strategies for full quantum gravity. I will summarize some of them in broad terms.

Several models have been defined to study the broad framework of loop quantum gravity. We present a common perspective for cosmology where configurations are close to being isotropic. Effective techniques are then used to illustrate important features of the semiclassical limit and to show examples of correction terms to Einstein's equation. Applications include corrections to the Newton potential and to cosmological perturbation equations relevant for the CMB power spectrum.

I will review the foundations and recent progress in the use of effective field theory to elucidate the quantum predictions of general relativity.

Recently, a generally covariant formulation of quantum field theory was found which covers axiomatic as well as renormalized perturbative quantum field theory. It will be discussed whether these results can be used for a background independent formulation of quantum gravity.

General Relativity is a deterministic theory with non-fixed causal structure. Quantum Theory is an inherently probabilistic theory with fixed causal structure. A theory of Quantum Gravity must reduce, in appropriate limits, to General Relativity and Quantum Theory. It seems likely, therefore, that Quantum Gravity will inherit the radical aspects of these two less fundamental theories – namely that it will be a probabilistic theory with non-fixed causal structure. In this talk I will present the causaloid formalism which is a general framework for such theories. I will show how Quantum Theory can be formulated in the causaloid framework and provide tentative results for how it may be possible to formulate General Relativity in the framework. Finally, I will discuss the issue of formulating a theory of Quantum Gravity using the causaloid formalism.

The search for a satisfying theory that unifies general relativity with quantum field theory is surely one of the major tasks for physicists in the 21st century. During the last decade, the phenomenology of quantum gravity and string theory has been examined from various points of view, opening new perspectives and testable predictions. I will give a short introduction into these effective models which allow to extend the standard model and include the expected effects of the underlying fundamental theory. I will talk about models with extra dimensions, models with a minimal length scale and those with a deformation of Lorentz invariance. The focus is on observable consequences, such as black hole and graviton production and modifications of standard-model cross-sections.

Nonlocality in quantum mechanics is a foundational concept which has seen renewed recent interest in response to needs in exploiting quantum entanglement as a resource for quantum information processing. Nonlocality may also have a major presence in quantum gravity. In this talk I examine where nonlocality resides in some of the forerunning candidate theories of quantum gravity while trying to place them in relation to the issue of quantum - classical correspondence and the challenge to uncover the microscopic structure from observed macroscopic phenomena. Some authors recently suggest that microscopic locality leads to macroscopic locality and residual random dislocations, placing stochasticity as a price for gaining macroscopic locality. One context where I understand a little how these two issues - nonlocality and stochasticity - originate together is in nonequilibrium statistical mechanics. We see nonlocal dissipation and nonlocal fluctuations (colored noise) arising naturally in the open system dynamics of Langevin and the effectively open system dynamics of Boltzmann. They stem from the two key processes of coarse-graining and backreaction. Another quantity which may provide us with some clue to these two issues is correlation. The hierarchy of successively higher order correlations provides a measure of the gradations of nonlocality. Correlation is also related to fluctuations in one form of the fluctuation-dissipation theorem. If we agree that the primary goal of quantum gravity is to uncover the microscopic structure of spacetime, and if we allow for the possibility that quantizing the macroscopic collective variables of spacetime in general relativity may not yield such a theory, then coarse-graining, backreaction, fluctuations and correlation may play an essential role in such a quest. I also mention some characteristics of emergent theories, and suggest a few avenues how to start this journey to uncover the microscopic structure of spacetime from the known macroscopic features given by the theory of general relativity.

An important aspect of the desired low energy limit of a background independent quantum theory of gravity is to show how locality arises in the emergent low energy theory. We discuss why we should not expect the high energy theory to have our geometric notions of locality. We present two models of a high energy breakdown of locality: a disordered locality perturbation of a flat geometry and a fully pre-geometric system with no notion of locality which has a local ground state.

Group field theories are non-local quantum field theories on group manifolds, and a generalization of matrix models. Having been first introduced in the context of simplicial quantum gravity, have gained attention as being potentially of much interest in the context of loop quantum gravity and spin foam models. After a brief introduction to the group field theory formalism, I review some of the results already obtained in this approach. I will then try to offer a new perspective on how group field theories should be interpreted and used towards a complete theory of quantum gravity. In particular, I will argue that group field theories can represent on the one hand a common unifying framework for loop quantum gravity, spin foam models and simplicial approaches, like quantum Regge calculus and dynamical triangulations, and on the other hand a consistent microscopic description of spacetime considered as a condensed matter system. From this, a novel approach to the issues of the emergence of the continuum and of General Relativity as an effective description of spacetime, in this approximation, is proposed. Finally, I will briefly report on some recent results and work in progress inspired by and supporting this new perspective.

Topological field theories are simple examples of background independent field theories which are non perturbatively renormalizable in the sense that regularization ambiguities have no effect on physical quantities. Three dimensional vacuum general relativity coupled to point particles is an important example of such theory. The corresponding generalization to higher dimensions has been studied recently. In four dimensions, one-dimensional extended objects (strings) are the natural form of matter that couples to four dimensional BF theory. After briefly reviewing these models, we will show how these topological theories of extended objects can accommodate physically interesting degrees of freedom maintaining their topological nature. We will propose a way by which topological theories of 2d world-sheet matter (of the kind presented in the first part of the talk) might be used to construct background independent quantum field theories with local degrees of freedom and no regulator dependence.

We discuss the application of the uniform discretization procedure to handle the dynamics of loop quantum gravity with spherical symmetry.

Free (unconstrained) gravitational initial data variables are known for initial hypersurfaces consisting of two intersecting null hypersurfaces. Recently the Poisson bracket on functions of such data has been obtained. This opens the prospect of a constraint free canonical formulation of general relativity. The reasons for looking for such a formulation, and some of its features will be discussed.

The basic ideas and main results of the asymptotic safety scenario in Quantum Einstein Gravity (QEG) are reviewed and possible implications for the cosmology of the early universe are discussed.

Machine computation is a relatively unutilized tool in quantum gravity, in part because of the enormous scale of the problems which arise, and the corresponding substantial initial investment which must be made to write code for supercomputers. However, the advent of computational frameworks, such as Cactus, is changing this situation, by drastically decreasing the time required to develop code for supercomputers. I will briefly describe the Cactus framework, and present three insights that have arisen from its use. One is with regard to how entropy bounds may arise from discrete gravity, a second regarding how continuum topology may emerge from an underlying causal set, and third how the nature of the spectrum of the Ashtekar-Lewandowski volume operator of Loop Quantum Gravity depends crucially on the nature of the embedding of a spin network vertex.

It has recently been possible to begin the computation of n-points functions in loop quantum gravity. I review the basic ideas and the present state of these calculations. The Barrett-Crane vertex appears to yield some n-point functions with the correct low-energy limit, but there are also indications of a wrong behavior. The problem can be traced to the way intertwiner quantum numbers are treated in the Barrett-Crane model. This is also the source of the general discrepancy between the Barrett-Crane vertex and LQG. I present a new vertex for loop quantum gravity, introduced in collaboration with Jonathan Engle and Roberto Pereira, which may correct the problem and is fully consistent with the LQG kinematics.

I review several notion of background independence in quantum gravity: full or partial, manifest or not etc., and discuss the evidence for and against taking the notion seriously as a guide for research. I review the current status in string theory with regard to this question and discuss the extent to which existing non-perturbative formulations of string theory are background independent.

Histories-based forms of quantum mechanics seem better suited to the needs of quantum gravity than the more familiar alternative based on state-vectors and selfadjoint operators. Such a re-casting of the theory opens up the possibility to dispense with "external observers" by introducing the concept of an "event" and locating the predictive content of the formalism in the "preclusion" of certain events. The most straightforward such preclusion rule engenders contradictions which are fatal if one sticks with classical logic, but which can be accommodated by an "anhomomorphic logic" that effectively identifies reality with a Z_2 valued function on the space of possible events.

Our goal is to contribute to the development of a background-independent, non-perturbative approach to quantization of the gravitational field based on the conformal and projective structures. Physically the conformal structure is determined by the local behavior of null wave-fronts and rays; and the projective structure by the local behavior of freely falling massive test particles, respectively. In general relativity, these structures may be taken as fundamental, and the pseudo-Riemannian metric and affine connection derived from them. Various initial value problems in GR may be reformulated on the basis of the conformal-projective breakup-- in particular, null-initial value problems and the 2+2 decomposition of the field equations-- with the aim of investigating how best to isolate the two degrees of freedom of the gravitational field, a question of crucial importance for their quantization. The quantum of action sets limits on the co-measurability of various physically measurable quantities, and thereby determines their commutation relations. Hence, the co-measurability of quantities derived from the conformal and projective structures, such as the conformal 2-structure, will be analyzed as a heuristic guide to their quantization. We have already derived the connection and metric from a Palatini-type variational principle utilizing the conformal and projective structures, and will explore this and various other variational principles that could form the starting point for a Feynman type quantization of the gravitational field.

We describe the conceptual and mathematical setup of Loop Quantum Gravity (LQG).

Last modified 31 March 2008.