The basic problem of quantum cosmology is the definition of the quantum state of the universe, with appropriate boundary conditions on Riemannian three-geometries. This paper describes recent progress in the corresponding analysis of quantum amplitudes for Euclidean Maxwell theory and linearized gravity. Within the framework of FaddeevPopov formalism and zeta-function regularization, various ch...

We report on a new approach to the calculation of Chern-Simons theory expectation values, using the mathematical underpinnings of loop quantum gravity, as well as the Duflo map, a quantization map for functions on Lie algebras. These new developments can be used in the quantum theory for certain types of black hole horizons, and they may offer new insights for loop quantum gravity, Chern-Simons...

Quantum geometrodynamics is canonical quantum gravity with the three-metric as the configuration variable. Its central equation is the Wheeler–DeWitt equation. Here I give an overview of the status of this approach. The issues discussed include the problem of time, the relation to the covariant theory, the semiclassical approximation as well as applications to black holes and cosmology. I concl...

Given a symmetry group acting on a principal fibre bundle, symmetric states of the quantum theory of a diffeomorphism invariant theory of connections on this fibre bundle are defined. These symmetric states, equipped with a scalar product derived from the AshtekarLewandowski measure for loop quantum gravity, form a Hilbert space of their own. Restriction to this Hilbert space yields a quantum s...

Using the coherent-state representation we show that the classical Einstein equation for the FRW cosmological model with a general minimal scalar field can be derived from the semiclassical quantum Einstein equation. E-Mail: [email protected] E-Mail: [email protected] E-Mail: [email protected] 1 Canonical quantum gravity has been initiated by DeWitt in the seminal paper [1]. As ...

All attempts to quantize gravity face several difficult problems. Among these problems are: (i) metric positivity (positivity of the spatial distance between distinct points), (ii) the presence of anomalies (partial second-class nature of the quantum constraints), and (iii) perturbative nonrenormalizability (the need for infinitely many distinct counterterms). In this report, a relatively nonte...

It is a general belief that the existence of an initial singularity can be remove through the employment of a quantum theory of gravity. However, there is no consistent theory of gravity until now, and in this sense the problem of the initial singularity remains of actuality. It is well known that is possible to construct a quantum model for the universe as a whole, through the Wheeler-DeWitt (...

Quantum gravity is analyzed from the viewpoint of the renormalization group. The analysis is based on methods introduced by J. Polchinski concerning the perturbative renormalization with flow equations. In the first part of this work, the program of renormalization with flow equations is reviewed and then extended to effective field theories that have a finite UV cutoff. This is done for a scal...