Table of contents

The formulation of general relativity discovered by Ashtekar (1986, 1987) and the recent results obtained in non-perturbative quantum gravity using loop-space techniques are reviewed. The new formulation is based on the choice of a set of Lagrangian (and Hamiltonian) variables, instead of the spacetime metric. In terms of these new variables, the dynamical equations are remarkably simplified and a structural identity between general relativity and the Yang-Mills theories is revealed. The formalism has proven to be useful in numerous problems in gravitational physics. In quantum gravity, the new formalism has overcome long-standing difficulties and led to unexpected results. A nonperturbative approach to quantum theory has been constructed in terms of the Wilson loops of the Ashtekar connection. This approach, denoted as loop-space representation, has led to the complete solution of the quantum diffeomorphism constraint in terms of knot states, to the discovery of an infinite-dimensional class of solutions to the quantum gravitational dynamics, and to certain surprising indications on the existence of a discrete structure of spacetime around the Planck length. These results are presented in a compact self-contained form. The basic Ashtekar formalism is presented and its applications are outlined. The loop-space representation and the non-perturbative knot states of quantum gravity are described in detail, with particular regard to their physical interpretation and to the information they may provide on the microstructure of spacetime.

The authors give a brief review of some of the recent developments in W gravity coupled to matter theories. They then extend the discussion to the case of minimal W conformal field theory, realized in terms of free bosons with background charges, and use this action to investigate some aspects of the anomaly structure and the covariantization of W gravity.

A metric-affine generalization of the Riemannian geometry is considered. In the torsion-free case the only characteristic object indicating the deviation from the Riemannian spacetime is a non-metricity tensor. Spacetime geometry with a generalized Weyl vector non-metricity is proposed and characteristic types of this geometry are systematically studied. Lagrangians and associated equations of motion of spinor and electromagnetic fields on the background of the spacetime with vector non-metricity (SVN) are considered in detail. The one-loop approximation in quantizing the non-metricity (metron) and electromagnetic fields on the Minkowskian background is treated to estimate the renormalized metron constant.

The author obtains cosmological solutions of the low energy heterotic string assuming Calabi-Yau compactification with non-minimal curvature terms (R²_ABCD-4R²_AB+R²). In particular he addresses the problem of which initial conditions to choose, since on emergence from the quantum gravity era, they are as yet undeterminable. He finds that specific constraints are necessary on the initial-value phase space to be compatible with acceptable late-time behaviour.

A very general solution of Einstein-Maxwell equations, of the form g_ab= eta _ab-2Vk_ak_b has been obtained. The congruence k_a is null, geodesic and shear-free and the 4-current J^a is null. The Debney-Kerr-Schild solution and several other known solutions have been derived as particular cases.

Through a detailed calculation of the propagator of a free massive scalar particle, the authors show that the Batalin-Fradkin-Vilkovisky and Faddeev path integrals are different. The analysis also provides another counter-example to the usual statement of the Fradkin-Vilkovisky theorem.

The authors separate the linearized general fourth-order field equation into three second-order normal equations. The special coordinate condition of Teyssandier (1988) is not used; rather the normal harmonic coordinate condition is used. As an example, the special case of a static spherically symmetric mass distribution is discussed in detail. The resulting metric g_ii is obtained in a different gauge from that in Teyssandier. Some conclusions are also discussed.

The authors describe an application of the method of the effective potential to a model theory of gravity with independent vierbein and O(1, 3) connection. Adopting a kind of mean field approximation, they find that a Coleman-Weinberg effects occurs: radiative corrections due to loops of the gauge field generate an effective potential for the vierbein whose absolute minimum corresponds to Minkowski space.