Nonlinear quantum gravity on the constant mean curvature foliation

A new approach to quantum gravity is presented based on a nonlinear quantization scheme for canonical field theories with an implicitly defined Hamiltonian. The constant mean curvature foliation is employed to eliminate the momentum constraints in canonical general relativity. It is, however, argued that the Hamiltonian constraint may be advantageously retained in the reduced classical system to be quantized. This permits the Hamiltonian constraint equation to be consistently turned into an expectation value equation on quantization that describes the scale factor on each spatial hypersurface characterized by a constant mean exterior curvature. This expectation value equation augments the dynamical quantum evolution of the unconstrained conformal three-geometry with a transverse traceless momentum tensor density. The resulting quantum theory is inherently nonlinear. Nonetheless, it is unitary and free from a nonlocal and implicit description of the Hamiltonian operator. Finally, by imposing additional homogeneity symmetries, a broad class of Bianchi cosmological models are analyzed as nonlinear quantum minisuperspaces in the context of the proposed theory.Comment: 14 pages. Classical and Quantum Gravity (To appear

Semiclassical States in Quantum Cosmology: Bianchi I Coherent States

We study coherent states for Bianchi type I cosmological models, as examples of semiclassical states for time-reparametrization invariant systems. This simple model allows us to study explicitly the relationship between exact semiclassical states in the kinematical Hilbert space and corresponding ones in the physical Hilbert space, which we construct here using the group averaging technique. We find that it is possible to construct good semiclassical physical states by such a procedure in this model; we also discuss the sense in which the original kinematical states may be a good approximation to the physical ones, and the situations in which this is the case. In addition, these models can be deparametrized in a natural way, and we study the effect of time evolution on an "intrinsic" coherent state in the reduced phase space, in order to estimate the time for this state to spread significantly.Comment: 21 pages, 1 figure; Version to be published in CQG; The discussion has been slightly reorganized, two references added, and some typos correcte

An action principle for the quantization of parametric theories and nonlinear quantum cosmology

By parametrizing the action integral for the standard Schrodinger equation we present a derivation of the recently proposed method for quantizing a parametrized theory. The reformulation suggests a natural extension from conventional to nonlinear quantum mechanics. This generalization enables a unitary description of the quantum evolution for a broad class of constrained Hamiltonian systems with a nonlinear kinematic structure. In particular, the new theory is applicable to the quantization of cosmological models where a chosen gravitational degree of freedom acts as geometric time. This is demonstrated explicitly using three cosmological models: the Friedmann universe with a massless scalar field and Bianchi type I and IX models. Based on these investigations, the prospect of further developing the proposed quantization scheme in the context of quantum gravity is discussed.Comment: 14 page

Essential Constants for Spatially Homogeneous Ricci-flat manifolds of dimension 4+1

The present work considers (4+1)-dimensional spatially homogeneous vacuum cosmological models. Exact solutions -- some already existing in the literature, and others believed to be new -- are exhibited. Some of them are the most general for the corresponding Lie group with which each homogeneous slice is endowed, and some others are quite general. The characterization ``general'' is given based on the counting of the essential constants, the line-element of each model must contain; indeed, this is the basic contribution of the work. We give two different ways of calculating the number of essential constants for the simply transitive spatially homogeneous (4+1)-dimensional models. The first uses the initial value theorem; the second uses, through Peano's theorem, the so-called time-dependent automorphism inducing diffeomorphismsComment: 26 Pages, 2 Tables, latex2

Nucleation of (4)R^{(4)}R Brane Universes

The creation of brane universes induced by a totally antisymmetric tensor living in a fixed background spacetime is presented, where a term involving the intrinsic curvature of the brane is considered. A canonical quantum mechanical approach employing Wheeler-DeWitt equation is done. The probability nucleation for the brane is calculated taking into account both an instanton method and a WKB approximation. Some cosmological implications arose from the model are presented.Comment: 19 pages, 2 figure

Nontrivial Dynamics in the Early Stages of Inflation

Inflationary cosmologies, regarded as dynamical systems, have rather simple asymptotic behavior, insofar as the cosmic baldness principle holds. Nevertheless, in the early stages of an inflationary process, the dynamical behavior may be very complex. In this paper, we show how even a simple inflationary scenario, based on Linde's ``chaotic inflation'' proposal, manifests nontrivial dynamical effects such as the breakup of invariant tori, formation of cantori and Arnol'd's diffusion. The relevance of such effects is highlighted by the fact that even the occurrence or not of inflation in a given Universe is dependent upon them.Comment: 26 pages, Latex, 9 Figures available on request, GTCRG-94-1

The Bohm Interpretation of Quantum Cosmology

I make a review on the aplications of the Bohm-De Broglie interpretation of quantum mechanics to quantum cosmology. In the framework of minisuperspaces models, I show how quantum cosmological effects in Bohm's view can avoid the initial singularity, isotropize the Universe, and even be a cause for the present observed acceleration of the Universe. In the general case, we enumerate the possible structures of quantum space and time.Comment: 28 pages, 1 figure, contribution to the James Cushing festschrift to appear in Foundations of Physic

Background Independent Quantum Gravity: A Status Report

The goal of this article is to present an introduction to loop quantum gravity -a background independent, non-perturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to providing a bird's eye view of the present status of the subject, the article should also serve as a vehicle to enter the field and explore it in detail. To aid non-experts, very little is assumed beyond elements of general relativity, gauge theories and quantum field theory. While the article is essentially self-contained, the emphasis is on communicating the underlying ideas and the significance of results rather than on presenting systematic derivations and detailed proofs. (These can be found in the listed references.) The subject can be approached in different ways. We have chosen one which is deeply rooted in well established physics and also has sufficient mathematical precision to ensure that there are no hidden infinities. In order to keep the article to a reasonable size, and to avoid overwhelming non-experts, we have had to leave out several interesting topics, results and viewpoints; this is meant to be an introduction to the subject rather than an exhaustive review of it.Comment: 125 pages, 5 figures (eps format), the final version published in CQ