Background independent quantizations: the scalar field I

We are concerned with the issue of quantization of a scalar field in a diffeomorphism invariant manner. We apply the method used in Loop Quantum Gravity. It relies on the specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the `quantum' polymer *-star algebra and looking for positive linear functionals, called states. The assumed in our paper homeomorphism invariance allows to determine a complete class of the states. Except one, all of them are new. In this letter we outline the main steps and conclusions, and present the results: the GNS representations, characterization of those states which lead to essentially self adjoint momentum operators (unbounded), identification of the equivalence classes of the representations as well as of the irreducible ones. The algebra and topology of the problem, the derivation, all the technical details and more are contained in the paper-part II.Comment: 13 pages, minor corrections were made in the revised versio

Automorphism covariant representations of the holonomy-flux *-algebra

We continue an analysis of representations of cylindrical functions and fluxes which are commonly used as elementary variables of Loop Quantum Gravity. We consider an arbitrary principal bundle of a compact connected structure group and following Sahlmann's ideas define a holonomy-flux *-algebra whose elements correspond to the elementary variables. There exists a natural action of automorphisms of the bundle on the algebra; the action generalizes the action of analytic diffeomorphisms and gauge transformations on the algebra considered in earlier works. We define the automorphism covariance of a *-representation of the algebra on a Hilbert space and prove that the only Hilbert space admitting such a representation is a direct sum of spaces L^2 given by a unique measure on the space of generalized connections. This result is a generalization of our previous work (Class. Quantum. Grav. 20 (2003) 3543-3567, gr-qc/0302059) where we assumed that the principal bundle is trivial, and its base manifold is R^d.Comment: 34 pages, 1 figure, LaTeX2e, minor clarifying remark

Background independent quantizations: the scalar field II

We are concerned with the issue of quantization of a scalar field in a diffeomorphism invariant manner. We apply the method used in Loop Quantum Gravity. It relies on the specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the `quantum' polymer *-star algebra and looking for positive linear functionals, called states. Assumed in our paper homeomorphism invariance allows to derive the complete class of the states. They are determined by the homeomorphism invariant states defined on the CW-complex *-algebra. The corresponding GNS representations of the polymer *-algebra and their self-adjoint extensions are derived, the equivalence classes are found and invariant subspaces characterized. In the preceding letter (the part I) we outlined those results. Here, we present the technical details.Comment: 51 pages, LaTeX, no figures, revised versio

Hilbert space built over connections with a non-compact structure group

Quantization of general relativity in terms of SL(2,C)-connections (i.e. in terms of the complex Ashtekar variables) is technically difficult because of the non-compactness of SL(2,C). The difficulties concern the construction of a diffeomorphism invariant Hilbert space structure on the space of cylindrical functions of the connections. We present here a 'toy' model of such a Hilbert space built over connections whose structure group is the group of real numbers. We show that in the case of any Hilbert space built analogously over connections with any non-compact structure group (this includes some models presented in the literature) there exists an obstacle which does not allow to define a *-representation of cylindrical functions on the Hilbert space by the multiplication map which is the only known way to define a diffeomorphism invariant representation of the functions.Comment: 45 pages, no figures, LaTeX2e, the discussion of results extende

Spherically Symmetric Quantum Geometry: Hamiltonian Constraint

Variables adapted to the quantum dynamics of spherically symmetric models are introduced, which further simplify the spherically symmetric volume operator and allow an explicit computation of all matrix elements of the Euclidean and Lorentzian Hamiltonian constraints. The construction fits completely into the general scheme available in loop quantum gravity for the quantization of the full theory as well as symmetric models. This then presents a further consistency check of the whole scheme in inhomogeneous situations, lending further credence to the physical results obtained so far mainly in homogeneous models. New applications in particular of the spherically symmetric model in the context of black hole physics are discussed.Comment: 33 page

Loop Quantum Cosmology

Quantum gravity is expected to be necessary in order to understand situations where classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e. the fact that the backward evolution of a classical space-time inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding space-time is then modified. One particular realization is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. Main effects are introduced into effective classical equations which allow to avoid interpretational problems of quantum theory. They give rise to new kinds of early universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function which allows to extend space-time beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of space-time arising in loop quantum gravity and its application to cosmology sheds new light on more general issues such as time.Comment: 104 pages, 10 figures; online version, containing 6 movies, available at http://relativity.livingreviews.org/Articles/lrr-2005-11