Ashtekar Constraint Surface as Projection of Hilbert-Palatini One

The Hilbert-Palatini (HP) Lagrangian of general relativity being written in terms of selfdual and antiselfdual variables contains Ashtekar Lagrangian (which governs the dynamics of the selfdual sector of the theory on condition that the dynamics of antiselfdual sector is not fixed). We show that nonequivalence of the Ashtekar and HP quantum theories is due to the specific form (of the "loose relation" type) of constraints which relate self- and antiselfdual variables so that the procedure of (canonical) quantisation of such the theory is noncommutative with the procedure of excluding antiselfdual variables.Comment: 9 pages of LaTeX fil

Non-minimal couplings, quantum geometry and black hole entropy

The black hole entropy calculation for type I isolated horizons, based on loop quantum gravity, is extended to include non-minimally coupled scalar fields. Although the non-minimal coupling significantly modifies quantum geometry, the highly non-trivial consistency checks for the emergence of a coherent description of the quantum horizon continue to be met. The resulting expression of black hole entropy now depends also on the scalar field precisely in the fashion predicted by the first law in the classical theory (with the same value of the Barbero-Immirzi parameter as in the case of minimal coupling).Comment: 14 pages, no figures, revtex4. Section III expanded and typos correcte

The weaving of curved geometries

In the physical interpretation of states in non-perturbative loop quantum gravity the so-called weave states play an important role. Until now only weaves representing flat geometries have been introduced explicitly. In this paper the construction of weaves for non-flat geometries is described; in particular, weaves representing the Schwarzschild solution are constructed.Comment: 9 pages, THU-92/2

Quantum geometry and black hole entropy: inclusion of distortion and rotation

Equilibrium states of black holes can be modelled by isolated horizons. If the intrinsic geometry is spherical, they are called type I while if it is axi-symmetric, they are called type II. The detailed theory of geometry of quantum type I horizons and the calculation of their entropy can be generalized to type II, thereby including arbitrary distortions and rotations. The leading term in entropy of large horizons is again given by 1/4th of the horizon area for the same value of the Barbero-Immirzi parameter as in the type I case. Ideas and constructions underlying this extension are summarized.Comment: Text based on parallel talk given at the VI Mexican School on Gravitation and Mathematical Physics: ``Approaches to Quantum Gravity'', held in Playa del Carmen, Mexico, in November of 2004; IGPG preprint number added; metadata abstract correcte

2+1 Gravity without dynamics

A three dimensional generally covariant theory is described that has a 2+1 canonical decomposition in which the Hamiltonian constraint, which generates the dynamics, is absent. Physical observables for the theory are described and the classical and quantum theories are compared with ordinary 2+1 gravity.Comment: 9 page

On diffeomorphism invariance for lattice theories

We consider the role of the diffeomorphism constraint in the quantization of lattice formulations of diffeomorphism invariant theories of connections. It has been argued that in working with abstract lattices, one automatically takes care of the diffeomorphism constraint in the quantum theory. We use two systems in order to show that imposing the diffeomorphism constraint is imperative to obtain a physically acceptable quantum theory. First, we consider $2+1$ gravity where an exact lattice formulation is available. Next, general theories of connections for compact gauge groups are treated, where the quantum theories are known --for both the continuum and the lattice-- and can be compared.Comment: 11 Pages, Revtex, 3 figure

On obtaining classical mechanics from quantum mechanics

Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum mechanical system naturally has the structure of an infinite dimensional symplectic manifold (`quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straight forwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and recovers the linear classical phase space $\mathbb{R}^{\mathrm{2N}}$. We report on how the procedure also allows extraction of non-linear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (Polymer quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality.Comment: revtex4, 24 pages, no figures. In the version 2 certain technical errors in section I-B are corrected, the part on WKB (and section II-B) is removed, discussion of dynamics and semiclassicality is extended and references are added. Accepted for publication on Classical and Quantum Gravit

Spherically Symmetric Quantum Horizons

Isolated horizon conditions specialized to spherical symmetry can be imposed directly at the quantum level. This answers several questions concerning horizon degrees of freedom, which are seen to be related to orientation, and its fluctuations at the kinematical as well as dynamical level. In particular, in the absence of scalar or fermionic matter the horizon area is an approximate quantum observable. Including different kinds of matter fields allows to probe several aspects of the Hamiltonian constraint of quantum geometry that are important in inhomogeneous situations.Comment: 4 pages, RevTeX

The reduced phase space of spherically symmetric Einstein-Maxwell theory including a cosmological constant

We extend here the canonical treatment of spherically symmetric (quantum) gravity to the most simple matter coupling, namely spherically symmetric Maxwell theory with or without a cosmological constant. The quantization is based on the reduced phase space which is coordinatized by the mass and the electric charge as well as their canonically conjugate momenta, whose geometrical interpretation is explored. The dimension of the reduced phase space depends on the topology chosen, quite similar to the case of pure (2+1) gravity. We investigate several conceptual and technical details that might be of interest for full (3+1) gravity. We use the new canonical variables introduced by Ashtekar, which simplifies the analysis tremendously.Comment: 37p, LATE