Towards new background independent representations for Loop Quantum Gravity

Recently, uniqueness theorems were constructed for the representation used in Loop Quantum Gravity. We explore the existence of alternate representations by weakening the assumptions of the so called LOST uniqueness theorem. The weakened assumptions seem physically reasonable and retain the key requirement of explicit background independence. For simplicity, we restrict attention to the case of gauge group U(1).Comment: 22 pages, minor change

Quantum Spin Dynamics VIII. The Master Constraint

Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single Master constraint. The MCP is designed to overcome the complications associated with the non -- Lie -- algebra structure of the Dirac algebra of Hamiltonian constraints and was successfully tested in various field theory models. For the case of 3+1 gravity itself, so far only a positive quadratic form for the Master Constraint Operator was derived. In this paper we close this gap and prove that the quadratic form is closable and thus stems from a unique self -- adjoint Master Constraint Operator. The proof rests on a simple feature of the general pattern according to which Hamiltonian constraints in LQG are constructed and thus extends to arbitrary matter coupling and holds for any metric signature. With this result the existence of a physical Hilbert space for LQG is established by standard spectral analysis.Comment: 19p, no figure

Loop quantization of spherically symmetric midi-superspaces

We quantize the exterior of spherically symmetric vacuum space-times using a midi-superspace reduction within the Ashtekar new variables. Through a partial gauge fixing we eliminate the diffeomorphism constraint and are left with a Hamiltonian constraint that is first class. We complete the quantization in the loop representation. We also use the model to discuss the issues that will arise in more general contexts in the ``uniform discretization'' approach to the dynamics.Comment: 18 pages, RevTex, no figures, some typos corrected, published version, for some reason a series of figures were incorrectly added to the previous versio

Regge calculus from a new angle

In Regge calculus space time is usually approximated by a triangulation with flat simplices. We present a formulation using simplices with constant sectional curvature adjusted to the presence of a cosmological constant. As we will show such a formulation allows to replace the length variables by 3d or 4d dihedral angles as basic variables. Moreover we will introduce a first order formulation, which in contrast to using flat simplices, does not require any constraints. These considerations could be useful for the construction of quantum gravity models with a cosmological constant.Comment: 8 page

Loop Quantum Cosmology: Recent Progress

Aspects of the full theory of loop quantum gravity can be studied in a simpler context by reducing to symmetric models like cosmological ones. This leads to several applications where loop effects play a significant role when one is sensitive to the quantum regime. As a consequence, the structure of and the approach to classical singularities are very different from general relativity: The quantum theory is free of singularities, and there are new phenomenological scenarios for the evolution of the very early universe including inflation. We give an overview of the main effects, focussing on recent results obtained by several different groups.Comment: 17 pages, 2 figures, Plenary talk at ICGC 04, Cochin, Indi

Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework

Recently the Master Constraint Programme for Loop Quantum Gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Wheeler -- DeWitt constraint equations in terms of a single Master Equation. While the proposal has some promising abstract features, it was until now barely tested in known models. In this series of five papers we fill this gap, thereby adding confidence to the proposal. We consider a wide range of models with increasingly more complicated constraint algebras, beginning with a finite dimensional, Abelean algebra of constraint operators which are linear in the momenta and ending with an infinite dimensional, non-Abelean algebra of constraint operators which closes with structure functions only and which are not even polynomial in the momenta. In all these models we apply the Master Constraint Programme successfully, however, the full flexibility of the method must be exploited in order to complete our task. This shows that the Master Constraint Programme has a wide range of applicability but that there are many, physically interesting subtleties that must be taken care of in doing so. In this first paper we prepare the analysis of our test models by outlining the general framework of the Master Constraint Programme. The models themselves will be studied in the remaining four papers. As a side result we develop the Direct Integral Decomposition (DID) for solving quantum constraints as an alternative to Refined Algebraic Quantization (RAQ).Comment: 42 pages, no figure

Timelike surfaces in Lorentz covariant loop gravity and spin foam models

We construct a canonical formulation of general relativity for the case of a timelike foliation of spacetime. The formulation possesses explicit covariance with respect to Lorentz transformations in the tangent space. Applying the loop approach to quantize the theory we derive the spectrum of the area operator of a two-dimensional surface. Its different branches are naturally associated to spacelike and timelike surfaces. The results are compared with the predictions of Lorentzian spin foam models. A restriction of the representations labeling spin networks leads to perfect agreement between the states as well as the area spectra in the two approaches.Comment: a wrong sign corrected in equation (65

Group Field Theory: An overview

We give a brief overview of the properties of a higher dimensional generalization of matrix model which arises naturally in the context of a background independent approach to quantum gravity, the so called group field theory. We show that this theory leads to a natural proposal for the physical scalar product of quantum gravity. We also show in which sense this theory provides a third quantization point of view on quantum gravity.Comment: 10 page

On (Cosmological) Singularity Avoidance in Loop Quantum Gravity

Loop Quantum Cosmology (LQC), mainly due to Bojowald, is not the cosmological sector of Loop Quantum Gravity (LQG). Rather, LQC consists of a truncation of the phase space of classical General Relativity to spatially homogeneous situations which is then quantized by the methods of LQG. Thus, LQC is a quantum mechanical toy model (finite number of degrees of freedom) for LQG(a genuine QFT with an infinite number of degrees of freedom) which provides important consistency checks. However, it is a non trivial question whether the predictions of LQC are robust after switching on the inhomogeneous fluctuations present in full LQG. Two of the most spectacular findings of LQC are that 1. the inverse scale factor is bounded from above on zero volume eigenstates which hints at the avoidance of the local curvature singularity and 2. that the Quantum Einstein Equations are non -- singular which hints at the avoidance of the global initial singularity. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is {\it not} bounded from above on zero volume eigenstates. Thus, in full LQG, if curvature singularity avoidance is realized, then not in this simple way. In fact, it turns out that the boundedness of the inverse scale factor is neither necessary nor sufficient for curvature singularity avoidance and that non -- singular evolution equations are neither necessary nor sufficient for initial singularity avoidance because none of these criteria are formulated in terms of observable quantities.After outlining what would be required, we present the results of a calculation for LQG which could be a first indication that our criteria at least for curvature singularity avoidance are satisfied in LQG.Comment: 34 pages, 16 figure