Exploring the diffeomorphism invariant Hilbert space of a scalar field

As a toy model for the implementation of the diffeomorphism constraint, the interpretation of the resulting states, and the treatment of ordering ambiguities in loop quantum gravity, we consider the Hilbert space of spatially diffeomorphism invariant states for a scalar field. We give a very explicit formula for the scalar product on this space, and discuss its structure. Then we turn to the quantization of a certain class of diffeomorphism invariant quantities on that space, and discuss in detail the ordering issues involved. On a technical level these issues bear some similarity to those encountered in full loop quantum gravity.Comment: 20 pages, no figures; v3: corrected typos, added reference, some clarifications added; version as published in CQ

The Proca-field in Loop Quantum Gravity

In this paper we investigate the Proca-field in the framework of Loop Quantum Gravity. It turns out that the methods developed there can be applied to the symplectically embedded Proca-field, giving a rigorous, consistent, non-perturbative quantization of the theory. This can be achieved by introducing a scalar field, which has completely different properties than the one used in spontaneous symmetry breaking. The analysis of the kernel of the Hamiltonian suggests that the mass term in the quantum theory has a different role than in the classical theory.Comment: 15 pages. v2: 19 pages, amended sections 2 and 6, references added v3: 20 pages, amended section 6 and minor correction

Dynamics of Scalar Field in Polymer-like Representation

In recent twenty years, loop quantum gravity, a background independent approach to unify general relativity and quantum mechanics, has been widely investigated. We consider the quantum dynamics of a real massless scalar field coupled to gravity in this framework. A Hamiltonian operator for the scalar field can be well defined in the coupled diffeomorphism invariant Hilbert space, which is both self-adjoint and positive. On the other hand, the Hamiltonian constraint operator for the scalar field coupled to gravity can be well defined in the coupled kinematical Hilbert space. There are 1-parameter ambiguities due to scalar field in the construction of both operators. The results heighten our confidence that there is no divergence within this background independent and diffeomorphism invariant quantization approach of matter coupled to gravity. Moreover, to avoid possible quantum anomaly, the master constraint programme can be carried out in this coupled system by employing a self-adjoint master constraint operator on the diffeomorphism invariant Hilbert space.Comment: 24 pages, accepted for pubilcation in Class. Quant. Gra

Gauge Field Theory Coherent States (GCS) : I. General Properties

In this article we outline a rather general construction of diffeomorphism covariant coherent states for quantum gauge theories. By this we mean states $\psi_{(A,E)}$, labelled by a point (A,E) in the classical phase space, consisting of canonically conjugate pairs of connections A and electric fields E respectively, such that (a) they are eigenstates of a corresponding annihilation operator which is a generalization of A-iE smeared in a suitable way, (b) normal ordered polynomials of generalized annihilation and creation operators have the correct expectation value, (c) they saturate the Heisenberg uncertainty bound for the fluctuations of $\hat{A},\hat{E}$ and (d) they do not use any background structure for their definition, that is, they are diffeomorphism covariant. This is the first paper in a series of articles entitled ``Gauge Field Theory Coherent States (GCS)'' which aim at connecting non-perturbative quantum general relativity with the low energy physics of the standard model. In particular, coherent states enable us for the first time to take into account quantum metrics which are excited {\it everywhere} in an asymptotically flat spacetime manifold. The formalism introduced in this paper is immediately applicable also to lattice gauge theory in the presence of a (Minkowski) background structure on a possibly {\it infinite lattice}.Comment: 40 pages, LATEX, no figure

Gauge Field Theory Coherent States (GCS) : II. Peakedness Properties

In this article we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mour\~ao and Thiemann to arbitrary, finite, piecewise analytic graphs. However, both of these works were incomplete with respect to the following two issues : (a) The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched. (b) While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space. In this paper we resolve these issues, in particular, we prove that this family of states satisfies all the usual properties : i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation, ii) Saturation of the unquenched Heisenberg uncertainty bound. iii) (Over)completeness. These states therefore comprise a candidate family for the semi-classical analysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity, enable error-controlled approximations and set a new starting point for {\it numerical canonical quantum general relativity and gauge theory}. The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed.Comment: 70 pages, LATEX, 29 figure

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

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

Electric fields and double layers in plasmas

Various mechanisms for driving double layers in plasmas are briefly described, including applied potential drops, currents, contact potentials, and plasma expansions. Some dynamical features of the double layers are discussed. These features, as seen in simulations, laboratory experiments, and theory, indicate that double layers and the currents through them undergo slow oscillations which are determined by the ion transit time across an effective length of the system in which double layers form. It is shown that a localized potential dip forms at the low potential end of a double layer, which interrupts the electron current through it according to the Langmuir criterion, whenever the ion flux into the double is disrupted. The generation of electric fields perpendicular to the ambient magnetic field by contact potentials is also discussed. Two different situations were considered; in one, a low-density hot plasma is sandwiched between high-density cold plasmas, while in the other a high-density current sheet permeates a low-density background plasma. Perpendicular electric fields develop near the contact surfaces. In the case of the current sheet, the creation of parallel electric fields and the formation of double layers are also discussed when the current sheet thickness is varied. Finally, the generation of electric fields and double layers in an expanding plasma is discussed

Regularized Hamiltonians and Spinfoams

We review a recent proposal for the regularization of the scalar constraint of General Relativity in the context of LQG. The resulting constraint presents strengths and weaknesses compared to Thiemann's prescription. The main improvement is that it can generate the 1-4 Pachner moves and its matrix elements contain 15j Wigner symbols, it is therefore compatible with the spinfoam formalism: the drawback is that Thiemann anomaly free proof is spoiled because the nodes that the constraint creates have volume.Comment: 4 pages, based on a talk given at Loops '11 in Madrid, to appear in Journal of Physics: Conference Series (JPCS

Towards the QFT on Curved Spacetime Limit of QGR. I: A General Scheme

In this article and a companion paper we address the question of how one might obtain the semiclassical limit of ordinary matter quantum fields (QFT) propagating on curved spacetimes (CST) from full fledged Quantum General Relativity (QGR), starting from first principles. We stress that we do not claim to have a satisfactory answer to this question, rather our intention is to ignite a discussion by displaying the problems that have to be solved when carrying out such a program. In the present paper we propose a scheme that one might follow in order to arrive at such a limit. We discuss the technical and conceptual problems that arise in doing so and how they can be solved in principle. As to be expected, completely new issues arise due to the fact that QGR is a background independent theory. For instance, fundamentally the notion of a photon involves not only the Maxwell quantum field but also the metric operator - in a sense, there is no photon vacuum state but a "photon vacuum operator"! While in this first paper we focus on conceptual and abstract aspects, for instance the definition of (fundamental) n-particle states (e.g. photons), in the second paper we perform detailed calculations including, among other things, coherent state expectation values and propagation on random lattices. These calculations serve as an illustration of how far one can get with present mathematical techniques. Although they result in detailed predictions for the size of first quantum corrections such as the gamma-ray burst effect, these predictions should not be taken too seriously because a) the calculations are carried out at the kinematical level only and b) while we can classify the amount of freedom in our constructions, the analysis of the physical significance of possible choices has just begun.Comment: LaTeX, 47 p., 3 figure