203 research outputs found
Unboundedness of triad-like operators 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) the quantum Einstein equations are non-singular which hints at the avoidance of the global initial singularity. This rests on (1) a key technique developed for LQG and (2) the fact that there are no inhomogeneous excitations. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is 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 the 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
Properties of the Volume Operator in Loop Quantum Gravity II: Detailed Presentation
The properties of the Volume operator in Loop Quantum Gravity, as constructed
by Ashtekar and Lewandowski, are analyzed for the first time at generic
vertices of valence greater than four. The present analysis benefits from the
general simplified formula for matrix elements of the Volume operator derived
in gr-qc/0405060, making it feasible to implement it on a computer as a matrix
which is then diagonalized numerically. The resulting eigenvalues serve as a
database to investigate the spectral properties of the volume operator.
Analytical results on the spectrum at 4-valent vertices are included. This is a
companion paper to arXiv:0706.0469, providing details of the analysis presented
there.Comment: Companion to arXiv:0706.0469. Version as published in CQG in 2008.
More compact presentation. Sign factor combinatorics now much better
understood in context of oriented matroids, see arXiv:1003.2348, where also
important remarks given regarding sigma configurations. Subsequent
computations revealed some minor errors, which do not change qualitative
results but modify some numbers presented her
Shape in an Atom of Space: Exploring quantum geometry phenomenology
A phenomenology for the deep spatial geometry of loop quantum gravity is
introduced. In the context of a simple model, an atom of space, it is shown how
purely combinatorial structures can affect observations. The angle operator is
used to develop a model of angular corrections to local, continuum flat-space
3-geometries. The physical effects involve neither breaking of local Lorentz
invariance nor Planck scale suppression, but rather reply on only the
combinatorics of SU(2) recoupling. Bhabha scattering is discussed as an example
of how the effects might be observationally accessible.Comment: 14 pages, 7 figures; v2 references adde
On the resolution of the big bang singularity in isotropic Loop Quantum Cosmology
In contrast to previous work in the field, we construct the Loop Quantum
Cosmology (LQC) of the flat isotropic model with a massless scalar field in the
absence of higher order curvature corrections to the gravitational part of the
Hamiltonian constraint. The matter part of the constraint contains the inverse
triad operator which can be quantized with or without the use of a Thiemann-
like procedure. With the latter choice, we show that the LQC quantization is
identical to that of the standard Wheeler DeWitt theory (WDW) wherein there is
no singularity resolution. We argue that the former choice leads to singularity
resolution in the sense of a well defined, regular (backward) evolution through
and beyond the epoch where the size of the universe vanishes.
Our work along with that of the seminal work of Ashtekar, Pawlowski and Singh
(APS) clarifies the role, in singularity resolution, of the three `exotic'
structures in this LQC model, namely: curvature corrections, inverse triad
definitions and the `polymer' nature of the kinematic representation. We also
critically examine certain technical assumptions made by APS in their analysis
of WDW semiclassical states and point out some problems stemming from the
infrared behaviour of their wave functionsComment: 26 pages, no figure
Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity
We analyze combinatorial structures which play a central role in determining
spectral properties of the volume operator in loop quantum gravity (LQG). These
structures encode geometrical information of the embedding of arbitrary valence
vertices of a graph in 3-dimensional Riemannian space, and can be represented
by sign strings containing relative orientations of embedded edges. We
demonstrate that these signature factors are a special representation of the
general mathematical concept of an oriented matroid. Moreover, we show that
oriented matroids can also be used to describe the topology (connectedness) of
directed graphs. Hence the mathematical methods developed for oriented matroids
can be applied to the difficult combinatorics of embedded graphs underlying the
construction of LQG. As a first application we revisit the analysis of [4-5],
and find that enumeration of all possible sign configurations used there is
equivalent to enumerating all realizable oriented matroids of rank 3, and thus
can be greatly simplified. We find that for 7-valent vertices having no
coplanar triples of edge tangents, the smallest non-zero eigenvalue of the
volume spectrum does not grow as one increases the maximum spin \jmax at the
vertex, for any orientation of the edge tangents. This indicates that, in
contrast to the area operator, considering large \jmax does not necessarily
imply large volume eigenvalues. In addition we give an outlook to possible
starting points for rewriting the combinatorics of LQG in terms of oriented
matroids.Comment: 43 pages, 26 figures, LaTeX. Version published in CQG. Typos
corrected, presentation slightly extende
Properties of the Volume Operator in Loop Quantum Gravity I: Results
We analyze the spectral properties of the volume operator of Ashtekar and
Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the
classical volume expression for regions in three dimensional Riemannian space.
Our analysis considers for the first time generic graph vertices of valence
greater than four. Here we find that the geometry of the underlying vertex
characterizes the spectral properties of the volume operator, in particular the
presence of a `volume gap' (a smallest non-zero eigenvalue in the spectrum) is
found to depend on the vertex embedding. We compute the set of all
non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of
valence 5--7, and argue that these sets can be used to label spatial
diffeomorphism invariant states. We observe how gauge invariance connects
vertex geometry and representation properties of the underlying gauge group in
a natural way. Analytical results on the spectrum on 4-valent vertices are
included, for which the presence of a volume gap is proved. This paper presents
our main results; details are provided by a companion paper arXiv:0706.0382v1.Comment: 36 pages, 7 figures, LaTeX. See also companion paper
arXiv:0706.0382v1. Version as published in CQG in 2008. See arXiv:1003.2348
for important remarks regarding the sigma configurations. Subsequent
computations have revealed some minor errors, which do not change the
qualitative results but modify some of the numbers presented her
The status of Quantum Geometry in the dynamical sector of Loop Quantum Cosmology
This letter is motivated by the recent papers by Dittrich and Thiemann and,
respectively, by Rovelli discussing the status of Quantum Geometry in the
dynamical sector of Loop Quantum Gravity. Since the papers consider model
examples, we also study the issue in the case of an example, namely on the Loop
Quantum Cosmology model of space-isotropic universe. We derive the
Rovelli-Thiemann-Ditrich partial observables corresponding to the quantum
geometry operators of LQC in both Hilbert spaces: the kinematical one and,
respectively, the physical Hilbert space of solutions to the quantum
constraints. We find, that Quantum Geometry can be used to characterize the
physical solutions, and the operators of quantum geometry preserve many of
their kinematical properties.Comment: Latex, 12 page
Quantum resolution of black hole singularities
We study the classical and quantum theory of spherically symmetric spacetimes
with scalar field coupling in general relativity. We utilise the canonical
formalism of geometrodynamics adapted to the Painleve-Gullstrand coordinates,
and present a new quantisation of the resulting field theory. We give an
explicit construction of operators that capture curvature properties of the
spacetime and use these to show that the black hole curvature singularity is
avoided in the quantum theory.Comment: 5 pages, version to appear in CQ
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
Degenerate geometrical configurations in quantum gravity are important to
understand if the fate of classical singularities is to be revealed. However,
not all degenerate configurations arise on an equal footing, and one must take
into account dynamical aspects when interpreting results: While there are many
degenerate spatial metrics, not all of them are approached along the dynamical
evolution of general relativity or a candidate theory for quantum gravity. For
loop quantum gravity, relevant properties and steps in an analysis are
summarized and evaluated critically with the currently available information,
also elucidating the role of degrees of freedom captured in the sector provided
by loop quantum cosmology. This allows an outlook on how singularity removal
might be analyzed in a general setting and also in the full theory. The general
mechanism of loop quantum cosmology will be shown to be insensitive to recently
observed unbounded behavior of inverse volume in the full theory. Moreover,
significant features of this unboundedness are not a consequence of
inhomogeneities but of non-Abelian effects which can also be included in
homogeneous models.Comment: 28 pages, 1 figure; v2: extended discussion of singularity removal
and summar
New insights in quantum geometry
Quantum geometry, i.e., the quantum theory of intrinsic and extrinsic spatial
geometry, is a cornerstone of loop quantum gravity. Recently, there have been
many new ideas in this field, and I will review some of them. In particular,
after a brief description of the main structures and results of quantum
geometry, I review a new description of the quantized geometry in terms of
polyhedra, new results on the volume operator, and a way to incorporate a
classical background metric into the quantum description. Finally I describe a
new type of exponentiated flux operator, and its application to Chern-Simons
theory and black holes.Comment: 10 pages, 3 figures; Proceedings of Loops'11, Madrid, submitted to
Journal of Physics: Conference Series (JPCS
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