151 research outputs found
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
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
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
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
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
Three-geometry and reformulation of the Wheeler-DeWitt equation
A reformulation of the Wheeler-DeWitt equation which highlights the role of
gauge-invariant three-geometry elements is presented. It is noted that the
classical super-Hamiltonian of four-dimensional gravity as simplified by
Ashtekar through the use of gauge potential and densitized triad variables can
furthermore be succinctly expressed as a vanishing Poisson bracket involving
three-geometry elements. This is discussed in the general setting of the
Barbero extension of the theory with arbitrary non-vanishing value of the
Immirzi parameter, and when a cosmological constant is also present. A proposed
quantum constraint of density weight two which is polynomial in the basic
conjugate variables is also demonstrated to correspond to a precise simple
ordering of the operators, and may thus help to resolve the factor ordering
ambiguity in the extrapolation from classical to quantum gravity. Alternative
expression of a density weight one quantum constraint which may be more useful
in the spin network context is also discussed, but this constraint is
non-polynomial and is not motivated by factor ordering. The article also
highlights the fact that while the volume operator has become a preeminient
object in the current manifestation of loop quantum gravity, the volume element
and the Chern-Simons functional can be of equal significance, and need not be
mutually exclusive. Both these fundamental objects appear explicitly in the
reformulation of the Wheeler-DeWitt constraint.Comment: 10 pages, LaTeX fil
Grasping rules and semiclassical limit of the geometry in the Ponzano-Regge model
We show how the expectation values of geometrical quantities in 3d quantum
gravity can be explicitly computed using grasping rules. We compute the volume
of a labelled tetrahedron using the triple grasping. We show that the large
spin expansion of this value is dominated by the classical expression, and we
study the next to leading order quantum corrections.Comment: 18 pages, 1 figur
A semiclassical tetrahedron
We construct a macroscopic semiclassical state state for a quantum
tetrahedron. The expectation values of the geometrical operators representing
the volume, areas and dihedral angles are peaked around assigned classical
values, with vanishing relative uncertainties.Comment: 10 pages; v2 revised versio
From the discrete to the continuous - towards a cylindrically consistent dynamics
Discrete models usually represent approximations to continuum physics.
Cylindrical consistency provides a framework in which discretizations mirror
exactly the continuum limit. Being a standard tool for the kinematics of loop
quantum gravity we propose a coarse graining procedure that aims at
constructing a cylindrically consistent dynamics in the form of transition
amplitudes and Hamilton's principal functions. The coarse graining procedure,
which is motivated by tensor network renormalization methods, provides a
systematic approximation scheme towards this end. A crucial role in this coarse
graining scheme is played by embedding maps that allow the interpretation of
discrete boundary data as continuum configurations. These embedding maps should
be selected according to the dynamics of the system, as a choice of embedding
maps will determine a truncation of the renormalization flow.Comment: 22 page
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