10 research outputs found
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
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
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
Loop Quantum Cosmology: A Status Report
The goal of this article is to provide an overview of the current state of
the art in loop quantum cosmology for three sets of audiences: young
researchers interested in entering this area; the quantum gravity community in
general; and, cosmologists who wish to apply loop quantum cosmology to probe
modifications in the standard paradigm of the early universe. An effort has
been made to streamline the material so that, as described at the end of
section I, each of these communities can read only the sections they are most
interested in, without a loss of continuity.Comment: 138 pages, 15 figures. Invited Topical Review, To appear in Classical
and Quantum Gravity. Typos corrected, clarifications and references adde
Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity
We perform a canonical, reduced phase space quantisation of General
Relativity by Loop Quantum Gravity (LQG) methods. The explicit construction of
the reduced phase space is made possible by the combination of 1. the Brown --
Kuchar mechanism in the presence of pressure free dust fields which allows to
deparametrise the theory and 2. Rovelli's relational formalism in the extended
version developed by Dittrich to construct the algebra of gauge invariant
observables. Since the resulting algebra of observables is very simple, one can
quantise it using the methods of LQG. Basically, the kinematical Hilbert space
of non reduced LQG now becomes a physical Hilbert space and the kinematical
results of LQG such as discreteness of spectra of geometrical operators now
have physical meaning. The constraints have disappeared, however, the dynamics
of the observables is driven by a physical Hamiltonian which is related to the
Hamiltonian of the standard model (without dust) and which we quantise in this
paper.Comment: 31 pages, no figure