17 research outputs found
Born--Oppenheimer decomposition for quantum fields on quantum spacetimes
Quantum Field Theory on Curved Spacetime (QFT on CS) is a well established
theoretical framework which intuitively should be a an extremely effective
description of the quantum nature of matter when propagating on a given
background spacetime. If one wants to take care of backreaction effects, then a
theory of quantum gravity is needed. It is now widely believed that such a
theory should be formulated in a non-perturbative and therefore background
independent fashion. Hence, it is a priori a puzzle how a background dependent
QFT on CS should emerge as a semiclassical limit out of a background
independent quantum gravity theory. In this article we point out that the
Born-Oppenheimer decomposition (BOD) of the Hilbert space is ideally suited in
order to establish such a link, provided that the Hilbert space representation
of the gravitational field algebra satisfies an important condition. If the
condition is satisfied, then the framework of QFT on CS can be, in a certain
sense, embedded into a theory of quantum gravity. The unique representation of
the holonomy-flux algebra underlying Loop Quantum Gravity (LQG) violates that
condition. While it is conceivable that the condition on the representation can
be relaxed, for convenience in this article we consider a new classical
gravitational field algebra and a Hilbert space representation of its
restriction to an algebraic graph for which the condition is satisfied. An
important question that remains and for which we have only partial answers is
how to construct eigenstates of the full gravity-matter Hamiltonian whose BOD
is confined to a small neighbourhood of a physically interesting vacuum
spacetime.Comment: 38 pages, 2 figure
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
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
Identifying Users and Use of (Electric-) Free-Floating Carsharing in Berlin and Munich
In the last decade the attractiveness of carsharing increased rapidly, not least because of the introduction of free-floating carsharing in 2009. In those kind of carsharing systems the rental vehicle does not need to be returned to a particular station but can be parked in any part of the operating area.
There have been hardly any empirical findings on the use and effects of free-floating carsharing so far. Thus, this work presents results of user surveys (onCar questionnaire, online survey and discussions with focus groups) with customers of the free-floating carsharing operator DriveNow. Next to the analysis of the user the usage of such a carsharing system is evaluated by booking data of trips in Berlin and Munich from 2013. The Getis-Ord-Gi*-test is used for analyzing the spatial distribution of booking starts.
The operator launched 60 electric vehicles in the fleet that makes an additional analysis for this special kind of free-floating carsharing possible. All approaches want to draw an informative picture of a typical free-floating carsharing user on the one side and about how this new mobility service is used in urban areas on the other side. By the discussions in the focus groups one obtains furthermore an impression about the acceptance of electric vehicles by the customers.
One clear conclusion is that free-floating carsharing is mostly used by young well-educated people with an over-average income. Two main purposes of the trips are the way home and leisure time activities. The system is well-working in city or district centers while there are considerably less bookings in peripheral areas. This is also correct for electric free-floating carsharing that is principally accepted by the customers